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Title: The English works of Thomas Hobbes of Malmesbury, Volume 07 (of 11)
Author: Thomas Hobbes
Editor: Sir William Molesworth
Release date: May 13, 2026 [eBook #78674]
Language: English
Original publication: London: John Bohn, 1839
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THE
ENGLISH WORKS
OF
THOMAS HOBBES
OF MALMESBURY;
NOW FIRST COLLECTED AND EDITED
BY
SIR WILLIAM MOLESWORTH, BART.
-------
VOL. VII.
-------
LONDON:
LONGMAN, BROWN, GREEN, AND LONGMANS,
PATERNOSTER-ROW.
--
MDCCCXLV.
LONDON:
RICHARDS, PRINTER, 100, ST. MARTIN’S LANE.
CONTENTS.
---
PAGE
SEVEN PHILOSOPHICAL PROBLEMS 1
DECAMERON PHYSIOLOGICUM 69
PROPORTION OF A STRAIGHT LINE TO HALF THE ARC OF A QUADRANT 178
SIX LESSONS TO THE SAVILIAN PROFESSORS OF THE MATHEMATICS 181
ΣΤΙΓΜΑΙ, OR MARKS OF THE ABSURD GEOMETRY ETC. OF DR. WALLIS 357
EXTRACT OF A LETTER FROM HENRY STUBBE 401
THREE PAPERS PRESENTED TO THE ROYAL SOCIETY AGAINST DR. WALLIS 429
CONSIDERATIONS ON THE ANSWER OF DR. WALLIS 443
LETTERS AND OTHER PIECES 449
SEVEN
PHILOSOPHICAL PROBLEMS
AND
TWO PROPOSITIONS OF GEOMETRY.
BY
THOMAS HOBBES
OF MALMESBURY.
WITH
AN APOLOGY FOR HIMSELF AND HIS WRITINGS.
DEDICATED TO THE KING IN THE YEAR 1662.
TO THE KING.
That which I do here most humbly present to your sacred Majesty, is the
best part of my meditations upon the natural causes of events, both of
such as are commonly known, and of such as have been of late
artificially exhibited by the curious.
They are ranged under seven heads. 1. Problems of gravity. 2. Problems
of tides. 3. Problems of vacuum. 4. Problems of heat. 5. Problems of
hard and soft. 6. Problems of wind and weather. 7. Problems of motion
perpendicular and oblique, &c. To which I have added two propositions of
Geometry: one is, the duplication of the cube, hitherto sought in vain;
the other, a detection of the absurd use of arithmetic, as it is now
applied to geometry.
The doctrine of natural causes hath not infallible and evident
principles. For there is no effect which the power of God cannot produce
by many several ways.
But seeing all effects are produced by motion, he that supposing some
one or more motions, can derive from them the necessity of that effect
whose cause is required, has done all that is to be expected from
natural reason. And though he prove not that the thing was thus
produced, yet he proves that thus it may be produced when the materials
and the power of moving are in our hands: which is as useful as if the
causes themselves were known. And notwithstanding the absence of
rigorous demonstration, this contemplation of nature (if not rendered
obscure by empty terms) is the most noble employment of the mind that
can be, to such as are at leisure from their necessary business.
This that I have done I know is an unworthy present to be offered to a
king: though considered, as God considers offerings, together with the
mind and fortune of the offerer, I hope will not be to your Majesty
unacceptable.
But that which I chiefly consider in it is, that my writing should be
tried by your Majesty’s excellent reason, untainted with the language
that has been invented or made use of by men when they were puzzled; and
who is acquainted with all the experiments of the time; and whose
approbation, if I have the good fortune to obtain it, will protect my
reasoning from the contempt of my adversaries.
I will not break the custom of joining to my offering a prayer; and it
is, that your Majesty will be pleased to pardon this following short
apology for my _Leviathan_ . Not that I rely upon apologies, but upon
your Majesty’s most gracious general pardon.
That which is in it of theology, contrary to the general current of
divines, is not put there as my opinion, but propounded with submission
to those that have the power ecclesiastical.
I did never after, either in writing or discourse, maintain it.
There is nothing in it against episcopacy; I cannot therefore imagine
what reason any episcopal man can have to speak of me, as I hear some of
them do, as of an atheist, or man of no religion, unless it be for
making the authority of the Church wholly upon the regal power; which I
hope your Majesty will think is neither atheism nor heresy.
But what had I to do to meddle with matters of that nature, seeing
religion is not philosophy, but law?
It was written in a time when the pretence of Christ’s kingdom was made
use of for the most horrid actions that can be imagined; and it was in
just indignation of that, that I desired to see the bottom of that
doctrine of the kingdom of Christ, which divers ministers then preached
for a pretence to their rebellion: which may reasonably extenuate,
though not excuse the writing of it.
There is therefore no ground for so great a calumny in my writing. There
is no sign of it in my life; and for my religion, when I was at the
point of death at St. Germain’s, the Bishop of Durham can bear witness
of it, if he be asked. Therefore I most humbly beseech your sacred
Majesty not to believe so ill of me upon reports, that proceed often,
and may do so now, from the displeasure which commonly ariseth from
difference in opinion; nor to think the worse of me, if snatching up all
the weapons to fight against your enemies, I lighted upon one that had a
double edge.
Your Majesty’s poor and
most loyal subject,
THOMAS HOBBES.
==========
[Illustration: _Seven Philosophical Problems. English Works, Vol. 7._]
PHILOSOPHICAL PROBLEMS.
---
CHAPTER I.
PROBLEMS OF GRAVITY.
_A._ What may be the cause, think you, that stones and other bodies
thrown upward, or carried up and left to their liberty, fall down again,
for aught a man can see, of their own accord? I do not think with the
old philosophers, that they have any love to the earth; or are sullen,
that they will neither go nor stay. And yet I cannot imagine, what body
there is above that should drive them back.
_B._ For my part, I believe the cause of their descending is not in any
natural appetite of the bodies that descend; but rather that the globe
of the earth hath some special motion, by which it more easily casteth
off the air than it doth other bodies. And then this descent of those we
call heavy bodies must of necessity follow, unless there be some empty
spaces in the world to receive them. For when the air is thrown off from
the earth, somewhat must come into the place of it, in case the world be
full: and it must be those things which are hardliest cast off, that is,
those things which we say are heavy.
_A._ But suppose there be no place empty, (for I will defer the question
till anon), how can the earth cast off either the air or anything else?
_B._ I shall show you how, and that by a familiar example. If you lay
both your hands upon a basin with water in it, how little soever, and
move it circularly, and continue that motion for a while; and you shall
see the water rise upon the sides, and fly over. By which you may be
assured that there is a kind of circulating motion, which would cast off
such bodies as are contiguous to the body so moved.
_A._ I know very well there is; and it is the same motion which country
people use to purge their corn; for the chaff and straws, by casting the
grain to the sides of the sieve, will come towards the middle. But I
would see the figure.
_B._ Here it is. There is a circle pricked out, whose centre is A, and
three less circles, whose centres are B, C, D. Let every one of them
represent the earth, as it goeth from B to C, and from C to D, always
touching the uttermost circle and throwing off the air, as is marked at
E and F. And if the world were not full, there would follow by this
scattering of the air, a great deal of space left empty. But supposing
the world full, there must be a perpetual shifting of the air, one part
into the place of another.
_A._ But what makes a stone come down, suppose from G?
_B._ If the air be thrown up beyond G, it will follow that at the last,
if the motion be continued, all the air will be above G, that is, above
the stone; which cannot be, till the stone be at the earth.
_A._ But why comes it down still with increasing swiftness?
_B._ Because as it descends and is already in motion, it receiveth a new
impression from the same cause, which is the air, whereof as part
mounteth, part also must descend, supposing as we have done the
plentitude of the world. For, as you may observe by the figure, the
motion of the earth, according to the diameter of the uttermost circle,
is progressive; and so the whole motion is compounded of two motions,
one circular and the other progressive; and consequently the air ascends
and circulates at once. And because the stone descending receiveth a new
pressure in every point of its way, the motion thereof must needs be
accelerated.
_A._ It is true; for it will be accelerated equally in equal times; and
the way it makes will increase in a double proportion to the times, as
hath heretofore been demonstrated by Galileo. I see the solution now of
an experiment, which before did not a little puzzle me. You know that if
two plummets hang by two strings of equal length, and you remove them
from the perpendicular equally, I mean in equal angles, and then let
them go, they will make their turns and returns together and in equal
times; and though the arches they describe grow continually less and
less, yet the times they spend in the greater arches will still be equal
to the time they spend in the lesser.
_B._ It is true. Do you find any experiment to the contrary?
_A._ Yes; for if you remove one of the plummets from the perpendicular,
so as, for example, to make an angle with the perpendicular of eighty
degrees, and the other so as to make an angle of sixty degrees; they
will not make their turns and returns in equal times.
_B._ And what say you is the cause of this?
_A._ Because the arches are the spaces which these two motions describe,
they must be in double proportion to their own times: which cannot be,
unless they be let go from equal altitudes, that is, from equal angles.
_B._ It is right; and the experiment does not cross, but confirm the
equality of the times in all the arches they describe, even from ninety
degrees to the least part of one degree.
_A._ But is it not too bold, if not extravagant an assertion, to say the
earth is moved as a man shakes a basin or a sieve? Does not the earth
move from west to east every day once, upon its own centre; and in the
ecliptic circle once a year? And now you give it another odd motion. How
can all these consist in one and the same body?
_B._ Well enough. If you be a shipboard under sail, do not you go with
the ship? Cannot you also walk upon the deck? Cannot every drop of blood
move at the same time in your veins? How many motions now do you assign
to one and the same drop of blood? Nor is it so extravagant a thing to
attribute to the earth this kind of motion; but that I believe, if we
certainly knew what motion it is that causeth the descent of bodies, we
should find it either the same, or more extravagant. But seeing it can
be nothing above that worketh this effect, it must be the earth itself
that does it; and if the earth, then you can imagine no other motion to
do it withal but this. And you will wonder more, when by the same motion
I shall give you a probable account of the causes of very many other
works of nature.
_A._ But what part of the heaven do you suppose the poles of your
pricked circle point to?
_B._ I suppose them to be the same with the poles of the ecliptic. For,
seeing the axis of the earth in this motion and in the annual motion,
keeps parallel to itself, the axis must in both motions be parallel as
to sense. For the circle which the earth describes, is not of visible
magnitude at the distance it is from the sun.
_A._ Though I understand well enough how the earth may make a stone
descend very swiftly under the ecliptic, or not far from it, where it
throws off the air perpendicularly; yet about the poles of the circle
methinks, it should cast off the air very weakly. I hope you will not
say, that bodies descend faster in places remote from the poles, than
nearer to them.
_B._ No; but I ascribe it to the like motion in the sun and moon. For
such motions meeting, must needs cast the stream of the air towards the
poles; and then there will be the same necessity for the descent there,
that there is in other places, though perhaps a little more slowly. For
you may have observed, that when it snows in the south parts, the flakes
of snow are not so great as in the north: which is a probable sign they
fall in the south from a greater height, and consequently disperse
themselves more, as water does that falls down from a high and steep
rock.
_A._ It is not improbable.
_B._ In natural causes all you are to expect, is but probability; which
is better yet, than making gravity the cause, when the cause of gravity
is that which you desire to know; and better than saying the earth draws
it, when the question is, how it draws.
_A._ Why does the earth cast off air more easily than it does water, or
any other heavy bodies?
_B._ It is indeed the earth that casteth off that air which is next unto
it; but it is that air which casteth off the next air; and so
continually, air moveth air; which it can more easily do than any other
thing, because like bodies are more susceptible of one another’s
motions: as you may see in two lute-strings equally strained, what
motion one string being stricken communicates to the air, the same will
the other receive from the air; but strained to a differing note, will
be less or not at all moved. For there is no body but air, that hath not
some internal, though invisible, motion of its parts: and it is that
internal motion which distinguisheth all natural bodies one from
another.
_A._ What is the cause why certain squibs, though their substance be
either wood or other heavy matter, made hollow and filled with
gunpowder, which is also heavy; do nevertheless, when the gunpowder is
kindled, fly upwards?
_B._ The same that keeps a man that swims from sinking, though he be
heavier than so much water. He keeps himself up, and goes forward, by
beating back the water with his feet; and so does a squib, by beating
down the air with the stream of fired gunpowder, that proceeding from
its tail makes it recoil.
_A._ Why does any brass or iron vessel, if it be hollow, float upon the
water, being so very heavy?
_B._ Because the vessel and the air in it, taken as one body, is more
easily cast off than a body of water equal to it.
_A._ How comes it to pass, that a fish, (especially such a broad fish as
a turbot or a plaice, which are broad and thin), in the bottom of the
sea, perhaps a mile deep, is not pressed to death with the weight of
water that lies upon the back of it?
_B._ Because all heavy bodies descend towards one point, which is the
centre of the earth: and consequently the whole sea, descending at once,
does arch itself so, as that the upper parts cannot press the parts next
below them.
_A._ It is evident; nor can there possibly be any weight, as some
suppose there is, of a cylinder of air or water or any other liquid
thing, while it remains in its own element, or is sustained and inclosed
in a vessel by which one part cannot press the other.
==========
CHAPTER II.
PROBLEMS OF TIDES.
_A._ What makes the flux and reflux of the sea, twice in a natural day?
_B._ We must come again to our basin of water; wherein you have seen,
whilst it was moved, how the water mounteth up by the sides, and withal
goes circling round about. Now if you should fasten to the inside of the
basin some bar from the bottom to the top, you would see the water,
instead of going on, go back again from that bar ebbing, and the water
on the other side of the bar to do the same, but in counter-time; and
consequently to be highest where the contrary streams meet together; and
then return again, marking out four quarters of the vessel; two by their
meeting, which are the high waters; and two by their retiring, which are
the low waters.
_A._ What bar is that you find in the ocean that stops the current of
the water, like that you make in the basin?
_B._ You know that the main ocean lies east and west, between India and
the coast of America; and again on the other side, between America and
India. If therefore the earth have such a motion as I have supposed, it
must needs carry the current of the sea east and west: in which course,
the bar that stoppeth it, is the south part of America, which leaves no
passage for the water but the narrow strait of Magellan. The tide rises
therefore upon the coast of America; and the rising of the same in this
part of the world, proceedeth from the swelling chiefly of the water
there, and partly also from the North Sea; which lieth also east and
west, and has a passage out of the South Sea by the strait of Anian,
between America and Asia.
_A._ Does not the Mediterranean Sea lie also east and west? Why are
there not the like tides there?
_B._ So there are, proportionable to their lengths and quantity of
water.
_A._ At Genoa, at Ancona, there are none at all, or not sensible.
_B._ At Venice there are, and in the bottom of the straits, and a
current all along both the Mediterranean Sea and the Gulf of Venice: and
it is the current that makes the tides insensible at the sides; but the
check makes them visible at the bottom.
_A._ How comes it about that the moon hath such a stroke in the
business, as so sensibly to increase the tides at full and change?
_B._ The motion I have hitherto supposed but in the earth, I suppose
also in the moon, and in all those great bodies that hang in the air
constantly, I mean the stars, both fixed and errant. And for the sun and
moon, I suppose the poles of their motion to be the poles of the
equinoxial. Which supposed, it will follow (because the sun, the earth,
and the moon, at every full and change are almost in one straight line)
that this motion of the earth will then be made swifter than in the
quarters. For this motion of the sun and moon being communicated to the
earth, that hath already the like motion, maketh the same greater; and
much greater when they are all three in one straight line, which is only
at the full and change, whose tides are therefore called spring tides.
_A._ But what then is the cause that the spring tides themselves are
twice a-year, namely, when the sun is in the equinoxial, greater than at
any other times?
_B._ At other times of the year, the earth being out of the equinoxial,
the motion thereof, by which the tides are made, will be less augmented,
by so much as a motion in the obliquity of twenty-three degrees, or
thereabout, which is the distance between the equinoxial and ecliptic
circles, is weaker than the motion which is without obliquity.
_A._ All this is reasonable enough, if it be possible that such motions
as you suppose in these bodies, be really there. But that is a thing I
have some reason to doubt of. For the throwing off of air, consequent to
these motions, is the cause, you say, that other things come to the
earth; and therefore the like motions in the sun and moon and stars,
casting off the air, should also cause all other things to come to every
one of them. From whence it will follow, that the sun, moon, and earth,
and all other bodies but air, should presently come together into one
heap.
_B._ That does not follow. For if two bodies cast off the air, the
motion of that air will be repressed both ways, and diverted into a
course towards the poles on both sides; and then the two bodies cannot
possibly come together.
_A._ It is true. And besides, this driving of the air on both sides,
north and south, makes the like motion of air there also. And this may
answer the question, how a stone could fall to the earth under the poles
of the ecliptic, by the only casting off of air?
_B._ It follows from hence, that there is a certain and determinate
distance of one of these bodies, the stars, from another, without any
very sensible variation.
_A._ All this is probable enough, if it be true that there is no vacuum,
no place empty in all the world. And supposing this motion of the sun
and moon to be in the plain of the equinoxial, methinks that this should
be the cause of the diurnal motion of the earth; and because this motion
of the earth is, you say, in the plain of the equinoxial, the same
should cause also a motion in the moon on her own centre, answerable to
the diurnal motion of the earth.
_B._ Why not? What else can you think makes the diurnal motion of the
earth but the sun? And for the moon, if it did not turn upon its own
centre, we should see sometimes one, sometimes another face of the moon,
which we do not.
==========
CHAPTER III.
PROBLEMS OF VACUUM.
_A._ What convincing argument is there to prove, that in all the world
there is no empty place?
_B._ Many; but I will name but one; and that is, the difficulty of
separating two bodies hard and flat laid one upon another. I say the
difficulty, not the impossibility. It is possible, without introducing
vacuum, to pull asunder any two bodies, how hard and flat soever they
be, if the force used be greater than the resistance of the hardness.
And in case there be any greater difficulty to part them, besides what
proceeds from their hardness, than there is to pull them further asunder
when they are parted, that difficulty is argument enough to prove there
is no vacuum.
_A._ These assertions need demonstration. And first, how does the
difficulty of separation argue the plenitude of all the rest of the
world?
_B._ If two flat polished marbles lie one upon another, you see they are
hardly separated in all points at one and the same instant; and yet the
weight of either of them is enough to make them slide off one from the
other. Is not the cause of this, that the air succeeds the marble that
so slides, and fills up the place it leaves?
_A._ Yes, certainly. What then?
_B._ But when you pull the whole superficies asunder, not without great
difficulty, what is the cause of that difficulty?
_A._ I think, as most men do, that the air cannot fill up the space
between in an instant; for the parting is in an instant.
_B._ Suppose there be vacuum in that air into which the marble you pull
off is to succeed, shall there be no vacuum in the air that was round
about the two marbles when they touched? Why cannot that vacuum come
into the place between? Air cannot succeed in an instant, because a
body, and consequently cannot be moved through the least space in an
instant. But emptiness is not a body, nor is moved, but is made by the
act itself of separation. There is therefore, if you admit vacuum, no
necessity at all for the air to fill the space left in an instant. And
therefore, with what ease the marble coming off presseth out the vacuum
of the air behind it, with the same ease will the marbles be pulled
asunder. Seeing then, if there were vacuum, there would be no difficulty
of separation, it follows, because there is difficulty of separation,
that there is no vacuum.
_A._ Well, now, supposing the world full, how do you prove it possible
to pull those marbles asunder?
_B._ Take a piece of soft wax; do not you think the one half touches the
other half as close as the smoothest marbles? Yet you can pull them
asunder. But how? Still as you pull, the wax grows continually more and
more slender; there being a perpetual parting or discession of the
outermost part of the wax one from another, which the air presently
fills; and so there is a continual lessening of the wax, till it be no
bigger than a hair, and at last separation. If you can do the same to a
pillar of marble, till the outside give way, the effect will be the
same, but much quicker, after it once begins to break in the
superficies; because the force that can master the first resistance of
the hardness, will quickly dispatch the rest.
_A._ It seems so by the brittleness of some hard bodies. But I shall
afterward put some questions to you, touching the nature of hardness.
But now to return to our subject. What reason can you render (without
supposing vacuum) of the effects produced in the engine they use at
Gresham college?
_B._ That engine produceth the same effects that a strong wind would
produce in a narrow room.
_A._ How comes the wind in? You know the engine is a hollow round pipe
of brass, into which is thrust a cylinder of wood covered with leather,
and fitted to the cylinder so exactly as no air can possibly pass
between the leather and the brass?
_B._ I know it; and that they may thrust it up, there is a hole left in
the cylinder to let the air out before it, which they can stop when they
please. There is also in the bottom of the cylinder a passage into a
hollow globe of glass, which passage they can also open and shut at
pleasure. And at the top of that globe there is a wide mouth to put in
what they please to try conclusions on, and that also to be opened and
shut as shall be needful. It is of the nature of a pop-gun which
children use, but great, costly, and more ingenious. They thrust forward
and pull back the wooden cylinder (because it requires much strength)
with an iron screw. What is there in all this to prove the possibility
of vacuum.
_A._ When this wooden cylinder covered with leather, fit and close, is
thrust home to the bottom, and the holes in the hollow cylinder of brass
close stopped, how can it be drawn back, as with the screw they draw it,
but that the space it leaves must needs be empty: for it is impossible
that any air can pass into the place to fill it?
_B._ Truly I think it close enough to keep out straw and feathers, but
not to keep out air, nor yet matter. For suppose they were not so
exactly close but that there were round about a difference for a small
hair to lie between; then will the pulling back of the cylinder of wood
force so much air in, as in retiring it forces back, and that without
any sensible difficulty. And the air will so much more swiftly enter as
the passage is left more narrow. Or if they touch, and the contact be in
some points and not in all, the air will enter as before, in case the
force be augmented accordingly. Lastly, though they touch exactly, if
either the leather yield, or the brass, which it may do, to the force of
a strong screw, the air will again enter. Do you think it possible to
make two superficies so exquisitely touch in all points as you suppose,
or leather so hard as not to yield to the force of a screw? The body of
leather will give passage both to air and water, as you will confess
when you ride in rainy and windy weather. You may therefore be assured
that in drawing out their wooden leather cylinder, they force in as much
air as will fill the place it leaves, and that with as much swiftness as
is answerable to the strength that drives it in. The effect therefore of
their pumping is nothing else but a vehement wind, a very vehement wind,
coming in on all sides of the cylinder at once into the hollow of the
brass pipe, and into the hollow of the glass globe joined to it.
_A._ I see the reason already of one of their wonders, which is, that
the cylinder they pump with, if it be left to itself, after it is pulled
back, will swiftly go up again. You will say the air comes out again
with the same violence by reflection, and I believe it.
_B._ This is argument enough that the place was not empty. For what can
fetch or drive up the sucker, as they call it, if the place within were
empty? For that there is any weight in the air to do it, I have already
demonstrated to be impossible. Besides, you know, when they have sucked
out, as they think, all the air from the glass globe, they can
nevertheless both see through it what is done, and hear a sound from
within when there is any made; which, if there were no other, but there
are many other, is argument enough that the place is still full of air.
_A._ What say you to the swelling of a bladder even to bursting, if it
be a little blown when it is put into the receiver, for so they call the
globe of glass?
_B._ The streams of air that from every side meeting together, and
turning in an infinite number of small points, do pierce the bladder in
innumerable places with great violence at once, like so many invisible
small wimbles, especially if the bladder be a little blown before it be
put in, that it may make a little resistance. And when the air has once
pierced it, it is easy to conceive, that it must afterward by the same
violent motion be extended till it break. If before it break you let in
fresh air upon it, the violence of the motion will thereby be tempered,
and the bladder be less extended; for that also they have observed. Can
you imagine how a bladder should be extended and broken by being too
full of emptiness?
_A._ How come living creatures to be killed in this receiver, in so
little a time as three or four minutes of an hour?
_B._ If they suck into their lungs so violent a wind thus made, you must
needs think it will presently stop the passage of their blood; and that
is death; though they may recover if taken out before they be too cold.
And so likewise will it put out fire; but the coals taken out whilst
they are hot will revive again. It is an ordinary thing in many
coal-pits, whereof I have seen the experience, that a wind proceeding
from the sides of the pit every way, will extinguish any fire let down
into it, and kill the workmen, unless they be quickly taken out.
_A._ If you put a vessel of water into the receiver, and then suck out
the air, the water will boil; what say you to that?
_B._ It is like enough it will dance in so great a bustling of the air;
but I never heard it would be hot. Nor can I imagine how vacuum should
make anything dance. I hope you are by this time satisfied, that no
experiment made with the engine at Gresham College, is sufficient to
prove that there is, or that there may be vacuum.
_A._ The world you know is finite, and consequently, all that infinite
space without it is empty. Why may not some of that vacuum be brought
in, and mingled with the air here?
_B._ I know nothing in matters without the world.
_A._ What say you to Torricellio’s experiment in quicksilver, which is
this: there is a basin at A filled with quicksilver, suppose to B, and C
D a hollow glass pipe filled with the same, which if you stop with your
finger at B, and so set it upright, and then if you take away your
finger, the quicksilver will fall from C downwards but not to the
bottom, for it will stop by the way, suppose at D. Is it not therefore
necessary that that space between C and D be left empty? Or will you say
the quicksilver does not exactly touch the sides of the glass pipe?
_B._ I will say neither. If a man thrust down into a vessel of
quicksilver a blown bladder, will not that bladder come up to the top?
_A._ Yes, certainly, or a bladder of iron, or anything else but gold.
_B._ You see then that air can pierce quicksilver.
_A._ Yes, with so much force as the weight of quicksilver comes to.
_B._ When the quicksilver is fallen to D, there is so much the more in
the basin, and that takes up the place which so much air took up before.
Whither can this air go if all the world without that glass pipe B C
were full? There must needs be the same or as much air come into that
space, which only is empty, between C and D: by what force? By the
weight of the quicksilver between D and B. Which quicksilver weigheth
now upward, or else it could never have raised that part higher, which
was at first in the basin. So you see the weight of quicksilver can
press the air through quicksilver up into the pipe, till it come to an
equality of force as in D, where the weight of the quicksilver is equal
to the force which is required in air to go through it.
_A._ If a man suck a phial that has nothing in it but air, and presently
dip the mouth of it into water, the water will ascend into the phial. Is
not that an argument that part of the air had been sucked out, and part
of the room within the phial left empty?
_B._ No. If there were empty space in the world, why should not there be
also some empty space in the phial before it was sucked? And then why
does not the water rise to fill that? When a man sucks the phial he
draws nothing out, neither into his belly, nor into his lungs, nor into
his mouth; only he sets the air within the glass into a circular motion,
giving it at once an endeavour to go forth by the sucking, and an
endeavour to go back by not receiving it into his mouth; and so with a
great deal of labour glues his lips to the neck of the phial. Then
taking it off, and dipping the neck of the phial into the water before
the circulation ceases, the air, with the endeavour it hath now gotten,
pierces the water and goes out: and so much air as goes out, so much
matter comes up into the room of it.
==========
CHAPTER IV.
PROBLEMS OF HEAT AND LIGHT.
_A._ What is the cause of heat?
_B._ How know you, that any thing is hot but yourself?
_A._ Because I perceive by sense it heats me.
_B._ It is no good argument, the thing heats me; therefore it is hot.
But what alteration do you find in your body at any time by being hot?
_A._ I find my skin more extended in summer than in winter; and am
sometimes fainter and weaker then ordinary, as if my spirits were
exhaled; and I sweat.
_B._ Then that is it you would know the cause of. I have told you before
that by the motion I suppose both in the sun, and in the earth, the air
is dissipated, and consequently that there would be an infinite number
of small empty places, but that the world being full, there comes from
the next parts other air into the spaces they would else make empty.
When therefore this motion of the sun is exercised upon the superficies
of the earth, if there do not come out of the earth itself some corporal
substance to supply that tearing of the air, we must return again to the
admission of vacuum. If there do, then you see how by this motion fluid
bodies are made to exhale out of the earth. The like happens to a man’s
body or hand, which when he perceives, he says he is hot. And so of the
earth when it sendeth forth water and earth together in plants, we say
it does it by heat from the sun.
_A._ It is very probable, and no less probable, that the same action of
the sun is that which from the sea and moist places of the earth, but
especially from the sea, fetcheth up the water into the clouds. But
there be many ways of heating besides the action of the sun or of fire.
Two pieces of wood will take fire if in turning they be pressed
together.
_B._ Here again you have a manifest laceration of the air by the
reciprocal and contrary motions of the two pieces of wood, which
necessarily causeth a coming forth of whatsoever is aereal or fluid
within them, and (the motion pursued) a dissipation also of the other
more solid parts into ashes.
_A._ How comes it to pass that a man is warmed even to sweating, almost
with every extraordinary labour of his body?
_B._ It is easy to understand, how by that labour all that is liquid in
his body is tossed up and down, and thereby part of it also cast forth.
_A._ There be some things that make a man hot without sweat or other
evaporation, as caustics, nettles, and other things.
_B._ No doubt. But they touch the part they so heat, and cannot work
that effect at any distance.
_A._ How does heat cause light, and that partially, in some bodies more,
in some less, though the heat be equal?
_B._ Heat does not cause light at all. But in many bodies, the same
cause, that is to say, the same motion, causeth both together; so that
they are not to one another as cause and effect, but are concomitant
effects sometimes of one and the same motion.
_A._ How?
_B._ You know the rubbing or hard pressing of the eye, or a stroke upon
it, makes an apparition of light without and before it, which way soever
you look. This can proceed from nothing else but from the restitution of
the organ pressed or stricken, unto its former ordinary situation of
parts. Does not the sun by his thrusting back the air upon your eyes
press them? Or do not those bodies whereon the sun shines, though by
reflection, do the same, though not so strongly? And do not the organs
of sight, the eye, the heart, and brains, resist that pressure by an
endeavour of restitution outwards? Why then should there not be without
and before the eye, an apparition of light in this case as well as in
the other?
_A._ I grant there must. But what is that which appears after the
pressing of the eye? For there is nothing without that was not there
before; or if there were, methinks another should see it better, or as
well as he; or if in the dark, methinks it should enlighten the place.
_B._ It is a fancy, such as is the appearance of your face in a
looking-glass; such as is a dream; such as is a ghost; such as is a spot
before the eye that hath stared upon the sun or fire. For all these are
of the regiment of fancy, without any body concealed under them, or
behind them, by which they are produced.
_A._ And when you look towards the sun or moon, why is not that also
which appears before your eyes at that time a fancy?
_B._ So it is. Though the sun itself be a real body, yet that bright
circle of about a foot diameter cannot be the sun, unless there be two
suns, a greater and a lesser. And because you may see that which you
call the sun, both above you in the sky, and before you in the water,
and two suns, by distorting your eye, in two places in the sky, one of
them must needs be fancy. And if one, both. All sense is fancy, though
the cause be always in a real body.
_A._ I see by this that those things which the learned call the
accidents of bodies, are indeed nothing else but diversity of fancy, and
are inherent in the sentient, and not in the objects, except motion and
quantity. And I perceive by your doctrine you have been tampering with
_Leviathan_ . But how comes wood with a certain degree of heat to shine,
and iron also with a greater degree; but no heat at all to be able to
make water shine?
_B._ That which shineth hath the same motion in its parts that I have
all this while supposed in the sun and earth. In which motion there must
needs be a competent degree of swiftness to move the sense, that is, to
make it visible. All bodies that are not fluid will shine with heat, if
the heat be very great. Iron will shine and gold will shine; but water
will not, because the parts are carried away before they attain to that
degree of swiftness which is requisite.
_A._ There are many fluid bodies whose parts evaporate, and yet they
make a flame, as oil, and wine, and other strong drinks.
_B._ As for oil I never saw any inflamed by itself, how much soever
heated, therefore I do not think they are the parts of the oil, but of
the combustible body oiled that shine; but the parts of wine and strong
drinks have partly a strong motion of themselves, and may be made to
shine, but not with boiling, but by adding to them as they rise the
flame of some other body.
_A._ How can it be known that the particles of wine have such a motion
as you suppose?
_B._ Have you ever been so much distempered with drinking wine, as to
think the windows and table move?
_A._ I confess, though you be not my confessor, I have; but very seldom;
and I remember the window seemed to go and come in a kind of circling
motion, such as you have described. But what of that?
_B._ Nothing, but that it was the wine that caused it; which having a
good degree of that motion before, did, when it was heated in the veins,
give that concussion, which you thought was in the window, to the veins
themselves, and, by the continuation of the parts of man’s body, to the
brain; and that was it which made the window seem to move.
_A._ What is flame? For I have often thought the flame that comes out of
a small heap of straw to be more, before it hath done flaming, than a
hundred times the straw itself.
_B._ It was but your fancy. If you take a stick in your hand by one end,
the other end burning, and move it swiftly, the burning end, if the
motion be circular, shall seem a circle; if straight, a straight line of
fire, longer or shorter, according to the swiftness of the motion, or
the space it moves in. You know the cause of that.
_A._ I think it is, because the impression of that visible object, which
was made at the first instant of the motion, did last till it was ended.
For then it will follow that it must be visible all the way, the
impressions in all points of the time being equal.
_B._ The cause can be no other. The smallest spark of fire flying up
seems a line drawn upward; and again by that swift circular motion which
we have supposed for the cause of light, seems also broader than it is.
And consequently the flame of every thing must needs seem much greater
than it is.
_A._ What are those sparks that fly out of the fire?
_B._ They are small pieces of the wood or coals, or other fuel loosened
and carried away with the air that cometh up with them. And being
extinguished before their parts be quite dissipated into others, are so
much soot, and black, and may be fired again.
_A._ A spark of fire may be stricken out of a cold stone. It is not
therefore heat that makes this shining.
_B._ No it is the motion that makes both the heat and shining; and the
stroke makes the motion. For every of those sparks, is a little parcel
of the stone, which swiftly moved, imprinteth the same motion into the
matter prepared, or fit to receive it.
_A._ How comes the light of the sun to burn almost any combustible
matter by refraction through a convex glass, and by reflection from a
concave?
_B._ The air moved by the sun presseth the convex glass in such manner
as the action continued through it, proceedeth not in the same straight
line by which it proceeded from the sun, but tendeth more toward the
centre of the body it enters. Also when the action is continued through
the convex body, it bendeth again the same way. By which means the whole
action of the sun-beams are enclosed within a very small compass: in
which place therefore there must be a very vehement motion; and
consequently, if there be in that place combustible matter, such as is
not very hard to kindle, the parts of it will be dissipated, and receive
that motion which worketh on the eye as other fire does.
The same reason is to be given for burning by reflection. For there also
the beams are collected into almost a point.
_A._ Why may not the sun-beams be such a body as we call fire, and pass
through the pores of the glass so disposed as to carry them to a point,
or very near?
_B._ Can there be a glass that is all pores? if there cannot, then
cannot this effect be produced by the passing of fire through the pores.
You have seen men light their tobacco at the sun with a burning glass,
or with a ball of crystal, held which way they will indifferently. Which
must be impossible, unless the ball were all pores. Again, neither you
nor I can conceive any other fire than we have seen, nor than such as
water will put out. But not only a solid globe of glass or crystal will
serve for a burning-glass, but also a hollow one filled with water. How
then does the fire from the sun pass through the glass of water without
being put out before it come to the matter they would have it burn?
_A._ I know not. There comes nothing from the sun. If there did, there
is come so much from it already, that at this day we had had no sun.
==========
CHAPTER V.
PROBLEMS OF HARD AND SOFT.
_A._ What call you hard, and what soft?
_B._ That body whereof no one part is easily put out of its place,
without removing the whole, is that which I and all men call hard; and
the contrary soft. So that they are but degrees one of another.
_A._ What is the cause that makes one body harder than another, or,
seeing you say they are but degrees of one another, what makes one body
softer than another, and the same body sometimes harder, sometimes
softer?
_B._ The same motion which we have supposed from the beginning for the
cause of so many other effects. Which motion not being upon the centre
of the part moved, but the part itself going in another circle to and
again, it is not necessary that the motion be perfectly circular. For it
is not circulation, but the reciprocation, I mean the to and again, that
does cast off, and lacerate the air, and consequently produce the
fore-mentioned effects.
For the cause therefore of hardness, I suppose the reciprocation of
motion in those things which are hard, to be very swift, and in very
small circles.
_A._ This is somewhat hard to believe. I would you could supply it with
some visible experience.
_B._ When you see, for example, a cross-bow bent, do you think the parts
of it stir?
_A._ No. I am sure they do not.
_B._ How are you sure? You have no argument for it, but that you do not
see the motion. When I see you sitting still, must I believe there is no
motion in your parts within, when there are so many arguments to
convince me there is.
_A._ What argument have you to convince me that there is motion in a
cross-bow when it stands bent?
_B._ If you cut the string, or any way set the bow at liberty, it will
have then a very visible motion. What can be the cause of that?
_A._ Why the setting of the bow at liberty.
_B._ If the bow had been crooked before it was bent, and the string tied
to both ends, and then cut asunder, the bow would not have stirred.
Where lies the difference?
_A._ The bow bent has a spring; unbent it has none, how crooked soever.
_B._ What mean you by spring?
_A._ An endeavour of restitution to its former posture.
_B._ I understand spring as well as I do endeavour.
_A._ I mean a principle or beginning of motion in a contrary way to that
of the force which bent it.
_B._ But the beginning of motion is also motion, how insensible soever
it be. And you know that nothing can give a beginning of motion to
itself. What is it therefore that gives the bow (which you say you are
sure was at rest when it stood bent) its first endeavour to return to
its former posture?
_A._ It was he that bent it.
_B._ That cannot be. For he gave it an endeavour to come forward, and
the bow endeavours to go backward.
_A._ Well, grant that endeavour be motion, and motion in the bow unbent,
how do you derive from thence, that being set at liberty it must return
to its former posture?
_B._ Thus there being within the bow a swift (though invisible) motion
of all the parts, and consequently of the whole; the bending causeth
that motion, which was along the bow (that was beaten out when it was
hot into that length) to operate across the length in every part of it,
and the more by how much it is more bent; and consequently endeavours to
unbend it all the while it stands bent. And therefore when the force
which kept it bent is removed, it must of necessity return to the
posture it had before.
_A._ But has that endeavour no effect at all before the impediment be
removed? For if endeavour be motion, and every motion have some effect
more or less, methinks this endeavour should in time produce something.
_B._ So it does. For in time (in a long time) the course of this
internal motion will lie along the bow, not according to the former, but
to the new acquired posture. And then it well be as uneasy to return it
to its former posture, as it was before to bend it.
_A._ That is true. For bows long bent lose their appetite to
restitution, long custom becoming nature. But from this internal
reciprocation of the parts, how do you infer the hardness of the whole
body?
_B._ If you apply force to any single part of such a body, you must
needs disorder the motion of the next parts to it before it yield, and
there disordered, the motion of the next again must also be disordered;
and consequently no one part can yield without force sufficient to
disorder all: but then the whole body must also yield. Now when a body
is of such a nature as no single part can be removed without removing
the whole, men say that body is hard.
_A._ Why does the fire melt divers hard bodies, and yet not all?
_B._ The hardest bodies are those wherein the motion of the parts are
the most swift, and yet in the least circles. Wherefore if the fire, the
motion of whose parts are swift, and in greater circles, be made so
swift, as to be strong enough to master the motion of the parts of the
hard body, it will make those parts to move in a greater compass, and
thereby weaken their resistance, that is to say, soften them, which is a
degree of liquefaction. And when the motion is so weakened, as that the
parts lose their coherence by the force of their own weight, then we
count the body melted.
_A._ Why are the hardest things the most brittle, insomuch that what
force soever is enough to bend them, is enough also to break them?
_B._ In bending a hard body, as (for example) a rod of iron, you do not
enlarge the space of the internal motion of the parts of iron, as the
fire does; but you master and interrupt the motion, and that chiefly in
one place. In which place the motion that makes the iron hard being once
overcome, the prosecution of that bending must needs suddenly master the
motions of the parts next unto it, being almost mastered before.
_A._ I have seen a small piece of glass, the figure whereof is this, A A
B C. Which piece of glass if you bend toward the top, as in C, the whole
body will shatter asunder into a million of pieces, and be like to so
much dust. I would fain see you give a probable reason of that.
_B._ I have seen the experiment. The making of the glass is thus: they
dip an iron rod into the molten glass that stands in a vessel within the
furnace. Upon which iron rod taken out, there will hang a drop of molten
but tough metal of the figure you have described, which they let fall
into the water. So that the main drop comes first to the water, and
after it the tail, which though straight whilst it hung on the end of
the rod, yet by falling into the water becomes crooked. Now you know the
making of it, you may consider what must be the consequence of it.
Because the main drop A comes first to the water, it is therefore first
quenched, and consequently the motion of the parts of that drop, which
by the fire were made to be moved in a larger compass, is by the water
made to shrink into lesser circles towards the other end B, but with the
same or not much less swiftness.
_A._ Why so?
_B._ If you take any long piece of iron, glass, or other uniform and
continued body; and having heated one end thereof, you hold the other
end in your hand, and so quench it suddenly, though before you held it
easily enough, yet now it will burn your fingers.
_A._ It will so.
_B._ You see then how the motion of the parts from A toward C is made
more violent and in less compass by quenching the other parts first.
Besides, the whole motion that was in all the parts of the main drop A,
is now united in the small end B C. And this I take to be the cause why
that small part B C is so exceeding stiff. Seeing also this motion in
every small part of the glass, is not only circular, but proceeds also
all along the glass from A to B, the whole motion compounded will be
such as the motion of spinning any soft matter into thread, and will
dispose the whole body of the glass in threads, which in other hard
bodies are called the grain. Therefore if you bend this body (for
example) in C (which to do will require more force than a man would
think that has not tried) those threads of glass must needs be all bent
at the same time, and stand so, till by the breaking of the glass at C,
they be all at once set at liberty; and then all at once being suddenly
unbent, like so many brittle and overbent bows, their strings breaking,
be shivered in pieces.
_A._ It is like enough to be so. And if nature have betrayed herself in
any thing, I think it is in this, and in that other experience of the
crossbow; which strongly and evidently demonstrates the internal
reciprocation of the motion, which you suppose to be in the internal
parts of every hard body. And I have observed somewhat in
looking-glasses which much confirms that there is some such motion in
the internal parts of glass, as you have supposed for the cause of
hardness. For let the glass be A B, and let the object at C be a candle,
and the eye at D. Now by divers reflections and refractions in the two
superficies of the glass, if the lines of vision be very oblique, you
shall see many images of the candle, as E, F, G, in such order and
position as is here described. But if you remove your eye to C, and the
candle to D, they will appear in a situation manifestly different from
this. Which you will yet more plainly perceive if the looking-glass be
coloured, as I have observed in red and blue glasses; and could never
conceive any probable cause of it, till now you tell me of this secret
motion of the parts across the grain of the glass, acquired by cooling
it this or that way.
_B._ There be very many kinds of hard bodies, metals, stones, and other
kinds, in the bowels of the earth, that have been there ever since the
beginning of the world; and I believe also many different sorts of
juices that may be made hard. But for one general cause of hardness it
can be no other than such an internal motion of parts as I have already
described, whatsoever may be the cause of the several concomitant
qualities of their hardness in particular.
_A._ We see water hardened every frosty day. It is likely therefore you
may give a probable cause of ice. What is the cause of freezing of the
ocean towards the poles of the earth?
_B._ You know the sun being always between the tropics, and (as we have
supposed) always casting off the air; and the earth likewise casting it
off from itself, there must needs on both sides be a great stream of air
towards the poles, shaving the superficies of the earth and sea, in the
northern and southern climates. This shaving of the earth and sea by the
stream of air must needs contract and make to shrink those little
circles of the internal parts of earth and water, and consequently
harden them, first at the superficies, into a thin skin, which is the
first ice; and afterwards the same motion continuing, and the first ice
co-operating, the ice becomes thicker. And this I conceive to be the
cause of the freezing of the ocean.
_A._ If that be the cause, I need not ask how a bottle of water is made
to freeze in warm weather with snow, or ice mingled with salt. For when
the bottle is in the midst of it, the wind that goeth out both of the
salt and of the ice as they dissolve, must needs shave the superficies
of the bottle, and the bottle work accordingly on the water without it,
and so give it first a thin skin, and at last thicken it into a solid
piece of ice. But how comes it to pass that water does not use to freeze
in a deep pit?
_B._ A deep pit is a very thick bottle, and such as the air cannot come
at but only at the top, or where the earth is very loose and spungy.
_A._ Why will not wine freeze as well as water?
_B._ So it will when the frost is great enough. But the internal motion
of the parts of wine and other heating liquors is in greater circles and
stronger than the motion of the parts of water; and therefore less
easily to be frozen, especially quite through, because those parts that
have the strongest motion retire to the centre of the vessel.
==========
CHAPTER VI.
PROBLEMS OF RAIN, WIND, AND OTHER WEATHER.
_A._ What is the original cause of rain? And how is it generated?
_B._ The motion of the air (such as I have described to you already)
tending to the disunion of the parts of the air, must needs cause a
continual endeavour (there being no possibility of vacuum) of whatsoever
fluid parts there are upon the face of the earth and sea, to supply the
place which would else be empty. This makes the water, and also very
small and loose parts of the earth and sea to rise, and mingle
themselves with the air, and to become mist and clouds. Of which the
greatest quantity arise there, where there is most water, namely, from
the large parts of the ocean; which are the South Sea, the Indian Sea,
and the sea that divided Europe and Africa from America; over which the
sun for the greatest part of the year is perpendicular, and consequently
raiseth a greater quantity of water; which afterwards gathered into
clouds, falls down in rain.
_A._ If the sun can thus draw up the water, though but in small drops,
why can it not as easily hold it up?
_B._ It is likely it would also hold them up, if they did not grow
greater by meeting together, nor were carried away by the air towards
the poles.
_A._ What makes them gather together?
_B._ It is not improbable that they are carried against hills, and there
stopt till more overtake them. And when they are carried towards the
North or South where the force of the sun is more oblique and thereby
weaker, they descend gently by their own weight. And because they tend
all to the centre of the earth, they must needs be united in their way
for want of room, and so grow bigger. And then it rains.
_A._ What is the reason it rains so seldom, but snows so often upon very
high mountains?
_B._ Because, perhaps, when the water is drawn up higher than the
highest mountains, where the course of the air between the equator and
the poles is free from stoppage, the stream of the air freezeth it into
snow. And it is in those places only where the hills shelter it from
that stream, that it falls in rain.
_A._ Why is there so little rain in Egypt, and yet so much in other
parts nearer the equinoxial, as to make the Nile overflow the country?
_B._ The cause of the falling of rain I told you was the stopping, and
consequently the collection of clouds about great mountains, especially
when the sun is near the equinoxial, and thereby draws up the water more
potently, and from greater seas. If you consider therefore that the
mountains in which are the springs of Nile, lie near the equinoxial and
are exceedingly great, and near the Indian Sea, you will not think it
strange there should be great store of snow. This as it melts makes the
rain of Nile to rise, which in April and May going on toward Egypt
arrive there about the time of the solstice, and overflow the country.
_A._ Why should not the Nile then overflow that country twice a year,
for it comes twice a-year to the equinoxial.
_B._ From the autumnal equinox, the sun goeth on toward the southern
tropic, and therefore cannot dissolve the snow on that side of the hills
that looks towards Egypt.
_A._ But then there ought to be such another inundation southward.
_B._ No doubt but there is a greater descent of water there in their
summer than at other times, as there must be wheresoever there is much
snow melted. But what should that inundate, unless it should overflow
the sea that comes close to the foot of those mountains? And for the
cause why it seldom rains in Egypt, it may be this, that there are no
very high hills near it to collect the clouds. The mountains whence Nile
riseth being near two thousand miles off. The nearest on one side are
the mountains of Nubia, and on the other side Sina and the mountains of
Arabia.
_A._ Whence think you proceed the winds?
_B._ From the motion, I think, especially of the clouds, partly also
from whatsoever is moved in the air.
_A._ It is manifest that the clouds are moved by the winds; so that
there were winds before any clouds could be moved. Therefore I think you
make the effect before the cause.
_B._ If nothing could move a cloud but wind, your objection were good.
But you allow a cloud to descend by its own weight. But when it so
descends, it must needs move the air before it, even to the earth, and
the earth again repel it, and so make lateral winds every way, which
will carry forward other clouds if there be any in their way, but not
the cloud that made them. The vapour of the water rising into clouds,
must needs also, as they rise, raise a wind.
_A._ I grant it. But how can the slow motion of a cloud make so swift a
wind as it does?
_B._ It is not one or two little clouds, but many and great ones that do
it. Besides, when the air is driven into places already covered, it
cannot but be much the swifter for the narrowness of the passage.
_A._ Why does the south wind more often than any other bring rain with
it?
_B._ Where the sun hath most power, and where the seas are greatest,
that is in the south, there is most water in the air; which a south wind
can only bring to us. But I have seen great showers of rain sometimes
also when the wind hath been north, but it was in summer, and came
first, I think, from the south or west, and was brought back from the
north.
_A._ I have seen at sea very great waves when there was no wind at all.
What was it then that troubled the water?
_B._ But had you not wind enough presently after?
_A._ We had a storm within a little more than a quarter of an hour
after.
_B._ That storm was then coming and had moved the water before it. But
the wind you could not perceive, for it came downwards with the
descending of the clouds, and pressing the water bounded above your sail
till it came very near. And that was it that made you think there was no
wind at all.
_A._ How comes it to pass that a ship should go against the wind which
moves it, even almost point blank, as if it were not driven but drawn?
_B._ You are to know first, that what body soever is carried against
another body, whether perpendicularly or obliquely, it drives it in a
perpendicular to the superficies it lighteth on. As for example, a
bullet shot against a flat wall, maketh the stone, or other matter it
hits, to retire in a perpendicular to that flat; or, if the wall be
round, towards the centre, that is to say, perpendicularly. For if the
way of the motion be oblique to the wall, the motion is compounded of
two motions, one parallel to the wall, and the other perpendicular. By
the former whereof the bullet is carried along the wall side, by the
other it approacheth to it. Now the former of these motions can have no
effect upon it; all the battery is from the motion perpendicular, in
which it approacheth, and therefore the part it hits must also retire
perpendicularly. If it were not so, a bullet with the same swiftness
would execute as much obliquely shot, as perpendicularly, which you know
it does not.
_A._ How do you apply this to a ship?
_B._ Let A B be the ship, the head of it A. If the wind blow just from A
towards B, it is true the ship cannot go forward howsoever the sail be
set. Let C D be perpendicular to the ship, and let the sail E C be never
so little oblique to it, and F C perpendicular to E C, and then you see
the ship will gain the space D F to the headward.
_A._ It will so; but when it is very near to the wind it will go forward
very slowly, and make more way with her side to the leeward.
_B._ It will indeed go slower in the proportion of the line A E to the
line C E. But the ship will not go so fast as you think sideward: one
cause is the force of that wind which lights on the side of the ship
itself; the other is the bellying of the sail; for the former, it is not
much, because the ship does not easily put from her the water with her
side; and bellying of the sail gives some little hold for the wind to
drive the ship astern.
_A._ For the motion sideward I agree with you; but I had thought the
bellying of the sail had made the ship go faster.
_B._ But it does not; only in a fore wind it hinders least.
_A._ By this reason a broad thin board should make the best sail.
_B._ You may easily foresee the great incommodities of such a sail. But
I have seen tried in little what such a wind can do in such a case. For
I have seen a board set upon four truckles, with a staff set up in the
midst of it for a mast, and another very thin and broad board fastened
to that staff in the stead of a sail, and so set as to receive the wind
very obliquely, I mean so as to be within a point of the compass
directly opposite to it, and so placed upon a reasonable smooth pavement
where the wind blew somewhat strongly. The event was first, that it
stood doubting whether it should stir at all or no, but that was not
long, and then it ran a-head extreme swiftly, till it was overthrown by
a rub.
_A._ Before you leave the ship, tell me how it comes about that so small
a thing as a rudder can so easily turn the greatest ship.
_B._ It is not the rudder only, there must also be a stream to do it;
you shall never turn a ship with a rudder in a standing pool, nor in a
natural current. You must make a stream from head to stern, either with
oars or with sails; when you have made such a stream, the turning of the
rudder obliquely holds the water from passing freely, and the ship or
boat cannot go on directly, but as the rudder inclines to the stern, so
will the ship turn; but this is too well known to insist upon. You have
observed that the rudders of the greatest ships are not very broad, but
go deep into the water, whereas western barges, though but small
vessels, have their rudders much broader, which argues that the holding
of water from passing is the true office of a rudder; and therefore to a
ship that draws much water the rudder is made deep accordingly; and in
barges that draw little water, the rudders being less deep, must so much
the more be extended in breadth.
_A._ What makes snow?
_B._ The same cause which, speaking of hardness, I supposed for the
cause of ice. For the stream of air proceeding from that both the earth
and the sun cast off the air, consequently maketh a stream of air from
the equinoxial towards the poles, passing amongst the clouds, shaving
those small drops of water whereof the clouds consist, and congeals them
as they do the water of the sea, or of a river. And these small frozen
drops are that which we call snow.
_A._ But then how are great drops frozen into hailstones, and that
especially (as we see they are) in summer?
_B._ It is especially in summer, and hot weather, that the drops of
water which make the clouds, are great enough; but it is then also that
clouds are sooner and more plentifully carried up. And therefore the
current of the air strengthened between the earth and the clouds,
becomes more swift; and thereby freezeth the drops of water, not in the
cloud itself, but as they are falling. Nor does it freeze them
thoroughly, the time of their falling not permitting it, but gives them
only a thin coat of ice; as is manifest by their sudden dissolving.
_A._ Why are not sometimes also whole clouds when pregnant and ready to
drop, frozen into one piece of ice?
_B._ I believe they are so whensoever it thunders.
_A._ But upon what ground do you believe it?
_B._ From the manner or kind of noise they make, namely a crack; which I
see not how it can possibly be made by water or any other soft bodies
whatsoever.
_A._ Yes, the powder they call _aurum fulminans_, when thoroughly warm,
gives just such another crack as thunder.
_B._ But why may not every small grain of that _aurum fulminans_ by
itself be heard, though a heap of them together be soft, as is any heap
of sand. Salts of all sorts are of the nature of ice. But gold is
dissolved into _aurum fulminans_ by nitre and other salts. And the least
grain of it gives a little crack in the fire by itself. And therefore
when they are so warmed by degrees, the crack cannot choose but be very
great.
_A._ But before it be _aurum fulminans_ they use to wash away the salt
(which they call dulcifying it), and then they dry it gently by degrees.
_B._ That is, they exhale the pure water that is left in the powder, and
leave the salt behind to harden with drying. Other powder made of salts
without any gold in them will give a crack as great as _aurum
fulminans_. A very great chemist of our times hath written, that salt of
tartar, saltpetre, and a little brimstone ground together into a powder,
and dried, a few grains of that powder will be made by the fire to give
as great a clap as a musket.
_A._ Methinks it were worth your trial to see what effect a quart or a
pint of _aurum fulminans_ would produce, being put into a great gun made
strong enough on purpose, and the breech of the gun set in hot cinders,
so as to heat by degrees, till the powder fly.
_B._ I pray you try it yourself; I cannot spare so much money.
_A._ What is it that breaketh the clouds when they are frozen?
_B._ In very hot weather the sun raiseth from the sea and all moist
places abundance of water, and to a great height. And whilst this water
hangs over us in clouds, or is again descending, it raiseth other
clouds, and it happens very often that they press the air between them,
and squeeze it through the clouds themselves very violently; which as it
passes shaves and hardens them in the manner declared.
_A._ That has already been granted; my question is what breaks them?
_B._ I must here take in one supposition more.
_A._ Then your basin, it seems, holds not all you have need of.
_B._ It may for all this, for the supposition I add is no more but this;
that what internal motion I ascribe to the earth, and the other concrete
parts of the world, is to be supposed also in every of their parts how
small soever; for what reason is there to think, in case the whole earth
have in truth the motion I have ascribed to it, that one part of it
taken away, the remaining part should lose that motion. If you break a
loadstone, both parts will retain their virtue, though weakened
according to the diminution of their quantity; I suppose therefore in
every small part of the earth the same kind of motion, which I have
supposed in the whole: and so I recede not yet from my basin.
_A._ Let it be supposed, and withal, that abundance of earth, (which I
see you aim at), be drawn up together with the water. What then?
_B._ Then if many pregnant clouds, some ascending and some descending
meet together, and make concavities between, and by the pressing out of
the air, as I have said before, become ice; those atoms, as I may call
them, of earth will, by the straining of the air through the water of
the clouds, be left behind, and remain in the cavities of the clouds,
and be more in number than for the proportion of the air therein.
Therefore for want of liberty they must needs justle one another, and
become, as they are more and more straightened of room, more and more
swift, and consequently at last break the ice suddenly and violently,
now in one place, and by and by in another; and make thereby so many
claps of thunder, and so many flashes of lightning. For the air
recoiling upon our eyes, is that which maketh those flashes to our
fancy.
_A._ But I have seen lightning in a very clear evening, when there has
been neither thunder nor clouds.
_B._ Yes, in a clear evening; because the clouds and the rain were below
the horizon, perhaps forty or fifty miles off; so that you could not see
the clouds nor hear the thunder.
_A._ If the clouds be indeed frozen into ice, I shall not wonder if they
be sometimes also so situated, as, like looking-glasses, to make us see
sometimes three or more suns by refraction and reflection.
==========
CHAPTER VII.
PROBLEMS OF MOTION PERPENDICULAR, AND OBLIQUE;
OF PRESSION AND PERCUSSION; REFLECTION AND
REFRACTION; ATTRACTION AND REPULSION.
_A._ If a bullet from a certain point given, be shot against a wall
perpendicularly, and again from the same point obliquely, what will be
the proportion of the forces wherewith they urge the wall? For example,
let the wall be A B, a point given E, a gun C E, that carries the bullet
perpendicularly to F, and another gun D E, that carries the like bullet
with the same swiftness obliquely to G; in what proportion will their
forces be upon the wall?
_B._ The force of the stroke perpendicular from E to F will be greater
then the oblique force from E to G, in the proportion of the line E G to
the line E F.
_A._ How can the difference be so much? Can the bullet lose so much of
its force in the way from E to G?
_B._ No; we will suppose it loseth nothing of its swiftness. But the
cause is, that their swiftness being equal, the one is longer in coming
to the wall than the other, in proportion of time, as E G to E F. For
though their swiftness be the same, considered in themselves, yet the
swiftness of their approach to the wall is greater in E F than in E G,
in proportion of the lines themselves.
_A._ When a bullet enters not, but rebounds from the wall, does it make
the same angle going off, which it did falling on, as the sun-beams do?
_B._ If you measure the angles close by the wall their difference will
not be sensible; otherwise it will be great enough, for the motion of
the bullet grows continually weaker. But it is not so with the sun-beams
which press continually and equally.
_A._ What is the cause of reflection? When a body can go no further on,
it has lost its motion. Whence then comes the motion by which it
reboundeth?
_B._ This motion of rebounding or reflecting proceedeth from the
resistance. There is a difference to be considered between the
reflection of light, and of a bullet, answerable to their different
motions, pressing and striking. For the action which makes reflection of
light, is the pressure of the air upon the reflecting body, caused by
the sun, or other shining body, and is but a contrary endeavour; as if
two men should press with their breasts upon the two ends of a staff,
though they did not remove one another, yet they would find in
themselves a great disposition to press backward upon whatsoever is
behind them, though not a total going out of their places. Such is the
way of reflecting light. Now, when the falling on of the sun-beams is
oblique, the action of them is nevertheless perpendicular to the
superficies it falls on. And therefore the reflecting body, by
resisting, turneth back that motion perpendicularly, as from F to E; but
taketh nothing from the force that goes on parallel in the line of E H,
because the motion never presses. And thus of the two motions from F to
E, and from E to H, is a compounded motion in the line F H, which maketh
an angle in B G, equal to the angle F G E.
But in percussion (which is the motion of the bullet against a wall,)
the bullet no sooner goeth off than it loseth of its swiftness, and
inclineth to the earth by its weight. So that the angles made in falling
on and going off, cannot be equal, unless they be measured close to the
point where the stroke is made.
_A._ If a man set a board upright upon its edge, though it may very
easily be cast down with a little pressure of one’s finger, yet a bullet
from a musket shall not throw it down, but go through it. What is the
cause of that?
_B._ In pressing with your finger you spend time to throw it down. For
the motion you give to the part you touch is communicated to every other
part before it fall. For the whole cannot fall till every part be moved.
But the stroke of a bullet is so swift, as it breaks through, before the
motion of the part it hits can be communicated to all the other parts
that must fall with it.
_A._ The stroke of a hammer will drive a nail a great way into a piece
of wood on a sudden. What weight laid upon the head of a nail, and in
how much time will do the same? It is a question I have heard propounded
amongst naturalists.
_B._ The different manner of the operation of weight from the operation
of a stroke, makes it incalculable. The suddenness of the stroke upon
one point of the wood takes away the time of resistance from the rest.
Therefore the nail enters so far as it does. But the weight not only
gives them time, but also augments the resistance; but how much, and in
how much time, is, I think, impossible to determine.
_A._ What is the difference between reflection and recoiling?
_B._ Any reflection may, and not unproperly, be called recoiling; but
not contrariwise every recoiling reflection. Reflection is always made
by the reaction of a body pressed or stricken; but recoiling not always.
The recoiling of a gun is not caused by its own pressing upon the
gunpowder, but by the force of the powder itself, inflamed and moved
every way alike.
_A._ I had thought it had been by the sudden re-entering of the air
after the flame and bullet were gone out. For it is impossible that so
much room as is left empty by the discharging of the gun, should be so
suddenly filled with the air that entereth at the touchhole.
_B._ The flame is nothing but the powder itself, which scattered into
its smallest parts, seems of greater bulk by much, than in truth it is,
because they shine. And as the parts scatter more and more, so still
more air gets between them, entering not only at the touchhole, but also
at the mouth of the gun, which two ways being opposite, it will be much
too weak to make the gun recoil.
_A._ I have heard that a great gun charged too much or too little, will
shoot, not above, nor below, but beside the mark; and charged with one
certain charge between both, will hit it.
_B._ How that should be I cannot imagine. For when all things in the
cause are equal, the effects cannot be unequal. As soon as fire is
given, and before the bullet be out, the gun begins to recoil. If then
there be any unevenness or rub in the ground more on one side than on
the other, it shall shoot beside the mark, whether too much, or too
little, or justly charged; because if the line wherein the gun recoileth
decline, the way of the bullet will also decline to the contrary side of
the mark. Therefore I can imagine no cause of this event, but either in
the ground it recoils on, or in the unequal weight of the parts of the
breech.
_A._ How comes refraction?
_B._ When the action is in a line perpendicular to the superficies of
the body wrought upon, there will be no refraction at all. The action
will proceed still in the same straight line, whether it be pression as
in light, or percussion as in the shooting of a bullet. But when the
pression is oblique, then will the refraction be that way which the
nature of the bodies through which the action proceeds shall determine.
_A._ How is light refracted?
_B._ If it pass through a body of less, into a body of greater
resistance, and to the point of the superficies it falleth on, you draw
a line perpendicular to the same superficies, the action will proceed
not in the same line by which it fell on, but in another line bending
toward that perpendicular.
_A._ What is the reason of that?
_B._ I told you before, that the falling on worketh only in the
perpendicular; but as soon as the action proceedeth farther inward than
a mere touch, it worketh partly in the perpendicular, and partly
forward, and would proceed in the same line in which it fell on, but for
the greater resistance which now weakeneth the motion forward, and makes
it to incline towards the perpendicular.
_A._ In transparent bodies it may be so; but there be bodies through
which the light cannot pass at all.
_B._ But the action by which light is made, passeth through all bodies.
For this action is pression; and whatsoever is pressed, presseth that
which is next behind, and so continually. But the cause why there is no
light seen through it, is the unevenness of the parts within, whereby
the action is by an infinite number of reflections so diverted and
weakened, that before it hath proceeded through, it hath not strength
left to work upon the eye strongly enough to produce sight.
_A._ If the body being transparent, the action proceed quite through,
into a body again of less resistance, as out of glass into the air,
which way shall it then proceed in the air?
_B._ From the point where it goeth forth, draw a perpendicular to the
superficies of the glass, the action now freed from the resistance it
suffered, will go from that perpendicular, as much as it did before come
towards it.
_A._ When a bullet from out of the air entereth into a wall of earth,
will that also be refracted towards the perpendicular.
_B._ If the earth be all of one kind, it will. For the parallel motion,
will there also at the first entrance be resisted, which it was not
before it entered.
_A._ How then comes a bullet, when shot very obliquely into any broad
water, and having entered, yet to rise again into the air?
_B._ When a bullet is shot very obliquely, though the motion be never so
swift, yet the approach downwards to the water is very slow, and when it
cometh to it, it casteth up much water before it, which with its weight
presseth downwards again, and maketh the water to rise under the bullet
with force enough to master the weak motion of the bullet downwards, and
to make it rise in such manner as bodies use to rise by reflection.
_A._ By what motion (seeing you ascribe all effects to motion) can a
loadstone draw iron to it?
_B._ By the same motion hitherto supposed. But though all the smallest
parts of the earth have this motion, yet it is not supposed that their
motions are in equal circles; nor that they keep just time with one
another; nor that they have all the same poles. If they had, all bodies
would draw one another alike. For such an agreement of motion, of way,
of swiftness, and of poles, cannot be maintained, without the
conjunction of the bodies themselves in the centre of their common
motion, but by violence. If therefore the iron have but so much of the
nature of the loadstone as readily to receive from it the like motion,
as one string of a lute doth from another string strained to the same
note, (as it is like enough it hath, the loadstone being but one kind of
iron ore), it must needs after that motion received from it, unless the
greatness of the weight hinder, come nearer to it, because at distance
their motions will differ in time, and oppose each other, whereby they
will be forced to a common centre. If the iron be lifted up from the
earth, the motion of the loadstone must be stronger, or the body of it
nearer, to overcome the weight; and then the iron will leap up to the
loadstone as swiftly, as from the same distance it would fall down to
the earth; but if both the stone and the iron be set floating upon the
water, the attraction will begin to be manifest at a greater distance,
because the hindrance of the weight is in part removed.
_A._ But why does the loadstone, if it float on a calm water, never fail
to place itself at last in the meridian just north and south.
_B._ Not so, just in the meridian, but almost in all places with some
variations. But the cause I think is, that the axis of this magnetical
motion is parallel to the axis of the ecliptic, which is the axis of the
like motion in the earth, and consequently that it cannot freely
exercise its natural motion in any other situation.
_A._ Whence may this consent of motion in the loadstone and the earth
proceed? Do you think, as some have written, that the earth is a great
loadstone?
_B._ Dr. Gilbert, that was the first that wrote anything of this subject
rationally, inclines to that opinion. Descartes thought the earth,
excepting this upper crust of a few miles depth, to be of the same
nature with all other stars, and bright. For my part, I am content to be
ignorant; but I believe the loadstone hath been given its virtue by a
long habitude in the mine, the vein of it lying in the plane of some of
the meridians, or rather of some of the great circles that pass through
the poles of the ecliptic, which are the same with the poles of the like
motion supposed in the earth.
_A._ If that be true, I need not ask why the filings of iron laid on a
loadstone equally distant from its poles will lie parallel to the axis,
but on each side will incline to the pole that is next. Nor why by
drawing a loadstone all along a needle of iron, the needle will receive
the same poles. Nor why when the loadstone and iron, or two loadstones,
are put together floating upon water, will fall one of them astern of
the other, that their like parts may look the same way, and their unlike
touch, in which action they are commonly said to repel one another. For
all this may be derived from the union of their motions. One thing more
I desire to know, and that is; what are those things they call spirits?
I mean ghosts, fairies, hobgoblins, and the like apparitions.
_B._ They are no part of the subject of natural philosophy.
_A._ That which in all ages, and all places is commonly seen (as those
have been, unless a great part of mankind be liars) cannot, I think, be
supernatural.
_B._ All this that I have hitherto said, though upon better ground than
can be had for a discourse of ghosts, you ought to take but for a dream.
_A._ I do so. But there be some dreams more like sense then others. And
that which is like sense pleases me as well in natural philosophy, as if
it were the very truth.
_B._ I was dreaming also once of these things; but was wakened by their
noise. And they never came into any dream of mine since, unless
apparitions in dreams and ghost be all one.
==========
CHAPTER VIII.
THE DELPHIC PROBLEM, OR DUPLICATION OF THE
CUBE.
_A._ Have you seen a printed paper sent from Paris, containing the
duplication of the cube, written in French?
_B._ Yes. It was I that writ it, and sent it thither to be printed, on
purpose to see what objections would be made to it by our professors of
algebra here.
_A._ Then you have also seen the confutations of it by algebra.
_B._ I have seen some of them; and have one by me. For there was but one
that was rightly calculated, and that is it which I have kept.
_A._ Your demonstration then is confuted though but by one.
_B._ That does not follow. For though an arithmetical calculation be
true in numbers, yet the same may be, or rather must be false, if the
units be not constantly the same.
_A._ Is their calculation so inconstant, or rather so foolish as you
make it?
_B._ Yes. For the same number is sometimes so many lines, sometimes so
many planes, and sometimes so many solids; as you shall plainly see, if
you will take the pains to examine first a demonstration I have to prove
the said duplication, and after that, the algebraic calculation which is
pretended to confute it. And not only that this one is false, but also
any other arithmetical account used in geometry, unless the numbers be
always so many lines, or always so many superficies, or always so many
solids.
_A._ Let me see the geometrical demonstration.
_B._ There it is. Read it.
TO FIND A CUBE DOUBLE TO A CUBE GIVEN:
Let the side of the cube given be V D. Produce V D to A, till A D be
double to D V. Then make the square of A D, namely A B C D. Divide A B
and C D in the middle at E and F. Draw E F. Draw also A C cutting E F in
I. Then in the sides B C and A D take B R and A S, each of them equal to
A I or I C.
Lastly, divide S D in the middle at T, and upon the centre T, with the
distance T V, describe a semi-circle cutting A D in Y, and D C in X.
I say the cube of D X is double to the cube of D V. For the three lines
D Y, D X, D V are in continual proportion. And continuing the
semi-circle V X Y till it cut the line R S, drawn and produced in Z, the
line S Z will be equal to D X. And drawing X Z it will pass through T.
And the four lines T V, T X, T Y and T Z will be equal. And therefore
joining Y X and Y Z, the figure V X Y Z will be a rectangle.
[Illustration:
_Delphic Problem.
Vol. VII. Eng. p. 60._
]
Produce C D to P so as D P be equal to A D. Now if Y Z produced fall on
P, there will be three rectangle equiangled triangles, D P Y, D Y X, and
D X V; and consequently four continual proportionals, D P, D Y, D X, and
D V, whereof D X is the least of the means. And therefore the cube of D
X will be double to the cube of D V.
_A._ That is true; and the cube of D Y will be double to the cube of D
X; and the cube of D P double to the cube of D Y. But that Y Z produced,
falls upon P, is the thing they deny, and which you ought to
demonstrate.
_B._ If Y Z produced fall not on P, then draw P Y, and from V let fall a
perpendicular upon P Y, suppose at _u._. Divide P V in the midst at
_a._, and join _a u._; which done _a u._ will be equal to _a._ V or _a._
P. For because V _u._ P is a right angle, the point _u._ will be in the
semi-circle whereof P V is the diameter.
Therefore drawing V R, the angle _u._ V R will be a right angle.
_A._ Why so?
_B._ Because T V and T Y are equal; and T D, T S equal; S Y will also be
equal to D V. And because D P and R S are equal and parallel, R Y will
be equal and parallel to P V. And therefore V R and P Y that join them
will be equal and parallel. And the angles P _u._ V, R V _u._ will be
alternate, and consequently equal. But P _u._ V is a right angle;
therefore also R V _u._ will be a right angle.
_A._ Hitherto all is evident. Proceed.
_B._ From the point Y raise a perpendicular cutting V R wheresoever in
_t._, and then (because P Y and V R are parallel) the angle Y _t_ V will
be a right angle. And the figure _u_ Y _t_ V a rectangle, and _u t_
equal to Y V. But Y V is equal to Z X; and therefore Z X is equal to _u
t_. And _u t_ must pass through the point T (for the diameters of any
rectangle divide each other in the middle), therefore Z and _u_ are the
same point, and X and _t_ the same point. Therefore Y Z produced falls
upon P. And D X is the lesser of the two means between A D and D V. And
the cube of D X double to the cube of D V, which was to be demonstated
_A._ I cannot imagine what fault there can be in this demonstration, and
yet there is one thing which seems a little strange to me. And it is
this. You take B R, which is half the diagonal, and which is the sine of
forty-five degrees, and which is also the mean proportional between the
two extremes; and yet you bring none of these proprieties into your
demonstration. So that though you argue from the construction, yet you
do not argue from the cause. And this perhaps your adversaries will
object, at least, against the art of your demonstration, or enquire by
what luck you pitched upon half the diagonal for your foundation.
_B._ I see you let nothing pass. But for answer you must know, that if a
man argue from the negative of the truth, though he know not that it is
the truth which is denied, yet he will fall at last, after many
consequences, into one absurdity or another. For though false do often
produce truth, yet it produces also absurdity, as it hath done here. But
truth produceth nothing but truth. Therefore in demonstrations that tend
to absurdity, it is no good logic to require all along the operation of
the cause.
_A._ Have you drawn from hence no corollaries?
_B._ No. I leave that for others that will; unless you take this for a
corollary, that, what arithmetical calculation soever contradicts it, is
false.
_A._ Let me see now the algebraical demonstration against it.
_B._ Here it is:
Let A B or A D be equal to 2
Then D F or D V is equal to 1
And B R or A S is equal to the square root of 2
And D Y equal to 3
want the square root of 2
The cube of A B is equal to 8
The cube of D Y is equal to 45
want the square root of 1682 that is almost equal to 4
For 45 want the square root of 1681 is equal to 4
Therefore D Y is a little less then the greater of the two means between
A D and D V.
_A._ There is I see some little difference between this arithmetical and
your geometrical demonstration. And though it be insensible, yet if his
calculation be true, yours must needs be false, which I am sure cannot
be.
_B._ His calculation is so true, that there is never a proposition in it
false, till he come to the conclusion, that the cube of D Y is equal to
45, want the square root of 1682. But that, and the rest, is false.
_A._ I shall easily see that A D is certainly 2, whereof D V is 1, and A
V is certainly 3, whereof D V is 1.
_B._ Right.
_A._ And B R is without doubt the square root of 2.
_B._ Why, what is 2?
_A._ 2 is the line A D as being double to D V which is 1.
_B._ And so, the line B R is the square root of the line A D.
_A._ Out upon it, it is absurd. Why do you grant it to be true in
arithmetic?
_B._ In arithmetic the numbers consist of so many units, and are never
considered there as nothings. And therefore every one line has some
latitude, and if you allow to B I, the semi-diagonal, the same latitude
you do to A B, or to B R, you will quickly see the square of half the
diagonal to be equal to twice the square of half A B.
_A._ Well, but then your demonstration is not confuted; for the point Y
will have latitude enough to take in that little difference which is
between the root of 1681 and the root of 1682. This putting off an unit
sometimes for one line, sometimes for one square, must needs mar the
reckoning. Again he says, the cube of A B is equal to 8; but seeing A B
is 2, the cube of A B must be just equal to four of its own sides; so
that the unit which was before sometimes a line, sometimes a square, is
now a cube.
_B._ It can be no otherwise when you so apply arithmetic to geometry, as
to number the lines of a plane, or the planes of a cube.
_A._ In the next place, I find that the cube of D Y is equal to 45, want
the square root of 1682. What is that 45? Lines, or squares, or cubes?
_B._ Cubes; cubes of D V.
_A._ Then if you add to 45 cubes of D V the square root of 1682, the sum
will be 45 cubes of D V; and if you add to the cube of D Y the same root
of 1682, the sum will be the cube of D Y, plus the square root of 1682,
and these two sums must be equal.
_B._ They must so.
_A._ But the square root of 1682, being a line, adds nothing to a cube;
therefore the cube alone of D Y, which he says is equal almost to 4
cubes of D V, is equal to 45 cubes of the same D V.
_B._ All these impossibilities do necessarily follow the confounding of
arithmetic and geometry.
_A._ I pray you let me see the operation by which the cube of D Y (that
is, the cube of 3, want the root of 2) is found equal to 45, want the
square root of 1682.
_B._ Here it is.
A DETECTION OF THE ABSURD USE OF ARITHMETIC AS IT IS NOW APPLIED TO
GEOMETRY.
3————√2
3————√2
———————
—√18 + 2
9—√18
—————————
9—√72 + 2
3——√2
——————————————
——√162 + 12——√8
27——√648 + 6
———————————————————
27—√658—√162 + 18—√8
======================
_A._ Why for two roots of 18 do you put the root of 72.
_B._ Because 2 roots of 18 are equal to one root of four times 18, which
is 72.
_A._ Next we have, that the root of 2 multiplied into 2 makes the root
of 8. How is that true?
_B._ Does it not make 2 roots of two? And is not B R the root of 2, and
2 B R equal to the diagonal? And is not the square of the diagonal equal
to 8 squares of D V?
_A._ It is true. But here the root of 8 is put for the cube of the root
of 2. Can a line be equal to a cube?
_B._ No. But here we are in arithmetic again, and 8 is a cubic number.
_A._ How does the root of 2 multiplied into the root of 72 make 12?
_B._ Because it makes the root of 2 times 72, that is to say the root of
144 which is 12.
_A._ How does 9 roots of 2 make the root of 162?
_B._ Because it makes the root of 2 squares of 9, that is the root of
162.
_A._ How does 3 roots of 72 make the root of 648?
_B._ Because it makes the root of 9 times 72, that is of 648.
_A._ For the total sum I see 27 and 18, which make 45. Therefore the
root of 648 together with the root of 162 and of 8, which are to be
deducted from 45, ought to be equal to the root of 1682.
_B._ So they are. For 648 multiplied by 162 makes 104976, of
which the double root is 648
and 648 and 162 added together make 810
Therefore the root of 648, added to the root of 162, makes
the root of 1458
Again 1458 into 8 is 11664. The double root whereof is 216
The sum of 1458 and 8 added together 1466
The sum of 1466 and 216 is 1682, and the root, the root of 1682
_A._ I see the calculation in numbers is right, though false in lines.
The reason whereof can be no other than some difference between
multiplying numbers into lines or planes, and multiplying lines into the
same lines or planes.
_B._ The difference is manifest. For when you multiply a number into
lines, the product is lines; as the number 2 multiplied into 3 lines is
no more than 3 lines 2 times told. But if you multiply lines into lines
you make planes, and if you multiply lines into planes you make solid
bodies. In geometry there are but three dimensions, lengths,
superficies, and body. In arithmetic there is but one, and that is
number or length which you will. And though there be some numbers called
plane, other solids, others plano-solid, others square, others cubic,
others square-square, others quadrato-cubic, others cubi-cubic, &c., yet
are all these but one dimension, namely number, or a file of things
numbered.
_A._ But seeing this way of calculation by numbers is so apparently
false, what is the reason this calculation came so near the truth?
_B._ It is because in arithmetic units are not nothings, and therefore
have breadth. And therefore many lines set together make a superficies
though their breadth be insensible. And the greater the number is into
which you divide your line, the less sensible will be your error.
_A._ Archimedes, to find a straight line equal to the circumference of a
circle, used this way of extracting roots. And it is the way also by
which the table of sines, secants, and tangents have been calculated.
Are they all out?
_B._ As for Archimedes, there is no man that does more admire him than I
do: but there is no man that cannot err. His reasoning is good. But he,
as all other geometricians before and after him, have had two principles
that cross one another when they are applied to one and the same
science. One is, that a point is no part of a line, which is true in
geometry, where a part of a line when it is called a point, is not
reckoned; another is, that a unit is part of a number; which is also
true; but when they reckon by arithmetic in geometry, there a unit is
sometimes part of a line, sometimes a part of a square, and sometimes
part of a cube. As for the table of sines, secants, and tangents, I am
not the first that find fault with them. Yet I deny not but they are
true enough for the reckoning of acres in a map of land.
_A._ What a deal of labour has been lost by them that being professors
of geometry have read nothing else to their auditors but such stuff as
this you have here seen. And some of them have written great books of it
in strange characters, such as in troublesome times, a man would suspect
to be a cypher.
_B._ I think you have seen enough to satisfy you, that what I have
written heretofore concerning the quadrature of the circle, and of other
figures made in imitation of the parabola, has not been yet confuted.
_A._ I see you have wrested out of the hands of our antagonists this
weapon of algebra, so as they can never make use of it again. Which I
consider as a thing of much more consequence to the science of geometry,
than either of the duplication of the cube, or the finding of two mean
proportionals, or the quadrature of a circle, or all these problems put
together.
FINIS.
DECAMERON PHYSIOLOGICUM;
OR,
TEN DIALOGUES OF NATURAL PHILOSOPHY.
BY
THOMAS HOBBES
OF MALMESBURY.
TO WHICH IS ADDED
THE PROPORTION OF A STRAIGHT LINE TO
HALF THE ARC OF A QUADRANT,
BY THE SAME AUTHOR.
[Illustration]
DECAMERON PHYSIOLOGICUM.
CHAPTER I.
OF THE ORIGINAL OF NATURAL PHILOSOPHY.
_A._ I have heard exceeding highly commended a kind of thing which I do
not well understand, though it be much talked of, by such as have not
otherwise much to do, by the name of philosophy; and the same again by
others as much despised and derided: so that I cannot tell whether it be
good or ill, nor what to make of it, though I see many other men that
thrive by it.
_B._ I doubt not, but what so many do so highly praise must be very
admirable, and what is derided and scorned by many, foolish and
ridiculous. The honour and scorn falleth finally not upon philosophy,
but upon the professors. Philosophy is _the knowledge of natural
causes_. And there is no knowledge but of truth. And to know the true
causes of things, was never in contempt, but in admiration. Scorn can
never fasten upon truth. But the difference is all in the writers and
teachers. Whereof some have neither studied, nor care for it, otherwise
than as a trade to maintain themselves or gain preferment; and some for
fashion, and to make themselves fit for ingenious company: and their
study hath not been meditation, but acquiescence in the authority of
those authors whom they have heard commended. And some, but few, there
be, that have studied it for curiosity, and the delight which commonly
men have in the acquisition of science, and in the mastery of difficult
and subtil doctrines. Of this last sort I count Aristotle, and a few
others of the ancients, and some few moderns: and to these it is that
properly belong the praises which are given to philosophy.
_A._ If I have a mind to study, for example natural philosophy, must I
then needs read Aristotle, or some of those that now are in request?
_B._ There is no necessity of it. But if in your own meditation you
light upon a difficulty, I think it is no loss of time, to enquire what
other men say of it, but to rely only upon reason. For though there be
some few effects of nature, especially concerning the heavens, whereof
the philosophers of old time have assigned very rational causes, such as
any man may acquiesce in, as of eclipses of the sun and moon by long
observation, and by the calculation of their visible motions; yet what
is that to the numberless and quotidian phenomena of nature? Who is
there amongst them or their successors, that has satisfied you with the
causes of gravity, heat, cold, light, sense, colour, noise, rain, snow,
frost, winds, tides of the sea, and a thousand other things which a few
men’s lives are too short to go through, and which you and other curious
spirits admire (as quotidian as they are), and fain would know the
causes of them, but shall not find them in the books of naturalists; and
when you ask what are the causes of any of them, of a philosopher now,
he will put you off with mere words; which words, examined to the
bottom, signify not a jot more than I cannot tell, or because it is:
such as are intrinsical quality, occult quality, sympathy, antipathy,
antiperistasis, and the like. Which pass well enough with those that
care not much for such wisdom, though wise enough in their own ways; but
will not pass with you that ask not simply what is the cause, but in
what manner it comes about that such effects are produced.
_A._ That is cozening. What need had they of that? When began they thus
to play the charlatans?
_B._ Need had they none. But know you not that men from their very
birth, and naturally, scramble for every thing they covet, and would
have all the world, if they could, to fear and obey them? If by fortune
or industry one light upon a secret in nature, and thereby obtain the
credit of an extraordinary knowing man, should he not make use of it to
his own benefit? There is scarce one of a thousand but would live upon
the charges of the people as far as he dares. What poor geometrician is
there, but takes pride to be thought a conjurer? What mountebank would
not make a living out of a false opinion that he were a great physician?
And when many of them are once engaged in the maintenance of an error,
they will join together for the saving of their authority to decry the
truth.
_A._ I pray, tell me, if you can, how and where the study of philosophy
first began.
_B._ If we may give credit to old histories, the first that studied any
of the natural sciences were the astronomers of Ethiopia. My author is
Diodorus Siculus, accounted a very faithful writer, who begins his
history as high as is possible, and tells us that in Ethiopia were the
first astronomers; and that for their predictions of eclipses, and other
conjunctions and aspects of the planets, they obtained of their king not
only towns and fields to a third part of the whole land, but were also
in such veneration with the people, that they were thought to have
discourse with their gods, which were the stars; and made their kings
thereby to stand in awe of them, that they durst not either eat or drink
but what and when they prescribed; no nor live, if they said the gods
commanded them to die. And thus they continued in subjection to their
false prophets, till by one of their kings, called Ergamenes, (about the
time of the Ptolemies), they were put to the sword. But long before the
time of Ergamenes, the race of these astrologers (for they had no
disciples but their own children) was so numerous, that abundance of
them (whether sent for or no I cannot tell) transplanted themselves into
Egypt, and there also had their cities and lands allowed them, and were
in request not only for astronomy and astrology, but also for geometry.
And Egypt was then as it were an university to all the world, and
thither went the curious Greeks, as Pythagoras, Plato, Thales, and
others, to fetch philosophy into Greece. But long before that time,
abundance of them went into Assyria, and had their towns and lands
assigned them also there; and were by the Hebrews called Chaldees.
_A._ Why so?
_B._ I cannot tell; but I find in Martinius’s Lexicon they are called
Chasdim, and Chesdim, and (as he saith) from one Chesed the son of
Nachor; but I find no such man as Chesed amongst the issue of Noah in
the scripture. Nor do I find that there was any certain country called
Chaldæa; though a town where any of them inhabited were called a town of
the Chaldees. Martinius saith farther, that the same word Chasdim did
signify also Demons.
_A._ By this reckoning I should conjecture they were called Chusdim, as
being a race of Ethiopians. For the land of Chus is Ethiopia; and so the
name degenerated first into Chuldim, and then into Chaldim; so that they
were such another kind of people as we call gipsies; saving that they
were admired and feared for their knavery, and the gipsies counted
rogues.
_B._ Nay pray, except Claudius Ptolomæus, author of that great work of
astronomy, the Almegest.
_A._ I grant he was excellent both in astronomy and geometry, and to be
commended for his _Almegest_ ; but then for his _Judiciar Astrologie_
annexed to it, he is again a gipsy. But the Greeks that travelled, you
say, into Egypt, what philosophy did they carry home?
_B._ The mathematics and astronomy. But for that sublunary physics,
which is commonly called natural philosophy, I have not read of any
nation that studied it earlier than the Greeks, from whom it proceeded
to the Romans. Yet both Greeks and Romans were more addicted to moral
than to natural philosophy; in which kind we have their writings, but
loosely and incoherently, written upon no other principles than their
own passions and presumptions, without any respect to the laws of
commonwealth, which are the ground and measure of all true morality. So
that their books tend rather to teach men to censure than to obey the
laws; which has been a great hindrance to the peace of the western world
ever since. But they that seriously applied themselves to natural
philosophy were but few, as Plato and Aristotle, whose works we have;
and Epicurus whose doctrine we have in Lucretius. The writings of
Philolaus and many other curious students being by fire or negligence
now lost: though the doctrines of Philolaus concerning the motion of the
earth have been revived by Copernicus, and explained and confirmed by
Galileo now of late.
_A._ But methinks the natural philosophy of Plato, and Aristotle, and
the rest, should have been cultivated and made to flourish by their
disciples.
_B._ Whom do you mean, the successors of Plato, Epicurus, Aristotle, and
the other first philosophers? It may be some of them may have been
learned and worthy men. But not long after, and down to the time of our
Saviour and his Apostles, they were for the most part a sort of needy,
ignorant, impudent, cheating fellows, who by the profession of the
doctrine of those first philosophers got their living. For at that time,
the name of philosophy was so much in fashion and honour amongst great
persons, that every rich man had a philosopher of one sect or another to
be a schoolmaster to his children. And these were they that feigning
Christianity, with their disputing and readiness of talking got
themselves into Christian commons, and brought so many heresies into the
primitive Church, every one retaining still a tang of what they had been
used to teach.
_A._ But those heresies were all condemned in the first Council of Nice.
_B._ Yes. But the Arian heresy for a long time flourished no less than
the Roman, and was upheld by divers Emperors, and never fully
extinguished as long as there were Vandals in Christendom. Besides,
there arose daily other sects, opposing their philosophy to the doctrine
of the Councils concerning the divinity of our Saviour; as how many
persons he was, how many natures he had. And thus it continued till the
time of Charlemagne, when he and Pope Leo the third divided the power of
the empire into temporal and spiritual.
_A._ A very unequal division.
_B._ Why? Which of them think you had the greater share?
_A._ No doubt, the Emperor: for he only had the sword.
_B._ When the swords are in the hands of men, whether had you rather
command the men or the swords?
_A._ I understand you. For he that hath the hands of the men, has also
the use both of their swords and strength.
_B._ The empire thus divided into spiritual and temporal, the freedom of
philosophy was to the power spiritual very dangerous. And for that cause
it behoved the Pope to get schools set up not only for divinity, but
also for other sciences, especially for natural philosophy. Which when
by the power of the Emperor he had effected, out of the mixture of
Aristotle’s metaphysics with the Scripture, there arose a new science
called School-divinity; which has been the principal learning of these
western parts from the time of Charlemagne till of very late.
_A._ But I find not in any of the writings of the Schoolmen in what
manner, from the causes they assign, the effect is naturally and
necessarily produced.
_B._ You must not wonder at that. For you enquire not so much, when you
see a change of anything, what may be said to be the cause of it, as how
the same is generated; which generation is the entire progress of nature
from the efficient cause to the effect produced. Which is always a hard
question, and for the most part impossible for a man to answer to. For
the alterations of the things we perceive by our five senses are made by
the motion of bodies, for the most part, either for distance, smallness,
or transparence, invisible.
_A._ But what need had they then to assign any cause at all, seeing that
they could not show the effect was to follow from it?
_B._ The Schools, as I said, were erected by the Pope and Emperor, but
directed by the Pope only, to answer and confute the heresies of the
philosophers. Would you have them then betray their profession and
authority, that is to say, their livelihood, by confessing their
ignorance? Or rather uphold the same, by putting for causes, strange and
unintelligible words; which might serve well enough not only to satisfy
the people whom they relied on, but also to trouble the philosophers
themselves to find a fault in.
_A._ Seeing you say that alteration is wrought by the motion of bodies,
pray tell me first what I am to understand by the word body.
_B._ It is a hard question, though most men think they can easily answer
it, as that it is whatsoever they can see, feel, or take notice of by
their senses. But if you will know indeed what is body, we must enquire
first what there is that is not body. You have seen, I suppose, the
effects of glasses, how they multiply and magnify the object of our
sight; as when a glass of a certain figure will make a counter or a
shilling seem twenty, though you be well assured there is but one. And
if you set a mark upon it, you will find the mark upon them all. The
counter is certainly one of those things we call bodies: are not the
others so too?
_A._ No, without doubt. For looking through a glass cannot make them
really more than they are.
_B._ What then be they but fancies, so many fancies of one and the same
thing in several places?
_A._ It is manifest they are so many idols, mere nothings.
_B._ When you have looked upon a star or candle with both your eyes, but
one of them a little turned awry with your finger, has not there
appeared two stars, or two candles? And though you call it a deception
of the sight, you cannot deny but there were two images of the object.
_A._ It is true, and observed by all men. And the same I say of our
faces seen in looking-glasses, and of all dreams, and of all apparitions
of dead men’s ghosts; and wonder, since it is so manifest, I never
thought upon it before, for it is a very happy encounter, and such as
being by everybody well understood, would utterly destroy both idolatry
and superstition, and defeat abundance of knaves that cheat and trouble
the world with their devices.
_B._ But you must not hence conclude that whosoever tells his dream, or
sometimes takes his direction from it, is therefore an idolater, or
superstitious, or a cheater. For God doth often admonish men by dreams
of what they ought to do; yet men must be wary in this case that they
trust not dreams with the conduct of their lives farther than by the
laws of their country is allowed: for you know what God says, Deut.
xiii: _If a prophet or a dreamer of dreams give thee a sign or a wonder,
and the sign come to pass, yet if he bid thee serve other Gods let him
be put to death_. Here by serving other Gods (since they have chosen God
for their King) we are to understand revolting from their King, or
disobeying of his laws. Otherwise I see no idolatry nor superstition in
following a dream, as many of the Patriarchs in the Old Testament, and
of the Saints in the New Testament did.
_A._ Yes: their own dreams. But when another man shall dream, or say
that he has dreamed, and require me to follow that, he must pardon me if
I ask him by what authority, especially if he look I should pay him for
it.
_B._ But if commanded by the laws you live under, you ought to follow
it. But when there proceed from one sound divers echoes, what are those
echoes? And when with fingers crossed you touch a small bullet, and
think it two; and when the same herb or flower smells well to one and
ill to another, and the same at several times, well and ill to yourself,
and the like of tastes, what are those echoes, feelings, odours, and
tastes?
_A._ It is manifest they are all but fancies. But certainly when the sun
seems to my eye no bigger than a dish, there is behind it somewhere
somewhat else, I suppose a real sun, which creates those fancies, by
working, one way or other, upon my eyes, and other organs of my senses,
to cause that diversity of fancy.
_B._ You say right; and that is it I mean by the word body, which
briefly I define to be any thing that hath a being in itself, without
the help of sense.
_A._ Aristotle, I think, meaneth by body, _substance_, or _subjectum_,
wherein colour, sound, and other fancies are, as he says, inherent. For
the word essence has no affinity with substance. And Seneca says, he
understands it not. And no wonder: for essence is no part of the
language of mankind, but a word devised by philosophers out of the
copulation of two names, as if a man having two hounds could make a
third, if it were need, of their couples.
_B._ It is just so. For having said in themselves, (for example): _a
tree is a plant_, and conceiving well enough what is the signification
of those names, knew not what to make of the word _is_, that couples
those names; nor daring to call it a body, they called it by a new name
(derived from the word _est_), _essentia_, and _substantia_, deceived by
the idiom of their own language. For in many other tongues, and namely
in the Hebrew, there is no such copulative. They thought the names of
things sufficiently connected, when they are placed in their natural
consequence; and were therefore never troubled with essences, nor other
fallacy from the copulative _est_.
==========
CHAPTER II.
OF THE PRINCIPLES AND METHOD OF NATURAL
PHILOSOPHY.
_A._ This history of the old philosophers has not put me out of love,
but out of hope of philosophy from any of their writings. I would
therefore try if I could attain any knowledge therein by my own
meditation: but I know neither where to begin, nor which way to proceed.
_B._ Your desire, you say, is to know the causes of the effects or
phenomena of nature; and you confess they are fancies, and,
consequently, that they are in yourself; so that the causes you seek for
only are without you, and now you would know how those external bodies
work upon you to produce those phenomena. The beginning therefore of
your enquiry ought to be at; _What it is you call a cause?_ I mean an
efficient cause: for the philosophers make four kinds of causes, whereof
the efficient is one. Another they call the formal cause, or simply the
form or essence of the thing caused; as when they say, four equal angles
and four equal sides are the cause of a square figure; or that heaviness
is the cause that makes heavy bodies to descend; but that is not the
cause you seek for, nor any thing but this: _It descends because it
descends_. The third is the material cause, as when they say, the walls
and roof, &c. of a house are the cause of a house. The fourth is the
final cause, and hath place only in moral philosophy.
_A._ We will think of final causes upon some other occasion; of formal
and material not at all: I seek only the efficient, and how it acteth
from the beginning to the production of the effect.
_B._ I say then, that in the first place you are to enquire diligently
into the nature of motion. For the variations of fancies, or (which is
the same thing) of the phenomena of nature, have all of them one
universal efficient cause, namely the variety of motion. For if all
things in the world were absolutely at rest, there could be no variety
of fancy; but living creatures would be without sense of all objects,
which is little less than to be dead.
_A._ What if a child new taken from the womb should with open eyes be
exposed to the azure sky, do not you think it would have some sense of
the light, but that all would seem unto him darkness?
_B._ Truly, if he had no memory of any thing formerly seen, or by any
other sense perceived, (which is my supposition), I think he would be in
the dark. For darkness is darkness, whether it be black or blue, to him
that cannot distinguish.
_A._ Howsoever that be, it is evident enough that whatsoever worketh is
moved: for action is motion.
_B._ Having well considered the nature of motion, you must thence take
your principles for the foundation and beginning of your enquiry.
_A._ As how?
_B._ Explain as fully and as briefly as you can what you constantly mean
by motion; which will save yourself as well as others from being seduced
by equivocation.
_A._ Then I say, motion is nothing but change of place for all the
effect of a body upon the organs of our senses is nothing but fancy.
Therefore we can fancy nothing from seeing it moved, but change of
place.
_B._ It is right. But you must then tell me also what you understand by
place: for all men are not yet agreed on that.
_A._ Well then; seeing we fancy a body, we cannot but fancy it
somewhere. And therefore I think place is the fancy of here or there.
_B._ That is not enough. Here and there are not understood by any but
yourself, except you point towards it. But pointing is no part of a
definition. Besides, though it help him to find the place, it will never
bring him to it.
_A._ But seeing sense is fancy, when we fancy a body, we fancy also the
figure of it, and the space it fills up. And then I may define place to
be the precise space within which the body is contained. For space is
also part of the image we have of the object seen.
_B._ And how define you time?
_A._ As place is to a body, so, I think, is time to the motion of it;
and consequently I take time to be our fancy or image of the motion. But
is there any necessity of so much niceness?
_B._ Yes. The want of it is the greatest, if not the only, cause of all
the discord amongst philosophers, as may easily be perceived by their
abusing and confounding the names of things that differ in their nature;
as you shall see when there is occasion to recite some of the tenets of
divers philosophers.
_A._ I will avoid equivocation as much as I can. And for the nature of
motion, I suppose I understand it by the definition. What is next to be
done?
_B._ You are to draw from these definitions, and from whatsoever truth
else you know by the light of nature, such general consequences as may
serve for axioms, or principles of your ratiocination.
_A._ That is hard to do.
_B._ I will draw them myself, as many as for our present discourse of
natural causes we shall have need of; so that your part will be no more
than to take heed I do not deceive you.
_A._ I will look to that.
_B._ My first axiom then shall be this: Two bodies, at the same time,
cannot be in one place.
_A._ That is true: for we number bodies as we fancy them distinct, and
distinguish them by their places. You may therefore add: nor one body at
the same time in two places. And philosophers mean the same, when they
say: there is no penetration of bodies.
_B._ But they understand not their own words: for penetration signifies
it not. My second axiom is, that nothing can begin, change, or put an
end to, its own motion. For supposing it begin just now, or being now in
motion, change its way or stop; I require the cause why now rather than
before or after, having all that is necessary to such motion, change, or
rest, alike at all times?
_A._ I do not doubt but the argument is good in bodies inanimate; but
perhaps in voluntary agents it does not hold.
_B._ How it holds in voluntary agents we will then consider when our
method hath brought us to the powers and passions of the mind. A third
axiom shall be this: whatsoever body being at rest is afterwards moved,
hath for its immediate movement some other body which is in motion and
toucheth it. For, since nothing can move itself, the movent must be
external. And because motion is change of place, the movent must put it
from its place, which it cannot do till it touch it.
_A._ That is manifest, and that it must more than touch it; it must also
follow it. And if more parts of the body are moved than are by the
movent touched, the movent is not immediate. And by this reason, a
continued body, though never so great, if the first superficies be
pressed never so little back, the motion will proceed through it.
_B._ Do you think that to be impossible? I will prove it from your own
words: for you say that the movent does then touch the body which it
moveth. Therefore it puts it back; but that which is put back, puts back
the next behind, and that again the next; and so onward to any distance,
the body being continued. The same is also manifest by experience,
seeing one that walks with a staff can distinguish, though blind,
between stone and glass; which were impossible, if the parts of his
staff between the ground and his hand made no resistance. So also he
that in the silence of the night lays his ear to the ground, shall hear
the treading of men’s feet farther than if he stood upright.
_A._ This is certainly true of a staff or other hard body, because it
keeps the motion in a straight line from diffusion. But in such a fluid
body as the air, which being put back must fill an orb, and the farther
it is put back, the greater orb, the motion will decrease, and in time,
by the resistance of air to air, come to an end.
_B._ That any body in the world is absolutely at rest, I think not true:
but I grant, that in a space filled everywhere with body, though never
so fluid, if you give motion to any part thereof, that motion will by
resistance of the parts moved, grow less and less, and at last cease;
but if you suppose the space utterly void, and nothing in it, then
whatsoever is once moved shall go on eternally: or else that which you
have granted is not true, viz., that nothing can put an end to its own
motion.
_A._ But what mean you by resistance?
_B._ Resistance is the motion of a body in a way wholly or partly
contrary to the way of its movent, and thereby repelling or retarding
it. As when a man runs swiftly, he shall feel the motion of the air in
his face. But when two hard bodies meet, much more may you see how they
abate each other’s motion, and rebound from one another. For in a space
already full, the motion cannot, in an instant, be communicated through
the whole depth of the body that is to be moved.
_A._ What other definitions have I need of?
_B._ In all motion, as in all quantity, you must take the beginning of
your reckoning from the least supposed motion. And this I call the first
endeavour of the movent; which endeavour, how weak soever, is also
motion. For if it have no effect at all, neither will it do anything
though doubled, trebled, or by what number soever multiplied: for
nothing, though multiplied, is still nothing. Other axioms and
definitions we will take in, as we need them, by the way.
_A._ Is this all the preparation I am to make?
_B._ No, you are to consider also the several kinds and properties of
motion, viz., when a body being moved by one or more movents at once, in
what way it is carried, straight, circular, or otherwise crooked; and
what degree of swiftness; as also the action of the movent, whether
trusion, vection, percussion, reflection, or refraction; and farther you
must furnish yourself with as many experiments (which they call
phenomenon) as you can. And supposing some motion for the cause of your
phenomenon, try, if by evident consequence, without contradiction to any
other manifest truth or experiment, you can derive the cause you seek
for from your supposition. If you can, it is all that is expected, as to
that one question, from philosophy. For there is no effect in nature
which the Author of nature cannot bring to pass by more ways than one.
_A._ What I want of experiments you may supply out of your own store, or
such natural history as you know to be true; though I can be well
content with the knowledge of the causes of those things which everybody
sees commonly produced. Let us therefore now enquire the cause of some
effect particular.
_B._ We will begin with that which is the most universal, the universe;
and enquire in the first place, if any place be absolutely empty, that
is to say in the language of philosophers, whether there be any vacuum
in nature?
==========
CHAPTER III.
OF VACUUM.
_A._ It is hard to suppose, and harder to believe, that the infinite and
omnipotent Creator of all things should make a work so vast as is the
world we see, and not leave a few little spaces with nothing at all in
them; which put altogether in respect of the whole creation, would be
insensible.
_B._ Why say you that? Do you think any argument can be drawn from it to
prove there is vacuum?
_A._ Why not? For in so great an agitation of natural bodies, may not
some small parts of them be cast out, and leave the places empty from
whence they were thrown?
_B._ Because He that created them is not a fancy, but the most real
substance that is; who being infinite, there can be no place empty where
He is, nor full where He is not.
_A._ It is hard to answer this argument, because I do not remember that
there is any argument for the maintenance of vacuum in the writings of
divines: therefore I will quit that argument, and come to another. If
you take a glass vial with a narrow neck, and having sucked it, dip it
presently at the neck into a basin of water, you shall manifestly see
the water rise into the vial. Is not this a certain sign that you had
sucked out some of the air, and consequently that some part of the vial
was left empty?
_B._ No; for when I am about to suck, and have air in my mouth,
contracting my checks I drive the same against the air in the glass, and
thereby against every part of the sides of the hard glass. And this
gives to the air within an endeavour outward, by which, if it be
presently dipped into the water, it will penetrate and enter into it.
For air if it be pressed will enter into any fluid, much more into
water. Therefore there shall rise into the vial so much water as there
was air forced into the basin.
_A._ This I confess is possible, and not improbable.
_B._ If sucking would make vacuum, what would become of those women that
are nurses? Should they not be in a very few days exhausted, were it not
that either the air which is in the child’s mouth penetrateth the milk
as it descends, and passeth through it, or the breast is contracted?
_A._ From what experiment can you evidently infer that there is no
vacuum?
_B._ From many, and such as to almost all men are known and familiar. If
two hard bodies, flat and smooth, be joined together in a common
superficies parallel to the horizontal plane, you cannot without great
force pull them asunder, if you apply your force perpendicularly to the
common superficies: but if you place that common superficies erect to
the horizon, they will fall asunder with their own weight. From whence I
argue thus: since their contiguity, in what posture soever, is the same,
and that they cannot be pulled asunder by a perpendicular force without
letting in the ambient air in an instant, which is impossible; or almost
in an instant, which is difficult: and on the other side, when the
common superficies is erect, the weight of the same hard bodies is able
to break the contiguity, and let in the air successively; it is manifest
that the difficulty of separation proceeds from this, that neither air
nor any other body can be moved to any, how small soever, distance in an
instant; but may easily be moved (the hardness at the sides once
mastered) successively. So that the cause of this difficulty of
separation is this, that they cannot be parted except the air or other
matter can enter and fill the space made by their diremption. And if
they were infinitely hard, not at all. And hence also you may understand
the cause why any hard body, when it is suddenly broken, is heard to
crack; which is the swift motion of the air to fill the space between.
Another experiment, and commonly known, is of a barrel of liquor, whose
tap-hole is very little, and the bung so stopped as to admit no air; for
then the liquor will not run: but if the tap-hole be large it will,
because the air pressed by a heavier body will pierce through it into
the barrel. The like reason holds of a gardener’s watering-pot, when the
holes in the bottom are not too great. A third experiment is this: turn
a thin brass kettle the bottom upwards, and lay it flat upon the water.
It will sink till the water rise within to a certain height, but no
higher: yet let the bottom be perforated, and the kettle will be full
and sink, and the air rise again through the water without. But if a
bell were so laid on, it would be filled and sink, though it were not
perforated, because the weight is greater than the weight of the same
bulk of water.
_A._ By these experiments, without any more, I am convinced, that there
is not actually in nature any vacuum; but I am not sure but that there
may be made some little place empty, and this from two experiments, one
whereof is Toricellius’ experiment, which is this: take a cylinder of
glass, hollow throughout, but close at the end, in form of a sack.
_B._ How long?
_A._ As long as you will, so it be more than twenty-nine inches.
_B._ And how broad?
_A._ As broad as you will, so it be broad enough to pour into it
quicksilver. And fill it with quicksilver, and stop up the entrance with
your finger, so as to unstop it again at your pleasure. Then set down a
basin, or, if you will, a sea of quicksilver, and inverting the cylinder
full as it is, dip the end into the quicksilver, and remove your finger,
that the cylinder may empt itself. Do you conceive me? For there is so
many passing by, that I cannot paint it.
_B._ Yes, I conceive you well enough. What follows?
_A._ The quicksilver will descend in the cylinder, not till it be level
with that in the basin, according to the nature of heavy fluids, but
stay and stand above it, at the height of twenty-nine inches or very
near it, the bottom being now uppermost, that no air can get in.
_B._ What do you infer from this?
_A._ That all the cavity above twenty-nine inches is filled with vacuum.
_B._ It is very strange that I, from this same experiment, should infer,
and I think evidently, that it is filled with air. I pray, tell me, when
you had inverted the cylinder, full as it was, and stopped with your
finger, dipped into the basin, if you had then removed your finger,
whether you think the quicksilver would not all have fallen out?
_A._ No sure. The air would have been pressed upward through the
quicksilver itself: for a man with his hand can easily thrust a bladder
of air to the bottom of a basin of quicksilver.
_B._ It is therefore manifest that quicksilver can press the air through
the same quicksilver.
_A._ It is manifest; and also itself rise into the air.
_B._ What cause then can there be, why it should stand still at twenty
nine inches above the level of the basin, rather than any place else?
_A._ It is not hard to assign the cause of that. For so much quicksilver
as was above the twenty-nine inches, will rise the first level of that
in the basin, as much as if you had poured it on; and thereby bring it
to an equilibrium. So that I see plainly now, that there is no necessity
of vacuum from this experiment. For I considered only that naturally
quicksilver cannot ascend in air, nor air descend in quicksilver, though
by force it may.
_B._ Nor do I think that Torricellius or any other vacuist thought of it
more than you. But what is the second experiment?
_A._ There is a sphere of glass, which they call a recipient, of the
capacity of three or four gallons. And there is inserted into it the end
of a hollow cylinder of brass above a foot long; so that the whole is
one vessel, and the bore of the cylinder three inches diameter. Into
which is thrust by force a solid cylinder of wood, covered with leather
so just, as it may in every point exactly touch the concave superficies
of the brass. There is also, to let out the air which the wooden
cylinder as it enters (called the sucker) drives before it, a flap to
keep out the external air while they are pulling the sucker. Besides, at
the top of the recipient there is a hole to put into it anything for
experiment. The sucker being now forced up into the cylinder, what do
you think must follow?
_B._ I think it will require as much strength to pull it back, as it did
to force it in.
_A._ That is not it I ask, but what would happen to the recipient?
_B._ I think so much air as would fill the place the sucker leaves,
would descend into it out of the recipient; and also that just so much
from the external air would enter into the recipient, between the brass
and the wood, at first very swiftly, but, as the place increased, more
leisurely.
_A._ Why may not so much air rather descend into the place forsaken, and
leave as much vacuum as that comes to in the recipient? For otherwise no
air will be pumped out, nor can that wooden pestle be called a sucker.
_B._ That is it I say. There is no air either pumped or sucked out.
_A._ How can the air pass between the leather and the brass, or between
the leather and the wood, being so exactly contiguous, or through the
leather itself?
_B._ I conceive no such exact contiguity, nor such fastness of the
leather: for I never yet had any that in a storm would keep out either
air or water.
_A._ But how then could there be made in the recipient such strange
alteration both on animate and inanimate bodies?
_B._ I will tell you how. The air descends out of the recipient, because
the air which the sucker removeth from behind itself, as it is pulling
out, has no place to retire into without, and therefore is driven into
the engine between the wood of the sucker and the brass of the cylinder,
and causes as much air to come into the place forsaken by the retiring
sucker; which causeth, by oft repetition of the force, a violent
circulation of the air within the recipient, which is able quickly to
kill anything that lives by respiration, and make all the alterations
that have appeared in the engine.
==========
CHAPTER IV.
OF THE SYSTEM OF THE WORLD.
_B._ You are come in good time; let us therefore sit down. There is ink,
paper, ruler, and compass. Draw a little circle to represent the body of
the sun.
_A._ It is done. The centre is A, the circumference is L M.
_B._ Upon the same centre A, draw a larger circle to stand for the
ecliptic: for you know the sun is always in the plane of the ecliptic.
_A._ There it is. The diameters of it at right angles are B Z.
_B._ Draw the diameter of the equator.
_A._ How?
_B._ Through the centre A (for the earth is also always in the plane of
the equator or of some of its parallels) so as to be distant from B
twenty-three degrees and a half.
_A._ Let it be H I: and let C G be equal to B H; and so C will be one of
the poles of the ecliptic, suppose the north-pole; and then H will be
east, and I west. And C A produced to the circumference in E, makes E
the south-pole.
_B._ Take C K equal to C G, and the chord G K will be the diameter of
the arctic circle, and parallel to H I, the diameter of the equator.
Lastly, upon the point B, draw a little circle wherein I suppose to be
the globe of the earth.
_A._ It is drawn, and marked with _l m_. And B D and K G joined will be
parallel; and as H and I are east and west, and so are B and D, and G
and K.
_B._ True; but producing Z B to the circumference _l m_ in _b_, the line
B _b_ will be in the diameter of the ecliptic of the earth, and B _m_ in
the diameter of the equator of the earth. In like manner, if you produce
K G cutting the circle, whose centre is G, in _d_ and _e_, and make an
angle _n_ G _d_ equal to _b_ B _m_, the line _n_ G will be in the
ecliptic of the earth, because G _d_ is in the equator of the earth. So
that in the annual motion of the earth through the ecliptic, every
straight line drawn in the earth, is perpetually kept parallel to the
place from whence it is removed.
_A._ It is true; and it is the doctrine of Copernicus. But I cannot yet
conceive by what one motion this circle can be described otherwise than
we are taught by Euclid. And then I am sure that all the diameters shall
cross one another in the centre, which in this figure is A.
_B._ I do not say that the diameters of a sphere or circle can be
parallel; but that if a circle of a lesser sphere be moved upon the
circumference of a great circle of a greater sphere, that the straight
lines that are in the lesser sphere may be kept parallel perpetually to
the places they proceed from.
_A._ How? And by what motion?
_B._ Take into your hand any straight line (as in this figure), the line
L A M, which we suppose to be the diameter of the sun’s body; and moving
it parallelly with the ends in the circumference, so as that the end M
may withal describe a small circle, as M _a_. It is manifest that all
the other points of the same line L M will, by the same motion, at the
same time, describe equal circles to it. Likewise if you take in your
hand any two diameters fastened together, the same parallel motion of
the line L M, shall cause all the points of the other diameter to make
equal circles to the same M _a_.
_A._ It is evident; as also that every point of the sun’s body shall do
the like. And not only so, but also if one end describe any other
figure, all the other points of the body shall describe like and equal
figures to it.
_B._ You see by this, that this parallel motion is compounded of two
motions, one circular upon the superficies of a sphere, the other a
straight motion from the centre to every point of the same superficies,
and beyond it.
_A._ I see it.
_B._ It follows hence, that the sun by this motion must every way repel
the air; and since there is no empty place for retiring, the air must
turn about in a circular stream; but slower or swifter according as it
is more or less remote from the sun; and that according to the nature of
fluids, the particles of the air must continually change place with one
another; and also that the stream of the air shall be the contrary way
to that of the motion, for else the air cannot be repelled.
_A._ All this is certain.
_B._ Well; then if you suppose the globe of the earth to be in this
stream which is made by the motion of the sun’s body from east to west,
the stream of air wherein is the earth’s annual motion will be from west
to east.
_A._ It is certain.
_B._ Well. Then if you suppose the globe of the earth, whose circle is
moved annually, to be _l m_, the stream of the air without the ecliptic
falling upon the superficies of the earth _l m_ without the ecliptic,
being slower, and the stream that falleth within swifter, the earth
shall be turned upon its own centre proportionally to the greatness of
the circles; and consequently their diameters shall be parallel; as also
are other straight lines correspondent.
_A._ I deny not but the streams are as you say; and confess that the
proportion of the swiftness without, is to the swiftness within, as the
sun’s ecliptic to the ecliptic of the earth; that is to say, as the
angle H A B to the angle _m_ B _b_. And I like your argument the better,
because it is drawn from Copernicus his foundation. I mean the
compounded motion of straight and circular.
_B._ I think I shall not offer you many demonstrations of physical
conclusions that are not derived from the motions supposed or proved by
Copernicus. For those conclusions in natural philosophy I most suspect
of falshood, which require most variety of suppositions for their
demonstrations.
_A._ The next thing I would know, is how great or little you suppose
that circle _a_ M?
_B._ I suppose it less than you can make it: for there appears in the
sun no such motion sensible. It is the first endeavour of the sun’s
motion. But for all that, as small as the circle is, the motion may be
as swift, and of as great strength as it is possible to be named. It is
but a kind of trembling that necessarily happeneth in those bodies,
which with great resistance press upon one another.
_A._ I understand now from what cause proceedeth the annual motion. Is
the sun the cause also of the diurnal motion?
_B._ Not the immediate cause. For the diurnal motion of the earth is
upon its own centre, and therefore the sun’s motion cannot describe it.
But it proceedeth as a necessary consequence from the annual motion. For
which I have both experience and demonstration. The experiment is this:
into a large hemisphere of wood, spherically concave, put in a globe of
lead, and with your hands hold it fast by the brim, moving your hand
circularly, but in a very small compass; you shall see the globe
circulate about the concave vessel, just in the same manner as the earth
doth every year in the air; and you shall see withal, that as it goes,
it turns perpetually upon its own centre, and very swiftly.
_A._ I have seen it: and it is used in some great kitchens to grind
mustard.
_B._ Is it so? Therefore take a hemisphere of gold, if you have it, the
greater the better, and a bullet of gold, and, without mustard, you
shall see the same effect.
_A._ I doubt it not. But the cause of it is evident. For any spherical
body being in motion upon the sides of a concave and hard sphere, is all
the way turned upon its own centre by the resistance of the hard wood or
metal. But the earth is a bullet without weight, and meeteth only with
air, without any harder body in the way to resist it.
_B._ Do you think the air makes no resistance, especially to so swift a
motion as is the annual motion of the earth? If it do make any
resistance, you cannot doubt but that it shall turn the earth
circularly, and in a contrary way to its annual motion; that is to say,
from east to west, because the annual motion is from west to east.
_A._ I confess it. But what deduce you from these motions of the sun?
_B._ I deduce, first, that the air must of necessity be moved both
circularly about the body of the sun according to the ecliptic, and also
every way directly from it. For the motion of the sun’s body is
compounded of this circular motion upon the sphere L M, and of the
straight motion of its semi-diameters from the centre A to the
superficies of the sun’s body, which is L M. And therefore the air must
needs be repelled every way, and also continually change place to fill
up the places forsaken by other parts of the air, which else would be
empty, there being no vacuum to retire unto. So that there would be a
perpetual stream of air, and in a contrary way to the motion of the
sun’s body, such as is the motion of water by the sides of a ship under
sail.
_A._ But this motion of the earth from west to east is only circular,
such as is described by a compass about a centre; and cannot therefore
repel the air as the sun does. And the disciples of Copernicus will have
it to be the cause of the moon’s monthly motion about the earth.
_B._ And I think Copernicus himself would have said the same, if his
purpose had been to have shown the natural causes of the motions of the
stars. But that was no part of his design; which was only from his own
observations, and those of former astronomers, to compute the times of
their motions; partly to foretel the conjunctions, oppositions, and
other aspects of the planets; and partly to regulate the times of the
Church’s festivals. But his followers, Kepler and Galileo, make the
earth’s motion to be the efficient cause of the monthly motion of the
moon about the earth; which without the like motion to that of the sun
in L M, is impossible. Let us therefore for the present take it in as a
necessary hypothesis; which from some experiment that I shall produce in
our following discourses, may prove to be a certain truth.
_A._ But seeing A is the centre both of the sun’s body and of the annual
motion of the earth, how can it be (as all astronomers say it is) that
the orb of the annual motion of the earth should be eccentric to the
sun’s body? For you know that from the vernal equinox to the autumnal,
there be one hundred and eighty-seven days; but from the autumnal
equinox to the vernal, there be but one hundred and seventy-eight days.
What natural cause can you assign for this eccentricity?
_B._ Kepler ascribes it to a magnetic virtue, viz. that one part of the
earth’s superficies has a greater kindness for the sun than the other
part.
_A._ I am not satisfied with that. It is magical rather than natural,
and unworthy of Kepler. Tell me your own opinion of it.
_B._ I think that the magnetical virtue he speaks of, consisteth in
this: that the southern hemisphere of the earth is for the greatest part
sea, and that the greatest part of the northern hemisphere is dry land.
But how it is possible that from thence should proceed the eccentricity
(the sun being nearest to the earth, when he is in the winter solstice),
I shall show you when we come to speak of the motions of air and water.
_A._ That is time enough: for I intend it for our next meeting. In the
mean time I pray you tell me what you think to be the cause why the
equinoctial, and consequently the solstitial, points are not always in
one and the same point of the ecliptic of the fixed stars. I know they
are not, because the sun does not rise and set in points diametrically
opposite: for if it did, there would be no difference of the seasons of
the year.
_B._ The cause of that can be no other, than that the earth, which is _l
m_, hath the like motion to that which I suppose the sun to have in L M,
compounded of straight and circular from west to east in a day, as the
annual motion hath in a year; so that, not reckoning the eccentricity,
it will be moved through the ecliptics in one revolution, as Copernicus
proveth, about one degree. Suppose then the whole earth moved from H to
I, (which is half the year) circularly, but falling from I to _i_ in the
same time about thirty minutes, and as much in the other hemisphere from
H to _k_; then draw the line _i k_, which will be equal and parallel to
H I, and be the diameter of the equator for the next year. But it shall
not cut the diameter of the ecliptic B Z in A, which was the equinoctial
of the former year, but in _o_ thirty-six seconds from the first degree
of Aries. Suppose the same done in the hemisphere under the plane of the
paper, and so you have the double of thirty-six seconds, that is
seventy-two seconds, or very near, for the progress of the vernal
equinox in a year. The cause why I suppose the arch I _i_ to be half a
degree in the ecliptic of the earth, is, that Copernicus and other
astronomers, and experience, agree in this, that the equinoctial points
proceed according to the order of the signs, Aries, Taurus, Gemini, &c.
from west to east every hundredth year one degree or very near.
_A._ In what time do they make the whole revolution through the ecliptic
of the sky?
_B._ That you may reckon. For we know by experience that it hath
proceeded about one degree, that is sixty minutes, constantly a long
time in a hundred years. But as one hundred years to one degree, so is
thirty-six thousand years to three hundred and sixty degrees. Also as
one hundred years to one degree, so is one year to the hundredth part of
one degree, or sixty minutes; which is (60)/(100), or thirty-six seconds
for the progress of one year; which must be somewhat more than a degree
according to Copernicus, who, (lib. iii. cap. 2) saith, that for four
hundred years before Ptolomy it was one degree almost constantly. Which
is well enough as to the natural cause of the precession of the
equinoctial points, which is the often-said compounded motion, though
not an exact astronomical calculation.
_A._ And it is a great sign that his supposition is true. But what is
the cause that the obliquity of the ecliptic, that is, the distance
between the equinoctial and the solstice, is not always the same?
_B._ The necessity of the obliquity of the ecliptic is but a consequence
to the precession of the equinoctial points. And therefore, if from C,
the north pole, you make a little circle, C _u_, equal to fifteen
minutes of a degree upon the earth, and another, _u s_, equal to the
same, which will appear like this figure 8, that is, (as Copernicus
calls it), a circle twined, the pole C will be moved half the time of
the equinoctial points, in the arc C _u_, and as much in the alternate
arc _u s_ descending to _s_. But in the arc _s u_, and its alternate
rising to C, the cause of the twining is the earth’s annual motion the
same way in the ecliptic, and makes the four quarters of it; and makes
also their revolution twice as slow as that of the equinoctial points.
And, therefore, the motion of it is the same compounded motion which
Copernicus takes for his supposition, and is the cause of the precession
of the equinoctial points, and consequently of the variation of the
obliquity, adding to it or taking from it somewhere more, somewhere
less; so as that one with another the addition is not much more, nor the
subtraction much less than thirty minutes. But as for the natural
efficient cause of this compounded motion, either in the sun, or the
earth, or any other natural body, it can be none but the immediate hand
of the Creator.
_A._ By this it seems that the poles of the earth are always the same,
but make this 8 in the sphere of the fixed stars near that which is
called Cynosura.
_B._ No: it is described on the earth, but the annual motion describes a
circle in the sphere of the fixed stars. Though I think it improper to
say a sphere of the fixed stars, when it is so unlikely that all the
fixed stars should be in the superficies of one and the same globe.
_A._ I do not believe they are.
_B._ Nor I, since they may seem less one than another, as well by their
different distances, as by their different magnitudes. Nor is it likely
that the sun (which is a fixed star) is the efficient cause of the
motion of those remoter planets, Mars, Jupiter, and Saturn; seeing the
whole sphere, whose diameter is the distance between the sun and the
earth, is but a point in respect of the distance between the sun and any
other fixed star. Which I say only to excite those that value the
knowledge of the cause of comets, to look for it in the dominion of some
other sun than that which moveth the earth. For why may not there be
some other fixed star, nearer to some planet than is the sun, and cause
such a light in it as we call a comet?
_A._ As how?
_B._ You have seen how in high and thin clouds above the earth, the
sun-beams piercing them have appeared like a beard; and why might not
such a beard have appeared to you like a comet, if you had looked upon
it from as high as some of the fixed stars?
_A._ But because it is a thing impossible for me to know, I will proceed
in my own way of inquiry. And seeing you ascribe this compounded motion
to the sun and earth, I would grant you that the earth (whose annual
motion is from west to east) shall give the moon her monthly motion from
east to west. But then I ask you whether the moon have also that
compounded motion of the earth, and with it a motion upon its own
centre, as hath the earth? For seeing the moon has no other planet to
carry about her, she needs it not.
_B._ I see reason enough, and some necessity, that the moon should have
both those motions. For you cannot think that the Creator of the stars,
when he gave them their circular motion, did first take a centre, and
then describe a circle with a chain or compass, as men do? No; he moved
all the parts of a star together and equally in the creation: and that
is the reason I give you. The necessity of it comes from this
phenomenon, that the moon doth turn one and the same face towards the
earth; which cannot be by being moved about the earth parallelly, unless
also it turn about its own centre. Besides, we know by experience, that
the motion of the moon doth add not a little to the motion of the sea:
which were impossible if it did not add to the stream of the air, and by
consequence to that of the water.
_A._ If you could get a piece of the true and intimate substance of the
earth, of the bigness of a musket-bullet, do you believe that the bullet
would have the like compounded motion to that which you attribute to the
sun, earth, and moon?
_B._ Yes, truly; but with less strength, according to its magnitude;
saving that by its gravity falling to the earth, the activity of it
would be unperceived.
_A._ I will trouble you no more with the nature of celestial
appearances; but I pray you tell me by what art a man may find what part
of a circle the diameter of the sun’s body doth subtend in the ecliptic
circle?
_B._ Kepler says it subtends thirty minutes, which is half a degree. His
way to find it is by letting in the sun-beams into a close room through
a small hole, and receiving the image of it upon a plane
perpendicularly. For by this means he hath a triangle, whose sides and
angles he can know by measure; and the vertical angle he seeks for, and
the substance of the arc of the sun’s body.
_A._ But I think it impossible to distinguish where the part illuminate
toucheth the part not illuminate.
_B._ Another way is this: upon the equinoctial day, with a watch that
shows the minutes standing by you, observe when the lower brim of the
sun’s setting first comes to the horizon, and set the index to some
minute of the watch; and observe again the upper brim when it comes to
the horizon: then count the minutes, and you have what you look for.
Other way I know none.
==========
CHAPTER V.
OF THE MOTIONS OF WATER AND AIR.
_A._ I have considered, as you bad me, this compounded motion with great
admiration. First, it is that which makes the difference between
_continuum_ and _contiguum_, which till now I never could distinguish.
For bodies that are but contiguous, with any little force are parted;
but by this compounded motion (because every point of the body makes an
equal line in equal time, and every line crosses all the rest) one part
cannot be separated from another, without disturbing the motion of all
the other parts at once. And is not that the cause, think you, that some
bodies when they are pressed or bent, as soon as the force is removed,
return again of themselves to their former figure?
_B._ Yes, sure; saving that it is not of themselves that they return,
(for we were agreed that nothing can move itself), but it is the motion
of the parts which are not pressed, that delivers those that are. And
this restitution the learned now call the spring of a body. The Greeks
called it _antitypia_.
_A._ When I considered this motion in the sun and the earth and planets,
I fancied them as so many bodies of the army of the Almighty in an
immense field of air, marching swiftly, and commanded (under God) by his
glorious officer the sun, or rather forced so to keep their order in
every part of every of those bodies, as never to go out from the
distance in which he had set them.
_B._ But the parts of the air and other fluids keep not their places so.
_A._ No: you told me that this motion is not natural in the air, but
received from the sun.
_B._ True: but since we seek the natural causes of sublunary effects,
where shall we begin?
_A._ I would fain know what makes the sea to ebb and flow at certain
periods, and what causeth such variety in the tides.
_B._ Remember that the earth turneth every day upon its own axis from
west to east; and all the while it so turneth, every point thereof by
its compounded motion makes other circlings, but not on the same centre,
which is (you know) a rising in one part of the day, and a falling in
the other part. What think you must happen to the sea, which resteth on
it, and is a fluid body?
_A._ I think it must make the sea rise and fall. And the same happeneth
also to the air, from the motion of the sun.
_B._ Remember, also, in what manner the sea is situated in respect of
the dry land.
_A._ Is not there a great sea that reacheth from the straits of Magellan
eastward to the Indies, and thence to the same straits again? And is not
there a great sea called the Atlantic sea that runneth northward to us?
And does not the great south sea run also up into the northern seas? But
I think the Indian and the South sea of themselves to be greater than
all the rest of the surface of the globe.
_B._ How lieth the water in those two seas?
_A._ East and west, and rises and falls a little, as it is forced to do
by this compounded motion, which is a kind of succussion of the earth,
and fills both the Atlantic and Northern seas.
_B._ All this would not make a visible difference between high and low
water, because this motion being so regular, the unevenness would not be
great enough to be seen. For though in a basin the water would be thrown
into the air, yet the earth cannot throw the sea into the air.
_A._ Yes; the basin, if gently moved, will make the water so move, that
you shall hardly see it rise.
_B._ It may be so. But you should never see it rise as it doth, if it
were not checked. For at the straits of Magellan, the great South sea is
checked by the shore of the continent of Peru and Chili, and forced to
rise to a great height, and made to run up into the northern seas on
that side by the coast of China; and at the return is checked again and
forced through the Atlantic into the British and German seas. And this
is done every day. For we have supposed that the earth’s motion in the
ecliptic caused by the sun is annual; and that its motion in the
equinoctial is diurnal. It followeth therefore from this compounded
motion of the earth, the sea must ebb and flow twice in the space of
twenty-four hours, or thereabout.
_A._ Has the moon nothing to do in this business?
_B._ Yes. For she hath also the like motion. And is, though less swift,
yet much nearer to the earth. And therefore when the sun and moon are in
conjunction or opposition, the earth, as from two agents at once, must
needs have a greater succussion. And if it chance at the same time the
moon also be in the ecliptic, it will be yet greater, because the moon
then worketh on the earth less obliquely.
_A._ But when the full or new moon happen to be then when the earth is
in the equinoctial points, the tides are greater than ordinary. Why is
that?
_B._ Because then the force by which they move the sea, is at that time,
to the force by which they move the same at other times, as the
equinoctial circle to one of its parallels, which is a lesser circle.
_A._ It is evident. And it is pleasant to see the concord of so many and
various motions, when they proceed from one and the same hypothesis. But
what say you to the stupendous tides which happen on the coasts of
Lincolnshire on the east, and in the river of Severn on the west?
_B._ The cause of that, is their proper situation. For the current of
the ocean through the Atlantic sea, and the current of the south sea
through the northern seas, meeting together, rise the water in the Irish
and British seas a great deal higher than ordinary. Therefore the mouth
of the Severn being directly opposite to the current from the Atlantic
sea, and those sands on the coast of Lincolnshire directly opposite to
the current of the German sea, those tides must needs fall furiously
into them, by this succussion of the water.
_A._ Does, when the tide runs up into a river, the water all rise
together, and fall together when it goes out?
_B._ No: one part riseth and another falleth at the same time; because
the motion of the earth rising and falling, is that which makes the
tide.
_A._ Have you any experiment that shows it?
_B._ Yes. You know that in the Thames, it is high water at Greenwich
before it is high water at London-bridge. The water therefore falls at
Greenwich whilst it riseth all the way to London. But except the top of
the water went up, and the lower part downward, it were impossible.
_A._ It is certain. It is strange that this one motion should salve so
many appearances, and so easily. But I will produce one experiment of
water, not in the sea, but in a glass. If you can show me that the cause
of it is this compounded motion, I shall go near to think it the cause
of all other effects of nature hitherto disputed of. The experiment is
common, and described by the Lord Chancellor Bacon, in the third page of
his natural history. Take, saith he, a glass of water, and draw your
finger round about the lip of the glass, pressing it somewhat hard;
after you have done so a few times, it will make the water frisk up into
a fine dew. After I had read this, I tried the same with all diligence
myself, and found true not only the frisking of the water to above an
inch high, but also the whole superficies to circulate, and withal to
make a pleasant sound. The cause of the frisking he attributes to a
tumult of the inward parts of the substance of the glass striving to
free itself from the pressure.
_B._ I have tried and found both the sound and motion; and do not doubt
but the pressure of the parts of the glass was part of the cause. But
the motion of my finger about the glass was always parallel; and when it
chanced to be otherwise, both sound and motion ceased.
_A._ I found the same. And being satisfied, I proceed to other
questions. How is the water, being a heavy body, made to ascend in small
particles into the air, and be there for a time sustained in form of a
cloud, and then fall down again in rain?
_B._ I have shown already, that this compounded motion of the sun, in
one part of its circumlation, drives the air one way, and in the other
part, the contrary way; and that it cannot draw it back again, no more
than he that sets a stone a flying can pull it back. The air therefore,
which is contiguous to the water, being thus distracted, must either
leave a vacuum, or else some part of the water must rise and fill the
spaces continually forsaken by the air. But, that there is no vacuum,
you have granted. Therefore the water riseth into the air, and maketh
the clouds; and seeing they are very small and invisible parts of the
water, they are, though naturally heavy, easily carried up and down with
the wind, till, meeting with some mountain or other clouds, they be
pressed together into greater drops, and fall by their weight. So also
it is forced up in moist ground, and with it many small atoms of the
earth, which are either twisted with the rising water into plants, or
are carried up and down in the air incertainly. But the greatest
quantity of water is forced up from the great South and Indian Seas,
that lie under the tropic of capricorn. And this climate is that which
makes the sun’s perigæum to be always on the winter-solstice. And that
is the part of the terrestrial globe which Kepler says is kind to the
sun; whereas the other part of the globe, which is almost all dry land,
has an antipathy to the sun. And so you see where this magnetical virtue
of the earth lies. For the globe of the earth having no natural appetite
to any place, may be drawn by this motion of the sun a little nearer to
it, together with the water which it raiseth.
_A._ Can you guess what may be the cause of wind?
_B._ I think it manifest that the unconstant winds proceed from the
uncertain motion of the clouds ascending and descending, or meeting with
one another. For the winds after they are generated in any place by the
descent of a cloud, they drive other clouds this way and that way before
them, the air seeking to free itself from being pent up in a strait. For
when a cloud descendeth, it makes no wind sensible directly under
itself. But the air between it and the earth is pressed and forced to
move violently outward. For it is a certain experiment of mariners, that
if the sea go high when they are becalmed, they say they shall have more
wind than they would; and take in their sails all but what is necessary
for steering. They know, it seems, that the sea is moved by the descent
of clouds at some distance off: which presseth the water, and makes it
come to them in great waves. For a horizontal wind does but curl the
water.
_A._ From whence come the rivers?
_B._ From the rain, or from the falling of snow on the higher ground.
But when it descendeth under ground, the place where it again ariseth is
called the spring.
_A._ How then can there be a spring upon the top of a hill?
_B._ There is no spring upon the very top of a hill, unless some natural
pipe bring it thither from a higher hill.
_A._ Julius Scaliger says, there is a river, and in it a lake, upon the
top of Mount Cenis in Savoy; and will therefore have the springs to be
ingendered in the caverns of the earth by condensation of the air.
_B._ I wonder he should say that. I have passed over that hill twice
since the time I read that in Scaliger, and found that river as I
passed, and went by the side of it in plain ground almost two miles;
where I saw the water from two great hills, one on one side, the other
on the other, in a thousand small rillets of melting snow fall down into
it. Which has made me never to use any experiment the which I have not
myself seen. As for the conversion of air into water by condensation,
and of water into air by rarefaction, though it be the doctrine of the
Peripatetics, it is a thing incogitable, and the words are
insignificant. For by densum is signified only frequency and closeness
of parts; and by rarum the contrary. As when we say a town is thick with
houses, or a wood with trees, we mean not that one house or tree is
thicker than another, but that the spaces between are not so great. But,
since there is no vacuum, the spaces between the parts of air are no
larger than between the parts of water, or of any thing else.
_A._ What think you of those things which mariners that have sailed
through the Atlantic Sea, called _spouts_, which pour down water enough
at once to drown a great ship?
_B._ It is a thing I have not seen: and therefore can say nothing to it;
though I doubt not but when two very large and heavy clouds shall be
driven together by two great and contrary winds, the thing is possible.
_A._ I think your reason good. And now I will propound to you another
experiment. I have seen an exceeding small tube of glass with both ends
open, set upright in a vessel of water, and that the superficies of the
water within the tube was higher a good deal than of that in the vessel;
but I see no reason for it.
_B._ Was not part of the glass under water? Must not then the water in
the vessel rise? Must not the air that lay upon it rise with it? Whither
should this rising air go, since there is no place empty to receive it?
It is therefore no wonder if the water, pressed by the substance of the
glass which is dipt into it, do rather rise into a very small pipe, than
come about a longer way into the open air.
_A._ It is very probable. I observed also that the top of the inclosed
water was a concave superficies; which I never saw in other fluids.
_B._ The water hath some degree of tenacity, though not so great but
that it will yield a little to the motion of the air; as is manifest in
the bubbles of water, where the concavity is always towards the air. And
this I think the cause why the air and water meeting in the tube make
the superficies towards the air concave, which it cannot do to a fluid
of greater tenacity.
_A._ If you put into a basin of water a long rag of cloth, first
drenched in water, and let the longer part of it hang out, it is known
by experience, that the water will drop out as long as there is any part
of the other end under water.
_B._ The cause of it is, that water, as I told you, hath a degree of
tenacity. And therefore being continued in the rag till it be lower
without than within, the weight will make it continue dropping, though
not only because it is heavy (for if the rag lay higher without than
within, and were made heavier by the breadth, it would not descend), but
it is because all heavy bodies naturally descend with proportion of
swiftness duplicate to that of the time; whereof I shall say more when
we talk of gravity.
_A._ You see how despicable experiments I trouble you with. But I hope
you will pardon me.
_B._ As for mean and common experiments, I think them a great deal
better witnesses of nature, than those that are forced by fire, and
known but to very few.
==========
CHAPTER VI.
OF THE CAUSES AND EFFECTS OF HEAT AND COLD.
_A._ It is a fine day, and pleasant walking through the fields, but that
the sun is a little too hot.
_B._ How know you that the sun is hot?
_A._ I feel it.
_B._ That is to say, you know that yourself, but not that the sun is
hot. But when you find yourself hot, what body do you feel?
_A._ None.
_B._ How then can you infer your heat from the sense of feeling? Your
walking may have made you hot: is motion therefore hot? No. You are to
consider the concomitants of your heat; as, that you are more faint, or
more ruddy, or that you sweat, or feel some endeavour of moisture or
spirits tending outward; and when you have found the causes of those
accidents, you have found the causes of heat, which in a living
creature, and especially in a man, is many times the motion of the parts
within him, such as happen in sickness, anger, and other passions of the
mind; which are not in the sun nor in fire.
_A._ That which I desire now to know, is what motions and of what bodies
without me are the efficient causes of my heat.
_B._ I showed you yesterday, in discoursing of rain, how by this
compounded motion of the sun’s body, the air was every way at once
thrust off west and east; so that where it was contiguous, the small
parts of the water were forced to rise, for the avoiding of vacuum.
Think then that your hand were in the place of water so exposed to the
sun. Must not the sun work upon it as it did upon the water? Though it
break not the skin, yet it will give to the inner fluids and looser
parts of your hand, an endeavour to get forth, which will extend the
skin, and in some climates fetch up the blood, and in time make the skin
black. The fire also will do the same to them that often sit with their
naked skins too near it. Nay, one may sit so near, without touching it,
as it shall blister or break the skin, and fetch up both spirits and
blood mixt into a putrid oily matter, sooner than in a furnace oil can
be extracted out of a plant.
_A._ But if the water be above the fire in a kettle, what then will it
do? Shall the particles of water go toward the fire, as it did toward
the sun?
_B._ No. For it cannot. But the motion of the parts of the kettle which
are caused by the fire, shall dissipate the water into vapour till it be
all cast out.
_A._ What is that you call fire? Is it a hard or fluid body?
_B._ It is not any other body but that of the shining coal; which coal,
though extinguished with water, is still the same body. So also in a
very hot furnace, the hollow spaces between the shining coals, though
they burn that you put into them, are no other body than air moved.
_A._ Is it not flame?
_B._ No. For flame is nothing but a multitude of sparks, and sparks are
but the atoms of the fuel dissipated by the incredible swift motion of
the movent, which makes every spark to seem a hundred times greater than
it is, as appears by this; that, when a man swings in the air a small
stick fired at one end, though the motion cannot be very swift, yet the
fire will appear to the eye to be a long, straight, or crooked line.
Therefore a great many sparks together flying upward, must needs appear
unto the sight as one continued flame. Nor are the sparks stricken out
of a flint any thing else but small particles of the stone, which by
their swift motion are made to shine. But that fire is not a substance
of itself, is evident enough by this, that the sun-beams passing through
a globe of water will burn as other fire does. Which beams, if they were
indeed fire, would be quenched in the passage.
_A._ This is so evident, that I wonder so wise men as Aristotle and his
followers, for so long a time could hold it for an element, and one of
the primary parts of the universe. But the natural heat of a man or
other living creature, whence proceedeth it? Is there anything within
their bodies that hath this compounded motion?
_B._ At the breaking up of a deer I have seen it plainly in his bowels
as long as they were warm. And it is called the peristaltic motion, and
in the heart of a beast newly taken out of his body; and this motion is
called systole and diastole. But they are both of them this compounded
motion, whereof the former causeth the food to wind up and down through
the guts, and the latter makes the circulation of the blood.
_A._ What kind of motion is the cause of cold? Methinks it should be
contrary to that which causeth heat.
_B._ So it is in some respect. For seeing the motion that begets heat,
tendeth to the separation of the parts of the body whereon it acteth, it
stands with reason, that the motion which maketh cold, should be such as
sets them closer together. But contrary motions are, to speak properly,
when upon two ends of a line two bodies move towards each other, the
effect whereof is to make them meet. But each of them, as to this
question, is the same.
_A._ Do you think (as many philosophers have held and now hold), that
cold is nothing but a privation of heat?
_B._ No. Have you never heard the fable of the satyr that dwelling with
a husbandman, and seeing him blow his fingers to warm them, and his
pottage to cool it, was so scandalized, that he ran from him, saying he
would no longer dwell with one that could blow both hot and cold with
one breath? Yet the cause is evident enough. For the air which had
gotten a calefactive power from his vital parts, was from his mouth and
throat gently diffused on his fingers, and retained still that power.
But to cool his pottage he straightened the passage at his lips, which
extinguished the calefactive motion.
_A._ Do you think wind the general cause of cold? If that were true, in
the greatest winds we should have the greatest frosts.
_B._ I mean not any of those uncertain winds which, I said, were made by
the clouds, but such as a body moved in the air makes to and against
itself; (for it is all one motion of the air whether it be carried
against the body, or the body against it); such a wind as is constant,
if no other be stirring, from east to west; and made by the earth
turning daily upon its own centre; which is so swift, as, except it be
kept off by some hill, to kill a man, as by experience hath been found
by those who have passed over great mountains, and specially over the
Andes which are opposed to the east. And such is the wind which the
earth maketh in the air by her annual motion, which is so swift, as
that, by the calculation of astronomers, to go sixty miles in a minute
of an hour. And therefore this must be the motion which makes it so cold
about the poles of the ecliptic.
_A._ Does not the earth make the wind as great in one part of the
ecliptic as in another?
_B._ Yes. But when the sun is in Cancer, it tempers the cold, and still
less and less, but least of all in the winter-solstice, where his beams
are most oblique to the superficies of the earth.
_A._ I thought the greatest cold had been about the poles of the
equator.
_B._ And so did I once. But the reason commonly given for it is so
improbable, that I do not think so now. For the cause they render of it
is only, that the motion of the earth is swiftest in the equinoctial,
and slowest about the poles; and consequently, since motion is the cause
of heat, and cold is but, as it was thought, a want of the same, they
inferred that the greatest cold must be about the poles of the
equinoctial. Wherein they miscounted. For not every motion causeth heat,
but this agitation only, which we call compounded motion; though some
have alleged experience for that opinion; as that a bullet out of a gun
will with its own swiftness melt. Which I never shall believe.
_A._ It is a common thing with many philosophers to maintain their
fancies with any rash report, and sometimes with a lie. But how is it
possible that so soft a substance as water should be turned into so hard
a substance as ice?
_B._ When the air shaves the globe of the earth with such swiftness, as
that of sixty miles in a minute of an hour, it cannot, where it meets
with still water, but beat it up into small and undistinguishable
bubbles, and involve itself in them as in so many bladders or skins of
water. And ice is nothing else but the smallest imaginable parts of air
and water mixed; which is made hard by this compounded motion, that
keeps the parts so close together, as not to be separated in one place
without disordering the motion of them all. For when a body will not
easily yield to the impression of an external movent in one place
without yielding in all, we call it hard; and when it does, we say it is
soft.
_A._ Why is not ice as well made in a moved as in a still water? Are
there not great seas of ice in the northern parts of the earth?
_B._ Yes, and perhaps also in the southern parts. But I cannot imagine
how ice can be made in such agitation as is always in the open sea, made
by the tides and by the winds. But how it may be made at the shore, it
is not hard to imagine. For in a river or current, though swift, the
water that adhereth to the banks is quiet, and easily by the motion of
the air driven into small insensible bubbles; and so may the water that
adhereth to those bubbles, and so forwards till it come into a stream
that breaks it, and then it is no wonder though the fragments be driven
into the open sea, and freeze together into greater lumps. But when in
the open sea, or at the shore, the tide or a great wave shall arise,
this young and tender ice will presently be washed away. And therefore I
think it evident, that as in the Thames the ice is first made at the
banks where the tide is weak or none, and, broken by the stream, comes
down to London, and part goes to the sea floating till it dissolve, and
part, being too great to pass the bridge, stoppeth there and sustains
that which follows, till the river be quite frozen over; so also the ice
in the northern seas begins first at the banks of the continent and
islands which are situated in that climate, and then broken off, are
carried up and down, and one against another, till they become great
bodies.
_A._ But what if there be islands, and narrow inlets of the sea, or
rivers also about the pole of the equinoctial?
_B._ If there be, it is very likely the sea may also there be covered
all over with ice. But for the truth of this, we must stay for some
farther discovery.
_A._ When the ice is once made and hard, what dissolves it?
_B._ The principal cause of it, is the weight of the water itself; but
not without some abatement in the stream of the air that hardeneth it;
as when the sunbeams are less oblique to the earth, or some contrary
wind resisteth the stream of the air. For when the impediment is
removed, then the nature of the water only worketh, and, being a heavy
body, downward.
_A._ I forgot to ask you, why two pieces of wood rubbed swiftly one
against another, will at length set on fire.
_B._ Not only at length, but quickly, if the wood be dry. And the cause
is evident, viz. the compounded motion which dissipates the external
small parts of the wood. And then the inner parts must of necessity, to
preserve the plentitude of the universe, come after; first the most
fluid, and then those also of greater consistence, which are first
erected, and the motion continued, made to fly swiftly out; whereby the
air driven to the eye of the beholder, maketh that fancy which is called
light.
_A._ Yes; I remember you told me before, that upon any strong pressure
of the eye, the resistance from within would appear a light. But to
return to the enquiry of heat and cold, there be two things that beyond
all other put me into admiration. One is the swiftness of kindling in
gunpowder. The other is the freezing of water in a vessel, though not
far from the fire, set about with other water with ice and snow in it.
When paper or flax is flaming, the flame creeps gently on; and if a
house full of paper were to be burnt with putting a candle to it, it
will be long in burning; whereas a spark of fire would set on flame a
mountain of gunpowder in almost an instant.
_B._ Know you not gunpowder is made of the powder of charcoal,
brimstone, and saltpetre? Whereof the first will kindle with a spark,
the second flame as soon as touched with fire; and the third blows it,
as being composed of many orbs of salt filled with air, and as it
dissolveth in the flame, furiously blowing increaseth it. And as for
making ice by the fireside; it is manifest that whilst the snow is
dissolving in the external vessel, the air must in the like manner break
forth, and shave the superficies of the inner vessel, and work through
the water till it be frozen.
_A._ I could easily assent to this, if I could conceive how the air that
shaves, as you say, the outside of the vessel, could work through it. I
conceive well enough a pail of water with ice or snow dissolving in it,
and how it causeth wind. But how that wind should communicate itself
through the vessel of wood or metal, so as to make it shave the
superficies of the water which is within it, I do not so well
understand.
_B._ I do not say the inner superficies of the vessel shaves the water
within it. But it is manifest that the wind made in the pail of water by
the melting snow or ice presseth the sides of the vessel that standeth
in it; and that the pressure worketh clean through, how hard soever the
vessel be; and that again worketh on the water within, by restitution of
its parts, and so hardeneth the water by degrees.
_A._ I understand you now. The ice in the pail by its dissolution
transfers its hardness to the water within.
_B._ You are merry. But supposing, as I do, that the ice in the pail is
more than the water in the vessel, you will find no absurdity in the
argument. Besides, the experiment, you know, is common.
_A._ I confess it is probable. The Greeks have the word φρίκη (whence
the Latins have their word _frigus_) to signify the curling of the water
by the wind; and use the same also for horror, which is the passion of
one that cometh suddenly into a cold air, or is put into a sudden
affright, whereby he shrinks, and his hair stands upright. Which
manifestly shows that the motion which causeth cold, is that which
pressing the superficies of a body, sets the parts of it closer
together. But to proceed in my queries. Monsieur Des Cartes, whom you
know, hath written somewhere, that the noise we hear in thunder,
proceeds from breaking of the ice in the clouds; what think you of it?
Can a cloud be turned into ice?
_B._ Why not? A cloud is but water in the air?
_A._ But how? For he has not told us that.
_B._ You know that it is only in summer, and in hot weather, that it
thunders; or if in winter, it is taken for a prodigy. You know also,
that of clouds, some are higher, some lower, and many in number, as you
cannot but have oftentimes observed, with spaces between them.
Therefore, as in all currents of water, the water is there swiftest
where it is straitened with islands, so must the current of air made by
the annual motion be swiftest there, where it is checked with many
clouds, through which it must, as it were, be strained, and leave behind
it many small particles of earth always in it, and in hot weather more
than ordinary.
_A._ This I understand, and that it may cause ice. But when the ice is
made, how is it broken? And why falls it not down in shivers?
_B._ The particles are enclosed in small caverns of the ice; and their
natural motion being the same which we have ascribed to the globe of the
earth, requires a sufficient space to move in. But when it is imprisoned
in a less room than that, then a great part of the ice breaks: and this
is the thunder-clap. The murmur following is from the settling of the
air. The lightning is the fancy made by the recoiling of the air against
the eye. The fall is in rain, not in shivers; because the prisons which
they break are extreme narrow, and the shivers being small, are
dissolved by the heat. But in less heat they would fall in drops of
hail, that is to say, half frozen by the shaving of the air as they
fall, and be in a very little time, much less than snow or ice,
dissolved.
_A._ Will not that lightning burn?
_B._ No. But it hath often killed men with cold. But this extraordinary
swiftness of lightning consisteth not in the expansion of the air, but
in a straight and direct stream from where it breaks forth; which is in
many places successively, according to the motion of the cloud.
_A._ Experience tells us that. I have now done with my problems
concerning the great bodies of the world, the stars, and element of air
in which they are moved, and am therein satisfied, and the rather,
because you have answered me by the supposition of one only motion, and
commonly known, and the same with that of Copernicus, whose opinion is
received by all the learned; and because you have not used any of these
empty terms, sympathy, antipathy, antiperistasis, etc., for a natural
cause, as the old philosophers have done to save their credit. For
though they were many of them wise men, as Plato, Aristotle, Seneca, and
others, and have written excellently of morals and politics, yet there
is very little natural philosophy to be gathered out of their writings.
_B._ Their ethics and politics are pleasant reading, but I find not any
argument in their discourses of justice or virtue drawn from the supreme
authority, on whose laws all justice, virtue, and good politics depend.
_A._ Concerning this cover, or, as some have called it, the scurf or
scab of the terrestrial star, I will begin with you tomorrow. For it is
a large subject, containing animals, vegetables, metals, stones, and
many other kinds of bodies, the knowledge whereof is desired by most
men, and of the greatest and most general profit.
_B._ And this is it, in which I shall give you the least satisfaction;
so great is the variety of motion, and so concealed from human senses.
==========
CHAPTER VII.
OF HARD AND SOFT, AND OF THE ATOMS THAT FLY IN
THE AIR.
_A._ Concerning this cover of the earth, made up of an infinite number
of parts of different natures, I had much ado to find any tolerable
method of enquiry. But I resolved at last to begin with the questions
concerning hard and soft, and what kind of motion it is that makes them
so. I know that in any pulsion of air, the parts of it go innumerable
and inexplicable ways; but I ask only if every point of it be moved?
_B._ No. If you mean a mathematical point, you know it is impossible.
For nothing is movable but body. But I suppose it divisible, as all
other bodies, into parts divisible. For no substance can be divided into
nothings.
_A._ Why may not that substance within our bodies, which are called
animal spirits, be another kind of body, and more subtile than the
common air?
_B._ I know not why, no more than you or any man else knows why it is
not very air, though purer perhaps than the common air, as being
strained through the blood into the brain and nerves. But howsoever that
be, there is no doubt, but the least parts of the common air,
respectively to the whole, will easilier pierce, with equal motion, the
body that resisteth them, than the least parts of water. For it is by
motion only that any mutation is made in any thing; and all things
standing as they did, will appear as they did. And that which changeth
soft into hard, must be such as makes the parts not easily to be moved
without being moved all together; which cannot be done but by some
motion compounded. And we call hard, that whereof no part can be put out
of order without disordering all the rest; which is not easily done.
_A._ How water and air beaten into extreme small bubbles is hardened
into ice, you have told me already, and I understand it. But how a soft
homogeneous body, as air or water, should be so hardened, I cannot
imagine.
_B._ There is no hard body that hath not also some degree of gravity;
and consequently, being loose, there must be some efficient cause, that
is, some motion, when it is severed from the earth, to bring the same to
it again. And seeing this compounded motion gives to the air and water
an endeavour from the earth, the motion which must hinder it, must be in
a way contrary to the compounded motion of the earth. For whatsoever,
having been asunder, comes together again, must come contrary ways, as
those that follow one another go the same way, though both move upon the
same line.
_A._ What experiment have you seen to this purpose?
_B._ I have seen a drop of glass like that of the second figure, newly
taken out of the furnace, and hanging at the end of an iron rod, and yet
fluid, and let fall into the water and hardened. The club-end of it A A
coming first to the water, the tail B C following it. It is proved
before, that the motion that makes it is a compounded motion, and gives
an endeavour outward to every part of it; and that the motion which
maketh cold, is such as shaving the body in every point of contact, and
turning it, gives them all an endeavour inward. Such is this motion made
by the sinking of the hot and fluid glass into the water. It is
therefore manifest that the motion which hardeneth a soft body, must in
every point of contact be in the contrary way to that which makes a hard
body soft. And farther, that slender tail B C shall be made much more
hard than common glass. For towards the upper end, in C, you cannot
easily break it, as small as it is. And when you have broken it, the
whole body will fall into dust, as it must do, seeing the bending is so
difficult. For all the parts are bent with such force, that upon the
breaking at D, by their sudden restitution to their liberty, they will
break together. And the cause why the tail B C, being so slender,
becomes so hard, is, that all the endeavour in the great part A B, is
propagated to the small part B C, in the same manner as the force of the
sun-beams is derived almost to a point by a burning-glass. But the cause
why, when it is broken in D, it breaks also in so many other places, is,
that the endeavour in all the other parts, which is called the spring,
unbends it; from whence a motion is caused the contrary way, and that
motion continued bends it more the other way and breaks it, as a bow
over-bent is broken into shivers by a sudden breaking of the string.
_A._ I conceive now how a body which having been hard and softened
again, may be rehardened; but how a fluid and mere homogeneous body, as
air or water, may be so, I see not yet. For the hardening of water is
making a hard body of two fluids, whereof one, which is the water, hath
some tenacity; and so a man may make a bladder hard with blowing into
it.
_B._ As for mere air, which hath no natural motion of itself, but is
moved only by other bodies of a greater consistence, I think it
impossible to be hardened. For the parts of it so easily change places,
that they can never be fixed by any motion. No more I think can water,
which though somewhat less fluid, is with an insensible force very
easily broken.
_A._ It is the opinion of many learned men, that ice, in long time, will
be turned into crystal; and they allege experience for it. For they say
that crystal is found hanging on high rocks in the Alps, like icicles on
the eaves of a house; and why may not that have formerly been ice, and
in many years have lost the power of being reduced?
_B._ If that were so, it would still be ice, though also crystal: which
cannot be, because crystal is heavier than water, and therefore much
heavier than ice.
_A._ Is there then no transubstantiation of bodies but by mixture?
_B._ Mixture is no transubstantiation.
_A._ Have you never seen a stone that seemed to have been formerly wood,
and some like shells, and some like serpents, and others like other
things?
_B._ Yes. I have seen such things, and particularly I saw at Rome, in a
stone-cutter’s workhouse, a billet of wood, as I thought it, partly
covered with bark, and partly with the grain bare, as long as a man’s
arm, and as thick as the calf of a man’s leg; which handling I found
extreme heavy, and saw a small part of it which was polished, and had a
very fine gloss, and thought it a substance between stone and metal, but
nearest to stone. I have seen also a kind of slate painted naturally
with forest-work. And I have seen in the hands of a chemist of my
acquaintance at Paris, a broken glass, part of a retort, in which had
been the rosin of turpentine, wherein though there were left no rosin,
yet there appeared in the piece of glass many trees; and plants in the
ground about them, such as grow in woods; and better designed than they
could be done by any painter; and continued so for a long time. These be
great wonders of nature, but I will not undertake to show their causes.
But yet this is most certain, that nothing can make a hard body of a
soft, but by some motion of its parts. For the parts of the hardest body
in the world can be no closer together than to touch; and so close are
the parts of air and water, and consequently they should be equally
hard, if their smallest parts had not different natural motions.
Therefore if you ask me the causes of these effects, I answer, they are
different motions. But if you expect from me how and by what motions, I
shall fail you. For there is no kind of substance in the world now, that
was not at the first creation, when the Creator gave to all things what
natural and special motion he thought good. And as he made some bodies
wondrous great, so he made others wondrous little. For all his works are
wondrous. Man can but guess, nor guess farther, than he hath knowledge
of the variety of motion. I am therefore of opinion, that whatsoever
perfectly homogeneous is hard, consisteth of the smallest parts, or, as
some call them, atoms, that were made hard in the beginning, and
consequently by an eternal cause; and that the hardness of the whole
body is caused only by the contact of the parts by pressure.
_A._ What motion is it that maketh a hard body to melt?
_B._ The same compounded motion that heats, namely, that of fire, if it
be strong enough. For all motion compounded is an endeavour to
dissipate, as I have said before, the parts of the body to be moved by
it. If therefore hardness consist only in the pressing contact of the
least parts, this motion will make the same parts slide off from one
another, and the whole to take such a figure as the weight of the parts
shall dispose them to, as in lead, iron, gold, and other things melted
with heat. But if the small parts have such figures as they cannot
exactly touch, but must leave spaces between them filled with air or
other fluids, then this motion of the fire, will dissipate those parts
some one way, some another, the hard part still hard; as in the burning
of wood or stone into ashes or lime. For this motion is that which
maketh fermentation, scattering dissimilar parts, and congregating
similar.
_A._ Why do some hard bodies resist breaking more one way than another?
_B._ The bodies that do so, are for the most part wood, and receive that
quality from their generation. For the heat of the sun in the
spring-time draweth up the moisture at the root, and together with it
the small parts of the earth, and twisteth it into a small twig, by its
motion upwards, to some length, but to very little other dimensions, and
so leaves it to dry till the spring following; and then does the same to
that, and to every small part round about it; so that upward the
strength is doubled, and the next year trebled, &c. And these are called
the grain of the wood, and but touch one another, like sticks with
little or no binding, and therefore can hardly be broken across the
grain, but easily all-along it. Also some other hard bodies have this
quality of being more fragile one way than another, as we see in
quarrels of a glass window, that are aptest many times to break in some
crooked line. The cause of this may be, that when the glass, hot from
the furnace, is poured out upon a plain, any small stones in or under it
will break the stream of it into divers lines, and not only weaken it,
but also cause it falsely to represent the object you look on through
it.
_A._ What is the cause why a bow of wood or steel, or other very hard
body, being bent, but not broken, will recover its former degree of
straightness?
_B._ I have told you already, how the smallest parts of a hard body have
every one, by the generation of hardness, a circular, or other
compounded motion; such motion is that of the smallest parts of the bow.
Which circles in the bending you press into narrower figures, as a
circle into an ellipsis, and an ellipsis into a narrower but longer
ellipsis with violence; which turns their natural motion against the
outward parts of the bow so bent, and is an endeavour to stretch the bow
into its former posture. Therefore if the impediment be removed, the bow
must needs recover its former figure.
_A._ It is manifest; and the cause can be no other but that, except the
bow have sense.
_B._ And though the bow had sense, and appetite to boot, the cause will
be still the same.
_A._ Do you think air and water to be pure and homogeneous bodies?
_B._ Yes, and many bodies both hard and heavy to be so too, and many
liquors also besides water.
_A._ Why then do men say they find one air healthy, another infectious?
_B._ Not because the nature of the air varies, but because there are in
the air, drawn, or rather, beaten up by the sun, many little bodies,
whereof some have such motion as is healthful, others such as is hurtful
to the life of man. For the sun, as you see in the generation of plants,
can fetch up earth as well as water: and from the driest ground any kind
of body that lieth loose, so it be small enough, rather than admit any
emptiness. By some of these small bodies it is that we live; which being
taken in with our breath, pass into our blood, and cause it, by their
compounded motion, to circulate through the veins and arteries; which
the blood of itself, being a heavy body, without it cannot do. What kind
of substance these atoms are, I cannot tell. Some suppose them to be
nitre. As for those infectious creatures in the air, whereof so many die
in the plague, I have heard that Monsieur Des Cartes, a very ingenious
man, was of opinion, that they were little flies. But what grounds he
had for it, I know not, though there be many experiments that invite me
to believe it. For first, we know that the air is never universally
infected over a whole country, but only in or near to some populous
town. And therefore the cause must also be partly ascribed to the
multitude thronged together, and constrained to carry their excrements
into the fields round about and near to their habitation, which in time
fermenting breed worms, which commonly in a month or little more,
naturally become flies; and though engendered at one town, may fly to
another. Secondly, in the beginning of a plague, those that dwell in the
suburbs, that is to say, nearest to this corruption, are the poorest of
the people, that are nourished for the most part with the roots and
herbs which grow in that corrupted dirt; so that the same filth makes
both the blood of poor people, and the substance of the fly. And it is
said by Aristotle, that everything is nourished by the matter whereof it
is generated. Thirdly, when a town is infected, the gentlemen, and those
that live on wholsomest food, scarce one of five hundred die of the
plague. It seems therefore, whatsoever creatures they be that invade us
from the air, they can discern their proper nourishment, and do not
enter into the mouth and nostrils with the breath of every man alike, as
they would do if they were inanimate. Fourthly, a man may carry the
infection with him a great way into the country in his clothes, and
infect a village. Shall another man there draw the infection from the
clothes only by his breath? Or from the hangings of a chamber wherein a
man hath died? It is impossible. Therefore whatsoever killing thing is
in the clothes or hangings, it must rise and go into his mouth or
nostrils before it can do him hurt. It must therefore be a fly, whereof
great numbers get into the blood, and there feeding and breeding worms,
obstruct the circulation of blood, and kill the man.
_A._ I would we knew the palate of those little animals; we might
perhaps find some medicine to fright them from mingling with our breath.
But what is that which kills men that lie asleep too near a
charcoal-fire? Is it another kind of fly? Or is charcoal venomous?
_B._ It is neither fly nor venom, but the effect of a flameless glowing
fire, which dissipates those atoms that maintain the circulation of the
blood; so that for want of it, by degrees they faint, and being asleep
cannot remove, but in short time, there sleeping die; as is evident by
this, that being brought into the open air, without other help, they
recover.
_A._ It is very likely. The next thing I would be informed of, is the
nature of gravity. But for that, if you please, we will take another
day.
==========
CHAPTER VIII.
OF GRAVITY AND GRAVITATION.
_B._ What books are those?
_A._ Two books written by two learned men concerning gravity. I brought
them with me, because they furnish me with some material questions about
that doctrine; though of the nature of gravity, I find no more in either
of them than this, that gravity is an intrinsical quality, by which a
body so qualified descendeth perpendicularly towards the superficies of
the earth.
_B._ Did neither of them consider that descending is local motion; that
they might have called it an intrinsical motion rather than an
intrinsical quality?
_A._ Yes. But not how motion should be intrinsical to the special
individual body moved. For how should they, when you are the first that
ever sought the differences of qualities in local motion, except your
authority in philosophy were greater with them than it is? For it is
hard for a man to conceive, except he see it, how there should be motion
within a body, otherwise than as it is in living creatures.
_B._ But it may be they never sought, or despaired of finding what
natural motion could make any inanimate thing tend one way rather than
another.
_A._ So it seems. But the first of them inquires no farther than, why so
much water, being a heavy body, as lies perpendicularly on a fish’s back
in the bottom of the sea, should not kill it. The other, whereof the
author is Dr. Wallis, treateth universally of gravity.
_B._ Well; but what are the questions which from these books you intend
to ask me?
_A._ The author of the first book tells me, that water and other fluids
are bodies continued, and act, as to gravity, as a piece of ice would do
of the same figure and quantity. Is that true?
_B._ That the universe, supposing there is no place empty, is one entire
body, and also, as he saith it is, a continual body, is very true. And
yet the parts thereof may be contiguous, without any other cohesion but
touch. And it is also true, that a vessel of water will descend in a
medium less heavy, but fluid, as ice would do.
_A._ But he means that water in a tub would have the same effect upon a
fish in the bottom of the tub, as so much ice would have.
_B._ That also would be true, if the water were frozen to the sides of
it. Otherwise the ice, if there be enough, will crush the fish to death.
But how applies he this, to prove that the water cannot hurt a fish in
the sea by its weight?
_A._ It plainly appears that water does not gravitate on any part of
itself beneath it.
_B._ It appears by experience, but not by this argument, though instead
of water the tub were filled with quicksilver.
_A._ I thought so. But how it comes to pass that the fish remains
uncrushed, I cannot tell.
_B._ The endeavour of the quicksilver downward is stopped by the
resistance of the hard bottom. But all resistance is a contrary
endeavour; that is, an endeavour upwards, which gives the like endeavour
to the quicksilver, which is also heavy, and thereby the endeavour of
the quicksilver is diverted to the sides round about, where stopped
again by the resistance of the sides, it receives an endeavour upwards,
which carries the fish to the top, lying all the way upon a soft bed of
quicksilver. This is the true manner how the fish is saved harmless. But
your author, I believe, either wanted age, or had too much business, to
study the doctrine of motion; and never considered that resistance is
not an impediment only, or privation, but a contrary motion; and that
when a man claps two pieces of wax together, their contrary endeavour
will turn both the pieces into one cake of wax.
_A._ I know not the author; but it seems he has deeplier considered this
question than other men; for in the introduction to his book he saith,
“that men have pre-engaged themselves to maintain certain principles of
their own invention, and are therefore unwilling to receive anything
that may render their labour fruitless;” and, “that they have not
strictly enough considered the several interventions that abate, impede,
advance, or direct the gravitation of bodies.”
_B._ This is true enough; and he himself is one of those men, in that he
considered not, that resistance is one of those interventions which
abate, impede, and direct gravitation. But what are his suppositions for
the questions he handles?
_A._ His first is, that as in a pyramid of brick, wherein the bricks are
so joined that the uppermost lies everywhere over the joint or cement of
the two next below it, you may break down a part and leave a cavity, and
yet the bricks above will stand firm and sustain one another by their
cross posture: so also it is in wheat, hailshot, sand, or water; and so
they arch themselves, and thereby the fish is every way secured by an
arch of water over it.
_B._ That the cause why fishes are not crushed nor hurt in the bottom of
the sea by the weight of the water, is the water’s arching itself, is
very manifest. For if the uppermost orb of the water should descend by
its gravity, it would tend toward the centre of the earth, and place
itself all the way in a less and lesser orb, which is impossible. For
the places of the same body are always equal. But that wheat, sand,
hailshot, or loose stones should make a firm arch, is not credible.
_A._ The author therefore, it seems, quits it, and taketh a second
hypothesis for the true cause, though the former, he saith, be not
useless, but contributes its part to it.
_B._ I see, though he depart from his hypothesis, he looks back upon it
with some kindness. What is his second hypothesis?
_A._ It is, that air and water have an endeavour to motion upward,
downward, directly, obliquely, and every way. For air, he saith, will
come down his chimney, and in at his door, and up his stairs.
_B._ Yes, and mine too; and so would water, if I dwelt under water,
rather than admit of vacuum. But what of that?
_A._ Why then it would follow, that those several tendencies or
endeavours would so abate, impede, and correct one another, as none of
them should gravitate. Which being granted, the fish can take no harm;
wherein I find one difficulty, which is this: the water having an
endeavour to motion every way at once, methinks it should go no way, but
lie at rest; which, he saith, was the opinion of Stevinus, and rejecteth
it, saying, it would crush the fish into pieces.
_B._ I think the water in this case would neither rest nor crush. For
the endeavour being, as he saith, intrinsical, and every way, must needs
drive the water perpetually outward; that is to say, as to this
question, upwards; and seeing the same endeavour in one individual body
cannot be more ways at once than one, it will carry it on perpetually
without limit, beyond the fixed stars; and so we shall never more have
rain.
_A._ As ridiculous as it is, it necessarily follows.
_B._ What are Dr. Wallis’s suppositions?
_A._ He goes upon experiments. And, first, he allegeth this, that water
left to itself without disturbance, does naturally settle itself into a
horizontal plane.
_B._ He does not then, as your author and all other men, take gravity
for that quality whereby a body tendeth to the centre of the earth.
_A._ Yes, he defines gravity to be a natural propension towards the
centre of the earth.
_B._ Then he contradicteth himself. For if all heavy bodies tend
naturally to one centre, they shall never settle in a plane, but in a
spherical superficies. But against this, that such an horizontal plane
is found in water by experience, I say it is impossible. For the
experiment cannot be made in a basin, but in half a mile at sea
experience visibly shows the contrary. According to this, he should
think also that a pair of scales should hang parallel.
_A._ He thinks that too.
_B._ Let us then leave this experiment. What says he farther concerning
gravity?
_A._ He takes for granted, not as an experiment but an axiom, that
nature worketh not by election, but _ad ultimum virium_, with all the
power it can.
_B._ I think he means, (for it is a very obscure passage), that every
inanimate body by nature worketh all it can without election; which may
be true. But it is certain that men, and beasts, work often by election,
and often without election; as when he goes by election, and falls
without it. In this sense I grant him, that nature does all it can. But
what infers he from it?
_A._ That naturally every body has every way, if the ways oppose not one
another, an endeavour to motion; and consequently, that if a vessel have
two holes, one at the side, another at the bottom, the water will run
out at both.
_B._ Does he think the body of water that runs out at the side, and that
which runs out at the bottom, is but one and the same body of water?
_A._ No, sure; he cannot think but that they are two several parts of
the whole water in the vessel.
_B._ What wonder is it then, if two parts of water run two ways at once,
or a thousand parts a thousand ways? Does it follow thence that one body
can go more than one way at once? Why is he still meddling with things
of such difficulty? He will find at last that he has not a genius either
for natural philosophy or for geometry. What other suppositions has he?
_A._ My first author had affirmed, that a lighter body does not
gravitate on a heavier; against this Dr. Wallis thus argueth: Let there
be a siphon, A B C D, filled with quicksilver to the level A D; if then
you pour oil upon A as high as to E, he asketh if the oil in A E, as
being heavy, shall not press down the quicksilver a little at A, and
make it rise a little at D, suppose to F; and answers himself, that
certainly it will; so that it is neither an experiment nor an
hypothesis, but only his opinion.
_B._ Whatsoever it be, it is not true; though the doctor may be
pardoned, because the contrary was never proved.
_A._ Can you prove the contrary?
_B._ Yes; for the endeavour of the quicksilver both from A and D
downward, is stronger than that of the oil downward. If, therefore, the
endeavour of the quicksilver were not resisted by the bottom B C, it
would fall so, by reason of the acceleration of heavy bodies in their
descending, as to leave the oil, so that it should not only not press,
but also not touch the quicksilver. It is true, in a pair of scales
equally charged with quicksilver, that the addition of a little oil to
either scale will make it preponderate. And that was it deceived him.
_A._ It is evident. The last experiment he cites is the weighing of air
in a pair of scales, where it is found manifestly that it has some
little weight. For if you weigh a bladder, and put the weight into one
scale, and then blow the bladder full of air, and put it into the other
scale, the full bladder will outweigh the empty. Must not then the air
gravitate?
_B._ It does not follow. I have seen the experiment just as you describe
it, but it can never be thence demonstrated that air has any weight. For
as much air as is pressed downward by the weight of the blown bladder,
so much will rise from below, and lay itself spherically at the altitude
of the centre of gravity of the bladder so blown. So that all the air
within the bladder above that centre is carried thither imprisoned, and
by violence: and the force that carries it up is equal to that which
presseth it down. There must, therefore, be allowed some little
counterpoise in the other scale to balance it. Therefore, the experiment
proves nothing to his purpose. And whereas they say there be small heavy
bodies in the air, which make it gravitate, do they think the force
which brought them thither cannot hold them there?
_A._ I leave this question of the fish as clearly resolved, because the
water tending every way to one point, which is the centre of the earth,
must of necessity arch itself. And now tell me your own opinion
concerning the cause of gravity, and why all bodies descend or ascend
not all alike. For there can be no more matter in one place than another
if the places be equal.
_B._ I have already showed you in general, that the difference of motion
in the parts of several bodies makes the difference of their natures.
And all the difference of motions consisteth either in swiftness, or in
the way, or in the duration. But to tell you in special why gold is
heaviest, and then quicksilver, and then, perhaps, lead, is more than I
hope to know, or mean to enquire; for I doubt not but that the species
of heavy, hard, opaque, and diaphanous, were all made so at their
creation, and at the same time separated from different species. So that
I cannot guess at any particular motions that should constitute their
natures, farther than I am guided by the experiments made by fire or
mixture.
_A._ You hope not then to make gold by art?
_B._ No, unless I could make one and the same thing heavier than it was.
God hath from the beginning made all the kinds of hard, and heavy, and
diaphanous bodies that are, and of such figure and magnitude as he
thought fit; but how small soever, they may by accretion become greater
in the mine, or perhaps by generation, though we know not how. But that
gold, by the art of man, should be made of not gold, I cannot
understand; nor can they that pretend to show how. For the heaviest of
all bodies, by what mixture soever of other bodies, will be made
lighter, and not to be received for gold.
_A._ Why, when the cause of gravity consisteth in motion, should you
despair of finding it?
_B._ It is certain that when any two bodies meet, as the earth and any
heavy body will, the motion that brings them to or towards one another,
must be upon two contrary ways; and so also it is when two bodies press
each other in order to make them hard; so that one contrariety of motion
might cause both hard and heavy, but it doth not, for the hardest bodies
are not always the heaviest; therefore I find no access that way to
compare the causes of different endeavours of heavy bodies to descend.
_A._ But show me at least how any heavy body that is once above in the
air, can descend to the earth, when there is no visible movent to thrust
or pull it down.
_B._ It is already granted, that the earth hath this compounded motion
supposed by Copernicus, and that thereby it casteth the contiguous air
from itself every way round about. Which air so cast off, must
continually, by its nature, range itself in a spherical orb. Suppose a
stone, for instance, were taken up from the ground, and held up in the
air by a man’s hand, what shall come into the place it filled when it
lay upon the earth?
_A._ So much air as is equal to the stone in magnitude, must descend and
place itself in an orb upon the earth. But then I see that to avoid
vacuum, another orb of air of the same magnitude must descend, and place
itself in that, and so perpetually to the man’s hand; and then so much
air as would fill the place must descend in the same manner, and bring
the stone down with it. For the stone having no endeavour upward, the
least motion of the air, the hand being removed, will thrust it
downward.
_B._ It is just so. And farther, the motion of the stone downward shall
continually be accelerated according to the odd numbers from unity; as
you know hath been demonstrated by Galileo. But we are nothing the
nearer, by this, to the knowledge of why one body should have a greater
endeavour downward than another. You see the cause of gravity is this
compounded motion with exclusion of vacuum.
_A._ It may be it is the figure that makes the difference. For though
figure be not motion, yet it may facilitate motion, as you see commonly
the breadth of a heavy body retardeth the sinking of it. And the cause
of it is, that it makes the air have farther to go laterally, before it
can rise from under it. For suppose a body of quicksilver falling in the
air from a certain height, must it not, going as it does toward the
centre of the earth, as it draws nearer and nearer to the earth, become
more and more slender, in the form of a solid sector? And if it have far
to go, divide itself into drops? This figure of a solid sector is like a
needle with the point downward, and therefore I should think that
facilitating the motion of it does the same that would be done by
increasing the endeavour.
_B._ Do not you see that this way of facilitating is the same in water,
and in all other fluid heavy bodies? Besides, your argument ought to be
applicable to the weighing of bodies in a pair of scales, which it is
not, for there they have no such figure; it should also hold in the
comparison of gravity in hard and fluid bodies.
_A._ I had not sufficiently considered it. But supposing now, as you do,
that both heavy and hard bodies, in their smallest parts, were made so
in the creation; yet, because quicksilver is harder than water, a drop
of water shall in descending be pressed into a more slender sector than
a drop of quicksilver, and consequently the earth shall more easily cast
off any quantity of water than the same quantity of quicksilver.
_B._ This one would think were true; as also that of simple fluid
bodies, those whose smallest parts, naturally, without the force of
fire, do strongliest cohere, are generally the heaviest. But why then
should quicksilver be heavier than stone or steel? Fluidity and hardness
are but degrees between greater fluidity and greater hardness. Therefore
to the knowledge of what it is that causeth the difference, in different
bodies, of their endeavour downward, there are required, if it can be
known at all, a great many more experiments than have been yet made. It
is not difficult to find why water is heavier than ice, or other body
mixed of air and water. But to believe that all bodies are heavier or
lighter according to the quantity of air within them, is very hard.
_A._ I see by this, that the Creator of the world, as by his power he
ordered it, so by his wisdom he provided it should be never disordered.
Therefore leaving this question, I desire to know whether if a heavy
body were as high as a fixed star, it would return to the earth.
_B._ It is hard to try. But if there be this compounded motion in the
great bodies so high, such as is in the earth, it is very likely that
some heavy bodies will be carried to them. But we shall never know it
till we be at the like height.
_A._ What think you is the reason why a drop of water, though heavy,
will stand upon a horizontal plane of dry or unctuous wood, and not
spread itself upon it? For let A B, in the sixth figure, be the dry
plane, D the drop of water, and D C perpendicular to A B. The drop D,
though higher, will not descend and spread itself upon it.
_B._ The reason I think is manifest. For those bodies which are made by
beating of water and air together, show plainly that the parts of water
have a great degree of cohesion. For the skin of the bubble is water,
and yet it can keep the air, though moved, from getting out. Therefore
the whole drop of water at D, hath a good deal of cohesion of parts. And
seeing A B is an horizontal plane, the way from the contact in D either
to A or B is upwards, and consequently there is no endeavour in D either
of those ways, but what proceeds from so much weight of water as is able
to break that cohesion, which so small a drop is too weak to do. But the
cohesion being once broken, as with your finger, the water will follow.
_A._ Seeing the descent of a heavy body increaseth according to the odd
numbers 1, 3, 5, 7, &c. and the aggregates of those numbers, viz. of 1
and 3; and 1 and 3 and 5; and of 1 and 3 and 5 and 7, &c. are square
numbers, namely 4, 9, 16; the whole swiftness of the descent will be, I
think, to the aggregate of so many swiftnesses equal to the first
endeavour, as square numbers are to their sides, 1, 2, 3, 4. Is it so?
_B._ Yes, you know it hath been demonstrated by Galileo.
_A._ Then if, for instance, you put into a pair of scales equal
quantities of quicksilver and water, seeing they are both accelerated in
the same proportion, why should not the weight of quicksilver to the
weight of water be in duplicate proportions to their first endeavours?
_B._ Because they are in a pair of scales. For there the motion of
neither of them is accelerated. And therefore it will be, as the first
endeavour of the quicksilver to the first endeavour of the water, so the
whole weight to the whole weight. By which you may see, that the cause
which takes away the gravitation of liquid bodies from fish or other
lighter bodies within them, can never be derived from the weight.
_A._ I have one question more to ask concerning gravity. If gravity be,
as some define it, an intrinsical quality, whereby a body descendeth
towards the centre of the earth, how is it possible that a piece of iron
that hath this intrinsical quality should rise from the earth, to go to
a loadstone? Hath it also an intrinsical quality to go from the earth?
It cannot be. The cause therefore must be extrinsical. And because when
they are come together in the air, if you leave them to their own
nature, they will fall down together, they must also have some like
extrinsical cause. And so this magnetic virtue will be such another
virtue as makes all heavy bodies to descend, in this our world, to the
earth. If therefore you can from this your hypothesis of compounded
motion, by which you have so probably salved the problem of gravity,
salve also this of the loadstone, I shall acknowledge both your
hypothesis to be true, and your conclusion to be well deduced.
_B._ I think it not impossible. But I will proceed no farther in it now,
than, for the facilitating of the demonstrations, to tell you the
several proprieties of the magnet, whereof I am to show the causes. As
first, that iron, and no other body, at some little distance, though
heavy, will rise to it. Secondly, that if it be laid upon a still water
in a floating vessel, and left to itself, it will turn itself till it
lie in a meridian, that is to say, with one and the same line still
north and south. Thirdly, if you take a long slender piece of iron, and
apply the loadstone to it, and, according to the position of the poles
of the loadstone, draw it over to the end of the iron, the iron will
have the same poles with the magnet, so it be drawn with some pressure;
but the poles will lie in a contrary position; and also this long iron
will draw other iron to it as the magnet doth.
Fourthly, this long iron, if it be so small as that poised upon a pin,
the weight of it have no visible effect, the navigators use it for the
needle of their compass, because it points north and south; saving that
in most places by particular accidents it is diverted; which diversion
is called the variation of the horizontal needle. Fifthly, the same
needle placed in a plane perpendicular to the horizon, hath another
motion called the inclination. Which that you may the better conceive,
draw a fourth figure; wherein let there be a circle to represent the
terrella, that is to say, a spherical magnet.
_A._ Let this be it, whose centre is A, the north pole B, the south pole
C.
_B._ Join B C, and cross it at right angles with the diameter D E.
_A._ It is done.
_B._ Upon the point D set the needle parallel to B C, with the cross of
the south pole, and the barb for the north; and describe a square about
the circle B D C E, and divide the arch D B into four equal parts in
_a_, _b_, _c._
_A._ It is done.
_B._ Then place the middle of the needle on the points _a_, _b_, _c_, so
that they may freely turn; and set the barb which is at D towards the
north, and that which is at C towards the south. You see plainly by
this, that the angles of inclination through the arch D C taken
altogether, are double to a right angle. For when the south point of the
needle, looking north, as at D, comes to look south, as at C, it must
make half a circle.
_A._ That is true. And if you draw the sine of the arch D _a_, which is
_d a_, and the sine of the arch B _a_, which is _a e_, and the sine of
the arch D _b_, which is _b f_, and the sine of the arch B _c_, which is
_c g_, the needle will lie upon _b f_ with the north-point downwards, so
that the needle will be parallel to A D. Then from _a_ draw the line _a
h_, making the angle _e a h_ equal to the angle D A _a_. And then the
needle at _a_ shall lie in the line _a h_ with the south point toward
_h_. Finally, draw the line _c h_, which, with _c g_, will also make a
quarter of a right angle; and therefore if the needle be placed on the
point _c_, it will lie in _c h_ with the south point toward _h_. And
thus you see by what degrees the needle inclines or dips under the
horizon more and more from D till it come to the north pole at B; where
it will lie parallel to the needle in D; but with their barbs looking
contrary ways. And this is certain by experience, and by none
contradicted.
You see then why the degrees of the inclinatory needle, in coming from D
to B, are double to the degrees of a quadrant. It is found also by
experience, that iron both of the mine and of the furnace put into a
vessel so as to float, will lay itself (if some accident in the earth
hinder it not) exactly north and south. And now I am, from this
compounded motion supposed by Copernicus, to derive the causes why a
loadstone draws iron; why it makes iron to do the same; why naturally it
placeth itself in a parallel to the axis of the earth; why by passing it
over the needle it changes its poles; and what is the cause that it
inclines. But it is your part to remember what I told you of motion at
our second meeting; and what I told you of this compounded motion
supposed by Copernicus, at our fourth meeting.
==========
CHAPTER IX.
OF THE LOADSTONE AND ITS POLES, AND WHETHER
THEY SHOW THE LONGITUDE OF PLACES ON THE EARTH.
_A._ I come now to hear what natural causes you can assign of the
virtues of the magnet; and first, why it draws iron to it, and only
iron.
_B._ You know I have no other cause to assign but some local motion, and
that I never approved of any argument drawn from sympathy, influence,
substantial forms, or incorporeal effluvia. For I am not, nor am
accounted by my antagonists for a witch. But to answer this question, I
should describe the globe of the earth greater than it is at B in the
first figure, but that the terrella in the fourth figure will serve our
turn. For it is but calling B and C the poles of the earth, and D E the
diameter of the equinoctial circle, and making D the east, and E the
west. And then you must remember that the annual motion of the earth is
from west to east, and compounded of a straight and circular motion, so
as that every point of it shall describe a small circle from west to
east, as is done by the whole globe. And let the circles about _a b c_
be three of those small circles.
_A._ Before you go any farther, I pray you show me how I must
distinguish east and west in every part of this figure. For wheresoever
I am on earth, suppose at London, and see the sun rise suppose in
Cancer, is not a straight line from my eye to the sun terminated in the
east?
_B._ It is not due east, but partly east, partly south. For the earth,
being but a point compared to the sun, all the parallels to D E the
equator, such as are _e a_, _f b_, _c g_, if they be produced, will fall
upon the body of the sun. And therefore A _b_ is north-east; A _a_ east
north-east; and A _c_ north north-east.
_A._ Proceed now to the cause of attraction.
_B._ Suppose now that the internal parts of the loadstone had the same
motion with that of the internal parts of the sun which make the annual
motion of the earth from west to east, but in a contrary way, for
otherwise the loadstone and the iron can never be made to meet. Then set
the loadstone at a little distance from the earth, marked with _z_; and
the iron marked with _x_ upon the superficies of the earth. Now that
which makes _x_ rise to _z_, can be nothing else but air; for nothing
touches it but air. And that which makes the air to rise, can be nothing
but those small circles made by the parts of the earth, such are at _a b
c_, for nothing else touches the air. Seeing then the motion of each
point of the loadstone is from east to west in circles, and the motion
of each point of the iron from west to east; it follows, that the air
between the loadstone and the iron shall be cast off both east and west;
and consequently the place left empty, if the iron did not rise up and
fill it. Thus you see the cause that maketh the loadstone and the iron
to meet.
_A._ Hitherto I assent. But why they should meet when some heterogeneous
body lies in the air between them, I cannot imagine. And yet I have seen
a knife, though within the sheath, attract one end of the needle of a
mariner’s compass; and have heard it will do the same though a
stone-wall were between.
_B._ Such iron were indeed a very vigorous loadstone. But the cause of
it is the same that causeth fire or hot water, which have the same
compounded motion, to work through a vessel of brass. For though the
motion be altered by restraint within the heterogeneous body, yet being
continued quite through, it restores itself.
_A._ What is the cause why the iron rubbed over by a loadstone will
receive the virtue which the loadstone hath of drawing iron to it?
_B._ Since the motion that brings two bodies to meet must have contrary
ways, and that the motions of the internal parts of the magnet and of
the iron are contrary; the rubbing of them together does not give the
iron the first endeavour to rise, but multiplies it. For the iron
untouched will rise to a loadstone; but if touched, it becomes a
loadstone to other iron. For when they touch a piece of iron, they pass
the loadstone over it only one way, viz. from pole to pole; not back
again, for that would undo what before had been done; also they press it
in passing to the very end of the iron, and somewhat hard. So that by
this pressing motion all the small circles about the points _a b c_, are
turned the contrary way; and the halves of those small circles made on
the arch D B will be taken away and the poles changed, so as that the
north poles shall point south, and the south poles north, as in the
figure.
_A._ But how comes it to pass, that when a loadstone hath drawn a piece
of iron, you may add to it another, as if they begat one another? Is
there the like motion in the generation of animals?
_B._ I have told you that iron of itself will rise to the loadstone;
much more then will it adhere to it when it is armed with iron, and both
it and the iron have a plain superficies. For then not only the points
of contact will be many, which make the coherence stronger, but also the
iron wherewith it is armed is now another loadstone, differing a little,
which you perhaps think, as male and female. But whether this compounded
motion and confrication causeth the generation of animals, how should I
know, that never had so much leisure as to make any observation which
might conduce to that?
_A._ My next question is, seeing you say the loadstone, or a needle
touched with it, naturally respecteth the poles of the earth, but that
the variation of it proceedeth from some accidents in the superficies of
the earth; what are those accidents?
_B._ Suppose there be a hill upon the earth, for example, at _r_; then
the stream of the air which was between _z_ and _x_ westward, coming to
the hill, shall go up the hill’s side, and so down to the other side,
according to the crooked line which I have marked about the hill by
points; and this infallibly will turn the north point of the needle,
being on the east side, more towards the east, and that on the other
side more towards the west, than if there had been no hill. And where
upon the earth are there not eminences and depressions, except in some
wide sea, and a great way from land.
_A._ But if that be true, the variation in the same place should be
always the same, for the hills are not removed.
_B._ The variation of the needle at the same place is still the same;
but the variation of the variation is partly from the motion of the pole
itself, which by the astronomers is called _motus trepidationis_; and
partly from that, that the variation cannot be truly observed, for the
horizontal needle and the inclinatory needle incline alike, but cannot
incline in due quantity. For whether set upon a pin or an axis, their
inclination is hindered, in the horizontal needle, by the pin itself: if
upon an axis, if the axis be just, it cannot move; if slack, the weight
will hinder it; but chiefly because the north pole of the earth draws
away from it the north pole of the needle, for two like poles cannot
come together. And this is the cause why the variation in one place is
east, and another west.
_A._ This is indeed the most probable reason why the variation varies
that ever I heard given; and I should presently acknowledge that this
parallel motion of the axis of the earth in the ecliptic, supposed by
Copernicus, is the true annual motion of the earth, but that there is
lately come forth a book called _Longitude Found_ , which makes the
magnetical poles distant from the poles of the earth eight degrees and a
half.
_B._ I have the book. It is far from being demonstrated, as you shall
find, if you have the patience to see it examined. For wheresoever his
demonstration is true, the conclusion, if rightly inferred, will be
this, that the poles of the loadstone and the poles of the earth are the
same. And where, on the contrary, his demonstrations are fallacies, it
is because sometimes he fancieth the lines he hath drawn, not where they
are; sometimes because he mistakes his station; and sometimes because he
goes on some false principle of natural philosophy; and sometimes also
because he knoweth not sufficiently the doctrine of spherical triangles.
_A._ I think that is the book there which lies at your elbow. Pray you
read.
_B._ I find first (p. 4), that the grounds of his argument are the two
observations made by Mr. Burroughs, one at Vaygates, in 1576, where the
variation from the pole of the earth he found to be 11 deg. 15 min.
east; the other at Limehouse, near London, in 1580, where the variation
from the pole of the earth was 8 deg. 38 min. west, by which, he saith,
he might _find out the magnetical pole_.
_A._ Where is Vaygates?
_B._ In 70 degrees of north latitude; the difference of longitude
between London and it being 58 degrees.
_A._ The longitude of places being yet to seek, how came he to know this
difference of 58 degrees, except the poles of the magnet and the earth
be the same?
_B._ I believe he trusted to the globe for that. For the distance
between the places is not above 2000 miles the nearest way. But we will
pass by that, and come to his demonstration, and to his diagram, wherein
L is London, P the north-pole of the earth, V Vaygates. So that L P is
38 deg. 28 min.; P V 20 deg.; the angle L P V 58 deg. for the difference
between the longitudes of Vaygates and London. This is the construction.
But before I come to the demonstration, I have an inference to draw from
these observations, which is this. Because in the same year the
variation at London was 11 deg. 15 min. east, and at Vaygates 8 deg. 38
min. west; if you subtract 11 deg. 15 min. from the arc L P; and 8 deg.
38 min. from the arc L V, the variation on both sides will be taken
away; so that P V being the meridian of Vaygates, and L P the meridian
of London, they shall both of them meet in P the pole of the earth. And
if the pole of the magnet be nearer to the zenith of London than is the
pole of the earth, it shall be just as much nearer to the zenith of
Vaygates in the meridian of Vaygates, which is P V; as is manifest by
the diurnal motion of the earth.
_A._ All this I conceive without difficulty. Proceed to the
demonstration.
_B._ Mark well now. His words are these (page 5): From P L V subtract 11
deg. 15 min., and there remains the angle V L M. Consider now which is
the angle P L V, and which is the remaining angle V L M, and tell what
you understand by it.
_A._ He has marked the angle P L V with two numbers, 11 deg. 15 min. and
21 deg. 50 min., which together make 33 deg. 5 min. And the angle 11
deg. 15 min. being subtracted from P L V, there will remain 21 deg. 50
min. for the angle V L M. I know not what to say to it. For I thought
the arc P V, which is 20 deg., had been the arc of the spherical angle P
L V; and that the arc L V had been 58 deg., because he says the angle L
P V is so; and that the arc L M had been 46 deg., because the angle L P
M is so; and lastly, that the angle P L M had been 8 deg. 30 min.,
because the arc P M is so.
_B._ And what you thought had been true, if a spherical angle were a
very angle. For all men that have written of spherical triangles take
for the ground of their calculation, as Regiomontanus, Copernicus, and
Clavius, that the arch of a spherical angle is the side opposite to the
angle. You should have considered also that he makes the angle V P M 12
deg., but sets down no arc to answer it. But that you may find I am in
the right, look into the definitions which Clavius hath put down before
his treatise of spherical triangles, and amongst them is this; “the arc
of a spherical triangle is a part of a great circle intercepted between
the two sides drawn from the pole of the said great circle.”
_A._ The book is nothing worth; for it is impossible to subtract an arc
of a circle out of a spherical angle. And I see besides that he takes
the superficies that lieth between the sides L P and L M for an arch,
which is the quantity of an angle; and is a line, and cannot be taken
out of a superficies. I wonder how any man that pretends to mathematics
could be so much mistaken.
_B._ It is no great wonder. For Clavius himself striving to maintain
that a right angle is greater than the angle made by the diameter and
the circumference, fell into the same error. A corner, in vulgar speech,
and an angle, in the language of geometry, are not the same thing. But
it is easy even for a learned man sometimes to take them for the same,
as this author now has done; and proceeding he saith, subtract 8 deg. 38
min. from the angle P V L, and there remains the angle L V M.
_A._ That again is false, because impossible. What was it that deceived
him now?
_B._ The same misunderstanding of the nature of a spherical angle. Which
appears farther in this, that when he knew the arc V P was part of a
great circle, he thought V M, which he maketh 8 deg. 30 min., were also
parts of a great circle; which is manifestly false. For two great
circles, because they pass through the centre, do cut each other into
halves. But V P is not half a circle. He sure thought himself at
Vaygates, and that P M V was equal to P V, although in the same
hemisphere.
_A._ But how proves he that the arc P M is 8 degrees 30 minutes?
_B._ Thus. We have in two triangles, P L M and P V M, two sides and one
angle included, to find P M the distance of the magnetical pole from the
pole of the earth 8 deg. 30 min.
_A._ Is that all? It is very short for a demonstration of two so
difficult problems, as the quantity of 8 deg. 30 min.; and of the place
of the magnetical pole. But he has proved nothing till he has showed how
he found it. And though P M be 8 deg. 30 min., it follows not that M is
the magnetical pole.
_B._ Nor is it true. For if P M be 8 deg. 30 min., and V M 8 deg. 38
min., the whole arc P M V will be 17 deg. 8 min., which should be 20
deg. Besides, whereas the variations were east and west, the subtracting
of them should be also east and west, but they are north and south.
_A._ I am satisfied that the magnetical poles and the poles of the earth
are the same. But thus much I confess, if they were not the same, the
longitude were found. For the difference of the latitudes of the earth’s
equator and of the magnetical equator, is the difference of the
longitude. But proceed.
_B._ “The earth being a solid body, and the magnetic sphere that
encompasseth the earth being a substance that hath not solidity to keep
pace with the earth, loseth in its motion: and that may be the cause of
the motion of the magnetic poles from east to west.”
_A._ This is very fine and unexpected. The magnetic sphere, which I took
for a globe made of a magnet, has not solidity to keep pace with the
earth, though it be one of the hardest stones that are. It encompasseth
the earth; yet I thought nothing had encompassed the earth but air in
which I breath and move. By this also the whole earth must be a
loadstone. For two bodies cannot be in one place. So that he is yet no
farther than Dr. Gilbert whom he slights. And if the sphere be a magnet,
then the earth and loadstone have the same poles. See the force of
truth! which though it could not draw to it his reason, hath drawn his
words to it.
_B._ But perhaps he meant that the magnetic virtue encompasseth the
earth, and not the magnetic body.
_A._ But that helpeth him not. For if the body of the magnet be not
there, the virtue then is the virtue of the earth; and so again the
poles of the earth are magnetic poles.
_B._ You see how unsafe it is to boast of doctrines as of God’s gifts,
till we are sure that they are true. For God giveth and denieth as he
pleaseth, not as ourselves wish; as now to him he hath given confidence
enough, but hath denied him, at least hitherto, the finding of the
longitudes. In the next place (p. 8) he seems much pleased that his
doctrine agrees with an opinion of Keplerus, that from the creation to
the year of our Lord, it is to the year 1657 now 5650 years; and with
that which he saith some divines have held in times past, that as this
world was created in six days, so it should continue six thousand years.
By which account the world will be at an end three hundred and fifty
years hence; though the Scripture tells us it shall come as a thief in
the night. O what advantage three hundred and forty years hence will
they have that know this, over them that know it not, by taking up money
at interest, or selling lands at twenty years’ purchase!
_A._ But he says he will not meddle with that.
_B._ Yes, when he had meddled with it too much already.
_A._ But you have not told me wherein consisteth this agreement between
him and Keplerus.
_B._ I forgot it. It is in the motion of the magnetic poles. For
precedently (p. 7), he had said “that their period or revolution was six
hundred years; their yearly motion thirty-six minutes; and (p. 8) that
their motion is by sixes. Six tenths of a degree in one year; six
degrees in ten years; sixty degrees in a hundred years; and six times
sixty degrees in six hundred years.”
_A._ But what natural cause doth he assign of this revolution of six
hundred years?
_B._ None at all. For the magnet lying upon the earth, can have no
motion at all but what the earth and the air give it. And because it is
always at 8 deg. 30 min. distance from the pole of the earth, the earth
can give it no other motion than what it gives to its own poles by the
precession of the equinoctial points. Nor can the air give it any motion
but by its stream; which must needs vary when the stream varieth. But
what a vast difference does he make between the period of the motion of
the equinoctial points, which is about or near thirty-six thousand years
according to Copernicus (lib. iii. cap. 6), which makes the annual
precession to be 36 seconds, and the period of the magnetical poles’
motion, which is but six hundred years.
_A_. Go on.
_B_. He comes now (p. 15) to the inclinatory needle upon a spherical
loadstone. Where he shows, by diagram, that the needle and the
instrument together moved towards the magnetical pole, make the sum of
the inclinations equal to two quadrants, setting the north-point of the
needle southward: which I confess is true. But, in the same page, he
ascribeth the same motion to the earth in these words: “as the
horizontal needle hath a double motion about the round loadstone or
terrella, so also the inclinatory needle hath a double motion about the
earth.” What is this, but a confession that the poles of the magnet and
of the earth are the same?
_A._ It is plain enough.
_B._ Besides, seeing he placeth the magnetical pole at M in the meridian
of Vaygates, the needle being touched shall incline to the pole of the
earth which is P, as well there as at London, and make the north-pole of
the earth point south.
_A._ It is certain, because he puts both the magnetical pole and the
pole of the earth in the same meridian of the earth. Nor see I any cause
why, the needle being the same, it should not be as subject to
variation, and to variation of variation, and to all accidents of the
earth there, as in any other part.
_B._ He putteth (p. 16) a question, “at what distance from the earth are
the magnetic poles? and answers to it, they are very near the earth,
because the nearer the earth, the greater the strength.” What think you
of this?
_A._ I think they are in the superficies of the magnet, as the pole of
the earth is in the superficies of the earth. And consequently, that
then the earth must be a part of the magnet, and their poles the same.
For the body of the magnet and the body of the earth, if they be two,
cannot be in one place.
_B._ His next words are, “some things are to be considered concerning
those variations of the horizontal needle which are not according to the
situation of the place from the magnetic poles, but are contrary; as all
the West Indies according to the poles should be easterly, and they are
westerly. Which is by some accidental cause in the earth; and their
motion, as I formerly said, is a forced motion, and not natural.”
_A._ He has clearly overthrown his main doctrine. For to say the motion
of the needle is forced and unnatural, is a most pitiful shift, and
manifestly false, no motion being more constant or less accidental,
notwithstanding the variation, to which the inclinatory needle is no
less subject than the horizontal needle.
_B._ That which deceived him, was, that he thought them two sorts of
needles, forgetting what he had said of Norman’s invention of the
inclinatory needle by the inclining of the horizontal needle (p. 11).
For I will show you that what he says is easterly and should be
westerly, should be easterly as it is. Consider the fourth figure, in
which B is the north-pole, and B _c_ 11 deg. 15 min. easterly, which was
the variation at London in 1576 easterly. Suppose A _c_ to be the
needle, shall it not incline, as well here as at D _a_, and the
variation B _c_ be easterly? Again, D _a_ is 11 deg. 15 min., and the
needle in D parallel to A B, and at _a_ inclining also 11 deg. 15 min.
westerly. And is not the variation there D _a_ westerly, with the north
point of the needle in the line _a h_?
_A._ But the West-Indies are not in this hemisphere B C D E. The
variation therefore will proceed in an arc of the opposite hemisphere,
which is westerly.
_B._ I believe he might think so, forgetting that he and his compass
were on the superficies of the earth, and fancying them in the centre at
A.
_A._ It is like enough. If we had a straight line exactly equal to the
arc of a quadrant, I think it would very much facilitate the doctrine of
spherical triangles.
_B._ When you have done with your questions of natural philosophy, I
will give you a clear demonstration of the equality of a straight line
to the arc of a quadrant, which, if it satisfy you, you may carry with
you, and try thereby if you can find the angle of a spherical triangle
given.
_A._ It is time now to give over. And at our next meeting I desire your
opinion concerning the causes of diaphaniety, and refraction. This
Copernicus has done much more than he thought of. For he has not only
restored to us astronomy, but also made the way open to physiology.
==========
CHAPTER X.
OF TRANSPARENCE, REFRACTION, AND OF THE POWER
OF THE EARTH TO PRODUCE LIVING CREATURES.
_A._ Thinking upon what you said yesterday, it looked like a generation
of living creatures. I saw the love between the loadstone and the iron
in their mutual attraction, their engendering in their close and
contrary motion, and their issue in the iron, which being touched, hath
the same attractive virtue. Now seeing they have the same internal
motion of parts with that of the earth, why should not their substance
be the same, or very near a-kin?
_B._ The most of them, if not all, that have written on this subject,
when they call the loadstone a terrella, seem to think as you do. But I,
except I could find proof for it, will not affirm it. For the earth
attracteth all kind of bodies but air, and the loadstone none but iron.
The earth is a star, and it were too bold to pronounce any sentence of
its substance, especially of the planets, that are so lapt up in their
several coats, as that they cannot work on our eyes, or any organ of our
other senses.
_A._ I come therefore now to the business of the day. Seeing all
generation, augmentation, and alteration is local motion, how can a body
not transparent be made transparent?
_B._ I think it can never be done by the art of man. For as I said of
hard and heavy bodies in the creation, so I think of diaphanous, that
the very same individual body which was not transparent then, shall
never be made transparent by human art.
_A._ Do not you see that every day men make glass, and other diaphanous
bodies not much inferior in beauty to the fairest gems?
_B._ It is one thing to make one transparent of many by mixture, and
another to make transparent of not transparent. Any very hard stone, if
it be beaten into small sands, such as is used for hour-glasses, every
one of those sands, if you look upon it with a microscope, you will find
to be transparent; and the harder and whiter a stone is, so much the
more transparent, as I have seen in the stone of which are made
millstones, which stone is here called greet. And I doubt not but the
sands of white marble must be more transparent. But there are no sands
so transparent that they have not a scurf upon them, as hard, perhaps,
as the stone itself; which they whose profession it is to make glass,
have the art to scour and wash away. And therefore I think it no great
wonder to bring those sands into one lump, though I know not how they do
it.
_A._ I know they do it with lye made with a salt extracted from the
ashes of an herb, of which salt they make a strong lye, and mingle it
with the sand, and then bake it.
_B._ Like enough. But still it is a compound of two transparent bodies,
whereof one is the natural stone, the other is the mortar. This
therefore doth not prove, that one and the same body of not transparent
can be made transparent.
_A._ Since they can make one transparent body of many, why do they not
of a great many small sparks of natural diamond compound one great one?
It would bear the charges of all the materials, and beside, enrich them.
_B._ It is probable it would. But it may be they know no salt that
howsoever prepared, which, with how great a fire soever, can make them
melt. And, it may be, the true crystal of the mountain, which is found
in great pieces in the Alps, is but a compound of many small ones, and
made by the earth’s annual motion; for it is a very swift motion.
Suppose now that within a very small cavern of those rocks whose
smallest atoms are crystal, and the cavity filled with air; and consider
what a tumult would be made by the swift reciprocation of that air;
whether it would not in time separate those atoms from the rock, and
jumbling them together make them rub off their scurf from one another,
and by little and little to touch one another in polished planes, and
consequently stick together, till in length of time they become one lump
of clean crystal.
_A._ I believe that the least parts of created substances lay mingled
together at first, till it pleased God to separate all dissimilar
natures, and congregate the similar, to which this annual motion is
proper. But they say that crystal is found in the open air hanging like
icicles upon the rocks, which, if true, defeats this supposition of a
narrow cavern, and therefore I must have some farther experience of it
before I make it my opinion. But howsoever, it still holds true that
diaphanous bodies of all sorts, in their least parts, were made by God
in the beginning of the world. But it may be true, notwithstanding those
icicles. For the force of the air that could break off those diaphanous
atoms in a cavern, can do the same in the open air. And I know that a
less force of air can break some bodies into small pieces, not much less
hard than crystal, by corrupting them.
_B._ That which you now have said is somewhat. But I deny not the
possibility, but only doubt of the operation. You may therefore pass to
some other question.
_A._ Well, I will ask you then a question about refraction. I know
already that for the cause of refraction, when the light falleth through
a thinner medium upon a thicker, you assign the resistance of the
thicker body; but you do not mean there, by _rarum_ and _densum_, two
bodies whereof in equal spaces one has more substance in it than the
other.
_B._ No; for equal spaces contain equal bodies. But I mean by _densum_
any body which more resisteth the motion of the air, and by _rarum_ that
which resisteth less.
_A._ But you have not declared in what that resistance consisteth.
_B._ I suppose it proceedeth from the hardness.
_A._ But from thence it will follow, that all transparent bodies that
equally refract are equally hard, which I think is not true, because the
refraction of glass is not greater, at least in comparison of their
hardnesses, than that of water.
_B._ I confess it. Therefore I think we must take in gravity to a share
in the production of this refraction. For I never considered refraction
but in glass, because my business then was only to find the causes of
the phenomena of telescopes and microscopes. Let therefore A B (in fig.
7) be a hard, and consequently, a heavy body; and from above, as from
the sun, let C A be the line of incidence, and produced to D; and draw A
E perpendicular to A B. It is manifest that the hardness in A B shall
turn the stream of the light inwards toward A E, suppose in the line A
_e_. It is also evident that the endeavour in B, which is, being heavy,
downward, shall turn the stream again inward, towards A E, as in A _b_.
Thus it is in refraction from the sun downwards. In like manner, if the
light come from below, as from a candle in the point D, the line of
incidence will be D A, and produced will pass to C. And the resistance
of the hardness in A will turn the stream A C inward, suppose into A
_l_, and make C _l_ equal to D _e_. For passing into a thinner medium,
it will depart from the perpendicular in an angle equal to the angle D A
_e_, by which it came nearer to it in A _e_. So also the resistance of
the gravity in the point A shall turn the stream of the light into the
line A _i_, and make the angle _l_ A _i_ equal to the angle _e_ A _b_.
And thus you see in what manner, though not in what proportion, hardness
and gravity conjoin their resistance in the causing of refraction.
_A._ But you proved yesterday, that a heavy body does not gravitate upon
a body equally heavy. Now this A B has upper parts and lower parts; and
if the upper parts do not gravitate upon the lower parts, how can there
be any endeavour at all downward to contribute to the refraction?
_B._ I told you yesterday, that when a heavy body was set upon another
body heavier or harder than itself, the endeavour of it downward was
diverted another way, but not that it was extinguished. But in this
case, where it lieth upon air, the first endeavour of the lowest part
worketh downward. For neither motion nor body can be utterly
extinguished by a less than an omnipotent power. All bodies, as long as
they are bodies, are in motion one way or other, though the farther it
be communicated, so much the less.
_A._ But since you hold that motion is propagated through all bodies,
how hard or heavy soever they be, I see no cause but that all bodies
should be transparent.
_B._ There are divers causes that take away transparency. First, if the
body be not perfectly homogeneous, that is to say, if the smallest parts
of it be not all precisely of the same nature, or do not so touch one
another as to leave no vacuum within it; or though they touch, if they
be not as hard in the contact as in any other line. For then the
refractions will be so changed both in their direction, and in their
strength, as that no light shall come through it to the eye; as in wood
and ordinary stone and metal. Secondly, the gravity and hardness may be
so great, as to make the angle refracted so great, as the second
refraction shall not direct the beam of light to the eye; as if the
angle of refraction were D A E, the refracted line would be
perpendicular to A B, and never come to the line A D, in which is the
eye.
_A._ To know how much of the refraction is due to the hardness, and how
much to the gravity, I believe it is impossible, though the quantity of
the whole be easily measured in a diaphanous body given. And both you
and Mr. Warner have demonstrated, that as the sine of the angle
refracted in one inclination is to the sine of the angle refracted in
another inclination, so is the sine of one inclination to the sine of
the angle of the other inclination. Which demonstrations are both
published by Mersennus in the end of the first volume of his _Cogitata
Physico-Mathematica_. But since there be many bodies, through which
though there pass light enough, yet no object appear through them to the
eye, what is the reason of that?
_B._ You mean paper. For paper windows will enlighten a room, and yet
not show the image of an object without the room. But it is because
there are in paper abundance of pores, through which the air passing
moveth the air within; by the reflections whereof anything within may be
seen. And in the same paper there are again as many parts not
transparent, through which the air cannot pass, but must be reflected
first to all parts of the object, and from them again to the paper; and
at the paper either reflected again or transmitted, according as it
falls upon pores or not pores; so that the light from the object can
never come together at the eye.
_A._ There belongs yet to this subject the causes of the diversity of
colours. But I am so well satisfied with that which you have written of
it in the twenty-fourth chapter of your book _de Corpore_, that I need
not trouble you farther in it. And now I have but one question more to
ask you, which I thought upon last night. I have read in an ancient
historian, that living creatures after a great deluge were produced by
the earth, which being then very soft, there were bred in it, it may be
by the rapid motion of the sun, many blisters, which in time breaking,
brought forth, like so many eggs, all manner of living creatures great
and small, which since it is grown hard it cannot do. What think you of
it?
_B._ It is true that the earth produced the first living creatures of
all sorts but man. For God said (Gen. i. 24), _Let the earth produce
every living creature, cattle, and creeping thing, &c._ But then again
(ver. 25) it is said that _God made the beast of the earth, &c._ So that
it is evident that God gave unto the earth that virtue. Which virtue
must needs consist in motion, because all generation is motion. But man,
though the same day, was made afterward.
_A._ Why hath not the earth the same virtue now? Is not the sun the same
as it was? Or is there no earth now soft enough?
_B._ Yes. And it may be the earth may yet produce some very small living
creatures: and perhaps male and female. For the smallest creatures which
we take notice of, do engender, though they do not all by conjunction;
therefore if the earth produce living creatures at this day, God did not
absolutely rest from all his works on the seventh day, but (as it is
chap. ii. 2) _he rested from all the work he had made_. And therefore it
is no harm to think that God worketh still, and when and where and what
he pleaseth. Beside, it is very hard to believe, that to produce male
and female, and all that belongs thereto, as also the several and
curious organs of sense and memory, could be the work of anything that
had not understanding. From whence, I think we may conclude, that
whatsoever was made after the creation, was a new creature made by God
no otherwise than the first creatures were, excepting only man.
_A._ They are then in an error that think there are no more different
kinds of animals in the world now, than there were in the ark of Noah.
_B._ Yes, doubtless. For they have no text of Scripture from which it
can be proved.
_A._ The questions of nature which I could yet propound are innumerable.
And since I cannot go through them, I must give over somewhere, and why
not here? For I have troubled you enough, though I hope you will forgive
me.
_B._ So God forgive us both as we do one another. But forget not to take
with you the demonstration of a straight line equal to an arc of a
circle.
THE PROPORTION OF A STRAIGHT LINE TO HALF THE ARC OF A QUADRANT.
[Illustration]
Describe the square A B C D, and divide it by the diagonals A C and B D,
as also by the straight lines E G, F H, meeting in the centre I at right
angles, into four equal parts. Then with the radius A B describe the
quadrant B D cutting E G in K, and the diagonal A C in L; and so B L
will be half the arc B D, equal to which we are to find a straight line.
Divide I C into halves at M, and draw B M cutting E G in _a_. I say B M
is equal to the arc B L. For the demonstration whereof we are to assume
certain known truths and dictates of common-sense.
1. That the arc B K is the third part of the arc B D, and consequently
two-thirds of the arc B L, and B K to K L as two to one.
2. That if a straight line be equal to the arc B L, and one end in B,
the other will be somewhere in I C, and higher than the point L.
3. That wheresoever it be, two-thirds of it must be equal to the arc B
K, and one-fifth to the arc K L.
4. That the arc of a quadrant described in the third part of the radius,
or of E G, is equal to the third part of the arc B D, viz. to the arc B
K. I may therefore call a third part of E G, the radius of B K; and a
sixth part of E G, the radius of the arc K L, &c.
5. And lastly, that any straight line drawn from B to I C, if it be
equal to the arc B L, it must cut the half radius I G, whose quadrantal
arc is B L, into the proportion of two to one. For as the whole arc to
the whole E G, so are the parts of it to the parts of E G.
These premises granted, which I think cannot be denied, I say again,
that the straight line B M is equal to the arc B L.
DEMONSTRATION.
[Illustration]
Because B I is to I M, by construction, as two to one, and the line I G
divides the angle B I C in the midst, B _a_ will be to _a_ M as two to
one, that is to say, as the arc B K to the arc K L. From the point M to
the side B C erect a perpendicular M N. And because C M is half C I, the
line M N will be half G C; and B N will be three-quarters of B C; and
the square of B M equal to ten squares of a quarter of B C; and because
B M is to B _a_ as three to two, M N will be to _a_ G as three to two.
But M N is a quarter of E G, therefore _a_ G is two-thirds of a quarter
of E G; that is, one-third of I G; that is, one-sixth of the whole E G.
And I _a_ one-third of E G. Therefore I _a_ is the radius of the arc B
K; and _a_ G the radius of the arc K L; and E G the radius of the whole
arc B L D. Lastly, if a straight line be drawn from B to any other point
of the line I C, though any line may be divided into the proportion of
two to one, it shall not pass through the point _a_, and therefore not
divide the radius of B L, which is I G, into the proportion of two to
one. Therefore no straight line can be drawn from B to I C, except B M,
so as to be equal to the arc B L. Therefore the straight line B M and
the arc B L are equal.
Hence it follows, that seeing the square of B M is equal to ten squares
of a quarter of B C, that a straight line equal to the quadrantal arc B
L D is equal to ten squares of half the radius, as I have divers ways
demonstrated heretofore.
SIX LESSONS
TO THE
PROFESSORS OF THE MATHEMATICS,
ONE OF GEOMETRY, THE OTHER OF ASTRONOMY,
IN THE CHAIRS SET UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE, IN THE
UNIVERSITY OF OXFORD.
TO THE RIGHT HONOURABLE
HENRY LORD PIERREPONT,
VISCOUNT NEWARK, EARL OF KINGSTON, AND
MARQUIS OF DORCHESTER.
MY MOST NOBLE LORD,
Not knowing on my own part any cause of the favour your Lordship has
been pleased to express towards me, unless it be the principles, method,
and manners you have observed and approved in my writings; and seeing
these have all been very much reprehended by men, to whom the name of
public professors hath procured reputation in the university of Oxford,
I thought it would be a forfeiture of your Lordship’s good opinion, not
to justify myself in public also against them, which, whether I have
sufficiently performed or not in the six following Lessons addressed to
the same professors, I humbly pray your Lordship to consider. The volume
itself is too small to be offered to you as a present, but to be brought
before you as a controversy it is perhaps the better for being short. Of
arts, some are demonstrable, others indemonstrable; and demonstrable are
those the construction of the subject whereof is in the power of the
artist himself, who, in his demonstration, does no more but deduce the
consequences of his own operation. The reason whereof is this, that the
science of every subject is derived from a precognition of the causes,
generation, and construction of the same; and consequently where the
causes are known, there is place for demonstration, but not where the
causes are to seek for. Geometry therefore is demonstrable, for the
lines and figures from which we reason are drawn and described by
ourselves; and civil philosophy is demonstrable, because we make the
commonwealth ourselves. But because of natural bodies we know not the
construction, but seek it from the effects, there lies no demonstration
of what the causes be we seek for, but only of what they may be.
And where there is place for demonstration, if the first principles,
that is to say, the definitions contain not the generation of the
subject, there can be nothing demonstrated as it ought to be. And this
in the three first definitions of Euclid sufficiently appeareth. For
seeing he maketh not, nor could make any use of them in his
demonstrations, they ought not to be numbered among the principles of
geometry. And Sextus Empiricus maketh use of them (misunderstood, yet so
understood as the said professors understand them) to the overthrow of
that so much renowned evidence of geometry. In that part therefore of my
book where I treat of geometry, I thought it necessary in my definitions
to express those motions by which lines, superficies, solids, and
figures, were drawn and described, little expecting that any professor
of geometry should find fault therewith, but on the contrary supposing I
might thereby not only avoid the cavils of the sceptics, but also
demonstrate divers propositions which on other principles are
indemonstrable. And truly, if you shall find those my principles of
motion made good, you shall find also that I have added something to
that which was formerly extant in geometry.
For first, from the seventh chapter of my book _De Corpore_, to the
thirteenth, I have rectified and explained the principles of the
science; _id est_, I have done that business for which Dr. Wallis
receives the wages. In the seventh, I have exhibited and demonstrated
the proportion of the parabola and parabolasters to the parallelograms
of the same height and base; which, though some of the propositions were
extant without that demonstration, were never before demonstrated, nor
are by any other than this method demonstrable.
In the eighteenth, as it is now in English, I have demonstrated, for
anything I yet perceive, equation between the crooked line of a parabola
or any parabolaster and a straight line.
In the twenty-third I have exhibited the centre of gravity of any sector
of a sphere.
Lastly, the twenty-fourth, which is of the nature of refraction and
reflection, is almost all new.
But your Lordship will ask me what I have done in the twentieth, about
the quadrature of the circle. Truly, my Lord, not much more than before.
I have let stand there that which I did before condemn, not that I think
it exact, but partly because the division of angles may be more exactly
performed by it than by any organical way whatsoever; and I have
attempted the same by another method, which seemeth to me very natural,
but of calculation difficult and slippery. I call them only aggressions,
retaining nevertheless the formal manner of assertion used in
demonstration. For I dare not use such a doubtful word as _videtur_,
because the professors are presently ready to oppose me with a _videtur
quod non_. Nor am I willing to leave those aggressions out, but rather
to try if it may be made pass for lawful, (in spite of them that seek
honour, not from their own performances, but from other men’s failings),
amongst many difficult undertakings carried through at once to leave one
and the greatest for a time behind; and partly because the method is
such as may hereafter give farther light to the finding out of the exact
truth.
But the principles of the professors that reprehend these of mine, are
some of them so void of sense, that a man at the first hearing, whether
geometrician or not geometrician, must abhor them. As for example:
1. That two equal proportions are not double to one of the same
proportions.
2. That a proportion is double, triple, &c. of a number, but not of a
proportion.
3. That the same body, without adding to it, or taking from it, is
sometimes greater, and sometimes less.
4. That a quantity may grow less and less eternally, so as at last to be
equal to another quantity; or, which is all one, that there is a last in
eternity.
5. That the nature of an angle consisteth in that which lies between the
lines that comprehend the angle in the very point of their concourse,
that is to say, an angle is the superficies which lies between the two
points which touch, or, as they understand a point, the superficies that
lies between the two nothings which touch.
6. That the quotient is the proportion of the division to the dividend.
Upon these and some such other principles is grounded all that Dr.
Wallis has said, not only in his _Elenchus_ of my geometry, but also in
his treatises of the _Angle of Contact_, and in his _Arithmetica
Infinitorum_; which two last I have here in two or three leaves wholly
and clearly confuted. And I verily believe that since the beginning of
the world, there has not been, nor ever shall be, so much absurdity
written in geometry, as is to be found in those books of his; with which
there is so much presumption joined, that an ἀποκατάϛασις of the like
conjunction cannot be expected in less than a Platonic year. The cause
whereof I imagine to be this, that he mistook the study of _symbols_ for
the study of _geometry_, and thought symbolical writing to be a new kind
of method, and other men’s demonstrations set down in symbols new
demonstrations. The way of analysis by squares, cubes, &c., is very
ancient, and useful for the finding out whatsoever is contained in the
nature and generation of rectangled planes, which also may be found
without it, and was at the highest in Vieta; but I never saw anything
added thereby to the science of geometry, as being a way wherein men go
round from the equality of rectangled planes to the equality of
proportion, and thence again to the equality of rectangled planes,
wherein the symbols serve only to make men go faster about, as greater
wind to a windmill.
It is in sciences as in plants; growth and branching is but the
generation of the root continued; nor is the invention of theorems
anything else but the knowledge of the construction of the subject
prosecuted. The unsoundness of the branches are no prejudice to the
roots, nor the faults of theorems to the principles. And active
principles will correct false theorems if the reasoning be good; but no
logic in the world is good enough to draw evidence out of false or
unactive principles. But I detain your Lordship too long. For all this
will be much more manifest in the following discourses, wherein I have
not only explained and rectified many of the most important principles
of geometry, but also by the examples of those errors which have been
committed by my reprehenders, made manifest the evil consequence of the
principles they now proceed on. So that it is not only my own defence
that I here bring before you, but also a positive doctrine concerning
the true grounds, or rather atoms of geometry, which I dare only say are
very singular, but whether they be very good or not, I submit to your
Lordship’s judgment. And seeing you have been pleased to bestow so much
time, with great success, in the reading of what has been written by
other men in all kinds of learning, I humbly pray your Lordship to
bestow also a little time upon the reading of these few and short
lessons; and if your Lordship find them agreeable to your reason and
judgment, let me, notwithstanding the clamour of my adversaries, be
continued in your good opinion, and still retain the honour of being,
My most noble Lord,
Your Lordship’s most
humble and obliged servant,
THOMAS HOBBES.
LONDON, _June 10, 1656_.
LESSONS
OF
THE PRINCIPLES OF GEOMETRY, &c.
TO THE EGREGIOUS PROFESSORS OF THE MATHEMATICS, ONE OF
GEOMETRY, THE OTHER OF ASTRONOMY, IN THE CHAIRS SET
UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE,
IN THE UNIVERSITY OF OXFORD.
LESSON I.
I suppose, most egregious professors, you know already that by geometry,
though the word import no more but the measuring of land, is understood
no less the measuring of all other quantity than that of bodies. And
though the definition of geometry serve not for proof, nor enter into
any geometrical demonstration, yet for understanding of the principles
of the science, and for a rule to judge by, who is a geometrician, and
who is not, I hold it necessary to begin therewith.
Geometry is the science of determining the quantity of anything, not
measured, by comparing it with some other quantity or quantities
measured. Which science therefore whosoever shall go about to teach,
must first be able to tell his disciple what measuring or dimension is;
by what each several kind of quantity is measured; what quantity is, and
what are the several kinds thereof. Therefore as they, who handle any
one part of geometry, determine by definition the signification of every
word which they make the subject or predicate of any theorem they
undertake to demonstrate; so must he which intendeth to write a whole
body of geometry, define and determine the meaning of whatsoever word
belongeth to the whole science. The design of Euclid was to demonstrate
the properties of the five regular bodies mentioned by Plato; in which
demonstrations there was no need to allege for argument the definition
of quantity, which it may be was the cause he hath not anywhere defined
it, but done what he undertook without it. And though having perpetually
occasion to speak of measure, he hath not defined measure; yet instead
thereof he hath, in the beginning of his first elements, assumed an
axiom which serveth his turn sufficiently as to the measure of lines,
which is the eighth axiom; that those things which lie upon one another
all the way (called by him ἐφαρμόζοντα) are equal. Which axiom is
nothing else but a description of the art of measuring length and
superficies. For this ἐφάρμοσις can have no place in solid bodies,
unless two bodies could at the same time be in one place. But amongst
the principles of geometry universal, the definitions are necessary,
both of quantity and dimensions.
Quantity is that which is signified by what we answer to him that
asketh, _how much_ any thing is? and thereby determine the magnitude
thereof. For magnitude being a word indefinite, if a man ask of a thing,
_quantum est?_ that is, _how much_ it is, we do not satisfy him by
saying it is magnitude or quantity, but by saying it is _tantum_, _so
much_. And they that first called it in Greek, πηλικότης, and in Latin
_quantity_, might more properly have called it in Latin _tantity_, and
in Greek τηλικότης; and we, if we allowed ourselves the eloquence of the
Greeks and Latins, should call it the _so-muchness_.
There is therefore to everything concerning which a man may ask without
absurdity, _how much it is_, a certain quantity belonging, determining
the magnitude to be _so much_. Also wheresoever there is _more_ and
_less_, there is one kind of quantity or other. And first there is the
quantity of bodies, and that of three kinds: length, which is by one way
of measuring; superficies, made of the complication of two lengths, or
the measure taken two ways; and solid, which is the complication of
three lengths, or of the measure taken three ways, for breadth or
thickness are but other lengths. And the science of geometry, so far
forth as it contemplateth bodies only, is no more but by measuring the
length of one or more lines, and by the position of others known in one
and the same figure, to determine by ratiocination, how much is the
superficies; and by measuring length, breadth, and thickness, to
determine the quantity of the whole body. Of this kind of magnitudes and
quantities the subject is body.
And because for the computing of the magnitudes of bodies, it is not
necessary that the bodies themselves should be present, the ideas and
memory of them supplying their presence, we reckon upon those imaginary
bodies, which are the quantities themselves, and say the length is so
great, the breadth so great, &c. which in truth is no better than to say
the length is so long, or the breadth so broad, &c. But in the mind of
an intelligent man it breedeth no mistake.
Besides the quantity of bodies, there is a quantity of time. For seeing
men, without absurdity, do ask how much it is; by answering _tantum_,
_so much_, they make manifest there is a quantity that belongeth unto
time, namely, a length. And because length cannot be an accident of
time, which is itself an accident, it is the accident of a body; and
consequently the length of the time, is the length of the body; by which
length or line, we determine how much the time is, supposing some body
to be moved over it.
Also we not improperly ask with _how much_ swiftness a body is moved;
and consequently there is a quantity of motion or swiftness, and the
measure of that quantity is also a line. But then again, we must suppose
another motion, which determineth the time of the former. Also of force,
there is a question of _how much_, which is to be answered by _so much_;
that is, by quantity. If the force consist in swiftness, the
determination is the same with that of swiftness, namely, by a line; if
in swiftness and quantity of body jointly, then by a line and a solid;
or if in quantity of body only, as weight, by a solid only.
So also is number quantity; but in no other sense than as a line is
quantity divided into equal parts.
Of an angle, which is of two lines whatsoever they be, meeting in one
point, the digression or openness in other points, it may be asked how
great is that digression? This question is answered also by quantity. An
angle therefore hath quantity, though it be not the subject of quantity;
for the body only can be the subject, in which body those straddling
lines are marked.
And because two lines may be made to divaricate by two causes; one, when
having one end common and immoveable, they depart one from another at
the other ends circularly, and this is called simply an angle; and the
quantity thereof is the quantity of the arch, which the two lines
intercept.
The other cause is the bending of a straight line into a circular or
other crooked line, till it touch the place of the same line, whilst it
was straight, in one only point. And this is called an angle of
contingence. And because the more it is bent, the more it digresseth
from the tangent, it may be asked _how much_ more? And therefore the
answer must be made by quantity; and consequently an angle of
contingence hath its quantity as well as that which is called simply an
angle. And in case the digression of two such crooked lines from the
tangent be uniform, as in circles, the quantity of their digression may
be determined. For, if a straight line be drawn from the point of
contact, the digression of the lesser circle will be to the digression
of the greater circle, as the part of the line drawn from the point of
contact, and intercepted by the circumference of the greater circle is
to the part of the same line intercepted by the circumference of the
lesser circle, or, which is all one, as the greater radius is to the
lesser radius. You may guess by this what will become of that part of
your last book, where you handle the question of the quantity of an
angle of contingence.
Also there lieth a question of _how much comparatively_ one magnitude is
to another magnitude, as how much water is in a tun in respect of the
ocean, how much in respect of a pint; _little_ in the first respect,
_much_ in the latter. Therefore the answer must be made by some
respective quantity. This respective quantity is called _ratio_ and
proportion, and is determined by the quantity of their differences; and
if their differences be compared also with the quantities themselves
that differ, it is called simply proportion, or proportion geometrical.
But if the differences be not so compared, then it is called proportion
arithmetical. And where the difference is none, there is no quantity of
the proportion, which in this case is but a bare comparison.
Also concerning heat, light, and divers other qualities, which have
degrees, there lieth a question of _how much_, to be answered by a _so
much_, and consequently they have their quantities, though the same with
the quantity of swiftness: because the intensions and remissions of such
qualities are but the intensions and remissions of the swiftness of that
motion by which the agent produceth such a quality. And as quantity may
be considered in all the operations of nature, so also doth geometry run
quite through the whole body of natural philosophy.
To the principles of geometry the definition appertaineth also of a
_measure_, which is this, _one quantity is the measure of another
quantity, when it, or the multiple of it, is coincident in all points
with the other quantity_. In which definition, instead of that ἐφαρμογὴ
of Euclid, I put coincidence. For the superposition of quantities, by
which they render the word ἐφαρμογὴ, cannot be understood of bodies, but
only of lines and superficies. Nevertheless many bodies may be
coincident successively with one and the same place, and that place will
be their measure, as we see practised continually in the measuring of
liquid bodies, which art of measuring may properly be called ἐφάρμοσις,
but not superposition.
Also the definitions of _greater_, _less_, and _equal_, are necessary
principles of geometry. For it cannot be imagined than any geometrician
should demonstrate to us the equality and inequality of magnitudes,
except he tell us first what those words do signify. And it is a wonder
to me, that Euclid hath not anywhere defined what are equals, or at
least, what are equal bodies, but serveth his turn throughout with that
forementioned ἐφάρμοσις, which hath no place in solids, nor in time, nor
in swiftness, nor in circular, or other crooked lines; and therefore no
wonder to me, why geometry hath not proceeded to the calculation neither
of crooked lines, nor sufficiently of motion, nor of many other things,
that have proportion to one another.
Equal bodies, superficies, and lines, are those of which every one is
capable of being coincident with the place of every one of the rest: and
equal times, wherein with one and the same motion equal lines are
described. And equally swift are those motions by which we run over
equal spaces in any time determined by any other motion. And universally
all quantities are equal, that are measured by the same number of the
same measures.
It is necessary also to the science of geometry, to define what
quantities are of one and the same kind, which they call _homogeneous_,
the want of which definitions hath produced those wranglings (which your
book _De Angulo Contactus_ will not make to cease) about the angle of
contingence.
_Homogeneous_ quantities are those which may be compared by (ἐφάρμοσις)
application of their measures to one another; so that solids and
superficies are heterogeneous quantities, because there is no
coincidence or application of those two dimensions.
No more is there of line and superficies, nor of line and solid, which
are therefore heterogeneous. But lines and lines, superficies and
superficies, solids and solids, are homogeneous.
Homogeneous also are line, and the quantity of time; because the
quantity of time is measured by the application of a line to a line; for
though time be no line, yet the quantity of time is a line, and the
length of two times is compared by the length of two lines.
Weight and solid have their quantity homogeneous, because they measure
one another by application, to the beam of a balance. Line and angle
simply so called, have their quantity homogeneous, because their measure
is an arch or arches of a circle applicable in every point to one
another.
The quantity of an angle simply so called, and the quantity of an angle
of contingence are heterogeneous. For the measures by which two angles
simply so called are compared, are in two coincident arches of the same
circle; but the measure by which an angle of contingence is measured, is
a straight line intercepted between the point of contact and the
circumference of the circle; and therefore one of them is not applicable
to the other; and consequently of these two sorts of angles the
quantities are heterogeneous. The quantities of two angles of
contingence are homogeneous; for they may be measured by the ἐφάρμοσις
of two lines, whereof one extreme is common, namely, the point of
contact, the other extremes are in the arches of the two circles.
Besides this knowledge of what is quantity and measure, and their
several sorts, it behoveth a geometrician to know why, and of what, they
are called principles. For not every proposition that is evident is
therefore a principle. A principle is the beginning of something. And
because definitions are the beginnings or first propositions of
demonstration, they are therefore called principles, principles, I say,
of demonstration. But there be also necessary to the teaching of
geometry other principles, which are not the beginnings of
demonstration, but of construction, commonly called petitions; as that
it may be granted _that a man can draw a straight line, and produce it;
and with any radius, on any centre describe a circle_, and the like. For
that a man may be able to describe a square, he must first be able to
draw a straight line; and before he can describe an equilateral
triangle, he must be able first to describe a circle. And these
petitions are therefore properly called principles, not of
demonstration, but of operation. As for the commonly received third sort
of principles, called _common notions_, they are principles, only by
permission of him that is the disciple; who being ingenuous, and coming
not to cavil but to learn, is content to receive them, though
demonstrable, without their demonstrations. And though definitions be
the only principles of demonstration, yet it is not true that every
definition is a principle. For a man may so precisely determine the
signification of a word as not to be mistaken, yet may his definition be
such as shall never serve for proof of any theorem, nor ever enter into
any demonstration, such as are some of the definitions of Euclid, and
consequently can be no beginnings of demonstration, that is to say, no
principles.
All that hitherto hath been said, is so plain and easy to be understood,
that you cannot, most egregious professors, without discovering your
ignorance to all men of reason, though no geometricians, deny it. And
the same (saving that the words are all to be found in dictionaries)
new; also to him that means to learn, not only the practice, but also
the science of geometry necessary, and, though it grieve you, mine. And
now I come to the definitions of Euclid.
The first is of a point: Σημεῖον, &c. “_Signum est, cujus est pars
nulla_,” that is to say, _a mark is that of which there is no part_.
Which definition, not only to a candid, but also to a rigid construer,
is sound and useful. But to one that neither will interpret candidly,
nor can interpret accurately, is neither useful nor true. Theologers say
the soul hath no part, and that an angel hath no part, yet do not think
that soul or angel is a point. A mark or as some put instead of it
ϛίγμη, which is a mark with a hot iron, is visible; if visible, then it
hath quantity, and consequently may be divided into parts innumerable.
That which is indivisible is no quantity; and if a point be not
quantity, seeing it is neither substance nor quality, it is nothing. And
if Euclid had meant it so in his definition, as you pretend he did, he
might have defined it more briefly, but ridiculously, thus, _a point is
nothing_. Sir Henry Savile was better pleased with the candid
interpretation of Proclus, that would have it understood respectively to
the matter of geometry. But what meaneth this _respectively to the
matter of geometry_? It meaneth this, that no argument in any
geometrical demonstration should be taken from the division, quantity,
or any part of a point; which is as much as to say, a point is that
whose quantity is not drawn into the demonstration of any geometrical
conclusion; or, which is all one, whose quantity is not considered.
An accurate interpreter might make good the definition thus, _a point is
that which is undivided_; and this is properly the same with _cujus non
est pars_: for there is a great difference between _undivided_ and
_indivisible_, that is, between _cujus non est pars_, and _cujus non
potest esse pars_. Division is an act of the understanding; the
understanding is therefore that which maketh parts, and there is no part
where there is no consideration but of one. And consequently Euclid’s
definition of a point is accurately true, and the same with mine, which
is, that _a point is that body whose quantity is not considered_. And
_considered_ is that, as I have defined it chap. I. at the end of the
third article, which is not put to account in demonstration.
Euclid therefore seemeth not to be of your opinion, that say a point is
nothing. But why then doth he never use this definition in the
demonstration of any proposition? Whether he useth it expressly or no, I
remember not; but the sixteenth proposition of the third book without
the force of this definition is undemonstrated.
The second definition is of a line: γραμμὴ δὲ μῆκος ἂπλατες. “_Linea est
longitudo latitudinis expers_; _a line is length which hath no
breadth_;” and if candidly interpreted, sound enough, though rigorously
not so. For to what purpose is it to say _length not broad_, when there
is no such thing as a _broad length_. One path may be broader than
another path, but not one mile than another mile; and it is not the path
but the mile which is the way’s length. If therefore a man have any
ingenuity he will understand it thus, _that a line is a body whose
length is considered without its breadth_, else we must say absurdly a
_broad length_; or untruly, that there be bodies which have length and
yet no breadth; and this is the very sense which Apollonius, saith
Proclus, makes of this definition; “when we measure,” says he, “the
length of a way, we take not in the breadth or depth, but consider only
one dimension.” See this of Proclus cited by Sir Henry Savile, where you
shall find the very word _consider_.
The fourth definition is of a straight line, thus Ἐυθεῖα γραμμή ἐϛιν,
&c. “_Recta linea est quæ ex æquo sua ipsius puncta inter jacet._” _A
straight line is that which lieth equally (or perhaps evenly) between
its own points._ This definition is inexcusable. Between what points of
its own can a straight line lie but between its extremes? And how lies
it evenly between them, unless it swerve no more from some other line
which hath the same extremes, one way than another? And then why are not
between the same points both the lines straight? How bitterly, and with
what insipid jests would you have reviled Euclid for this, if living now
he had written a _Leviathan_ . And yet there is somewhat in this
definition to help a man, not only to conceive the nature of a straight
line (for who doth not conceive it?) but also to express it. For he
meant perhaps to call a straight line that which is all the way from one
extreme to another, equally distant from any two or more such lines as
being like and equal have the same extremes. So the axis of the earth is
all the way equally distant from the circumference of any two or more
meridians. But then before he had defined a straight line, he should
have defined what lines are _like_, and what are _equal_. But it had
been best of all, first to have defined crooked lines, by the
possibility of a deduction or setting further asunder of their extremes;
and then straight lines, by the impossibility of the same.
The seventh definition, which is that of a plain superficies, hath the
same faults.
The eighth is of a plane angle, Ἑπὶπεδος γωνία ἑϛὶν ἡ ἐν ἐπιπέδω, &c.
“_Angulus planus est duarum linearum in plano se mutuo tangentium, et
non in directum jacentium, alterius ad alteram inclinatio._” _A plane
angle is the inclination one towards another of two lines that touch one
another in the same plane, and lie not in the same straight line._
Besides the faults here observed by Sir Henry Savile, as the clause of
not lying in the same straight line, and the obscurity or equivocation
of the word _inclination_, there is yet another, which is, that by this
definition two right angles together taken, are no angle; which is a
fault which you somewhere (asking leave to use the word _angle_,
καταχριϛικώς acknowledge, but avoid not. For in geometry, where you
confess there is required all possible accurateness, every καταχρῆσις is
a fault. Besides you see by this definition, that Euclid is not of your,
but of Clavius’s opinion. For it is manifest that the two lines which
contain an angle of contact incline one towards another, and come
together in a point, and lie not in the same straight line, and
consequently make an angle.
The thirteenth definition is exact, but makes against your doctrine,
that a point is nothing. Examine it. Ὅρος ἐϛὴν ὅ τινός ἐϛῖ πέρας.
“_Terminus est quod alicujus extremum est._” _A term or bound is that
which is the extreme of anything._ We had before, _the extremes of a
line are points_. But what is in a line the extreme, but the first or
last _part_, though you may make that part as small as you will? A point
is therefore a part, and nothing is no extreme.
The fourteenth, Σχῆμα ἐϛὶ τὸ ὑπὸ τινος ἤ τινῶν ὅρων περιεχόμενον.
“_Figura est (subaudi quantitas) quæ ab aliquo, vel aliquibus terminis
undique continetur sive clauditur._” _A figure is quantity contained
within some bound or bounds._ Or shortly thus, _a figure is quantity
every way determined_, is in my opinion as exact a definition of a
figure as can possibly be given, though it must not be so in yours. For
this _determination_ is the same thing with _circumscription_; and
whatsoever is anywhere _(ubicunque) definitivè_ is there also
_circumscriptivè_; and by this means the distinction is lost, by which
theologers, when they deny God to be in any place, save themselves from
being accused of saying he is nowhere; for that which is nowhere is
nothing. This definition of Euclid cannot therefore possibly be embraced
by you that carry double, namely, mathematics and theology. For if you
reject it, you will be cast out of all mathematic schools; and if you
maintain it, from the society of all school-divines, and lose the thanks
of the favour you have shown (you the astronomer) to Bishop Bramhall.
The fifteenth is of a circle. Κοὐκλος ἐστὶ σχῆμα ἐπίπεδον, &c. _A circle
is a plain figure comprehended by one line which is called the
circumference, to which circumference all the straight lines drawn from
one of the points within the figure are equal to one another._ This is
true. But if a man had never seen the generation of a circle by the
motion of a compass or other equivalent means, it would have been hard
to persuade him that there was any such figure possible. It had been
therefore not amiss first to have let him see that such a figure might
be described. Therefore so much of geometry is no part of philosophy,
which seeketh the proper passions of all things in the generation of the
things themselves.
After the fifteenth till the last or thirty-fifth definition, all are
most accurate, but the last which is this, _parallel straight lines are
those which being in the same plane, though infinitely produced both
ways, shall never meet_. Which is less accurate. For how shall a man
know that there be straight lines which shall never meet, though both
ways infinitely produced? Or how is the definition of parallels, that
is, of lines perpetually equidistant, good, wherein the nature of
equidistance is not signified? Or if it were signified, why should it
not comprehend as well the parallelism of circular and other crooked
lines, as of straight, and as well of superficies, as of lines? By
parallels is meant equidistant both lines and superficies, and the word
is therefore not well defined without defining first equality of
distance. And because the distance between two lines or superficies, is
the shortest line that can join them, there either ought to be in the
definition the _shortest distance_, which is that of the perpendicular
and without inclination, or the distance in equal inclination, that is,
in equal angles. Therefore if parallels be defined to be those lines or
superficies, where the lines drawn from one to another in equal angles
be equal, the definition, as to like lines, or like superficies, will be
universal and convertible. And if we add to this definition, that the
equal angles be drawn not opposite ways, it will be absolute, and
convertible in all lines and superficies; and the definition will be
this: _parallels are those lines and superficies between which every
line drawn, in any angle, is equal to any other line drawn in the same
angle the same way_. For by this definition the distance between them
will perpetually be equal, and consequently they will never come nearer
together, how much, or which way soever they be produced. And the
converse of it will be also true, _if two lines, or two superficies be
parallel, and a straight line be drawn from one to the other, any other
straight line, drawn from one to the other in the same angle, and the
same way, will be equal to it_. This is manifestly true, and, most
egregious professors, new, at least to you.
And thus much for the definitions placed before the first of Euclid’s
Elements.
Before the third of his Elements is this definition: “_In circulo
æqualiter distare a centro rectæ lineæ dicuntur, cum perpendiculares quæ
a centro in ipsas ducuntur sunt æquales_.” _In a circle two straight
lines are said to be equally distant from the centre, upon which the
perpendiculars drawn from the centre are equal._ This is true; but it is
rather an axiom than a definition, as being demonstrable that the
perpendicular is the measure of the distance between a point and a
straight or a crooked line.
Before the fifth Element the first definition is of a part: _Pars est
magnitudo magnitudinis, minor majoris, cum minor metitur majorem_. _A
part is one magnitude of another, the less of the greater, when the less
measureth the greater._ From which definition it followeth, that more
than a half is not a part of the whole. But because Euclid meaneth here
an aliquot part, as a half, a third, or a fourth, &c., it may pass for
the definition of a measure under the name of part, as thus: _a measure
is a part of the whole, when multiplied it may be equal to the whole_,
though properly a measure is external to the thing measured, and not the
aliquot part itself, but equal to an aliquot part.
But the third definition is intolerable; it is the definition of λόγος,
in Latin _ratio_, in English, _proportion_, in this manner, λόγος ἐςὶ
δύο μεγεθῶν ὁμογενῶν ῆ κατὰ πηλικότητα προς ἄλληλα ποιὰ σχέσις. “_Ratio
est duarum magnitudinum ejusdem generis mutua quædam secundum
quantitatem habitudo._” _Proportion is a certain mutual habitude in
quantity, of two magnitudes of the same kind, one to another._ First, we
have here _ignotum per ignotius_; for every man understandeth better
what is meant by _proportion_ than by habitude. But it was the phrase of
the Greeks when they named like proportions, to say, the first to the
second, οὕτως ἔχει, _id est, ita se habet_, and in English, _is as_, the
third to the fourth. As for example, in the proportions of two to four,
and three to six, to say two to four, οὕτως ἔχει, _id est, ita se habet,
id est_, _is as_, three to six. From which phrase Euclid made this his
definition of proportion by ποιὰ σχέσις, which the Latins translate
_quædam habitudo_. _Quædam_ in a definition is a most certain note of
not understanding the word _defined_; and in Greek, ποιὰ σχέσις is much
worse; for to render rightly the Greek definition, we are to say in
English, that proportion is a what-shall-I-call-it-_isness_, or _soness_
of two magnitudes, &c.; than which nothing can be more unworthy of
Euclid. It is as bad as anything was ever said in geometry by Orontius,
or by Dr. Wallis. That proportion is quantity compared, that is to say,
little or great in respect of some other quantity, as I have above
defined it, is I think intelligible.
The fourth is, Ἀναλογία δέ ὲστιν ῆ των λόγων ὁμοιότης. “_Proportio vero
est rationum similitudo._” Here we have no one word by which to render
Ἀναλογία; for our word _proportion_ is already bestowed upon the
rendering of λόγος. Nevertheless the Greek may be translated into
English thus, _iterated proportions_. But iterated proportion is the
same with _eadem ratio_. To what purpose then serveth the sixth
definition, which is of _eadem ratio_? For Ἀναλογία and _eadem ratio_
and _similitudo rationum_, are the same thing, as appeareth by Euclid
himself, where he defines those quantities, that are in the same
proportion by ἀνάλογον. Therefore the sixth definition is but a _lemma_,
and assumed without demonstration.
The fourteenth, “_Compositio rationis est sumptio antecedentis cum
consequente, ceu unius, ad ipsum consequentem_,” _To compound
proportion, is to take both antecedent and consequent together as one
magnitude, and compare it to the consequent_, is good; though he might
have compared it as well with the antecedent; for both ways it had been
a composition of proportion. We are to note here, that the composition
defined in this place by Euclid is not adding together of proportions,
but of two quantities that have proportion. And therefore it is not the
same composition which he defineth in the fourth place before the sixth
element, for there he defineth the addition of one proportion to another
proportion in this manner: λόγος ἐκ λόγων συγκεῖσθαι λέγεται, &c. _A
proportion is said to be compounded of proportions, when their
quantities multiplied into one another make a proportion_; as when we
would compound or add together the proportions of three to two, and of
four to five, we must multiply three and four, which maketh twelve, and
two and five, which maketh ten. And then the proportion of twelve to ten
is the sum of the proportions of three to two, and of four to five,
which is true, but not a definition; for it may and ought to be
demonstrated. For to define what is addition of two proportions (which
are always in four quantities, though sometimes one of them be twice
named) we are to say, that they are then added together when we make the
second to another in the same proportion, which the third hath to the
fourth.
And thus much of the definitions; of which some, very few, you see are
faulty; the rest either accurate, or good enough if well interpreted.
For the rest of the elements all are accurate, notwithstanding that you
allow not for good any definition in geometry that hath in it the word
_motion_, of which there be divers before the eleventh Element. But I
must here put you in mind, that geometry being a science, and all
science proceeding from a precognition of causes, the definition of a
sphere, and also of a circle, by the generation of it, that is to say,
by motion, is better than by the equality of distance from a point
within.
The second sort of principles are those of construction, usually called
_postulata_, or petitions. As for those _notiones communes_, called
_axioms_, they are from the definitions of their terms demonstrable,
though they be so evident as they need not demonstration. These
petitions are by Euclid called Ἀιτήματα, such as are granted by favour,
that is, simply petitions, whereas by axiom is understood that which is
claimed as due. So that between Ἀξίωμα and Ἀίτημα there is this other
difference, that this latter is simply a petition, the former a petition
of right.
Of petitions simply, the first is, _that from any point to any point may
be drawn a straight line_. The second, _that a finite straight line may
be produced_. The third, _that upon any centre at any distance may be
described a circle_. All which are both evident and necessary to be
granted.
And by all these a man may easily perceive that Euclid in the
definitions of a point, a line, and a superficies, did not intend that a
point should be nothing, or a line be without latitude, or a superficies
without thickness; for if he did, his petitions are not only
unreasonable to be granted, but also impossible to be performed. For
lines are not drawn but by motion, and motion is of body only. And
therefore his meaning was, that the quantity of a point, the breadth of
a line, and the thickness of a superficies were not to be _considered_,
that is to say, not to be reckoned in the demonstration of any theorems
concerning the quantity of bodies, either in length, superficies, or
solid.
==========
OF THE FAULTS THAT OCCUR IN
DEMONSTRATION.
TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.
LESSON II.
There be but two causes from which can spring an error in the
demonstration of any conclusion in any science whatsoever; and those are
ignorance or want of understanding, and negligence. For as in the adding
together of many and great numbers, he cannot fail that knoweth the
rules of addition, and is also all the way so careful, as not to mistake
one number or one place for another; so in any other science, he that is
perfect in the rules of logic, and is so watchful over his pen, as not
to put one word for another, can never fail of making a true, though not
perhaps the shortest and easiest demonstration.
The rules of demonstration are but of two kinds: one, that the
principles be true and evident definitions; the other, that the
inferences be necessary. And of true and evident definitions, the best
are those which declare the cause or generation of that subject, whereof
the proper passions are to be demonstrated. For science is that
knowledge which is derived from the comprehension of the cause. But when
the cause appeareth not, then may, or rather must we define some known
property of the subject, and from thence derive some possible way, or
ways, of the generation. And the more ways of generation are explicated,
the more easy will be the derivation of the properties; whereof some are
more immediate to one, some to another generation. He therefore that
proceedeth from untrue, or not understood definitions, is ignorant of
that he goes about; which is an ill-favoured fault, be the matter he
undertaketh easy or difficult, because he was not forced to undergo a
greater charge than he could carry through. But he that from right
definitions maketh a false conclusion, erreth through human frailty, as
being less awake, more troubled with other thoughts, or more in haste
when he was in writing. Such faults, unless they be very frequent, are
not attended with shame, as being common to all men, or are at least
less ugly than the former, except then, when he that committeth them
reprehendeth the same in other men. For that is in every man
intolerable, which he cannot tolerate in another. But to the end that
the faults of both kinds may by every man be well understood, it will
not be amiss to examine them by some such demonstrations as are publicly
extant. And for this purpose I will take such as are in mine and in your
books, and begin with your _Elenchus_ of the geometry contained in my
book _De Corpore_; to which I will also join your book lately set forth
concerning the _Angle of Contact, Conic Sections_ , and your
_Arithmetica Infinitorum_; and then examine the rest of my philosophy,
and yours that oppugn it. For I will take leave to consider you both
everywhere as one author, because you publicly declare your approbation
of one another’s doctrine.
My first definition is of a line, of length, and of a point. “The way,”
say I, “of a body moved, in which magnitude (though it always have some
magnitude) is not considered, is called a line; and the space gone over
by that motion, length, or one and a simple dimension.” To this
definition you say, first, “what mathematician did ever thus define a
line or length?” Whether you call in others for help or testimony, it is
not done like a geometrician; for they use not to prove their
conclusions by witnesses, but rely upon the strength of their own
reason; and when your witnesses appear, they will not take your part.
Secondly, you grant that what I say is true, but not a definition. But
to tell you truly what it is which we call a line, is to define a line.
Why then is not this a definition? “Because,” say you in the first
place, “it is not a reciprocal proposition.” But by your favour it is
reciprocal. For not only the way of a body whose quantity is not
considered is a line, but also every line is, or may be conceived to be,
the way of a body so moved. And if you object that there is a difference
between _is_ and _may be conceived to be_, Euclid, whom you call to your
aid, will be against you in the fourteenth definition before his
eleventh Element; where he defines a sphere just as convertibly as I
define a line; except you think the globes of the sun and stars cannot
be globes, unless they were made by the circumduction of a semicircle;
and again in the eighteenth definition, which is of a cone, unless you
admit no figure for a cone, which is not generated by the revolution of
a triangle; and again, in the twentieth definition, which is of a
cylinder, except it be generated by the circumvolution of a
parallelogram. Euclid saw that what proper passion soever should be
derived from these his definitions, would be true of any other cylinder,
sphere, or cone, though it were otherwise generated; and the description
of the generation of any one being by the imagination applicable to all,
which is equivalent to convertible, he did not believe that any rational
man could be misled by learning logic to be offended with it. Therefore
this exception proceedeth from want of understanding, that is, from
ignorance of the nature, and use of a definition.
Again, you object and ask: “What need is there of motion, or of body
moved, to make a man understand what is a line? Are not lines in a body
at rest, as well as in a body moved? And is not the distance of two
resting points length, as well as the measure of the passage? Is not
length one and a simple dimension, and one and a simple dimension line?
Why then is not line and length all one?” See how impertinent these
questions are. Euclid defines a sphere to be a solid figure described by
the revolution of a semicircle about the unmoved diameter. Why do you
not ask, what need there is to the understanding of what a sphere is, to
bring in the motion of a semicircle? Is not a sphere to be understood
without such motion? Is not the figure so made a sphere without this
motion? And where he defines the axis of a sphere to be that unmoved
diameter, may not you ask, whether there be no axis of a sphere, when
the whole sphere, diameter and all, is in motion? But it is not to my
purpose to defend my definition by the example of that of Euclid.
Therefore first, I say, to me, howsoever it may be to others, it was fit
to define a line by motion. For the generation of a line is the motion
that describes it. And having defined philosophy in the beginning, to be
the knowledge of the properties from the generation, it was fit to
define it by its generation. And to your question, _is not distance
length?_ I answer, that though sometimes distance be equivalent to
length, yet certainly the distance between the two ends of a thread
wound up into a clue is not the length of the thread; for the length of
the thread is equal to all the windings whereof the clue is made. But if
you will needs have distance and length to be all one, tell me of what
the distance between any two points is the length. Is it not the length
of the way? And how is that called way, which is not defined by some
motion? And have not several ways between the same places, as by land
and by water, several lengths? But they have but one distance, because
the distance is the shortest way. Therefore between the length of the
path, and the distance of places, there is a real difference in this
case, and in all cases a difference of the consideration. Your
objection, that line is longitude, proceeds from want of understanding
English. Do men ever ask what is the line of a thread, or the line of a
table, or of any other body? Do they not always ask what is the length
of it? And why, but because they use their own judgments, not yet
corrupted by the subtlety of mistaken professors. Euclid defines a line
to _be length without breadth_. If those terms be all one, why said he
not that a _line is a line without breadth_? But what definition of a
line give you? None. Be contented then with such as you receive, and
with this of mine, which you shall presently see is not amiss.
Your next objections are to my definition of a point. Which definition
adhereth to the former in these words, “and the body itself is called a
point.” Here again you call for help: “_Quis unquum mortalium, etc._
What mortal man, what sober man, did ever so define a point?” It is
well, and I take it to be an honour to be the first that do so. But what
objection do you bring against it. This: “That a point added to a point,
if it have magnitude, makes it greater.” I say it doth so, but then
presently it loseth the name of a point, which name was given to signify
that it was not the meaning of him that used it in demonstration to add,
subtract, multiply, divide, or any way compute it. Then you come in
with, “perhaps you will say though it have magnitude, that magnitude is
not considered.” You need not say _perhaps_. You know I affirm it; and
therefore your argument might have been left out, but that it gave you
an occasion of a digression into scurvy language.
And whereas you ask why I defined not a point thus: “_Punctum est corpus
quod non consideratur esse corpus, et magnum quod non consideratur esse
magnum_.” I will tell you why. First, because it is not Latin. Secondly,
because when I had defined it by _corpus_, there was no need to define
it again by _magnum_. I understand very well this language, “_punctum
est corpus, quod non consideratur ut corpus_.” A point is a body not
considered as body. But _punctum est corpus, quod non consideratur esse
corpus, vel esse magnum_, is not Latin; nor the version of it, _a point
is a body which is not considered to be a body_, English. My definition
was, that a point is that body whose magnitude is not considered, not
reckoned, not put to account in demonstration. And I exemplified the
same by the body of the earth describing the ecliptic line; because the
magnitude is not there reckoned nor chargeth the ecliptic line with any
breadth. But I perceive you understand not what the word _consideration_
signifieth, but take it for comparison or relation; and say I ought to
define a point simply, and not by relation to a great body; as if to
reckon and to compare were the same thing. “_Omnia mihi_,” saith Cicero,
“_provisa et considerata sunt_.” I have provided and reckoned
everything. There is a great difference between reckoning and relation.
Again, you ask, why _corpus motum_, a body moved? I will tell you;
because the motion was necessary for the generation of a line. And
though after the generation of the line the point should rest, yet it is
not necessary from this definition that it should be no more a point;
nor when Euclid defines a sphere by the circumduction of a semicircle
upon an axis that resteth, doth it follow from thence when the sphere,
axis, centre and all, as that of the earth, is moved from place to
place, that it is no more an axis.
Lastly, you object “that motion is accidentary to a point, and
consequently not essential, nor to be put into the definition.” And is
not the circumduction of a semicircle accidentary to a sphere? Or do you
think the sphere of the sun was generated by the revolution of a
semicircle? And yet it was thought no fault in Euclid to put the motion
into the definition of a sphere.
The conceit you have concerning definitions, that they must explicate
the essence of the thing defined, and must consist of a _genus_ and a
_difference_, is not so universally true as you are made believe, or
else there be very many insufficient definitions that pass for good with
you in Euclid. You are much deceived if you think these woful notions of
yours, and the language that doth everywhere accompany them, show
handsomely together. Or that such grounds as these be able to sustain so
many, and so haughty reproaches as you advance upon them, so as they
fall not, as you shall see immediately, upon your own head. I say a
point hath quantity, but not to be reckoned in demonstrating the
properties of lines, solids, or superficies; you say it hath no quantity
at all, but is plainly nothing.
The first of the petitions of Euclid is, “that a line may be drawn from
point to point at any distance.” The second, “that a straight line may
be produced.” The third, “that on any centre a circle may be described
at any distance.” And the eighth axiom (which Sir H. Savile observes to
be the foundation of all geometry) is this, “_Quæ sibi mutuo congruunt,
etc._ Those things that are applied to one another in all points are
equal.” All or any of these principles being taken away, there is not in
Euclid one proposition demonstrated or demonstrable. If a point have no
quantity, a line can have no latitude; and because a line is not drawn
but by motion, by motion of a body, and body imprinteth latitude all the
way, it is impossible to draw or produce a straight line, or to describe
a circular line without latitude. Also if a line have no latitude, one
straight line cannot be applied to another. To them therefore that deny
a point to have quantity, that is, a line to have latitude, the
forenamed principles are not possible, and consequently no proposition
in geometry is demonstrated or demonstrable. You therefore that deny a
point to have quantity, and a line to have breadth, have nothing at all
of the science of geometry. The practice you may have, but so hath any
man that hath learned the bare propositions by heart; but they are not
fit to be professors either of geometry or of any other science that
dependeth on it. Some man perhaps may say that this controversy is not
much worth, and that we both mean the same thing. But that man, though
in other things prudent enough, knoweth little of science and
demonstration. For definitions are not only used to give us the notions
of those things whose appellations are defined, for many times they that
have no science have the ideas of things more perfect than such as are
raised by definitions. As who is there that understandeth not better
what a straight line is, or what proportion is, and what many other
things are, without definition, than some that set down the definitions
of them. But their use is, when they are truly and clearly made, to draw
arguments from them for the conclusions to be proved. And therefore you
that in your following censures of my geometry, take your argument so
often from this, that a point is nothing, and so often revile me for the
contrary, are not to be allowed such an excuse. He that is here
mistaken, is not to be called negligent in his expression, but ignorant
of the science.
In the next place, you take exceptions to my definition of _equal
bodies_, which is this: “_Corpora æqualia sunt quæ eundem locum
possidere possunt_. Equal bodies are those which may have the same
place.” To which you object impertinently, that I may as well define a
man to be, _he that may be prince of Transylvania_, wittily, as you
count wit. Formerly in every definition, you exacted an explication of
the essence. You are therefore of opinion that the possibility of being
prince of Transylvania is no less essential to _a man_, than the
possibility of the being of two bodies successively in the same place,
is essential to _bodies equal_.
You take no notice of the twenty-third article of this same chapter,
where I define what it is we call essence, namely, that accident for
which we give the thing its name. As the essence of a man is his
capacity of reasoning; the essence of a white body, whiteness, &c.,
because we give the name of _man_ to such bodies as are capable of
reasoning, for that their capacity; and the name of _white_ to such
bodies as have that colour, for that colour. Let us now examine why it
is that men say bodies are one to another equal; and thereby we shall be
able to determine whether the _possibility of having the same place_ be
essential or not to _bodies equal_, and consequently whether this
definition be so like to the defining of a man by the _possibility of
being prince of Transylvania_ as you say it is. There is no man, besides
such egregious geometricians as yourselves, that inquireth the equality
of two bodies, but by measure. And for liquid bodies, or the aggregates
of innumerable small bodies, men (men, I say) measure them by putting
them one after another into the same vessel, that is to say, into the
same place, as Aristotle defines place, or into the space determined by
the vessel, as I define place. And the bodies that so fill the vessel,
they acknowledge and receive for equal. But though, when hard bodies
cannot be so measured, without the incommodity or trouble of altering
their figure, they then enquire, if the bodies are both of the same
kind, their equality by weight, knowing, without your teaching, that
equal bodies of the same nature weigh proportionably to their
magnitudes; yet they do it not for fear of missing of the equality, but
to avoid inconvenience or trouble. But you (you, I say), that have no
definition of equals, neither received from others, nor framed by
yourselves, out of your shallow meditation and deep conceit of your own
wits, contend against the common light of nature. So much is unheedy
learning a hinderance to the knowledge of the truth, and changeth into
elves those that were beginning to be men.
Again, when men inquire the equality of two bodies in length, they
measure them by a common measure; in which measure they consider neither
breadth nor thickness, but how the length of it agreeth, first with the
length of one of the bodies, then with the length of the other. And both
the bodies whose lengths are measured, are successively in the same
place under their common measure. _Place_ therefore in lines also, is
the proper index and discoverer of equality and inequality. And as in
length, so it is in breadth and thickness, which are but lengths
otherwise taken in the same solid body. But now when we come from this
equality and inequality of lengths known by measure, to determine the
proportions of superficies and of solids, by ratiocination, then it is
that we enter into geometry; for the making of definitions, in
whatsoever science they are to be used, is that which we call
_philosophia prima_. It is not the work of a geometrician, as a
geometrician, to define what is equality, or proportion, or any other
word he useth, though it be the work of the same man, as a man. His
geometrical part is, to draw from them as many true and useful theorems
as he can.
You object secondly, that a pyramis may be equal to a cube whilst it is
a pyramis. True. And so also whilst it is a pyramis it hath a
possibility by flexion and transposition of parts to become a cube, and
to be put into the place where another cube equal to it was before. This
is to argue like a child that hath not yet the perfect understanding of
any language.
In the third and fourth objection, you teach me to define equal bodies
(if I will needs define them by place) by the _equality of place_, and
to say, _that bodies are equal that have equal places_. Teach others, if
you can, to measure their grain, not by the same, but equal bushels.
In the fifth objection, you except against the the word _can_, in that I
say that bodies are equal which _can_ fill the same place. For the
greater body _can_, you say, fill the place of the less, though not
reciprocally the less of the greater. It is true, that though the place
of the less can never be the place of the greater, yet it may be filled
by a part of the greater. But it is not then the greater body that
filleth the place of the less, but a part of it, that is to say, a less
body. Howsoever, to take away from simple men this straw they stumble
at, I have now put the definition of equal bodies into these words:
_equal bodies are those whereof every one can fill the place of every
other_. And if my definition displease you, propound your own, either of
_equal bodies_, or of _equals_ simply. But you have none. Take therefore
this of mine.
The sixth is a very admirable exception. “What,” say you, “if the same
body can sometimes take up a greater, sometimes a lesser place, as by
rarefaction and condensation?” I understand very well that bodies may be
sometimes thin and sometimes thick, as they chance to stand closer
together or further from one another. So in the mathematic schools, when
you read your learned lectures, you have a thick or thronging audience
of disciples, which in a great church would be but a very thin company.
I understand how thick and thin may be attributed to bodies in the
plural, as to a company; but I understand not how any one of them is
thicker in the school than in the church; or how any one of them taketh
up a greater room in the school, when he can get in, than in the street.
For I conceive the dimensions of the body, and of the place, whether the
place be filled with gold or with air, to be coincident and the same;
and consequently both the quantity of the air, and the quantity of the
gold, to be severally equal to the quantity of the place; and therefore
also, by the first axiom of Euclid, equal to one another; insomuch as if
the same air should be by condensation contained in a part of the place
it had, the dimensions of it would be the same with the dimensions of
part of the place, that is, should be less than they were, and by
consequence the quantity less. And then either the same body must be
less also, or we must make a difference between greater bodies and
bodies of greater quantity; which no man doth that hath not lost his
wits by trusting them with absurd teachers. When you receive salary, if
the steward give you for every shilling a piece of sixpence, and then
say, every shilling is condensed into the room of sixpence, I believe
you would like this doctrine of yours much the worse. You see how by
your ignorance you confound the affairs of mankind, as far forth as they
give credit to your opinions, though it be but little. For nature abhors
even empty words, such as are (in the meaning you assign them),
_rarefying_ and _condensing_. And you would be as well understood if you
should say (coining words by your own power), that the same body might
take up sometimes a greater, sometimes a lesser place, by wallifaction
and wardensation, as by rarefaction and condensation. You see how
admirable this your objection is.
In the seventh objection you bewray another kind of ignorance, which is
the ignorance of what are the proper works of the several parts of
philosophy. “Though it were out of doubt,” say you, “that the same body
cannot have several magnitudes, yet seeing it is matter of natural
philosophy, nor hath anything to do with the present business, to what
purpose is it to mention it in a mathematical definition?” It seems by
this, that all this while you think it is a piece of the geometry of
Euclid, no less to make the definitions he useth, than to infer from
them the theorems he demonstrateth. Which is not true. For he that
telleth you in what sense you are to take the appellations of those
things which he nameth in his discourse, teacheth you but his language,
that afterwards he may teach you his art. But teaching of language is
not mathematic, nor logic, nor physic, nor any other science; and
therefore to call a definition, as you do, mathematical, or physical, is
a mark of ignorance, in a professor inexcusable. All doctrine begins at
the understanding of words, and proceeds by reasoning till it conclude
in science. He that will learn geometry must understand the terms before
he begin, which that he may do, the master demonstrateth nothing, but
useth his natural prudence only, as all men do when they endeavour to
make their meaning clearly known. For words understood are but the seed,
and no part of the harvest of philosophy. And this seed was it, which
Aristotle went about to sow in his twelve books of _metaphysics_, and in
his eight books concerning the hearing of _natural philosophy_. And in
these books he defineth time, place, substance or essence, quantity,
relation, &c., that from thence might be taken the definitions of the
most general words for principles in the several parts of science. So
that all definitions proceed from common understanding; of which, if any
man rightly write, he may properly call his writing _philosophia prima_,
that is, the seeds, or the grounds of philosophy. And this is the method
I have used, defining place, magnitude, and the other the most general
appellations in that part which I entitle _philosophia prima_. But you
now, not understanding this, talk of mathematical definitions. You will
say perhaps that others do the same as well as you. It may be so. But
the appeaching of others does not make your ignorance the less.
In the eighth place you object not, but ask me _why I define equal
bodies apart_? I will tell you. Because all other things which are said
to be equal, are said to be so from the equality of bodies; as two lines
are said to be equal, when they be coincident with the length of one and
the same body; and equal times, which are measured by equal lengths of
body, by the same motion. And the reason is, because there is no subject
of quantity, or of equality, or of any other accident but body; all
which I thought certainly was evident enough to any uncorrupted
judgment; and therefore that I needed first to define equality in the
subject thereof, which is body, and then to declare in what sense it was
attributed to time, motion, and other things that are not body.
The ninth objection is an egregious cavil. Having set down the
definition of _equal bodies_, I considered that some men might not allow
the attribute of equality to any things but those which are the subjects
of quantity, because there is no equality, but in respect of quantity.
And to speak rigidly, _magnum et magnitudo_ are not the same thing; for
that which is great, is properly a body, whereof greatness is an
accident. In what sense therefore, might you object, can an accident
have quantity? For their sakes therefore that have not judgment enough
to perceive in what sense men say the length is so long, or the
superficies so broad, &c. I added these words: “_Eadem ratione (qua
scilicet corpora dicuntur æqualia) magnitudo magnitudini æqualis
dicitur_,” that is, _in the same manner, as bodies are said to be equal,
their magnitudes also are said to be equal_. Which is no more than to
say, _when bodies are equal, their magnitudes also are called equal.
When bodies are equal in length, their lengths are also called equal.
And when bodies are equal in superficies, their superficies are also
called equal._ All which is common speech, as well amongst
mathematicians, as amongst common people; and, though improper, cannot
be altered, nor needeth to be altered to intelligent men. Nevertheless I
did think fit to put in that clause, that men might know what it is we
call equality, as well in magnitudes as in _magnis_, that is, in bodies.
Which you so interpret, as if it bore this sense, _that when bodies are
equal their superficies also must be equal_, contrary to your own
knowledge, only to take hold of a new occasion of reviling. How
unhandsome and unmanly this is, I leave to be judged by any reader that
hath had the fortune to see the world, and converse with honest men.
Against the fourteenth article, where I prove that the same body hath
always the same magnitude, you object nothing but this, _that though it
be granted, that the same body hath the same magnitude, while it
resteth, yet I bring nothing to prove that when it changeth place, it
may not also change its magnitude by being enlarged or contracted_.
There is no doubt but to a body, whether at rest or in motion, more body
may be added, or part of it taken away. But then it is not the same
body, unless the whole and the part be all one. If the schools had not
set your wit awry, you could never have been so stupid as not to see the
weakness of such objections. That which you add in the end of your
objections to this eighth chapter, _that I allow not Euclid this axiom
gratis, that the whole is greater than a part_, you know to be untrue.
At my eleventh chapter, you enter into dispute with me about the nature
of proportion. Upon the truth of your doctrine therein, and partly upon
the truth of your opinions concerning the definitions of a point, and of
a line, dependeth the question whether you have any geometry or none;
and the truth of all the demonstrations you have in your other books,
namely of the _Angle of Contact_ , and _Arithmetica Infinitorum_. Here I
say you enter, how you will get out, your reputation saved, we shall see
hereafter.
When a man asketh what proportion one quantity hath to another, he
asketh how great or how little the one is comparatively to, or in
respect of the other. When a geometrician prefixeth before his
demonstrations a definition, he doth it not as a part of his geometry,
but of natural evidence, not to be demonstrated by argument, but to be
understood in understanding the language wherein it is set down; though
the matter may nevertheless, if besides geometry he have wit, be of some
help to his disciple to make him understand it the sooner. But when
there is no significant definition prefixed, as in this case, where
Euclid’s definition of proportion, that it is a _whatshicalt habitude of
two quantities, &c._, is insignificant, and you allege no other, every
one that will learn geometry, must gather the definition from observing
how the word to be defined is most constantly used in common speech. But
in common speech if a man shall ask how much, for example, is six in
respect of four, and one man answer that it is greater by two, and
another that it is greater by half of four, or by a third of six, he
that asked the question will be satisfied by one of them, though perhaps
by one of them now, and by the other another time, as being the only man
that knoweth why he himself did ask the question. But if a man should
answer, as you would do, that the proportion of six to two is of those
numbers a certain quotient, he would receive but little satisfaction.
Between the said answers to this question, how much is six in respect of
four? there is this difference. He that answereth that it is more by
two, compareth not two with four, nor with six, for two is the name of a
quantity absolute. But he that answereth it is more by half of four, or
by a third of six, compareth the difference with one of the differing
quantities. For halfs and thirds, &c. are names of quantity compared.
From hence there ariseth two species or kinds of (_ratio_) proportion,
into which the general word _proportion_ may be divided. The one
whereof, namely, that wherein the difference is not compared with either
of the differing quantities, is called _ratio arithmetica_, arithmetical
proportion; the other _ratio geometrica_, geometrical proportion; and,
because this latter is only taken notice of by the name of proportion,
simply _proportion_. Having considered this, I defined proportion,
chapter II. article 3, in this manner: “_Ratio est relatio antecedentis
ad consequens secundum magnitudinem_:” _Proportion is the relation of
the antecedent to the consequent in magnitude_; having immediately
before defined relatives, antecedent, and consequent, in the same
article, and by way of explication added, that such relation was nothing
else but that one of the quantities was equal to the other, or exceeded
it by some quantity, or was by some quantity exceeded by it. And for
exemplification of the same, I added further, that the proportion of
three to two was, that three exceeded two by a unity; but said not that
the unity, or the difference whatsoever it were, was their proportion,
_for unity, and to exceed another by unity_, is not the same thing. This
is clear enough to others; let us therefore see why it is not so to you.
You say I make proportion to consist in that which remaineth after the
lesser quantity is subtracted out of the greater; and that you make it
to consist in the quotient, when one number is divided by the other.
Wherein you are mistaken; first, in that you say, I make the proportion
to consist in the remainder. For I make it to consist in the act of
exceeding, or of being exceeded, or of being equal; whereas the
remainder is always an absolute quantity, and never a proportion. To be
more or less than another number by two, is not the number two; likewise
to be equal to two, where the difference is _nothing_, is not that
_nothing_? Again, you mistake in saying the proportion consisteth in the
quotient. For divide twenty by five, the quotient is four. Is it not
absurd to say that the proportion of five to twenty, or of twenty to
five, is four? You may say the proportion of five to twenty, is the
proportion of one to four. And so say I. And you may therefore also say,
that the proportion of one to four is a measure of the proportion of
five to twenty, as being equal. And so say I. But that is only in
geometrical proportion, and not in proportion universally. For though
the _species_ obtain the denomination of the genus, yet it is not the
_genus_. But as the quotient giveth us a measure of the proportion of
the dividend to the divisor in geometrical proportion, so also the
remainder after subtraction is the measure of proportion arithmetical.
You object in the next place, “that if the proportion of one quantity to
another be nothing but the excess or defect, then, wheresoever the
excess or defect is the same, there the proportion is the same.” This
you say follows in your logic, and from thence, that the proportion of
three to two, and five to four is the same. But is not three to two, and
five to four, where the excess is the same number, the same proportion
arithmetical? And is not arithmetical proportion, proportion? You take
here (_ratio_) proportion, which is the _genus_, for that _species_ of
it which is called geometrical, because usually this species has the
name of proportion simply. Also that the proportion of three to two, is
the same with that of nine to six; is it not because the excesses are
one and three, the same portions of three and nine, that is to say the
same excesses comparatively? I wonder you ask me not what is the _genus_
of arithmetical and geometrical proportions, and what the _difference_?
The _genus_ is (_ratio_) proportion, or comparison in magnitude, and the
_difference_ is that one comparison is made by the absolute quantity,
the other by the comparative quantity, of the excess or defect, if there
be any. Can anything be clearer than this? You after come in with
_ignosce habitudini_ to no purpose. I am not so inhuman as not to pardon
dulness or madness: they are not voluntary faults. But when men
adventure voluntarily to talk of that they understand not censoriously
and scornfully, I may tell them of it.
This difference between the excesses or defects, as they are simply or
comparatively reckoned, being thus explained, all the rest of that you
say in your objections to this eleventh chapter (saving that art. 5 for
_ratio binarii ad quinarium est superari ternario_, as it is in other
places, I have put too hastily _ratio binarii ad quinarium est
ternarius_), will be understood by every reader to be frivolous, and to
proceed from the ignorance of what proportion is.
At the twelfth chapter you only note that I say, _that the proportion of
inequality is quantity, but the proportion of equality not quantity_,
and refer that which you have to say against it to the chapter
following; to which place I shall also come in the following lesson.
==========
OF THE FAULTS THAT OCCUR IN
DEMONSTRATION.
TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.
LESSON III.
You begin your reprehension of my thirteenth chapter with a question;
whereas _I_ divide proportion into arithmetical and geometrical. You ask
me what _proportion it is I so divide_. Euclid divides an angle into
right, obtuse, and acute. I may ask you as pertinently, what angle it is
he so divides? Or, when you divide _animal_ into _homo_ and _brutum_,
what animal that is, which you so divide? You see by this, how absurd
your question is. But you say the definition of proportion which I make
at Chap. II. art. 3., namely, that proportion is the comparison of two
magnitudes, one to another, agreeth not, neither with arithmetical, nor
with geometrical proportion. I believe you thought so then, but having
read what I have said in the end of the last lesson, if you think so
still, your fault will be too great to be pardoned easily. But why did
you think so before? Is it not because there was no definition in Euclid
of proportion universal, and because he maketh no mention of proportion
arithmetical, and because you had not in your minds a sufficient notion
thereof yourselves to supply that defect? And is not this the cause
also, why you put in this parenthesis (if arithmetical proportion ought
to be called proportion)? Which is a confession that you know not
whether there be such a thing as arithmetical proportion or not,
notwithstanding that on all occasions you speak of arithmetical
proportionals. Yes, this was it that made you think that proportion
universally, and proportion geometrical, is the same, and yet to say you
cannot tell whether they be the same or not. It is no wonder, therefore,
if in such confusion of the understanding, you apprehend not that the
proportions of two to five, and nine to twelve, are the same; so you are
blinded by seeing that they are not the same proportions geometrical.
Nor doth it help you that I say the difference is the proportion; for by
difference you might, if you would, have understood the act of
differing.
At the second article you note for a fault in method, that _after I had
used the words antecedent and consequent of a proportion in some of the
precedent chapters, I define them afterwards_. I do not believe you say
this against your knowledge, but that the eagerness of your malice made
you oversee; therefore go back again to the third article of chapter II.
where, having defined correlatives, I add these words, _of which the
first is called the_ antecedent, _the second the_ consequent. This is
but an oversight, though such as in me you would not have excused.
At the thirteenth article you find fault with, that I say _that the
proportion of inequality, whether it be of excess or of defect, is
quantity, but the proportion of equality is not quantity_. Whether that
which you say, or that which I say, be the truth, is a question worthy
of a very strict examination. The first time I heard it argued, was in
Mersennus’ chamber at Paris, at such time as the first volume of his
_Cogitata Physico-Mathematica_ was almost printed; in which, because he
had not said all he would say of proportion, he was forced to put the
rest into a general preface, which, as was his custom, he did read to
his friends before he sent it to the press. In that general preface,
under the title _De Rationibus atque Proportionibus_, at the numbers
twelve, thirteen, fourteen, he maintaineth against Clavius, _that the
composition of proportion is_ (as of all other things) _a composition of
the parts to make a total_, and _that the proportion of equality
answereth in quantity to_ non-ens, _or nothing; the proportion of
excess, to_ ens, _or quantity; and the proportion of defect, to less
than nothing; because equality_ (he says) _is a term of middle
signification between excess and defect_. And there also he refuteth the
arguments which Clavius, at the end of the ninth Element of Euclid,
bringeth to the contrary. And though this were approved by divers good
geometricians then present, and never gainsaid by any since, yet do not
I say it upon the credit of them, but upon sufficient grounds. For it
hath been demonstrated by Eutocius, that _if there be three magnitudes,
the proportion of the first to the third is compounded of the
proportions of the first to the second, and of the second to the third_;
which also I demonstrate in this article. And if there were never so
many magnitudes ranked, it might be likewise demonstrated, that the
proportion of the first to the last is compounded of the proportions of
the first to the second, and of the second to the third, and of the
third to the fourth, and so on to the last. If, therefore, we put in
order any three numbers, whereof the two last be equal, as four, seven,
seven, the proportion of four the first to seven the last, will be
compounded of the proportions of four the first to seven the second, and
of seven the second to seven the third. Wherefore the proportion of
seven to seven (which is of equality) addeth nothing to the proportion
of four the first, to seven the second; and consequently the proportion
of seven to seven hath no quantity; but that the proportion of
inequality hath quantity, I prove it from this, that one inequality may
be greater than another.
But for the clearing of this doctrine (which Mersennus calls intricate)
of the composition of proportions, I observed, first, that any two
quantities, being exposed to sense, their proportion was also exposed;
which is not intricate. Again, I observed that if besides the two
exposed quantities, there were exposed a third, so as the first were the
least, and the third the greatest, or the first the greatest, and the
third the least, that not only the proportions of the first to the
second, but also (because the differences and the quantities proceed the
same way) the proportion of the first to the last is exposed by
composition, or addition of the differences; nor is there any intricacy
in this. But when the first is less than the second, and the second
greater than the third, or the first greater than the second, and the
second less than the third, so that to make the first and second equal,
if we use addition, we must, to make the second and third equal, use
subtraction; then comes in the intricacy, which cannot be extricated,
but by such as know the truth of this doctrine which I now delivered out
of Mersennus, namely, that the proportions of excess, equality, and
defect, are as _quantity_, _not-quantity_, _nothing want quantity_; or
as symbolists mark them 0+1 . 0 . 0-1. And upon this ground I thought
depended the universal truth of this proposition, that in any rank of
magnitudes of the same kind, the proportion of the first to the last,
was compounded of all the proportions (in order) of the intermediate
quantities; the want of the proof thereof, Sir Henry Savile calls
(_nævus_) a mole in the body of geometry. This proposition is
demonstrated at the thirteenth article of this chapter.
But before we come thither, I must examine the arguments you bring to
confute this proposition, that the _proportion of inequality is
quantity, of equality, not quantity_.
And first, you object that equality and inequality are in the same
predicament: a pretty argument to flesh a young scholar in the logic
school, that but now begins to learn the predicaments. But what do you
mean by _æquale_ and _inequale_? Do you mean _corpus æquale_, and
_corpus inequale_? They are both in the predicament of substance,
neither of them in that of quantity. Or do you mean _æqualitas_ and
_inæqualitas_? They are both in the predicament of relation, neither of
them in that of quantity; and yet both _corpus_ and _inæqualitas_,
though neither of them be quantity, may be _quanta_, that is, both of
them have quantity. And when men say body is quantity, or inequality is
quantity, they are no otherwise understood, than if they had said
_corpus est tantum_, and _inæqualitas tanta_, not _tantitas_; that is,
bodies and inequalities are _so much_, not _somuchness_; and all
intelligent men are contented with that expression, and yourselves use
it. And the quantity of inequality is in the predicament of quantity,
because the measure of it is in that line by which one quantity exceeds
the other. But when neither exceedeth the other, then there is no line
of excess, or defect by which the equality can be measured, or said to
be _so much_, or be called quantity. Philosophy teacheth us how to range
our words; but Aristotle’s ranging them in his predicaments doth not
teach philosophy; and therefore no argument taken from thence, can
become a doctor and a professor of geometry.
To prove that the proportion of inequality was quantity, but the
proportion of equality not quantity, my argument was this: that _because
one inequality may be greater or less than another, but one equality
cannot be greater nor less than another: therefore inequality hath
quantity, or is tanta, and equality not_. Here you come in again with
your predicaments, and object, that to be susceptible of _magis_ and
_minus_, belongs not to quantity, but to quality; but without any proof,
as if you took it for an axiom. But whether true or false, you
understand not in what sense it is true or false. It is true that one
inequality is inequality, _as well_ as another; as one heat is heat _as
well_ as another, but not _as great_. _Tam_, but not _tantus_. But so it
is also in the predicament of quantity; one line is as well a line as
another, but not so great. All degrees, intentions, and remissions of
quality, are greater or less quantity of force, and measured by lines,
superficies, or solid quantity, which are properly in the predicament of
quantity. You see how wise a thing it is to argue from the predicaments
of Aristotle, which you understand not; and yet you pretend to be less
addicted to the authority of Aristotle now than heretofore.
In the next place you say, I may as well conclude from the not
susception of _greater_ and _less_, that a right angle is not quantity,
but an oblique one is. Very learnedly. As if to be _greater_ or _less_,
could be attributed to a quantity once determined. Number (that is,
number indefinitively taken) is susceptible of _greater_ and _less_,
because one number may be greater than another; and this is a good
argument to prove that number is quantity. And do you think the argument
the worse for this, that one six cannot be greater than another six?
After all these childish arguments which you have hitherto urged, can
you persuade any man, or yourselves, that you are logicians?
To the fifth and sixth article you object, first, _that if I had before
sufficiently defined_ (ratio) _proportion, I needed not again define
what is_ (eadem ratio) _the same proportion_; and ask me _whether when I
have defined_ man, _I use to define anew what is the_ same man? You
think when you have the definition of _homo_, you have also the
definition of _idem homo_, when it is harder to conceive what _idem_
signifies, than what _homo_. Besides, _idem_ hath not the same
signification always, and with whatsoever it be joined; it doth not
signify the same with _homo_, that it doth with _ratio_. For with _homo_
it signifies the same _individual man_, but with _ratio_ it signifies a
like, or an equal proportion: and both (_ratio_) _proportion_ and
(_idem_) _the same_, being defined, there will still be need of another
definition for (_eadem ratio_) _the same proportion_; and this is enough
to defend both myself and Euclid, against this objection: for Euclid
also, after he had defined (_ratio_) _proportion_, and that
sufficiently, as he believed, yet he defines _the same proportion_ again
apart. I know you did not mean in this place to object anything against
Euclid, but you saw not what you were doing. There is within you some
special cause of intenebration, which you should do well to look to.
In the next place you say, when I had defined arithmetical proportions
to be the same when the difference is the same; it was to be expected I
should define geometrical proportions to be then the same, when the
antecedents are of their consequents _totuple_ or _tantuple_, that is,
equimultiple (for _tantuplum_ signifies nothing). In plain words, you
expected, that as I defined one by subtraction, I should define the
other by the quotient in division. But why should you expect a
definition of the same proportion by the quotient? Neither reason nor
the authority of Euclid could move you to expect it. Or why should you
say _it was to be expected_? But it seems you have the vanity to place
the measure of truth in your own learning. In lines incommensurable
there may be the same proportion, when, nevertheless, there is no
quotient; for setting their symbols one above another doth not make a
quotient: for quotient there is none, but in _aliquot parts_. It is
therefore impossible to define proportion universally, by comparing
quotients. This incommensurability of magnitudes was it that confounded
Euclid in the framing of his definition of proportion at the fifth
Element. For when he came to numbers, he defined the _same proportion_
irreprehensibly thus: _numbers are then proportional, when the first of
the second and the third of the fourth are equimultiple, or the same
part, or the same parts_; and yet there is in this definition no mention
at all of a quotient. For though it be true, that if in dividing two
numbers you make the same quotient, the dividends and the divisors are
proportional, yet that is not the definition of the same proportion, but
a theorem demonstrable from it. But this definition Euclid could not
accommodate to proportion in general, because of incommensurability.
To supply this want, I thought it necessary to seek out some way,
whereby the proportion of two lines, commensurable or incommensurable,
might be continued perpetually the same. And this I found might be done
by the proportion of two lines described by some uniform motion, as by
an efficient cause both of the said lines, and also of their
proportions; which motions continuing, the proportions must needs be all
the way the same. And therefore I defined those magnitudes to have the
same geometrical proportion, _when some cause producing in equal times
equal effects, did determine both the proportions_. This, you say, needs
an Œdipus to make it understood. You are, I see, no Œdipus; but I do not
see any difficulty, neither in the definition nor in the demonstration.
That which you call perplexity in the explication, is your prejudice,
arising from the symbols in your fancy. For men that pretend no less to
natural philosophy than to geometry, to find fault with bringing motion
and time into a definition, when there is no effect in nature which is
not produced in time by motion, is a shame. But you swim upon other
men’s bladders in the superficies of geometry, without being able to
endure diving, which is no fault of mine; and therefore I shall, without
your leave, be bold to say, I am the first that hath made the grounds of
geometry firm and coherent. Whether I have added anything to the edifice
or not, I leave to be judged by the readers. You see, you that profess
with the pricking of bladders the letting out of their vapour, how much
you are deceived. You make them swell more than ever.
For the corollaries that follow this sixth article, you say they contain
nothing new. Which is not true. For the ninth is new, and the
demonstrations of all the rest are new, being grounded upon a new
definition of proportion; and the corollaries themselves, for want of a
good definition of proportion, were never before exactly demonstrated.
For the truth of the sixth definition of the fifth Element of Euclid
cannot be known but by this definition of mine; because it requires a
trial in all numbers possible, that is to say, an infinite time of
trial, whether the quimultiples of the first and third, and of the
second and fourth, in all multiplications, do together exceed, together
come short, and are together equal; which trial is impossible.
In objecting against the thirteenth and sixteenth article, I observe
that you bewray together, both the greatest ignorance and the greatest
malice; and it is well, for they are suitable to one another, and fit
for one and the same man. In the thirteenth article my proposition is
this: _If there be three magnitudes that have proportion one to another,
the proportions of the first to the second, and of the second to the
third, taken together_ (as one proportion), _are equal to the proportion
of the first to the third_. This demonstrated, there is taken away one
of those moles which Sir Henry Savile complaineth of in the body of
geometry. Let us see now what you say, both against the enunciation and
against the demonstration.
Against the enunciation you object, _that other men would say_ (not the
proportions of the first to the second, and of the second to the third,
taken together, &c. but) _the proportion which is compounded of the
proportion of the first to the second, and of the second to the third_,
&c. Is not the compounding of any two things whatsoever the finding of
the sum of them both, or the taking of them together as one total? This
is that absurdity of which Mersennus, in the general preface to his
_Cogitata Physico-Mathematica_, hath convinced Clavius, who, at the end
of Euclid’s ninth Element, denieth the composition of proportion to be a
composition of parts to make a total; which, therefore, he denied,
because he did not observe, that the addition of a proportion of defect
to a proportion of excess, was a subtraction of magnitude; and because
he understood not that to say, composition is not the making a whole of
parts, was contradiction; which all but too learned men would as soon as
they heard abhor. Therefore, in saying that other men would not speak in
that manner, you say in effect they would speak absurdly. You do well to
mark what other geometricians say; but you would do better if you could
by your own meditation upon the things themselves, examine the truth of
what they say. But you have no mind, you say, to contend about the
phrase. Let us see, therefore, what it is you contend about.
_The proportion_, you say, _which is compounded of double and triple
proportion, is not_, as I would have it, _quintuple, but sextuple_, as
in these numbers, six, three, one; where the proportion of six to three
is double, the proportion of three to one triple, and the proportion of
six to one sextuple, not quintuple. Tell me, egregious professors, how
is six to three double proportion? Is six to three the double of a
number, or the double of some proportion? All men know the number six is
double to the number three, and the number three triple to an unity. But
is the question here of compounding numbers, or of compounding
proportions? Euclid, at the last proposition of his ninth Element, says
indeed, that these numbers, one, two, four, eight, are ἐν διπλασίονι
ἀναλογία, in double proportion; yet there is no man that understands it
otherwise, than if he had said in proportion of the single quantity to
the double quantity; and after the same rate, if he had said three,
nine, twenty-seven, &c. had been in triple proportion, all men would
have understood it, of the proportion of any quantity to its triple.
Your instance, therefore, of six, three, one, is here impertinent, there
being in them no doubling, no tripling, no sextupling of proportions,
but of numbers. You may observe also, that Euclid never distinguished
between double and duplicate, as you do. One word διπλάσιον serves him
every where for either. Though, I confess, some curious grammarians take
διπλάσιον for duplicate in number, and διπλοῦν for double in quantity;
which will not serve your turn. Your geometry is not your own, but you
case yourselves with Euclid’s; in which, as I have showed you, there be
some few great holes; and you by misunderstanding him, as in this place,
have made them greater. Though the beasts that think your railing
roaring, have for a time admired you; yet now that through these holes
of your case I have showed them your ears, they will be less affrighted.
But to exemplify the composition of proportions, take these numbers,
thirty-two, eight, one, and then you shall see that the proportion of
thirty-two to one is the sum of the proportions of thirty-two to eight,
and of eight to one. For the proportion of thirty-two to eight is double
the proportion of thirty-two to sixteen; and the proportion of eight to
one, is triple the proportion of thirty-two to sixteen; and the
proportion of thirty-two to one is quintuple of thirty-two to sixteen;
but double and triple added together maketh quintuple. What can be here
denied?
My demonstration consisteth of three cases: the first is when both the
proportions are of defect, which is then when the first quantity is the
least; as in these three quantities, A B, A C, A D. The first case I
demonstrated thus: (A B C D)/(a) Let it be supposed that the point A
were moved uniformly through the whole line A D. The proportions,
therefore, of A B to A C, and of A C to A D, are determined by the
difference of the times in which they are described. And the proportion
also of A B to A D, is that which is determined by the difference of the
times in which they are described; but the difference of the times in
which A B and A C are described, together with the difference of the
times wherein A C and A D are described, is the same with the difference
of the times wherein are described A B and A D. The same cause,
therefore, which determines both the proportions of A B to A C, and of A
C to A D, determines also the proportion of A B to A D. Wherefore, by
the definition of _the same proportion_, article six, the proportion of
A B to A C, together with the proportion of A C to A D, is the same with
the proportion of A B to A D.
Consider now your argumentation against it. “_Let there be taken_,” say
you, “_between A and B the point_ a; and then in your own words, I argue
thus: _The difference of the times wherein are described A B and A C,
together with the difference of the times wherein are described A C and
A D, is the same with the difference of the times in which are
described_ a _B and_ a _C (namely, B D, or B C + C D_); wherefore, the
same cause which determines the two proportions of A B to A C, and of A
C to A D, determines also the proportion of a _B to_ a _D_.” Let me ask
you here whether you suppose the motion from _a_ to B, or from _a_ to D,
to have the same swiftness with the motion from A to B, or from A to D?
If you do not, then you deny the supposition. If you do, then B C, which
is the difference of the times A B and A C, cannot be the difference of
the times in which are described _a_ B and _a_ C, except A B and _a_ B
are equal. Let any man judge now whether there be any paralogism in
Orontius that can equal this. And whether all that follows in the rest
of this, and the next two whole pages, be not all a kind of raving upon
the ignorance of what is the meaning of proportion, which you also make
more ill-favoured by writing it; not in language, but in _gambols_; I
mean in the symbols, which have made you call those demonstrations
short, which put into words so many as a true demonstration requires,
would be longer than any of those of Clavius upon the twelfth Element of
Euclid.
To the sixteenth article you bring no argument, but fall into a loud
_oncethmus_ (the special figure wherewith you grace your oratory),
offended with my unexpected crossing of the doctrine you teach, that
proportion consisteth in a quotient. For that being denied you, your
_a/b - c/d + e/f - g/h + i/k_ comes to nothing, that is, to just as much
as it is worth. But are not you very simple men, to say that all
mathematicians speak so, when it is not speaking? When did you see any
man but yourselves publish his demonstrations by signs not generally
received, except it were not with intention to demonstrate, but to teach
the use of signs? Had Pappus no analytics? or wanted he the wit to
shorten his reckoning by signs? Or has he not proceeded analytically in
a hundred problems (especially in his seventh book), and never used
symbols? Symbols are poor unhandsome, though necessary, scaffolds of
demonstration; and ought no more to appear in public, than the most
deformed necessary business which you do in your chambers. “_But why_,”
say you, “_is this limitation to the proportion of the greater to the
less?_” I will tell you; because iterating of the proportion of the less
to the greater, is a making of the proportion less, and the defect
greater. And it is absurd to say that the taking of the same quantity
twice should make it less. And thence it is, that in quantities which
begin with the less, as one, two, four, the proportion of one to two is
greater than that of one to four, as is demonstrated by Euclid, Elem. 5,
prop. 8; and by consequent the proportion of one to four, is a
proportion of greater littleness than that of one to two. And who is
there, that when he knoweth that the respective greatness of four to
one, is double to that of the respective greatness of four to two, or of
two to one, will not presently acknowledge that the respective greatness
of one to two, or two to four, is double to the respective greatness of
one to four? But this was too deep for such men as take their opinions,
not from weighing, but from reading.
Lastly you object against the corollary of art. 28; which you make
absurd enough by rehearsing it thus: _Si quantitas aliqua divisa
supponatur in partes aliquot æquales numero infinitas_, &c. Do you think
that of _partes aliquot_, or of _partes aliquotæ_, it can be said
without absurdity, that they are _numero infinitæ_? And then you say I
seem to mean, that if of the quantity A B, there be supposed a part C B,
infinitely little; and that between A C and A B be taken two means, one
arithmetical, A E, the other geometrical, A D, the difference between A
D and A E, will be infinitely little. My meaning is, and is sufficiently
expressed, that the said means taken everywhere (not in one place only)
will be the same throughout: and you that say there needed not so much
pains to prove it, and think you do it shorter, prove it not at all. For
why may not I pretend against your demonstration, that B E, the
arithmetical difference, is greater than B D, the geometrical
difference. You bring nothing to prove it; and if you suppose it, you
suppose the thing you are to prove. Hitherto you have proceeded in such
manner with your _Elenchus_, as that so many objections as you have
made, so many false propositions you have advanced. Which is a peculiar
excellence of yours, that for so great a stipend as you receive, you
will give place to no man living for the number and grossness of errors
you teach your scholars.
At the fourteenth chapter your first exception is to the second article;
where I define a plane in this manner: _A plane superficies is that
which is described by a straight line so moved, as that every point
thereof describe a several straight line_. In which you require, first,
that instead of _describe_, I should have said _can describe_. Why do
you not require of Euclid, in the definition of a cone, instead of
_continetur_, _is contained_, he say _contineri potest_, _can be
contained_ ? If I tell you how one plane is generated, cannot you apply
the same generation to any other plane? But you object, that the plane
of a circle may be generated by the motion of the _radius_, whose every
point describeth, not a straight, but a crooked line, wherein you are
deceived; for you cannot draw a circle (though you can draw the
perimeter of a circle) but in a plane already generated. For the motion
of a straight line, whose one point resting, describeth with the other
points several perimeters of circles, may as well describe a conic
superficies, as a plane. The question, therefore, is, how you will, in
your definition, take in the plane which must be generated before you
begin to describe your circle, and before you know what point to make
your centre. This objection, therefore, is to no purpose; and besides,
that it reflecteth upon the perfect definitions of Euclid before the
eleventh Element, it cannot make good his definition (which is nothing
worth) of a plane superficies, before his first Element.
In the next place, you reprehend briefly this _corollary, that two
planes cannot enclose a solid_. I should, indeed, have added, _with the
base on whose extremes they insist_: but this is not a fault to be
ashamed of; for any man, by his own understanding, might have mended my
expression without departing from my meaning. But from your doctrine,
_that a superficies has no thickness_, it is impossible to include a
solid, with any number of planes whatsoever, unless you say that solid
is included which nothing at all includes.
At the third article, where I say _of crooked lines, some are everywhere
crooked, and some have parts not crooked_. You ask me what crooked line
has parts not crooked; and I answer, it is that line which with a
straight line makes a rectilineal triangle. But this, you say, cannot
stand with what I said before, namely, that a straight and crooked line
cannot be coincident; which is true, nor is there any contradiction; for
that part of a crooked line which is straight, may with a straight line
be coincident.
To the fourth article, where I define _the centre of a circle to be that
point of the radius, which in the description of the circle is unmoved_;
you object as a contradiction, that I had before defined a point to be
the body which is moved in the description of a line: foolishly, as I
have already shown at your objection to Chap. VIII. art. 12.
But at the sixth article, where I say, that _crooked and incongruous
lines touch one another but in one point_, you make a cavil from this,
that _a circle may touch a parabola in two points_. Tell me truly, did
you read and understand these words that followed? “_A crooked line
cannot be congruent with a straight line; because if it could, one and
the same line should be both straight and crooked._” If you did, you
could not but understand the sense of my words to be this: _when two
crooked lines which are incongruous, or a crooked and a straight line
touch one another, the contact is not in a line, but only in one point_;
and then your instance of a circle and a parabola was a wilful cavil,
not befitting a doctor. If you either read them not, or understood them
not, it is your own fault. In the rest that followeth upon this article,
with your diagram, there is nothing against me, nor anything of use,
novelty, subtlety, or learning.
At the seventh article, where I define both an _angle_, simply so
called, and an _angle of contingence_, by their several generations;
namely, that the former is generated _when two straight lines are
coincident, and one of them is moved, and distracted from the other by
circular motion upon one common point resting, &c._; you ask me “_to
which of these kinds of angle I refer the angle made by a straight line
when it cuts a crooked line_?” I answer easily and truly, To that kind
of angle which is called simply an angle. This you understand not. “For
how”, will you say, “can that angle which is generated by the divergence
of two straight lines, be other than rectilineal? or how can that angle
which is not comprehended by two straight lines, be other than
curvilineal?” I see what it is that troubles you; namely, the same which
made you say before, that if the body which describes a line be a point,
then there is nothing which is not moved that can be called a point. So
you say here, “If an angle be generated by the motion of a straight
line, then no angle so generated can be curvilineal;” which is as well
argued, as if a man should say, the house was built by the carriage and
motion of stone and timber, therefore, when the carriage and that motion
is ended, it is no more a house. Rectilineal and curvilineal hath
nothing to do with the nature of an angle simply so called, though it be
essential to an angle of contact. The measure of an angle, simply so
called, is a circumference of a circle; and the measure is always the
same kind of quantity with the thing measured. The rectitude or curvity
of the lines, which drawn from the centre, intercept the arch, is
accidentary to the angle, which is the same, whether it be drawn by the
motion circular of a straight line or of a crooked. The diameter and the
circumference of a circle make a right angle, and the same which is made
by the diameter and the tangent. And because the point of contact is
not, as you think, nothing, but a line unreckoned, and common both to
the tangent and the circumference; the same angle computed in the
tangent is rectilineal, but computed in the circumference, not
rectilineal, but mixed: or, if two circles cut one another, curvilineal.
For every chord maketh the same angle with the circumference which it
maketh with the line that toucheth the circumference at the end of the
chord. And, therefore, when I divide an angle, simply so called, into
rectilineal and curvilineal, I respect no more the generation of it,
than when I divide it into right and oblique. I then respect the
generation, when I divide an angle into an angle simply so called, and
an angle of contact. This that I have now said, if the reader remember
when he reads your objections to this, and to the ninth article, he will
need no more to make him see that you are utterly ignorant of the nature
of an angle; and that if ignorance be madness, not I, but you, are mad:
and when an angle is comprehended between a straight and a crooked line
(if I may compute the same angle as comprehended between the same
straight line and the point of contact), that it is consonant to my
definition of a point by a _magnitude not considered_. But when you, in
your treatise, _De Angulo Contactus_ (chap. III. p. 6, l. 8) have these
words: “_Though the whole concurrent lines incline to one another, yet
they form no angle anywhere but in the very point of concourse_:” you,
that deny a point to be anything, tell me how two nothings can form an
angle; or if the angle be not formed, neither before the concurrent
lines meet, nor in the point of concourse, how can you apprehend that
any angle can possibly be framed? But I wonder not at this absurdity;
because this whole treatise of yours is but one absurdity, continued
from the beginning to the end, as shall then appear when I come to
answer your objections to that which I have briefly and fully said of
that subject in my 14th chapter.
At the twelfth article, I confess your exception to my universal
definition of parallels to be just, though insolently set down. For it
is no fault of ignorance (though it also infect the demonstration next
it), but of too much security. The definition is this: _Parallels are
those lines or superficies, upon which two straight lines falling, and
wheresoever they fall, making equal angles with them both, are equal_;
which is not, as it stands, universally true. But inserting these words
_the same way_, and making it stand thus: _parallel lines or
superficies, are those upon which two straight lines falling the same
way, and wheresoever they fall, making equal angles, are equal_, it is
both true and universal; and the following consectary, with very little
change, as you may see in the translation, perspicuously demonstrated.
The same fault occurreth once or twice more; and you triumph
unreasonably, as if you had given therein a very great proof of your
geometry.
The same was observed also upon this place by one of the prime
geometricians of Paris, and noted in a letter to his friend in these
words (Chap. XIV. art. 12): “_The definition of parallels wanteth
somewhat to be supplied_.” And of the consectary he says, “_It
concludeth not, because it is grounded on the definition of parallels_.”
Truly and severely enough, though without any such words as savour of
arrogance, or of malice, or of the clown.
At the thirteenth article you recite the demonstration by which I prove
the perimeters of two circles to be proportional to their semidiameters;
and with _esto_, _fortasse_, _recte_, _omnino_, noddying to the several
parts thereof, you come at length to my last inference: _Therefore, by_
Chap. XIII. art. 6, _the perimeters and semidiameters of circles are
proportional_; which you deny; and therefore deny, because you say it
followeth by the same ratiocination, that _circles also and spheres are
proportional to their semidiameters_. “_For the same distance_, you say,
_of the perimeter from the centre which determines the magnitude of the
semidiameter, determines also the magnitude both of the circle and of
the sphere_.” You acknowledge that perimeters and semidiameters have the
cause of their determination such as in equal times make equal spaces.
Suppose now a sphere generated by the semidiameters, whilst the
semicircle is turned about. There is but one _radius_ of an infinite
number of _radii_, which describes a great circle; all the rest describe
lesser circles parallel to it, in one and the same time of revolution.
Would you have men believe, that describing greater and lesser circles,
is according to the supposition (_temporibus æqualibus æqualia facere_)
to make equal spaces in equal times? Or, when by the turning about of
the semidiameter is described the plane of a circle, does it, think you,
in equal times make the planes of the interior circles equal to the
planes of the exterior? Or is the _radius_ that describes the inner
circles equal to the _radius_ that describes the exterior? It does not,
therefore, follow from anything I have said in this demonstration, that
either spheres or planes of circles, are proportional to their _radii_;
and consequently, all that you have said, triumphing in your own
incapacity, is said imprudently by yourselves to your own disgrace. They
that have applauded you, have reason by this time to doubt of all the
rest that follows, and if they can, to dissemble the opinion they had
before of your geometry. But they shall see before I have done, that not
only your whole _Elenchus_ , but also your other books of the _Angle of
Contact_ , &c. are mere ignorance and gibberish.
To the fourteenth article you object, that (in the sixth figure) I
assume gratis, that F G, D E, B C, are proportional to A F, A D, A B;
and you refer it to be judged by the reader: and to the reader I refer
it also. The not exact drawing of the figure (which is now amended) is
it that deceived you. For A F, F D, D B, are equal by construction.
Also, A G, G E, E C, are equal by construction. And F G, D K, B H, K E,
H I, I C, are equal by parallelism. And because A F, F G, are as the
velocities wherewith they are described; also 2 A F (that is A D) and 2
F G (that is D E) are as the same velocities. And finally, 3 A F (that
is A B) and 3 F G (that is B C) are as the same velocities. It is not
therefore assumed gratis, that F G, D E, B C are proportional to A F, A
D, A B, but grounded upon the sixth article of the thirteenth chapter;
and consequently your objection is nothing worth. You might better have
excepted to the placing of D E, first at adventure, and then making A D
two-thirds of A B; for that was a fault, though not great enough to
trouble a candid reader; yet great enough to be a ground, to a malicious
reader, of a cavil.
That which you object to the third _corollary_ of art. 15, was certainly
a dream. There is no assuming of an angle C D E, for an angle H D E, or
B D E, neither in the demonstration, nor in any of the corollaries. It
may be you dreamt of somewhat in the twentieth article of chapter XVI.
But because that article, though once printed, was afterwards left out,
as not serving to the use I had designed it for, I cannot guess what it
is: for I have no copy of that article, neither printed nor written; but
am very sure, though it were not useful, it was true.
Article the sixteenth. Here we come to the controversy concerning the
_angle of contact, which_, you say, _you have handled, in a special
treatise published; and that you have clearly demonstrated, in your
public lectures, that Peletarius was in the right. But that I agree not
sufficiently, neither with Peletarius nor with Clavius._ I confess I
agree not in all points with Peletarius, nor in all points with Clavius.
It does not thence follow that I agree not with the truth. I am not, as
you, of any faction, neither in geometry nor in politics. If I think
that you, or Peletarius, or Clavius, or Euclid, have erred, or been too
obscure, I see no cause for which I ought to dissemble it. And in this
same question I am of opinion that Peletarius did not well in denying
the _angle of contingence_ to be _an angle_. And that Clavius did not
well to say, _the angle of a semicircle_ was less than _a right-lined
right angle_. And that Euclid did not well to leave it so obscure what
he meant by _inclination_ in the definition of a _plane angle_, seeing
elsewhere he attributeth inclination only to acute angles; and scarce
any man ever acknowledged inclination in a straight line, to any other
line to which it was perpendicular. But you, in this question of what is
inclination, though you pretend not to depart from Euclid, are,
nevertheless, more obscure than he; and also are contrary to him. For
Euclid by inclination meaneth the inclination of one line _to_ another;
and you understand it of the inclination of one line _from_ another;
which is not inclination, but declination. For you make two straight
lines, when they lie one on another, to lie ἁκλινῶς, that is, without
any inclination (because it serves your turn); not observing that it
followeth thence, that inclination is a digression of one line _from_
another. This is in your first argument in the behalf of Peletarius (p.
10, l. 22), and destroys his opinion. For, according to Euclid, the
greatest angle is the greatest inclination; and an angle equal to two
right angles by this ἀκλισία, should not be the greatest inclination, as
it is, but the least that can be. But if by the inclination of two
lines, we understand that proceeding of them to a common point, which is
caused by their generation, which, I believe, was Euclid’s meaning; then
will the _angle of contact_ be no less an _angle_ than a _rectilineal_
angle, but only (as Clavius truly says it is) heterogeneous to it; and
the doctrine of Clavius more conformable to Euclid than that of
Peletarius. Besides, if it be granted you, that there is no inclination
of the circumference to the tangent, yet it does not follow that their
concourse doth not form some kind of angle; for Euclid defineth there
but one of the kinds of a plane angle. And then you may as much in vain
seek for the proportion of such angle to the angle of contact, as seek
for the _focus_ or _parameter of the parabola of Dives and Lazarus_.
Your first argument therefore is nothing worth, except you make good
that which in your second argument you affirm, namely, that all plane
angles, not excepting the angle of contact, are (_homogeneous_) of the
same kind. You prove it well enough of other curvilineal angles; but
when you should prove the same of an angle of contact, you have nothing
to say but (p. 17, l. 15), “_Unde autem illa quam somniet heterogenia
oriatur, neque potest ille ullatenus ostendere, neque ego vel
somniare_:” “_Whence should arise that diversity of kind which he dreams
of, neither can he at all show, nor I dream_;” as if you knew what he
could do if he were to answer you; or all were false which you cannot
dream of. So that besides your customary vanity, here is nothing
hitherto proved, neither for the opinion of Peletarius, nor against that
of Clavius. I have, I think, sufficiently explicated, in the first
lesson, that the angle of contact is quantity, namely, that it is the
quantity of that crookedness or flexion, by which a straight line is
bent into an arch of a circle equal to it; and that because the
crookedness of one arch may be greater than the crookedness of another
arch of another circle equal to it; therefore the question _quanta est
curvitas_, how much is the crookedness, is pertinent, and to be answered
by _quantity_. And I have also shown you in the same lesson, that the
quantity of one angle of contact is compared with that of another angle
of contact by a line drawn from the point of contact, and intercepted by
their circumferences; and that it cannot be compared by any measure with
a rectilineal angle.
[Illustration]
But let us see how you answer to that which Clavius has objected
already. “_They are heterogeneous_,” says he, “_because the angle of
contact, how oft soever multiplied, can never exceed a rectilineal
angle_.” To answer which, you allege _it is no angle at all; and that
therefore, it is no angle at all, because the lines have no inclination
one to another_. How can lines that have no inclination one to another,
ever come together? But you answer, _at least they have no inclination
in the point of contact_. And why have two straight lines inclination
before they come to touch, more than a straight line and an arch of a
circle? And in the point of contact itself, how can it be that there is
less inclination of the two points of a straight line and an arch of a
circle, than of the points of two straight lines? But the straight
lines, you say, will cut; which is nothing to the question; and yet this
also is not so evident, but that it may receive an objection. Suppose
two circles, A G B and C F B, to touch in B, and have a common tangent
through B. Is not the line C F B G A a crooked line? and is it not cut
by the common tangent D B E? What is the quantity of the two angles F B
E and G B D, seeing you say neither D B G nor E B F is an angle? It is
not, therefore, the cutting of a crooked line, and the touching of it,
that distinguisheth an angle simply, from an angle of contact. That
which makes them differ, and in kind, is, that the one is the quantity
of a _revolution_, and the other, the quantity of _flexion_.
In the seventh chapter of the same treatise, you think you prove the
angle of contact, if it be an angle, and a rectilineal angle to be
(_homogeneous_) of the same kind; when you prove nothing but that you
understand not what you say. Those quantities which can be added
together, or subtracted one from another, are of the same kind; but an
angle of contact may be subtracted from a right angle, and the remainder
will be the angle of a semicircle, &c. So you say, but prove it not,
unless you think a man must grant you that the superficies contained
between the tangent and the arch, which is it you subtract, is the angle
of contact; and that the plane of the semicircle is the angle of the
semicircle, which is absurd; though, as absurd as it is, you say it
directly in your _Elenchus_ , p. 35, l. 14, in these words: “_When
Euclid defines a plane angle to be the inclination of two lines, he
meaneth not their aggregate, but that which lies between them_.” It is
true, he meaneth not the aggregate of the two lines; but that he means
that which lies between them, which is nothing else but an indeterminate
superficies, is false, or Euclid was as foolish a geometrician as either
of you two.
Again, you would prove the angle of contact, if it be an angle, to be of
the same kind with a rectilineal angle, out of Euclid (III. 16); where
he says, _it is less than any acute angle_. And it follows well, that if
it be an angle, and less than any rectilineal angle, it is also of the
same kind with it. But, to my understanding, Euclid meant no more, but
that it was neither greater nor equal; which is as truly said of
heterogeneous, as of homogeneous quantities. If he meant otherwise, he
confirms the opinion of Clavius against you, or makes the quantity of an
angle to be a superficies, and indefinite. But I wonder how you dare
venture to determine whether two quantities be homogeneous or not,
without some definition of homogeneous (which is a hard word), that men
may understand what it meaneth.
In your eighth chapter you have nothing but Sir H. Savile’s authority,
who had not then resolved what to hold; but esteeming the angle of
contact, first, as others falsely did, by the superficies that lies
between the tangent and the arch, makes the angle of contact and a
rectilineal angle homogeneous; and afterwards, because no multiplication
of the angle of contact can make it equal to the least rectilineal
angle, with great ingenuity returneth to his former uncertainty.
In your ninth and tenth chapters you prove with much ado, that the
angles of like segments are equal; as if that might not have been taken
gratis by Peletarius, without demonstration. And yet your argument,
contained in the ninth chapter, is not a demonstration, but a
conjectural discourse upon the word _similitude_. And in the eleventh
chapter, wherein you answer to an objection, which might be made to your
argument in the precedent page, taken from the parallelism of two
concentric circles, though objection be of no moment, yet you have in
the same treatise of yours that which is much more foolish, which is
this, (p. 38, l. 12): “_Non enim magnitudo anguli_,” _&c._ _“_The_
magnitude of an angle is not to be estimated by that straddling of the
legs, which it hath without the point of concourse, but by that
straddling which it hath in the point of the concourse itself._” I pray
you tell me what straddling there is of two coincident points,
especially such points as you say are nothing? When did you ever see two
nothings straddle?
The arguments in your twelfth and thirteenth chapters are grounded all
on this untruth, that an angle is that which is contained between the
lines that make it; that is to say, is a plane superficies, which is
manifestly false; because the measure of an angle is an arch of a
circle, that is to say, a line; which is no measure of a superficies.
Besides this gross ignorance, your way of demonstration, by putting N
for a great number of sides of an equilateral polygon, is not to be
admitted; for, though you understand something by it, you demonstrate
nothing to anybody but those who understand your symbolic tongue, which
is a very narrow language. If you had demonstrated it in Irish or Welsh,
though I had not read it, yet I should not have blamed you, because you
had written to a considerable number of mankind, which now you do not.
In your last chapters you defend Vitellio without need; for there is no
doubt but that whatsoever crooked line be touched by a straight line,
the angle of contingence will neither add anything to, nor take anything
from, a rectilineal right angle; but that it is because the angle of
contact is no angle, or no quantity, is not true. For it is therefore an
angle, because an angle of contact; and therefore quantity, because one
angle of contact may be greater than another; and therefore
heterogeneal, because the measure of an angle of contact cannot
(_congruere_) be applied to the measure of a rectilineal angle, as they
think it may, who affirm with you that the nature of an angle consisteth
in that which is contained between the lines that comprehend it, viz.,
in a plane superficies. And thus you see in how few lines, and without
brachygraphy, your treatise of the angle of contingence is discovered
for the greatest part to be false, and for the rest, nothing but a
detection of some errors of Clavius grounded on the same false
principles with your own. To return now from your treatise of the angle
of contact back again to your _Elenchus_ .
The fault you find at art. 18, is, that I understand not that Euclid
makes a _plane angle_ to be that which is contained between the two
lines that form it. It is true, that I do not understand that Euclid was
so absurd, as to think the nature of an angle to consist in superficies;
but I understand that you have not had the wit to understand Euclid.
The nineteenth article of mine in this fourteenth chapter, is this:
“_All respect or variety of position of two lines, seemeth to be
comprehended in four kinds_. For they are either _parallel_, or (_being
if need be produced_) _make an angle_; or, (if drawn out far enough)
_touch_; or, lastly, they are _asymptotes_”; in which you are first
offended with the word _It seems_. But I allow you, that never err, to
be more peremptory than I am. For to me it seemed (I say again seemed)
that such a phrase, in case I should leave out something in the
enumeration of the several kinds of position, would save me from being
censured for untruth; and yet your instance of two straight lines in
divers planes, does not make my enumeration insufficient. For those
lines, though not parallels, nor cutting both the planes, yet being
moved parallelly from one plane to another, will fall into one or other
of the kinds of position by me enumerated; and consequently, are as much
that position, as two straight lines in the same plane, not parallel,
make the same angle, though not produced till they meet, which they
would make if they were so produced: for you have nowhere proved, nor
can prove, that two such lines do not make an angle. It is not the
actual concurrence of the lines, but the arch of a circle, drawn upon
that point for centre, in which they would meet if they were produced,
and intercepted between them, that constitutes the angle.
Also your objection concerning asymptotes _in general_ is absurd. You
would have me add, that _their distance shall at last be less than any
distance that can be assigned_; and so make the definition of the
_genus_ the same with that of the _species_. But because you are not
professors of logic, it is not necessary for me to follow your counsel.
In like manner, if we understand one line to be moved towards another
always parallelly to itself, which is, though not actually, yet
potentially the same position, all the rest of your instances will come
to nothing.
At the two-and-twentieth article you object to me the use of the word
_figure_, before I had defined it: wherein also you do absurdly; for I
have nowhere before made such use of the word _figure_, as to argue
anything from it; and therefore your objection is just as wise as if you
had found fault with putting the word figure in the titles of the
chapters placed before the book. If you had known the nature of
demonstration, you had not objected this.
You add further, that by my definition of _figure_, a solid sphere, and
a sphere made hollow within, is the same figure; but you say not why,
nor can you derive any such thing from my definition. That which
deceived your shallowness, is, that you take those points that are in
the concave superficies of a hollowed sphere, not to be contiguous to
anything without it, because that whole concave superficies is within
the whole sphere. Lastly, for the fault you find with the definition of
_like figures in like positions_, I confess there wants the same word
which was wanting in the definition of parallels; namely, _ad easdem
partes_ (_the same way_) which should have been added in the end of the
definition of like figures, &c., and may easily be supplied by any
student of geometry, that is not otherwise a fool.
At the fifteenth chapter, art. 1, number 6, you object as a
contradiction, that _I make motion to be the measure of time; and yet,
in other places, do usually measure motion and the affections thereof by
time_. If your thoughts were your own, and not taken rashly out of
books, you could not but, (with all men else that see time measured by
clocks, dials, hour-glasses, and the like), have conceived sufficiently,
that there cannot be of time any other measure besides motion; and that
the most universal measure of motion, is a line described by some other
motion; which line being once exposed to sense, and the motion whereby
it was described sufficiently explicated, will serve to measure all
other motions and their time: for time and motion (time being but the
mental image or remembrance of the motion) have but one and the same
dimension, which is a line. But you, that would have me measure
_swiftness_ and _slowness_ by longer and shorter motion, what do you
mean by _longer_ and _shorter motion_? Is _longer_ and _shorter_ in the
motion, or in the duration of the motion, which is time? Or is the
motion, or the duration of the motion, that which is exposed, or
designed by a line? Geometricians say often, _let the line A B be the
time_; but never say, _let the line A B be the motion_. There is no
unlearned man that understandeth not what is time, and motion, and
measure; and yet you, that undertake to teach it (most egregious
professors) understand it not.
At the second article you bring another argument (which it seems in its
proper place you had forgotten), to prove that a point is not quantity
not considered, but absolutely nothing; which is this, _That if a point
be not nothing, then the whole is greater than its two halves_. How does
that follow? Is it impossible when a line is divided into two halves,
that the middle point should be divided into two halves also, being
quantity?
At the seventh article, I have sufficiently demonstrated, that all
motion is infinitely propagated, as far as space is filled with body.
You allege no fault in the demonstration, but object from sense, _that
the skipping of a flea is not propagated to the Indies_. If I ask you
how you know it, you may wonder perhaps, but answer you cannot. Are you
philosophers, or geometricians, or logicians, more than are the simplest
of rural people? or are you not rather less, by as much as he that
standeth still in ignorance, is nearer to knowledge, than he that
runneth from it by erroneous learning?
And, lastly, what an absurd objection is it which you make to the eighth
article, where I say that _when two bodies of equal magnitude fall upon
a third body, that which falls with greater velocity, imprints the
greater motion_? You object, _that not so much the magnitude is to be
considered as the weight_; as if the weight made no difference in the
velocity, when notwithstanding weight is nothing else but motion
downward. Tell me, when a weighty body thrown upwards worketh on the
body it meeteth with, do you not then think it worketh the more for the
greatness, and the less for the weight.
==========
OF THE FAULTS THAT OCCUR IN
DEMONSTRATION.
TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.
LESSON IV.
Of twenty articles which you say (of nineteen which I say) make the
sixteenth chapter, you except but three, and confidently affirm the rest
are false. On the contrary, except three or four faults, such as any
geometrician may see proceed not from ignorance of the subject, or from
want of the art of demonstration, (and such as any man might have mended
of himself) but from security; I affirm that they are all true, and
truly demonstrated; and that all your objections proceed from mere
ignorance of the mathematics.
The first fault you find is this, that I express not (art. 1) what
_impetus_ it is, which I would have to be multiplied into the time.
The last article of my thirteenth chapter was this, “_If there be a
number of quantities propounded, howsoever equal or unequal to one
another; and there be another quantity which so often taken as there be
quantities propounded, is equal to their whole sum; that quantity I call
the mean arithmetical of them all_.” Which definition I did there insert
to serve me in the explication of those propositions of which the
sixteenth chapter consisted, but did not use it here as I intended. My
first proposition therefore as it standeth yet in the Latin, being this,
“_the velocity of any body moved during any time, is so much as is the
product of the impetus in one point of time, multiplied into the whole
time_;” to a man that hath not skill enough to supply what is wanting,
is not intelligible. Therefore I have caused it in the English to go
thus: “_the velocity of any body in whatsoever time moved, hath its
quantity determined by the sum of all the several_ (impetus)
_quicknesses, which it hath in the several points of the time of the
body’s motion_. And added, _that all the_ impetus _together taken
through the whole time is the same thing with the mean_ impetus (which
mean is defined (Chapter XIII. art. 29) _multiplied into the whole
time_.” To this first article, as it is uncorrected in the Latin, you
object, _that meaning by_ impetus _some middle_ impetus, _and assigning
none, I determine nothing_. And it is true. But if you had been
geometricians sufficient to be professors, you would have shewed your
skill much better, by making it appear that this middle _impetus_ could
be none but that, which being taken so often, as there be points in the
line of time, would be equal to the sum of all the several _impetus_
taken in the points of time respectively; which you could not do.
To the _corollary_, you ask first how _impetus_ can be ordinately
applied to a line; absurdly. For does not Archimedes sometimes say, and
with him many other excellent geometricians, _let such a line be the
time_? And do they not mean, that that line, or the motion over it, is
the measure of the time? And may not also a line serve to measure the
swiftness of a motion? _You thought_, you say, _only lines ought to be
said to be ordinately applied to lines_. Which I easily believe; for I
see you understand not that a line, though it be not the time itself,
may be the quantity of a time. You thought also, all you have said in
your _Elenchus_ , in your doctrine of the _angle of contact_, in your
_Arithmetica Infinitorum_, and in your _Conics_ , is true; and yet it is
almost all proved false, and the rest nothing worth.
Secondly, you object, that _I design a parallelogram by one only side_.
It was indeed a great oversight, and argueth somewhat against the man,
but nothing against his art. For he is not worthy to be thought a
geometrician that cannot supply such a fault as that, and correct his
book himself. Though you could not do it, yet another from beyond sea
took notice of the same fault in this manner, “_He maketh a
parallelogram of but one side_; it should be thus: _vel denique per
parallelogrammum cujus unum latus est medium proportionale inter impetum
maximum (sive ultimo acquisitum) et impetus ejusdem maximi semissem;
alterum vero latus, medium proportionale, inter totum tempus, et ejusdem
totius temporis semissem_.” Which I therefore repeat, that you may learn
good manners; and know, that they who reprehend, ought also, when they
can, to add to their reprehension the correction.
At the second article, you are pleased to advise me, instead of _in omni
motu uniformi_, to put in _in omnibus motibus uniformibus_. You have a
strange opinion of your own judgment, to think you know to what end
another man useth any word, better than himself. My intention was only
to consider motions uniform, and motions from rest uniformly, or
regularly accelerated, that I might thereby compute the lengths of
crooked lines, such as are described by any of those motions. And
therefore it was enough to prove this theorem to be true in all uniform
or uniformly accelerated _motion_, not _motions_; though it be true also
in the plural. It seems you think a man must write all he knows, whether
it conduce, or not, to his intended purpose. But that you may know that
I was not (as you think), ignorant how far it might be extended, you may
read it demonstrated at the same article in the English universally.
Against the demonstration itself you run into another article, namely,
the thirteenth, which is this problem: “_the length being given, which
is passed over in a given time by uniform motion, to find the length
which shall be passed over by motion uniformly accelerated in the same
time, so as that the_ impetus _last acquired be equal to the time_.”
Which you recite imperfectly, thereby to make it seem that such a length
is not determined. Whether you did this out of ignorance, or on purpose,
thinking it a piece of wit, as your pretended mystery which goes
immediately before, I cannot tell, for in neither place can any wit be
espied by any but yourselves. To imagine motions with their times and
ways, is a new business, and requires a steady brain, and a man that can
constantly read in his own thoughts, without being diverted by the noise
of words. The want of this ability, made you stumble and fall
unhandsomely in the very first place (that is in Chap. XIII. art. 13),
where you venture to reckon both motion and time at once; and hath made
you in this chapter to stumble in the like manner at every step you go.
As, for example, when I say, _as the product of the time, and impetus,
to the product of the time and impetus, so the space to the space when
the motion is uniform_; you come in with, _nay, rather as the time to
the time_; as if the parallelograms A I, and A H, were not also as the
times A B, and A F. Thus it is, when men venture upon ways they never
had been in before, without a guide.
In the corollary, you are offended with the permutation of the
proportion of times and lines, because you think, (you that have scarce
one right thought of the principles of geometry), that line and time are
heterogeneous quantities. I know time and line are of divers natures;
and more, that neither of them is _quantity_. Yet they may be both of
them _quanta_, that is, they may _have quantity_; but that their
quantities are heterogeneous is false. For they are compared and
measured both of them by straight lines. And to this there is nothing
contrary in the place cited by you out of Clavius; or if there were, it
were not to be valued. And to your question, what is the proportion of
an _hour_ to an _ell_? I answer, it is the same proportion that _two
hours_ have to _two ells_. You see your question is not so subtle as you
thought it. By and bye you confess that in times and lines there is
_quid homogeneum_ (this _quid_ is an infallible sign of not fully
understanding what you say); which is false if you take it of the lines
themselves; though if you take it of their quantities, it is true
without a _quid_. Lastly, you tell m”e how I might have expressed myself
so as it might have been true. But because your expressions please me
not, I have not followed your advice.
To the third article, which is this: “_In motu uniformiter a quiete
accelerato_,” _etc._ “_In motion uniformly accelerated from rest, that
is, when the impetus increaseth in proportion to the times, the length
run over in one time is to the length run over in another time, as the
product of the impetus multiplied by the time, to the product of the
impetus multiplied by the time_;” you object, “_that the lengths run
over are in that proportion which the impetus hath to the impetus; not
that which the impetus hath to the time, because impetus to time has no
proportion, as being heterogeneous_.” First, when you say the impetus,
do you mean some one impetus designed by some one of the unequal
straight lines parallel to the base B I? That is manifestly false. You
mean the aggregate of all those unequal parallels. But that is the same
thing with the time multiplied into the mean impetus. And so you say the
same that I do. Again, I ask, where it is that I say or dream that the
lengths run over are in the proportion of the impetus to the times? Is
it you or I that dream? And for your heterogeneity of the quantities of
time and of swiftness, I have already in divers places showed you your
error. Again, why do you make B I represent the lengths run over, which
I make to be represented by D E, a line taken at pleasure; and you also
a few lines before make the same B I to design the greatest acquired
impetus? These are things which show that you are puzzled and entangled
with the unusual speculation of time and motion, and yet are thrust on
with pride and spite to adventure upon the examination of this chapter.
Secondly, you grant the demonstration to be good, supposing I mean it,
as I seem to speak, of one and the same motion. But why do I not mean it
of one and the same motion, when I say not in _motions_, but in _motion_
uniform? _Because_, say you, _in that which follows, I draw it also to
different motions_. You should have given at least one instance of it;
but there is no such matter. And yet the proposition is in that case
also true; though then it must not be demonstrated by the similitude of
triangles, as in the case present. And therefore the objections you make
from different impetus acquired in the same time, and from other cases
which you mention, are nothing worth.
At the fourth article, you allow the demonstration all the way (except
the faults of the third, which I have already proved to be none) till I
come to say, “_that because the proportion of F K to B I is double to
the proportion of A F to A B, therefore the proportion of A B to A F is
double to the proportion of B I to F K_.” This you deny, and wonder at
as strange, (for it is indeed strange to you), and in many places you
exclaim against it as extreme ignorance in geometry. In this place you
only say, “_no such matter; for though one proportion be double to
another, yet it does not follow that the converse is the double of the
converse_.” So that this is the issue to which the question is reduced,
whether you have any or no geometry. I say, if there be three quantities
in continual proportion, and the first be the least, the proportion of
the first to the second is double to the proportion of the first to the
third; and you deny it. The reason of our dissent consisteth in this,
that you think the doubling of a proportion to be the doubling of the
quantity of the proportion, as well in proportions of defect, as in
proportions of excess; and I think that the doubling of a proportion of
defect, is the doubling of the defect of the quantity of the same. As
for example in these three numbers, 1, 2, 4, which are in continual
proportion, I say the quantity of the proportion of one to two, is
double the quantity of the proportion of one to four. And the quantity
of the proportion of one to four, is half the quantity of the proportion
of one to two. And yet deny not but that the quantity of the defect in
the proportion of one to two is doubled in the proportion of one to
four. But because the doubling of defect makes greater defect, it maketh
the quantity of the proportion less. And as for the part which I hold in
this question, first, there is thus much demonstrated by Euclid, El. v.
prop. 8; that the proportion of one to two, is greater than the
proportion of one to four, though how much it is greater be not there
demonstrated. Secondly, I have demonstrated (Chap, XIII. art. 16); that
it is twice as great, that is to say, (to a man that speaks English),
double. The introducing of _duplicate_, _triplicate_, &c. instead of
_double_, _triple_, &c. (though now they be words well understood by
such as understand what proportion is), proceeded at first from such as
durst not for fear of absurdity, call the half of any thing double to
the whole, though it be manifest that the half of any defect is a double
quantity to the whole defect; for want added to want maketh greater
want, that is, a less positive quantity. This difference between
_double_ and _duplicate_, lighting upon weak understandings, has put men
out of the way of true reasoning in very many questions of geometry.
Euclid never used but one word both for _double_ and _duplicate_. It is
the same fault when men call half a quantity _subduplicate_, and a third
part _subtriplicate_ of the whole, with intention (as in this case) to
make them pass for words of signification different from the _half_ and
the _third part_. Besides, from my definition of proportion (which is
clear, and easy to be understood by all men, but such as have read the
geometry of others unluckily) I can demonstrate the same evidently and
briefly thus. My definition is this, _proportion is the quantity of one
magnitude taken comparatively to another_. Let there be therefore three
quantities, 1, 2, 4, in continual proportion. Seeing therefore the
quantity of four in respect of one, is twice as great as the quantity of
the same four in respect of 2, it followeth manifestly that the quantity
of 1 in respect of 4, is twice as little as the quantity of the same 1
in respect of 2; and consequently the quantity of 1 in respect of 2, is
twice as great as the quantity of the same 1 in respect of 4; which is
the thing I maintain in this question. Would not a man that employs his
time at bowls, choose rather to have the advantage given him of three in
nine, than of one in nine? And why, but that three is a greater quantity
in respect of nine, than is one? Which is as much as to say, three to
nine hath a greater proportion than one to nine; as is demonstrated by
Euclid, El. v. prop. 8. Is it not therefore (you that profess
mathematics, and theology, and practise the depression of the truth in
both) well owled of you, to teach the contrary? But where you say “_that
the point K_ (in the second figure of the table belonging to this
sixteenth chapter) _is not in the parabolical line whose diameter is A
B, and base B I, but in the parabolical line of the complement of my
semiparabola_ (_as I may learn from the twenty-third proposition of
your_ Arithmetica Infinitorum) _whose diameter is A C, and base I C_.”
What line is that? Is it the same line with that of my semiparabola, or
not the same? If the same, why find you fault? If not the same, you
ought to have made a semiparabola on the diameter A C, and base I C, and
following my construction made it appear that K is not in the line
wherein I say it is; which you have not done, nor could do.
Then again, running on in the same blindness of passion, you pretend I
make the proportion of B I to F K double to that of A B to A F, and then
confute it; when you knew I made the proportion of F K to B I, double to
that of F N, to B I, that is, of A F to A B; and this was it you should
have confuted. That which followeth is but a triumphing in your own
ignorance, wherein you also say, “_that all that I afterwards build upon
this doctrine is false_.” You see whether it be like to prove so or not.
As for your _Arithmetica Infinitorum_, I shall then read to you a piece
of a lesson on it when I come to your objections against the next
Chapter. In the mean time let me tell you, it is not likely you should
be great geometricians, that know not what is quantity, nor measure, nor
straight, nor angle, nor homogeneous, nor heterogeneous, nor proportion,
as I have already made appear in this and the former lessons.
To the first corollary of this fourth article your exception I confess
is just, and (which I wonder at) without any incivility. But this argues
not ignorance, but security. For who is there that ever read any thing
in the Conics, that knows not that the parts of a parabola cut off by
lines parallel to the base, are in triplicate proportion to their bases?
But having hitherto designed the time by the diameter, and the impetus
by the base; and in the next chapter (where I was to calculate the
proportion of the parabola, to the parallelogram) intending to design
the time by the base, I mistook and put the diameter again for the time;
which any man but you might as easily have corrected as reprehended.
To the second corollary, which is this, _that the lengths run over in
equal times by motion so accelerated, as that the impetus increase in
double proportion to their times, are as the differences of the cubic
numbers beginning at unity, that is, as seven, nineteen, thirty-seven,
&c._ you say it is false. But why? “_Because_” say you “_portions of the
parabola of equal altitude, taken from the beginning, are not as those
numbers seven, nineteen, thirty-seven, &c._” Does this, think you,
contradict any thing in this proposition of mine? Yes, because, you
think, the lengths gone over in equal times, are the same with the parts
of the diameter cut off from the vertex, and proportional to the numbers
one, two, three, &c. Whereas the lengths run over, are as the aggregates
of their velocities, that is, as the parts of the parabola itself, that
is, as the cubes of their bases, that is, as the numbers one, eight,
twenty-seven, sixty-four, &c., and consequently the lengths run over in
equal times, are as the differences of those cubic numbers, one, eight,
twenty-seven, sixty-four, whose differences are seven, nineteen,
thirty-seven, &c. The cause of your mistake was, that you cannot yet,
nor perhaps ever will, contemplate time and motion (which requireth a
steady brain) without confusion.
The third corollary you also say is false, “_whether it be meant of
motion uniformly accelerated_ (as the words are) _or_ (_as perhaps_, you
say, _I meant it_) _of such motion as is accelerated in double
proportion to the time_.” You need not say perhaps I meant it. The words
of the proposition are enough to make the meaning of the corollary
understood. But so also you say it is false. Methinks you should have
offered some little proof to make it seem so. You think your authority
will carry it. But on the contrary I believe rather that they that shall
see how your other objections hitherto have sped, will the rather think
it true, because you think it false. The demonstration as it is, is
evident enough; and therefore I saw no cause to change a word of it.
To the fifth article you object nothing, but that it dependeth on this
proposition (Chap. XIII. art. 16): “_That when three quantities are in
continual proportion, and the first is the least, as in these numbers,
four, six, nine, the proportion of the first to the second, is double to
the proportion of the same first to the last_;” which is there
demonstrated, and in the former lessons so amply explicated, as no man
can make any further doubt of the truth of it. And you will, I doubt
not, assent unto it. But in what estate of mind will you be then? A man
of a tender forehead after so much insolence, and so much contumelious
language grounded upon arrogance and ignorance, would hardly endure to
outlive it. In this vanity of yours, you ask me whether I be angry, or
blush, or can endure to hear you. I have some reason to be angry; for
what man can be so patient as not to be moved with so many injuries? And
I have some reason to blush, considering the opinion men will have
beyond sea, (when they shall see this in Latin) of the geometry taught
in Oxford. But to read the worst you can say against me, I can endure,
as easily at least, as to read any thing you have written in your
treatises of the _Angle of Contact_, of the _Conic Sections_, or your
_Arithmetica Infinitorum_.
The sixth, seventh, eighth articles, you say are sound. True. But never
the more to be thought so for your approbation, but the less; because
you are not fit, neither to reprehend, nor praise; and because all that
you have hitherto condemned as false, hath been proved true. Then you
show me how you could demonstrate the sixth and seventh articles a
shorter way. But though there be your symbols, yet no man is obliged to
take them for demonstration. And though they be granted to be dumb
demonstrations, yet when they are taught to speak as they ought to do,
they will be longer demonstrations than these of mine.
To the ninth article, which is this, “_If a body be moved by two movents
at once, concurring in what angle soever, of which, one is moved
uniformly, the other, with motion uniformly accelerated from rest, till
it acquire an impetus equal to that of the uniform motion, the line in
which the body is carried, shall be the crooked line of a
semiparabola_,” you lift up your voice again, and ask, _what latitude?
what diameter? what inclination_ of the diameter to the ordinate lines?
If your founder should see this, or the like objections of yours, he
would think his money ill bestowed. When I say, _in what angle soever_,
you ask, _in what angle?_ When I say _two movents, one uniform, the
other uniformly accelerated, make the body describe a semiparabolical
line_; you ask, _which is the diameter?_ as not knowing that the
accelerated motion describes the diameter, and the other a parallel to
the base. And when I say _the two movents meet in a point, from which
point both the motions begin, and one of them from rest_, you ask me
_what is the altitude?_ As if that point where the motion begins from
rest were not the vertex; or that the vertex and base being given, you
had not wit enough to see that the altitude of the parabola is
determined? When Galileo’s proposition, which is the same with this of
mine, supposed no more but a body moved by these two motions, to prove
the line described to be the crooked line of a semiparabola, I never
thought of asking him what altitude, nor what diameter, nor what angle,
nor what base, had his parabola. And when Archimedes said, let the line
A B be the time, I should never have said to him, _do you think time to
be a line_, as you ask me whether I think impetus can be the base of a
parabola. And why, but because I am not so egregious a mathematician, as
you are. In this giddiness of yours, caused by looking upon this
intricate business of motion, and of time, and the concourse of motion
uniform, and uniformly accelerated, you rave upon the numbers 1, 4, 9,
16, &c. without reference to any thing that I had said; insomuch as any
one that had seen how much you have been deceived in them before, in
your scurvy book of _Arithmetica Infinitorum_, would presently conclude,
that this objection was nothing else but a fit of the same madness which
possessed you there.
My tenth article is like my ninth; and your objections to it are the
same which are to the former. Therefore you must take for answer just
the same which I have given to your objection there.
To the eleventh, you say first, you have done it better at the
sixty-fourth article of your _Arithmetica Infinitorum_. But what you
have done there, shall be examined when I come to the defence of my next
chapter. And whereas I direct the reader for the finding of the
proportions of the complements of those figures to the figures
themselves, to the table of art. 3, Chap, XVII., you say that if the
increase of the _spaces_, were to the increase of the times, as one to
two, then the complement should be to the parallelogram as one to three,
and say you find not (1)/(3) in the table. Did you not see that the
table is only of those figures which are described by the concourse of a
motion uniform with a motion accelerated? You had no reason therefore to
look for (1)/(3) in that table; for your case is of motion uniform
concurring with motion retarded, because you make not the proportions of
the spaces to the proportions of the times as two to one, but the
contrary; so that your objection ariseth from want of observing what you
read. But I “_may learn_” you say, “_these, and greater matters than
these, in your twenty-third and sixty-fourth propositions of your_
Arithmetica Infinitorum.” This, which you say here is a great absurdity;
but if you mean I shall find greater there, I will not say against you.
This (1)/(3) you looked for, belongs to the complements of the figures
calculated in that table; which because you are not able to find out of
yourselves, I will direct you to them. Your case is of (1)/(3) for the
complement of a parabola. Take the denominator of the fraction which
belongs to the parabola, namely three, and for numerator take the
numerator of the fraction which belongs to the triangle, namely one, and
you have the fraction sought. And in like manner for the complement of
any other figure. As, for example, of the second parabolaster, whose
fraction hath for denominator five, take the numerator of the fraction
of the same triangle which is one, and you have (1)/(3) for the fraction
sought for; and so of the rest, taking always one for the numerator.
The twelfth article, which you say is miserably false, I have left
standing unaltered. For not comprehending the sense of the proposition,
you make a figure of your own, and fight against your own fancied
motions, different from mine. Other geometricians that understand the
construction better, find no fault. And if you had in your own fifth
figure drawn a line through N parallel to A E, and upon that line
supposed your accelerated motion, you would quickly have seen that in
the time A E, the body moved from rest in A, would have fallen short of
the diagonal A D; and that all your extravagant pursuing of your own
mistake had been absurd.
My thirteenth article you say is ridiculous. But why? “_The impetus last
acquired cannot_” you say, “_be equal to a time_.” But the quantity of
the impetus may be equal to the quantity of a time, seeing they are both
measured by line. And when they are measured by the same described line,
each of their quantities is equal to that same line, and consequently to
one another. But when I meet with this kind of objection again, since I
have so often already shown it to be frivolous, and no less to be
objected against all the ancients that ever demonstrated any thing by
motion, than against me, I purpose to neglect it.
Secondly, you object “_that motion uniformly accelerated does no more
determine swiftness, than motion uniform_.” True; you needed not have
used sixteen lines to set down that. But suppose I add, as I do, so as
the last acquired impetus be equal to the time. _But that_, you say, _is
not sense_; which is the objection I am to neglect. But, you say again,
supposing it sense, this limitation helps me nothing. Why? _Because_,
you say, _a parabola may be described upon a base given, and yet have
any altitude, or any diameter one will_. Who doubts it? But how follows
it from thence, that when a parabolical line is described by two
motions, one uniform, the other uniformly accelerated from rest, that
the determining of the base does not also determine the whole parabola?
But fifthly, you say, _that this equality of the impetus to the time
does not determine the base_. Why not? _Because_, you say, _it is an
error proceeding from this, that I understand not what is_ ratio
subduplicata. I looked for this. I have shown and inculcated
sufficiently before, but the error is on your side; and therefore must
tell you, that this objection, and also a great part of the rest of your
errors in geometry, proceedeth from this, that you know not what
proportion is. But see how wisely you argue about this duplication of
proportion. For thus you say _verbatim_. “_Stay a little. What
proportion has duplicate proportion to single proportion? Is it always
the same? I think not for example, duplicate proportion_ (4)/(1) = (2 in
2)/(1 in 1) _is double to the single_ (2)/(1). _Duplicate proportion_
(9)/(1) = (3 in 3)/(1 in 1) _is triple to its single_ (3)/(1).” Let any
man, even of them that are most ready in your symbols, say in your
behalf (if he be not ashamed) that the proportion of nine to one is
triple to the proportion of three to one, as you do.
In the fourteenth, fifteenth, and sixteenth articles, you bid me repeat
your objections to the thirteenth. I have done it; and find that what
you have objected to the thirteenth, may as well be objected to these;
and consequently, that my answer there will also serve me here.
Therefore, if you can endure it, read the same answer over again.
But you have not yet done, you say, with these articles. Therefore
(after you had for a while spoken perplextly, conjecturing, not without
just cause, that I could not understand you) you say that to the end I
may the better perceive your meaning, I should take the example
following. “_Let a movent (in the first figure of this chapter) be moved
uniformly in the time A B, with the continual impetus A C, or B I, whose
whole velocity shall therefore be the parallelogram A C I B. And another
movent be uniformly accelerated, so as in the time A B it acquire the
same impetus B I. Now as the whole velocity, is to the whole velocity,
so is the length run over, to the length run over._” All this I
acknowledge to be according to my sense, saving that your putting your
word _movens_ instead of my word _mobile_ hath corrupted this article.
For in the first article, I meddle not with motion by concourse, wherein
only I have to do with two movents to make one motion; but in this I do,
wherein my word is not _movens_ but _mobile_; by which it is easy to
perceive you understand not this proposition. Then you proceed: “_But
the length run over by that accelerated motion is greater than the
length run over by that uniform motion._” Where do I say that? You
answer, “_in the ninth and thirteenth article, in making A B (in the
fifth figure) greater than A C; and A H (in the eighth figure) greater
than A B; and consequently, the triangle A B I, greater than the
parallelogram A C I B_.” That consequently is without consequence; for
it importeth nothing at all in this demonstration, whether A B, or A C
in the fifth figure be the greater. Besides I speak there of the
concourse of two movents, that describe the parabolical line A G D;
where the increasing impetus (because it increaseth as the times) will
be designed by the ordinate lines in the parabola A G D B. And if both
the motions in A B and A C were uniform, the aggregate of the impetus
would be designed by the triangle A B D, which is less than the
parallelogram A C D B. But you thought that the motion made by A C
uniformly, is the same with the motion made uniformly in the same time
by the motions in A B and A C concurring; so likewise, in the eighth
figure, there is nothing hinders A H from being greater than A B, unless
I had said that A B had been described in the time A C with the whole
impetus A C maintained entire; of which there is nothing in the
proposition, nor would at all have been pertinent to it. Therefore all
this new undertaking of the thirteenth, fourteenth, fifteenth, and
sixteenth articles, is to as little purpose as your former objections.
But I perceive that these new and hard speculations, though they turn
the edge of your wit, turn not the edge of your malice.
At the seventeenth article, you show again the same confusion. Return to
the eighth figure: “_if in a time given a body run over two lengths, one
with uniform, the other with accelerated motion_”; as for example, if in
the same time A C, a body, run over the line A B with uniform motion,
and the line A H with motion accelerated; “_and again in a part of that
time it run over a part of the length A H, with uniform motion, and
another part of the same with motion accelerated_;” as for example, in
the time A M it run over with uniform motion the line A I, and with
motion accelerated the line A B. _I say the excess of the whole A H
above the part A B, is to the excess of the whole A B above the part A
I, as the whole A H to the whole A B._ But first you will say, that
these words _as the whole A H to the whole A B_, are left out in the
proposition. But you acknowledge that it was my meaning; and you see it
is expressed before I come to the demonstration. And therefore it was
absurdly done to reprehend it. Let us therefore pass to the
demonstration. Draw I K parallel to A C, and make up the parallelogram A
I K M. And supposing first the acceleration to be uniform, divide I K in
the midst at N; and between I N, and I K, take a mean proportional I L.
_And the straight line A L, drawn and produced, shall cut the line B D
in F, and the line C G in G_ (which lines C G, and B D, as also H G and
B F, are determined, though you could not carry it so long in memory, by
the demonstration of the thirteenth article). _For seeing A B is
described by motion uniformly accelerated, and A I by motion uniform in
the same time A M; and I L is a mean proportional between I N (the half
of I K) and I K; therefore by the demonstration of the thirteenth
article, A I is a mean proportional between A B and the half of A B,
namely A O. Again, because A B is described by uniform motion, and A H
by motion uniformly accelerated, both of them in the same time A C, B F
is a mean proportional between B D and half B D, namely B E; therefore
by the demonstration of the same thirteenth article, the straight line A
L F produced will fall on G; and the line A H will be to the line A B,
as the line A B to the line A I. And consequently as A H to A B, so H B
to B I; which was to be demonstrated._ And by the like demonstration the
same may be proved, where the acceleration is in any other proportion
that can be assigned in numbers, saving that whereas this demonstration
dependeth on the construction of the thirteenth article, if the motion
had been accelerated in double proportion to the times, it would have
depended on the fourteenth, where the lines are determined. Which
determinations being not repeated, but declared before, in the
thirteenth article, to which this diagram belongeth, you take no notice
of, but go back to a figure belonging to another article, where there
was no use of these determinations. But because I see that the words of
the proposition, are as of four motions, and not of two motions made by
twice two movents, I must pardon them that have not rightly understood
my meaning; and I have now made the proposition according to the
demonstration. Which being done, all that you have said in very near two
leaves of your _Elenchus_ comes to nothing; and the fault you find comes
to no more than a too much trusting to the skill and diligence of the
reader. And whereas after you had sufficiently troubled yourself upon
this occasion, you add, “_that if Sir H. Savile had read my Geometry, he
had never given that censure of Joseph Scaliger, in his lecture upon
Euclid, that he was the worst geometrician of all mortal men, not
exceptioning so much as Orontius, but that praise should have been kept
for me_.” You see by this time, at least others do, how little I ought
to value that opinion; and that though I be the least of geometricians,
yet my geometry is to yours as 1 to 0. I recite these words of yours, to
let the world see your indiscretion in mentioning so needlessly that
passage of your founder. It is well known that Joseph Scaliger deserved
as well of the state of learning, as any man before or since him; and
that though he failed in his ratiocination concerning the quadrature of
the circle, yet there appears in that very failing so much knowledge of
geometry, that Sir H. Savile could not but see that there were mortal
men very many that had less; and consequently he knew that that censure
of his in a rigid sense (without the license of an hyperbole) was
unjust. But who is there that will approve of such hyperboles to the
dishonour of any but of unworthy persons, or think Joseph Scaliger
unworthy of honour from learned men? Besides, it was not Sir H. Savile
that confuted that false quadrature, but Clavius. What honour was it
then for him to triumph in the victory of another? When a beast is slain
by a lion, is it not easy for any of the fowls of the air to settle
upon, and peck him? Lastly, though it were a great error in Scaliger,
yet it was not so great a fault as the least sin; and I believe that a
public contumely done to any worthy person after his death, is not the
least of sins. Judge therefore whether you have not done indiscreetly,
in reviving the only fault, perhaps that any man living can lay to your
founder’s charge; and yet this error of Scaliger’s was no greater than
one of your own of the like nature, in making the true spiral of
Archimedes equal to half the circumference of the circle of the first
revolution; and then thinking to cover your fault by calling it
afterwards an aggregate of arches of circles (which is no spiral at all
of any kind) you do not repair but double the absurdity. What would Sir
Henry Savile have said to this?
The eighteenth article is this, “_in any parallelogram, if the two sides
that contain the angle be moved to their opposite sides, the one
uniformly, the other uniformly accelerated; the side that is moved
uniformly, by its concourse through all its longitude, hath the same
effect which it would have if the other motion were also uniform, and
the line described were a mean proportional between the whole length,
and the half of the same_.”
To the proposition you object first, “_that it is all one whether the
other motion be uniform or not, because the effect of each of their
motions, is but to carry the body to the opposite side_.” But do you
think that whatsoever be the motions, the body shall be carried by their
concourse always to the same point of the opposite side? If not, then
the effect is not all one when a motion is made by the concourse of two
motions uniform and accelerated, and when it is made by the concourse of
two uniform or of two accelerated motions.
Secondly, you say that these words, _and the line described were a mean
proportional between the whole length, and the half of the same_, have
no sense, or that you are deceived. True. For you are deceived; or
rather you have not understanding enough distinctly to conceive variety
of motions though distinctly expressed. For when a line is gone over
with motion uniformly accelerated, you cannot understand how a mean
proportional can be taken between it and its half; or if you can, you
cannot conceive that that mean can be gone over with uniform motion in
the same time that the whole line was run over by motion uniformly
accelerated. Yet these are things conceivable, and your want of
understanding must be made my fault.
My demonstration is this, _in the parallelogram A B C D, (Fig. 11). Let
the side A B be conceived to be moved uniformly till it lie in C D; and
let the time of that motion be A C, or B D. And in the same time let it
be conceived that A C is moved with uniform acceleration, till it lie in
B D._ To which you object, _that then the acceleration last acquired
must be far greater than that wherewith A B is moved uniformly: else it
shall never come to the place you would have it in the same time_. What
proof bring you for this? None here. Where then? Nowhere that I
remember. On the contrary I have proved (Art. 9 of the chapter) that the
line described by the concourse of those two motions, namely, uniform
from A B to C D, and uniformly accelerated from A C to B D, is the
crooked line of the semiparabola A H D. And though I had not, yet it is
well known that the same is demonstrated by Galileo. And seeing it is
manifest that in what proportion the motion is accelerated in the line A
B, in the same proportion the impetus beginning from rest in A is
increased in the same times (which impetus is designed all the way by
the ordinate lines of the semiparabola), the greatest impetus acquired
must needs be the base of the semiparabola, namely B D, equal to A C,
which designs the whole time. I cannot therefore imagine what should
make you say without proof, that the greatest acquired impetus is
greater than that which is designed by the base B D. Next you say, “you
see not to what end I divide A B in the middle at E.” No wonder; for you
have seen nothing all the way. Others would see it is necessary for the
demonstration; as also that the point F is not to be taken arbitrarily;
and likewise that the thirteenth article, which you admit not for proof,
is sufficiently demonstrated, and your objections to it answered. By the
way you advise me, where I say _percursam eodem motu uniformi, cum
impetu ubique_, &c. to blot out _cum_; because the _impetus_ is not a
_companion_ in the way, but the _cause_. Pardon me in that I cannot take
your learned counsel; for the word _motu uniformi_ is the ablative of
the _cause_, and _impetu_ the ablative of the _manner_. But to come
again to your objections, you say, I make “_a greater space run over in
the same time by the slower motion than by the swifter_.” How does that
appear? _because there is no doubt, but the swiftness is greater where
the greatest impetus is always maintained, than where it is attained to
in the same time from rest_. True, but that is, when they are considered
asunder without concourse, but not then when by the concourse they
debilitate one another, and describe a third line different from both
the lines, which they would describe singly. In this place I compare
their effects as contributing to the description of the parabolical line
A H D. What the effects of their several motions are, when they are
considered asunder, is sufficiently shown before in the first article.
You should first have gotten into your minds the perfect and distinct
ideas of all the motions mentioned in this chapter, and then have
ventured upon the censure of them, but not before. And then you would
have seen that the body moved from A, describeth not the line A C, nor
the line A B, but a third, namely the semiparabolical line A H D.
Again, where I say, _Wherefore, if the whole A B be uniformly moved to C
D, in the same time wherein A C is moved uniformly to F G_; you ask me
“_whether with the same impetus or not?_” How is it possible that in the
same time two unequal lengths should be passed over the same impetus?
“_But why_,” say you, “_do you not tell us with what impetus A C comes
to F G?_” What need is there of that, when all men know that in uniform
motion and the same time, impetus is to impetus, as length to length?
Which to have expressed had not been pertinent to the demonstration.
That which follows in the demonstration, _rursus suppono quod latus A
C_, &c. to these words, _ut ostensum est_, _Art. 12_, you confute with
saying you have proved that article to be false. But you may see now, if
you please, at the same place that I have proved your objection to be
frivolous.
After this you run on without any argument against the rest of the
demonstration, showing nothing all the way, but that the variety and
concourse of motions, the speculations whereof you have not been used
to, have made you giddy.
To the nineteenth article you apply the same objection which you made to
the eighteenth. Which having been answered, it appears that from the
very beginning of your Elenchus to this place all your objections
(except such as are made to three or four mistakes of small importance
in setting down my mind), are mere paralogisms, and such are less
pardonable than any paralogism in Orontius, both because the subject as
less difficult is more easily mastered, and because the same faults are
most shamefully committed by a reprehender than by any other man.
I had once added to these nineteen articles a twentieth, which was this:
“_If from a point in the circumference there be drawn a cord, and a
tangent equal to it, the angle which they make shall be double to the
aggregate of all the angles made by the cords of all the equal arches
into which the arch given can possibly be divided_.” Which proposition
is true, and I did when I writ it think I might have use of it. But be
it, or the demonstration of it true or false, seeing it was not
published by me, it is somewhat barbarous to charge me with the faults
thereof. No doctor of humanity but would have thought it a poor and
wretched malice, publicly to examine and censure papers of geometry
never published, by what means soever they came into his hands. I must
confess that in these words, _in such kind of progression arithmetical_
(that is, which begins with 0) _the sum of all the numbers taken
together, is equal to half the number that is made by multiplying the
greatest into the least_, there is a great error; for by this account
these numbers, 0, 1, 2, 3, 4, taken together, should be equal to
nothing. I should have said they are equal to that number which is made
by multiplying half the greatest into the number of the terms. There was
therefore, if those words were mine (for truly I have no copy of them,
nor have had since the book was printed, and I have no great reason, as
any man may see, to trust your faith) a great error in the writing, but
not an erroneous opinion in the writer. The demonstration so corrected
is true. And the angles that have the proportions of the numbers 1, 2,
3, 4, are in the table of your _Elenchus_ , fig. 12, the angles G A D, H
D E, I E F, K F B. And if the divisions were infinite, so that the first
were not to be reckoned but as a cypher, the angle C A B would be double
to them altogether. This mistake of mine, and the finding that I had
made no use of it in the whole book, was the cause why I thought fit to
leave it quite out. But your professorships, could not forbear to take
occasion thereby, to commend your zeal against _Leviathan_ to your
doctorships of divinity, by censuring it.
==========
OF THE FAULTS THAT OCCUR IN
DEMONSTRATION.
TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.
LESSON V.
At the seventeenth chapter, your first exception is to the definition of
proportional proportions, which is this: “_Four proportions are then
proportional, when the first is to the second, as the third to the
fourth_.” The reader will hardly believe that your exception is in
earnest. You say, I mean not by proportionality the “_quantity of the
proportions_.” Yes I do. Therefore I say again, that _four proportions
are then proportional, when the quantity of the first proportion, is to
the quantity of the second proportion, as the quantity of the third
proportion, to the quantity of the fourth proportion_. Is not my meaning
now plainly enough expressed? Or is it not the same definition with the
former. But what do I mean, you will say, by the quantity of a
proportion? I mean the determined greatness of it, that is, for example,
in these numbers, the quantity of the proportion of two to three, is the
same with the quantity of the proportion of four to six, or six to nine;
and again, the quantity of the proportion of six to four, is the same
with the quantity of the proportion of nine to six, or of three to two.
But now what do you mean by the quantity of a proportion? You mean that
two and three, are the quantities of the proportion of two to three (for
so Euclid calls them) and that six and four are the quantities of the
proportion of six to four, which is the same with the proportion of
three to two. And by this rule, one and the same proportion shall have
an infinite number of quantities; and consequently the quantity of a
proportion can never be determined. I call one proportion double to
another, when one is equal to twice the other; as the proportion of four
to one, is double to the proportion of two to one. You call that
proportion double where one number, line, or quantity absolute, is
double to the other; so that with you the proportion of two to one is a
double proportion. It is easy to understand how the number two is double
to one, but to what, I pray you, is double the proportion of two to one,
or of one to two? Is not every double proportion double to some
proportion? See whether this geometry of yours can be taken by any man
of sound mind for sense. “_But it is known_,” you say, “_that in
proportions, double is one thing, and duplicate another_;” so that it
seems to you, that in talking of proportion men are allowed to speak
senselessly. “_It is known_,” you say. To whom? It is indeed in use at
this day to call _double duplicate_, and _triple triplicate_. And it is
well enough; for they are words that signify the same thing, but that
they differ (in what subject soever) I never heard till now. I am sure
that Euclid, whom you have undertaken to expound, maketh no such
difference. And even there where he putteth these numbers, one, two,
four, eight, &c. for numbers in _double_ proportion (which is the last
proposition of the ninth element) he meaneth not that one to two, or two
to one, is a _double_ proportion, but that every number in that
progression is _double_ to the number next before it; and yet he does
not call it _analogia dupla_, but _duplicate_. This distinction in
proportions between _double_ and _duplicate_, proceeded long after from
want of knowledge that the proportion of one to two is _double_ to the
proportion of one to four; and this from ignorance of the different
nature of proportions of _excess_, and proportions of _defect_. And you
that have nothing but by tradition saw not the absurdities that did hang
thereon.
In the second article I make E K, (fig. 1) the third part of L K, which
you say is false; and consequently the proposition undemonstrated. And
thus you prove it false: “_Let A C be to G C, or G K to G L, as eight to
one_ (_for seeing the point G is taken arbitrarily, we may place it
where we will, &c._)” and upon this placing of G arbitrarily, you prove
well enough that E K is not a third part of L K. But you did not then
observe, that I make _the altitude A G, less than any quantity given_,
and by consequence E K to differ from a third part by a less difference
than any quantity that can be given. Therefore as yet the demonstration
proceedeth well enough. But perceiving your oversight, you thought fit
(though before, you thought this confutation sufficient) to endeavour to
confute it another way; but with much more evidence of ignorance. For
when I come to say, _the proportion therefore between A C and G C is
triple, in arithmetical proportion, to the proportion between G K and G
E, &c._ you say, “_the proportion of A C to G C is the proportion of
identity, as also that of G K to G E.”_ But why? Does my construction
make it so? Do not I make G C less than A C, though with less difference
than any quantity that can be assigned? And then where I say, _therefore
E K is the third part of L K_, you come in, by parenthesis, with (_or a
fourth, or a fifth, &c._). Upon what ground? Because you think it will
pass for current, without proof, that a point is nothing. Which if it
do, geometry also shall pass for nothing, as having no ground nor
beginning but in nothing. But I have already in a former lesson
sufficiently showed you the consequence of that opinion. To which I may
add, that it destroys the method of _indivisibles_, invented by
Bonaventura; and upon which, not well understood, you have grounded all
your scurvy book of _Arithmetica Infinitorum_; where your indivisibles
have nothing to do, but as they are supposed to have quantity, that is
to say, to be _divisibles_. You allow, it seems, your own nothings to be
somethings, and yet will not allow my somethings to be considered as
nothing. The rest of your objections having no other ground than this,
“_that a point is nothing_,” my whole demonstration standeth firm; and
so do the demonstrations of all such geometricians, ancient and modern,
as have inferred any thing in the manner following, viz. _If it be not
greater nor less, then it is equal. But it is neither greater nor less.
Therefore, &c. If it be greater, say by how much. By so much. It is not
greater by so much. Therefore it is not greater. If it be less, say how
much, &c._ Which being good demonstrations are together with mine
overthrown by the nothingness of your _point_, or rather of your
understanding; upon which you nevertheless have the vanity of advising
me what to do, if I demonstrate the same again; meaning I should come to
your false, impossible, and absurd method of _Arithmetica Infinitorum_,
worthy to be gilded, I do not mean with gold.
And for your question, why I set the base of my figure upwards, you may
be sure it was not because I was afraid to say, that the proportions of
the ordinate lines beginning at the vertex were triplicate, or otherwise
multiplicate of the proportions of the intercepted parts of the
diameter. For I never doubted to call double duplicate, nor triple
triplicate, &c., or if I had, I should have avoided it afterwards at the
tenth article of the same chapter. But because when I went about to
compare the proportions of the ordinate lines with those of their
contiguous diameters, the first thing I considered in them was in what
manner the base grew less and less till it vanished into a point. And
though the base had been placed below, it had not therefore required any
change in the demonstration. But I was the more apt to place the base
uppermost, because the motion began at the base, and ended at the
vertex. To proceed which way I pleased was in my own choice; and it is
of grace that I give you any account of it at all.
To the third article, together with its table, you say, “_it falls in
the ruin of the second; and that the same is to be understood of the
sixth, seventh, eighth, and ninth_.” For confutation whereof I need to
say no more, but that they all stand good by the confutation of your
objections to the second.
To the fourth article you say, “_the description of those curvilineal
figures is easy_.” True, to some men; and now that I have showed you the
way, it is easy enough for you also. For the way you propound is wholly
transcribed out of the figure of the second article, which article you
had before rejected. For seeing the lines H F, G E, A B, &c. are equal
to the lines C Q, C O, C D; and the lines Q F, O E, B D, equal to the
lines C H, C G, C A; the proportion of D B to O E, will be triple (that
is, triplicate) to the proportion of C O to G E; and the proportion of D
B to Q F, triple to the proportion of C D to C Q; and consequently,
because the complement B D C F E B is made by the decrease of A C in
triple proportion to that of the decrease of C D, it will be (by the
second article) a third part of the figure A B E F C A. So that it comes
all to one pass, whether we take triple proportion in decreasing to make
the complement, or triple proportion in increasing to make the figure;
for the proportion of H F to B A, is triple to the proportion of C H to
C A. Wherefore you have done no more but what you have seen first done,
saving that from your construction you prove not the figure to be triple
to the complement; perhaps because you have proved the contrary in your
_Arithmetica Infinitorum_. But your way differs from mine, in that you
call the proportion subtriplicate, which I call triplicate; as if the
divers naming of the same thing made it differ from itself. You might as
well have said briefly, the proposition is true, but ill proved, because
I call the proportion of one or two triple, or triplicate of that of one
to eight; which you say is false, and hath infected the fourth, fifth,
ninth, tenth, eleventh, thirteenth, fourteenth, fifteenth, sixteenth,
seventeenth, and nineteenth articles of the sixteenth chapter. But I
say, and you know now, that it is true; and that all those articles are
demonstrated.
Lastly you add, “_Tu vero, in presente articulo, &c. id est, you bid
find as many mean proportionals as one will, between two given lines; as
if that could not be done by the geometry of planes, &c._” You might
have left out _Tu vero_ to seek an _Ego quidem_. But tell me, do you
think that you can find two mean proportionals (which is less than as
many as one will) by the geometry of planes? We shall see anon how you
go about it. I never said it was impossible, and if you look upon the
places cited by you more attentively, you will find yourself mistaken.
But I say, the way to do it has not been yet found out, and therefore it
may prove a solid problem for anything you know.
The fifth article you reject, because it citeth the corollary of the
twenty-eighth article of the thirteenth chapter, where there is never a
word to that purpose. But there is in the twenty-sixth article; which
was my own fault, though you knew not but it might have been the
printer’s.
To the tenth you object for almost three leaves together, against these
words of mine, _because_, in the sixth figure, _B C is to B F in
triplicate proportion of C D to F E, therefore inverting, F E is to C D
in triplicate proportion of B F to B C_. This you objected then. But now
that I have taught you so much geometry, as to know _that of three
quantities, beginning at the least, if the third be to the first in
triplicate proportion of the second to the first, also by conversion the
first to the second shall be in triplicate proportion of the first to
the third_; if it were to do again, you would not object it.
My eleventh article you would allow for demonstrated, if my second had
been demonstrated, upon which it dependeth. Therefore seeing your
objections to that article are sufficiently answered, this article also
is to be allowed.
The twelfth also is allowed upon the same reason. What falsities you
shall find in such following propositions as depend upon the same second
article, we shall then see when I come to the places where you object
against them.
To the thirteenth article you object, “_that the same demonstration may
be as well applied to a portion of any conoeides, parabolical,
hyperbolical, elliptical, or any other, as to the portion of a sphere_.”
By the truth of this let any man judge of your and my geometry. Your
comparison of the sphere and conoeides, so far holds good, as to prove
that the superficies of the conoeides is greater than the superficies of
the cone described by the subtense of the parabolical, hyperbolical, or
elliptical line. But when I come to say, that _the cause of the excess
of the superficies of the portion of the sphere above the superficies of
the cone, consists in the angle D A B, and the cause of the excess of
the circle made upon the tangent A D, above the superficies of the same
cone, consists in the magnitude of the same angle D A B_, how will you
apply this to your conoeides? For suppose that the crooked line A B (in
the seventh figure) were not an arch of a circle, do you think that the
angles which it maketh with the subtense A B, at the points A and B,
must needs be equal? Or if they be not, does the excess of the
superficies of the circle upon A D above the superficies of the cone, or
the excess of the superficies of the portion of the conoeides above the
superficies of the same cone, consist in the angle D A B, or rather in
the magnitude of the two unequal angles D A B, and A B A? You should
have drawn some other crooked line, and made tangents to it through A
and B, and you would presently have seen your error. See how you can
answer this; for if this demonstration of mine stand firm, I may be bold
to say, though the same be well demonstrated by Archimedes, that this
way of mine is more natural, as proceeding immediately from the natural
efficient causes of the effect contained in the conclusion; and besides,
more brief and more easy to be followed by the fancy of the reader.
To the fourteenth article you say that I “_commit a circle in that I
require in the fourth article the finding of two mean proportionals, and
come not till now to show how it is to be done_.” Nor now neither. But
in the mean time you commit two mistakes in saying so. The place cited
by you in the fourth article is, in the Latin, p. 215, line 26, in the
English, p. 255, line 24. Let any reader judge whether that be a
requiring it, or a supposing it to be done; this is your first mistake.
The second is, that in this place the proportion itself, which is, “_If
these deficient figures could be described in a parallelogram
exquisitely, there might be found thereby between any two lines given,
as many mean proportionals as one would_,” is a theorem, upon
supposition of these crooked lines exquisitely drawn; but you take it
for a problem.
And proceeding in that error, you undertake the invention of two mean
proportionals, using therein my first figure, which is of the same
construction with the eighth that belongeth to this fourteenth article.
Your construction is, “_Let there be taken in the diameter C A, (fig. 1)
the two given lines, or two others proportional to them, as C H, C G,
and their ordinate lines H F, G E (which by construction are in
subtriplicate proportion of the intercepted diameters). These lines will
show the proportions which those four proportionals are to have._” But
how will you find the length of H F or G E, the ordinate lines? Will you
not do it by so drawing the crooked line C F E, as it may pass through
both the points F and E? You may make it pass through one of them, but
to make it pass through the other, you must find two mean proportionals
between G K and G L, or between H I and H P; which you cannot do, unless
the crooked line be exactly drawn; which it cannot be by the geometry of
planes. Go shew this demonstration of yours to Orontius, and see what he
will say to it.
I am now come to an end of your objections to the seventeenth chapter,
where you have an epiphonema not to be passed over in silence. But
because you pretend to the demonstration of some of these propositions
by another method in your _Arithmetica Infinitorum_, I shall first try
whether you be able to defend those demonstrations as well as I have
done these of mine by the method of motion.
The first proposition of your _Arithmetica Infinitorum_ is this lemma:
“_In a series, or row of quantities, arithmetically proportional,
beginning at a point or cypher, as 0, 1, 2, 3, 4, &c. to find the
proportion of the aggregate of them all, to the aggregate of so many
times the greatest, as there are terms_.” This is to be done by
multiplying the greatest into half the number of the terms.
The demonstration is easy. But how do you demonstrate the same? “_The
most simple way_,” say you, “_of finding this and some other problems,
is to do the thing itself a little way, and to observe and compare the
appearing proportions, and then by induction to conclude it
universally_.” Egregious logicians and geometricians, that think an
induction, without a numeration of all the particulars sufficient, to
infer a conclusion universal, and fit to be received for a geometrical
demonstration! But why do you limit it to the natural consecution of the
numbers, 0, 1, 2, 3, 4, &c? Is it not also true in these numbers, 0, 2,
4, 6, &c. or in these, 0, 7, 14, 21, &c? Or in any numbers where the
difference of nothing and the first number is equal to the difference
between the first and second, and between the second and third, &c.?
Again, are not these quantities, 1, 3, 5, 7, &c. in continual proportion
arithmetical? And if you put before them a cypher thus, 0, 1, 3, 5, 7,
do you think that the sum of them is equal to the half of five times
seven? Therefore though your lemma be true, and by me (Chap. XIII. art.
5) demonstrated; yet you did not know why it is true; which also appears
most evidently in the first proposition of your _Conic Sections_ , where
first you have this, “_that a parallelogram whose altitude is infinitely
little, that is to say, none, is scarce anything else but a line_.” Is
this the language of geometry? How do you determine this word _scarce_?
The least altitude, is somewhat or nothing. If somewhat, then the first
character of your arithmetical progression must not be a cypher; and
consequently the first eighteen propositions of this your _Arithmetica
Infinitorum_ are all nought. If nothing, then your whole figure is
without altitude, and consequently your understanding nought. Again, in
the same proposition, you say thus: “_We will sometimes call those
parallelograms rather by the name of lines than of parallelograms, at
least when there is no consideration of a determinate altitude; but
where there is a consideration of a determinate altitude (which will
happen sometimes) there that little altitude shall be so far considered,
as that being infinitely multiplied it may be equal to the altitude of
the whole figure._” See here in what a confusion you are when you resist
the truth. When you consider no determinate altitude, that is no
quantity of altitude, then you say your parallelogram shall be called a
line. But when the altitude is determined, that is, when it is quantity,
then you will call it a parallelogram. Is not this the very same
doctrine which you so much wonder at and reprehend in me, in your
objections to my eighth chapter, and your word _considered_ used as I
used it? It is very ugly in one that so bitterly reprehendeth a doctrine
in another, to be driven upon the same himself by the force of truth
when he thinks not on it. Again, seeing you admit in any case those
infinitely little altitudes to be quantity, what need you this
limitation of yours, “_so far forth as that by multiplication they may
be made equal to the altitude of the whole figure_?” May not the half,
the third, the fourth, or the fifth part, &c. be made equal to the whole
by multiplication? Why could you not have said plainly, _so far forth as
that every one of those infinitely little altitudes be not only
something but an aliquot part of the whole_? So you will have an
_infinitely little_ altitude, that is to say, _a point to be both
nothing and something and an aliquot part_. And all this proceeds from
not understanding the ground of your profession. Well, the lemma is
true. Let us see the theorems you draw from it. The first is (p. 3)
“_that a triangle to a parallelogram of equal base and altitude is as
one to two_.” The conclusion is true, but how know you that?
“_Because_,” say you, “_the triangle consists as it were_ [_as it were_,
is no phrase of a geometrician] _of an infinite number of straight
parallel lines_.” Does it so? Then by your own doctrine, which is, that
“_lines have no breadth_,” the altitude of your triangle consisteth of
an infinite number of no altitudes, that is of an infinite number of
nothings, and consequently the area of your triangle has no quantity. If
you say that by the parallels you mean infinitely little parallelograms,
you are never the better; for if infinitely little, either they are
nothing, or if somewhat, yet seeing that no two sides of a triangle are
parallel, those parallels cannot be parallelograms. I see they may be
counted for parallelograms by not considering the quantity of their
altitudes in the demonstration. But you are barred of that plea, by your
spiteful arguing against it in your _Elenchus_ . Therefore this third
proposition, and with it the fourth, is undemonstrated.
Your fifth proposition is, “_the spiral line is equal to half the circle
of the first revolution_.” But what spiral line? We shall understand
that by your construction, which is this: “_The straight line M A_ [in
your figure which I have placed at the end of the fifth lesson] _turned
round (the point M remaining unmoved) is supposed to describe with its
point A the circle A O A, whilst some point, in the same M A, whilst it
goes about, is supposed to be moved uniformly from M to A, describing
the spiral line_.” This therefore, is the spiral line of Archimedes; and
your proposition affirms it to be equal to the half of the circle A O A;
which you perceived not long after to be false. But thinking it had been
true, you go about to prove it, “_by inscribing in the circle an
infinite multitude of equal angles, and consequently an infinite number
of sectors, whose arches will therefore be in arithmetical proportion_;”
which is true. “_And the aggregate of those arches equal to half the
circumference A O A_;” which is true also. And thence you conclude
“_that the spiral line is equal to half the circumference of the circle
A O A_;” which is false. For the aggregate of that infinite number of
infinitely little arches, is not the spiral line made by your
construction, seeing by your construction the line you make is
manifestly the spiral of Archimedes; whereas no number, though infinite,
of arches of circles, how little soever, is any kind of spiral at all;
and though you call it a spiral, that is but a patch to cover your
fault, and deceiveth no man but yourself. Besides, you saw not how
absurd it was, for you that hold a point to be absolutely nothing, to
make an infinite number of equal angles (the radius increasing as the
number of angles increaseth) and then to say, “_that the arches of the
sectors whose angles they are, are as_ 0, 1, 2, 3, 4, &c.” For you make
the first angle 0, and all the rest equal to it; and so make 0, 0, 0, 0,
0, &c. to be the same progression with 0, 1, 2, 3, 4, &c. The influence
of this absurdity reacheth to the end of the eighteenth proposition. So
many are therefore false, or nothing worth. And you needed not to wonder
that the doctrine contained in them was omitted by Archimedes, who never
was so senseless as to think a spiral line was compounded of arches of
circles.
Your nineteenth proposition is this other lemma: “_In a series, or a
row, of quantities, beginning from a point, or cypher, and proceeding
according to the order of the square numbers, as_ 0, 1, 4, 9, 16, _&c.
to find what proportion the whole series hath to so many times the
greatest_.” And you conclude “_the proportions to be that of 1 to 3_.”
Which is false, as you shall presently see. First, let the series of
squares with the prefixed cypher, and under every one of them the
greatest 4 be (0 . 1 . 4)/(4 . 4 . 4). And you have for the sum of the
squares 5, and for thrice the greatest 12, the third part whereof is 4.
But 5 is greater than 4, by 1, that is, by one twelfth of 12; which
quantity is somewhat, let it be called A. Again, let the row of squares
be lengthened one term further, and the greatepm divst set under every
one of them as (0 . 1 . 4 . 9)/(9 . 9 . 9 . 9). The sum of the squares
is 14, and the sum of four times the greatest is 36, whereof the third
part is 12. But 14 is greater than 12 by two unities, that is, by two
twelfths of 12, that is, by 2 A. The difference therefore between the
sum of the squares, and the sum of so many times the greatest square, is
greater, when the cypher is followed by three squares, than when by but
two. Again, let the row have five terms, as in these numbers (0 . 1
. 4 . 9 . 16)/(16 . 16 . 16 . 16 . 16) with the greatest five times
described, and the sum of the squares will be 30, the sum of all the
greatest will be 80. The third part whereof is 26(2)/(3). But 30 is
greater than 26(2)/(3) by 3(1)/(3), that is, by three twelfths of
twelve, and (1)/(3) of a twelfth, that is, by 3(1)/(3) A. Likewise in
the series continued to six places with the greatest six times
subscribed, as ( 0 . 1 . 4 . 9 . 16 . 25)/(25 . 25 . 25 . 25 . 25
. 25) the sum of the squares is 55, and the sum of the greatest six
times taken is 150, the third part whereof is 50. But 55 is greater than
50 by 5, that is, by five-twelfths of 12, that is by 5 A. And so
continually as the row groweth longer, the excess also of the aggregate
of the squares above the third part of the aggregate of so many times
the greatest square, growing greater. And consequently if the number of
the squares were infinite, their sum would be so far from being equal to
the third part of the aggregate of the greatest as often taken, as that
it would be greater than it by a quantity greater than any that can be
given or named.
That which deceived you was partly this, that you think, as you do in
your _Elenchus_ , that these fractions (1)/(12) (1)/(18) (1)/(24)
(1)/(30) (1)/(36) &c. are proportions, as if (1)/(12) were the
proportion of one to twelve, and consequently (2)/(12) double the
proportion of one to twelve; which is as unintelligible as
school-divinity; and I assure you, far from the meaning of Mr. Ougthred
in the sixth chapter of his _Clavis Mathematica_, where he says that
4(3)/(7) is the proportion of 31 to 7; for his meaning is, that the
proportion of 4(3)/(7) to one, is the proportion of 31 to 7; whereas if
he meant as you do, then 8(6)/(7) should be double the proportion of 31
to 7. Partly also because you think (as in the end of the twentieth
proposition) that if the proportion of the numerators of these fractions
(1)/(12) (1)/(18) (1)/(24) (1)/(30) (1)/(36) to their denominators
decrease eternally, they shall so vanish at last as to leave the
proportion of the sum of all the squares to the sum of the greatest so
often taken, (that is, an infinite number of times), as one to three, or
the sum of the greatest to the sum of the increasing squares, as three
to one; for which there is no more reason than for four to one, or five
to one, or any other such proportion. For if the proportions come
eternally nearer and nearer to the subtriple, they must needs also come
nearer and nearer to subquadruple; and you may as well conclude thence
that the upper quantities shall be to the lower quantities as one to
four, or as one to five, &c. as conclude they are as one to three. You
can see without admonition, what effect this false ground of yours will
produce in the whole structure of your _Arithmetica Infinitorum_; and
how it makes all that you have said unto the end of your thirty-eighth
proposition, undemonstrated, and much of it false.
The thirty-ninth is this other lemma: “_In a series of quantities
beginning with a point or cypher, and proceeding according to the series
of the cubic numbers, as O. 1. 8. 27. 64, &c. to find the proportion of
the sum of the cubes to the sum of the greatest cube, so many times
taken as there be terms_.” And you conclude that “_they have a
proportion of 1 to 4_;” which is false.
Let the first series be of three terms subscribed with the greatest (0.
1. 8.)/(8. 8. 8.); the sum of the cubes is nine; the sum of all the
greatest is 24; a quarter whereof is 6. But 9 is greater than 6 by three
unities. An unity is something. Let it be therefore A. Therefore the row
of cubes is greater than a quarter of three times eight, by three A.
Again, let the series have four terms, as (0. 1. 8. 27)/(27. 27. 27.
27); the sum of the cubes is 36; a quarter of the sum of all the
greatest is twenty-seven. But thirty-six is greater than twenty-seven by
nine, that is, by 9 A. The excess therefore of the sum of the cubes
above the fourth part of the sum of all the greatest, is increased by
the increase of the number of terms. Again, let the terms be five, as
(0. 1. 8. 27. 64)/(64. 64. 64. 64. 64), the sum of the cubes is one
hundred; the sum of all the greatest three hundred and twenty; a quarter
whereof is eighty. But one hundred is greater than eighty by twenty,
that is, by 20 A. So you see that this lemma also is false. And yet
there is grounded upon it all that which you have of comparing parabolas
and paraboloeides with the parallelograms wherein they are accommodated.
And therefore though it be true, that the parabola is (2)/(3) and the
cubical paraboloeides (3)/(4) of their parallelograms respectively, yet
it is more than you were certain of when you referred me, for the
learning of geometry, to this book of yours. Besides, any man may
perceive that without these two lemmas (which are mingled with all your
compounded series with their excesses) there is nothing demonstrated to
the end of your book: which to prosecute particularly, were but a vain
expense of time. Truly, were it not that I must defend my reputation, I
should not have showed the world how little there is of sound doctrine
in any of your books. For when I think how dejected you will be for the
future, and how the grief of so much time irrecoverably lost, together
with the conscience of taking so great a stipend, for mis-teaching the
young men of the University, and the consideration of how much your
friends will be ashamed of you, will accompany you for the rest of your
life, I have more compassion for you than you have deserved. Your
treatise of the _Angle of Contact_ , I have before confuted in a very
few leaves. And for that of your _Conic Sections_ , it is so covered
over with the scab of symbols, that I had not the patience to examine
whether it be well or ill demonstrated.
Yet I observed thus much, that you find a tangent to a point given in
the section by a diameter given; and in the next chapter after, you
teach the finding of a diameter, which is not artificially done.
I observe also, that you call the _parameter_ an imaginary line, as if
the place thereof were less determined than the diameter itself; and
then you take a mean proportional between the intercepted diameter, and
its contiguous ordinate line, to find it. And it is true, you find it:
but the parameter has a determined quantity, to be found without taking
a mean proportional. For the diameter and half the section being given,
draw a tangent through the vertex, and dividing the angle in the midst
which is made by the diameter and tangent, the line that so divideth the
angle, will cut the crooked line. From the intersection draw a line (if
it be a parabola) parallel to the diameter, and that line shall cut off
in the tangent from the vertex the parameter sought. But if the section
be an ellipsis, or an hyperbole, you may use the same method, saving
that the line drawn from the intersection must not be parallel, but must
pass through the end of the transverse diameter, and then also it shall
cut off a part of the tangent, which measured from the vertex is the
parameter. So that there is no more reason to call the parameter an
imaginary line than the diameter.
Lastly, I observe that in all this your new method of conics, you show
not how to find the _burning points_, which writers call the _foci_ and
_umbilici_ of the section, which are of all other things belonging to
the conics most useful in philosophy. Why therefore were they not as
worthy of your pains as the rest, for the rest also have already been
demonstrated by others? You know the focus of the parabola is in the
axis distant from the vertex a quarter of the parameter. Know also that
the focus of an hyperbole, is in the axis, distant from the vertex, as
much as the hypotenusal of a rectangled triangle, whose one side is half
the transverse axis, the other side half the mean proportional between
the whole transverse axis and the parameter, is greater than half the
transverse axis.
The cause why you have performed nothing in any of your books (saving
that in your _Elenchus_ you have spied a few negligences of mine, which
I need not be ashamed of) is this, that you understood not what is
_quantity_, _line_, _superficies_, _angle_, and _proportion_; without
which you cannot have the science of any one proposition in geometry.
From this one and first definition of Euclid, “_a point is that whereof
there is no part_,” understood by Sextus Empiricus, as you understand
it, that is to say misunderstood, Sextus Empiricus had utterly destroyed
most of the rest, and demonstrated, that in geometry there is no
science, and by that means you have betrayed the most evident of the
sciences to the sceptics. But as I understand it for _that whereof no
part is reckoned_, his arguments have no force at all, and geometry is
redeemed. If a line have no latitude, how shall a cylinder rolling on a
plane, which it toucheth not but in a line, describe a superficies? How
can you affirm that any of those things can be without quantity, whereof
the one may be greater or less than the other? But in the common contact
of divers circles the external circle maketh with the common tangent a
less angle of contact than the internal. Why then is it not quantity? An
angle is made by the concourse of two lines from several regions,
concurring, by their generation, in one and the same point. How then can
you say the angle of contact is no angle? One measure cannot be
applicable at once to the angle of contact, and angle of conversion. How
then can you infer, if they be both angles, that they must be
homogeneous? Proportion is the relation of two quantities. How then can
a quotient or fraction, which is quantity absolute, be a proportion? But
to come at last to your _Epiphonema_ , wherein, though I have perfectly
demonstrated all those propositions concerning the proportion of
parabolasters to their parallelograms, and you have demonstrated none of
them (as you cannot now but plainly see), but committed most gross
paralogisms, how could you be so transported with pride, as insolently
to compare the setting of them forth as mine, to the act of him that
steals a horse, and comes to the gallows for it. You have read, I think,
of the gallows set up by Haman. Remember therefore also who was hanged
upon it.
After your dejection I shall comfort you a little, a very little, with
this, that whereas this eighteenth chapter containeth two problems, one,
“_the finding of a straight line equal to the crooked line of a
semi-parabola_;” the other, “_the finding of straight lines equal to the
crooked lines of the parabolasters, in the table of the third article of
the seventeenth chapter_;” you have truly demonstrated that they are
both false; and another hath also demonstrated the same another way.
Nevertheless, the fault was not in my method, but in a mistake of one
line for another and such as was not hard to correct; and is now so
corrected in the English as you shall not be able (if you can
sufficiently imagine motions) to reprehend. The fault was this, that in
the triangles which have the same base and altitude with the parabola
and parabolaster, I take for designation of the mean uniform impetus, a
mean proportional, in the first figure, between the whole diameter and
its half, and, in the second figure, a mean proportional between the
whole diameter and its third part; which was manifestly false, and
contrary to what I had shown in the sixteenth chapter. Whereas I ought
to have taken the half of the base, as now I have done, and thereby
exhibited the straight lines equal to those crooked lines, as I
undertook to do. Which error therefore proceeded not from want of skill,
but from want of care; and what I promised (as bold as you say the
promise was), I have now performed.
The rest of your exceptions to this chapter, are to these words in the
end: “_There be some that say, that though there be equality between a
straight and crooked line, yet now, they say, after the fall of Adam, it
cannot be found without the especial help of divine grace_.” And you say
you think there be none that say so. I am not bound to tell you who they
are. Nevertheless, that other men may see the spirit of an ambitious
part of the clergy, I will tell you where I read it. It is in the
_Prolegomena_ of Lalovera, a Jesuit, to his Quadrature of the Circle, p.
13 and 14, in these words: “_Quamvis circuli tetragonismus sit_ φύσει
_possibilis, an tamen etiam_ πρός ἡμᾶς, _hoc est, post Adæ lapsum homo
ejus scientiam absque speciali divinæ gratiæ auxilio, possit comparare,
jure merito inquirunt theologi, pronunciantque; hanc veritatem tanta
esse caligine involutam ut illam videre nemo possit, nisi ignorantiæ ex
primi parentis prævaricatione propagatas tenebras indebitus divinæ lucis
radius dissipet; quod verissimum esse sentio_.” Wherein I observed that
he, supposing he had found that quadrature, would have us believe it was
not by the ordinary and natural help of God (whereby one man reasoneth,
judgeth and remembereth better than another), but by a special (which
must be a supernatural) help of God, that he hath given to him of the
order of Jesus above others that have attempted the same in vain.
Insinuating thereby, as handsomely as he could, a special love of God
towards the Jesuits. But you taking no notice of the word _special_,
would have men think I held, that human sciences might be acquired
without any help of God. And thereupon proceed in a great deal of ill
language to the end of your objections to this chapter. But I shall take
notice of your manners for altogether in my next lesson.
At the nineteenth chapter you see not, you say, the method. Like enough.
In this chapter I consider not the cause of reflection, which consisteth
in the resistance of bodies natural; but I consider the consequences,
arising from the supposition of the equality of the angle of reflection,
to that of incidence; leaving the causes both of reflection, and of
refraction, to be handled together in the twenty-fourth chapter. Which
method, think what you will, I still think best.
Secondly, you say I define not, here, but many chapters after, what an
angle of incidence, and what an angle of reflection is. Had you not been
more hasty than diligent readers, you had found that those definitions
of the angle of incidence, and of reflection, were here set down in the
first article, and not deferred to the twenty-fourth. Let not therefore
your own oversight be any more brought in for an objection.
Thirdly, you say there is no great difficulty in the business of this
chapter. It may be so, now it is down; but before it was done, I doubt
not but you that are a professor would have done the same, as well as
you have done that of the _Angle of Contact_ , or the business of your
_Arithmetica Infinitorum_ . But what a novice in geometry would have
done I cannot tell.
To the third, fourth, and fifth article, you object a want of
determination; and show it by instance, as to the third article. But
what those determinations should be, you determine not, because you
could not. The words in the third article, are first these, _if there
fall two straight lines parallel, &c._ which is too general. It should
be, _if there fall the same way two straight lines parallel, &c._ Next
these, _their reflected lines produced inwards shall make an angle, &c._
This also is too general. I should have said, _their reflected lines
produced inwards, if they meet within, shall make an angle, &c._ Which
done, both this article and the fourth and fifth are fully demonstrated.
And without it, an intelligent reader had been satisfied, supplying the
want himself by the construction.
To the eighth, you object only the too great length and labour of it,
because you can do it a shorter way. Perhaps so now, as being easy to
shorten many of the demonstrations both of Euclid, and other the best
geometricians that are or have been. And this is all you had to say to
my nineteenth chapter. Before I proceed, I must put you in mind that
these words of yours, “_adducis malleum, ut occidas muscam_,” are not
good Latin, _malleum affers_, _malleum adhibes_, _malleo uteris_, are
good. When you speak of bringing bodies animate, _ducere_ and _adducere_
are good, for there _to bring_, is _to guide or lead_. And of bodies
inanimate, _adducere_ is good for _attrahere_, which is to draw to. But
when you bring a hammer, will you say _adduco malleum_, _I lead a
hammer_? A man may lead another man, and a ninny may be said to lead
another ninny, but not a hammer. Nevertheless, I should not have thought
fit to reprehend this fault upon this occasion in an Englishman, nor to
take notice of it, but that I find you in some places nibbling, but
causelessly, at my Latin.
Concerning the twentieth chapter, before I answer to the objections
against the propositions themselves, I must answer to the exception you
first take to these words of mine, “_Quæ de dimensione circuli et
angulorum pronuntiata sunt tanquam exactè inventa, accipiat lector
tanquam dicta_ _problematicè._” To which you say thus: “_We are wont in
geometry to call some propositions theorems, others problems, &c. of
which a theorem is that wherein some assertion is propounded to be
proved; a problem that wherein something is commanded to be done_.” Do
you mean _to be done_, and not proved? By your favour, a problem in all
ancient writers signifies no more but a proposition uttered, to the end
to have it, by them to whom it is uttered, examined whether it be true
or not true, faisable or not faisable; and differs not amongst
geometricians from a theorem but in the manner of propounding. For this
proposition, _to make an equilateral triangle_, so propounded they call
a problem. But if propounded thus: _If upon the ends of a straight line
given be described two circles, whose radius is the same straight line,
and there be drawn from the intersection of the circles to their two
centres, two straight lines, there will be made an equilateral
triangle_, then they call it a theorem; and yet the proposition is the
same. Therefore these words, _accipiat lector tanquam dicta
problematicè_ signify plainly this, that I would have the reader, take
for propounded to him to examine, whether from my construction the
quadrature of the circle can be truly inferred or not; and this is not
to bid him, as you interpret it, to square the circle. And if you
believe that _problematicè_ signifies probably, you have been very
negligent in observing the sense of the ancient Greek philosophers in
the word problem. Therefore your _solemus in geometria_, &c. is nothing
to the purpose; nor had it been though you had spoken more properly, and
said _solent_, leaving out yourselves.
[Illustration:
_Six Lessons._
_Vol. VII. Eng. p.310_II. 325_
]
My first article hath this title, “_from a false supposition, a false
quadrature of the circle_.” Seeing therefore you were resolved to show
where I erred, you should have proved either that the supposition was
true, and the conclusion falsely inferred, or contrarily, that though
the supposition be false, yet the conclusion is true; for else you
object nothing to my geometry, but only to my judgment, in thinking fit
to publish it; which nevertheless you cannot justly do, seeing it was
likely to give occasion to ingenious men (the practice of it being so
accurate to sense) to inquire wherein the fallacy did consist. And for
the problem as it was first printed, but never published, and
consequently ought to have passed for a private paper stolen out of my
study, your public objecting against it (in the opinion of all men that
have conversed so much with honest company as to know what belongs to
civil conversation), was sufficiently barbarous in divines. And seeing
you knew I had rejected that proposition, it was but a poor ambition to
take wing as you thought to do, like beetles from my egestions. But let
that be as it will, you will think strange now I should resume, and make
good, at least against your objection, that very same proposition. So
much of the figure as is needful you will find noted with the same
letters, and placed at the end of this fifth lesson. Wherein let B I, be
an arch not greater than the radius of the circle, and divided into four
equal parts, in L, N, O. Draw S N, the sine of the arch B N, and produce
it to T, so as S T be double to S N, that is, equal to the chord B I.
Draw likewise _a_ L, the sine of the arch B L, and produce it to _c_, so
as _a c_ be quadruple to _a_ L, that is, equal to the two chords B N, N
I. Upon the centre N with the radius N I, draw the arch I _d_, cutting B
U the tangent in _d_. Then will B N produced cut the arch I _d_, in the
midst at _o_. In the line B S produced take S _b_, equal to B S; then
draw and produce _b_ N, and it will fall on the point _d_. And B _d_, S
T, will be equal; and _d_ T joined and produced will fall upon _o_, the
midst of the arch I _d_. Join I T, and produce it to the tangent B U in
U. I say, that the straight line I T U shall pass through _c_. For
seeing B S, S _b_, are equal, and the angle at S a right angle, the
straight lines B N, and _b_ N, are also equal, and the triangles B N
_b_, _d_ N _o_ like and equal; and the lines _d_ T, T _o_ equal. Draw _o
i_ parallel to _d_ U, cutting I U in _i_; and the triangles _d_ T U, _o_
T _i_ will also be like and equal. Produce S T to the arch _d o_ I in
_e_, and produce it further to _f_, so that the line _e f_ be equal to T
_e_; and then S _f_ will be equal to _a c_. Therefore _f c_ joined will
be parallel to B S. In _c f_ produced take _f g_ equal to _c f_; and
draw _g m_ parallel to _d_ U, cutting I U in _m_, and _d o_ in _n_; and
let the intersection of the two lines _a c_ and _d o_ be in _r_; which
being done, the triangles _m n_ T, _r c_ T will be like and equal.
Therefore _m n_ and _r c_ are equal; and consequently the straight line
I _m_ T U shall pass through _c_. Dividing therefore _a c_ in the midst
at _t_, and S N in the midst at _l_, and joining _t_ N, L _l_, the lines
L _l_, _t_ N, and _c_ T produced, will all meet in one and the same
point of B S produced; suppose at _q_. Therefore the point _q_ being
given by the two known points T and I, the lines drawn from _q_ through
equal parts of the sine of the arch B I, (for example through the points
P, Q, R, of the sine M I), shall cut off equal arches, as B L, L N, N O,
O I. And this is enough to make good that problem, as to your objection.
The straight line therefore B U, for any thing you have said, is proved
equal to the arch B I, and the division of any angle given into any
proportion given, the quadrature of any sector, and the construction of
any equilateral polygon is also given. And though in this also I should
have erred, yet it cannot be denied but that I have used a more natural,
a more geometrical, and a more perspicuous method in the search of this
so difficult a problem, than you have done in your _Arithmetica
Infinitorum_. For though it be true that the aggregate of all the mean
proportionals between the radius, together with an infinitely little
part of the same, and the radius wanting an infinitely little part of
the same; and again, between the radius, together with two infinitely
little parts, and the radius wanting two infinitely little parts, and so
on eternally, will be equal to the quadrant (a thing which every mean
geometrician knew before); yet it was absurd to think those means could
be calculated in numbers by interpoling of a symbol; especially when you
make that symbol to stand for a number neither true nor surd; as if
there were a number that could neither be uttered in words, nor not be
uttered in words. For what else is surd, but that which cannot be
spoken?
To the fifth article, though your discourse be long, you object but two
things. One is, that “_Whereas the spiral of Archimedes is made of two
motions, one straight, the other circular, both uniform, I taking the
motion compounded of them both for one of those that are compounded,
conclude falsely, that the generation of the spiral is like to the
generation of the parabola_.” What heed you use to take in your
reprehensions, appears most manifestly in this objection. For I say in
that demonstration of mine, that _the velocity of the point A in
describing the spiral increaseth continually in proportion to the
times_. For seeing it goes on uniformly in the semidiameter, it is
impossible it should not pass into greater and greater circles,
proportionally to the times, and consequently it must have a swifter and
swifter motion circular, to be compounded with the uniform motion in
every point of the radius as it turneth about. This objection therefore
is nothing but an effect of a will, without cause, to contradict.
The other objection is, that “_Granting all to be true hitherto, yet
because it depends upon the finding of a straight line equal to a
parabolical line in the eighteenth chapter, where I was deceived, I am
also deceived here_.” True. But because in the eighteenth chapter of
this English edition I have found a straight line equal to the spiral
line of Archimedes. I must here put you in mind that by these words in
your objections to the fifth article at your number two, _Quatenus verum
est, etc._, _we have demonstrated prop. 10, 11, 13_, _Arithmetica
Infinitorum_; you make it appear that you thought your spiral (made of
arches or circles) was the true spiral of Archimedes; which is fully as
absurd as the quadrature of Joseph Scaliger, whose geometry you so much
despise.
To the sixth article, which is a digression concerning the analytics of
geometricians, you deny _that the efficient cause of the construction
ought to be contained in the demonstration_. As if any problem could be
known to be truly done, otherwise than by knowing first how, that is to
say, by what efficient cause, and in what manner, it is to be done.
Whatsoever is done without that knowledge, cannot be demonstrated to be
done; as you see in your computation of the parabola, and paraboloeides,
in your _Arithmetica Infinitorum_.
And whereas I said that _the ends of all straight lines drawn from a
straight line, and passing through one and the same point, if their
parts be proportional, shall be in a straight line_; is true and
accurate; as also, _if they begin in the circumference of a circle, they
shall also be in the circumference of another circle_. And so is this:
_if the proportion be duplicate, they shall be in a parabola_. All this
I say is true and accurately spoken. But this was no place for the
demonstration of it. Others have done it. And I perceive by that you put
in by parenthesis (“_Intelligis credo inter duas peripherias
concentricas_”) that you understand not what I mean.
Hitherto reach your objections to my geometry: for the rest of your
book, it containeth nothing but a collection of lies, wherewith you do
what you can, to extenuate as vulgar, and disgrace as false, that which
followeth, and to which you have made no special objection.
I shall therefore only add in this place concerning your _Analytica per
Potestates_, that it is no art. For the rule, both in Mr. Ougthred, and
in Des Cartes, is this: “_When a problem or question is propounded,
suppose the thing required done, and then using a fit ratiocination, put
A or some other vowel for the magnitude sought_.” How is a man the
better for this rule without another rule, how to know when the
ratiocination is fit? There may therefore be in this kind of analysis
more or less natural prudence, according as the analyst is more or less
wise, or as one man in choosing of the unknown quantity with which he
will begin, or in choosing the way of the consequences which he will
draw from the hypothesis, may have better luck than another. But this is
nothing to art. A man may sometimes spend a whole day in deriving of
consequences in vain, and perhaps another time solve the same problem in
a few minutes.
I shall also add, that symbols, though they shorten the writing, yet
they do not make the reader understand it sooner than if it were written
in words. For the conception of the lines and figures (without which a
man learneth nothing) must proceed from words either spoken or thought
upon. So that there is a double labour of the mind, one to reduce your
symbols to words, which are also symbols, another to attend to the ideas
which they signify. Besides, if you but consider how none of the
ancients ever used any of them in their published demonstrations of
geometry, nor in their books of arithmetic, more than for the roots and
potestates themselves; and how bad success you have had yourself in the
unskilful using of them, you will not, I think, for the future be so
much in love with them as to demonstrate by them that first part you
promise of your _Opera Mathematica_. In which, if you make not amends
for that which you have already published, you will much disgrace those
mathematicians you address your epistles to, or otherwise have
commended; as also the Universities, as to this kind of learning, in the
sight of learned men beyond sea. And thus having examined your pannier
of Mathematics, and finding in it no knowledge, neither of quantity, nor
of measure, nor of proportion, nor of time, nor of motion, nor of any
thing, but only of certain characters, as if a hen had been scraping
there; I take out my hand again, to put it into your other pannier of
theology, and good manners. In the mean time I will trust the objections
made by you the astronomer (wherein there is neither close reasoning,
nor good style, nor sharpness of wit, to impose upon any man) to the
discretion of all sorts of readers.
==========
OF MANNERS.
TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.
LESSON VI.
Having in the precedent lessons maintained the truth of my geometry, and
sufficiently made appear that your objections against it are but so many
errors of your own, proceeding from misunderstanding of the propositions
you have read in Euclid, and other masters of geometry; I leave it to
your consideration to whom belong, according to your own sentence, the
unhandsome attributes you so often give me upon supposition, that you
yourselves are in the right, and I mistaken; and come now to purge
myself of those greater accusations which concern my manners. It cannot
be expected that there should be much science of any kind in a man that
wanteth judgment; nor judgment in a man that knoweth not the manners due
to a public disputation in writing; wherein the scope of either party
ought to be no other than the examination and manifestation of the
truth. For whatsoever is added of contumely, either directly or
_scommatically_, is want of charity and uncivil, unless it be done by
way of reddition from him that is first provoked to it. I say unless it
be by way of reddition; for so was the judgment given by the emperor
Vespasian in a quarrel between a senator and a knight of Rome which had
given him ill language. For when the knight had proved that the first
ill language proceeded from the senator, the emperor acquitted him in
these words: “_Maledici senatoribus non oportere; remaledicere, fas et
civile esse_.” Nevertheless, now-a-days, uncivil words are commonly and
bitterly used by all that write in matter of controversy, especially in
divinity, excepting now and then such writers as have been more than
ordinarily well bred, and have observed how heinous and hazardous a
thing such contumely is amongst some sorts of men, whether that which is
said in disgrace be true or false. For evil words by all men of
understanding are taken for a defiance, and a challenge to open war. But
that you should have observed so much, who are yet in your mother’s
belly, was not a thing to be much expected.
The faults in manners you lay to my charge are these: 1. _Self-conceit._
2. _That I will be very angry with all men that do not presently submit
to my dictates._ 3. _That I had my doctrine concerning Vision, out of
papers which I had in my hands of Mr. Warner’s._ 4. _That I have injured
the universities._ 5. _That I am an enemy to religion._ These are great
faults; but such as I cannot yet confess. And therefore I must, as well
as I can, seek out the grounds upon which you build your accusation.
Which grounds (seeing you are not acquainted with my conversation) must
be either in my published writings, or reported to you by honest men,
and without suspicion of interest in reporting it. As for my
self-conceit and ostentation, you shall find no such matter in my
writings. That which you allege from thence is first, that in the
epistle dedicatory I say of my book _De Corpore_, “_though it be little,
yet it is full; and if good may go for great, great enough._” When a man
presenting a gift great or small to his betters, adorneth it the best he
can to make it the more acceptable; he that thinks this to be
ostentation and self-conceit, is little versed in the common actions of
human life. And in the same epistle, where I say of civil philosophy:
“_It is no ancienter than my book De Cive_;” these words are added: “_I
say it provoked, and that my detractors may see they lose their
labour_.” But that which is truly said, and upon provocation, is not
boasting, but defence. A short sum of that book of mine, now publicly in
French, done by a gentleman I never saw, carrieth the title of _Ethics
Demonstrated_ . The book itself translated into French, hath not only a
great testimony from the translator Sorberius, but also from Gassendus,
and Mersennus, who being both of the Roman religion had no cause to
praise it, or the divines of England have no cause to find fault with
it. Besides, you know that the doctrine therein contained is generally
received by all but those of the clergy, who think their interest
concerned in being made subordinate to the civil power; whose
testimonies therefore are invalid. Why therefore, if I commend it also
against them that dispraise it publicly, do you call it boasting? “_You
have heard_,” you say, “_that I had promised the quadrature of the
circle, &c._” You heard then that which was not true. I have been asked
sometimes, by such as saw the figure before me, what I was doing, and I
was not afraid to say I was seeking for the solution of that problem;
but not that I had done it. And afterwards being asked of the success, I
have said, I thought it done. This is not boasting; and yet it was
enough, when told again, to make a fool believe it was boasting. But
you, the astronomer, in the epistle before your philosophical essay, say
“_You had a great expectation of my philosophical and mathematical
works, before they were published_.” It may be so. Is that my fault? Can
a man raise a great expectation of himself by boasting? If he could,
neither of you would be long before you raised it of yourselves; saving
that what you have already published, has made it now too late. For I
verily believe there was never seen worse reasoning than in that
philosophical essay; which any judicious reader would believe proceeded
from a prevaricator, rather than from a man that believed himself; nor
worse principles, than those in your books of Geometry. The expectation
of that which should be written by me, was raised partly by the
_Cogitata Physica-Mathematica_ of Mersennus, wherein I am often named
with honour; and partly by others with whom I then conversed in Paris,
without any ostentation. That no man has a great expectation of any
thing that shall proceed from either of you two, I am content to let it
be your praise.
Another argument of my self-conceit, you take from my contempt of the
writers of metaphysics and school-divinity. If that be a sign of
self-conceit, I must confess I am guilty; and if your geometry had then
been published, I had contemned that as much. But yet I cannot see the
consequence (unless you lend me your better logic) from despising
insignificant and absurd language, to self-conceit.
And again, in your _Vindiciæ Academiarum_, you put for boasting, that in
my _Leviathan_ , page 331, I would have _that book by entire sovereignty
imposed upon the Universities_; and in my _Review_ , p. 713, that I say
of my _Leviathan_ , “_I think it may be profitably printed, and more
profitably taught in the University_.” The cause of my writing that
book, was the consideration of what the ministers before, and in the
beginning of, the civil war, by their preaching and writing did
contribute thereunto. Which I saw not only to tend to the abatement of
the then civil power, but also to the gaining of as much thereof as they
could (as did afterwards more plainly appear) unto themselves. I saw
also that those ministers, and many other gentlemen who were of their
opinion, brought their doctrines against the civil power from their
studies in the Universities. Seeing therefore that so much as could be
contributed to the peace of our country, and the settlement of sovereign
power without any army, must proceed from teaching; I had reason to
wish, that civil doctrine were truly taught in the Universities. And if
I had not thought that mine was such, I had never written it. And having
written it, if I had not recommended it to such as had the power to
cause it to be taught, I had written it to no purpose. To me therefore
that never did write anything in philosophy to show my wit, but, as I
thought at least, to benefit some part or other of mankind, it was very
necessary to commend my doctrine to such men as should have the power
and right to regulate the Universities. I say my doctrine; I say not my
_Leviathan_ . For wiser men may so digest the same doctrine as to fit it
better for a public teaching. But as it is, I believe it hath framed the
minds of a thousand gentlemen to a conscientious obedience to present
government, which otherwise would have wavered in that point. This
therefore was no vaunting, but a necessary part of the business I took
in hand. You ought also to have considered, that this was said in the
close of that part of my book which concerneth policy merely civil.
Which part, if you, the astronomer, that now think the doctrine unworthy
to be taught, were pleased once to honour with praises printed before
it, you are not very constant nor ingenuous. But whether you did so or
not, I am not certain, though it was told me for certain. If it were not
you, it was somebody else whose judgment has as much weight at least as
yours.
And for anything you have to say from your own knowledge, I remember not
that I ever saw either of your faces. Yet you, the professor of
geometry, go about obliquely to make me believe that Vindex hath
discoursed with me, once at least, though I remember it not. I suppose
it therefore true; but this I am sure is false, that either he or any
man living did ever hear me brag of my science, or praise myself, but
when my defence required it. Perhaps some of our philosophers that were
at Paris at the same time, and acquainted with the same learned men that
I was acquainted with, might take for bragging the maintaining of my
opinions, and the not yielding to the reasons alledged against them. If
that be ostentation, they tell you the truth. But you that are so wise
should have considered, that even such men as profess philosophy are
carried away with the passions of emulation and envy (the sole ground of
this your accusation) as well as other men, and instanced in yourselves.
And this is sufficient to shake off your aspersions of ostentation and
self-conceit. For if I added, that my acquaintance know that I am
naturally of modest rather than of boasting speech, you will not believe
it; because you distinguish not between that which is said upon
provocation, and that which is said without provocation, from vain
glory.
The next accusation is: “_That I will be very angry with all men that do
not presently submit to my dictates; and that for advancing the
reputation of my own skill, I care not what unworthy reflections I cast
on others_.” This is in the epistle placed before the _Vindiciæ
Academiarum_, subscribed by N S, as the plain song for H D in the rest
of the book to descant upon. I know well enough the authors’ names; and
am sorry that N S has lent his name to be abused to so ill a purpose.
But how does this appear? What argument, what witness is there of it?
You offer none; nor am I conscious of any. I begin to suspect since you,
the professor of geometry, have in your objections to the twentieth
chapter these words concerning “_Vindex, ocularis ille testis de quo hic
agitur, erat, ni fallor, ille ipse_,”--that Vindex himself, in other
company, has bestowed a visit on me. Seeing you will have me believe it,
let it be so; and, as it is likely, not long after my return into
England. At which time (for the reputation, it seems, I had gotten by my
boasting) divers persons that professed to love philosophy and
mathematics, came to see me; and some of them to let me see them, and
hear and applaud what they applauded in themselves. I see now it hath
happened to me with Vindex, as it happened to Dr. Harvey with Moranus.
Moranus, a jesuit, came out of Flanders hither, especially, as he says,
to see what learned men in divinity, ethics, physics, and geometry, were
here yet alive, to the end that by discoursing with them in these
sciences, he might correct either his own, or their errors. Amongst
others he was brought, he says, to that most civil and renowned old man
Dr. Harvey. That is very well. And in good earnest if he had made good
use of the time which was very patiently afforded him, he might have
learned of him (or of no man living) very much knowledge concerning the
circulation of the blood, the generation of living creatures, and many
other difficult points of natural philosophy. And if he had had anything
in him but common and childish learning, he could have showed it nowhere
more to his advantage, than before him that was so great a judge of such
matters. But what did he? That precious time (which was but little,
because he was to depart again presently for Flanders) he bestowed
wholly in venting his own childish opinions, not suffering the Doctor
scarce to speak; losing thereby the benefit he came for, and discovering
that he came not to hear what others could say, but to show to others
how learned he was himself already. Why else did he take so little time,
and so misspend it? Or why returned he not again? But when he had talked
away his time, and found (though patiently and civilly heard) he was not
much admired, he took occasion, writing against me, to be revenged of
Dr. Harvey, by slighting his learning publicly; and tells me that his
learning was only experiments; which he says I say have no more
certainty than civil histories. Which is false. My words are: “_Ante hos
nihil certi in physica erat præter experimenta cuique sua, et historias
naturales, si tamen et hæ dicendæ certæ sint, quæ civilibus historiis
certiores non sunt_.” Where I except expressly from uncertainty the
experiments that every man maketh to himself. But you see the near cut,
by which vain glory joined with ignorance passeth quickly over to envy
and contumely.
Thus it seems by your own confession I was used by Vindex. He comes with
some of my acquaintance in a visit. What he said I know not, but if he
discoursed then, as in his philosophical essay he writeth, I will be
bold to say of myself, I was so far from morosity, or, to use his
phrase, from being tetrical, as I may very well have a good opinion of
my own patience. And if there passed between us the discourse you
mention in your _Elenchus_, page 116, it was an incivility in him so
great, that without great civility I could not have abstained from
bidding him be gone. That which passed between us you say was this: “_I
complained that whereas I made sense, nothing but a perception of motion
in the organ, nevertheless, the philosophy schools through all Europe,
led by the text of Aristotle, teach another doctrine, namely, that
sensation is performed by species_.” This is a little mistaken. For I do
glory, not complain, that whereas all the Universities of Europe hold
sensation to proceed from species, I hold it to be a perception of
motion in the organ. The answer of Vindex, you say, was: “_That the
other hypothesis, whereby sense was explicated by the principles of
motion, was commonly admitted here before my book came out, as having
been sufficiently delivered by Des Cartes, Gassendus, and Sir Kenelm
Digby, before I had published anything in this kind_.” This then, it
seems, was it that made me angry. Truly I remember not an angry word
that ever I uttered in all my life to any man that came to see me,
though some of them have troubled me with very impertinent discourse;
and with those that argued with me, how impertinently soever, I always
thought it more civility to be somewhat earnest in the defence of my
opinion, than by obstinate and affected silence to let them see I
contemned them, or hearkened not to what they said. If I were earnest in
making good, that the manner of sensation by such motion as I had
explicated in my _Leviathan_ , is in none of the authors by him named,
it was not anger, but a care of not offending him, with any sign of the
contempt which his discourse deserved. But it was incivility in him to
make use of a visit, which all men take for a profession of friendship,
to tell me that that which I had already published for my own, was found
before by Des Cartes, Gassendus, and Sir Kenelm Digby. But let any man
read Des Cartes; he shall find, that he attributeth no motion at all to
the object of sense, but an inclination to action, which inclination no
man can imagine what it meaneth. And for Gassendus, and Sir Kenelm
Digby, it is manifest by their writings, that their opinions are not
different from that of Epicurus, which is very different from mine. Or
if these two, or any of those I conversed with at Paris, had prevented
me in publishing my own doctrine, yet since it was there known and
declared for mine by Mersennus in the preface to his _Ballistica_ (of
which the three first leaves are employed wholly in the setting forth of
my opinion concerning sense, and the rest of the faculties of the soul)
they ought not therefore to be said to have found it out before me. And
consequently this answer which you say was given me by Vindex was
nothing else but untruth and envy; and, because it was done by way of
visit, incivility. But you have not alleged, nor can allege, any words
of mine, from which can be drawn that I am so angry as you say I am with
those that submit not to my dictates. Though the discipline of the
University be never so good; yet certainly this behaviour of yours and
his are no good arguments to make it thought so. But you the professor
of geometry, that out of my words spoken against Vindex in my twentieth
chapter, argue my angry humour, do just as well, as when (in your
_Arithmetica Infinitorum_) from the continual increase of the excess of
the row of squares above the third part of the aggregate of the
greatest, you conclude they shall at last be equal to it. For though you
knew that Vindex had given me first the worst words that possibly can be
given, yet you would have that return of mine to be a demonstration of
an angry humour; not then knowing what I told you even now in the
beginning of this lesson, of the sentence given by Vespasian. But to
this point I shall speak again hereafter.
Your third accusation is: “_That I had my doctrine of vision, which I
pretended to be my own, out of papers which I had a long time in my
hands of Mr. Warner’s_.” I never had sight of Mr. Warner’s papers in all
my life, but that of _Vision by Refraction_ (which by his approbation I
carried with me to Park, and caused it to be printed under his own name,
at the end of Mersennus his _Cogitata Physico-Mathematica_, which you
may have there seen, and another treatise of the proportions of alloy in
gold and silver coin; which is nothing to the present purpose). In all
my conversation with him, I never heard him speak of anything he had
written, or was writing, _De penicillo optico_. And it was from me that
he first heard it mentioned that light and colour were but fancy. Which
he embraced presently as a truth, and told me it would remove a rub he
was then come to in the discovery of the place of the image. If after my
going hence he made any use of it (though he had it from me, and not I
from him), it was well done. But wheresoever you find my principles,
make use of them, if you can, to demonstrate all the symptoms of vision;
and I will do (or rather have done and mean to publish) the same; and
let it be judged by that, whether those principles be of mine, or other
men’s invention. I give you time enough, and this advantage besides,
that much of my optics hath been privately read by others. For I never
refused to lend my papers to my friends, as knowing it to be a thing of
no prejudice to the advancement of philosophy, though it be, as I have
found it since, some prejudice to the advancement of my own reputation
in those sciences; which reputation I have always postposed to the
common benefit of the studious.
You say further (you the geometrician) that I had the proposition of the
spiral line equal to a parabolical line from Mr. Robervall: true. And if
I had remembered it, I would have taken also his demonstration; though
if I had published his, I would have suppressed mine. I was comparing in
my thoughts those two lines, spiral and parabolical, by the motions
wherewith they were described; and considering those motions as uniform,
and the lines from the centre to the circumference, not to be little
parallelograms, but little sectors, I saw that to compound the true
motion of that point which described the spiral, I must have one line
equal to half the perimeter, the other equal to half the diameter. But
of all this I had not one word written. But being with Mersennus and Mr.
Robervall in the cloister of the convent, I drew a figure on the wall,
and Mr. Robervall perceiving the deduction I made, told me that since
the motions which make the parabolical line, are one uniform, the other
accelerated, the motions that make the spiral must be so also; which I
presently acknowledged; and he the next day, from this very method,
brought to Mersennus the demonstration of their equality. And this is
the story mentioned by Mersennus, prop. 25, corol. 2, of his
_Hydraulica_; which I know not who hath most magnanimously interpreted
to you in my disgrace.
The fourth accusation is: “_That I have injured the Universities_.”
Wherein? First, “_In that I would have the doctrine of my Leviathan by
entire sovereignty be imposed on them_.” You often upbraid me with
thinking well of my own doctrine; and grant by consequence, that I
thought this doctrine good; I desired not therefore that anything should
be imposed upon them, but what (at least in my opinion) was good both
for the Commonwealth and them. Nay more, I would have the state make use
of them to uphold the civil power, as the Pope did to uphold the
ecclesiastical. Is it not absurdly done to call this an injury? But to
question, you will say, whether the civil doctrine there taught be such
as it ought to be, or not, is a disgrace to the Universities. If that be
certain, it is certain also that those sermons and books, which have
been preached and published, both against the former and the present
government, directly or obliquely, were not made by such ministers and
others as had their breeding in the Universities; though all men know
the contrary. But the doctrine which I would have to be taught there,
what is it? It is this: “_That all men that live in a Commonwealth, and
receive protection of their lives and fortunes from the supreme governor
thereof, are reciprocally bound, as far as they are able, and shall be
required, to protect that governor_.” Is it, think you, an unreasonable
thing to impose the teaching of such doctrine upon the Universities? Or
will you say they taught it before, when you know that so many men which
came from the Universities to preach to the people, and so many others
that were not ministers, did stir the people up to resist the then
supreme civil power? And was it not truly therefore said, that the
Universities receiving their discipline from the authority of the pope,
were the shops and operatories of the clergy? Though the competition of
the papal and civil power be taken now away, yet the competition between
the ecclesiastical and the civil power hath manifestly enough appeared
very lately. But neither is this an upbraiding of an University (which
is a corporation or body artificial), but of particular men, that desire
to uphold the authority of a Church, as of a distinct thing from the
Commonwealth. How would you have exclaimed, if, instead of recommending
my _Leviathan_ to be taught in the Universities, I had recommended the
erecting of a new and lay-university, wherein lay-men should have the
reading of physics, mathematics, moral philosophy, and politics, as the
clergy have now the sole teaching of divinity? Yet the thing would be
profitable, and tend much to the polishing of man’s nature, without much
public charge. There will need but one house, and the endowment of a few
professions. And to make some learn the better, it would do very well
that none should come thither sent by their parents, as to a trade to
get their living by, but that it should be a place for such ingenuous
men, as being free to dispose of their own time, love truth for itself.
In the mean time divinity may go on in Oxford and Cambridge to furnish
the pulpit with men to cry down the civil power, if they continue to do
as they did. If I had, I say, made such a motion in my _Leviathan_ ,
though it would have offended the divines, yet it had been no injury.
But it is an injury, you will say, to deny in general the utility of the
ancient schools, and to deny that we have received from them our
geometry. True, if I had not spoken distinctly of the schools of
philosophy, and said expressly, that the geometricians passed not then
under the name of philosophers; and that in the school of Plato (the
best of the ancient philosophers) none were received that were not
already in some measure geometricians. Euclid taught geometry; but I
never heard of a sect of philosophers called Euclidians, or
Alexandrians, or ranged with any of the other sects, as Peripatetics,
Stoics, Academics, Epicureans, Pyrrhonians, &c. But what is this to the
Universities of Christendom? Or why are we beholden for geometry to our
universities, more than to Gresham College, or to private men in London,
Paris, and other places, which never taught or learned it in a public
school? For even those men that living in our Universities have most
advanced the mathematics, attained their knowledge by other means than
that of public lectures, where few auditors, and those of unequal
proficiency, cannot make benefit by one and the same lesson. And the
true use of public professors, especially in the mathematics, being to
resolve the doubts, and problems, as far as they can, of such as come
unto them with desire to be informed.
That the Universities now are not regulated by the Pope, but by the
civil power, is true, and well. But where say I the contrary? And thus
much for the first injury.
Another, you say, is this, that in my _Leviathan_ , p. 670, I say: “_The
principal schools were ordained for the three professions of Roman
religion, Roman law, and the art of medicine_.” Thirdly, that I say:
“_Philosophy had no otherwise place there than as a hand-maid to Roman
religion_.” Fourthly: “_Since the authority of Aristotle was received
there, that study is not properly philosophy, but Aristotelity_.”
Fifthly: “_That for geometry, till of late times it had no place there
at all_.” As for the second, it is too evident to be denied; the
fellowships having been all ordained for those professions; and (saving
the change of religion) being so yet. Nor hath this any reflection upon
the Universities, either as they now are, or as they then were, seeing
it was not in their own power to endow themselves, or to receive other
laws and discipline than their founder and the state was pleased to
ordain. For the third, it is also evident. For all men know that none
but the Roman religion had any stipend or preferment in any university,
where that religion was established? No, nor for a great while, in their
commonwealths; but were everywhere persecuted as heretics. But you will
say, the words of my _Leviathan_ are not, philosophy “_had no place_,”
but “_hath no place_.” Are you not ashamed to lay to my charge a mistake
of the word _hath_ for _had_? which was either a mistake of the printer,
or if it were so in the copy, it could be no other than the mistake of a
letter in the writing, unless you think you can make men believe that
after fifty years being acquainted with what was publicly professed and
practised in Oxford and Cambridge, I knew not what religion they were
of. This taking of advantage from the mistake of a word, or of a letter,
I find also in the _Elenchus_, where for _prætendit se scire_, there is
_prætendit scire_, which you the geometrician sufficiently mumble,
mistaking it I think for an anglicism, not for a fault of the
impression.
To the fourth, you pretend, that men are not now so tied to Aristotle as
not to _enjoy a liberty of philosophising, though it were otherwise when
I was conversant in Magdalen Hall_. Was it so then? Then am I absolved,
unless you can show some public act of the university made since that
time to alter it. For it is not enough to name some few particular
ingenuous men that usurp that liberty in their private discourses, or,
with connivance, in their public disputations. And your doctrine, that
even here you avow, of _abstracted essences_, _immaterial substances_,
and of _nunc-stans_; and your improper language in using the word (not
as mine, for I have it nowhere) _successive eternity_; as also your
doctrine of _condensation_, and your arguing from natural reason the
incomprehensible mysteries of religion, and your malicious writing, are
very shrewd signs that you yourselves are none of those which you say do
_freely philosophise_; but that both your philosophy and your language
are under the servitude, not of the Roman religion, but of the ambition
of some other doctors, that seek, as the Roman clergy did, to draw all
human learning to the upholding of their power ecclesiastical. Hitherto
therefore there is no injury done to the universities. For the fifth,
you grant it, namely, “_that till of late there was no allowance for the
teaching of geometry_.” But lest you should be thought to grant me
anything, you say, you the astronomer, “_geometry hath now so much place
in the universities, that when Mr. Hobbes shall have published his
philosophical and geometrical pieces, you assure yourself you shall be
able to find a greater number in the university who will understand as
much, or more, of them than he desired they should_,” &c. But though
this be true of the _now_, yet it maketh nothing against my _then_. I
know well enough that Sir Henry Savile’s lectures were founded and
endowed since. Did I deny _then_ that there were in Oxford many good
geometricians? But I deny _now_, that either of you is of the number.
For my philosophical and geometrical pieces are published, and you have
understood only so much in them, as all men will easily see by your
objections to them, and by your own published geometry, that neither of
you understand anything either in philosophy or in geometry. And yet you
would have those books of yours to stand for an argument, and to be an
index of the philosophy and geometry to be found in the universities.
Which is a greater injury and disgrace to them, than any words of mine,
though interpreted by yourselves.
Your last and greatest accusation, or rather railing (for an accusation
should contain, whether true or false, some particular fact, or certain
words, out of which it might seem at least to be inferred), is, that I
am an enemy to religion. Your words are: “_It is said that Mr. Hobbes is
no otherwise an enemy to the Roman religion, saving only as it hath the
name of religion_.” This is said by Vindex. You, the geometrician, in
your epistle dedicatory, say thus: “_With what pride and imperiousness
he tramples on all things both human and divine, uttering fearful and
horrible words of God_, (peace), _of sin, of the holy Scripture, of all
incorporeal substances in general, of the immortal soul of man, and of
the rest of the weighty points of religion_ (down), _it is not so much
to be doubted as lamented_.” And at the end of your objections to the
eighteenth chapter, “_Perhaps you take the whole history of the fall of
Adam for a fable, which is no wonder, when you say the rules of
honouring and worshipping of God are to be taken from the laws_.” Down,
I say; you bark now at the supreme legislative power. Therefore it is
not I, but the laws which must rate you off. But do not many other men,
as well as you, read my _Leviathan_ , and my other books? And yet they
all find not such enmity in them against religion. Take heed of calling
them all atheists that have read and approved my _Leviathan_ . Do you
think I can be an atheist and not know it? Or knowing it, durst have
offered my atheism to the press? Or do you think him an atheist, or a
contemner of the Holy Scripture, that sayeth nothing of the Deity but
what he proveth by the Scripture? You that take so heinously that I
would have the rules of God’s worship in a Christian commonwealth taken
from the laws, tell me, from whom you would have them taken? From
yourselves? Why so, more than from me? From the bishops? Right, if the
supreme power of the commonwealth will have it so; if not, why from them
rather than from me? From a consistory of presbyters by themselves, or
joined with lay-elders, whom they may sway as they please? Good, if the
supreme governor of the commonwealth will have it so; if not, why from
them, rather than from me, or from any man else? They are wiser and
learneder than I. It may be so; but it has not yet appeared. Howsoever,
let that be granted. Is there any man so very a fool as to subject
himself to the rules of other men in those things which so nearly
concern himself, for the title they assume of being wise and learned,
unless they also have the sword which must protect them. But it seems
you understand the sword as comprehended. If so, do you not then receive
the rules of God’s worship from the civil power? Yes, doubtless; and you
would expect, if your consistory had that sword, that no man should dare
to exercise or teach any rules concerning God’s worship which were not
by you allowed. See therefore how much you have been transported by your
malice towards me, to injure the civil power by which you live. If you
were not despised, you would in some places and times, where and when
the laws are more severely executed, be shipped away for this your
madness to America, I would say, to Anticyra. What luck have I, when
this, of the laws being the rules of God’s public worship, was by me
said and applied to the vindication of the Church of England from the
power of the Roman clergy, it should be followed with such a storm from
the ministers, presbyterian and episcopal, of the Church of England?
Again, for those other points, namely, that I approve not of incorporeal
bodies, nor of other immortality of the soul, than that which the
Scripture calleth eternal life, I do but as the Scripture leads me. To
the texts whereof by me alleged, you should either have answered, or
else forborne to revile me for the conclusions I derived from them.
Lastly, what an absurd question is it to ask me whether it be in the
power of the magistrate, whether the world be eternal or not? It were
fit you knew it is in the power of the supreme magistrate to make a law
for the punishment of them that shall pronounce publicly of that
question anything contrary to that which the law hath once pronounced.
The truth is, you are content that the papal power be cut off, and
declaimed against as much as any man will; but the ecclesiastical power,
which of late was aimed at by the clergy here, being a part thereof,
every violence done to the papal power is sensible to them yet; like
that which I have heard say of a man, whose leg being cut off for the
prevention of a gangrene that began in his toe, would nevertheless
complain of a pain in his toe, when his leg was cut off.
Thus much in my defence; which I believe if you had foreseen, this
accusation of yours had been left out. I come now to examine (though it
be done in part already) what manners those are which I find everywhere
in your writings.
And first, how came it into your minds that a man can be an atheist, I
mean an atheist in his conscience? I know that David confesseth of
himself, upon sight of the prosperity of the wicked, that his feet had
almost slipped, that is, that he had slipped into a short doubtfulness
of the Divine Providence. And if anything else can cause a man to slip
in the same kind, it is the seeing such as you (who though you write
nothing but what is dictated to each of you by a doctor of divinity) do
break the greatest of God’s commandments, which is charity, in every
line before his face. And though such forgettings of God be somewhat
more than short doubtings, and sudden transportations incident to human
passion, yet I do not for that cause think you atheists and enemies of
religion, but only ignorant and imprudent Christians. But how, I say,
could you think me an atheist, unless it were because finding your
doubts of the Deity more frequent than other men do, you are thereby the
apter to fall upon that kind of reproach? Wherein you are like women of
poor and evil education when they scold; amongst whom the readiest
disgraceful word is whore: why not thief, or any other ill name, but
because, when they remember themselves, they think that reproach the
likeliest to be true?
Secondly, tell me what crime it was which the Latins called by the name
of _scelus_? You think not, unless you be Stoics, that all crimes are
equal. _Scelus_ was never used but for a crime of greatest mischief, as
the taking away of life and honour; and besides, basely acted, as by
some clandestine way, or by such a way as might be covered with a lie.
But when you insinuate in a writing published that I am an atheist, you
make yourselves authors to the multitude, and do all you can to stir
them up to attempt upon my life; and if it succeed, then to sneak out of
it by leaving the fault on them that are but actors. This is to
endeavour great mischief basely, and therefore _scelus_. Again, to
deprive a man of the honour he hath merited, is no little wickedness;
and this you endeavour to do by publishing falsely that I challenge as
my own the inventions of other men. This is therefore _scelus_ publicly
to tell all the world that I will be angry with all men that do not
presently submit to my dictates; to deprive me of the friendship of all
the world; great damage, and a lie, and yours. For to publish any
untruth of another man to his disgrace, on hearsay from his enemy, is
the same fault as if he published it on his own credit. If I should say
I have heard that Dr. Wallis was esteemed at Oxford for a simple fellow,
and much inferior to his fellow-professor Dr. Ward (as indeed I have
heard, but do not believe it), though this be no great disgrace to Dr.
Wallis, yet he would think I did him injury. Therefore public accusation
upon hearsay is _scelus_. And whosoever does any of these things does
_sceleratè_. But you the professors of the mathematics at Oxford, by the
advice of two doctors of divinity have dealt thus with me. Therefore you
have done, I say not foolishly, though no wickedness be without folly,
but _sceleratè_, ὅπερ ἔδει δεῖξαι.
Thirdly, it is ill manners, in reprehending truth, to send a man in a
boasting way to your own errors; as you the professor of geometry have
often sent me to your two tractates of the _Angle of Contact_ and
_Arithmetica Infinitorum_.
Fourthly, it is ill manners, to diminish the just reputation of worthy
men after they be dead, as you the professor of geometry have done in
the case of Joseph Scaliger.
Fifthly, when I had in my _Leviathan_ suffered the clergy of the Church
of England to escape, you did imprudently in bringing any of them in
again. An Ulysses upon so light an occasion would not have ventured to
return again into the cave of Polyphemus.
Lastly, how ill does such levity and scurrility, which both of you have
shown so often in your writings, become the gravity and sanctity
requisite to the calling of the ministry? They are too many to be
repeated. Do but consider, you the geometrician, how unhandsome it is to
play upon my name, when both yours and mine are plebeian names; though
from Willis by Wallis, you go from yours in Wallisius. The jest of using
at every word _mi Hobbi_, is lost to them beyond sea. But this is not so
ill as some of the rest. I will write out one of them, as it is in the
fourth page of your _Elenchus_: “_Whence it appears that your Empusa was
of the number of those fairies which you call in English hob-goblins.
The word is made of_ ἕν and πους; _and thence comes the children’s play
called the play of Empusa, Anglicè_ (hitherto in Latin all but
_hob-goblins_, then follows in English) _fox, fox, come out of your
hole_ (then in Latin again), _in which the boy that is called the fox,
holds up one foot, and jumps with the other, which in English is to
hop_.” When a stranger shall read this, and hoping to find therein some
witty conceit, shall with much ado have gotten it interpreted and
explained to him, what will he think of our doctors of divinity at
Oxford, that will take so much pains as to go out of the language they
set forth in, for so ridiculous a purpose? You will say it is a pretty
_paranomasia_. How you call it there I know not, but it is commonly
called here a _clinch_; and such a one as is too insipid for a boy of
twelve years old, and very unfit for the sanctity of a minister, and
gravity of a doctor of divinity. But I pray you tell me where it was you
read the word _empusa_ for the boy’s play you speak of, or for any other
play amongst the Greeks? In this (as you have done throughout all your
other writings) you presume too much upon your first cogitations. There
be a hundred other scoffing passages, and ill-favoured attributes given
me in both your writings, which the reader will observe without my
pointing to them, as easily as you would have him; and which perhaps
some young students, finding them full of gall, will mistake for salt.
Therefore to disabuse those young men, and to the end they may not
admire such kind of wit, I have here and there been a little sharper
with you than else I would have been. If you think I did not spare you,
but that I had not wit enough to give you as scornful names as you give
me, are you content I should try? Yes (you the geometrician will say)
give me what names you please, so you call me not _Arithmetica
Infinitorum_. I will not. Nor _Angle of Contact_ ; nor _Arch Spiral_ ;
nor _Quotient_ . I will not. But I here dismiss you both together. So go
your ways, you _Uncivil Ecclesiastics, Inhuman Divines, Dedoctors of
morality, Unasinous Colleagues, Egregious pair of Issachars, most
wretched Vindices and Indices Academiarum_; and remember Vespasian’s
law, that it is uncivil to give ill language first, but civil and lawful
to return it. But much more remember the law of God, to obey your
sovereigns in all things; and not only not to derogate from them, but
also to pray for them, and as far as you can to maintain their
authority, and therein your own protection. And, do you hear? take heed
of speaking your mind so clearly in answering my _Leviathan_, as I have
done in writing it. You should do best not to meddle with it at all,
because it is undertaken, and in part published already, and will be
better performed, from term to term, by one Christopher Pike.
ΣΤΙΓΜΑΙ
Αγεωμετρίας, Αγροικίας, Αντίπολιτείας, Αμαθείας,
OR
MARKS
OF THE
ABSURD GEOMETRY, RURAL LANGUAGE, SCOTTISH
CHURCH POLITICS, AND BARBARISMS
OF
JOHN WALLIS,
PROFESSOR OF GEOMETRY AND DOCTOR OF DIVINITY.
BY
THOMAS HOBBES,
OF MALMESBURY.
TO THE RIGHT HONOURABLE
HENRY, LORD PIERREPONT,
VISCOUNT NEWARK, EARL OF KINGSTON, AND
MARQUIS OF DORCHESTER.
==========
MY MOST NOBLE LORD,
I did not intend to trouble your Lordship twice with this contention
between me and Dr. Wallis. But your Lordship sees how I am constrained
to it; which, whatsoever reply the Doctor makes, I shall be constrained
to no more. That which I have now said of his Geometry, Manners,
Divinity, and Grammar, altogether is not much, though enough. As for
that which I here have written concerning his Geometry, which you will
look for first, is so clear, that not only your Lordship, and such as
have proceeded far in that science, but also any man else that doth but
know how to add and subtract proportions, (which is taught at the
twenty-third proposition of the sixth of Euclid), may see the Doctor is
in the wrong. That which I say of his ill language and politics is yet
shorter. The rest, which concerneth grammar, is almost all another
man’s, but so full of learning of that kind, as no man that taketh
delight in knowing the proprieties of the Greek and Latin tongues, will
think his time ill bestowed in the reading it. I give the Doctor no more
ill words, but am returned from his manners to my own. Your Lordship may
perhaps say, my compliment in my title-page is somewhat coarse; and it
is true. But, my Lord, it is since the writing of the title-page, that I
am returned from the Doctor’s manners to my own; which are such as I
hope you will not be ashamed to own me, my Lord, for one of
Your Lordship’s most humble
and obedient servants,
THOMAS HOBBES.
==========
TO
DOCTOR WALLIS,
IN ANSWER TO HIS
SCHOOL DISCIPLINE
---
SIR,
When unprovoked you addressed unto me, in your _Elenchus_, your harsh
compliment with great security, wantonly to show your wit, I confess you
made me angry, and willing to put you into a better way of considering
your own forces, and to move you a little as you had moved me, which I
perceive my lessons to you have in some measure done; but here you shall
see how easily I can bear your reproaches, now they proceed from anger,
and how calmly I can argue with you about your geometry and other parts
of learning.
I shall in the first part confer with you about your _Arithmetica
Infinitorum_, and afterwards compare our manner of elocution; then your
politics; and last of all your grammar and critics. Your spiral line is
condemned by him whose authority you use to prove me a plagiary, (that
is, a man that stealeth other men’s inventions, and arrogates them to
himself), whether it be Roberval or not that writ that paper, I am not
certain. But I think I shall be shortly; but whosoever it be, his
authority will serve no less to show that your doctrine of the spiral
line, from the fifth to the eighteenth proposition of your _Arithmetica
Infinitorum_, is all false; and that the principal fault therein (if all
faults be not principal in geometry, when they proceed from ignorance of
the science) is the same that I objected to you in my _Lessons_. And for
the author of that paper, when I am certain who it is, it will be then
time enough to vindicate myself concerning that name of plagiary. And
whereas he challenges the invention of your method delivered in your
_Arithmetica Infinitorum_, to have been his before it was yours, I
shall, I think, by and by say that which shall make him ashamed to own
it; and those that writ those encomiastic epistles to you ashamed of the
honour they meant to you. I pass therefore to the nineteenth
proposition, which in Latin is this: your geometry!
“_Si proponatur series quantitatum in duplicata ratione arithmetice
proportionalium (sive juxta seriem numerorum quadraticorum) continue
crescentium, a puncto vel 0 inchoatarum, (puta ut 0. 1. 4. 9. 16. etc.),
propositum sit, inquirere quam habeat illa rationem ad seriem totidem
maximæ æqualium._
“_Fiat investigatio per modum inductionis ut_ (_in prop. 1_)
_Eritque_,
(0 + 1 = 1)/(1 + 1 = 2) = (1)/(3) + (1)/(6)
(0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (1)/(3) + (1)/(12)
(0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) = (1)/(3) + (1)/(18) _et sic
deinceps_.
“_Ratio proveniens est ubique major quam subtripla seu (1)/(3); excessus
autem perpetuo decrescit prout numerus terminorum augetur (puta (1)/(6)
(1)/(12) (1)/(18) (1)/(24) etc.) aucto nimirum fractionis denominatore
sive consequente rationis in singulis locis numero senario (ut patet) ut
sit rationis provenientis excessus supra subtriplam, ea quam habet
unitas ad sextuplum numeri terminorum post 0; adeoque._”
That is, if there be propounded a row of quantities in duplicate
proportion of the quantities arithmetically proportional (or proceeding
in the order of the square numbers) continually increasing; and
beginning at a point or 0; let it be propounded to find what proportion
the row hath; to as many quantities equal to the greatest;
Let it be sought by induction (as in the first proposition).
The proportion arising is everywhere greater than subtriple, or (1)/(3),
and the excess perpetually decreaseth as the number of terms is
augmented, as here, (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c.
denominator of the fraction being in every place augmented by the number
six, as is manifest; so that the excess of the rising proportion above
subtriple is the same which unity hath to six times the number of terms
after 0; and so.
Sir, in these your characters I understand by the cross + that the
quantities on each side of it are to be added together and make one
aggregate; and I understand by the two parallel lines = that the
quantities between which they are placed are one to another equal; this
is your meaning, or you should have told us what you meant else; I
understand also, that in the first row 0 + 1 is equal to 1, and 1 + 1
equal to 2; and that in the second row 0 + 1 + 4 is equal to 5; and 4 +
4 + 4 equal to 12; but (which you are too apt to grant) I understand
your symbols no further; but must confer with yourself about the rest.
And first I ask you (because fractions are commonly written in that
manner) whether in the uppermost row (which is (0 + 1 = 1)/(1 + 1 = 2) =
(1)/(3) + (1)/(6))(0)/(1) be a fraction, (1)/(1) be a fraction, (1)/(2)
be a fraction, that is to say, a part of an unit, and if you will, for
the cypher’s sake, whether (0)/(1), be an infinitely little part of 1;
and whether (1)/(1) or 1 divided by 1 signify an unity? if that be your
meaning, then the fraction (0)/(1) added to the fraction (1)/(1) is
equal to the fraction (1)/(2): But the fraction (0)/(1) is equal to O;
therefore the fraction (0)/(1) + (1)/(1) is equal to the fraction
(1)/(1); and (1)/(1) equal to (1)/(2) which you will confess to be an
absurd conclusion, and cannot own that meaning.
I ask you therefore again, if by (0)/(1) you mean the proportion of 0 to
1; and consequently by (1)/(1) the proportion of 1 to 1, and by (1)/(2)
the proportion of 1 to 2: if so, then it will follow, that if the
proportions of 0 to 1 and of 1 to 1 be compounded by addition, the
proportion arising will be the proportion of 1 to 2. But the proportion
of 0 to 1 is infinitely little, that is, none. Therefore the proposition
arising by composition will be that of 1 to 1, and equal (because of the
symbol =) to the proportion of 1 to 2, and so 1 = 2. This also is so
absurd that I dare say that you will not own it.
There may be another meaning yet: perhaps you mean that the uppermost
quantity 0 + 1 is equal to the uppermost quantity 1; and the lowermost
quantity 1 + 1 equal to the lowermost quantity 2: which is true. But how
then in this equation (1)/(2) = (1)/(3) + (1)/(6)? Is the uppermost
quantity 1 equal to the uppermost quantity 1 + 1; or the lowermost
quantity 2 equal to the lowermost quantity 3 + 6? Therefore neither can
this be your meaning. Unless you make your symbols more significant, you
must not blame me for want of understanding them.
Let us now try what better success we shall have where the places are
three, as here:
(0 + 1 + 4 = 5)/(4 + 4 + 4 = 12) = (5)/(12) = (1)/(3) + (1)/(12):
If your symbols be fractions, the compound of them by addition is
(5)/(4), for 0(1)/(4) and (4)/(4) make (5)/(4); and consequently
(because of the symbol = ) (5)/(4) equal to (5)/(12), which is not to be
allowed, and therefore that was not your meaning. If you meant that the
proportions of 0 to 4 and of 1 to 4 and of 4 to 4 compounded, is equal
to the proportion of 5 to 12, you will fall again into no less an
inconvenience. For the proportion arising out of that composition will
be the proportion of 1 to 4. For the proportion of 0 to 4 is infinitely
little. Then to compound the other two, set them in this order 1. 4. 4.
and you have a proportion compounded of 1 to 4 and of 4 to 4, namely,
the proportion of the first to the last, which is of 1 to 4, which must
be equal, by this your meaning, to the proportion of 5 to 12, and
consequently as 5 to 12, so is 1 to 4, which you must not own. Lastly,
if you mean that the uppermost quantities to the uppermost, and the
lowermost to the lowermost in the first equation are equal, it is
granted, but then again in the second equation it is false. It concerns
your fame in the mathematics to look about how to justify these
equations which are the premises to your conclusion following, namely,
that the proportion arising is every where greater than sub-triple, or a
third; and that the excess (that is, the excess above subtriple)
perpetually decreaseth as the number of terms is augmented, as here
(1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30), &c. which I will show you
plainly is false.
But first I wonder why you were so angry with me for saying you made
proportion to consist in the quotient, as to tell me it was abominably
false, and to justify it, cite your own words _penes quotientem_; do not
you say here, the proportion is everywhere greater than subtriple, or
(1)/(3)? And is not (1)/(3) the quotient of 1 divided by 3? You cannot
say in this place that _penes_ is understood; for if it were expressed
you would not be able to proceed.
But I return to your conclusion, that the excess of the proportion of
the increasing quantities above the third part of so many times the
greatest, decreaseth, as (1)/(6) (1)/(12) (1)/(18) (1)/(24) (1)/(30),
&c. For by this account in this row (0 + 1)/(1 + 1) = (1)/(2) where the
quantity above exceeds the third part of the quantities below by
(1)/(3), you make (1)/(3) equal to (1)/(6), which you do not mean. It
may be said your meaning is, that the proportion of 1 to the subtriple
of 2 which is (2)/(3), exceedeth what? I cannot imagine what, nor
proceed further where the terms be but two. Let us therefore take the
second row, that is, (0 + 1 + 4)/(4 + 4 + 4) = (5)/(12). The sum above
is 5, the sum below is 12, the third part whereof is 4; if you mean,
that the proportion of 5 to 4 exceeds the proportion of 4 to 12 (which
is subtriple) by (1)/(12), you are out again. For 5 exceeds 4 by unity,
which is (12)/(12). I do not think you will own such an equation as
(12)/(12) = (1)/(12) Therefore I believe you mean (and your next
proposition assures me of it), that the proportion of 5 to 4 exceeds
subtriple proportion by the proportion of 1 to 12; if you do so, you are
yet deceived.
For if the proportion of 5 to 4 exceeds subtriple proportion by the
proportion of 1 to 12, then subtriple proportion, that is, of 4 to 12
added to the proportion of 1 to 12 must make the proportion of 5 to 4.
But if you look on these quantities, 4, 12, 144, you will see, and must
not dissemble, that the proportion of 4 to 12 is subtriple, and the
proportion of 12 to 144 is the same with that of 1 to 12. Therefore by
your assertion it must be as 5 to 4 so 4 to 144, which you must not own.
And yet this is manifestly your meaning, as appeareth in these words:
“_Ut sit rationis provenientis excessus supra subtriplam ea quam habet
unitas ad sextuplum numeri terminorum post 0, adeoque_,” which cannot be
rendered in English, nor need to be. For you express yourself in the
twentieth proposition very clearly; I noted it only that you may be more
merciful hereafter to the stumblings of a hasty pen. For _excessus ea
quam_ does not well, nor is to be well excused by _subauditur ratio_.
Your twentieth proposition is this:
“_Si proponatur series quantitatum in duplicata ratione arithmetice
proportionalium (sive juxta seriem numerorum quadraticorum) continue
crescentium, a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem
totidem maximæ æqualium subtriplam superabit; eritque excessus ea ratio
quam habet unitas ad sextuplum numeri terminorum post 0, sive quam habet
radix quadratica termini primi post 0 ad sextuplum radicis quadraticæ
termini maximi._”
That is, if there be propounded a row of quantities in duplicate
proportion of arithmetically-proportionals (or according to the row of
square numbers) continually increasing, and beginning with a point or O.
The proportion of that row to a row of so many equals to the greatest,
shall be greater than subtriple proportion, and the excess shall be that
proportion which unity hath to the sextuple of the number of terms after
0, or the same which the square root of the first number after 0, hath
to the sextuple of the square root of the greatest.
For proof whereof you have no more here than _patet ex præcedentibus_;
and no more before but _adeoque_. You do not well to pass over such
curious propositions so slightly; none of the ancients did so, nor, that
I remember, any man before yourself. The proposition is false, as you
shall presently see.
Take, for example, any one of your rows: as (0 + 1 + 4)/(4 + 4 + 4). By
this proportion of yours 1 + 4, which makes 5, is to 12 in more than
subtriple proportion; by the proportion of 1 to the sextuple of 2 which
is 12. Put in order these three quantities 5, 4, 12, and you must see
the proportion of 5 to 12 is greater than the proportion of 4 to 12,
that is, subtriple proportion, by the proportion of 5 to 4. But by your
account the proportion of 5 to 4 is greater than that of 4 to 12 by the
proportion of 1 to 12. Therefore, as 5 to 4 so is 1 to 12, which is a
very strange paradox.
After this you bring in this consectary: “_Cum autem crescente numero
terminorum excessus ille supra rationem subtriplam continue minuatur, ut
tandem quovis assignabili minor evadat (ut patet) si in infinitum
producatur, prorsus evaniturus est. Adeoque._”
That is, seeing as the number of terms increaseth, that excess above
subtriple proportion continually decreaseth, so as at length it becomes
less than any assignable (as is manifest) if it be produced infinitely,
it shall utterly vanish, and so. And so what?
Sir, this consequence of yours is false. For two quantities being given,
and the excess of the greater above the less, that excess may
continually be decreased, and yet never quite vanish. Suppose any two
unequal quantities differing by more than an unit, as 3 and 6, the
excess 3, let 3 be diminished, first by an unit, and the excess will be
2, and the quantities will be 3 and 5; 5 is greater than 4, the excess
1. Again, let 1 be diminished and made (1)/(2), the excess 4 and the
quantities 3 and 4(1)/(2), 4(1)/(2) is yet greater than 4. Again
diminish the excess to (1)/(4), the quantities will be 3 and 4(1)/(4),
yet still 4(1)/(4) is greater than 4. In the same manner you may proceed
to (1)/(8) (1)/(16) (1)/(32), &c. infinitely; and yet you shall never
come within an unit (though your unit stand for 100 miles) of the lesser
quantity propounded 3, if that 3 stands for 300 miles. The excesses
above subtriple proportion do not decrease in the manner you say it
does, but in the manner which I now shall show you.
In the first row (0 + 1)/(1 + 1) a third of the quantities below is
(2)/(3), set in order these three quantities 1 (2)/(9) (2)/(3). The
first is 1, equal to the sum above, the last is (2)/(3), equal to the
subtriple of the sum below. The middlemost is (2)/(9) subtriple to the
last quantity (2)/(3). The excess of the proportion of 1 to (2)/(3)
above the subtriple proportion of (2)/(9) to (2)/(3) is the proportion
of 1 to (2)/(9) that is of 9 to 2, that is, of 18 to 4.
Secondly, in the second row, which is (0 + 1 + 4)/(4 + 4 + 4), a third
of the sum below is 4, the sum above is 5. Set in order these
quantities, 1, 5, 4, 12. There the proportion of 15 to 12 is the
proportion of 5 to 4. The proportion of 4 to 12 is subtriple; the excess
is the proportion of 15 to 4, which is less than the proportion of 18 to
4, as it ought to be; but not less by the proportion of (1)/(6) to
(1)/(12) as you would have it.
Thirdly, in the third row, which is (0 + 1 + 4 + 9)/(9 + 9 + 9 + 9). A
third of the sum below is 12, the sum above is 14. Set in order these
quantities, 42, 4, 12. There the proportion of 42 to 12 is the same with
that of 14 to 4. And the proportion of 4 to 12 subtriple, less than the
former excess of 15 to 4. And so it goes on decreasing all the way in
this manner, 18 to 4, 15 to 4, 14 to 4, &c. which differs very much from
your 1 to 6, 1 to 12, 1 to 18, &c. and the cause of your mistake is
this: you call the twelfth part of twelve (1)/(12), and the eighteenth
part of thirty-six you call (1)/(18), and so of the rest. But what need
of all those equations in symbols, to show that the proportion
decreases; is there any man can doubt, but that the proportion of 1 to 2
is greater than that of 5 to 12, or that of 5 to 12 greater than that of
14 to 36, and so on continually forwards; or could you have fallen into
this error, unless you had taken, as you have done in very many places
of your _Elenchus_, the fractions (1)/(6) and (1)/(12), &c. which are
the quotients of 1 divided by 6 and 12, for the very proportions of 1 to
6 and 1 to 12. But notwithstanding the excess of the proportions of the
increasing quantities, to subtriple proportion decrease, still, as the
number of terms increaseth, and that what proportions soever I shall
assign, the decrement will in time (in time, I say, without proceeding
_in infinitum_) produce a less, yet it does not follow that the row of
increasing quantities shall ever be equal to the third part of the row
of so many equals to the last or greatest. For it is not, I hope, a
paradox to you, that in two rows of quantities the proportion of the
excesses may decrease, and yet the excesses themselves increase, and do
perpetually.
For in the second and third rows, which are (0 + 1 + 4 = 5)/(4 + 4 + 4
= 12) and (0 + 1 + 4 + 9 = 14)/(9 + 9 + 9 + 9 = 36) 5 exceeds the third
part of 12 by a quarter of the square of 4, and 14 exceeds the third
part of 36 by 2 quarters of the square of 4, and proceeding on, the sum
of the increasing quantities where the terms are 5 (which sum is 30)
exceedeth the third part of those below, (those below are 80, and their
third part 26(2)/(3)) by 3 quarters and (1)/(2) a quarter of the square
of 4, and when the terms are 6, the quantities above will exceed the
third part of them below by 5 quarters of the square of 4. Would you
have men believe, that the further they go, the excess of the increasing
quantities above the third part of those below shall be so much the
less? And yet the proportions of those above, to the thirds of those
below, shall decrease eternally; and therefore your twenty-first
proposition is false, namely this:
“_Si proponatur series infinita quantitatum in duplicata ratione
arithmetice proportionalium (sive juxta seriem numerorum quadraticorum),
continue crescentium a puncto sive 0 inchoatarum; erit illa ad seriem
totidem maximæ æqualium, ut 1 ad 3._”
That is, if an infinite row of quantities be propounded in duplicate
proportion of arithmetically-proportionals (or according to the row of
quadratic numbers), continually increasing and beginning from a point or
0; that row shall be to the row of as many equals to the greatest, as 1
to 3. This is false, _ut patet ex præcedentibus_; and, consequently, all
that you say in proof of the proportion of your _parabola_ to a
_parallelogram_, or of the _spiral_ (the true _spiral_) to a _circle_ is
in vain.
But your spiral puts me in mind of what you have under-written to the
diagram of your proposition 5. _The spiral, in both figures, was to be
continued whole to the middle, but, by the carelessness of the graver,
it is in one figure_ manca, _in the other_ intercisa.
Truly, Sir, you will hardly make your reader believe that a graver could
commit those faults without the help of your own copy, nor that it had
been in your copy, if you had known how to describe a spiral line then
as now. This I had not said, though truth, but that you are pleased to
say, though not truth, that I attributed to the printer some faults of
mine.
I come now to the thirty-ninth proposition, which is this:
“_Si proponatur series quantitatum in triplicata ratione arithmetice
proportionalium (sive juxta seriem numerorum cubicorum), continue
crescentium a puncto sive 0 inchoatarum (puta ut 0, 1, 8, 27, etc.),
propositum sit inquirere quam habeat series illa rationem ad seriem
totidem maximæ æqualium_:
“_Fiat investigatio per modum inductionis_ (_ut in prop. 1, et prop.
19_):
_Eritque_
(0 + 1 = 1)/(1 + 1 = 2) = (2)/(4) = (1)/(4) + (1)/(4)
(0 + 1 + 8 = 9)/(8 + 8 + 8 = 24) = (1)/(4) + (1)/(8)
(0 + 1 + 8 + 27 = 36)/(27 + 27 + 27 + 27 = 108) = (4)/(12)
= (1)/(4) + (1)/(12)
_Et sic deinceps._
“_Ratio proveniens est ubique major quam subquadrupla, sive (1)/(4).
Excessus autem perpetuo decrescit, pro ut numerus terminorum augetur,
puta (1)/(4) (1)/(8) (1)/(12) (1)/(16) etc. Aucto nimirum fractionis
denominatore sive consequente rationis in singulis locis numero
quaternatio, ut patet, ut sit rationis provenientis excessus supra
subquadruplam ea quam habet unitas ad quadruplum numeri terminorum post
0 adeoque._”
That is, if a row of quantities be propounded in triplicate proportion
of arithmetically proportionals (or according to the row of cubic
numbers), continually increasing, and beginning from a point or 0, as 0,
1, 8, 27, 64, &c., let it be propounded to inquire, what proportion that
row hath to a row of as many equals to the greatest.
Be it sought by way of induction, as in proposition 1 and 19.
The proposition arising is everywhere greater than subquadruple, or
(1)/(4), and the excess perpetually decreaseth as the number of terms
increaseth, as (1)/(4) (1)/(8) (1)/(12) (1)/(16) (1)/(20) &c. The
denominator of the fraction, or consequent of the proportion, being in
every place augmented by the number 4, as is manifest, so that the
excess of the arising proportion above subquadruple is the same with
that which an unit hath to the quadruple of the number of the terms
after 0, and so. Here are just the same faults which are in proposition
19.
For, if (0)/(1) be a fraction, and (1)/(1) be a fraction, and (1)/(2) be
another fraction, then this equation (0 + 1 = 1)/(1 + 1 = 2) is false.
For this fraction (0)/(1) is equal to 0; and, therefore, we have (1)/(1)
= (1)/(2), that is, the whole equal to half. But perhaps you do not mean
them fractions, but proportions; and, consequently, that the proportion
of 0 to 1, and of 1 to 1, compounded by addition (I say by addition, not
that I, but that you think there is a composition of proportions by
multiplication, which I shall show you anon is false), must be equal to
the proportion of 1 to 2, which cannot be. For the proportion of 0 to 1
is infinitely little, that is, none at all; and, consequently, the
proportion of 1 to 1 is equal to the proportion of 1 to 2, which is
again absurd. There is no doubt but the whole number of 0 + 1 is equal
to 1, and the whole number of 1 + 1 equal to 2. But, reckoning them as
you do, not for whole numbers, but for fractions or proportions, the
equations are false.
Again, your second equation, (2)/(4) = (1)/(4) + (1)/(4), though meant
of fractions, that is, of quotients, it be true, and serve nothing to
your purpose, yet, if it be meant of proportions, it is false. For the
proportion of 1 to 4, and of 1 to 4 being compounded, are equal to the
proportion of 1 to 16, and so you make the proportion of 2 to 4 equal to
the proportion of 1 to 16, where, as it is but subquaduplicate, as you
call it, or the quarter of it, as I call it. And, in the same manner,
you may demonstrate to yourself the same fault in all the other rows of
how many terms soever they consist. Therefore, you may give for lost
this thirty-ninth proposition, as well as all the other thirty-eight
that went before. As for the conclusion of it, which is, _that the
excess of the arising proportion_, &c. They are the words of your
fortieth proposition, where you express yourself better, and make your
error more easy to be detected.
The proposition is this:
“_Si proponatur series quantitatum in triplicata ratione arithmetice
proportionalium (sive juxta seriem numerorum cubicorum) continue
crescentium a puncto vel 0 inchoatarum, ratio quam habet illa ad seriem
totidem maximæ æqualium subquadruplam superabit; eritque excessus ea
ratio quam habet unitas ad quadruplum numeri terminorum post 0; sive
quam habet radix cubica termini primi post 0 ad quadruplum radicis
cubicæ termini maximi. Patet ex præcedente._
“_Quum autem crescente numero terminorum excessus ille supra rationem
subquadruplam ita continuo minuatur, ut tandem quolibet assignabili
minor evadat, ut patet, si in infinitum procedatur, prorsus evaniturus
est, adeoque._
“_Patet ex propositione_ _præcedente._”
That is, if a row of quantities be propounded in triplicate proportion
of arithmetically proportionals (or according to the row of cubic
numbers), continually increasing, and beginning at a point or 0; the
proportion which that row hath to a row of as many equals to the
greatest, is greater than subquadruple proportion; and the excess is
that proportion which one unit hath to the quadruple of the number of
terms after 0; or, which the cubic root of the first term after 0 hath
to the quadruple of the root of the greatest term.
It is manifest by the precedent propositions.
And, seeing the number of terms increasing, that excess above quadruple
proportion doth so continually decrease, as that, at length, it becomes
less than any proportion that can be assigned, as is manifest, if the
proceeding be infinite, it shall quite vanish. And so
This conclusion was annexed to the end of your thirty-ninth proposition,
as there proved. What cause you had to make a new proposition of it,
without other proof than _patet ex præcedente_, I cannot imagine. But,
howsoever, the proposition is false.
For example, set forth any of your rows, as this of fewer terms:
(0 + 1 + 8 + 27 = 36)/((27 + 27 + 27 + 27 =
108)
The row above is 36, the fourth part of the row below is 27. The
quadruple of the number of terms after 0 is 12. Then, by your account,
the proportion of 36 to 108 is greater than subquadruple proportion by
the proportion of 1 to 12. For trial whereof, set in order these three
quantities, 36, 27, 108. The proportion of 36 (the uppermost row) to 108
(the lowermost row) is compounded by addition of the proportions 36 to
27, and 27 to 108. And the proportion of 36 to 108, exceedeth the
proportion of 27 to 108, by the proportion of 36 to 27. But the
proportion of 27 to 108 is subquadruple proportion. Therefore, the
proportion of 36 to 108 exceedeth subquadruple proportion, by the
proportion of 36 to 27. And, by your account, by the proportion of 1 to
12; and, consequently, as 36 to 27, so is 1 to 12. Did you think such
demonstrations as these should always pass?
Then, for your inference from the decrease of the proportions of the
excess, to the vanishing of the excess itself, I have already showed it
to be false; and by consequence that your next proposition, namely, the
fortieth, is also false.
The proposition is this:
“_Si proponatur series infinita quantitatum in triplicata ratione
arithmetice proportionalium (sive juxta seriem numerorum cubicorum),
continue crescentium a puncto sive 0 inchoatarum, erit illa ad seriem
totidem maximæ æqualium, ut 1 ad 4, patet ex præcedente._”
That is, if there be propounded an infinite row of quantities in
triplicate proportion of arithmetically proportionals (or according to
the row of cubic numbers), continually increasing, and beginning at a
point or 0; it shall be to the row of as many equals to the greatest as
1 to 4. Manifest out of the precedent proposition.
Even as manifest as that 36, 27, 1, 12, are proportionals. Seeing,
therefore, your doctrine of the spiral lines and the spaces is given by
yourself for lost, and a vain attempt, your first forty-one propositions
are undemonstrated, and the grounds of your demonstrations all false.
The cause whereof is partly your taking quotient for proportion, and a
point for 0, as you do in the first, sixteenth, and fortieth
propositions, and in other places where you say, _beginning at a point
or 0_, though now you deny you ever said either. There be very many
places in your _Elenchus_, where you say both; and have no excuse for
it, but that, in one of the places, you say the proportion is _penes
quotientem_, which is to the same or no sense.
Your forty-second proposition is grounded on the fortieth; and
therefore, though true, and demonstrated by others, is not demonstrated
by you.
Your forty-third is this:
“_Pari methodo invenietur ratio seriei infinitæ quantitatum arithmetice
proportionalium in ratione quadruplicata, quintuplicata, sextuplicata,
etc., arithmetice proportionalium a puncto seu 0 inchoatarum, ad seriem
totidem maximæ æqualium. Nempe in quadruplicata erit, ut 1 ad 5; in
quintuplicata, ut 1 ad 6; in sextuplicata, ut 1 ad 7. Et sic deinceps._”
That is, by the same method will be found, the proportion of an infinite
row of arithmetically proportionals, in proportion quadruplicate,
quintuplicate, sextuplicate, &c., of arithmetically proportionals,
beginning at a point or 0, to the row of as many equals to the greatest;
namely, in quadruplicate, it shall be as 1 to 5; in quintuplicate, as 1
to 6; in sextuplicate, as 1 to 7; and so forth.
But by the same method that I have demonstrated, that the propositions
19, 20, 21, 39, 40, and 41, are false: any man else, that will examine
the forty-third may find it false also. And, because all the rest of the
propositions of your _Arithmetica Infinitorum_ depend on these, they may
safely conclude, that there is nothing demonstrated in all that book,
though it consist of 194 propositions. The proportions of your
parabolocides to their parallelograms are true, but the demonstrations
false, and infer the contrary. Nor were they ever demonstrated (at least
the demonstrations are not extant) but by me; nor can they be
demonstrated, but upon the same grounds, concerning the nature of
proportion, which I have clearly laid, and you not understood. For, if
you had, you could never have fallen into so gross an error as is this
your book of _Arithmetica Infinitorum_, or that of the angle of contact.
You may see by this, that your symbolic method is not only not at all
inventive of new theorems, but also dangerous in expressing the old. If
the best masters of symbolics think for all this you are in the right,
let them declare it. I know how far the analysis by the powers of the
lines extendeth, as well as the best of your half-learnt epistlers, that
approve so easily of such analogisms as those, 5, 4, 1, 12, and 36, 27,
1, 12, &c.
It is well for you that they who have the disposing of the professors’
places take not upon them to be judges of geometry. For, if they did,
seeing you confess you have read these doctrines in your school, you had
been in danger of being put out of your place.
When the author of the paper wherein I am called Plagiary, and wherein
the honour is taken from you of being the first inventor of these fine
theorems, shall read this that I have here written, he will look to get
no credit by it; especially if it be Roberval, which methinks it should
not be. For he understands what proportion is, better than to make 5 to
4 the same with 1 to 12. Or to make, again, the proportion of 36 to 27
the same with that of 1 to 12; and innumerable _disproportionalites_
that may be inferred from the grounds you go on. But if it be Roberval
indeed, that snatches this invention from you, when he shall see this
burning coal hanging at it, he will let it fall again, for fear of
spoiling his reputation.
But what shall I answer to the authority of the three great
mathematicians that sent you those encomiastic letters. For the first,
whom you say I use to praise, I shall take better heed hereafter of
praising any man for his learning whilst he is young, further than that
he is in a good way. But it seems he was in too ready a way of thinking
very well of himself, as you do of yourself. For the muddiness of my
brain I must confess it; but, Sir, ought not you to confess the same of
yours? No, men of your tenets use not to do so. He wonders, say you, you
thought it worth the while to foul your fingers about such a piece. It
is well; every man abounds in his own sense. If you and I were to be
compared by the compliments that are given us in private letters, both
you and your complimentors would be out of countenance; which
compliments, besides that which has been printed and published in the
commendations of my writings, if it were put together, would make a
greater volume than either of your libels. And truly, Sir, I had never
answered your Elenchus as proceeding from Dr. Wallis, if I had not
considered you also as the minister to execute the malice of that sort
of people that are offended with my _Leviathan_.
As for the judgment of that public Professor that makes himself a
witness of the goodness of your geometry, a man may easily see by the
letter itself that he is a dunce. And for the English person of quality
whom I know not, I can say no more yet than I can say of all three, that
he is so ill a geometrician, as not to detect those gross paralogisms as
infer that 5 to 4 and 1 to 12 are the same proportion. He came into the
cry of those whom your title had deceived.
And now I shall let you see that the composition of proportion by
multiplication, as it is in the fifth definition of the sixth element,
is but another way of adding proportions one to another. Let the
proportions be of 2 to 3, and of 4 to 5. Multiply 2 into 4 and 3 into 5,
the proportion arising is of 8 to 15. Put in order these three
quantities, 8, 12, 15. The proportion therefore of 8 to 15, compounded
of the proportions of 8 to 12, (that is, of 2 to 3) and of 12 to 15,
that is, of 4 to 5 by addition. Again, let the proportion be of 2 to 3,
and of 4 to 5, multiply 2 into 5 and 3 into 4, the proportions arising
is of 10 to 12. Put in order these three numbers, 10, 8, 12. The
proportion 10 to 12 is compounded of the proportions of 10 to 8, that is
of 5 to 4, and of 8 to 12, that is, of 2 to 3 by addition. I wonder you
know not this.
I find not any more clamour against me for saying the proportion of 1 to
2 is double to that of 1 to 4.
Your book, you speak of, concerning proportion against _Meibomius_ is
like to be very useful when neither of you both do understand what
proportion is.
You take exceptions, as that I say, that _Euclid_ has but one word for
_double_ and _duplicate_; which nevertheless was said very truly, and
that word is sometimes διπλάσιος and sometimes διπλάσιων. And you think
you have come off handsomely with asking me whether διπλάσιος and
διπλασίων be one word.
Nor are you now of the mind you were, that a point is not _quantity
unconsidered_, but that in an infinite series it may be safely
neglected. What is _neglected_ but unconsidered.
Nor do you any more stand to it, that the _quotient_ is the
_proportion_. And yet were these the main grounds of your _Elenchus_.
But you will say, perhaps, I do answer to the defence you have now made
in this your _School Discipline_: ’tis true. But ’tis not because you
answer never a word to my former objections against these propositions
19, 89; but because you do so shift and wriggle, and throw out ink, that
I cannot perceive which way you go, nor need I, especially in your
vindication of your _Arithmetica Infinitorum_. Only I must take notice
that in the end of it, you have these words, “Well, _Arithmetica
Infinitorum_ _is come off clear_” You see the contrary. For sprawling is
no defence.
It is enough to me that I have clearly demonstrated both before
sufficiently, and now again abundantly, that your book of _Arithmetica
Infinitorum_ is all nought from the beginning to the end, and that
thereby I have effected that your authority shall never hereafter be
taken for a prejudice. And, therefore, they that have a desire to know
the truth in the questions between us, will henceforth, if they be wise,
examine my geometry, by attentive reading me in my own writings, and
then examine, whether this writing of yours confute or enervate mine.
There is in my fifth lesson a proposition, with a diagram to it, to make
good, I dare say, at least against you, my twentieth chapter concerning
the dimension of a circle. If that demonstration be not shown to be
false, your objections to that chapter, though by me rejected, come to
nothing. I wonder why you pass it over in silence. But you are not, you
say, bound to answer it. True, nor yet to defend what you have written
against me.
Before I give over the examination of your geometry, I must tell you
that your words, (p. 101 of your _School Discipline_), against the first
corollary are untrue.
Your words are these: “_you affirm that the proportion of the parabola A
B I to the parabola A F K is triplicate to the proportion of the time A
B to A F, as it is in the English_.” This is not so. Let the reader turn
to the place and judge. And going on you say, “_or of the impetus B I to
F K as it is in the Latin_.” Nay, as it is in the English, and the other
in the Latin. It is but your mistake; but a mistake is not easily
excused in a false accusation.
Your exception to my saying, “_that the differences of two quantities is
their proportion_,” (when they differ, as the no difference, when they
be equal), might have been put in amongst other marks of your not
sufficiently understanding the Latin tongue. _Differre_ and
_differentia_ differ no more than _vivere_ and _vita_, which is nothing
at all, but as the other words require that go with them, which other
words you do not much use to consider. But _differre_ and _the quantity
by which they differ_, are quite of another kind. _Differre_ (τὸ
διαφέρειν, τὸ ὑπερέχειν) _differing_, _exceeding_, is not quantity, but
relation. But the quantity by which they differ is always a certain and
determined quantity, yet the word _differentia_ serves for both, and is
to be understood by the coherence with that which went before. But I had
said before, and expressly to prevent cavil, that relation is nothing
but a comparison, and that proportion is nothing but relation of
quantities, and so defined them, and therefore I did there use the word
_differentia_ for _differing_, and not for the quantity which was left
by subtraction. For a quantity is not a differing. This I thought the
intelligent reader would of himself understand without putting me,
instead of _differentia_, to use (as some do, and I shall never do) the
mongrel word τὸ _differre_. And whereas in one only place for _differre
ternario_ I have writ _ternarius_, if you had understood what was
clearly expressed before, you might have been sure it was not my
meaning, and therefore the excepting against it was either want of
understanding, or want of candour, choose which you will.
You do not yet clear your doctrine of _condensation_ and _rarefaction_.
But I believe you will by degrees become satisfied that they who say the
same numerical body may be sometimes greater, sometimes less, speak
absurdly, and that _condensation_ and _rarefaction_ here, and
_definitive_ and _circumscriptive_, and some other of your distinctions
elsewhere are but snares, such as school divines have invented
——ᾥσπερ άράχνης
Ὀυλόμενος χέζει ἀλύσεις μυίαις ἀθαρέσσι,
to entangle shallow wits.
And that that distinction which you bring here, “_that it is of the same
quantity while it is in the same place, but it may be of a different
quantity when it goes out of its place_,” (as if the place added to, or
took any quantity from the body placed), is nothing but mere words. It
is true that the body which swells changeth place, but it is not by
becoming itself a greater body, but by admixtion of air or other body,
as when water riseth up in boiling, it taketh in some parts of air. But
seeing the first place of the body is to the body equal, and the second
place equal to the same body, the places must also be equal to one
another, and consequently the dimensions of the body remain equal in
both places.
Sir, when I said that such doctrine was taught in the Universities, I
did not speak against the Universities, but against such as you. I have
done with your geometry, which is one στιγμὴ.
RURAL LANGUAGE.
As for your eloquence, let the reader judge whether yours or mine be the
more _muddy_, though I in plain scolding should have outdone you, yet I
have this excuse which you have not, that I did but answer your
challenge at that weapon which you thought fit to choose. The catalogue
of the hard language which you put in at pages 3 and 4 of your _School
Discipline_, I acknowledge to be mine, and would have been content you
had put in all. The titles you say I give you of _fools_, _beasts_, and
_asses_, I do not give you, but drive back upon you, which is no more
than not to own them; for the rest of the catalogue, I like it so well
as you could not have pleased me better than by setting those passages
together to make them more conspicuous; that is all the defence I will
make to your accusations of that kind.
And now I would have you to consider whether you will make the like
defence against the faults that I shall find in the language of your
_School Discipline_.
I observe, first, the facetiousness of your title-page, “_Due correction
for Mr. Hobbes, or School Discipline, for not saying his Lessons
right_.” What a quibble is this upon the word lesson; besides, you know
it has taken wind; for you vented it amongst your acquaintance at Oxford
then when my _Lessons_ were but upon the press. Do you think if you had
pretermitted that piece of wit, the opinion of your judgment would have
been ere the less? But you were not content with this, but must make
this metaphor from the rod to take up a considerable part of your book,
in which there is scarce anything that yourself can think wittily said
besides it. Consider also these words of yours: “_It is to be hoped that
in time you may come to learn the language, for you be come to great_ A
_already_.” And presently after, “_were I great_ A, _before I would be
willing to be so used, I should wish myself little_ a _a hundred
times_.” Sir, you are a doctor of divinity and a professor of geometry,
but do not deceive yourself, this does not pass for wit in these parts,
no, nor generally at Oxford; I have acquaintance there that will blush
at the reading it.
Again, in another place you have these words: “_Then you catechize us_,
‘_what is your name? Are you geometricians? Who gave you that name_,’”
&c. Besides in other places such abundance of the like insipid conceits,
as would make men think, if they were no otherwise acquainted with the
University but by reading your books, that the dearth there of salt were
very great. If you have any passage more like to salt than these are
(excepting _now and anon_) you may do well to show it to your
acquaintance, lest they despise you; for, since the detection of your
geometry, you have nothing left you else to defend you from contempt.
But I pass over this kind of eloquence, and come to somewhat yet more
rural.
Page 27, line 1, you say I have given Euclid his _lurry_. And again,
page 129, line 11, “_and now he is left to learn his lurry_.” I
understand not the word _lurry_. I never read it before, nor heard it,
as I remember, but once, and that was when a clown threatening another
clown said he would give him _such a lurry come poop_, &c. Such words as
these do not become a learned mouth, much less are fit to be registered
in the public writings of a doctor of divinity. In another place you
have these words, “_just the same to a cow’s thumb_,” a pretty adage.
Page 2, “_But prithee tell me_.” And again, page 95, “_prithee tell me,
why dost thou ask me such a question_,” and the like in many other
places.
You cannot but know how easy it is and was for me to have spoken to you
in the same language. Why did I not? Because I thought that amongst men
that were civilly bred it would have redounded to my shame, as you have
cause to fear that this will redound to yours. But what moved you to
speak in that manner? Were you angry? If I thought that the cause, I
could pardon it the sooner, but it must be very great anger that can put
a man, that professeth to teach good manners, so much out of his wits as
to fall into such a language as this of yours. It was perhaps an
imagination that you were talking to your inferior, which I will not
grant you, nor will the heralds, I believe, trouble themselves to decide
the question. But, howsoever, I do not find that civil men use to speak
so to their inferiors. If you grant my learning but to be equal to
yours, (which you may certainly do without very much disparaging of
yourself abroad in the world), you may think it less insolence in me to
speak so to you in respect of my age, than for you to speak so to me in
respect of your young doctorship. You will find that for all your
doctorship, your elders, if otherwise of as good repute as you, will be
respected before you. But I am not sure that this language of yours
proceeded from that cause; I am rather inclined to think you have not
been enough in good company, and that there is still somewhat left in
your manners for which the honest youths of Hedington and Hincsey may
compare with you for good language, as great a doctor as you are.
For my verses of the Peak, though they be as ill in my opinion as I
believe they are in yours, and made long since, yet they are not so
obscene as that they ought to be blamed by Dr. Wallis. I pray you, sir,
whereas you have these words in your _School Discipline_, page 96,
“_unless you will say that one and the same motion may be now and anon
too_.” What was the reason you put these words, _now and anon too_, in a
different character, that makes them to be more taken notice of? Do you
think that the story of the minister that uttered his affection (if it
be not a slander) not unlawfully but unseasonably, is not known to
others as well as to you? What needed you then, when there was nothing
that I had said could give the occasion, to use those words; there is
nothing in my verses that do _olere hircum_ so much as this of yours. I
know what good you can receive by ruminating on such ideas, or
cherishing of such thoughts. But I go on to other words of mine by you
reproached, “_you may as well seek the focus of the parabola of Dives
and Lazarus_,” which you say is mocking the Scripture; to which I answer
only, that I intended not to mock the Scripture, but you, and that which
was not meant for mocking was none. And thus you have a second στιγμὴ.
GRAMMAR AND CRITIQUES.
I come now to the comparison of our Grammar and Critiques. You object
first against the signification I give of στιγμὴ, and say thus: “_What
should come into your cap_ (that, if you mark it, in a man that wears a
square cap to one that wears a hat, is very witty) _to make you think
that_ στιγμὴ _signifies a mark or brand with a hot iron? I perceive
where the business lies, it was_ στίγμα _run in your mind when you
talked of_ στιγμὴ; _and because the words are somewhat alike you jumble
them both together_.” Sir, I told you once before, you presume too much
upon your first cogitations. Aristophanes, in _Ranis_, Act. V. Scen. 5,
Κἄν μὴ ταχέως ἥκωσι
Νὴ τεν Ἀπόλλο στίξας ἀυτοὺς.
The old commentator upon the word στίξας saith thus, ϛίξας ἀντὶ τοῦ
ϛιγματίσας, ἠν γάρ ξένος. That is, στίξας for ϛιγματίσας, for he
(Adimantus) was not a citizen. I hope the commentator does not here mock
Aristophanes for jumbling ϛίξας and ϛιγματίσας together, for want of
understanding Greek. No, ϛίξας and στιγματίσας signify the same, save
that for branding I seldom read ϛιγματίσας but ϛίξας. For ϛίγμα does no
more signify a brand with a hot iron, than ϛιγμὴ a point made also with
a hot iron. They have both one common theme ϛίζω, which does not signify
_pungo_, nor _interpungo_, nor _inuro_, for all your Lexicon, but _notam
imprimere_, or _pungendo notare_, without any restriction to burning or
punching. It is therefore no less proper to say that ϛιγμὴ is a mark
with a hot iron, than to say the same of στίγμα. The difference is only
this, that when they marked a slave, or a rascal, as you are not
ignorant is usually done here at the assizes in the hand or shoulder
with a hot iron, they called that ϛίγμα, not for the burning, but for
the mark. And as it would have been called ϛίγμα that was imprinted on a
slave, though made by staining or incision, so it is ϛιγμὴ, though done
with a hot iron. And therefore there was no jumbling of those two words
together, as for want of reading Greek authors, and by trusting too much
to your dictionaries, which you say are proofs good enough for such a
business, you were made to imagine. The use I have made thereof was to
show that a point, both by the word Σημεῖον in Euclid, and by the word
στιγμὴ in some others, was not _nothing_, but a _visible_ mark, the
ignorance whereof hath thrown you into so many paralogisms in geometry.
But do you think you can defend your _Adducis Malleum_ as well as I have
now defended my ϛιγμὴ? You have brought, I confess, above a hundred
places of authors, where there is the word _duco_, or some of its
compounds, but none of them will justify _Adducis Malleum_, and,
excepting two of those places, you yourself seem to condemn them all,
comparing yours with none of the rest but with these two only, both out
of Plautus, by you not well understood. The first is in _Casina_, Act.
V. Scen. 2, “_Ubi intro hanc novam nuptam deduxi, via recta, clavem
abduxi_;” which you, presently presuming of your first thoughts, a
peculiar fault to men of your principles, assure yourself is right. But
if you look on the place as Scaliger reads it, cited by the commentator,
you will find it should be _obduxi_, and that _clavis_ is there used for
the bolt of the lock. Besides, he bolted it within. Whither then could
he carry away the key? The place is to be rendered thus, _when I had
brought in this new bride I presently locked the door_, and is this _as
bad every whit_ as _Adducis Malleum_? The second place is in
_Amphytruo_, Act. I. Scen. 1, “_Eam_ (cirneam), _ut a matre fuerat
natum, plenam vini eduxi meri_,” which you interpret _I brought out a
flagon of wine_, unlearnedly. They are the words of Mercury transformed
into Sosia. And to try whether Mercury were Sosia or not, Sosia asked
him where he was and what he did during the battle; to which Mercury
answered, who knew where Sosia then was and what he did, _I was in the
cellar, where I filled a cirnea, and brought it up full of wine, pure as
it came from its mother_. By the mother of the wine meaning the vine,
and alluding to the education of children, for _ebibi_ said _eduxi_, and
with an _emphasis_ in _meri_, because _cirnea_ (from Κφνάω, _misceo_)
was a vessel wherein they put water to temper to their wine. Intimating
that though the vessel was _cirnea_, yet the wine was _merum_. This is
the true sense of the place; but you will have _eduxi_ to be, _I brought
out_, though he came not out himself. You see, sir, that neither this is
so bad as _Adducis Malleum_.
But suppose out of some one place in some one blind author you had
paralleled your _Adducis Malleum_, do you think it must therefore
presently be held for good Latin? Why more than _learn his lurry_ must
be therefore thought good English a thousand years hence, because it
will be read in Dr. Wallis’s long-lived works. But how do you construe
this passage (1 Tim. ii. 15) of the Greek Testament: Σωθήσεται δὲ διὰ
τῆς τεκνογονίας, ἐὰν μείνωσιν ἐν πίστει? You construe it thus: _she
shall be saved notwithstanding child-bearing, if (the woman) remain in
the faith_. Is child-bearing any obstacle to the salvation of women? You
might as well have translated the first verse of the fifth of Romans in
this manner, _Being then justified by faith, we have peace with God
notwithstanding our Lord Jesus Christ_. I let pass your not finding in
τεκνογονίας, as good a grammarian as you are, a nominative case to
μείνωσιν. If you had remembered the place, 1 Pet. iii. 20, εσώθησαν δὶ
ὑδατος, that is, _they were saved in the waters_, you would have thought
your construction justified then very well; but you had been deceived,
for διὰ does not there signify _causam, ablationem impedimenti_, but
_transitum_; not _cause or removing an impediment_, but _passage_. Being
come thus far, I found a friend that hath eased me of this dispute; for
he showed me a letter written to himself from a learned man, that hath
out of very good authors collected enough to decide all the grammatical
questions between you and me, both Greek and Latin. He would not let me
know his name, nor anything of him but only this, that he had better
ornaments than to be willing to go clad abroad in the habit of a
grammarian. But he gave me leave to make use of so much of the letter as
I thought fit in this dispute, which I have done, and have added it to
the end of this writing. But before I come to that, you must not take it
ill, though I have done with your _School Discipline_, if I examine a
little some other of your printed writings as you have examined mine;
for neither you in geometry, nor such as you in church politics, cannot
expect to publish any unwholesome doctrine without some antidotes from
me, as long as I can hold a pen. But why did you answer nothing to my
sixth _Lesson_? Because, you say, it concerned your colleague only. No,
sir, it concerned you also, and chiefly, for I have not heard that your
colleague holdeth those dangerous principles which I take notice of in
you, in my sixth _Lesson_, page 350, upon the occasion of these words,
not his but yours: “_Perhaps you take the whole history of the fall of
Adam for a fable, which is no wonder, seeing you say the rules of
honouring and worshipping of God are to be taken from the laws_.” In
answer to which I said thus: “_You that take so heinously, that I would
have the rule of God’s worship in a Christian commonwealth to be taken
from the laws, tell me from whom you would have them taken? From
yourself? Why so, more than from me? From the bishops? Right, if the
supreme power of the commonwealth will have it so; if not, why from them
rather than from me? From a consistory of presbyters themselves, or
joined with lay elders, whom they may sway as they please? Good, if the
supreme governor of the commonwealth will have it so. If not, why from
them rather than from me, or from any man else? They are wiser and
learneder than I; it may be so, but it has not yet appeared. Howsoever,
let that be granted. Is there any man so very a fool as to subject
himself to the rules of other men in those things which do so nearly
concern himself, for the title they assume of being wise and learned,
unless they also have the sword which must protect them? But it seems
you understand the sword as comprehended. If so, do not you then receive
the rules of God’s worship from the civil power? Yes, doubtless; and you
would expect, if your consistory had that sword, that no man should dare
to exercise or teach any rules concerning God’s worship which were not
by you allowed._”
This will be thought strong arguing, if you do not answer it. But the
truth is, you could say nothing against it without too plainly
discovering your disaffection to the government. And yet you have
discovered it pretty well in your second _Thesis_, maintained in the Act
at Oxford, 1654, and since by yourself published. This _Thesis_ I shall
speak briefly to.
SCOTCH CHURCH POLITICS.
You define ministers of the Gospel to be those _to whom the preaching of
the Gospel by their office is enjoined by Christ_. Pray you, first, what
do you mean by saying preaching _ex officio is enjoined by Christ_? Are
they preachers _ex officio_, and afterwards enjoined to preach? _Ex
officio_ adds nothing to the definition; but a man may easily see your
purpose to disjoin yourself from the state by inserting it.
Secondly, I desire to know in what manner you will be able out of this
definition to prove yourself a minister? Did Christ himself immediately
enjoin you to preach, or give you orders? No. Who then, some bishop, or
minister, or ministers? Yes; by what authority? Are you sure they had
authority immediately from Christ? No. How then are you sure but that
they might have none? At least, some of them through whom your authority
is derived might have none. And therefore if you run back for your
authority towards the Apostles’ times but a matter of sixscore years,
you will find your authority derived from the Pope, which words have a
sound very unlike to the voice of the laws of England. And yet the Pope
will not own you. There is no man doubts but that you hold that your
office comes to you by successive imposition of hands from the time of
the Apostles; which opinion in those gentle terms passeth well enough;
but to say you derive your authority from thence, not through the
authority of the sovereign power civil, is too rude to be endured in a
state that would live in peace. In a word, you can never prove you are a
minister, but by the supreme authority of the commonwealth. Why then do
you not put some such clause into your definition? As thus, _ministers
of the Gospel are those to whom the preaching of the Gospel is enjoined
by the sovereign power in the name of Christ_. What harm is there in
this definition, saving only it crosses the ambition of many men that
hold your principles? Then you define the power of a minister thus:
“_The power of a minister is that which belongeth to a minister of the
Gospel in virtue of the office he holds, inasmuch as he holds a public
station, and is distinguished from private Christians. Such as is the
power of preaching the Gospel, administering the sacrament, the use of
ecclesiastical censures, and ordaining of ministers_,” _&c._
Again, how will you prove out of this definition that you, or any man
else, hath the power of a minister, if it be not given him by him that
is the sovereign of the commonwealth? For seeing, as I have now proved,
it is from him that you must derive your ministry, you can have no other
power than that which is limited in your orders, nor that neither longer
than he thinks fit. For if he give it you for the instruction of his
subjects in their duty, he may take it from you again whensoever he
shall see you instruct them with undutiful and seditious principles. And
if the sovereign power give me command, though without the ceremony of
imposition of hands, to teach the doctrine of my _Leviathan_ in the
pulpit, why am not I, if my doctrine and life be as good as yours, a
minister as well as you, and as public a person as you are? For _public
person_, primarily, is none but the civil sovereign, and so secondarily,
all that are employed in the execution of any part of the public charge.
For all are his ministers, and therefore also Christ’s ministers because
he is so; and other ministers are but his vicars, and ought not to do or
say anything to his people contrary to the intention of the sovereign in
giving them their commission.
Again, if you have in your commission a power to excommunicate, how can
you think that your sovereign who gave you that commission, intended it
for a commission to excommunicate himself? that is, as long as he stand
excommunicate, to deprive him of his kingdom. If all subjects were of
your mind, as I hope they will never be, they will have a very unquiet
life. And yet this has, as I have often heard, been practised in
Scotland, when ministers holding your principles had power enough,
though no right, to do it.
And for administration of the sacraments, if by the supreme power of the
commonwealth it were committed to such of the laity as know how it ought
to be done as well as you, they would _ipso facto_ be ministers as good
as you. Likewise the right of ordination of ministers depends not now on
the imposition of hands of a minister or presbytery, but on the
authority of the Christian sovereign, Christ’s immediate vicar and
supreme governor of all persons and judge of all causes, both spiritual
and temporal, in his own dominions, which I believe you will not deny.
This being evident, what acts are those of yours which you call
_authoritative_, and receive not from the authority of the civil power?
A constable does the acts of a constable _authoritatively_ in that
sense. Therefore you can no otherwise claim your power than a constable
claimeth his, who does not exercise his office in the constabulary of
another. But you forget that the Scribes and the Pharisees sit no more
in Moses’ chair.
You would have every minister to be a minister of the universal Church,
and that it be lawful for you to preach your doctrine at Rome; if you
would be pleased to try, you would find the contrary. You bring no
argument for it that looks like reason. Examples prove nothing, where
persons, times, and other circumstances differ; as they differ very much
now when kings are Christians, from what they were then when kings
persecuted Christians. It is easy to perceive what you aim at.
You would fain have market-day lectures set up by authority, (not by the
authority of the civil power, but by the authority of example of the
Apostles in the emission of preachers to the infidels), not knowing that
any Christian may lawfully preach to the infidels; that is to say,
proclaim unto them that Jesus is the Messiah, without need of being
otherways made a minister, as the deacons did in the Apostles’ time; nor
that many teachers, unless they can agree better, do anything else but
prepare men for faction, nay, rather you know it well enough, but it
conduces to your end upon the market-days to dispose at once both town
and country, under a false pretence of obedience to God, to a neglecting
of the commandments of the civil sovereign, and make the subject to be
wholly ruled by yourselves, wherein you have already found yourselves
deceived. You know how to trouble and sometimes undo a slack government,
and had need to be warily looked to, but are not fit to hold the reins.
And how should you, being men of so little judgment as not to see the
necessity of unity in the governor, and of absolute obedience in the
governed, as is manifest out of the place of your _Elenchus_ above
recited. The doctrine of the duty of private men in a commonwealth is
much more difficult, not only than the knowledge of your symbols, but
also than the knowledge of geometry itself. How then do you think, when
you err so grossly in a few equations, and in the use of most common
words, you should be fit to govern so great nations as England, Ireland,
and Scotland, or so much as to teach them? For it is not reading but
judgment that enables one man to teach another.
I have one thing more to add, and that is the disaffection I am charged
withal to the universities. Concerning the Universities of Oxford and
Cambridge, I ever held them for the greatest and noblest means of
advancing learning of all kinds, where they should be therein employed,
as being furnished with large endowments and other helps of study, and
frequented with abundance of young gentlemen of good families and good
breeding from their childhood. On the other side, in case the same means
and the same wits should be employed in the advancing of the doctrines
that tend to the weakening of the public, and strengthening of the power
of any private ambitious party, they would also be very effectual for
that; and consequently that if any doctrine tending to the diminishing
of the civil power were taught there, not that the Universities were to
blame, but only those men that in the universities, either in lectures,
sermons, printed books, or theses, did teach such doctrine to their
hearers or readers. Now you know very well that in the time of the Roman
religion, the power of the Pope in England was upheld principally by
such teachers in the universities. You know also how much the divines
that held the same principles in Church government with you, have
contributed to our late troubles. Can I therefore be justly taxed with
disaffection to the universities for wishing this to be reformed? And it
hath pleased God of late to reform it in a great measure, and indeed as
I thought totally, when out comes this your _Thesis_ boldly maintained
to show the contrary. Nor can I yet call this your doctrine the doctrine
of the university; but surely it will not be unreasonable to think so,
if by public act of the university it be not disavowed, which done, and
that as often as there shall be need, there can be no longer doubt but
that the universities of England are not only the noblest of all
Christian universities, but also absolutely, and of the greatest benefit
to this commonwealth that can be imagined, except that benefit of the
head itself that uniteth and ruleth all. I have not here particularized
at length all the ill consequences that may be deduced from this
_Thesis_ of yours, because I may, when further provoked, have somewhat
to say that is new. So much for the third ϛιγμὴ.
AN EXTRACT OF A LETTER CONCERNING THE
GRAMMATICAL PART OF THE CONTROVERSY
BETWEEN MR. HOBBES AND DR. WALLIS.
Mr. Hobbes hath these words: “_Longitudinem percursam motu uniformi, cum
impetu ubique ipsi B D æquali_.” Dr. Wallis saith _cum_ were better out,
unless you would have _impetus_ to be only a companion, not a _cause_.
Mr. Hobbes answered it was the _ablative case of the manner_. The truth
is the ablative case of the _manner_ and _cause_ both, may be used with
the conjunction _cum_, as may be justified. Cicero in Lib. II. _De Nat.
Deorum_: “_Moliri aliquid cum labore operoso ac molesto_;” and in his
oration for Cæcina: “_De se autem hoc prædicat, Antiocho Ebulii servo
imperasse ut in Cæcinam advenientem cum ferro invaderet_.” Let us see
then what Dr. Wallis objects against Tully, where a casualty is
imported, though we may use _with_ in English, yet not _cum_ in Latin;
to kill with a sword, importing this to have an instrumental or causal
influence, and not only that it hangs by the man’s side whilst some
other weapon is made use of, is not in Latin _occidere cum gladio_, but
_gladio occidere_. This shows that the Doctor hath not forgot his
grammar, for the subsequent examples as well as this rule are borrowed
thence. But yet he might have known that great personages have never
confined themselves to this pedantry, but have chosen to walk in a
greater latitude. Most of the elegancies and idioms of every language
are exceptions to his grammar. But since Mr. Hobbes saith it is the
ablative case of the manner, there is no doubt it may be expressed with
_cum_. The Doctor in the meantime knew no more than what Lilly had
taught him; Alvarez would have taught him more; and Vossius in his book,
_De Constructione_, _cap._ XLVII. expressly teacheth, “_Ablativos causæ,
instrumenti, vel modi, non a verbo regi sed a præpositione omissa, a vel
ab, de, e vel ex, præ, aut cum, ac præpositiones eas quandoque exprimi
nisi quod cum ablativis instrumenti haud temere invenias_;” and
afterwards he saith, “_non timere imitandum_.” If this be so, then did
Mr. Hobbes speak grammatically, and with Tully, but not _usually_. And
might not one retort upon the Doctor, that Vossius is as great a critic
as he?
His next reflection is upon _prætendit scire_, this he saith is an
Anglicism. If this be all his accusation, upon this score we shall lose
many expressions that are used by the best authors, which I take to be
good Latinisms, though they be also Anglicisms, the latter being but an
imitation of the former. The Doctor therefore was too fierce to condemn
upon so general an account, that which was not to have been censured for
being an Anglicism, unless also it had been no Latinism. Mr. Hobbes
replies, that the printer had omitted _se_. He saith, this mends the
matter a little. It is very likely, for then it is just such another
Anglicism as that of Quintilian: “_Cum loricatus in foro ambularet,
prætendebat se id metu facere_.” The Doctor certainly was very
negligent, or else he could not have missed this in Robert Stephen. Or
haply he was resolved to condemn Quintilian for this and that other
Anglicism, “_Ignorantia prætendi non potest_,” as all those that have
used _prætendo_, which are many and as good authors as Dr. Wallis, that
makes his own encomiasts (not an Englishman amongst them) to write
Anglicisms.
Then he blames “_Tractatus hujus partis tertiæ, in qua motus et
magnitudo per se et abstracte consideravimus, terminum hic statuo_.”
Here I must confess the exception is colourable, yet I can parallel it
with the like objection made by Erasmus against Tully, out of whom
Erasmus quotes this passage: “_Diutius commorans Athenis, quoniam venti
negabant solvendi facultatem, erat animus ad te scribere_;” and excuses
it thus, that Tully might have had at first in his thoughts _volebam_ or
_statuebam_, which he afterwards relinquished for _erat animus_, and did
not remember what he had antecedently written, which did not vary from
his succeeding thoughts, but words. And this excuse may pass with any
who know that Mr. Hobbes values not the study of words, but as it serves
to express his thoughts, which were the same whether he wrote _in qua
motus et magnitudo per se at abstracte considerati sunt_ or
_consideravimus_. And if the Doctor will make this so capital, he must
prove it _voluntary_, and show that it is greater than what is legible
in the puny letter of his encomiast, whom he would have to be beyond
exception.
Now follows his ridiculous apology for _adducis malleum, ut occidas
muscam_. The cause why he did use that proverb, of his own phrasing, was
this. Mr. Hobbes had taken a great deal of pains to demonstrate what Dr.
Wallis thought he could have proved in short; upon this occasion he
objects, _adducis malleum ut occidas muscam_, which I shall suppose he
intended to English thus, _you bring a beetle to kill a fly_. Mr. Hobbes
retorted, that _adduco_ was not used in that sense. The Doctor
vindicates himself thus: _duco_, _deduco_, _reduco_, _perduco_,
_produco_, &c. signify the same thing, ergo, _adduco_ may be used in
that sense; which is a most ridiculous kind of arguing, where we are but
to take up our language from others, and not to coin new phrases. It is
not the grammar that shall secure the Doctor, nor weak analogies, where
elegance comes in contest. To justify his expression he must have showed
it _usu tritum_, or alleged the authority of some author of great note
for it. I have not the leisure to examine his impertinent citations
about those other compounds, nor yet of that simple verb _duco_; nay, to
justify his saying he hath not brought one parallel example. He talks
indeed very high, that _duco_, with its compounds, is a word of a large
signification, and amongst the rest _to bring_, _fetch_, _carry_, &c. is
so exceeding frequent in all authors, Plautus, Terence, Tully, Cæsar,
Tacitus, Pliny, Seneca, Virgil, Horace, Ovid, Claudian, &c. that he must
needs be either maliciously blind, or a very stranger to the Latin
tongue, that doth not know it, or can have the face to deny it. I read,
what will be my doom for not allowing his Latin; yet I must profess I
dare secure the Doctor for having read all authors, notwithstanding his
assertion, and I hope he will do the like for me. And for those which he
hath read, had he brought no better proofs than these, he had, I am
sure, been whipped soundly in Westminster School, for his impudence as
well as ignorance, by the learned master thereof at present. But I dare
further affirm, the Doctor hath not read in this point any, but only
consulted with Robert Stephen’s _Thesaurus Linguæ Latinæ_, whence he
hath borrowed his allegations in _adduco_; and for the other, I had not
so much idle time as to compare them. And, lest the fact might be
discovered, he hath sophisticated those authors whence Stephen cites the
expressions, and imposed upon them others. If it be not so, or that the
Doctor could not write it right when the copy was right before him, let
him tell me where he did ever read in Plautus, _adducta res in
fastidium_. I find the whole sentence in Pliny’s preface to Vespasian
(out of whom in the precedent paragraph he cites it) about the middle:
_alia vero ita multis prodita, ut in factidium sint adducta_, which is
the very example Stephanus useth, although he doth premise his _adducta
res in fastidium_. Let the Doctor tell where he ever did read in Horace,
_Ova noctuæ_, &c. _tædium vini adducunt_. Did he, or any else, with the
interposition of an &c. make Trochaics? I say, and Stephanus says so,
too, that it is in Pliny, lib. xiii. cap. 15, near the end; the whole
sentence runs thus: _Ebriosis Ova noctuæ per triduum data in vino,
tædium ejus adducunt_. I doubt not but these are the places he aimed at,
although he disguised and minced the quotations; if they be not, I
should be glad to augment my Stephanus with his additions.
These things premised, I come to consider the Doctor’s proofs: _Res eo
adducta est_: _adducta vita in extremum_: _adducta res in fastidium_:
_rem ad mucrones et manus adducere_: _contractares et adducta in
augustum_: _res ad concordiam adduci potest_: _in ordinem adducerem_:
_adducere febres, sitim, tedium vini_ (all in Robert Stephen) betwixt
which and _adducere malleum_, what a vast difference there is, I leave
them to umpire _qui terretes et religiosas nacti sunt aures_, who are
the competent judges of elegancy, and only cast in the verdict of one or
two, who are in any place (where the purity of the Latin tongue
flourisheth) of great esteem. Losæus, in his _Scopæ Linguæ Latinæ, ad
purgandam Linguam a barbarie_, &c. (would any think that the Doctor’s
elegant expression, frequent in all authors, which none but the
malicious or ignorant can deny, should suffer so contumelious an
expurgation?) Losæus, I say, hath these words: _Adferre plerique minus
attenti utuntur pro adducere. Quod Plautus, in Pseudolo, insigni exemplo
notat_.
_CA._--Attuli hunc.
_PS._--Quid attulisti?
_CA._--Adduxi volui dicere.
_PS._--Quis istic est?
_CA._--Charinus.
_Satis igitur admonet discriminis inter ducere, reducere, adducere, et
abducere, quæ de persona; et ferre, adferre_, &c. _quæ de re dicuntur_.
Idem, _Demetrium, quem ego novi, adduce: argentum non moror quin feras_.
_Cavendum igitur est ne vulgi more_, (let the Doctor mark this, and know
that _this author is authentic amongst the Ciceronians_), _adferre de
persona, dicamus, sed adducere; licet et hoc de certis quibusdam rebus
non inepte dicatur_. In this last clause he saith as much as Mr. Hobbes
saith, and what the Doctor proves; but, that ever the Doctor brought an
example which might resemble _adducis malleum_, is denied; for I have
mentioned already his allegations, every one, of _adduco_. Another
author, (a fit antagonist for the elegant Doctor), is the _Farrago
sordidorum Verborum_, joined with the Epitome of L. Valla’s
_Elegancies_. He saith: _Accerse, adhuc Petrum, Latine dicitur, pro eo
quod pueri dicunt, adfer Petrum_. And this may suffice to justify Mr.
Hobbes’s exception who proceeded no further than this author to tell the
Doctor that _adduco_ was used of animals. But the Doctor replies, _this
signification is true, but so may the other be also_. I say if it never
have been used so, it cannot be so, for we cannot coin new Latin words,
no more than French or Spanish who are foreigners. Mr. Hobbes was upon
the negative, and not to disprove the contrary opinion. If the Doctor
would be believed, he must prove it by some example, (which is all the
proof of elegancy), and till he do so, not to believe him, it is
sufficient not to have cause. But, Doctor Wallis, _why not adduco for a
hammer as well as a tree?_ I answer yes, equally for either, and yet for
neither. Did ever anybody go about to mock his readers thus solemnly? I
do not find, to my best remembrance, any example of it in Stephen, and
the Doctor is not wiser than his book; if there be, it is strange the
Doctor should omit the only pertinent example, and trouble us with such
impertinences for three or four pages. In Stephen there are _adducere
habenas_ and _adducere lorum_, but in a different sense. It is not
impossible I may guess at the Doctor’s aim. In Tully _de Nat. Deor._ as
I remember, there is this passage: _Quum autem ille respondisset, in
agro ambulanti ramulum adductum ut remissus esset, in oculum suum
recidisse_, where it signifies nothing else but to be _bent_, _bowed_,
_pulled back_, and in that sense, _the hammer of a clock_, or that of a
_smith, when he fetcheth his stroke_, may be said _adduci_. And this, I
conceive, the Doctor would have us in the close think to have been his
meaning; else, what doth he drive at in these words? “When you have done
the best you can, you will not be able to find better words than
_adducere malleum_ and _reducere_, to signify the two contrary motions
of the _hammer_, the one when you strike with it (_excellently
trivial!_) the other when you take it back (_better and better_), _What
to do?_ to fetch another stroke. If any can believe that this was his
meaning, I shall justify his Latin, but must leave it to him to prove
its sense. If he intended no more, why did he go about to defend the
other meaning, and never meddle with this? Which yet might have been
proved by this one example of mine? May not, therefore, his own saying
be justly retorted upon him in this case, _Adducis malleum, ut occidas
muscam_?
Another exception is, _Falsæ sunt, et multa istiusmodi_
(_propositiones_). I wish the Doctor could bring so good parallels, and
so many, out of any author, for his _Adducis malleum_, as Tully affords
in this case. Take one for all, out of the beginning of his _Paradoxes_:
_Animadverti sæpe Catonem, cum in senatu sententiam diceret, Locos
graves ex Philosophia tractare, abhorrentes ab hoc usu forensi, et
publico, sed dicendo consequi tamen, ut illa etiam populo probabilia
viderentur_. This is but a _Solæcophanes_, and hath many precedents
more, as in the second book of his _Academical Questions_, &c.
I cannot now stay upon each particular passage; I do not see any
necessity of tracing the Doctor in all his vagaries. Now, he disallows
_tanquam diceremus_, _as if we should say_. But why is that less
tolerable than _tanquam feceris_, _as if you had done_? “It should be
_quasi_, (forsooth!) or _ac si_, or _tanquam si_, which is Tully’s own
word.” What is _tanquam si_ become but one word? _Tanquam si tua res
agatur_, &c. Good Doctor, leave out Tully and all _Ciceronians_, or you
will for ever suffer for this, and your _Adducis malleum_. Is not this
to put yourself on their verdict when you oppose Mr. Hobbes with Tully?
But the Doctor gives his reason. And though he hath had the luck in his
_Adducis malleum_, to follow the first part of that saying, _Loquendum
cum vulgo_, yet now it is, _sentiendum cum sapientibus_. For _tanquam_
without _si_ signifies but _as_, not _as if_. It is pity the Doctor
could not argue in symbols too, that so we might not understand him; but
suppose all his papers to carry evidence with them, because they are
_mathematically_ scratched. How does he construe this:--
“Plance tumes alto Drusorum sanguine, tanquam
Feceris ipse aliquid, propter quod nobilis esses.”
So Cœlius, one much esteemed by Cicero, who hath inserted his Epistles
into his works, saith, in his fifth Epistle (Tul. Epist. Fam. lib. viii.
ep. 5), _Omnia desiderantur ab eo tanquam nihil denegatum sit ei quo
minus paratissimus esset qui publico negotio præpositus est_. But it was
not possible the Doctor should know this, it not being in Stephen, where
his examples for _tanquam si_ are.
But, the Doctor having pitched upon this criticism, and penned it,
somebody, I believe, put him in mind of the absurdity thereof; and yet
the generous _Professor_, (who writes running hand and never transcribed
his papers, if I am not misinformed), presumed nobody else could be more
intelligent than he, who had perused Stephen. He would not retract
anything, but subjoins, “That he will allow it as passable, because
other modern writers, and some of the ancients, have so used it, as Mr.
Hobbes hath done.” I know not what authors the Doctor meant, for, if I
am not much mistaken, I do not find any in Stephen. His citation of
Columella is not right, (lib. v. cap. 5), nor can I deduce anything
thence till I have read the passage, but, if he take Juvenal and Cœlius
for modern authors, I hope he will admit of Accius, Nævius, and
Carmenta, for the only ancients. Let him think upon this criticism, and
never hope pardon for his _Adducis malleum_, which is not half so well
justified, and yet none but _madmen_ or _fools_ reject it.
But certainly the Doctor should not have made it his business to object
_Anglicisms_, in whose Elenchus I doubt not but there may be found such
phrases as may serve to convince him that he is an Englishman, however
Scottified in his principles. If the Doctor doubt of it, or but desire a
catalogue, let him but signify his mind, and he shall be furnished with
a _Florilegium_. But I am now come to the main controversy about Empusa.
The Doctor saith nothing in defence of his _quibble_, nor gives any
reason why he jumbled languages to make a silly clinch, which will not
pass for wit either at Oxford or at Cambridge; no, nor at Westminster.
It seems he had derived _Empusa_ from ἓν and ποῦς, and said it was a
kind of _Hobgoblin_ that hopped upon one leg: and hence it was that the
boys’ play (_Fox come out of thy hole_) came to be called _Empusa_. I
suppose he means _Ludus Empusæ_. This derivation he would have to be
good, and that we may know his reading, (though he hath scarce consulted
any of the authors), he saith Mr. Hobbes did laugh at it, until somebody
told him that it was in the Scholiast of Aristophanes (as good a critic
as Mr. Hobbes), Eustathius, Erasmus, Cœlius Rhodiginus, Stephanus,
Scapula, and Calepine. But sure he doth not think to scape so. To begin
with the last; Calepine doth indeed say, _uno incedit pede, unde et
nomen_. But he is a _Modern_, and I do not see why his authority should
outweigh mine if his author’s reasons do not. He refers to Erasmus and
Rhodiginus. Erasmus in the adage, _Proteo mutabilior_ hath these words
of Empusa: _Narrant autem uno videri pedi_--this is not to hop--_unde et
nomen inditum putant_, Ἔμπουσαν ὁιονεὶ ἑνίποδα. He doth not testify his
approbation of the derivation at all, only lets you know what
etymologies some have given before him. And doth anybody think that Dr.
Harmar was the first which began to show his wit, (or folly), in
etymologizing words? Cœlius Rhodiginus doth not own the derivation, only
saith, _Nominis ratio est, ut placet Eustathio, quia uno incedit
pede_;--is this to hop?--_sed nec desunt qui alterum interpretentur
habere æneum pedem, et inde appellatam Empusam; quod in Batrachis
Aristophanes expressit_. And then he recites the interpretation that
Aristophanes’s Scholiast doth give upon the text, of which by and by. If
any credit be to be attributed to this allegation, his last thoughts are
opposite to Dr. Wallis; and _Empusa_ must be so called, not because she
hopped upon one leg, but because she had but one, the other being brass.
But for the former derivation he refers to Eustathius.
As to Eustathius, I do easily conjecture that the reader doth believe
that Rhodiginus doth mean Eustathius upon Homer, for that is the book of
most repute and fame, his other piece being no way considerable for bulk
or repute. But it is not that book, nor yet his History of Ismenias, but
his notes upon the 725th verse of Dionysius Περῖηγητής. The poet had
said of the stone _Jaspis_, that it was
Ἐχθρηὶν Ἐμπούσησι καὶ ἄλλοις ἔιδώλοισιν,
Upon which Eustathius thus remarks: Δοκεῖ γαρ ἀλεξίκακος εἶναι ἡ λίθος
ἅυτη, καὶ ἀποτροπιαςτικὴ φασμἀτων, ὧν ἕν ἐςτι καὶ ἡ Ἔμπουσα, δαιμονιόν
τι τερί τἰὼ Ἑκάτην, ἑνὶ ποδὶ δοκοῦν δἰήκεσθαι· (_fortè_ διερείδεσθαι
_Steph._) ὄθεν καὶ παρονομάζεται, ὡς ἔι τις ἔιπη μονόπους ποδι ζωοῦ· ὡς
τοῦ ἑτέρου ποδος χαλκοῦ ὄντος, κατὰ τὸν μῦθον. This testimony doth not
prove anything of _hopping_, and, as to the derivation, I cannot but say
that Eustathius had too much of the grammarian in him, and this is not
the first time, neither in this book, nor elsewhere, wherein he hath
trifled. It is observable out of the place, that there were more
_Empusas_ than one, as, indeed, the name is applied by several men to
any kind of frightful phantasm. And so it is used by several authors,
and for as much as phantasms are various, according as the persons
affrighted have been severally educated, &c. every man did impose this
name upon his own apprehensions. This gave men occasion to fain _Empusa_
as such--for who will believe that she was not apprehended as having
four legs, when she appeared in the form of a cow, dog, &c.--but, as
apprehended by _Bacchus_ and his man at that time. I do not find that
she appeared in any shape but such as made use of legs in going, whence
I imagine that _Empusæ_ might be opposite to the θεοὶ νεποδες, which
appellation was anciently fixed upon the gods, (_propitious_) upon a
two-fold account; first, for that they were usually effigiated as having
no feet, which is evident from ancient sculpture, and secondly, for that
they are all said not to walk, but rather swim, if I may so express that
_non gradiuntur, sed fluunt_, which is the assertion of all the
commentators I have ever seen upon that verse of Virgil:--
“Et vera incessu patuit dea”----
This whole discourse may be much illustrated from a passage in
Heliodorus, Æthiop. lib. iii. sec. 12, 13. Calasiris told Cnemon that
the Gods Apollo and Diana did appear unto him; Cnemon replied, Ἀλλὰ τίνα
δὴ τρόπον ἒφασκες ἐνδεδεῖχθαἱ σοι τοῦς θεοῦς ὅτι μὴ ἐνύπνιον ἦλθον, ἀλλ’
ἐναργῶς ἐφᾶνησαν; upon this the old priest answered, that both gods and
demons, when they appear to men, may be discovered by the curious
observer, both in that they never shut their eyes, καὶ τῳ βαδίσματι
πλέον, οὐ κατὰ διάστησιν τῶν ποδῶν οὐδέ μετάθεσιν ἀνυομένω, ἀλλὰ κατὰ
τινα ρὕμην ἀέριον, καὶ ὁρμὴν ἀπαραπόδιστον, τεμνόντων μᾶλλον τὸ περιεχον
ἢ διαπορευομένων. Δὶο δὴ καὶ τὰ ἀγάλματα τῶν θεῶν Ἀιγύπτιοι τὼ πὸδέ
ζευγνύντες καὶ ὥσπερ ἑνοῦντες ἵστᾶσιν. ἅ δὴ καὶ Ὅμηρος ἐιδῶς, ἅτε
Ἀιγύπτιος, καὶ τὴν ἱερὰν πάιδευσιν ἐκδιδαχθείς, συμβολικῶς τοῖς ἔπεσιν
ἐναπεθετο, τοῖς δυναμένοις συνιέναι γνωριζειν καταλιπών, ἐπι τοῦ
ποσειδῶνος, το
Ἴχνια γὰρ μετόπισθε, ποδῶν ἠδέ κνημάων
Ῥεῖ ἔγνων ἀπιὸντος.
οἴον ῥέοντος ἐν τῆ πορεία, τοῦτο γάρ εστι τό ῥεῖ ἀπιόντος, καὶ οῦχ ὥς
τινες ἠπάτηνται, ῥᾳδίως ἔγνων ὑπολαμβάνοντες. Farnaby, upon the place in
Virgil, observes, that _Deorum incessus est continuus et æqualis, non
dimotis pedibus, neque transpositis_, ἀλλὰ κατὰ ῥύμην ἀέριον. Cornelius
Schrevelius in the new Leyden notes saith, _Antiquissima quæque Deorum
simulachra, quod observarunt viri magni, erant_ τοῦς πόδας συμβεβηκότα,
_diique ipsi non gradiuntur sed fluunt_. Their statues were said to
stand rather upon columns than upon legs, for they seem to have been
nothing but columns shaped out into this or that figure, the base
whereof carrying little of the representation of a foot. These things
being premised, I suppose it easy for the intelligent reader to find out
the true etymology of _Empusa, quasi_ ἐν ποσιν οῦσα, or βάινουσα, from
going on her feet, whereas the other _gods_ and _demons_ had a different
gait. If any can dislike this deduction, and think her so named from
ἑνιπους, whereas she always went upon two legs, (if her shape permitted
it) though she might draw the one after her, as a man doth a wooden leg:
I say, if any, notwithstanding what hath been said, can join issue with
the Doctor, my reply shall be Σοὶ μὲν ταῦτα δοκοῦντ’ ἐστὶν, ἐμὸι δὲ
τάδε.
Now, as to the words of Aristophanes upon which the Scholiast descants,
they are these:--speaking of an apparition strangely shaped, sometimes
like a camel, sometimes like an ox, a beautiful woman, a dog, &c.
Bacchus replies:
Ἔμπουσα τοινὺν γ’ἐστι.
ΞΑ. πυρὶ γοῦν λάμπεται
ἅπαν το προσωπον, καὶ σκελος χαλκοῦν ἔχει.
ΔΙ. Νὴ τὸν Ποσειδῶ, καὶ βολιτινον θάτερον.
ΞΑ. Σἁφ’ ἵσθι.
The Scholiast hereupon tells us that _Empusa_, was Φαντασμα δαιμονιῶδες
ὑπὸ Ἑκάτης ἐπιπεμπόμενον καὶ φαινόμενον τοἴς δυστυχοῦσιν, ὅ δοκεῖ πολλὰς
μορφας αλλασσεω καὶ ὁι μεν φασιν ἀυτην μονοποδα εῖναι, καὶ
ἐτυμολογοῦσιν’ ὁιονεὶ ἑνιποδα, διὰ το ἑνὶ ποδι κεχρῆσθαι. And this is
all that is material in the Scholiast, except that he adds by and by,
that βολιτινον σκελος is all one with the leg of an ass. And this very
text and Scholiast is that to which all the authors he names, and more,
do refer.
I come now to Stephen, who, in his index, and in the word ποδίζω, gives
the derivation of _Empusa_. Ποδιζω, _gradior, incedo_, (not to hop) _sic
Suidas_ Ἔμπουσαν _dictam ait_ παρὰ το ἑνὶ ποδιζειν. In the index thus:
_sunt qui dictam putent_ παρὰ τὸ ἑνὶ ποδὶζειν, _quod uno incedat pedi,
quasi_ Ἔμπουσαν, _alterum enim pedem æneum habet_. But neither Stephen,
nor any else, except _Suidas_, whom the hypercritical Doctor had not
seen, no, not the Scholiast of Aristophanes (a better critic than Mr.
Hobbes) doth relate the etymology as their own. Nay, there is not one
that saith _Empusa_ hopped on one leg, which is to be proved out of
them. The great Etymological Dictionary deriveth it παρὰ τὸ ἐμποδιζειν,
to _hinder_, _let_, &c. its apparition being a token of ill luck. But,
as to the Doctor’s deduction, it saith, Ἔμπουσα Ψιλοῦπαι, εἰ καὶ δοκεῖ
παρὰ τὸ ἕνα συγκεῖσθαι. It doth only _seem_ so. And it is strange that
ἑν should not alter only its _aspiration_, but change its ν into μ,
which I can hardly believe admittable in Greek, least there should be no
difference betwixt its derivatives and those of ἐν. When I consider the
several μορμόνες which the Grecians had, some whereof did fly, some had
no legs, &c., I can think that the origin of this name may have been
thus: some amazed person saw a _spectrum_, and, giving another notice of
it, his companion might answer, it is Βριμὼ, Μορμὼ Ἡκὰτη, but he,
meeting with a new phantasm, cries, ἐν ποσὶ βαίνει or βαδίζει, for which
apprehension of his, somebody coined this expression of Ἔμποῦσα. It may
also be possibly deduced from Ἐμποδὶζω, so that τύχη ἐμποδιζουσα might
afterwards be reduced to the single term of _Empusa_. Nor do I much
doubt but that those who are conversant in languages, and know how that
several expressions are often jumbled together to make up one word upon
such like cases, will think this a probable origination. I believe,
then, that Mr. Hobbes’s friend did never tell him it was in Eustathius,
or that _Empusa_ was an _hopping phantasm_. It had two legs and went
upon both, as a man may upon a wooden leg. Ἔμποῦσα is also a name for
Lamia, and such was that which Menippus might have married, which, I
suppose, did neither hop nor go upon one leg, for he might have
discovered it. But Mr. Hobbes did not except against the derivation,
(although he might justly, derivations made afterwards carrying more of
fancy than of truth, and the Doctor is not excused for asserting what
others barely relate, none approve), but asked him where that is, in
what authors _he read that boys’ play to be so called_. To which
question, the Doctor, to show his reading and the good authors he is
conversant in, replies, _in Junius’s Nomenclator, Rider and Thomas’s
Dictionary, sufficient authors in such a business_, which, methinks, no
man should say that were near to so copious a library. It is to be
remembered that the trial now is in Westminster School, and amongst
Ciceronians, neither whereof will allow those to be sufficient authors
of any Latin word. Alas, they are but _Vocabularies_; and, if they bring
no author for their allegation, all that may be allowed them is, that,
by way of allusion, our modern play may be called _Ludus Empusæ_. But
that it is so called we must expect, till some author do give it the
name. These are so good authors, that I have not either of them in my
library. But I have taken the pains to consult, first, Rider; I looked
in him, (who was only author of the English Dictionary) and I could not
find any such thing. It is true, in the Latin Dictionary, which is
joined with Rider, but made by Holyoke; (O that the Doctor would but
mark!) in the index of obsolete words, there is _Ascoliasmus, Ludus
Empusæ_, _Fox to thy hole_, for which word, not signification, he
quoteth Junius. The same is in Thomasius, who refers to Junius in like
manner. But could the Doctor think the word obsolete, when the play is
still in fashion? Or, doth he think that this play is so ancient as to
have had a name so long ago, that it should now be grown obsolete? As
for Junius’s interpretation of _Empusa_, it is this: _Empusa, spectrum,
quod se infelicibus ingerit, uno pede ingrediens_. Had the Doctor ever
read him, he would have quoted him for his derivation of _Empusa_, I
suppose. In Ascoliasmus, he saith, _Ascoliasmus, Empusæ Ludus, fit ubi,
altero pede in aere librato, unico subsiliunt pede:_ ἀσκολιασμὸς
_Pollux; Almanicè, Hinckelen; Belgicè,_ _Op een been springhen;
Hinckepincken, Flandris_. But what is it in English he doth not tell,
although he doth so in other places often. What the Doctor can pick out
of the Dutch I know not; but, if that do not justify him, as I think it
doth not, he hath wronged Junius, and greatly imposed upon his readers.
But, to illustrate this controversy further, I cannot be persuaded the
Doctor ever looked into Junius, for, if he had, I am confident,
according to his wonted accurateness, he would have cited Pollux’s
_Onomasticon_ into the bargain, for Junius refers to him, and I shall
set down his words, that so the reader may see what _Ascoliasmus_ was,
and all the Doctor’s authors say _Ludus Empusæ_ and _Ascoliasmus_ were
one and the same thing. Julius Pollux (lib. ix. cap. 7): Ὁ δε
Ἀσκολὶασμὸς, (old editions read it, Ἀ’σκολιασμὸς et ασκολιάζω) τοῦ
ἑτέρου ποδὸς αἰωρουμένου, κατὰ μόνου τοῦ ἑτέρου πηδᾶν ἔπόιει; ὅπερ
Ἀσκωλιάζὲιν ὠνόμαζον· ἤτοι εἰς μἢκος ἐνήλλαντο, ἢ ὁ μὲν ἐδίωκεν οὕτως,
οἱ δὲ ὑπέφευγον ἐπ’ ἀμφοῖν θὲοντες, ἕως τινὸς τῳ φερομένῳ ποδὶ ὁ διὼκων
δυνηθῇ τυχεῖν· ἤ καὶ στάντες ἐπήδων, ἀριθμοῦντες τὰ πηδήματα· προσέκειτο
γὰρ τῷ πλήθει τὸ νικᾶν. Ἀσκωλιάζειν δὲ ἐκαλεῖτο καὶ τὸ ἐπιπηδᾶν ἀσκῷ
κενῷ καὶ ὑποπλέω πνευματος, ἠλείμμένω, ἵναπερ ὀλισθάνοιεν περὶ τὴν
ἀλοιφὴν. “So that _Ascoliasmus_, and consequently, _Ludus Empusæ_, was a
certain sport which consisted in hopping, whether it were by striving
who could hop furthest, or whether only one did pursue the rest hopping,
and they fled before him on both legs, which game he was to continue
till he had caught one of his fellows, or whether it did consist in the
boys’ striving who could hop longest. Or, lastly, whether it did consist
in hopping upon a certain bladder, which, being blown up and well oiled
over, was placed upon the ground for them to hop upon, that so the
unctuous bladder might slip from under them and give them a fall.” And
this is all that Pollux holds forth. Now, of all these ways, there is
none that hath any resemblance with our _Fox to thy hole_; but the
second: and yet, in its description, there is no mention of beating him
with gloves, as they do now-a-days, and wherein the play consists as
well as in hopping. It might, notwithstanding, be called _Ludus Empusæ_,
but not in any sort our _Fox to thy hole_; so that the Doctor and his
authors are out, imposing that upon Junius and Pollux which they never
said. And thus much may suffice as to this point. I shall only add out
of Meursius’s _Ludi Græci_, that _Ascolia_ were not _Ludus Empusæ_ but
_Bacchisacra_, and he quotes Aristophanes’s Scholiast in Plutus, Ἀσκώλια
ἑορτὴ Διονύσου ἀσκὸν γαρ οἵνου πληροῦντες, ἑνὶ ποδὶ τοῦτον ἐπεπήδον, καὶ
ὁ πηδήσας ἆθλον εἶχε τὸν οἵνου. As also Hesychius, Ἀσκωλιάζειν, κυρίως
τὸ ἐπὶ τοῦς ἀσκοὺς ἅλλεσθαι.
But I could have told the Doctor where he might have read of _Empusa_ as
being the name of a certain sport or game, and that is, _in Turnebus
Adversaria_, lib. xxvii. cap. 33. There he speaks of several games
mentioned by Justinian in his _Code_, at the latter end of the third
book, one of which he takes to be named _Empusa_; adding withal, _that
the other are games, it is indisputable_, only _Empusa in lite et causa
erit, quod nemo nobis facile assensurus sit Ludum esse, cum constet
spectrum quoddam fuisse formas, varie mutans. Sed quid vetat eo nomine
Ludum fuisse? Certe ad vestigia vitiatæ Scripturæ quam proximo accedit._
Yet he only is satisfied in this conjecture, till somebody else shall
produce a better. And now what shall I say? Was not Turnebus as good a
critic, and of as great reading as Dr. Wallis, who had read over Pollux,
and yet is afraid that nobody will believe _Empusa_ to have been a game,
and all he allegeth for it is, _quid vetat_? Truly, all I shall say, and
so conclude this business, is, that he had read over an infinity of
books, yet, had not had the happiness, which the Doctor had, to consult
with _Junius’s Nomenclator, Thomasius and Rider’s Dictionary, authors
sufficient in such a case_.
I now come to the Doctor’s last and greatest triumph, at which I cannot
but stand in admiration, when I consider he hath not got the victory.
Had the Doctor been pleased to have conversed with some of the fifth
form in Westminster School, (for he needed not to have troubled the
learned master), he might have been better informed than to have exposed
himself thus.
Mr. Hobbes had said that στιγμὴ signified _a mark with a hot iron_; upon
which saying the Doctor is pleased to play the droll thus: “Prithee tell
me, good Thomas, before we leave this point, (O the wit of a divinity
doctor!) who it was told thee that στιγμὴ was a mark with an hot iron,
for it is a notion I never heard till now, and do not believe it yet.
Never believe him again that told thee that lie, for as sure as can be,
he did it to abuse thee; ϛιγμὴ signifies a distinctive point in writing,
made with a pen or quill, not a mark made with a _hot iron_, such as
they brand rogues withal; and, accordingly, ϛιζω δῖαϛιζω, _distinguo_,
_interstinguo_, are often so used. It is also used of a _mathematical_
point, or somewhat else that is very small, στιγμὴ χρὸνου, a moment, or
the like. What should come in your cap, to make you think that ϛῖγμὴ
signifies a mark or brand with a _hot iron_? I perceive where the
business lies; it was ϛίγμα ran in your mind when you talked of ϛιγμὴ,
and, because the words are somewhat alike, you jumbled them both
together, according to your usual care and accurateness, as if they had
been the same.”
When I read this I cannot but be astonished at the Doctor’s confidence,
and applaud him who said, ἀμάθεια θάρσὸς φέρει. That the Doctor should
never hear that ϛιγμὴ signifies _a mark with a hot iron_, is a manifest
argument of his ignorance. But, that he should advise Mr. Hobbes not to
believe his own readings, or any man’s else that should tell him it did
signify any such thing, is a piece of notorious impudence. That ϛιγμὴ
_signifies a distinctive point in writing made with a pen or quill_, (is
a pen one thing and a quill another to write with?) nobody denies. But,
it must be withal acknowledged it signifies many things else. I know the
Doctor is a _good historian_, else he should not presume to object the
want of history to another; let him tell us how long ago it is since men
have made use of pens or quills in writing; for, if that invention be of
no long standing, this signification must also be such, and so it could
not be that from any allusion thereunto the mathematicians used it for a
point. Another thing I would fain know of this great historian, how long
ago ϛίζω and διαϛίζω began to signify _interpungo_? For, if the
mathematics were studied before the mystery of printing was found out,
(as shall be proved whenever it shall please the Doctor, out of his no
reading, to maintain the contrary), then the _mathematical_ use thereof
should have been named before the _grammatical_. And, if this word be
translatitious, and that sciences were the effect of long contemplation,
the names used wherein are borrowed from talk, Mr. Hobbes did well to
say, that στιγμὴ precedaneously to that _indivisible_ signification
which it afterwards had, did signify a _visible mark_ made by a hot
iron, or the like. And, in this procedure, he did no more than any man
would have done, who considers that all our knowledge proceeds from our
senses; as also that words do, _primarily_, signify things obvious to
_sense_, and only _secondarily_, such as men call _incorporeal_. This
leads me to a further consideration of this word. Hesychius, (of whom it
is said that he is _Legendus non tanquam Lexicographus, sed tanquam
justus author_), interprets στιγμὴ, νυγμή, which is a point of a greater
or lesser size, made with any thing. So ϛίζω signifies to prick or mark
with anything in any manner, and hath no impropriated signification in
itself, but according to the writer that useth it. Thus, in a
_grammarian_ ϛίζω signifies to _distinguish_, by _pointing_ often;
sometimes, even in them, it is the same with ὀβελίζω; sometimes it
signifies to set a mark that something is wanting in that place, which
marks were called ϛιγμαί. In matters of policy, ϛίζω signifies to
_disallow_, because they used to put a ϛιγμὴ (not ϛίγμα) before his name
who was either disapproved or to be mulcted. In punishment it signifies
to _mark_ or _brand_, whereof I cannot at present remember any other
ways than that of an _hot iron_, which is most usual in authors, because
most practised by the ancients. But, that the mark which the _Turks_ and
others do imprint without burning may be said ϛίζεσθαι, I do not doubt,
no more than that Herodian did to give that term to the ancient Britons,
of whom he says, τὰ σώματα ἐϛίζοντο γραφαῖς ποικίλαις, καὶ ζώων
παντοδαπῶν εἰκόσι. Thus, horses that were branded with κάππα and σαν
(κοππἀτιαι and σαμφοραι) were said ϛίζεσθαι. Thus, in its origin, ϛιγμὴ
doth signify a _brand or mark with an hot iron_, or the like; and that
must be the proper signification of στιγμὴ, which is proper to ϛίζω,
none but such as Dr. Wallis can doubt. In its _descendants_ it is no
less evident, for, from στιγμὴ comes _stigmosus_, which signifies to be
branded; _Vitelliana cicatrice stigmosus_, not _stigmatosus_. So Pliny
in his Epistles, as Robert Stephen cites it. And στιγματιας (the
derivative of στιγμὴ, which signifies any mark, as well as a brand, even
such as remain after stripes, being black and blue), was a nickname
imposed upon the grammarian Nicanor, ὅτι περὶ στιγμῶν ἐπολυλόγησε. And,
though we had not any examples of στῖγμὴ being used in this sense, yet,
from thence, for any man to argue against it, (but he who knows no more
than Stephen tells him) is madness, unless he will deny that any word
hath lost its right signification, and is used only, by the authors we
have, although neither the Doctor nor I have read all them, in its
analogical signification. I have always been of opinion, that στιγμὴ
signified a _single point_, big or little, it matters not; and στίγμα, a
_composure of many_; as γραμμὴ signifies a _line_, and γράμμα a
_letter_, made of several lines. For στίγμα signified the _owl_, the
_sæmæna_, the letter K, yea, _whole words, lines, epigrams_ engraven in
men’s faces; and στιγμὴ, I doubt not, had signified _a single point_,
had such been used, and so it became translatitiously used by
grammarians and mathematicians. I could give grounds for this
conjecture, and not be so impertinent as the Doctor in his sermon, where
he told men that σοφός was not in Homer; that from ἄφρων came _ebrius_;
that _sobrietas_ was not bad Latin, and that _sobrius_ was once, as I
remember, in Tully. Is this to speak suitably to the oracles of God, or
rather to lash out into idle words? Hath the Doctor any ground to think
these are not impertinences? Or, are we, poor mortals, accountable for
such _idle_ words as fall from us in private discourses, whilst these
ambassadors from heaven _droll_ in the pulpit without any danger of an
after-reckoning?
But I proceed to a further survey of the Doctor’s intolerable ignorance.
His charge in the end of the _school-master’s_ rant is, that he should
_remember_ στίγμα and στιγμὴ _are not all one_. I complained before that
he hath not cited Robert Stephen aright; now I must tell him he hath
been negligent in the reading of Henry Stephen: for in him he might have
found that στίγμα was sometimes all one with στιγμὴ, though there be no
example in him wherein στιγμὴ is used for στίγμα. Hath not Hesiod, (as
Stephen rightly citeth it), in his _Scutum_, 166-67.
Στἴγματα δ’ ὥς ἐπέφαντο ἴδεῖν δεινοῖσι δράκουσι
Κυανέα κατὰ νῶτα
_ubi scholiastes_ ὥσπερ δὲ στιγμαὶ ἦσαν ἐπάνω, τῶν ῥάχεων τῶν δρακόντων,
κατάστίκτοι γὰρ καὶ ποικίλοι ὁι ὄφεις. So Johannes Diaconus upon the
place, a man who (if I may use the Doctor’s phrase) was _as good a
critic as_ the Geometry Professor.
Thus much for the _Doctor_. To the understanding _reader_, I say that
στιγμὴ is used for burning with a hot iron: _2 Macchab._ ix. 11, where
speaking of Antiochus’s lamentable death, his body putrefying and
breeding worms, he is said, ἐις ετίγνωσιν τοῦ θεοῦ ἔρχεθαι θείᾳ μάστιγι,
κατα στιγμὴν ἐπιτεινόμενος ταῖς ἀλγηδόσι; _being pained as if he had
been pricked or burned with hot irons_. And that this is the meaning of
that elegant writer, shall be made good against the Doctor, when he
shall please to defend the vulgar interpretation. Pausanias, in
_Bœoticis_, speaking of Epaminondas, who had taken a town belonging to
the Sicyonians, called Phœbia (Φουβία) wherein were many Bœotian
fugitives, who ought, by law, to have been put to death, saith he
dismissed them under other names, giving them only a _brand_ or _mark_.
Πόλισμα ἑλὼν Σικυωνἰων Φουβίαν, ἔνθὰ ἦσαν το πολὺ οἱ Βοιώτιοι φυγάδες,
στιγμήν ἀφίησι τοῦς ἐγκαταληφθέντας ἄλλην σφίσιν ἣν ετυχε πατρίδα
ἐπονομάζων ἐκάστω. It is true στιγμὴν is here put _adverbially_, but
that doth not alter the case. Again, Zonaras, in the third tome of his
History, in the life of the Emperor Theophilus, saith, that when
Theophanes and another monk had reproved the said emperor for
demolishing images, he took and _stigmatized_ each of them with twelve
_iambics_ in their faces: εἶτα καὶ τὰς ὄψεις ἀυτῶν κάτεστιξε καὶ ταῖς
στιγμαῖς μέλαν ἐπέχεε γράμματα δὲ ἐτύπουν τὰ στιγματα, τὰ δὲ ἦσαν ἴαμβοι
οὗτοι. A place so evident, that I know not what the Doctor can reply.
This place is just parallel to what the same author saith in the life of
Irene, τἀς ὄψέις σφών καταστιξας ἐν γράμμασι, μέλανος εγχεομένου τοῖς
στίγμασι. If the Doctor object that he is a modern author, he will never
be able to render him as inconsiderable as Adrianus Junius’s
_Nomenclator_, Thomasius and Rider. If any will deny that he writes good
Greek, Hieronymus Wolfius will tell them, his only fault is
περισσολογια, _redundancy_ in words, and not the use of _bad_ ones.
Another example of στιγμὴ used in this sense, is in the collections out
of Diodorus Siculus, lib. xxxiv. as they are to be found at the end of
his works, and as Photius hath transcribed them into his _Bibliotheca_.
He saith that the Romans did buy multitudes of servants and employ them
in Sicily: Οἷς, ἐκ τῶν σωματοτροφείων ἀγεληδὸν απαχθεῖσιν, ἐυθύς
χαρακτῆρα ἐπέβαλλον, καὶ στιγμὰς τοἴς σώμασιν. These are the words but
of one author, but ought to pass for the judgment of two, seeing
Photius, by inserting them, hath made them his own.
Besides, it is the judgment of a great _master_ of the Greek tongue,
that _stigmata non tam puncta ipsa quam punctis variatam superficiem
Græci vocaverunt_. I need not, I suppose, name him, so great a critic as
the Doctor cannot be ignorant of him.
Nor, were στίγματα commonly, but upon extraordinary occasions, imprinted
with an hot iron. The letters were first made by incision, then the
blood _pressed_, and the place filled up with ink, the composition
whereof is to be seen in Aetius. And thus they did use to _matriculate_
soldiers also in the hand. Thus, did the Grecian emperor, in the
precedent example of Zonaras. And if the Doctor would more, let him
repair to Vinetus’s comment upon the fifteenth Epigram of Ausonius.
And now I conceive enough hath been said to vindicate Mr. Hobbes, and to
show the insufferable ignorance of the puny professor, and unlearned
critic. If any more shall be thought necessary, I shall take the pains
to collect more examples and authorities, though I confess I had rather
spend time otherwise, than in matter of so little moment. As for some
other passages in his book, I am no competent judge of _symbolic
stenography_. The Doctor (Sir Reverence) might have used a cleanlier
expression than that of a _shitten piece_, when he censures Mr. Hobbes’s
book.
Hitherto the letter.[1] By which you may see _what came into my (not
square) cap to call_ στιγμὴ _a mark with a hot iron, and that they who
told me_ that, did no more tell me a lie than they told you a lie that
said the same of στίγμα; and, if στιγμὴ be not right as I use it now,
then call these notes not στιγρας, but στίγματα. I will not contend with
you for a trifle. For, howsoever you call them, you are like to be known
by them. Sir, the calling of a divine hath justly taken from you some
time that might have been employed in geometry. The study of algebra
hath taken from you another part, for algebra and geometry are not all
one; and you have cast away much time in practising and trusting to
symbolical writings; and for the authors of geometry you have read, you
have not examined their demonstrations to the bottom. Therefore, you
perhaps may be, but are not yet, a geometrician, much less a good
divine. I would you had but so much ethics as to be civil. But you are a
notable critic; so fare you well, and consider what honour you do,
either to the University where you are received for professor, or to the
University from whence you came thither, by your geometry; and what
honour you do to Emanuel College by your divinity; and what honour you
do to the degree of Doctor, with the manner of your language. And take
the counsel which you publish out of your encomiast his letter; think me
no more worthy of your pains, you see how I have fouled your fingers.
-----
Footnote 1:
Written by Henry Stubbe, M.A. of Christ Church, Oxford, who was,
according to Anthony a Wood, “the most noted personage of his age that
these late times have produced.”
-----
THREE PAPERS
PRESENTED TO THE ROYAL SOCIETY
AGAINST DR. WALLIS.
TOGETHER WITH
CONSIDERATIONS
ON DR. WALLIS’S ANSWER TO THEM,
BY
THOMAS HOBBES,
OF MALMESBURY.
THREE PAPERS
PRESENTED TO THE ROYAL SOCIETY.
==========
TO THE RIGHT HONOURABLE AND OTHERS, THE LEARNED MEMBERS OF THE ROYAL
SOCIETY, FOR THE ADVANCEMENT OF SCIENCES.
PRESENTETH _to your consideration, your most humble servant, Thomas
Hobbes, (who hath spent much time upon the same subject), two
propositions, whereof the one is lately published by Dr. Wallis, a
member of your Society, and Professor of Geometry; which if it should be
false, and pass for truth, would be a great obstruction in the way to
the design you have undertaken. The other is a problem, which, if well
demonstrated, will be a considerable advancement of geometry; and though
it should prove false, will in no wise be an impediment to the growth of
any other part of philosophy._
DR. WALLIS,
DE MOTU, _Cap._ v. _Prop._ 1.
If there be understood an infinite row of quantities beginning with 0 or
(1)/(0), and increasing continually according to the natural order of
numbers, 0, 1, 2, 3, &c. or according to the order of their squares, as,
0, 1, 4, 9, &c. or according to the order of their cubes, as, 0, 1, 8,
27, &c. whereof the last is given; the proportion of the whole, shall be
to a row of as many, that are equal to the last, in the first case, as 1
to 2; in the second case, as 1 to 3; in the third case, as 1 to 4, &c.
This proposition is the ground of all his doctrine concerning the
centres of gravity of all figures. Wherein may it please you to
consider:
First, whether there can be understood an infinite row of quantities,
whereof the last can be given. Secondly, whether a finite quantity can
be divided into an infinite number of lesser quantities, or a finite
quantity can consist of an infinite number of parts, which he buildeth
on as received from Cavallieri. Thirdly, whether (which in consequence
he maintaineth) there be any quantity greater than infinite. Fourthly,
whether there be, as he saith, any finite magnitude of which there is no
centre of gravity. Fifthly, whether there be any number infinite. For it
is one thing to say, that a quantity may be divided perpetually without
end, and another thing to say, that a quantity may be divided into an
infinite number of parts. Sixthly, if all this be false, whether that
whole book of _Arithmetica Infinitorum_, and that definition which he
buildeth on, and supposeth to be the doctrine of Cavallieri, be of any
use for the confirming or confuting of any propounded doctrine.
Humbly praying you would be pleased to declare herein your judgment, the
examination thereof being so easy, that there needs no skill either in
geometry, or in the Latin tongue, or in the art of logic, but only of
the common understanding of mankind to guide your judgment by.
THOMAS HOBBES,
ROSET. _Prop._ v.
_To find a straight line equal to two-fifths of the arc of a
quadrant._
I describe a square A B C D, and in it a quadrant D A C. Suppose D T be
two-fifths of D C, then will the quadrantal arc T V be two-fifths of the
arc C A. Again let D R be a mean proportional between D C and D T; then
will the quadrantal arc R S be a mean proportional between the arc C A
and the arc T V.
Suppose further a right line were given equal to the arc C A, and a
quadrantal arc therewith described; then will D C, C A, the arc on C A
be continually proportional. Set these proportionals in order by
themselves.
D C, C A, arc on C A∺
D R, R S, arc on R S∺
D T, T V, arc on T V∺
which are in continual proportion of the semi-diameter of the arc. And D
C, D R, D T are in a continual proportion by construction, and therefore
also C A, R S, T V, and arc on C A, arc on R S, arc on T V, in continual
proportion.
Therefore as D C to R S, so is R S to the arc on T V. And D C, R S, the
arc on T V will be continually proportional. And because D C, C A, the
arc on C A are also continually proportional, and have the first
antecedent D C common; the proportion of the arc on C A to the arc on T
V is (by Eucl. xiv. 28) duplicate of the proportion of C A to R S, and
the arc on R S a mean proportional between the arc on C A and the arc on
T V.
Now if D C be greater than R S, also R S must be greater than the arc on
T V; and the arc C A greater than the arc on R S. Therefore seeing D C,
C A, arc on C A, are continually proportional; the arc on T V, the arc
on R S, the arc on C A cannot be continually proportional, which is
contrary to what has been demonstrated. Therefore D C is not greater
than R S. Suppose, then, R S to be greater than D C, then will the arc
on R S be a mean proportional between the arc on T V, and a greater arc
than that on C A; and so the inconvenience returneth. Therefore the
semidiameter D C is equal to the arc R S, and D R equal to T V, that is
to say to two-fifths of the arc C A, which was to be demonstrated. Nor
needeth there much geometry for examining of this demonstration.
Therefore I submit them both to your censure, as also the whole
_Rosetum_, a copy whereof I have caused to be delivered to the secretary
of your society.
[Illustration]
TO THE
RIGHT HONOURABLE AND OTHERS,
THE LEARNED MEMBERS
OF
THE ROYAL SOCIETY,
FOR THE ADVANCEMENT OF THE SCIENCES.
-------
Presenteth to your consideration, your most humble servant Thomas
Hobbes, a confutation of a theorem which hath a long time passed for
truth; to the great hinderance of Geometry, and also of Natural
Philosophy, which thereon dependeth.
THE THEOREM.
_The four sides of a square being divided into any number of equal
parts, for example into 10; and straight lines drawn through the
opposite points, which will divide the square into 100 lesser squares;
the received opinion, and which Dr. Wallis commonly useth, is, that the
root of those 100, namely 10, is the side of the whole square._
THE CONFUTATION.
_The root 10 is a number of those squares, whereof the whole containeth
100, whereof one square is an unity; therefore the root 10, is 10
squares: Therefore the root of 100 squares is 10 squares, and not the
side of any square; because the side of a square is not a superficies,
but a line. For as the root of 100 unities is 10 unities, or of 100
soldiers 10 soldiers: so the root of 100 squares is 10 of those squares.
Therefore the theorem is false; and more false, when the root is
augmented by multiplying it by other greater numbers._
Hence it followeth, that no proposition can either be demonstrated or
confuted from this false theorem. Upon which, and upon the numeration of
infinites, is grounded all the geometry which Dr. Wallis hath hitherto
published.
And your said servant humbly prayeth to have your judgment hereupon: and
that if you find it to be false, you will be pleased to correct the
same: and not to suffer so necessary a science as geometry to be
stifled, to save the credit of a professor.
TO THE
RIGHT HONOURABLE AND OTHERS,
THE LEARNED MEMBERS
OF
THE ROYAL SOCIETY,
FOR THE ADVANCEMENT OF THE SCIENCES.
---
Your most humble servant Thomas Hobbes presenteth, that the quantity of
a line calculated by extraction of roots is not to be truly found. And
further presenteth to you the invention of a straight line equal to the
arc of a circle.
A square root is a number which multiplied into itself produced a
number.
DEFINITION.
And the number so produced is called a square number. For example:
Because 10 multiplied by 10 makes 100; the root is 10, and the square
number 100.
CONSEQUENT.
In the natural row of numbers, as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 16, &c. every one is the square of some number in the same
row. But square numbers (beginning at 1) intermit first two numbers,
then four, then six, &c. So that none of the intermitted numbers is a
square number, nor has any square root.
PROP. I.
A square root (speaking of quantity) is not a line, such as Euclid
defines, without latitude, but a rectangle.
[Illustration]
Suppose A B C D be the square, and A B, B C, C D, D A, be the sides, and
every side divided into 10 equal parts, and lines drawn through the
opposite points of division; there will then be made 100 lesser squares,
which taken altogether are equal to the square A B C D. Therefore the
whole square is 100, whereof one square is an unit; therefore 10 units,
which is the root, is ten of the lesser squares, and consequently has
latitude; and therefore it cannot be the side of a square, which,
according to Euclid, is a line without latitude.
CONSEQUENT.
It follows hence, that whosoever taketh for a principle, that a side of
a square is a mere line without latitude, and that the root of a square
is such a line (as Dr. Wallis continually does) demonstrates nothing.
But if a line be divided into what number of equal parts soever, so the
line have breadth allowed it (as all lines must, if they be drawn), and
the length be to the breadth as number to an unit; the side and the roof
will be all of one length.
PROP. II.
[Illustration]
Any number given is produced by the greatest root multiplied into
itself, and into the remaining fraction. Let the number given be two
hundred squares, the greatest root is 14(4)/(14) squares. I say that 200
is equal to the product of 14 into itself, together with 14 multiplied
into (4)/(14). For 14 multiplied into itself makes 196. And 14 into
(4)/(14) makes (56)/(14) which is equal to 4. And 4 added to 196 maketh
200; as was to be proved. Or take any other number 8, the greatest root
is 2; which multiplied into itself is 4, and the remainder (2)/(4)
multiplied into 2, is 4, and both together 8.
PROP. III.
But the same square calculated geometrically by the like parts,
consisteth (by Euclid II. 4) of the same numeral great square 196, and
of the two rectangles under the greatest side 14, and the remainder of
the side, or (which is all one) of one rectangle under the greatest
side, and double the remainder of the side; and further of the square of
the less segment; which altogether make 200, and moreover (1)/(49) of
those 200 squares, as by the operation itself appeareth thus:
The side of the greater segment is 14(4)/(14)
14(4)/(14)
Which multiplied into itself makes 200.
The product of 14, the greatest segment, into the two fractions
(4)/(14), that is, into (4)/(14) (or into twice (2)/(14)) is (56)/(14)
(that is 4); and that 4 added to 196 makes 200.
Lastly, the product of (2)/(14) into (2)/(14) or (1)/(7) into (1)/(7) is
(1)/(49). And so the same square calculated by roots is less by (1)/(49)
of one of those two hundred squares, than by the true and geometrical
calculation; as was to be demonstrated.
CONSEQUENT.
It is hence manifest, that whosoever calculates the length of an arc or
other line by the extraction of roots, must necessarily make it shorter
than the truth, unless the square have a true root.
-------
_The Radius of a Circle is a Mean Proportion between the Arc of a
Quadrant and two-fifths of the same._
Describe a square A B C D, and in it a quadrant D C A. In the side D C
take D T two-fifths of D C, and between D C and D T a mean proportional
D R, and describe the quadrantal arcs R S, T V. I say the arc R S is
equal to the straight line D C. For seeing the proportion of D C to D T
is duplicate of the proportion of D C to D R, it will be also duplicate
of the proportion of the arc C A to the arc R S, and likewise duplicate
of the proportion of the arc R S to the arc T V.
Suppose some other arc, less or greater than the arc R S, to be equal to
D C, as for example _r s_: then the proportion of the arc _r s_ to the
straight line D T will be duplicate of the proportion of R S to T V, or
D R to D T. Which is absurd; because D _r_ is by construction greater or
less than D R. Therefore the arc R S is equal to the side D C, which was
to be demonstrated.
COROL.
[Illustration]
Hence it follows that D R is equal to two-fifths of the arc C A. For R
S, T V, D T, being continually proportional, and the arc T V being
described by D T, the arc R S will be described by a straight line equal
to T V. But R S is described by the straight line D R. Therefore D R is
equal to T V, that is to two-fifths of C A.
And your said servant most humbly prayeth you to consider, if the
demonstration be true and evident, whether the way of objecting against
it by square root, used by Dr. Wallis; and whether all his geometry, as
being built upon it, and upon his supposition of an infinite number, be
not false.
CONSIDERATIONS
UPON THE ANSWER OF DOCTOR WALLIS
TO THE
THREE PAPERS OF MR. HOBBES.
Dr. Wallis says, all that is affirmed, is but _if we_ SUPPOSE _that,
this will follow_.
But it seemeth to me, that if the supposition be impossible, then that
which follows will either be false, or at least undemonstrated.
First, this proposition being founded upon his _Arithmetica
Infinitorum_, if there he affirm an absolute infiniteness, he must here
also be understood to affirm the same. But in his thirty-ninth
proposition he saith thus: “_Seeing that the number of terms increasing,
the excess above sub-quadruple is perpetually diminished, so at last it
becomes less than any proportion that can be assigned; if it proceed in
infinitum it must utterly vanish. And therefore if there be propounded
an infinite row of quantities in triplicate proportion of quantities
arithmetically proportioned (that is, according to the row of cubical
numbers) beginning from a point or 0; that row shall be to a row of as
many, equal to the greater, as 1 to 4._” It is therefore manifest that
he affirms, that in an infinite row of quantities the last is given; and
he knows well enough that this is but a shift.
Secondly, he says, that usually in Euclid, and all after him, by
_infinite_ is meant but, more than any assignable _finite_, or the
greatest possible. I am content it be so interpreted. But then from
thence he must demonstrate those his conclusions, which he hath not yet
done. And when he shall have done it, not only the conclusions, but also
the demonstration, will be the same with mine in Cap. XIV. Art. 2, 3,
&c. of my book _De Corpore_. And so he steals what he once condemned. A
fine quality.
Thirdly, he says, (by Euclid’s tenth proposition, but he tells not of
what book), that a line may be bisected, and the halves of it may again
be bisected, and so onwards infinitely; and that upon such supposed
section infinitely continued, the parts must be supposed infinitely
many.
I deny that; for Euclid, if he says a line may be divisible into parts
perpetually divisible, he means that all the divisions, and all the
parts arising from those divisions, are perpetually finite in number.
Fourthly, he says, that there may be supposed a row of quantities
infinitely many, and continually increasing, whereof the last is given.
It is true, a man may say, (if that be supposing) that white is black:
but, if _supposing_ be _thinking_, he cannot suppose an infinite row of
quantities whereof the last is given. And if he say it, he can
demonstrate nothing from it.
Fifthly, he says (for one absurdity begets another) _that a superficies
or solid may be supposed so constituted as to be_ infinitely long, _but_
finitely great, _(the breadth continually decreasing in greater
proportion than the length increaseth), and so as to have no centre of
gravity. Such is Toricellio’s Solidum Hyperbolicum acutum, and others
innumerable, discovered by Dr. Wallis, Monsieur Fermat, and others. But,
to determine this, requires more of geometry and logic, (whatsoever it
do of the Latin tongue), than Mr. Hobbes is master of._
I do not remember this of Toricellio, and I doubt Dr. Wallis does him
wrong and Monsieur Fermat too. For, to understand this for sense, it is
not required that a man should be a geometrician or a logician, but that
he should be mad.
In the next place, he puts to me a question as absurd as his answers are
to mine. Let him ask himself, saith he, if he be still of opinion, _that
there is no argument in natural philosophy to prove that the world had a
beginning_. First, whether, in case it had no beginning, there must not
have passed an infinite number of years before Mr. Hobbes was born.
Secondly, whether, at this time, there have not passed more, that is,
more than that infinite number. Thirdly, whether, in that infinite (or
more than infinite) number of years, there have not been a greater
number of days and hours, and of which, hitherto, the last is given.
Fourthly, whether, if this be an absurdity, we have not then, (contrary
to what Mr. Hobbes would persuade us), an argument in nature to prove
the world had a beginning.
To this I answer, not willingly, but in service to the truth, that, by
the same argument, he might as well prove that God had a beginning.
Thus, in case he had not, there must have passed an infinite length of
time before Mr. Hobbes was born; but there hath passed at this day more
than that infinite length, by eighty-four years. And this day, which is
the last, is given. If this be an absurdity, have we not then an
argument in nature to prove that God had a beginning? Thus it is when
men entangle themselves in a dispute of that which they cannot
comprehend. But, perhaps, he looks for a solution of his argument to
prove that there is somewhat greater than infinite; which I shall do so
far as to show it is not concluding. If from this day backwards to
eternity be more than infinite, and from Mr. Hobbes his birth backwards
to the same eternity be infinite, then take away from this day backwards
to the time of Adam, which is more than from this day to Mr. Hobbes his
birth, then that which remains backwards must be less than infinite. All
this arguing of infinites is but the ambition of school-boys.
TO THE LATTER PART OF THE FIRST PAPER.
There is no doubt if we give what proportion we will of the radius to
the arc, but that the arc upon that arc will have the same proportion.
But that is nothing to my demonstration. He knows it, and wrongs the
Royal Society in presuming they cannot find the impertinence of it.
My proof is this: that if the arc on T V, and the arc R S, and the
straight line C D, be not equal, then the arc on T V, the arc on R S,
and the arc on C A, cannot be proportional; which is manifest by
supposing in D C a less than the said D C, but equal to R S, and another
straight line, less than R S, equal to the arc on T V; and anybody may
examine it by himself.
I have been asked by some that think themselves logicians, why I
proceeded upon ⅖ rather than any other part of the radius. The reason I
had for it was, that, long ago, some Arabians had determined, that a
straight line, whose square is equal to 10 squares of half the radius,
is equal to a quarter of the perimeter; but their demonstrations are
lost. From that equality it follows, that the third proportional to the
quadrant and radius, must be a mean proportional between the radius and
⅖ of the same. But, my answer to the logicians was, that, though I took
any part of the radius to proceed on, and lighted on the truth by
chance, the truth itself would appear by the absurdity arising from the
denial of it. And this is it that Aristotle means, where he
distinguishes between a direct demonstration and a demonstration leading
to an absurdity. Hence it appears that Dr. Wallis’s objections to my
_Rosetum_ are invalid as built upon roots.
TO THE SECOND PAPER.
First, he says that it concerns him no more than other men, which is
true. I meant it against the whole herd of them who apply their algebra
to geometry. Secondly, he says that a bare number cannot be the side of
a square figure. I would know what he means by a bare number. Ten lines
may be the side of a square figure. Is there any number so bare, as by
it we are not to conceive or consider anything numbered? Or, by 10
nothings understands he bare 10? He struggles in vain, his conscience
puzzles him. Thirdly, he says 10 squares is the root of 100 square
squares. To which I answer, first, that there is no such figure as a
square square. Secondly, that it follows hence, that a root is a
superficies, for such is 10 squares. Lastly, he says that, neither the
number 10, nor 10 soldiers, is the root of 100 soldiers; because 100
soldiers is not the product of 10 soldiers into 10 soldiers. This last I
grant, because nothing but numbers can be multiplied into one another. A
soldier cannot be multiplied by a soldier. But no more can a square
figure by a square figure, though a square number may. Again, if a
captain will place his 100 men in a square form, must he not take the
root of 100 to make a rank or file? And are not those 10 men?
TO THE THIRD PAPER.
He objects nothing here, but that _the side of a square is not a
superficies, but a line_, and that a _square root (speaking of quantity)
is not a line, but a rectangle_, is a contradiction. The reader is to
judge of that.
To his scoffings I say no more, but that they may be retorted in the
same words, and are therefore childish.
And now I submit the whole to the Royal Society, with confidence that
they will never engage themselves in the maintenance of these
unintelligible doctrines of Dr. Wallis, that tend to the suppression of
the sciences which they endeavour to advance.
LETTERS
AND OTHER PIECES.
LETTERS AND OTHER PIECES.
==========
I.
A LETTER FROM MR. HOBBS TO MY MR.[2]
HONORABLE SIR,
Though I may goe whither and when I will for anie necessity you have of
my service, yet there is a necessity of good manners that obliges me as
yo^r servant to lett you knowe att all times where to find me. Wee goe
out of Paris 3 weekes hence, or sooner, towards Venice, but by what way
I knowe not, because the ordinary high way through the territory of
Milan is encumbered with the warre betweene the French and the
Spaniards. Howsoever, wee have to be there in October next. If you
require anie service that I can doe there, it may please you to convey
your command by Devonshire house. But if you command me nothing, I have
forbidden my letters to look for answer: their busines being only to
informe and to lett you knowe that the image of your noblenes decayes
not in my memory, but abides fresh to keepe me eternally
Your
THO. HOBBS.
-----
Footnote 2:
This letter is to be found in the British Museum, amongst the
Lansdowne MSS. 238, entitled “a collection of letters to and from
persons of eminence in the reigns of Elizabeth, James I, and Charles
I, made by some person in the service of Sir Gervas Clifton”. It is
without date: but the allusion to the war between France and Spain,
and the passage in the VITA THO. HOBBES, “Anno sequente qui erat
Christi 1629, rogatus a nobilissimo viro domino Gervasio Clifton”, &c.
(p. xiv), show that it must have been written in either 1629 or 1630.
-----
II.
TO A FRIEND IN ENGLAND.
WORTHY SIR,
I have been behind hand with you a long time for a letter I received of
yours at Angers, that place affording nothing wherewith to pay a debt of
that kind, all matter of news being sooner known in England than here:
and the news you writ me was of that kind, that none from England could
be more welcome, because it concerned the honour of Welbeck and Clifton,
two houses in which I am very much obliged.
Monsieur having given the slip to the Spaniards at Bruxelles, came to
the King about ten days ago at St. Germains, where he was received with
great joy. The next day the Cardinal entertained him at Ruelle: and the
day after that he went to Limours, where he is now, and from thence he
goes away shortly to Bloys, to stay there this winter. The Cardinal of
Lyons is going to Rome to treat about the annulling of Monsieur’s
marriage, which is here by Parliament declared void, but yet they
require the sentence of the Pope. There goes somebody thither on the
part of his wife, to get the marriage approved: but who that is, I ’know
not. The Swedish party in Germany is in low estate, but the French
prepare a great army for those parts, pretending to defend the places
which the Swedes have put into the King of France his protection,
whereof Philipsbourgh is one; a place of importance for the Lower
Palatinate. This is all the French news.
For your question, _why a man remembers less his own face, which he sees
often in a glass, than the face of a friend that he has not seen of a
great time_, my opinion in general is, that a man remembers best those
faces whereof he has had the greatest impressions, and that the
impressions are the greater for the oftener seeing them, and the longer
staying upon the sight of them. Now you know men look upon their own
faces but for short fits, but upon their friends’ faces long time
together, whilst they discourse or converse together; so that a man may
receive a greater impression from his friend’s face in a day, than from
his own in a year; and according to this impression, the image will be
fresher in his mind. Besides, the sight of one’s friend’s face two hours
together, is of greater force to imprint the image of it, than the same
quantity of time by intermissions. For the intermissions do easily
deface that which is but lightly imprinted. In general, I think that
lasteth longer in the memory which hath been stronglier received by the
sense.
This is my opinion of the question you propounded in your letter. Other
new truths I have none, at least they appear not new to me. Therefore if
this resolution of your first question seems probable, you may propound
another, wherein I will endeavour to satisfy you, as also in any thing
of any other nature you shall command me, to my utmost power; taking it
for an honour to be esteemed by you, as I am in effect,
Your humble and faithful servant
THO. HOBBES.
_Paris, Oct. 21/31, 1634._
My Lord Fielding and his Lady came to Paris on Saturday night last.
III.
TO MY WORTHY FRIEND MR. GLEN.[3]
WORTHY SIR,
I received here in Florence, two days since, a letter from you of the
19th of January. It was long by the way; but when it came it did
thoroughly recompence that delay. For it was worth all the pacquets I
had received a great while together. All that passeth in these parts is
equally news, and therefore no news; else I would labour to requite your
letter in that point, though in the handsome setting down of it, I
should still be your inferior.
I long infinitely to see those books of the Sabbaoth[4], and am of your
mind they will put such thoughts into the heads of vulgar people, as
will confer little to their good life. For when they see one of the ten
commandments to be _jus humanum_ merely, (as it must be if the Church
can alter it), they will hope also that the other nine may be so too.
For every man hitherto did believe that the ten commandments were the
moral, that is, the eternal law.
I desire also to see Selden’s _Mare Clausum_, having already a great
opinion of it.
You may perhaps, by some that go to Paris, send me those of the
Sabbaoth, for the other being in Latin, I doubt not to find it in the
Rue St. Jaques.
We are now come hither from Rome, and hope to be in Paris by the end of
June. I thank you for your letter, and desire you to believe that I can
never grow strange to one, the goodness of whose acquaintance I have
found by so much experience. But I have to write to so many, that I
write to you seldomer than I desire; which I pray pardon, and esteem me
Your most affectionate friend
and humble servant
THO. HOBBES.
_Florence, Apr. (6)/(16) 1636._
My Lord and Mr. Nicholls, and all our company commend them to you.
-----
Footnote 3:
Probably George Glen, who was installed Prebend of Worcester in 1660,
and died in 1669.
Footnote 4:
The History of the Sabbath. In two books. By Peter Heylyn. 4to. 1636.
-----
IV.
LETTER TO SIR CHARLES CAVENDISH.[5]
HONORABLE SIR,
[Illustration]
The last weeke I had the honor to receave two letters from you at once,
one of the 30 of Dec., the other of the 7^{th} of Jan., w^{ch} I
acknowledged, but could not answer in my last. In the first you begin
with a difficulty on the principle of Mons^r de Cartes, _that it is all
one to move a weight two spaces, or the double of that waight one
space_, and so on in other proportions: to w^{ch} you object the
difference of swiftnesse, w^{ch} is greater when a waight is moved two
spaces than when double waight is moved one space. Certenly de Cartes
his meaning was by force the same that mine, namely, a multiplication of
the weight of a body in to the swiftnesse wherew^{th} it is moved. So
that when I move a pound two foote at the rate of a mile an howre, I do
the same as if of 2 poundes I moved one pound a foote at y^e rate of a
mile an howre, the other pound another foote at the same rate, not in
directū, but parallell to the first pound. As if the wayt A B were moved
to C D at the rate of a mile an howre, ’tis all one as if the waight A E
were moved to F H at the same rate. Here is all the difference: this
swiftnesse or rate of a mile an howre is, in the first case, layd out in
the 2 spaces A G, G C, the latter, in the 2 spaces A G, E G. The first
case, as like as if a footman should run w^{th} double swiftnesse
endwayes, w^{ch} is y^e doubling of swiftnesse in one man: in the other,
it is as if you doubled the swiftnesse by doubling the man: for every
man has his owne swiftnesse. And so A H is the swiftnesse A G doubled,
as well as A D. For that, that Mons^r de Cartes will not have just twice
the force requisite to move the same weight twice as fast, I can say
nothing. The papers I have of his touching that are in my trunk, w^{ch}
hath bene taken by Dunkerkers, and taken againe from them by French, and
at length recovered by frends I made: but I shall not have it yet this
fortnight. In the meane time I am not in that opinion, but do assure
myself, the patient being the same, double force in the agent shall
worke upon it double effect.
In the same letter you require a better explication of y^e proportion I
gather betweene wayght and swiftnesse: wherein, because you have not my
figure, I imagine you have mistaken me very much. And first, you thinke,
I suppose, D E equall to A B: w^{ch} I am sure is a mistake. For I put A
B for any line you will to expresse a _minutū secundum_. I will,
therefore, go over againe the demonstration I sent you before, and see
if I can do it cleerer.
[Illustration]
Let A B stand for the time knowne wherein the waight D descendeth to E.
And let there bee a cylinder of the same matter the waight D consisteth
of, and let the altitude of that cylinder be D C: w^{ch} I shew before
was the swiftnesse wherew^{th} that cylinder _presseth_, not wherew^{th}
it _falleth_. And wee are now to enquire how farre such matter as the
cylinder is made of must descend from D, before it attayne a swiftnesse
equall to this pressing swiftnesse D C. And I say it must fall to L. For
in the time A B it is knowne that the waight in D will fall to E: and it
is demonstrated by Gallileo, that when such waight comes to E, it shall
be able to go twice the space it hath fallen in the same time. Therefore
the waight D being in E, hath velocity to carry him the space D K
(w^{ch} I put double to D E) in the same time A B. But I put B F equall
to D K. Therefore, in the time A B, the waight’s velocity acquired in E
shall be such as to go from B to F without decrease of velocity by the
way. Hence I go on to finde in what point the waight in D comes to where
it getteth a velocity equall to C D. Therefore, I apply D C to G H,
parallel to B F: and then it is, as the time A B to the time A G, so the
velocity acquired at the end of the time A B to the velocity acquired at
the end of the time A G. For the swiftnesse acquired from time to time
(I say, not from place to place, but from time to time) are
proportionable to the times wherein they are acquired: w^{ch} is the
postulate on w^{ch} Galileo builds all his doctrine. And as A B to A G,
so the line B F to the line G H. But, at the end of the time A B, the
waight D is by supposition in E, in that degree of velocity as to go B F
or D K in the same time A B. The question therefore is, where the waight
D shall be at the end of the time A G. For there it hath the velocity of
going G H or D C in the same time, because the velocity G H is to the
velocity B F as the line G H to the line B F, or as the time of descent
A G to the time A B. But, because the spaces of the descent are in
double the proportion of the times of descent, make it as B F to G H,
that is, D K to D C, so D C to another, D L. The velocity, therefore,
acquired in the point of descent E, namely the velocity D K or B F, is
to the velocity acquired in the point L, namely, the velocity G H or D
C, (w^{ch} is the velocity of the cylinder’s waight), as D K to D L. And
therefore in L the waight D has acquired a velocity equal to the
velocity of the waight of the cylinder.
In the same letter you desire to knowe, how any mediū, as water,
retardeth the motion of a stone that falls into it. To w^{ch} I answer
out of that you say afterwards, that nothing can hinder motion but
contrary motion: that the motion of the water, when a stone falls into
it, is point blanke contrary to the motion of the stone. For the stone
by descent causeth so much water to ascend as the bignesse of the stone
comes to. For imagine so much water taken out of the place w^{ch} the
stone occupies, and layd upon the superficies of the water: it presseth
downeward as the stone does, and maketh the water that is below to rise
upwards, and this rising upwards is contrary to the descent, and is no
other operation than we see in scales, when of two equal bullets in
magnitude that w^{ch} is of heavier metal maketh the other to rise. And
thus farre goes your letter of Dec. 30.
For the first quære in your second letter, concerning how we see in the
time the lucide body contracts itselfe, I have no other solution but
that w^{ch} your selfe hath given: w^{ch} is, that the reciprocation is
so quicke, that the effect of the first motion lasteth till the next
comes, and longer. For by experience we observe that the end of a
firebrand swiftely moved about in circle, maketh a circle of fire:
w^{ch} could not be, if the impression made at the beginning of the
circulation did not last till the end of it. For if the same firebrand
be moved slower, there will appear but a peece of a circle, bigger or
lesser according to the swiftnesse or slownesse of y^e motion. For the
cause of such reciprocation, it is hard to guess what it is. It may well
be the reaction of the medium. For though the mediū yeld, yet it
resisteth to: for there can be no passion w^{th}out reaction. And if a
man could make an hypothesis to salve that contraction of y^e sun, yet
such is the nature of naturall thinges, as a cause may be againe
demanded of such hypothesis: and never should one come to an end
w^{th}out assigning the immediate hand of God. Whereas in mathematicall
sciences wee come at last to a definition, w^{ch} is a beginning or
principle, made true by pact and consent amongst ourselves. Further, you
conceave a difficulty how the medium can be continually driven on, if
there be such an alternate contraction. To w^{ch}, first I answer, that
the motion forward is propagated to the utmost distance in an instant,
and the first push is therefore enough, and in another instant is made
the returne back in y^e like manner. And though it were not done in an
instant, yet we see by experience in rivers, as in y^e Thames, that the
tide goes upward towards London pushed by the water below, and yet at
the same instant the water below is going backe to the sea. For seeing
it is high-water at Blackwall before ’tis so at Greenwich, the water
goes backe from Blackwall when it goes on at Greenwich. And so it would
happen, though Blackwall and Greenwich were nerer together then that any
quantity given could come betweene.
[Illustration]
In my letter from London, speaking of the refraction of a bullet, I
thinke I delivered my opinion to be, that a bullet falling out of a
thinner medium into a thicker, looseth in the entring nothing but motion
perpendicular: but being entred, he looseth proportionably both of one
and the other. For suppose a bullet, whose diameter is A B, be in the
thiner mediū, and enter at C into the thicker medium. The thicker
medium, at the first touch of B in the point C, worketh nothing upon the
line A B. And when the diameter A B is entred, suppose halfe way, yet
the thicker medium operates laterally but on one halfe of it. So that in
the somme there is a losse of velocity perpendicular (to the quantity
that the diameter A B requires) without any offence to the motion
laterall, but so much of the diameter as is within the thicker mediū is
retarded both wayes, and looses of his absolute motion, w^{ch} is
compounded of perpendicular and laterall, and that proportionally.
Suppose now that a bullet passe from A to D, and receave a peculiar
losse of his perpendicular motion by entring at D, so great that he
proceed in the perpendicular but halfe so farre, as for example from D
to I: and then being in, the thicknesse of the medium take away more of
his velocity both perpendicular and laterall, suppose halfe that w^{ch}
was left of the perpendicular motion and halfe of his first laterall
motion, so that the perpendicular motion is but D K, and the laterall
motion D E. Then will the line of refraction be D G. As for that
argumentation of Des Cartes, it is, in my opinion, as I have heretofore
endeavored to shew you, a mere paralogism.
Lastly, you make this quære, why light hath not at severall inclinations
severall swiftnesses as well as a bullet. The bullet itself passeth
through the severall media: whereas in the motion of light, the body
moved, w^{ch} is the mediū, entreth not into the other medium, but
thrusteth it on: and so the parts of that medium thrust on one another,
whereby the laterall motion of the thicker medium hath nothing to worke
upon, because nothing enters, but stoppes onely and retardes, in oblique
_incidence_, that end w^{ch} comes first to it, and thereby causes a
refraction the contrary way to that of a bullet, in such manner as I set
forth to you in one of my letters from hence concerning the cause of
refraction. And this is all I can say for the present to the quæres of
y^r two last letters.
I have enquired concerning perspectives after the manner of De Cartes.
Mydorgius tells me there is none that goes about them, as a thing too
hard to do. And I believe it. For here is one Mons^r de Bosne in towne,
that dwells at Bloys, an excellent workman, but by profession a lawier,
and is counsellor of Bloys, and a better philosopher in my opinion then
De Cartes, and not inferior to him in the analytiques. I have his
acquaintance by Pere Mersenne. He tells me he hath tryed De Cartes his
way, but cannot do it: and now he workes upon a crooked line of his owne
invention. He sayes he shall have made one w^{th}in a moneth after he
shall returne to Bloys: after that he will see what he can discover in
the heavens himselfe, and then if he discover any new thing he will let
his way be publique together w^{th} the effects. This is all the hope I
can give you yet. So w^{th} my prayers to God to keepe you in prosperity
this troublesome time, I rest
Your most humble and obedient servant
TH. HOBBES.
_Paris, Feb. 8, stile no. 1641._
To the Right Honorable
Sr CHARLES CAVENDYSSHE
present these
/
/
/
/
at Wellinger.
-----
Footnote 5:
Harleian MS. 6796.
-----
V.
LETTER TO MR. BEALE.[6]
SIR,
The young woman at Over-Haddon hath been visited by divers persons of
this house. My Lord himself hunting the hare one day at the Town’s end,
with other gentlemen and some of his servants, went to see her on
purpose: and they all agree with the relation you say was made to
yourself. They further say on their own knowledge, that part of her
Belly touches her Back-bone. She began (as her Mother says) to loose her
appetite in December last, and had lost it quite in March following:
insomuch as that since that time she has not eaten nor drunk any thing
at all, but only wetts her lips with a feather dipt in water. They were
told also that her gutts (she alwayes keeps her bed) lye out by her at
her fundament shrunken. Some of the neighbouring ministers visit her
often: others that see her for curiosity give her mony, sixpence or a
shilling, which she refuseth, and her mother taketh. But it does not
appear they gain by it so much as to breed a suspition of a cheat. The
woman is manifestly sick, and ’tis thought she cannot last much longer.
Her talk (as the gentlewoman that went from this house told me) is most
heavenly. To know the certainty, there bee many things necessary which
cannot honestly be pryed into by a man. First, whether her gutts (as
’tis said) lye out. Secondly, whether any excrement pass that way, or
none at all. For if it pass, though in small quantity, yet it argues
food proportionable, which may, being little, bee given her secretly and
pass through the shrunken intestine, which may easily be kept clean.
Thirdly, whether no urine at all pass: for liquors also nourish as they
go. I think it were somewhat inhumane to examin these things too nearly,
when it so little concerneth the commonwealth: nor do I know of any law
that authoriseth a Justice of peace, or other subject, to restrain the
liberty of a sick person so farr as were needful for a discovery of this
nature. I cannot therefore deliver any judgment in the case. The
examining whether such a thing as this bee a miracle, belongs I think to
the Church. Besides, I myself in a sickness have been without all manner
of sustenance for more than six weeks together: which is enough to make
mee think that six months would not have made it a miracle. Nor do I
much wonder that a young woman of clear memory, hourely expecting death,
should bee more devout then at other times. ’Twas my own case. That
which I wonder at most, is how her piety without instruction should bee
so eloquent as ’tis reported.
THO. HOBBES.
_Chatsworth, Oct. 20. 68._
-----
Footnote 6:
Amongst the MSS. of the Royal Society.
-----
VI.
LETTER TO MR. OLDENBURG.[7]
WORTHY S^R
In the last Transactions for September and October I find a letter
addressed to you from D^r Wallis, in answer to my LUX MATHEMATICA. I
pray you tell me that are my old acquaintance, whether it be (his words
and characters supposed to be interpreted) intelligible. I know very
well you understand sense both in Latine, Greeke, and many other
languages. He shows you no ill consequence in any of my arguments.
Whereas I say there is no proportion of _infinite_ to _finite_. He
answers, he meant _indefinite_; but derives not his conclusion from any
other notion than simply _infinite_. I said the root of a square number
cannot be the length of the side of a square figure, because a root is
part of a square number, but length is no part of a square figure. To
which he answers nothing. In like manner, he shuffles off all my other
objections, though he know well enough that whatsoever he has written in
Geometry (except what he has taken from me and others) dependeth on the
truth of my objections. I perceive by many of his former writings that I
have reformed him somewhat as to the Principles of Geometry, though he
thanke me not. He shuffles and struggles in vaine, he has the hooke in
his guills, I will give him line enough: for (which I pray you tell him)
I will no more teach him by replying to any thing he shall hereafter
write, whatsoever they shall say that are confident of his Geometry.
_Qui volunt decipi, decipiuntur._ He tells you that I bring but _crambe
sæpe cocta_. For which I have a just excuse, and all men do the same;
they repeat the same words often when they talk with them that cannot
heare.
I desire also this reasonable favour from you: that, if hereafter I
shall send you any paper tending to the advancement of physiques or
mathematiques, and not too long, you will cause it to be printed by him
that is printer to the Society, as you have done often for D^r Wallis:
it will save me some charges.
I am, S^r,
Your affectionate frend and humble seruant
THOMAS HOBBES.
_November the 26th, 1672._
ffor my worthy and much honoured
frend M^r HENRY OLDENBURGH,
Secretary to the Royal Society.
-----
Footnote 7:
Amongst the MSS. of the Royal Society.
-----
VII.
TO THE RIGHT HONOURABLE
THE MARQUIS OF NEWCASTLE.[8]
The passions of man’s mind, except onely one, may bee observed all in
other living creatures. They have desires of all sorts, love, hatred,
feare, hope, anger, pitie, æmulation, and y^e like: onely of curiositie,
which is y^e desire to know y^e causes of thinges, I never saw signe in
any other living creature but in man. And where it is in man, I find
alwaies a defalcation or abatement for it of another passion, which in
beastes is commonly predominant, namely, a ravenous qualitie, which in
man is called _avarice_. The desire of knowledge and desire of needlesse
riches are incompatible, and destructive one of another. And therefore
as in the cognitive faculties reason, so in the motive curiositie, are
the markes that part y^e bounds of man’s nature from that of beastes.
Which makes mee, when I heare a man, upon the discovery of any new and
ingenious knowledge or invention, aske gravely, that is to say,
scornefully, _what ’tis good for_, meaning what monie it will bring in,
(when he knows as little, to one that hath sufficient what that overplus
of monie is good for), to esteeme that man not sufficiently removed bn
484.png from brutalitie. Which I thought fit to say by way of
anticipation to y^e grave scorners of philosophie, and that your
lordship, after having performed so noble and honourable acts for
defence of your countrie, may thinke it no dishonour in this unfortunate
leasure to have employed some thoughts in the speculation of the noblest
of the senses, _vision_.
That which I have written of it is grounded especially upon that w^{ch}
about 16 yeares since I affirmed to your Lo^{PP} at Welbeck, that light
is a fancy in the minde, caused by motion in the braine, which motion
againe is caused by the motion of y^e parts of such bodies as we call
_lucid_: such as are the sunne and y^e fixed stars, and such as here on
earth is fire. By putting you in mind hereof, I doe indeed call you to
witnesse of it: because, the same doctrine having since been published
by another, I might bee challenged for building on another man’s ground.
Yett philosophical ground I take to be of such a nature, that any man
may build upon it that will, especially if the owner himselfe will nott.
But upon this ground, with the helpe of some other speculations drawne
from the nature of motion and action, I have, I thinke, derived y^e
reason of all the phænomena I have mett with concerning light and
vision, both solidly enough nott to be confuted, and withall easilie
enough to be understood by such as can give that attention thereto which
the figures, whereby such motion as causeth vision is described, do
require. All that I shall bee ever able to adde to it, is polishing:
for, being the first draught, it could nott bee so perfect as I hope
hereafter to make it in Latin. Butt as it is, it will sufficiently give
your Lo^{PP} satisfaction in those _quæres_ you were pleased to make
concerning this subject. I am content that it passe, in respect of some
drosse that yett cleaves to it, for ore: w^{ch} is much better than old
ends raked out of the kennell of sophisters’ bookes. And for such I
commend it to your Lo^p, and myselfe to your accustomed good opinion:
which hath beene hitherto so greate honour to mee, as I am nott known to
the world by any thing so much as by being,
My most noble lord,
Your Lo^{p’s} most humble
and most obliged servant
THO. HOBBES.
The treatise ends with the following passage:--
To conclude, I shall doe like those that build a new house where an
old one stood before, that is to say, carry away the rubbish.
And first, away goes the old opinion that the _shewes_ (which they
call visible species) of all objects, are in all places, and all the
babble _de extramittendo et intromittendo_. For their species are
nothing else but fancie, made by the light proceeding directly or by
repurcussion or refraction made from the object to y^e eye, and so
moving the braine and other parts within.
Secondly, the opinion which Vitellio takes for an axiome and
foundation of his _Catoptricques_, that y^e place of y^e image by
reflexion is in the perpendicular drawn from the object to the glasse.
For it is false both in plaine glasses and in sphæricall, whether
convexe or concave.
Thirdly, the opinion that light is engendred faster in hard bodies, as
glasse, than in thin and fluid, as aire.
Fourthly, that objects are seen by _penicilli_ that have their common
base in the pupills: for y^e center of y^e eye is in their common
base.
Fifthly, the opinion that there bee other visuali lines by which wee
see distinctly besides y^e optique axis.
Sixthly, the opinion that perspective glasses and amplifying glasses
are best made of hyperbolicall figures.
Seventhly, the opinion that light is a bodie, or any other such thing
than such light as wee have in dreames.
Eighthly, that y^e object appeares greater and lesse in ye same
proportion that y^e angles have under which they are seene.
Lastly, is to be cast away the conceipt of millions of strings in ye
optique nerve, by which the object playes upon the braine, and makes
y^e soule listen unto it, and other innumerable such trash.
How doe I feare that y^e attentive reader will find that which I have
delivered concerning y^e _Optiques_ fitt to bee cast outt as rubbish
among the rest. If hee doe, hee will recede from y^e authoritie of
experience, which confirmeth all I have said. Butt if it bee found
true doctrine, (though yett it wanteth polishing), I shall deserve the
reputation of having beene y^e first to lay the grounds of two
sciences; this of _Optiques_, y^e most curious, and y^t other of
_Natural Justice_, which I have done in my booke DE CIVE, y^e most
profitable of all other.
-----
Footnote 8:
Harleian MS. 3360: a treatise on Optics, entitled “A minute or first
draught of the Optiques. In two parts. By Thomas Hobbes. At Paris,
1646.” The second part, _On Vision_, we have in Latin, in the DE
HOMINE: the first, _On Illumination_, was never published. The
dedication to the Marquis of Newcastle, and the concluding paragraph,
is all that is here given of the treatise.
-----
VIII.
TO THE KING’S MOST EXCELLENT MAJESTY
THE HUMBLE PETITION OF THOMAS HOBBES[9]
Sheweth, that though your Majesty hath been pleased to take off the
restraint of late years laid upon the pensions payable out of your privy
purse, yet your Majesty’s Officers refuse to pay the pension of your
petitioner without your Majesty’s express command.
And humbly beseaceth your Majesty, (considering his extreme age,
perpetual infirmity, frequent and long sickness, and the aptness of his
enemies to take any occasion to report that your petitioner by some ill
behaviour hath forfeited your wonted favour), that you would be pleased
to renew your order for the payment of it in such manner as to his great
comfort he hath for many years enjoyed it.[10]
And daily prayeth to God Almighty to bless your Majesty with long life,
constant health, and happinesse.
-----
Footnote 9:
Additional MSS. 4292. Brit. Mus.
Footnote 10:
Deinde redux mihi Rex concessit habere quotannis
Centum alias libras ipsius ex loculis:
Dulce mihi donum.
VITA _Carm. expres._ p. xcviii.
-----
END OF VOL. VII.
RICHARDS, PRINTER, 100, ST. MARTIN’S LANE.
------------------------------------------------------------------------
Transcriber’s Note
Some Greek passages employ the stigma ligature (‘st’). The available
Unicode character (ϛ) is nearly indistinguishable from the final form of
sigma (ς). Occasionally, the ‘ου’ ligature is employed. The only
available character (ᴕ) is a Latin, not a Greek character. It is
rendered here as ‘ου’. There were also a number of instances of improper
placement of diacritical marks, particularly in cases where the
breathing mark and accents appear on the first rather than the second of
two leading vowels, e.g. ‘ὅυτως’ rather than ‘οὕτως’.
At 93.4, the response ‘No sure’ is obviously incorrect. It is most
likely that it should have read ‘Not sure.’, but it may also have been
an unfinished line.
Other errors deemed most likely to be the printer’s have been corrected,
and are noted here. Given the age of the text, any corrections were made
sparingly. The references are to the page and line in the original.
8.35 with increasing [s]wiftness? Restored.
12.34 being so very heavy[?] Added.
54.8 too much or too[l i/ li]ttle Replaced.
62. was to be demonst[r]ated. Inserted.
65.33 which is 72[.] Added.
67.1 and multip[l]ying lines Inserted.
72.5 and subtil doc[t]rines. Inserted.
74.1 who begi[u/n]s his history Inverted.
91.16 the space be[t]ween. Inserted.
91.18 whose tap[-]hole is very little Inserted.
96.5 and th[a/e]n H will be east Replaced.
114.27 a spring[ ]upon the top Inserted.
158.19 the air which[ which] was Redundant.
163.11 8 degrees 30 minutes[.]? Removed.
192.19 (called by him [ε/ἐ]φαρμόζοντα) Replaced.
200.5 may so precisely deter[ter]mine Removed.
206.1 lines whi[e/c]h shall never meet Replaced.
208.10 [ὅυ/οὕ]τως ἔχει Replaced.
208.13 [ὅυ/οὕ]τως ἔχει Replaced.
225.8 as by raref[r]action and condensation. Removed.
263.17 [“]_The magnitude of an angle_ Added.
277.33 th[e/a]n of one in nine? Replaced.
297.26 At the seventee[n]th chapter Inserted.
300.22 _that of G K to G [E,/E.]_” Replaced.
323.7 _tanquam dicta problematicè._[”] Added.
338.31 and so mis[s]pend it? Inserted.
351.16 by which you live[,/.] Replaced.
376.4 _propositione præcedente._[”] Added.
376.22 it shall quite vanish. [And so] Missing?
342.12 to the present purpose[)]. Added (likely).
382.12 whether διπλάσιος and διπλασ[ι]ίων be one Removed.
391.5 by the word Σημ[~ε/εῖ]ον in Euclid Replaced.
412.17 κα[ι/ὶ] ἡ Ἔμπουσα Replaced.
413.26 τῶν θεῶν [Α/Ἀ]ιγύπτιοι Replaced.
413.27 κα[ἱ/ὶ] ἅτε Ἀιγύπτιος, Replaced.
413.30 ποδῶν [ὴ/ἠ]δέ κνημάων Replaced.
414.1 ῥᾳδίως[ ]ἔγνων ὑπολαμβάνοντες. Inserted.
414.4 ἀλλὰ κατὰ ῥύμην [ὰ/ἀ]έριον Replaced.
415.23 deriveth it [τ/π]αρὰ τὸ ἐμποδιζειν Replaced.
415.25 it saith, [Ἕ/Ἔ]μπουσα Ψιλοῦπαι Replaced.
415.33 some had no legs, &c[.] Added.
416.7 τύχη [ὲ/ἐ]μποδιζουσα Replaced.
416.16 [Ἕ/Ἔ]μποῦσα is also a name for Lamia Replaced.
418.16 [ὁι/οἱ] δὲ ὑπέφευγον Transposed.
418.20 καὶ τὸ [ὲ/ἐ]πιπηδᾶν Replaced.
419.13 γαρ [ὅι/οἵ]νου πληροῦντες, Replaced.
419.14.1 εἶχε τὸν [ὅι/οἵ]νου. Replaced.
419.14.2 [Α/Ἀ]σκωλιάζειν Replaced.
423.1 καὶ ζώων παντοδαπῶν [ἐι/εἰ]κόσι Replaced.
455.27 two spaces th[e/a]n when double Replaced.
456.3 a pound two foote a[t] the rate of Added.
456.6 rate of a mile an how[er/re] Transposed.
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