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Title: First lessons in Geometry
Author: Thomas Hill
Release date: November 8, 2026 [eBook #79951]
Language: English
Original publication: Boston: Hickeling, Swan, and Brown, 1855
Other information and formats: www.gutenberg.org/ebooks/79951
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*** START OF THE PROJECT GUTENBERG EBOOK FIRST LESSONS IN GEOMETRY ***
Transcriber’s Notes:
Underscores “_” before and after a word or phrase indicate _italics_
in the original text.
Small capitals have been converted to SOLID capitals.
Illustrations have been moved so they do not break up paragraphs.
Deprecated spellings have been preserved.
Typographical and punctuation errors have been silently corrected.
FIRST LESSONS IN GEOMETRY.
BY THOMAS HILL.
_FACTS BEFORE REASONING._
BOSTON:
HICKLING, SWAN, AND BROWN.
1855.
Entered according to Act of Congress,
in the year 1855, by
THOMAS HILL,
In the Clerk’s Office of the District Court
of the District of Massachusetts.
Stereotyped by
HOBART & ROBBINS,
NEW ENGLAND TYPE AND STEREOTYPE FOUNDRY,
BOSTON.
PREFACE.
I have long been seeking a Geometry for beginners, suited to my taste,
and to my convictions of what is a proper foundation for scientific
education. Finding that Mr. JOSIAH HOLBROOK agreed most cordially
with me, in my estimate of this study, I had hoped that his treatise
would satisfy me; but, although the best I had seen, it did not meet
my views. Meanwhile, my own children were in most urgent need of a
text-book, and the sense of their want has driven me to take the time
necessary for writing these pages. Two children, one of five, the other
of seven and a half, were before my mind’s eye all the time of my
writing; and _it will be found that children of this age are quicker
at comprehending first lessons in Geometry than those of fifteen_.
Many parts of this book will, however, be found adapted, not only to
children, but to pupils of adult age. The truths are sublime. I have
tried to present them in a simple and attractive dress.
I have addressed the child’s imagination, rather than his reason,
because I wished to teach him to conceive of forms. The child’s powers
of sensation are developed before his powers of conception, and these
before his reasoning powers. This is, therefore, the true order of
education; and a powerful logical drill, like Colburn’s admirable first
lessons of Arithmetic, is sadly out of place in the hands of a child
whose powers of observation and conception have, as yet, received no
training whatever. I have, therefore, avoided reasoning, and simply
given interesting geometrical facts, fitted, I hope, to arouse a child
to the observation of phenomena, and to the perception of forms as real
entities.
In the pronunciation of words at the foot of the page the notation of
Dr. Worcester’s Dictionaries has been followed.
WALTHAM, MASS., _Nov. 1854_.
CONTENTS.
CHAPTER. PAGE
I. WHAT IS THIS BOOK ABOUT? 7
II. POINTS, LINES, AND PLANES, 9
III. ABOUT STRAIGHT LINES AND CURVES, 11
IV. ABOUT ANGLES, 13
V. PARALLEL LINES, 16
VI. ABOUT TRIANGLES, 19
VII. MORE ABOUT TRIANGLES, 21
VIII. RIGHT TRIANGLES, 24
IX. SIMILARITY AND ISOPERIMETRY, 26
X. THE SIZE OF TRIANGLES, 30
XI. DIFFERENT KINDS OF TRIANGLES, 33
XII. QUADRANGLES, 36
XIII. PARALLELOGRAMS, 40
XIV. RECTANGLES AND SQUARES, 43
XV. TRIANGLES AND RECTANGLES, 46
XVI. CIRCLES, 52
XVII. MORE ABOUT CIRCLES, 55
XVIII. MEASURING ANGLES, 59
XIX. CHORDS, 62
XX. CHORDS AND TANGENTS, 66
XXI. MORE ABOUT CHORDS AND TANGENTS, 69
XXII. INSCRIBED POLYGONS, 73
XXIII. MORE ABOUT INSCRIBED POLYGONS, 77
XXIV. HOW MUCH FURTHER ROUND A HOOP THAN
ACROSS IT? 81
XXV. HOW TO MEASURE CIRCLES AND THEIR PARTS, 85
XXVI. ABOUT CURVATURE, 89
XXVII. ABOUT A WHEEL ROLLING, 92
XXVIII. MORE ABOUT THE ROLLING WHEEL, 95
XXIX. WHEELS ROLLING ROUND A WHEEL, 98
XXX. WHEEL ROLLING IN A HOOP, 101
XXXI. HANGING CHAIN, 105
XXXII. PATH OF A STONE IN THE AIR, 109
XXXIII. THE SHADOW OF A BALL, 112
XXXIV. THE SHADOW OF A REEL, 115
XXXV. THE COW’S FOOT IN A CUP OF MILK, 120
XXXVI. CUBES, 125
XXXVII. ABOUT CONES, 129
XXXVIII. THE SPHERE, 132
CHAPTER I.
WHAT THIS BOOK IS ABOUT.
1. I have written a little book for you about Geometry. You will find
a great many new words in it; but I have taken pains to explain them
all, and I think you will understand them all, if you will only begin
at the beginning, and read each chapter very carefully before you go
to another. And if you find any place in the book that you cannot
understand, I think you will do well to turn back, and read the whole
over again, from the second chapter. When you come again to the place
which you did not understand before, I think you will find it has grown
easier for you.
2. I hope you will find the book interesting. It tells about straight
lines, and circles, and many different curves, and a few solid bodies.
It will tell you curious things about the shadows of marbles, and
the rolling of hoops, and about tossing a ball, and other plays for
children. But, if some parts of the book do not seem interesting, you
should study those parts all the more carefully; for they may, perhaps,
be the most useful parts.
3. Geometry is the most useful of all the sciences. To understand
Geometry, will be a great help in learning all the other sciences; and
no other science can be learned unless you know something of Geometry.
To study it, will make your eye quicker in seeing things, and your hand
steadier in doing things. You can draw better, write better; cut out
clothes, make boots and shoes, work at any mechanical trade, or learn
any art, the better for understanding Geometry. And, if you want to
understand about plants and animals, and the wonderful way in which the
All-wise Creator has made them, you must learn a little Geometry, for
that explains the shapes of all things.
4. In this book I can teach you but little. I hope, however, that it
will be enough to make you want to know more. I shall tell you only the
easiest and most interesting things now; but when you are older you may
study what is more difficult. Many of the things that I shall tell you
will be very curious, and you will, perhaps, wonder how men can find
out such things. But when you are older I hope that you will be able to
find out such things yourselves.
CHAPTER II.
POINTS, LINES, AND PLANES.
1. A point is a place without any size. When I make a dot to mark the
place of a point, that is, to show where the point is, you must not
think that the point is so big as the dot. The point has no size at
all, but is only a place without any size. If I put my dot in the right
place, the point will be exactly in the middle of the dot. In common
talking, we sometimes call anything that is very small a point; and so
we talk of the point of a needle, or of the point of a lead-pencil. But
in Geometry a point is a place so small that it has no size at all;
neither width, nor length, nor depth.
[Illustration]
2. A line is a place that is long without having any breadth or
thickness. When I make a long, fine stroke with a pen or pencil, or
with a piece of chalk, you must not think that the stroke of pencil, or
ink, or chalk-mark, is itself the line. I only make it to show where
the line is, or to help you imagine a line; but the line itself is the
middle of the stroke; you cannot see it any more than you can see a
point, for it is only a place; and, although it has length, it has no
breadth nor thickness.
* * * * *
How large is a point? How can you mark the position of a point?
In what part of the dot is the point supposed to be? What is the
difference between the word point in common talk and in Geometry?
How wide is a line? How shall we mark a line? In what part of the
stroke should the line be? What is the end of a line? How can you
fancy this so as to make it like a needle-point? If a point could
move and leave a track behind it, what would that track be? How
may all that is described in the first thirty-five chapters of
the book be drawn?
3. The ends of a line are points. You may fancy a line to be the middle
of a very fine wire, and then you will easily see that the ends of it
are points.
[Illustration]
4. When the point of my pen or pencil moves along on paper, a fine
stroke of ink or of pencil-mark is left behind it. And that may help
you fancy a real point moving along and leaving a real line behind it.
It will, you know, be only fancy; because a real point is only a place,
and a place cannot move. But it is a good way to fancy a line as marked
out by the track of a moving point; that is, by the very centre of the
end of a pencil. It will help you very much in understanding Geometry,
if you fancy a line as the track of a moving point.
5. A plane is a flat surface, like the floor, or the top of the table,
or like your slate. I need not tell you any more exactly what a surface
is, and what a flat surface means; because I am going to be confined
to one plane for a long while. I mean that for a good many chapters I
shall tell you only about such lines as can be drawn upon your slate,
or upon the blackboard.
6. Now you have studied enough for one lesson. If you understand this
well, you have made a very good beginning in Geometry.
CHAPTER III.
ABOUT STRAIGHT LINES AND CURVES.
1. A line that is not bent in any part of it is called a straight line.
If we fancy a point moving in a straight line, we shall see it moving
always in the same direction. A straight line is the shortest path that
can be made from one point to another. So, when we wish to tell the
distance from one place to another, we measure how long the straight
line is that joins the two places. If a thread is stretched tight
across a table, it marks a straight line across the table. This is the
way that carpenters mark a straight line, by rubbing the thread first
with chalk; and gardeners lay out garden-paths and beds by stretching a
line.
* * * * *
What is a straight line? What is the direction in which a point
moves when moving in a straight line? What is the shortest path
from one place to another? How does a carpenter mark a straight
line? How does a gardener make his paths straight? What is a
curve line? How does a point move in a curve? In how many places
can one straight line cross another? Can a straight line always
cut a curve in more than one place? In how many places can a
straight line always cut a curve? Let the teacher draw a curve
upon the blackboard, cross it by a straight line, and then,
moving the chalk along the curve, require the scholars to say
“now,” whenever the chalk is moving in the same direction as the
straight line.
What is an angle? On what does the size of an angle depend? Let
the teacher draw angles on the blackboard, and ask which is the
larger, which the smaller, etc., being careful to make some of
the angles without vertices. What is the vertex of an angle?
Let the teacher call the scholar to the board to point out the
vertices of the angles he has drawn. When two straight lines
cross each other, what is always true about the angles they make?
What is a right angle? What examples can you give of a right
angle? What is the common name for the vertex of a right angle?
The teacher must be very careful not to let the child confound
the measure of an angle with either the length of the sides, or
area of the opening between them; but illustrate and explain it
only by differences of directions; such as points of compass,
arrows, turning your face about, etc. When two square corners are
put together, how do the outsides run?
Look about the room, or out of the window, and tell me what
straight lines you can see. Do you see any curve lines? Any lines
that make angles with each other? Can you show me any parallel
lines? Any concentric curves? What are parallel lines? What
can you say about the distance apart of parallel lines? When
a straight line crosses two parallel lines, what can you say
about the angles? (Let the teacher beware of forcing a child to
repeat the reasoning of sections two and five. To the teacher the
reasoning is easier than the conceptions; to the child it is just
the reverse.) How can you tell whether two lines are parallel?
[Illustration]
[Illustration]
2. A curve line bends in every part, but has no sharp corners in it.
And, if we fancy a point moving in a curve, we shall see it all the
time changing its direction, but never taking sudden turns. Perhaps it
will help you understand the difference between a straight line and a
curve, if I draw two lines at the side of the page. I think you will
understand, from what I have said, which is the straight line, and
which is the curve.
[Illustration]
[Illustration]
3. Now I want you to see that two straight lines can never cut across
each other in more than one place. If you draw only two lines on your
slate, and each line is straight, they cannot cross each other in two
places. But you cannot draw a curved line that you cannot cut, at
least, in two places. Try to draw, on your slate, a curve that cannot
be cut in two places by one straight line.
[Illustration]
4. There is one thing more that I want you to learn at this lesson.
Whenever a straight line joins two points of a curve, there is always
some point on the curve between the two points at which the curve goes
in the same direction as the straight line. So in my figure you see
that the curve at C goes in the same direction as the straight line
A B. Draw on your slate figures of curves, and cut them by straight
lines, and you will find it always so. This seems like a very simple
thing, and yet it is a very useful truth.
CHAPTER IV.
ABOUT ANGLES.
1. When two straight lines go in different directions, the difference
of their directions is called an angle. The size of the angle depends,
then, on the difference of the directions of the lines, and not on
their length. So that the angle which I have marked B, is larger than
that which I have marked A, because there is more difference in the
direction of the two lines at B, than of the two lines at A.
[Illustration]
2. The point where two straight lines meet, or where they would meet if
we fancied them drawn long enough, is called the vertex of the angle.
The vertex of the angles A and B is not marked down; but you may draw
two straight lines meeting, and the point where they meet will be the
vertex of the angle between them.
[Illustration]
3. When two straight lines cross each other, they make four angles. So,
in the figure in the margin, we have two straight lines making the four
angles, A E C, A E D, B E C, B E D. But when we say they make the angle
A E C, we have to fancy the straight line D C going in exactly the
opposite direction to that in which it goes to make the angle A E D.
4. If we fancy that the line A B, in the next figure, points from A to
B, while the line C D points from C to D, the two lines will make the
angle E. But if the line A B points from B to A, they will make the
angle F. When these two angles are equal, each is called a right angle.
[Illustration]
[Illustration]
5. When two straight lines cross each other, the opposite angles are
of the same size. I mean that in this figure the angle A E C is just
as large as the angle D E B, and the angle A E D just as large as the
angle C E B.
[Illustration]
6. When two straight lines crossing each other, make four equal angles,
each angle is called a right angle. Draw two lines on your slate at
right angles to each other. The side of your slate is at right angles
to the bottom. The top of a sheet of letter-paper makes a right angle
with the side. The sides of every square corner are at right angles
to each other. The vertex of a right angle is called a square corner.
When two square corners are put together, the outside edges will form a
straight line.
CHAPTER V.
PARALLEL LINES.
1. When two straight lines make no angle with each other, or when they
make an angle equal to two right angles with each other, they are
called parallel. That is to say, parallel lines are straight lines that
point in the same direction, or in exactly opposite directions. Try
whether you can draw such upon your slate.
[Illustration]
2. When two straight lines are parallel, they are just as far apart in
one place as in another. They could not come nearer and then go further
apart without bending; but a straight line does not bend in any part.
And, if they kept coming nearer until they met, they would make an
angle with each other, and the point where they met would be the vertex
of the angle. But parallel lines make no angle with each other.
3. If two straight lines are just as far apart in one place as in
another, they are parallel; they run in the same direction. Try whether
the sides of your slate are parallel, by measuring whether they are
just as far apart at the top of the slate as at the bottom.
[Illustration]
4. When two curves are everywhere at the same distance apart, they are
called concentric curves. Sometimes they are called parallel curves;
but this is not so good a name for them as concentric curves.
[Illustration]
5. When a straight line crosses two parallel lines, it makes the same
angles with the one as with the other. The direction of parallel lines
is alike, and so the difference of their directions from that of the
straight line must be alike.
6. If a straight line is parallel to one of two parallel lines, it
is parallel to the other. Draw now two parallel lines on your slate.
Draw a third line parallel to one of your first pair, and it will be
parallel to the other. All three of the lines will point in the same
direction.
CHAPTER VI.
A LITTLE ABOUT TRIANGLES.
[Illustration]
1. A triangle is a figure bounded by three straight lines. They are
very simple-looking things; and yet there are many curious things known
about them already. And those that know most about Geometry tell us
that no one has yet found out all that can be known about them.
[Illustration]
[Illustration]
2. The three angles of a triangle taken together will make two right
angles. You can try it, if you like, by cutting a triangle out of
paper, with a pair of scissors. Be very careful to make the edges
straight. Now cut off two of the corners by a waving line, and lay
the three corners of the triangle carefully together. The outer edges
will make one straight line, just as if you had put two square corners
together. You may make the triangle of any shape or size, and, if the
edges are straight, you will make two right angles.
* * * * *
Draw triangles, and ask “What is the name of these figures?” What
is a triangle? Draw a right triangle, and, pointing to the square
corner, ask, “What angle is this?” How many square corners can a
triangle ever have? How much do the three angles of a triangle
put together make? If one angle is a right angle, how much do the
other two put together make. If one side of a triangle is longer
than another, what do you know about the angles? If one angle is
larger than another, what do you know about the sides?
When we know that two sides of a triangle are equal, what do
we know of the angles? When we know that all the sides of a
triangle are of the same size, what do we know about the angles?
When we know that two angles are equal, what do we know about
the sides? When we know that the three angles are equal? What
is an equiangular triangle? What is an equilateral triangle?
Every equiangular triangle is also ______? And every equilateral
triangle is also ______? Let the teacher now copy the figure of
section four upon the blackboard, and ask what lines are in the
same proportion as C D and C A? What is the same as C D to C
E? What is the same as A D to D C? Let the teacher, also, draw
parallel lines at equal distances apart, like a staff of music,
and then, stretching a string across them, show the scholars that
the lines divide the string equally in whatever direction it is
held. Always find that the three corners put together make a
straight line with their outer edges, just as two square corners
would do. And this is what we mean by saying that the three
angles of a triangle taken together, will make two right angles.
3. A triangle cannot have more than one angle as large as a right angle.
4. If one angle in a triangle is a right angle, the other two, put
together, will, of course, just be equal to a right angle. You can
try this by cutting paper triangles with one square corner, and then
cutting off the other corners by a waving line, and putting them
together.
[Illustration]
5. If one side of a triangle is longer than another side, the angle
opposite the longer side is larger than that opposite the shorter side.
Now look at the figure. The side _a_ is opposite the angle A, and the
side _b_ opposite the angle B. The side _b_ is longer than the side
_a_; and from this we may know that the angle B is larger than the
angle A.
[Illustration]
6. Now, on the other hand, when one angle in a triangle is larger than
another, the side opposite the larger angle is longer than the side
opposite the smaller angle. So, if we know that B is larger than C, we
may know that _b_ is larger than _c_.
CHAPTER VII.
MORE ABOUT TRIANGLES.
1. Suppose that we found two sides of a triangle to be just equal to
each other, what should we know about the angles? We should know that
the angle opposite one side was just as large as the angle opposite the
other side. If the side _a_ is just as long as the side _c_, the angle
A is just as large as the angle C.
[Illustration]
2. But if, on the other hand, we know that the two angles are equal,
we shall know from that that the two sides are equal. If the angle A
is just as large as the angle C, then the side _a_ must be just as
long as the side _c_. Such a triangle is called isosceles, which means
equal-legged.
[Illustration]
3. Now if the three sides of a triangle are each equal to each other,
then the angles are equal to each other; and if, on the other hand, the
three angles are equal to each other, then the sides are equal to each
other. Such a triangle is called equiangular, or equilateral. Perhaps
this drawing will help you to imagine an equilateral triangle.
[Illustration]
4. If a line be drawn through a triangle parallel to one side of the
triangle, it divides the other two sides in the same proportion. I mean
that, if in such a triangle as A B C we draw D E parallel to A B, then
C D will be the same part of C A that C E is of C B. If C D is two
thirds of C A, then C E will be two thirds of C B.
5. Besides what I have already told you about the last figure,—that
is, about any triangle that is cut in two by a line parallel to one
side,—there are two other curious things for you to learn. In the first
place, A D will be in the same proportion to D C that B E is to E C;
and in the second place, C D will be in the same proportion to C E as A
C is to B C, and A D will also be in the same proportion to E B. If B C
is three quarters of A C, then C E will be three quarters of C D, and E
B will be three quarters of A D.
[Illustration]
6. If we divide one side of a triangle into equal parts, and then draw
lines through the points where we have divided the side, making these
lines parallel to another side of the triangle, the third side will be
divided into equal parts. Thus, if the side A B is divided into equal
parts, the side B C is also thus divided.
7. I am afraid this lesson will be difficult to understand; but in the
next I will try to tell you something that will be easier.
CHAPTER VIII.
RIGHT TRIANGLES.
[Illustration]
1. If you do not remember what a right angle is, you must turn back and
read chapter IV., section 6, and then you will be ready to go on with
this chapter.
When a triangle has a right angle for one of its angles, it is called a
right triangle. The angle A B C is a right angle, and the triangle A B
C a right triangle.
2. Any triangle may be divided into two right triangles, by drawing a
line through the vertex of the largest angle in such a way as to make
right angles with the longest side. Thus, if B is the largest angle in
the triangle A B C, we can draw B D in such a way as to make the angles
at D right angles; and this will divide the triangle into two right
triangles, A D B and C D B.
Now draw any triangles you please, upon your slate, and try whether you
cannot always divide them, in this way, into two right triangles.
* * * * *
What is a right angle? What is the common name for the vertex
of a right angle? What is a right triangle? How can you divide
any triangle into two right triangles? The teacher may call the
class to the blackboard, and allow them to draw triangles and
divide them in this way. To what kind of triangle can you add a
right triangle so as to make the whole figure a right triangle?
Let the pupils show this by drawing them on the blackboard. What
kind of triangle can be fancied as the difference between two
right triangles? If we divide a right triangle into two right
triangles, what do you know about them?
What is the meaning of “similar” figures? What is true of the
angles of similar figures? Let the teacher draw two similar
quadrilaterals on the board, make a little circle, star, cross
and accent, in the four figures of one, and bid a member of the
class come and put corresponding marks in the angles of the other
figure, thus: “Who will mark in that figure the angle which is
equal to this that I marked with a cross? Who will mark the one
equal to this one marked with a circle?” etc., etc. Then let the
teacher make a cross through two sides of one quadrilateral, and
say, “Who will mark the two sides in the other figure that are in
the same proportion as these?” Next let him draw a figure of a
liberty-pole, and a stake, and ask the child to explain in what
way he can measure the height of the liberty-pole by means of the
stake. When are two figures called isoperimetrical?
[Illustration]
3. If the largest angle in a triangle is larger than a right angle, we
can always fancy a right triangle added to it in such a way as to make
the whole figure a right triangle. If B be larger than a right angle,
we can make A B longer, and draw C D down in such a way as to make D
a right angle. Then the added triangle B C D is a right triangle, and
the whole figure A D C is also a right triangle. And, by taking away
B D C from A D C, we shall have A B C left. So that any triangle with
one angle larger than a right angle, like A B C, may be fancied as the
difference between two right triangles, like A D C and B D C.
[Illustration]
4. If we divide a _right_ triangle into two right triangles, as I have
told you how to do in the second section of this chapter, the two
little triangles will be of exactly the same shape as the whole large
triangle.
Thus, if B is a right angle, and the angles at D are both right angles,
then the three triangles, A B C, A B D and B D C, are all of exactly
the same shape.
CHAPTER IX.
SIMILARITY AND ISOPERIMETRY.
1. There are two very hard-looking words at the head of this chapter;
but they are not hard to understand. Similarity means likeness; and in
Geometry it means the having the same shape. Isoperimetry[1] means the
being of the same size round about.
[1] Īsọpẹrĭm’ẹtrỵ.
2. When two bodies, or two geometrical figures, are of exactly the same
shape, we call them similar. When two figures are similar, that is, of
exactly the same shape, the angles of one figure are exactly equal to
the angles of the other figure, and all the sides of the first figure
are in the same proportion to the corresponding sides of the other
figure. I told you that A D B and C D B are of the same shape. So that
the angle at A is just as large as the angle D B C, the angle at C just
as large as D B A, and the two angles at D are equal. And C B is the
same part of A B that C D is of D B, or that D B is of A D. All this is
meant by saying that the triangle C D B is of the same shape as A D B.
[Illustration]
[Illustration]
3. If we find that the three angles of one triangle are just equal to
the three angles of another triangle, we may know that the three sides
of the first triangle are in the same proportion to the corresponding
sides of the second triangle, and that the triangles are similar.
That is to say, that if the three angles of one triangle are equal to
the three angles of another triangle, the sides opposite to the equal
angles are in the same proportion to each other.
4. It is also true that when two figures are similar, any two sides of
the one are in the same proportion to each other that the corresponding
sides of the other are in to each other. Thus, C D is the same part of
C B that B D is of B A. And D B is the same part of D A that D C is of
D B.
[Illustration]
[Illustration]
5. All the nice calculations of engineers, and machinists, and
ship-builders, and navigators, and astronomers, are made by help of
similarity of triangles.
I will try to explain to you one single instance in which you can use
similarity of triangles. Suppose that a house stands on level ground,
and you wish to find out how high it is. Put a stake upright in the
ground, anywhere that you think best, say at B. Then lay your head
close to the ground, and move it until you can just see the top of the
house over the top of the stake. Then measure how far your eye is from
the bottom of the stake, and how far from the bottom of the house.
Then, as you will see by the figure, you will have two triangles of
the same shape, and the height of the house will be the same part of
the distance A C, that the height of the stake is of the distance A B.
If the height of the stake is equal to the distance of your eye from
the bottom of it, then the height of the house is just equal to the
distance of your eye from the foundation of the house. Another way of
finding similar triangles to measure a house, or tree, on level land,
is by using shadows. The shadow of the stake B, when the sun shone,
would be a triangle. The stake would be one side, the shadow on the
ground another, and the third side would be the edge of the shadow in
the air, going from the top of the stake to the end of the shadow on
the ground. A similar triangle would, at the same time, be made by the
shadow of the tree or house. So that, if you should measure at any time
the length of the shadow of the stake, and the length of the shadow of
the house, you could tell the height of the house; because the height
of the house would be the same part of the length of its shadow that
the stake was of its shadow.
6. When it is just as far round one figure as it is round another, the
two figures are called isoperimetrical[2], which means equal round
about.
[Illustration]
If one triangle measures, on its sides, two feet, and five feet, and
six feet, its perimeter[3], that is, the distance round it, will be
thirteen feet; because two and five and six make thirteen. And if
another triangle measures on its sides, three feet, and six feet,
and four feet, its perimeter will also be thirteen feet; because
three and six and four make thirteen. So, these two triangles will be
isoperimetrical; but they will not be similar, that is, they will be of
different shapes.
[2] Īsọpĕrịmĕt’rịcạl.
[3] Pẹrĭm’ẹtẹr.
7. Similar figures are those of the same shape; isoperimetrical figures
are those which measure equally round about.
CHAPTER X.
THE SIZE OF TRIANGLES.
1. If one side of a triangle can grow longer or shorter, while the
opposite angle opens and shuts, as though its vertex were a hinge, the
triangle will be largest when that angle is a right angle. Thus, if A B
and B C cannot be changed in length, but if A C is like an India-rubber
cord, and can be made longer or shorter by altering the angle at B,
then the triangle A B C will be largest when the angle at B is a right
angle.
[Illustration]
[Illustration]
2. You can show this very prettily in this way. Take two little
straight sticks and hold them together at one end with your thumb and
finger, while you spread the other two ends against the edge of the
table. The triangle made by the two sticks and the table-edge will be
largest when the sticks make a right angle with each other.
3. If one side of a triangle cannot change in length, and if the
other two can only change in such a way as to keep the triangle
isoperimetrical, the triangle will be largest when these two sides are
of equal length. Thus, if B C cannot grow either longer or shorter,
and if A B grows shorter exactly as fast as A C grows longer, or grows
longer exactly as fast as A C grows shorter, the triangle will be
largest when A B is equal to A C.
* * * * *
If two sticks lean their tops together, how must they be placed
to make the space between them and the ground largest? If two
boards be nailed together to make a pig-trough, what angle must
they make so as to have the trough hold most? If a tent is made
simply of two slant sides like the roof a house, how must we
pitch it so as to have most room in it?
Let the teacher provide sticks and strings and make triangles, as
directed in sections two, four and six, and ask, “When will this
triangle be largest?”
Set the class at the blackboard, and ask them to draw an
equilateral triangle, and tell what they know about its angles
and sides. An isosceles triangle. A scalene triangle. An
isosceles oblique triangle. Isosceles right triangle. Scalene
right triangle. Scalene oblique triangle. Ask them, also, to
point out all the triangles in objects within sight, and name
them according to this chapter.
[Illustration]
[Illustration]
4. You may show this in a very pretty way by tying the ends of a string
to the ends of a straight stick a good deal shorter than the string.
Then take the stick in one hand, and, putting one finger of the other
inside the string, pull it tight. The triangle formed between the
string and the stick will be largest when your finger is in the middle
of the string. But when you move your finger the triangle remains
isoperimetrical, if the string does not stretch.
5. If we suppose that all three sides of a triangle can change their
length, but only in such a way as to keep the triangle isoperimetrical,
then the triangle will be largest when the three sides are equilateral.
That is to say, that an equilateral triangle is the largest among
isoperimetrical triangles.
6. You can show this by taking a long piece of string and tying the
ends together. Put into this loop one finger of your right hand, and
get two playmates each to do the same. Stepping back, you can pull the
string into a large triangle, and you will see that it is largest when
your fingers are at equal distances, and the triangle equilateral. But
all the triangles made by moving the fingers will be isoperimetrical,
because the string remains of one length. When you grow older you can
prove these things; but now you can best show them to yourself in such
ways as this.
CHAPTER XI.
DIFFERENT KINDS OF TRIANGLES.
1. When a triangle has its three sides equal, it is called an
equilateral triangle. Equilateral means equal-sided. Every equilateral
triangle has its angles equal to each other; and is, therefore, called
equiangular. But equilateral is the more common name.
2. When a triangle has two of its sides of the same length it is called
an isosceles[4] triangle. Isosceles means equal-legged. The angles
opposite the equal legs are, as I hope you remember, equal to each
other.
[4] Īsŏs’cẹlēs.
An equilateral triangle must be isosceles; but an isosceles triangle
may not be equilateral. Every horse is an animal; but not every animal
is a horse.
3. Suppose that A B C is an isosceles triangle, and that _a_ is equal
to _c_. Now, if we draw a line B _b_ from B to the middle of A C, it
will divide the triangle into two equal right triangles. The angle at B
is divided exactly in the middle, the angles at _b_ are right angles,
and we could fold the triangle over on the line B _b_ as a hinge, and
the two pieces would fit each other exactly.
[Illustration]
[Illustration]
4. If a triangle has no two sides equal, it is called a scalene
triangle. Scalene means lame, or limping.
5. When a triangle has one right angle, it is, as I hope you remember,
called a right triangle. We can have isosceles right triangles, and
scalene right triangles; but, of course, we cannot have equilateral
right triangles.
6. When neither of the angles of a triangle is a right angle, the
triangle is called an oblique[5] triangle. Oblique triangles may be
equilateral, or isosceles, or scalene.
[5] Oblēk’.
* * * * *
REVIEW OF TRIANGLES.—Let the scholars now turn back to chapter
VI., and take six chapters as a review-lesson, and answer the
following questions, and others selected from the questions on
each chapter.
What is a triangle? If the three corners of a paper triangle are
placed together, what will their sum be? How will the outer edges
run? In a right triangle what will the sum of the two smaller
angles be? In an isosceles right triangle what part of a right
angle will each smaller angle be? How are the smallest angle in
a triangle and the shortest side in the same triangle placed? If
one corner of a triangle be cut off by a line parallel to the
opposite side, what is the shape of the little triangle that you
thus make? If we divide a right triangle into two right triangles
by a line through the vertex of the right angle, what will be the
shape of these little triangles? When are two triangles similar?
When isoperimetrical? When only one side of a triangle can alter
in length, how shall we make the triangle largest? When two sides
only can alter, but the triangle is kept isoperimetrical, how
is it made largest? What is the largest of all isoperimetrical
triangles?
CHAPTER XII.
QUADRANGLES.
[Illustration]
1. When a figure is bounded by four straight lines, it is called a
quadrangle. Sometimes it is called a quadrilateral, but generally a
quadrangle.
2. A line joining two opposite vertices is called a diagonal. In this
figure, the dotted line A C is a diagonal, and the dotted line B D is
another diagonal.
* * * * *
What is a quadrangle? What do you call a line that joins two
opposite vertices? Into what two figures is a quadrangle divided
by a diagonal? What is the sum of the angles of a triangle?
What is the sum of the angles of a quadrangle? How can you show
this by paper figures? Let now, the teacher draw quadrangles
and diagonals on the board, and ask, “What is this figure? What
is this line?” etc. What is the only figure that is strong and
stiff? In what figures can the angles be altered without altering
the sides? What is the use of a brace? Let the teacher provide
twigs or lamp-lighters for the illustration of section six.
3. Each diagonal divides the quadrangle into two triangles. If both
diagonals are drawn, the quadrangle is cut up into four little
triangles; but those are not the triangles of which I am speaking.
The dotted line A C, in the figure, divides the quadrangle into two
triangles, A B C and C D A. And I think you can easily see that, if we
put all the corners of a quadrangle together, they will make just as
large a sum as if we put all the angles of two triangles together.
[Illustration]
4. The sum of the four angles of any quadrangle is equal to four right
angles. But if we put the vertices of four right angles together, the
sides of the angles will make two straight lines crossing each other.
[Illustration]
[Illustration]
And if we make a paper quadrangle, cut off the corners (by a waving
line, so as not to become confused), and put these four corners
together, we shall find that they fill up all the space around the
corners. Compare now the letters in these four angles with the letters
at the corner of the first quadrangle that I drew for you, on page 36.
[Illustration]
[Illustration]
In cutting paper triangles and quadrangles, to show that the corners
put together make two or four right angles, you must be careful to make
the sides straight. Cut off the corners by a waving line, that you may
distinguish the edges of the cuts from the edges of the triangles or
quadrangles.
[Illustration]
5. The shape of a triangle cannot be altered, except by altering the
length of at least one side. But the shape of a quadrangle can be
altered without altering the length of a single side, by altering the
angles. The diagonals of a quadrangle can be made shorter or longer
without shortening or lengthening the sides. If you will look at the
figure you will see that B D might be pulled out, and A C crowded
together, so as to alter the shape of the quadrangle very much, without
altering the length of its sides at all.
6. Take two willow twigs, or two thin rolls of paper, such as are
used for lamp-lighters, and bend one into a triangle, and one into a
quadrangle. You will find that the triangle is stiff, and that the
quadrangle is not.
[Illustration]
[Illustration]
[Illustration]
7. If you have ever noticed the frame of a house, you have seen that
the carpenter puts braces in the corners. Without braces the timbers
would only make quadrangles, and so would have no stiffness except the
stiffness of the joints. But, by putting braces he makes triangles,
which cannot be pressed out of shape without being broken.
CHAPTER XIII.
PARALLELOGRAMS.
1. When a quadrangle has its opposite sides parallel it is called a
parallelogram. So that there are in every parallelogram two sets of
parallel sides.
2. The opposite angles of a parallelogram are equal. If this figure is
a parallelogram, so that A D is parallel to B C, and D C parallel to A
B, then the angle at D is equal to the angle at B, and the angle at A
is equal to that at C.
[Illustration]
3. A diagonal divides a parallelogram into two equal triangles. If you
could turn the triangle A B C round; so as to put the point B exactly
on the point D, and the line B C upon the line D A, then the line B
A would lie upon the line D C, and one triangle would exactly cover
the other. You can try this by cutting a parallelogram of paper, and
cutting it in two diagonally. But you must be very careful to have the
opposite sides exactly parallel.
* * * * *
What is a parallelogram? How many sets of parallel sides in a
parallelogram? How many sets of equal angles? Which angles of a
parallelogram are equal? Let the class draw parallelograms on
the blackboard, and show which angles are equal? Draw a diagonal
in your parallelogram. How does it divide the figure? How can
you tell whether a figure is a parallelogram? How must I lay
four sticks of unequal length on the floor so as to enclose the
largest space? How must I place them when the sticks can be
divided into two couples of equal length? Let the teacher show
the child that there are _two_ different answers to the last
question; one a parallelogram, the other not. And, giving a child
four unequal sticks, let him try how many quadrangles, all of the
largest possible size, he can make.
4. And this shows you that in every parallelogram the opposite sides
are equal. In the figure. A D is equal to B C, and A B to D C. If the
opposite sides of a quadrangle are parallel, they are equal.
[Illustration]
5. And if a quadrangle has its opposite sides equal, they are also
parallel. If I should find a quadrangle such as this figure, and, by
measuring, find that A D and B C are just equal, and also that A B and
D C are equal to each other, I should know that these equal lines are
parallel.
6. When a quadrangle is squeezed flatter without altering the length of
its sides, it alters the size, as well as the shape, of the quadrangle.
The quadrangle will be largest when the opposite angles, added
together, are equal to two right angles.
[Illustration]
In this figure, the angles B and D added together would be more
than two right angles. We could, then, make the quadrangle larger,
without altering the length of its sides, by drawing B and D apart,
and crowding A and C together, until the two angles, B and D, added
together, made two right angles. The angles A and C would, also, added
together, then be equal to two right angles, and the quadrangle would
be as large as we could make it without changing the length of the
sides.
[Illustration]
7. When a parallelogram is put into its largest form, as the opposite
angles are always equal, and the two are equal to two right angles,
each of the four angles will be a right angle.
CHAPTER XIV.
RECTANGLES AND SQUARES.
[Illustration]
1. When each angle of a parallelogram is a right angle, the figure
is called a rectangle. This is a very common figure in all sorts of
things that men make. The panes of glass in the windows, and the
windows themselves, the doors and the panels in them, the sides of the
room, the leaves of books, sheets of paper, and many other things, are
usually made in the shape of rectangles. But in the things that were
not made by men there are very few rectangles; they are scarcely to be
found even in crystals; coarse salt and iron pyrites[6] being the only
common things in which the Creator has used the rectangle.
[6] Pyrī’tēs.
2. When all the sides of a rectangle are equal, the figure is called
a square. And, as the corners of a square are all right angles, so a
right angle is sometimes called a square corner.
* * * * *
What is the geometrical name of the figure of a pane of glass?
Name all the objects you can think of which are rectangular in
shape. What natural objects present the form of a rectangle?
If you found a rectangular piece of stone or iron, would you
easily believe that men had not cut it into that shape? What
is a carpenter’s square? What is a square corner? What is the
geometrical meaning of the word square?
[Illustration]
3. When carpenters wish to mark a right angle on their boards or
timber, they use a simple tool, which I dare say you have seen, and
which is called a carpenter’s square.
4. But, in Geometry, the word square is only used to mean a rectangle
with equal sides.
5. I suppose you know the way in which men measure how long a thing is,
with a rule, or yard-stick, or tape. They measure how many inches, or
feet, or yards, or miles, it will take to stretch along by the side of
the thing they are measuring.
6. Men measure surfaces, such as painting, or carpeting, or fields, by
finding out how many squares, whose sides are each one inch, or one
foot, or one yard, it will take to cover the surface which they want
to measure. When a carpet-dealer sells so many yards of oil-cloth or
oil-carpeting, he means that the carpet could be cut in such a way as
to make just that number of squares, one yard on a side. When a man
says that there are so many feet of land in his door-yard, he means
that it would take just that number of square pieces of paper, one foot
on a side, to cover his yard.
[Illustration]
7. It is very easy for any one that knows the multiplication table, to
measure a rectangle. For the number of square inches, or yards, in a
rectangle is found by multiplying the number of inches, or yards, in
the breadth by the number in the width, and this will give the number
of squares in the rectangle.
If you have learned to count, you will see by this figure that a
rectangle four feet wide and seven feet long, could be divided into
twenty-eight squares, each being a foot on a side. We can divide it
into seven rows, of four square feet in a row, or into four rows, seven
square feet in a row.
* * * * *
How do men measure the length of things? What do they mean by
saying a hundred feet of land? How do they find out the number of
square feet or square inches in a rectangle? In a parallelogram?
Let the teacher copy the last figure on the blackboard, and ask
the children how, if the rectangle and parallelogram were made
of paper, they could cut the rectangle to make it cover the
parallelogram, and how the parallelogram to make it cover the
rectangle.
How do men measure triangles? How do they measure other surfaces
than parallelograms? How long ago did Pythagoras live? What did
he discover about right triangles?
How can you tell whether a triangle is a right triangle? How do
carpenters make the corner of a frame square? How do they cut
braces for square corners? How large is a square built on the
diagonal of a square? How large is a square whose corners are in
the middle of the sides of another square? Let the children draw
right triangles and squares upon the sides, and tell the relative
size of the squares according to the Pythagorean proposition. The
teacher should require this proposition, as written in italics,
to be committed to memory, as it is the most useful proposition
in the fifteen chapters.
8. When we wish to measure a parallelogram that is not a rectangle,
we have only to multiply the length by the breadth; because every
parallelogram is exactly the same size as a rectangle, of the same
length and breadth, would be.
You see in the figure that the rectangle A B C D is of the same length
and breadth as the parallelogram A B F E. And they are of exactly the
same size. If you cut the triangle C A E off one end of the rectangle
and place it at the other end, it will just cover the triangle B D F.
[Illustration]
9. When I say that we multiply the length of a parallelogram by its
breadth in order to find its measure, I mean, of course, that we must
multiply the _number_ of inches in the length by the _number_ of
inches in the breadth in order to find the _number_ of square inches,
that is, squares an inch on a side, that it would take to cover the
parallelogram. _Numbers_ are the only things that can be multiplied.
CHAPTER XV.
TRIANGLES AND RECTANGLES.
1. Every triangle may be imagined as half of a parallelogram. If we had
a triangle A B F, we could draw A E parallel to B F, and F E parallel
to B A, and that would make a parallelogram just twice as large as the
triangle.
[Illustration]
[Illustration]
2. And thus we may find the measure of a triangle; that is, we may find
how many squares of one inch on a side, or of one foot on a side, it
would take to cover the triangle exactly. We need only multiply any one
side of the triangle by the distance to a parallel line drawn through
the opposite vertex. This will give us the measure of a parallelogram,
and half of that will give us the size of the triangle.
3. In this way men measure surfaces of every shape, by dividing them
into triangles, and then finding out how large each triangle is. You
may draw upon your slates figures of any number of straight sides, and
then divide them into triangles by drawing diagonals. You can divide
the same figure into different sets of triangles by drawing different
diagonals.
[Illustration]
4. In every right triangle the square made on the side opposite the
right angle is just as large as the squares on the other two sides, put
together. If, in my figure, the triangle A B C has a right angle at C,
the square on the side A B will be just as large as the square on A C
added to the square on C B.
5. The side of a right triangle opposite the right angle is called the
hypotenuse, and the other sides are called the legs. So that what I
have already told you may be repeated in different words. _The square
on the hypotenuse is equivalent to the sum of the squares on the legs._
This is called the Pythagorean proposition, because it was found out by
a geometer, who lived more than two thousand years ago, whose name was
Pythagoras.
6. This Pythagorean proposition gives us a good way of trying whether
an angle is a right angle. If the sum of the squares on two sides of
a triangle is just equal to the square on the third side, we may know
that the angle opposite this third side is a right angle.
Now, a square is a rectangle as broad as it is long; so that, to find
how large the square on a leg would be, we need only multiply the
number of inches in the length of the leg by itself. If you should
measure the sides of a triangle, and find that one side was three
inches, and another four inches, and the third five inches; then, if
you know the multiplication table, you would know that the squares on
the sides would be nine inches, and sixteen inches, and twenty-five
inches. Moreover, if you add nine to sixteen, it makes twenty-five. So
that, in a triangle whose sides are three, four and five inches, the
side of five inches is a hypotenuse, and the opposite angle a right
angle.
7. If you understand the last section, you will also understand how
the Pythagorean proposition, that _the square on the hypotenuse is
equivalent to the sum of the squares on the legs_, is very useful to
carpenters and other persons who wish to make square work. Suppose you
were making a frame of a house, and wanted the timbers A B and A C to
make a right angle at A. You might measure three feet from A to B, and
four feet from A to C, and then alter the angle at A, by moving one of
the timbers until a stick of five feet long would just reach from C to
B. Instead of three, four and five, for large frames, they take six,
eight and ten feet.
[Illustration]
[Illustration]
8. Another use that carpenters make of the Pythagorean proposition is,
in cutting braces for frames. They measure the same number of feet from
A to B and from A to C; and then for the brace, B C, they measure just
as many times seventeen inches as there are feet in A B. This makes the
brace a very little too long; because a square on seventeen inches is a
very little more than twice as large as a square foot. But carpenters
like the brace to be a very little too long, because hammering the
timbers together makes the brace indent the timbers a little; and,
if the brace were not a very little too long at first, it would be a
little too short after it had indented the timbers; and that would
make A less than a right angle.
9. The square on the diagonal of a square is just twice as large as
that square. This you will see in the figure marked A.
[Illustration]
[Illustration]
[Illustration]
A square with its corners in the middle of the sides of another square
is just half as large as that square. This you will see in the figure
marked B.
If we fancy a square cut into four right triangles by two diagonals (as
at C), we can fancy each little triangle turned over on its own little
hypotenuse, which will make the figure at B a square just twice as
large as that at C.
You may take a square piece of paper, as at B, fold each corner over to
the centre of the square, as at C, then unfold it again, as at B, and I
think this will make you understand this section.
CHAPTER XVI.
CIRCLES.
1. We have had fifteen chapters about straight lines, and about figures
bounded by straight lines. I think, therefore, that you will be glad to
learn a little about curves.
[Illustration]
2. A curve is, as I have told you, a line that bends in every part of
it, but has no sharp corners. If a boy were wheeling a barrow in a
large, level field, just after a light snow, the middle of the track of
his wheel would be a line that you could easily follow with your eye.
If the boy went all the time in the same direction, he would make a
straight line; but if he kept turning a little all the while to the
right or left, he would make a curve.
* * * * *
What is a curved line? What is a straight line? How could you
mark a straight line with the wheel of a wheelbarrow? How would
you mark out a curve with the wheel? If a line bends equally in
every part, how long can it be? (Only long enough to return into
its own beginning.) What is a curve called that has no ends, and
bends equally in every part? But, if it bends equally in every
part, and yet has two ends, what is it called? When I swing the
door on its hinges, in what curve does the end of the latch
move? Where is the centre of the arc? When a stone hangs by a
thread, and swings backward and forward over a straight line
on the floor, what curve does the stone move in? Where is the
centre of the arc? What line does the thread mark out? Let the
teacher actually open and shut the door, and swing a plumb, while
asking these questions. Keep the child constantly in the habit of
seeing the perfect geometrical forms suggested by the material
appearances. Let them draw circles, and their radii, on the board.
What is the figure of a cent? What other things can you tell
me of in the shape of a circle? Is the circle or the rectangle
oftener found in natural objects? What instrument do men have to
draw circles with? How can you make a sort of compasses? What did
the old Greeks use for a blackboard and chalk? Did you ever see
a turning-lathe? A potter’s wheel? A tinman at work cutting out
round pieces? How did he cut them? Did you ever see a blacksmith
bend a wagon-tire? Why does passing the tire through the three
rollers make a circle of it? How does a cooper make a hoop round?
Let the teacher draw a circle with radius, cord, and diameter,
and ask what the name of each line is. What is the longest chord
we can have? If we divide the whole circumference into six equal
arcs, how long is the chord of each arc? How can you show it with
a pair of compasses?
3. If the boy kept on all the time turning in the same direction, just
as fast as he began to turn, the track of the wheel would come round
into itself, and make what is called the circumference of a circle. A
curve that bends equally in every part is called a circumference; and a
figure bounded by a circumference is called a circle.
[Illustration]
4. Parts of a circumference are called arcs. The word arc used to mean
a bow; and bows are shaped something like an arc.
5. If there was a tree in the middle of a field, and the boy should
keep all the time at the same distance from the tree, he would come
round to the place he started from, and the track of his wheel would
bend equally in every part. He would, in fact, go round in the
circumference of a circle.
6. You will see, in the figure of a circle in section three, a dot in
the middle. It shows the place where we suppose the tree to stand. And
in every circle there is a point in the middle, equally distant from
every part of the circumference. This point is called the centre of
the circle. It is not always marked by a dot, but the place is always
there whether it is marked or not.
7. A straight line drawn from the centre of a circle to its
circumference is called a radius. All the radii of the same circle are
equal. In this figure there are three radii marked by black lines.
[Illustration]
[Illustration]
8. We can fancy a circle made by a radius swinging all round a centre.
And you may draw very nice circles by holding, with one hand, a thread
fast to one spot on your slate, or blackboard, while, with your other
hand, you hold the other end of the string and your pencil together,
and draw a circumference, keeping the string stretched. You can also
draw them with one hand, by resting your little finger on the centre of
the circle, and being careful to keep the end of your crayon always at
the same distance from it. This way only answers for the blackboard.
CHAPTER XVII.
MORE ABOUT CIRCLES.
1. Rectangles and circles are the most common figures in all
manufactured things. They are the easiest figures to make exact, and
the most convenient when made. You have already noticed how common
rectangles are in what men make. Circles are almost as common. Buttons
and door-knobs, plates, pans and wheels, are circular. But in natural
things,—in things made by the great Creator,—circles are much more
common than rectangles. The sun, the moon and stars, the buds of many
flowers, the eyes of animals, and some little animals themselves, are
nearly in the shape of a circle.
2. To draw circles, men have what is called a pair of compasses, that
open and shut like a pair of tongs. They put one point down on the
paper hard enough to hold it still in the centre of the circle, and
then move the other leg round lightly, just bearing on hard enough to
mark the circumference. The circle will be larger or smaller, as the
compasses are opened wider, or are more nearly shut.
[Illustration]
[Illustration]
You can make a pair of compasses, to draw circles on the ground, by
putting one small nail through two bits of shingle, and whittling the
ends to a point. The old Greeks, of Pythagoras’ time, and afterwards,
used to study and teach Geometry with the help of figures drawn on
smoothly-spread sand.
3. To make things circular-shape, men sometimes draw a circle first,
with compasses, and then cut the thing to that shape. But generally
they use a different method. They use a turning-lathe, or something
that works on the same plan. In a turning-lathe a block of wood or
piece of iron is made to turn steadily round on a steel point that
marks the centre, while the chisel, that cuts the wood or iron, is held
steadily at the same distance from the steel point, so that the wood
or iron that is too far from the centre is cut off as it goes past the
chisel. On a potter’s wheel a lump of clay is made to turn round, while
its centre remains in one place; and thus the potter makes the crockery
round. The bottom of tinned pans is cut out by a sort of scissors that
is fastened at the right distance from a point around which the piece
of tinned plate is made to turn.
[Illustration]
Sometimes boys make a circle of leather, for a plaything. Put an awl
through a bit of leather into a board; then stick a sharp knife through
the leather into the board, slanting, with the cutting-edge down; the
blade at right angles to a line joining it to the awl. Now pull the
leather round the awl, under the knife, and you will cut out a circle.
Put a strong string through the awl-hole in the leather, with a large
knot at the end of it, and you will have a curious toy. Wet the leather
and press it under your foot upon a flat stone, and you can lift the
stone by the string. But be careful not to swing the stone about; for
nobody has a right to put other people in danger.
All these ways of making a circle are somewhat alike. But there is
another way,—to bend wire, or things of the kind, equally at every
part. Thus, the blacksmith passes an iron bar between three rollers,
that bend each part of the tire equally, and thus bring it into a
circular form for a wagon-tire.
4. A straight line joining the ends of an arc, is called a chord.
Any straight line going across a circle, having both ends in the
circumference, is a chord. All the lines marked A C and B C, in this
figure, are chords.
[Illustration]
[Illustration]
5. The longest chord that we can have is the one that goes through the
centre of the circle. It is called the diameter, and is just twice the
length of a radius.
[Illustration]
6. If we divide a circumference into six equal arcs, the chord of each
arc is just as long as a radius. So that, if you draw a circle on the
ground with your shingle compasses, you will find, if you are careful
neither to open nor close your compasses, that they will step round the
circumference in exactly six steps.
CHAPTER XVIII.
MEASURING ANGLES.
[Illustration]
1. If the vertex of a right angle is at the centre of a circle, the
lines forming the angle cut off one quarter of the circumference,
whether the circle is large or small. So with any angle whose vertex is
at the centre of a circle; its sides always cut off just that part of
a circumference which the angle is of four right angles. If it takes
eight of these little angles to make one right angle, it will take
eight of the little arcs to make a quarter of a circumference, whether
on a large or small circle.
* * * * *
How many degrees of arc make a quarter of a large circumference?
How many degrees make a quarter of a small circumference? If you
put the vertex of a right angle at the centre of a circle, how
many degrees of arc will its sides cut off? If an angle is an
angle of forty-five degrees, what part of a right angle is it?
Have you ever studied Arithmetic? Do you know how much nine times
ten make? Three times thirty? How many angles of 10° must be put
together to make a right angle? How many degrees in one third of
a right angle? What angle does a straight line make with itself?
How many degrees do the three angles of a triangle added together
make? How many degrees in each angle of an equilateral triangle?
If we want to tell how large an angle is, we tell what part of a
circumference it would cut off if its vertex were at the centre.
2. And, in order that we may easily tell the parts of a circumference,
we fancy every circumference as divided into quarters, and then each
of these quarters divided into ninety equal parts, which are called
degrees of arc. So that we tell how large an angle is by telling how
many of these degrees would be cut off between its sides if its vertex
were at the centre of a circle.
3. If the vertex of a right angle is placed at the centre of a circle,
its sides cut off a whole quarter circumference, and the right angle
is, therefore, sometimes called an angle of ninety degrees. Half a
right angle, like the angles of an isosceles right triangle, is called
an angle of forty-five degrees.
[Illustration]
4. As each right angle is an angle of ninety degrees, two right angles
together will make an angle of twice ninety, that is, of one hundred
and eighty degrees.
[Illustration]
5. You may take any point in a straight line, as the point C in the
line A B, and fancy the line as making an angle of one hundred and
eighty degrees with itself at that point. An angle is the difference of
direction of two lines. Now you cannot tell whether I moved my pen from
A to B, or from B to A; and so, if you like, you may fancy that I moved
it from A to C, and then from B to C; that is, you may fancy the line
meeting itself at C; that is, going in opposite directions on each side
of C; that is, making an angle of two right angles at C; that is, the
line makes an angle of one hundred and eighty degrees with itself at C.
6. You remember, I hope, that the three angles of a triangle taken
together are equivalent to two right angles. We can now say the same
thing in other words; we may say that the sum of the three angles of a
triangle is one hundred and eighty degrees.
[Illustration]
7. In an equilateral triangle, you remember that the three angles are
equal to each other. Each of them is then an angle of sixty degrees,
because three times sixty is one hundred and eighty.
CHAPTER XIX.
CHORDS.
[Illustration]
1. If, instead of putting the vertex of an angle at the centre of a
circle, we put it in the circumference, it will take in an arc just
twice as large as it would with its vertex in the centre.
2. If one angle has its vertex at the centre of a circle, as at C in
the figure, and another has its vertex in the circumference, as at A,
and if the sides of both these angles go through the circumference at
the same places, D and B, then one angle is just half as large as the
other. The angle at A is only half as large as that at C.
* * * * *
An angle, with its vertex at the centre, is measured by the arc
between its sides. How is the angle measured when its vertex
is in the circumference? How is an angle between two chords
measured, when the vertex is in the circumference?
Can any of the class draw a figure, and explain how a curve in a
railroad is laid out? Can you show how an arc may be drawn by a
card and two pins? When the card has a square corner, how large
is the arc? Can any one with a ruler and a piece of chalk show a
way of staking out a circular garden-path? (Use the ruler for the
stick of section eighth, and chalk dots for stakes.) Can you tell
me how to try whether an angle is a right angle without using the
Pythagorean proposition? Invite the scholar to draw a figure and
explain. If he cannot, draw it for him, and show, with shingle
compasses, if you have no other, how to divide the hypotenuse and
apply the test of section ten.
When two chords do not touch each other, how is the angle
measured? Can you draw a figure on the blackboard, and explain
this more fully? When two straight lines go through a circle and
meet outside the circle, how is the angle between them measured?
Can you draw G figure, and explain that? What is meant by a
tangent to a circle? How is the angle between a chord and a
tangent measured? Draw on the board a circle with a chord, and a
tangent at the end of it. Draw radii to the ends of the chord.
Show me which two angles in that figure should be the one just
double the other. What is the angle made by a diameter with a
tangent at the end of it? By a radius with a tangent at the end
of it? What is a tangent to any curve?
3. We can say the same thing in other words. Two chords starting from
the same point, A, in the circumference of a circle, make an angle of
just half as many degrees as there are in the arc D B between their
other ends. The arc D B is not drawn at all; but you can easily fancy
it there.
4. Now the arc D B would be of the same length, wherever the point A
were placed, and that will make you understand the next figure.
[Illustration]
5. All the angles that can be drawn in one arc, that is, with their
vertices in the arc, and their sides going through the ends of the arc,
are of the same size. The angles at C, C, C, C, are all four equal to
each other. Each one is measured by half the arc that is not drawn
between A and B.
[Illustration]
6. This gives you a curious way of drawing an arc of a circle. Drive
two pins in a board, as at A and B, and then move a bit of card, with
straight edges, in such a way as to keep the corner thrust, as far as
it will go, between the pins. The corner C will move in the arc of a
circle.
[Illustration]
7. If equal angles have their vertices at the same point in the
circumference, they will cut off equal arcs. That is, if the angles at
A are all equal to each other, the arcs at B, B, B, are equal to each
other.
This is the way in which railroads are laid out in curves. The engineer
measures equal angles from one point, as A, and equal chords, as at B,
B, B, and then the rails are laid to go round as arcs to those chords.
8. There are several other ways of drawing circles without using
compasses. I will tell you one more, which you will find very useful if
you ever want to lay out curved paths in a garden, or do anything of
that kind. Take a straight stick, A E B, and drive into the ground two
stakes, one at A, and the other at D, making A D about half the length
of the stick. Keeping one end of the stick at A, move the other end
round until the middle, E, is as far from D as you think best to put
it. Then drive a stake at B. Now put one end of the stick at D, and let
the middle, E, be just as far from the stake B as it was before from D.
Drive a new stake at the other end of the stick. Thus you can go on,
driving stakes as far as you wish to go. The size of the circle will be
made greater by making the distance D E smaller.
[Illustration]
[Illustration]
9. If the arc A C B is just half a circle, then the other arc A B is
a half-circle, and the angle A C B is measured by half a half-circle,
and is a right angle. If the corner of the card in section six of this
chapter is a square corner, then the arc will be a half-circumference.
10. If, in the last figure, A C is a right angle, A B will be a
diameter, and the middle of it, D, will be the centre of the circle,
and will be just as far from C as it is from A or B.
This gives us a very pretty way of trying whether an angle is a right
angle or not. If I want to know whether A C B is a right angle, I will
draw any line A B across it; and, if half A B will just reach from the
middle of A B to C, we may know that C is a right angle. If it does not
reach, C is less than a right angle; if it more than reaches, C is more
than a right angle.
CHAPTER XX.
CHORDS AND TANGENTS.
[Illustration]
1. When two chords do not touch each other, the angle between them is
measured by half the difference of the arcs between them. The angle
made by the chords A B and C D is measured by half the difference
between the arcs A D and B C. That is to say, the angle between A B and
C D is half as large as the difference of the two angles at E, A E
D and C E B.
2. We may imagine the chords lengthened into straight lines going
outside the circle until they meet; and it will not alter the size of
the angle. So that the angle A C B is measured by half the difference
of the arcs between its sides, wherever we place the circle.
[Illustration]
[Illustration]
3. If, in the last figure, we move the circle back until the
circumference touches the vertex C, then the smaller arc has become
of no size at all, and the difference between it and the large arc is
equal to the whole of the large arc. Then the angle is measured by half
the large arc, which is just the same thing that you learned in the
last chapter.
4. If we take the circle farthest from the vertex in the last figure,
and lift it up till it just touches the line A C, at the point D, as
in this figure, then the larger arc meets the smaller arc just at D.
5. The line A C is now called a tangent. Tangent means a toucher; and
the line A C just touches the circumference without cutting it.
6. If now we move the line B C without changing its direction, until it
stands in the place of E D, the small arc has become nothing; so that
the angle A D E is measured by half the arc between its sides.
[Illustration]
7. In other words, the angle between a chord and a tangent at one end
of the chord is measured by half the arc between them. That is to say,
it is half as large as the angle made by two radii to the ends of the
chord.
8. When the chord is a diameter, the angle is measured by half of half
the circle; that is, the angle is a right angle. A tangent at the end
of a diameter must always be at right angles to the diameter. And you
will easily see that it must be so with a radius. A tangent at the end
of a radius is at right angles to the radius.
9. There can be tangents to other curves as well as to circles. A
straight line that just touches a curve; or, if the curve winds, a
straight line going through a point in a curve in the same direction as
the curve at that point, is called a tangent to the curve.
CHAPTER XXI.
MORE ABOUT CHORDS AND TANGENTS.
[Illustration]
1. When two chords cross each other, the angle they make is measured by
half the _sum_ of the arcs between their ends. That is, if two straight
lines cross each other inside of a circle, their angle is measured by
half the _sum_ of the arcs between them; if they cross outside the
circle, the angle is measured by half the _difference_ of the arcs.
* * * * *
How do you measure the angle of two chords that cross each other?
How is it when the chords are diameters? If a chord of an arc
or any kind of curve be moved, keeping it parallel to its first
position, until it cuts off no arc, what does the chord become?
If I go to a place exactly north from me, and yet at no part of
my journey travel north, what must have been true of my road? (It
must have turned a corner in it.) If I travel over a winding road
without corners, and find myself at night in a place just east of
my starting, what do you know of the direction of my road? (It
must have gone east at some point of the way.) What can you tell
me about the radius that passes through the middle of a chord?
What of a straight line at right angles to a chord, through the
middle of it? If you find an arc drawn on the blackboard, how can
you find the centre of it?
[Illustration]
2. When the chords are both diameters, the arcs are equal, and half
the sum is just one arc; so that the angle is measured by one arc; and
that, you know, is the first thing that you learned about the measure
of an angle,—that an angle, with its vertex in the centre, is measured
by the arc between its sides.
3. Arcs are parts of circumferences. Parts of other curves are called
arcs of those curves; and straight lines joining the ends of those arcs
are called chords of those arcs. When we speak of an arc, we mean a
piece of a circle; and if we wish to speak of a piece of an ellipse, we
call it an arc of an ellipse. You will learn what an ellipse is, after
a while.
[Illustration]
4. In every arc of any curve there must be at least one place at which
the tangent is parallel to the chord of that arc. Now turn back to
chapter III., section fourth, and you will find that what I have now
told you is just the same as that, only in different language.
5. If the chord of any arc of any curve be moved, keeping it parallel
to its first position, it will cut off either a longer or else a
shorter arc. If moved one way, it cuts off a longer arc; if moved the
other way, it cuts off a shorter arc. If moved so as to cut off a
shorter arc, we can move it so far, keeping it still parallel to its
first position, that it will cut off no arc at all, but be tangent to
the curve at that point where the curve goes in the same direction as
the chord. You can easily perform this process of moving a chord, and
keeping it parallel to itself, by drawing any curve you choose on your
slate, and moving a stretched thread across it.
[Illustration]
6. Let us now go back to circles. If a radius be drawn through the
middle of a chord, it will be at right angles to the chord, and it
will end at the middle of the arc. Let us suppose that, in this figure,
the radius C D goes through the middle of the chord A B. Then, I say
that the angles at E will be right angles, and the arc A D will be
equal to the arc D B.
7. On the other hand, if we draw a straight line, through the middle of
a chord, at right angles to the chord, it will pass through the centre
of the circle.
[Illustration]
8. This gives you an easy way to find the centre of a circle when
you have an arc of the circle. You have only to draw two chords not
parallel to each other (the larger the angle they make the better), and
then draw lines through the middle of each chord at right angles to the
chord. As both these lines pass through the centre of the circle, the
centre must be at the point where the lines cross each other.
9. If you can get a card, such as business men have advertisements
printed on, you can use the longer side as a ruler by which to make a
straight line; and, by setting the short side carefully on this line,
you can draw, by the long side, a line at right angles to your first
one. You can probably find an old printed card by asking for it, and it
will serve as a ruler and square. Then you can draw short arcs by your
eye, and find the centres by section eight.
CHAPTER XXII.
INSCRIBED POLYGONS.
[Illustration]
1. When a triangle has its vertices in a circumference, it is said
to be inscribed in the circle. In other words, when the sides of a
triangle are chords in a circle, the triangle is said to be inscribed
in the circle. A B C is an inscribed polygon.
* * * * *
What is meant by an inscribed triangle? Where do you say that
the vertices of an inscribed triangle are? What are the sides?
If you find a triangle ready drawn, how can you find the centre
of a circle whose circumference will pass through the vertices?
How will you find the centre of a circle to which the sides of
the triangle will be tangent? What is meant by a circle inscribed
in a triangle? How can you divide an angle into halves? Let the
pupils illustrate as they recite, by drawings on the blackboard.
What is the largest triangle that can be put in a given circle;
that is, in a circle that you find already drawn? How long is the
chord of sixty degrees? What part of a circumference is sixty
degrees? How do you draw an equilateral triangle in a circle?
2. If we want to put a circle round a triangle, so that the triangle
shall be inscribed in the circle, we can easily do it by remembering
that the sides of the triangle will be chords in the circle, and so we
can find the centre of the circle, to put one leg of our compasses at,
by section eight of the last chapter. We have only to draw lines at
right angles to the middle of two sides of the triangle, and the point
where these lines cross, at C, will be the centre of a circle, whose
circumference will pass through the three vertices of the triangle.
Try this with triangles drawn on the ground, and you will then find
that, by putting one foot of your shingle compass at C, you can draw a
circumference through the vertices.
[Illustration]
[Illustration]
3. If, on the other hand, we wish to inscribe a circle in the triangle,
so that each side of the triangle shall be tangent to the circle, we
must draw lines dividing two of the angles of the triangle into halves,
and the point where these lines cross each other will be the centre of
the circle.
[Illustration]
4. But you do not know, perhaps, how to divide an angle into halves.
Suppose, then, that C A B is the angle you wish to divide. Measure A
B and A C of equal lengths. Put one foot of a pair of compasses at B,
and with the other foot scratch a little arc near what you think is the
middle of the angle. Now put one foot at C, and, with your compasses
open exactly as wide as before, make another arc crossing the first.
A straight line from A through the points where the arcs cross will
divide the angle into two equal parts.
5. The largest triangle that can be inscribed in a circle is an
equilateral triangle. So that there are two kinds of triangles in which
the equilateral triangle is largest; namely, the isoperimetrical, and
those inscribed in one circle.
[Illustration]
6. You remember, I hope, that the radius of a circle is the chord of
a sixth part of the circumference. If you draw an arc long enough to
draw a chord of the same length as a radius, and then draw radii to
the ends of this chord, you will make an equilateral, and, therefore,
equiangular triangle. Each angle will be one third of two right
angles; because the three are equal to each other, and the three
together are equal to two right angles. If three of them would be equal
to two right angles, six would be equal to four right angles. And so
the corners of six such triangles would fill up all the space about the
centre of a circle, and the six chords would just go round the circle.
7. If you wish to draw the largest triangle that you can in a circle,
you must open your compasses just as wide as they would be to draw the
circle, and then step six times round the circumference, marking the
points where the feet of the compasses step. Then join every other one
of these marks by three straight lines, and they will make an inscribed
equilateral triangle.
8. Do you understand what I mean when I say that the radius is equal to
the chord of sixty degrees?
CHAPTER XXIII.
MORE ABOUT INSCRIBED POLYGONS.
1. Any figure bounded by straight lines is a polygon; and, if its
vertices are in a circumference, it is an inscribed polygon.
2. A polygon of three sides is called a triangle; of four sides a
quadrangle; of five sides a pentagon; of six sides a hexagon.
[Illustration]
3. Any polygon of more than three sides can have its angles altered
without altering the length of its sides. You remember, I hope, how we
illustrated this, when we were studying quadrangles, by a bent twig,
or by a bent lamp-lighter. If the twig is bent into the form of a
triangle, and its ends held together, it cannot be altered in shape.
But, if in the form of any other polygon, you can flatten or stretch it
into different shapes, which will not only be isoperimetrical, but will
have the sides unchanged.
* * * * *
What is a polygon? An inscribed polygon? A polygon of three
sides? Of four sides? Of five sides? Of six sides? Suppose that
a man measures how long the sides of his field are, will that
tell him how large the field is? How can you show that it will
not, by a bent twig? But suppose his field has only three sides?
Suppose it has four sides, what angles must they make to have
his field the largest possible? When a quadrangle is in its
largest form, what can you say about its vertices? Suppose you
wish to lay seven sticks on the floor so as to enclose the most
space you can, how will you lay them? Suppose you take seven
other sticks more nearly equal in length, but whose lengths
added together make just as much as the first seven, with which
set can you enclose most space on the floor? If five children
take hold of a long loop of string, each taking hold with one
hand, how must they stand so as to make the opening in the loop
largest? If a sixth child comes in and takes hold, will they make
the loop larger or smaller? Suppose, now, that each child takes
hold with both hands? What is meant by a regular polygon? Two
isoperimetrical polygons? What is the largest of isoperimetrical
polygons of the same number of sides? What is the largest of
isoperimetrical regular polygons? Of all isoperimetrical figures
which is largest? Suppose you wish to make a string, lying on the
floor, enclose as much space on the floor as you can, how will
you arrange it?
How much further is it round a circle than across it? What is
the most common and roughest answer to this question? (3 times.)
What is a more exact answer? (²²/₇) How nearly would this give
the circumference of a circle as large as a plate? How nearly
the circumference of a large cistern? What still more exact
answer can you give? (3·1416.) Can the answer be exactly given
in figures? What Greek letter is used to express the exact
proportion of a circumference to a diameter?
If there are pupils in the class who have studied arithmetic, let
them answer such questions as the following, using twenty-two
sevenths in the head, or 3.1416 on the slate. The simplest and
best way of reading decimal fractions is simply to say “decimal
one four one six.”
What is the diameter of a hoop 44 inches in circumference? What
is the circumference of a hoop 21 inches in diameter? 14? 3·5?
10·5? 17·5? 11?
4. When a quadrangle is put into the largest form it can have without
altering its sides, it can be inscribed in a circle.
5. You can put a circle about any triangle you please; but the triangle
is the only polygon that can always be inscribed in a circle.
[Illustration]
6. When a quadrangle is inscribed in a circle, the sum of either two
opposite angles is equal to two right angles. You remember that I have
already told you that an angle with the vertex in a circumference is
measured by half the arc between its sides. But the arc between the
sides of one angle, in this quadrangle, added to the arc between the
sides of the opposite angle, makes up the whole circumference, and the
sum of the angles is measured by half the sum of the arcs; that is, by
half a circumference; that is, by one hundred and eighty degrees; that
is, the sum of two right angles.
7. Any polygon whatever, when it is inscribed in a circle, is in the
largest form that it can be put without altering the length of the
sides. If you had some sticks of different lengths that you wished to
lay down as a play-fence upon the ground, you could make the largest
field of them by laying them in such a position that the ends of the
sticks shall all be in the circumference of one circle.
[Illustration]
[Illustration]
8. When two isoperimetrical polygons, of the same number of sides,
are inscribed in circles, that polygon is largest which has its sides
most nearly equal. Thus, if the inscribed pentagons A and B were
isoperimetrical, A would be the larger, because its sides are equal,
although B would be in the larger circle.
9. And, therefore, of all isoperimetrical polygons of the same number
of sides, that one is the largest which has its sides exactly equal,
and which can be inscribed in a circle.
10. When a polygon with equal sides can be inscribed in a circle, it is
called a regular polygon. Not only are its sides equal to each other,
but its angles also are equal to each other. An equilateral triangle is
a regular triangle, and a square is a regular quadrangle.
11. A regular polygon is larger than any isoperimetrical polygon of the
same number of sides. This is just what I told you in the ninth section.
12. Of two isoperimetrical regular polygons, that is greater which has
the greater number of sides. That is to say, a square is greater than
its isoperimetrical equilateral triangle; a regular pentagon greater
than its isoperimetrical square; and so on.
13. Suppose you wish to enclose some land in the middle of a great
field, with sixty panels of fence. If the fence was put into a square
form, fifteen panels on a side, it would enclose more than if put into
a triangle with twenty panels on a side; it would enclose still more
if put round a regular pentagon, twelve panels on a side; still more
as a hexagon ten panels on a side; and most of all if put in a regular
polygon of sixty sides, one panel on a side.
14. We can fancy a circle to be a regular polygon, with more sides than
can be counted, and each side too short to be seen. So that, of all
isoperimetrical figures, the circle is the very largest. Sixty rods of
stone wall could not be put into any shape and enclose so much land as
if it were put in a circle.
CHAPTER XXIV.
HOW MUCH FURTHER IS IT ROUND A HOOP THAN ACROSS IT?
1. Everybody knows that it is about three times as far round a circle
as it is across it. But, if you measure how far it is across your hoop,
you will find that a string of three times that length will not quite
go around it.
And now the rest of this chapter will be too hard for children that
have not learned a little Arithmetic. If you have not learned how to
multiply and divide, you will have to go to chapter XXVI., without
understanding much about this chapter and the next. Still, I think you
will do well to study these chapters, and learn what you can out of
them, even if you do not know how to cipher.
2. The circumference of a circle and its diameter are nearly in the
same proportion as the numbers 22 and 7. So, if you measure across the
hoop, and take a string three and one seventh times that length, you
will find it just go round the hoop.
3. If the diameter of a circle is seven inches, the circumference will
lack only a hairbreadth of twenty-two inches; if the diameter is seven
feet, the circumference will not lack the breadth of your slate-pencil
of being twenty-two feet. Now ask some one to show you a circle about
seven inches in diameter, such as a breakfast-plate; and a circle seven
feet in diameter, such as a great hoop that a man can walk through
with his hat on. You can then judge how nearly the circumference and
diameter are in the same proportion as the numbers twenty-two and seven.
4. Nobody knows, and nobody ever can know, exactly how many times
further it is round a circle than across it. It is very often true
in Geometry that we cannot express by figures the lengths of two
lines. There are no two numbers in the same proportion as the side
and diagonal of a square. Nobody can tell exactly how many times
longer the diagonal is than the side of the square. And in like manner
there are no two numbers in the same proportion to each other as the
circumference and diameter of a circle.
5. But we very often want to speak of this proportion, and we want to
have some short name for it. Geometers have generally agreed to call it
_pi_; which is the name of the Greek letter for p, and it is written π.
6. I have told you that π is nearly twenty-two sevenths. And, in
almost every question about circles that you will want to answer, this
is exact enough. It is nearly enough exact for workmen to work by in
making tinned ware, or anything in which the diameter is less than
large wash-tubs.
7. So, if you know what the diameter of a circle is, and want to find
out how long the circumference is, you must multiply the diameter by
twenty-two, and divide the product by seven. If you know what the
circumference is, and want to find out what the diameter is, you
must multiply the circumference by seven, and divide the product by
twenty-two.
8. But, perhaps, you will at some time wish to be more exact, and then
it will be better to use a decimal fraction. Perhaps you have not
learned decimals yet; but, after you have learned them, you may want to
use a better value of π, and you can then turn back to this page, and
find that π is very nearly equal to 3·1416.
CHAPTER XXV.
HOW TO MEASURE THE SIZE OF A CIRCLE.
1. I have told you that men measure surfaces by squares. They find out,
if they can, how many squares it would take to cover the surfaces, if
the side of each square was just one inch, or one foot, or one yard. I
have told you, also, that we can easily find out how many such squares
it takes to cover a large square; that is, we can find the measure of a
square by multiplying the number of inches, feet or yards, on a side,
by itself; that is, by the same number.
[Illustration]
2. Now, if you draw a square with its sides tangent to a circle, the
sides of this square will be each equal to the diameter of the circle.
The measure of such a square is found, then, by multiplying the length
of the diameter by itself. And the measure of the circle can be found
by multiplying the measure of the square by one quarter of π.
* * * * *
If the class have not studied any arithmetic, this chapter must
be omitted until a review.
How do men measure surfaces? How do you find the measure of a
square? How do you measure a circle? Let the teacher draw a
square, and ask, What figure is this? Inscribe a circle, and
ask, What figure is this? How large a part of the square does it
enclose? (¾). More exactly? (¹¹/₁₄) Still more exactly? (·7854.)
By what part of π must you multiply the square in order to find
the measure of the circle? How many square inches in a circle
five inches in diameter. Suppose a little man, just one foot
high, and a man six feet high, how much more cloth will it take
to clothe one than the other? How much more yarn to knit his
stockings?
3. Since π is a little more than three, the circle is a little more
than three quarters of the square. If, for example, the diameter of a
circle is six inches, the square that will just enclose it contains
six times six, or thirty-six square inches; and the circle contains a
little more than three quarters of this. Three quarters of thirty-six
is twenty-seven; and, adding a little more, would make it about
twenty-eight inches.
4. If you wish to be more exact, you must multiply the thirty-six
square inches by one quarter of twenty-two sevenths; that is, by eleven
fourteenths. Or, in other words, we must multiply thirty-six by eleven,
and divide by fourteen, which will give us about twenty-eight and one
half square inches for the size of a circle six inches in diameter.
5. And, to be very exact in finding the size of a circle, you must
multiply the diameter by itself, and then by the decimal ·7854, which
is one quarter of 3·1416.
6. Circles are larger or smaller in the same proportion as the squares
built on their diameters. And something like this is true of all sorts
of surfaces. Two similar surfaces are always in proportion to the
squares of similar lines in those surfaces. If we have two polygons of
the same shape, they are of a size proportioned to the squares on their
corresponding sides or diagonals. If a side in one is twice as long
as a corresponding side in the other, then one polygon is four times
the size of the other, because twice two are four. If one side were
three times as long as the corresponding side in the other polygon, one
polygon would be nine times as large as the other, because three times
three are nine.
7. I am so anxious that you should remember this, that I will tell it
to you again in other words. All similar surfaces are in proportion to
the squares of corresponding lines; so that we may find the proportion
between the surfaces by multiplying the number that expresses the
proportion between the lines by itself.
Suppose two dogs were of exactly the same shape, but that one was
twice as high as the other. Then its tail would be twice as long as
the other’s, its ears would be twice as long, its eyes twice as wide
apart; and whatever line you chose to measure in one, it would be twice
as long as the same line in the other. But its skin would be four times
as large, the surface of its eye would be four times as large, it would
take four times as much leather to make boots for it, or four times as
much lather to shave it; that is, whatever surface you measured on the
one dog, you would find it four times as large as the same surface on
the other.
[Illustration]
Suppose that we had a foot-ball ten inches in diameter, and a little
batting-ball two inches in diameter. The diameter of the foot-ball
would be five times as much as that of the batting-ball, and it would
take twenty-five times as much leather to cover it, because five times
five is twenty-five.
CHAPTER XXVI.
ABOUT CURVATURE.
[Illustration]
1. Suppose that our boy, wheeling his barrow over the light fallen
snow, went winding about the field, making a curved track, which curved
in some places more than in others. Let us suppose that he began as
though he were going to make a large circle, but kept turning shorter
and shorter, and ended when he was turning, as though he would make a
very little circle. Then we should say that his track had, at first,
a large radius of curvature, but at the end had a small radius of
curvature.
2. Let us suppose that the boy was tied, by a long rope, to the trunk
of a large tree; and that, as he went round and round the tree, the
rope wound up upon the tree-trunk, shorter and shorter, and drew the
boy nearer and nearer to the tree. Then the rope would be the radius of
curvature of the boy’s path.
* * * * *
What is the name of a curve that bends equally in every part?
How would you draw such a curve upon the blackboard? If I unwrap
a thread from a spool, holding the spool still, and keeping the
thread tight, what sort of a curve shall I draw? What relation
will the circumference of the spool have to this curve? What
shall we call the straight part of the thread which runs between
my hand and the spool? What does the radius of curvature measure?
To what circle is the radius of curvature a radius? How do we
imagine all curves drawn? What is the evolute of a circle? Let
the teacher provide the board and pins to show the illustration
of section nine.
[Illustration]
3. Hold a spool of thread still, on your slate, and let it be the trunk
of the tree. Then tie the end of your slate-pencil to the end of the
thread, and, by keeping the thread tight as you unwind it, you may draw
a track like that of the boy’s wheelbarrow. The thread that is unwound
will be the radius of curvature of this mark. The radius of curvature
will be very short where the pencil is close to the spool, and grow
longer as you unwrap the thread. It will be different for every point
in the curve; because you cannot move the pencil without either
winding, or else unwinding, the thread.
4. We call this thread the _radius_ of curvature, because it is to
the curve like a radius to the circle. We call it the radius of
_curvature_, because it shows us how much a curve curves or bends.
When the radius of curvature is short, the curve bends very much; and
when the radius of curvature is long, the curve bends less; and so the
radius of curvature measures the bending or curvature of the curve.
5. If we draw a circle, with its centre at the point where the thread
is just leaving the spool, that is, where the thread is tangent to the
spool, and make the radius of the circle just equal to the thread that
has been unwound, that is, equal to the radius of curvature, then that
circle will exactly fit the curve at the point which the slate-pencil
is then marking. So that the radius of curvature, at any point of a
curve, is the radius of the circle that will exactly fit the curve at
that point.
6. Every curve can be imagined as made in a similar way, by unwrapping
a string off from some other curve; and this other curve is called the
evolute of the first curve.
7. But the evolute of a circle is a point; because the string that
makes a circumference must neither wind up nor unwind.
8. The evolute of the boy’s track is the circumference of the trunk of
the tree; and the evolute of the pencil-mark is the circumference of
the spool.
9. You may drive a row of pins into a soft pine board, making the row
curved. Then tie one end of a thread to the foot of the last pin, and
the other end of the thread to a lead-pencil near its point. By keeping
the string stretched, and sweeping it round so as to wrap up and unwrap
upon the fence of pins, you may draw a curve whose evolute will be the
row of pins. This pencil-mark, you will easily see, is made of little
arcs of circles, whose centres are the pins, and the length of thread
from a pin to its little arc is the radius of curvature at that place.
CHAPTER XXVII.
ABOUT A WHEEL ROLLING.
1. When a wagon is going upon a straight and level road, look at the
head of a spike in the tire of one of the wheels, and you will see that
it moves in beautiful curves, making a row of arches that is called a
cycloid.
[Illustration]
* * * * *
Let the teacher take a tin cup, a ribbon-block, or something
of the kind, and roll it carefully along the bottom of the
blackboard, watching and marking with chalk the path of a spot
on the side. Then ask, What is the name of this curve? What is
the height of the arch, compared with the _cup_, with which I
drew it? What is the breadth of the arch at the bottom? What is
the length of the curve of the arch? What is the space enclosed
between the arch and the bottom of the board? (still comparing
with the cup.) Let the teacher inscribe a circle, of the size
of the cup, and ask, Are these horns larger or smaller than the
circle? Suppose that your hoop has a spot on one side of it, in
what curve will the spot move when the hoop is rolling straight
forward? How high will it go from the ground? (Diameter of hoop.)
How far apart will the places be where it comes to the ground?
How far will the spot travel in going from one place to the next?
(Four times diameter of hoop.)
Which of you can tell me what a cycloid is? If you draw a line at
right angles to a cycloid, where will it pass? (Through the point
where the circle making the cycloid touched the line on which
it rolled when making that place in the cycloid.) The radius of
curvature is at right angles to a curve—what part of the radius
of a cycloid is cut off by a straight line joining the feet of
the arch? (One half.) How long is the radius of the cycloid at
the top of the arch? How long at the bottom? What is the evolute
of a cycloid? Explain what you mean by this?
Suppose two wires going from the north-east corner of the ceiling
to the south-west corner of the floor, one wire straight, the
other a part of a cycloid, down which wire would anything slide
the more quickly? Suppose one wire went from the north-east
corner of the ceiling to the south-west corner of the ceiling,
hanging down in the form of a whole arch of a cycloid, how much
longer would it take anything to slide from the ceiling to the
lowest part of the wire, than it would take for it to slide from
a place one foot from the middle? (No longer.) The teacher should
endeavor to obtain (from the prudential committee) the board
described in section eight.
2. That is to say, a cycloid is the path of a point in the
circumference of a circle rolling on a straight line. You can draw part
of a cycloid by putting the point of your pencil into a little notch in
the edge of a spool, and tying it fast, so that the point of the pencil
shall be kept just at the edge of the spool; and then rolling the spool
carefully and slowly against the inside of the frame of the slate.
3. You will see, I think, that each arch in the cycloid must be just as
high from C to D as the diameter of the circle that makes it; and just
as wide at the bottom, from A to B, as the whole circumference of the
circle.
4. But you will have to study Geometry a good while, before you can
prove the other interesting things which I am going to tell you. You
can easily understand what I am going to tell you; but you cannot
understand how I know it, as you can what I told you in the last
section.
5. The length of the curve, A D B, in each arch of a cycloid, is just
four times the height of the arch; that is, four times the diameter of
the circle that made the cycloid.
[Illustration]
6. The whole space that is enclosed between the arch of the cycloid
and the straight line on which it stands, is just three times as large
as the circle that made the cycloid. So, when a circle is inscribed
between the arch and the line, the curious three-cornered figures on
each side of the circle are each exactly as large as the circle itself.
7. Now, if you have studied Arithmetic, you will understand that, if
a wheel is three feet in diameter, the head of a spike in the tire
travels just twelve feet from where it leaves the ground until it
touches the ground again. The spots where it touches the earth will be
nine feet and three sevenths of a foot apart. And the space between its
path and the ground will be three times eleven fourteenths of nine
square feet; that is, twenty-one square feet and three fourteenths of a
square foot.
CHAPTER XXVIII.
MORE ABOUT A ROLLING WHEEL.
1. The head of a spike in the tire of a rolling wheel is moving, at
each instant, at right angles to a line joining it to the bottom of the
wheel.
2. That is to say, if a straight line is drawn from the bottom of the
rolling wheel to the head of the spike, and if a tangent to the cycloid
is drawn through the head of the spike, this straight line will be at
right angles to this tangent.
[Illustration]
3. And this straight line is exactly half of the radius of curvature
of that point in the cycloid. So that, at the top of an arch of the
cycloid the radius of curvature will be twice the diameter of the
circle, and as you go down the arch the radius of curvature will be
shorter and shorter, until just at the foot of the arch the radius of
curvature will be of no length at all.
4. The evolute of a cycloid is a cycloid of exactly the same size. That
is to say, if we should fasten a string in the point between two arches
of a cycloid, just long enough to wrap on the curve up to the middle
of the arches, its end, as it wrapped and unwrapped, would move in a
cycloid exactly like that to which it was fastened.
[Illustration]
[Illustration]
5. If a cycloid be turned upside down, and we fancy the inside of it to
be very exceedingly slippery, then there are two curious things about
it.
If I want to slide anything from A down to B, there is no curve, nor
straight line, down which a thing would slide so quickly as down the
cycloid. If a hill was hollowed out in that shape, sleds would run down
it faster than they could down any other shaped hill of the same height
and the same breadth at the bottom.
6. The second curious thing about sliding on the inside of a cycloid
is, that it takes always exactly the same time to slide to the bottom,
however high up or low down you start. If A, in the last figure, is
the top of such a hill, and C the lowest point, it will take a sled
exactly as long to go from B to C, as to go from A to C. But this, you
must remember, is only when we imagine the hill and the runners of the
sleds to be, both of them, perfectly slippery; so that there shall be
no rubbing. In that case, if the road from A to C was two miles long,
it would only take a sled twenty-eight seconds to come down the whole
length. And, if it starts from any other place on the road, say from
B, it will still take twenty-eight seconds to get to C.
7. If the road from A to C is half a mile long, a sled will come down
in fourteen seconds.
8. If a board is sawed out in the form of a cycloid, and a little
gutter made on the inside of the curve, you can try this by holding two
marbles, say one at A and the other at D, and letting go of them at the
same instant. They will meet exactly at C, one coming the whole way A
C, while the other is coming the short distance D C.
CHAPTER XXIX.
WHEELS ROLLING ROUND A WHEEL.
1. When one circle rolls around another, instead of rolling on a
straight line, any point in the circumference of the rolling circle
travels in a curve called an epicycloid.[7] You can draw an epicycloid
by rolling carefully the spool (with a pencil tied to it) around some
round thing held still on your slate.
[7] Ĕpịsī’clöĭd.
[Illustration]
[Illustration]
2. Set a lamp on a table in one corner of the room, and, in the
farthest corner of the room, on a table of nearly the same height,
set a bright tin cup, or a glass tumbler, nearly full of milk. On the
surface of the milk you will see a bright curve shaped like the inner
line in this figure. It is an epicycloid; such as would be made by a
circle of one quarter the diameter of the cup rolling on a circle half
the size of the cup. You can make it by daylight, by setting the cup of
milk in the sunshine, early in the morning or late in the afternoon.
3. Epicycloids will be of different shapes, according to the proportion
which the two circles bear to each other. The smaller the rolling
circle is in proportion to the other, the more nearly will an arch of
the epicycloid be like an arch of the cycloid.
* * * * *
What is an epicycloid? How does it differ from a cycloid? Let
the teacher draw an epicycloid as directed in section one, and
teach the children to do so. Have any of you seen the cow’s
foot in a cup of milk? What is the geometrical name of this
curve? What must be the proportion between the circles to make
this epicycloid? What is the evolute of any epicycloid? When
the circles are of the same size, what will be the shape of the
epicycloid?
4. The evolute of an epicycloid is a smaller epicycloid of the same
shape; and the evolute of that evolute must be a still smaller
epicycloid. So that we may fancy epicycloids packed one within another
like pill-boxes.
5. The epicycloid of section second is sometimes called by children the
cow’s foot in a cup of milk. The figure in the margin represents this
epicycloid with its nest of evolutes packed one within the other. If a
string is fastened at the point where the arches of the epicycloid come
together, and is just long enough to wrap round to the middle of the
arch, then, as it unwraps, the end will move in a larger epicycloid of
exactly the same shape.
[Illustration]
[Illustration]
6. When the circles are of the same size, the epicycloid will have but
one arch. The ends of the arch will come together at the same point.
The figure in the margin will show the shape of this epicycloid and its
evolutes.
CHAPTER XXX.
OF A WHEEL ROLLING ON THE INSIDE OF A HOOP.
1. When a circle rolls on the inside of another circle, instead of on
the outside, the curve is called a hypocycloid.[8]
[8] Hĭpọsī’clöid.
[Illustration]
[Illustration]
2. Suppose your slate-pencil were straight, so that it would lie flat
on the slate, and make a mark as broad as the pencil is long. Then
suppose you were to put your pencil across one corner of your slate
like a hypotenuse, and slide first one end up to the corner, and then
the other, keeping both ends all the time touching the slate-frame.
You would make a white mark in the corner, of a curved three-cornered
shape, like this figure. The curve inside would be a hypocycloid.
* * * * *
What is a hypocycloid? Suppose I were to draw a hundred right
angles, putting the vertices of the right angles together, one
exactly on another, making the hypotenuses of equal length,
but having no two of them make the same angle with the legs,
to what kind of a hypocycloid would all these hypotenuses be
tangent?—that is, what is the proportion between the radii of the
two circles? Suppose a man draws the foot of a ladder away from
the side of a house, letting the ladder slip down the side of the
house, to what curve in the air will the ladder be all the time a
tangent? If the diameter of the rolling circle is one fifth that
of the other circle, how many arches will the hypocycloid have?
If one fourth? If one third? But what does the hypocycloid become
when the diameter of the rolling circle is one half that of the
other? How do the evolutes of a hypocycloid differ from those of
an epicycloid? In what respect are they like them? What must be
the size of two spools that they may make the same hypocycloid in
the same hoop?
3. If you take the corner of your slate for a centre, and the length
of your pencil for a radius, and draw a quarter of a circle, as I have
done in the last figure; if you then roll on the inside of this arc a
circle whose diameter is one half the length of the pencil, it will
make the same hypocycloid. I have also drawn this circle in the figure.
4. You can draw a hypocycloid by rolling the spool and the pencil on
the inside of any little hoop hold firmly on the slate. The rim of the
cover of a large wooden pill-box will make a nice little hoop for this
purpose.
5. The evolute of a hypocycloid is a larger hypocycloid of the
same shape on the outside of it; and the hypocycloid itself may be
fancied as the evolute of a smaller hypocycloid within it; so that
hypocycloids, like epicycloids, are packed one within the other, like
nests of tubs or boxes.
[Illustration]
6. The hypocycloid that is made when the diameter of the rolling circle
is one quarter of the diameter of the circle that it rolls in, can be
made by sliding the hypotenuse backwards and forwards on the legs of a
right triangle. The hypotenuse must be kept of the same length, and it
will always be a tangent to the hypocycloid.
[Illustration]
7. This hypocycloid may be called a hypocycloid of four arches;
because, as you may see in the figure, both it and its evolutes have
each four arches.
[Illustration]
8. If the diameter of the spool is nearly half that of the hoop, the
pencil will move across the hoop in a very flat curve, almost like a
diameter of the hoop; and the evolute at the ends of the curve will be
almost like two parallel straight lines at right angles to the end of
the diameter; so that the string unwrapping from the evolute will be
very long. When the diameter of the spool is exactly half that of the
hoop, the hypocycloid is a straight line; and the evolute of it, if you
can fancy that there is any evolute, is two parallel straight lines at
right angles to its ends.
9. If the diameter of the spool is more than half that of the hoop, it
will make a hypocycloid like that made by a smaller spool. If you have
two spools, one of them as much wider than the radius of the hoop as
the other is smaller, so that the hoop will just let the two spools
stand in it side by side, then one spool will make exactly the same
hypocycloid as the other.
[Illustration]
10. These two spools cannot, of course, be both rolling in the hoop
at the same time; but we can easily imagine two circles of the same
size as the spools rolling in a circle as large as the hoop. Start the
circles from the position in which I have drawn them to rolling in
opposite directions, and if you roll the little circle faster than the
large one, so as to make them get round the hoop in the same time, the
points in the two circles which are now touching will keep together all
the time, making the same hypocycloid.
CHAPTER XXXI.
ABOUT A HANGING CHAIN.
1. When a chain hangs from two points not directly under each other, it
makes a beautiful curve called a catenary.[9] You must remember that,
in order to have a perfect catenary, we must take a very fine chain,
and then take only the middle line in it.
[9] Căt’ẹnạry.
[Illustration]
[Illustration]
2. Suppose we had four straight sticks joined together by the ends,
so as to have a sort of chain of four links. Suppose the middle two
were equal in length, and also that the end ones were equal to each
other. Hang them by two pins on a level, as you see them in the figure,
and notice exactly in what shape they hang. Now turn them upside
down, keeping them in the same shape, as you see them in the next
figure. They will exactly balance and stand like the rafters of a
double-pitched roof. If you set the sides more nearly perpendicular,
the top will fall in; crowding the sides apart until the point of the
top gets lower than the top of the sides, and then pulling the sides
together again till they touch at the top, and the two top pieces
hang straight down in the middle. But if, on the other hand, you lean
the sides together more than they should be, they will fall together,
crowding the top up, until the ends of the sides meet, and the top
pieces stand straight up, or fall to one side together. The four
sticks, hinged together at the ends, will not stand, like an arch,
unless they make the same angles with each other as they did when they
were hanging like a chain.
[Illustration]
[Illustration]
* * * * *
What is the geometrical name of the curve made by a hanging
chain? If the vertices of a polygon were perfectly limber hinges,
but the sides stiff, how should we place them to make them stand
as an arch? Did you ever see a gambrel roof? Did you like the
looks of it? What shape do you think a gambrel roof should have
to look well? (That in which the rafters would hang if inverted.)
If I draw a catenary on the blackboard, and tell you how long the
radius is at the bottom, can you show me how to find the radius
at any other part of the chain? Can you show me how to find a
straight line equal to any part of the catenary?
[Illustration]
3. And if we had a chain made of a great many short, stiff pieces of
wood or metal, hinged together by rivets, like the little chain inside
a watch, we could make it stand up like an arch, if we could put it
exactly in the same form as it hung; that is, in a catenary upside
down. If we arch it up too steep and pointed, the sides will fall in;
if we arch it too flat, the top will fall in. But arch it exactly as it
hung, and it will stand.
[Illustration]
[Illustration]
4. If we fasten one end of a chain to a post, and hang the other end by
a thread from the top of a higher post, the weight of the chain will
pull the thread inward, as in this figure.
But suppose another thread, tied to the end of the chain, should pass
over a little wheel, on a level with the end of a chain, as at C,
having a piece of the same kind of chain hung to it at B. Then you can
easily see that the weight of B would pull A out flatter, and make the
thread hang more nearly straight down by the side of the post; or,
if B were long enough, and thus heavy enough, it would even draw the
thread outward toward C.
5. If the piece of chain marked B is just long enough to pull the end
of A exactly under the top of the higher post, so as to make the thread
hang exactly straight down, then B will be just as long as the radius
of curvature of the catenary A at its lowest point.
6. Let A, in the next figure, be the lowest point of any catenary, and
C any other point in it you please.
Draw a horizontal line, E D, making the distance A E equal to the
radius of curvature at A. Now draw C D at right angles to the catenary
at the point C, and C D will be exactly the same length as the radius
of curvature at C. Draw A F parallel to C D, and the straight line E F
will be just as long as the piece of chain A C.
[Illustration]
CHAPTER XXXII.
THE PATH OF A STONE IN THE AIR.
1. When a boy tosses up his ball in the air, the centre of the ball
moves in a curve called a parabola. If you toss up the ball on the west
side of the house when the sun is setting, the shadow against the side
of the house will also move in a parabola.
2. If you hold a round ball in such a position that its upper edge is
just as high above the table as the blaze of a lamp is, then the edge
of the shadow on the table will be a parabola. A dinner-plate will also
make a parabola in the same manner.
[Illustration]
* * * * *
What is the geometrical name of the curve in which a ball moves
when tossed in the air? Which would make a more perfect parabola,
a ball of lead or a ball of cork? (Of lead, because least impeded
by the air.) How must I hold a plate so that the edge of its
shadow shall be a parabola? How can I draw a parabola with a
straight edge and square? What is the vertex of a parabola?
How near the vertex does the directrix pass? In what direction
does the directrix of a parabola lie? (Parallel to tangent at
the vertex.) How can you describe a parabola with reference to
its focus and directrix? Let the teacher copy the figure, and,
drawing a tangent at the point M, ask, How does this tangent
divide the angle A M P? How can you tell the length of the radius
of curvature at any point of a parabola?
3. But remember that, to be an exact parabola, the ball must be
perfectly round, as no ball can really be; the table perfectly flat, as
no table can really be; the blaze of the lamp a single bright point,
as no blaze of a lamp can be. It is easy to imagine exact figures, but
they can never be made. No line can be drawn so fine and true that a
microscope would not find a breadth to it, or waving irregularities in
it.
4. The parabola is a very useful curve; but it would be difficult to
explain to children how it is used. I shall tell you of one use, before
the end of this book.
[Illustration]
5. On a smooth board draw a straight line, such as B C. Near the
middle of the line, as at A, drive a small pin. Put one edge of the
square card against the pin, and one corner on the line B C, and draw
a pencil-line along the edge of the card, beginning at the corner on
B C, and going at right angles to the edge that is against the pin A.
Do this with the card in a great many different positions, only keeping
the edge against the pin and the corner on B C, and you will make a
place on the board nearly black with pencil-marks, with a curved edge
on the inside, around A, and the curve is a parabola.
6. The point A, in the last figure, is called the focus of the
parabola. The point in the parabola nearest the focus is called the
vertex of the parabola. The line B C is a tangent at the vertex.
[Illustration]
7. Let C M be a parabola, and let A be its focus. Draw D E parallel to
the tangent at the vertex, and as far from the vertex as the vertex is
from the focus. This line D E is called the directrix of the parabola.
8. Any point in the parabola is just as far from the focus as from the
directrix. That is to say, that, if we take any point, as M, and draw
a line M P at right angles to the directrix, and also a line M A, the
two lines M P and N A will be of exactly equal length.
9. A parabola may, therefore, be described as a curve, every part of
which is equally distant from a point called the focus, and from a
straight line called the directrix.
10. The parabola at the point M makes exactly the same angle with the
line M P that it does with the line M A.
11. If H C is the radius of curvature at the point C, and C F is in the
same straight line with H C, then C F is just half as long as H C. That
is to say, that a straight line drawn at right angles to any point in
a parabola, and ending in the directrix, is just half as long as the
radius of curvature at that point.
CHAPTER XXXIII.
THE SHADOW OF A BALL.
[Illustration]
1. Any curve that runs round into itself again encloses an oval. The
word oval really means egg-shaped; but, in geometry, we use it for any
figure bounded by one curve line, without any sharp corner.
* * * * *
What is an oval? What is an ellipse? How can you draw an ellipse
with a string and two pins? What is the name of the points
where the pins are? What angles do the two parts of the string
make with that part of the ellipse where your pencil is? (Equal
angles.) What other curve is the end of a very long ellipse
like? When the two foci of an ellipse are brought near together,
what curve does the ellipse become like? Can you explain how a
carpenter draws an ellipse by a “trammel and slots”? Did you ever
notice an elbow in a stove-pipe? What is the shape of the seam
around the elbow of the stove-pipe?
2. The shadow of a round ball falling on a flat surface, when all the
shadow can be seen, is either a circle or a particular kind of oval
called an ellipse. The shadow of a round plate is also an ellipse,
whenever the whole shadow can be seen on one flat surface.
[Illustration]
3. You can draw an ellipse by driving two pins into a board, as at A
and B in the figure, and tying a string, as A M B, one end to each pin,
then putting a pencil-point, as at M, inside the string, and stretching
it out, and moving it round.
4. The points where the pins are placed are called the foci of the
ellipse. Lines to the foci from any point in the ellipse, as at M, make
equal angles with the tangent at that point.
5. The nearer the foci are together, using the same string, A M B, the
more nearly a circle does the ellipse become; so that if the foci came
together, the ellipse would become a circle.
6. The further apart the foci are the longer and narrower is the
ellipse. When an ellipse is very long, and very narrow in proportion to
its length, each end of the ellipse becomes very much like a parabola.
7. When an ellipse is very exceedingly long, the ends are so much like
a parabola that even geometers call them parabolas. We call the path of
a ball tossed in the air a parabola, although in reality it is one end
of a very long ellipse, nearly four thousand miles long, with one focus
at the centre of the earth. But a real parabola is an ellipse so long
that it has no other end at all; it only has one end and one focus.
8. The moon goes round the earth in an ellipse; the earth goes round
the sun in an ellipse. And, if you were to cut the earth, or sun, or
moon, in two, with a straight cut, the cut surface would be either an
ellipse or a circle, according to the direction in which you cut it.
If the earth is cut in two from east to west, the section is a circle;
if in any other direction, an ellipse; for the earth is not perfectly
round.
[Illustration]
9. Carpenters sometimes draw ellipses by means of a board with two
narrow slits in it at right angles to each other. They have a ruler
with two pins in it, as at A and B, and a pencil in the end, as at C;
and, by moving one pin in one slit, and the other pin in the other
slit, the pencil C will move in an ellipse.
10. If you cut a round stick off slanting with a sharp knife, at one
cut, the cut end will be an ellipse.
CHAPTER XXXIV.
THE SHADOW OF A REEL.
1. If a reel for winding thread, such as is represented in the figure,
be held steadily in such a position that its shadow from a lamp will
fall on a flat wall, and then set to revolving, the sides of the shadow
will be curved, and the curve is called an hyperbola.
[Illustration]
[Illustration]
* * * * *
Did you ever see a “swift” for winding yarn? Did you ever see
it standing steady on a table, and spinning round very fast?
Did you notice that the sides looked curved? What curve was it?
Suppose a row of palings set in a straight line, the middle one
the shortest, and the others just long enough to reach the top of
the middle one, if they were leaned against it with their bottoms
standing where they now do; what curve would the top of such a
row of palings make? (Section two.) In what curve does a movable
pulley move, when the fixed pulley is not on a level with the
fixed end of the rope? How do a parabola and an hyperbola compare
with each other? How many foci has a parabola? An hyperbola? Can
you tell how to make an hyperbola by a chain and threads? Can you
describe how the shadow of a ball, or plate, may be made to grow
from a circle into an hyperbola? What other two curves does it
become before becoming an hyperbola?
[Illustration]
[Illustration]
2. Suppose we have strings tied, at various places, on a horizontal
wire, A B, and all drawn straight through one point, C. Cut them all
off on a line parallel to A B. Let the strings, after being thus
trimmed, hang straight down, and the ends will hang in a curve, as
shown in this figure, and that curve will be an hyperbola. This will
also be true if the strings are cut off exactly at the point C.
[Illustration]
3. If a rope is tied to a fixed point, say the hook A, and passes over
a fixed pulley, as B, not on a level with A, then a weight on a movable
pulley, C, will move, as you raise it by pulling the rope over B, in an
hyperbola.
[Illustration]
4. If we draw a circle round a centre, B, marking, also, some point
outside the circle, as at A, and then make a dot at every point which
we can find situated, like M, as far from the point A as from the
circumference of the circle round B, these dots will all be in an
hyperbola.
5. You see, then, that an hyperbola can be fancied as a parabola with a
circumference, instead of a straight line, for a directrix.
6. In the last figure the points A and B are called the foci of the
hyperbola. The curve is, at each point, as far from one focus as from
the directrix circle drawn round the other. That is, M P is of the same
length as M A.
[Illustration]
[Illustration]
7. If we hang light threads to each link of a hanging chain, such as Q
N, and cut their lower ends off on a level line, and then stretch the
chain out perfectly level, the lower ends of the thread will arch up
into an hyperbola.
8. If a round ball hangs exactly under a lamp, over a level table, its
shadow on the table will be a circle. But if the ball is moved to one
side, the shadow becomes an ellipse. Now raise the ball slowly, and
the shadow will begin to move away from the lamp. But one edge will
move away much faster than the other, so that the ellipse will grow
longer and longer. And if we imagine the table to be so large that we
cannot see the edges of it, then, when the upper edge of the ball is
just on a level with the lamp, the ellipse will be so long that it will
have no other end, and the end nearest the lamp will be a parabola. If
we raise the ball higher, the parabola becomes an hyperbola. And when
the ball is raised so high that its under side is as high as the lamp,
the shadow will not touch the table at all. The parabola and hyperbola
are made by the shadow of the lower side of the ball.
9. If you take a plate instead of a ball, you can make all the shadows,
circle, ellipse, parabola, and hyperbola, by a little pains-taking to
hold the plate at the proper angle with the table for the circle and
ellipse. For the parabola and hyperbola less care is required; only
that for a parabola the upper edge of the plate must be just as high as
the light; and for an hyperbola the plate must be higher. The shadow
of the lower edge of the plate makes the parabola or hyperbola.
CHAPTER XXXV.
THE COW’S FOOT IN A CUP OF MILK.
1. I have already told you how to make the bright curve called by
children the cow’s foot in a cup of milk. I have also told you how to
draw a parabola by drawing lines tangent to it until all the paper
outside the parabola is blackened by pencil-marks. I have also told you
how to draw a hypocycloid, by putting a short ruler across one corner
of your slate, keeping one end against the frame at the bottom, and
the other end against the frame at the side, and drawing pencil-marks
the whole length of the ruler on the side next the corner. The corner
will become, by making the marks at a great many different angles with
the frame, whitened with pencil-marks, all tangent to an arch of a
hypocycloid.
These curves are, in one respect, alike; they are made by drawing
tangents to them. For the cow’s foot is made by bright straight lines
of reflected light, all tangent to an epicycloid.
* * * * *
How are curves drawn by drawing only straight lines? What is the
geometrical name for all curves made by reflected light? What is
the caustic made by a circle? What is the child’s name for it?
How could we arrange a table and make caustics with the curves?
What is there peculiar about the caustic of a parabola? Of a
cycloid? Do you remember what parallel lines are? Concentric
curves? What is the caustic of a cycloid turned endwise to
the light? What caustic is always formed when the light falls
perpendicular to any part of a curve? When the light falls
parallel to it? Let the teacher copy the figures, and go over all
the peculiar points of section nine. What would be the caustic
formed by a lamp in the centre of a circle? In the focus of an
ellipse? In the focus of a parabola? Did you ever notice the
reflector in front of a locomotive engine? What is it for?
2. And whenever light is reflected from the inside of a polished curve,
the reflected light makes a bright curve of some kind, just as the
light reflected from the inside of a circle makes an epicycloid.
3. Curves made by reflected light are called caustics. The cow’s foot
in a cup is a caustic made by a circle. The caustic made by a circle is
an epicycloid.
4. Suppose you had a table so arranged that the setting sun should
shine over its surface. If on this table we should set narrow strips of
tin on their edges, they would reflect the sun-light and make bright
curves or caustics on the table.
5. If the tin were bent into a half of a circle, the caustic made by it
would be, as you already know, an epicycloid such as would be made by
one circle rolling on another of twice its diameter.
6. If another strip were bent into the form of a parabola, and turned
in such a direction that the sun-light fell at right angles to the
directrix of the parabola, then the caustic, instead of being a curve,
would be a single bright point at the focus of the parabola.
7. If you bend another strip into an arch of a cycloid, and turn the
straight line which joins the ends of the arch at right angles to the
sun-light, the caustic will be two arches of a cycloid of just half the
size, as shown in the figure.
[Illustration]
[Illustration]
8. If the cycloid be turned round at right angles to its last position,
so that the straight line joining the ends of the arch shall be
parallel to the sun-light, the caustic will not be a cycloid, but will
be curve concentric with a cycloid of half the size. In the figure, A B
represents the strip of tin in the form of a cycloid; C is the point of
the caustic, and D is one arch of the half-size cycloid with which the
caustic is concentric.
[Illustration]
9. Whatever curved form the tin may be bent into, the caustic will have
some curious properties, which I will now tell you.
Wherever the light falls at right angles to the curved tin, as at M,
the caustic will be at the middle of the radius of curvature, and the
radius of curvature will be tangent to the caustic, as the radius M D
is tangent to the caustic C, just half way from M to D. And the radius
of curvature of the caustic at this place, C, will be just one quarter
of the radius of curvature of the evolute of the tin at M. If E is the
radius of curvature of the evolute, then R, the radius of curvature of
the caustic, will be parallel to E, and be one quarter as long as E.
And both R and E will be at right angles to the radius M D.
[Illustration]
But wherever the sun-light falls as a tangent to the inside of the
curved strip of tin, the caustic will also be a curve tangent to the
inside of the tin, and its radius of curvature will be exactly three
quarters the radius of curvature of the tin curve. Thus, if M N is
a curve on the inside of which the sun-light falls parallel to the
curve at M, then M O will be the caustic, and its radius, M Q, will be
exactly three quarters of M P, the radius of M N.
10. All through this chapter on caustics I have spoken only of those
that are made when the light is at a great distance from the polished
tin. If we bring a lamp near the polished curve, the caustics made by
this lamp-light will be very different.
11. A lamp placed in the centre of a circle would not make a curved
caustic, but all the light would be thrown back to one point, in the
centre, where the lamp itself stood.
12. A lamp placed in focus of an ellipse would not make a curved
caustic, but all the light would be thrown to one point, the other
focus of the ellipse.
13. A lamp placed in the focus of a parabola would not make a curved
caustic; but all the light would be thrown straight out in parallel
lines perpendicular to the directrix of the parabola. For this reason
men have polished reflectors in the shape of a paraboloid to place
behind lamps when they want to throw out the light in one direction.
They use such mirrors in light-houses, to throw the light out over the
ocean; and they use them in front of locomotive engines, to throw light
straight forward on the track by night.
CHAPTER XXXVI.
SOLID GEOMETRY.
1. All the figures which I have told you about are such as could be
drawn on a flat sheet of paper. Before I finish my book I will tell you
a little about a few solid bodies.
[Illustration]
2. Cut out of a stiff piece of paper six equal squares, as I have drawn
them here; and then fold the paper, at each line where the squares join
each other, to a right angle. You will thus make the six equal squares
shut up a space like a box.
3. This solid figure, bounded by six equal squares, is called a cube.
* * * * *
What is the name of a solid bounded by six equal squares? What is
the difference between a yard of tape, a yard of oil-cloth, and
a yard of earth?—I mean between the three meanings of the word
yard in those phrases? Do you know how long a yard is? Did you
ever see a man two yards high? Build an earthen pyramid as tall
as such a man’s head, and how much higher would the great pyramid
of Egypt be? How much larger? What is the highest hill in this
neighborhood? How high is it? How many such hills set one on top
the other would it take to be as high as the White Hills? Suppose
Mount Washington were of the same shape as ______ hill, how many
such hills would it take to build Mount Washington? Suppose
Gulliver to be twelve times as high as the Lilliputs, but shaped
just like them, how many times larger would his nose be than
theirs? How many times larger would his thumb-nail be? How many
times larger would his finger be than theirs?
4. A cube is taken as the measure of all solids and fluids. A gallon,
for instance, is two hundred and thirty-one cubic inches; that is to
say, by a gallon of water we mean water enough to fill two hundred and
thirty-one little cubes, whose faces are square inches. Or, for another
instance, a cord of wood is one hundred and twenty-eight cubic feet;
that is, wood enough to make a pile as large as one hundred and twenty
boxes, whose sides are each a square foot.
5. You remember that the measure of a square is found by multiplying
the length of a side by itself. The measure of a cube is found by
multiplying the length of a side twice by itself. If I build a cube
with a side of three inches, one face will have nine square inches in
it, and the cube will be made up of three layers, each with nine cubic
inches in it. So that the cube of three inches is three times three
times three inches; that is, twenty-seven inches. A cube of four inches
would have sixteen square inches on a side, and consist of four layers
of sixteen cubic inches each; that is, the cube of four inches is
sixty-four inches.
6. Similar surfaces, you remember, are in proportion to the cubes on
corresponding lines. Similar solids are in proportion to the cubes on
corresponding lines.
7. And as you can find the proportion between two similar surfaces
by multiplying the number that expresses the proportion between the
lines by itself, so you can find the proportion between the solids by
multiplying the number that expresses the proportion between the lines
twice by itself.
8. Let us suppose, for instance, a heap of earth six feet high, built
in the shape of the great pyramid in Egypt, which is six hundred feet
high. The pyramid would be one hundred times as high as the heap of
earth; and, being of the same shape, would also be one hundred times as
wide at the bottom. But the pyramid would cover a hundred hundred, that
is, ten thousand times as much land as the heap of earth; and it would
be a hundred times ten thousand, that is to say, one million times, as
large as the heap of earth.
9. The highest mountains in the United States, east of the Mississippi
river, are about ten times as high as the great pyramid; and if one of
them, one of the White Hills, for example, were cut into the same shape
as the great pyramid, it would cover one hundred times as much land,
and have one thousand times as much stone in it, as one of the pyramids.
10. The highest mountains of Thibet are nearly five times as high as
the White Hills of New Hampshire. If one of the highest mountains of
Thibet were of the same shape as one of the White Hills, it would
have twenty-five times as much land on its sides, and would cover
twenty-five times as much space. And it would take one hundred and
twenty-five mountains like the White Hills to make one of the Himalaya
mountains.
11. I wish you to remember very carefully that when two things are of
the same shape, all corresponding lines are in the same proportion to
each other; all corresponding surfaces are in proportion to the squares
on those lines; and all corresponding solids in proportion to the cubes
on those lines.
12. I wish you also to remember that, if the number expressing the
proportion between the lines of two similar solids be multiplied
by itself, the product will express the proportion between the
corresponding surfaces; and, if it be again multiplied by itself,
the product will express the proportion between the corresponding
solidities. If the diameter of a foot-ball be five times that of a
batting-ball, the surface of the foot-ball will be twenty-five times as
much, and the size of the foot-ball will be one hundred and twenty-five
times as much, as that of the other ball.
CHAPTER XXXVII.
CONIC SECTIONS.
1. You will very often hear or read about conic sections; men began to
study them more than two thousand years ago, and have not yet learned
all the useful things that can be known about them. By conic sections,
we mean the circle, the ellipse, the parabola, and the hyperbola. I
have told you a little about these curves; enough, I hope, to make you
want to learn more; and I will, in this chapter, tell you why they are
called conic sections.
2. Cut out of pasteboard a figure bounded by an arc and two radii, such
as A B C. Curve it up equally, and join the edge A C to the edge B C.
It will then enclose (on all sides but one) a space; and the figure
thus formed is called a right cone.
[Illustration]
[Illustration]
3. Set a right cone up upon a plane, and the arc A B will become a
circle, such as D E in the figure. If we fancy a post standing straight
up in the centre of a circle, and a longer straight pole tied by one
end to the top of the post, while the other end just reaches the
circumference of the circle; if we then fancy this lower end carried
around the circumference, the pole will mark out in the air the surface
of a right cone.
* * * * *
If a right triangle could spin round on one leg, the other leg
will describe a circle, and the hypotenuse will go round a solid
body in the air; now what is its geometrical name? How shall
I cut a right cone so as to make the cut surface a circle? An
ellipse? A parabola? An hyperbola? How can you make part of a
cone in the air with a lamp and ball?
How can you show sections of such a cone? What is a right cone?
4. Cut a right cone in two by a plane parallel to the plane on which it
sits, and the cut surface will be a circle.
5. Cut a right cone in two by a plane inclined to the plane on which it
stands, and the cut surface will be an ellipse.
“Section” means a cut surface, and “conic” means belonging to a cone;
so that you can now understand why these curves are called “conic
sections”; it is because they can be made by cutting a cone.
[Illustration]
[Illustration]
[Illustration]
6. Cut the cone by a plane parallel to one side of the cone, and the
cut surface will be a parabola.
7. Cut the cone by a plane making a smaller angle with the centre-post
than the sides do, and the cut surface will be an hyperbola.
8. Now, if you will turn back and read chapter XXXIV., section eight,
again carefully, you will see that the shadow of a ball is part of a
cone in the air, with the vertex or point of the cone in the blaze of
the lamp, and that the flat table is a plane that makes conic sections
of the shadow.
[Illustration]
9. If you were to bend the pasteboard cone, so as to make D E some
other shape than a circle, the cone would no longer be a right cone.
A right cone has a circle for its base, and the vertex of the cone is
directly over the centre of the base.
CHAPTER XXXVIII.
THE SPHERE.
1. If a circle should spin round on one of its diameters, the
circumference would enclose a space called a sphere.
* * * * *
What is the geometrical name for a perfectly round solid? What
do we call a solid body that has the form of the geometrical
solid? What examples of balls or globes can you give me? How
must a circle move to have it describe a sphere? What shape is
the section of a sphere by a plane? What is a great circle on a
sphere? How large is the surface of a sphere?
How large is the solidity of a sphere? What is the largest body
of all having the same surface? Into what shape must I pack
anything to make it expose least surface? If you have studied
any arithmetic, you may now tell me how to calculate roughly the
circumference, the surface of a circle, the surface of a sphere,
and the solidity of a sphere, when you know the diameter. How
shall we calculate the same more exactly? How still more exactly?
Is there anymore Geometry to be learned than what is taught in
this book? Are there any new things yet to be discovered in
Geometry? Should you like to learn more about it when you grow
older?
2. Solid bodies in the shape of a sphere are called balls or globes.
The marbles with which boys play, are usually very perfect globes.
A soap-bubble blown thin, and free from any hanging drop of suds,
floating in still air, is a very perfect sphere.
[Illustration]
[Illustration]
3. Cut a sphere by any plane, and the cut surface will be a circle.
4. When the plane goes directly through the centre of the sphere,
the circle thus made is called a great circle of the sphere. All
great circles in a sphere are of the same size as the circle which we
imagined spinning to create the sphere.
5. The diameter of the great circle is called the diameter of the
sphere, and the radius of the great circle is called the radius of the
sphere.
6. The surface of the sphere is exactly four times the surface of
a great circle. A ball three inches in diameter would take as much
leather to cover it as would make four circles, each three inches in
diameter; or one circle six inches in diameter.
7. You remember that the measure of a circle is found by multiplying
the square of its diameter by one quarter of π. As the surface of the
sphere is four times as great, it is found by multiplying the square
of its diameter by π itself. The surface of a ball three inches in
diameter, for instance, will be nine square inches multiplied by π.
8. The solid measure of a sphere is found by multiplying the cube of
the diameter by one sixth of π. The cube of the diameter will just
enclose the sphere, and each of the six sides of the cube will be a
tangent plane to the sphere.
9. For roughly judging of circles and spheres we call π about three.
That is to say, a circumference is a little more than three times the
diameter; a circle is a little more than three fourths of the square on
the diameter; and a sphere is a little more than half the cube on the
diameter.
10. To be more exact, we call π twenty-two sevenths. That is to say,
a circumference is twenty-two sevenths of the diameter; a circle,
eleven fourteenths of the square on the diameter; and a sphere, eleven
twenty-firsts of the cube on the diameter.
11. To be still more exact, we call π, 3·1416. That is to say, to find
the circumference, multiply a diameter by 3·1416; to find the size of
the circle, multiply the diameter by itself, and then by ·7854; and to
find the contents of a sphere, multiply the diameter twice by itself,
and then by ·5236.
12. As the circle is the largest of isoperimetrical figures, so the
sphere is the largest of all bodies having the same amount of surface.
If you roll a piece of putty into a round ball, it will have less
surface than it could have in any other form.
13. But I think I have made my book long enough. I hope you have liked
it, and I hope that at some time you will study more Geometry, and
learn how to prove the truth of all I have told you. You will then find
that there is a great deal to be learned about what men already know of
Geometry, and that there is a great deal that is not known, at least
by any man. Of course, the great Creator, who has made all things in
number, weight and measure, knows everything. And the more we know, the
more clearly we shall see how great is His knowledge, how wonderful his
wisdom, and how beautiful the manner in which he has used what we call
Geometry in the forms he has given to all things on the earth or in the
sky.
THE END.
PRACTICAL QUESTIONS AND PROBLEMS FOR REVIEW BY THE OLDER SCHOLARS.
The chapters referred to may not always furnish a direct answer or
solution; but they will always suggest the true solution, which is not
always to be reasoned out, but is to be seen by the mind’s eye. Similar
questions can be multiplied indefinitely by a skilful teacher.
CHAPTER III.—What is the best way of making a garden-path straight?
How does a carpenter mark a long straight line? How will you make a
short straight line on paper? The railroad from my house to Boston is
about nine miles long, the carriage-road about eight; which is more
nearly straight? In travelling the carriage-road to Boston I cross the
Fitchburg railroad only once. I am on the north side of it at starting;
when I get into Boston, on which side of me must I look for the dépôt?
But I cross the Worcester railroad twice; on which side of me shall
I look for the dépôt of that road? If Boston lies exactly east of my
house, how can I manage to drive my horse there without having his head
once turned exactly to the east? Can I do it without having either head
or tail turned to the east?
CHAPTER VI.—Suppose a straight stick is made to turn upon a pin thrust
through the middle of it, which end will move the faster? (Neither.)
Which end will alter its direction most rapidly? If the pin is thrust
into a straight line that will not move, such as a crack in the floor,
which end of the stick will make the larger angle with the crack?
CHAPTER V.—If I hang two plumb-lines from the ceiling, from nails that
are just one foot apart, how far apart will the lines be six feet below
the ceiling?
CHAPTER VI.—In a triangle whose sides are three, four and five inches,
which is the largest and which the smallest angle? If a leaning pole
makes an angle equal to one third of a right angle, with a plumb-line,
what angle does it make with a level line passing through the foot of
the pole and under the bob hung from its summit? What angle will it
make with any other level line passing through its foot? What angle
will the pole make with a level line passing through its foot, at right
angles to one passing from the foot under the bob?
CHAPTERS VII., VIII., IX.—Suppose that I set a stake seven feet high
in a level piece of ground, and measure its shadow and the shadow
of other things on level ground as follows: When the shadow of the
stake was ten feet, that of the house was thirty feet; when that of
the stake was nine feet, that of my poplar-tree was forty-one; when
that of the stake was eight, that of my cherry-tree was twenty-six;
when that of the stake was six, that of the church-steeple was one
hundred and twenty-one. What is the height of the steeple, poplar-tree,
cherry-tree, and house?
CHAPTER X.—I have one post and two rails to make a fence to keep the
cattle from a young tree that has sprung up by the fence in my pasture;
how shall I make the largest pen for it? Another tree stands in the
middle of the field; my only materials, for making a defence around it,
are three posts, one rail, and a piece of rope longer than the rail. In
what is the largest triangle I can make?
CHAPTER XI.—How large is each angle in an isosceles right triangle?
How many such triangles will it take to make a square? What does a
carpenter mean by a mitre-joint? Do you know how a carpenter makes a
mitre-joint?
CHAPTER XII.—If, in a field with four straight sides, we find all the
sides of the same length, what may we know about the angles? If, in
such a field, we find the sides all equal, and two of the adjacent
angles equal, how large is each angle in the field? and what do you
call the shape of the field? I have seen braces that were of no use.
What is the proper way to make them? There is another use of three
points not exactly like this. Why is a three-footed table sure to stand
steady?
CHAPTERS XIII. and XIV.—How many yards of painting on the side of a
house forty-two feet long and twenty-one feet high? How many square
inches in a pane of seven by nine? How many in a pane of eight by ten?
How many feet of land in a lot with two sides of eighty feet each and
two of thirty-nine feet each, if the angles are such that the eighty
feet sides are only thirty-five feet apart? How do you know that this
lot is a parallelogram?
CHAPTER XV.—How many feet of land in a triangle whose sides are thirty,
forty and fifty feet? How many in a triangle whose sides are twelve,
five and thirteen feet? How do you know that these triangles are right
triangles? Suppose that a quadrangle has sides of three, four, twelve
and thirteen feet, and that the sides of three and four feet join in a
square corner, how many feet of land does it include? How do you know
that this quadrangle can be divided into two right triangles?
CHAPTERS XVI. to XXI.—If I have a piece of the felloes of a wheel and
want to find out how large the whole wheel is, what shall I do? How
would you lay out a garden-path in a circle? How would you make two
paths at right angles to each other? How will you make a circle on
the blackboard? How will you make two paths run at an angle of sixty
degrees with each other? If a steamboat’s tiller is lashed fast in
any position, in what curve will the boat run? How will you make the
circle larger? If there is a circle drawn on the blackboard, how can
I draw a tangent to it at any particular spot in the circumference?
There are four ways, one from XX. 7, one from XX. 8, one from XXI.
5, and one from XIX. 7. These ways have their special advantages and
disadvantages. Point them out.
CHAPTERS XXII. and XXIII.—How shall I find a point at an equal distance
from three given points? How can you find the place that is equally
distant from three of the corners of this room? How can you find a spot
equally distant from the east side, the south side, and the diagonal of
the room that runs south-west? What three kinds of polygons _with equal
sides and equal angles_ can be laid together like bricks in a pavement,
and fill up all the space?
CHAPTERS XXIV. and XXV.—Suppose the earth to be eight thousand miles
in diameter, what is its circumference? What is the length of an
arc of seventy degrees in a circle of one foot radius? Which weighs
most, a square sheet of tin eleven inches on a side, or a round piece
of the same thickness thirteen inches in diameter? If a sheet of tin
four inches square weighs an ounce, what will a sheet a foot square
weigh? What will a circle ten inches in diameter weigh? What is the
difference between a piece of land four rods square, and a piece of
four square rods? What is the difference between a foot square and a
square foot? If a church-spire is one hundred and twenty feet high, how
much more paint will it take to paint the church than to paint a model
of it, with a spire twelve inches high? Is a square foot of sheet-lead
necessarily in a square form? What proportion in the cloth required to
clothe a man five feet high, a man five feet ten inches, and a man six
feet, supposing the three men to be of the same form, and dressed in
the same fashion?
CHAPTERS XXVI., XXVII., XXVIII.—How far does the head of a spike in the
tire of a wheel four feet in diameter travel while the wagon goes four
miles on a level road? What is the radius of curvature of its path when
it is at the top of the wheel? When it is two feet from the ground?
When it is one foot from the ground? A pendulum-bob swings in an arc of
a circle; how, from XXVIII. 4, can you devise a plan to make it swing
in the arc of a cycloid? What is the shortest path from one point to
another? Is the shortest path always quickest? When is it not, for whom
or what is it not, and why not?
CHAPTERS XXIX. and XXX.—In a machine called a photometer, a bead is
placed on the rim of a wheel rolling inside of a hoop of just double
the diameter; in what path does the bead move? In a railroad curve the
cars cannot turn if the radius is too small; what objection to joining
two straight tracks which are at right angles to each other by a curved
track marked out by drawing many lines of equal length across the
corner?
CHAPTER XXXI.—If a rope weighs one pound for each yard, and I tie one
end of it to a staple in the wall, how hard must I pull _horizontally_
in order to make the radius of curvature ten feet at the lowest point
of the rope? How hard to make the radius of curvature twenty-one
feet? One hundred and eight feet? What is the radius of curvature of
a straight line? How hard must I pull horizontally to make the rope
straight? If a chain, weighing two pounds to the yard, hangs between
two posts of the same height, and the curvature of the chain in the
middle has a radius of five feet, with what force does it draw in each
post? What if the radius is thirty feet? If a piece of thread weighs
at the rate of an ounce to a thousand feet, what horizontal force is
required to make the radius of curvature a mile long? But what to draw
the thread straight? In answering any of these questions on chapter
XXXI., does it make any difference how long, or how short the rope,
chain or thread, is?
CHAPTERS XXXII., XXXIII., XXXIV.—What is the path of a rifle-ball in
the air? Can it then ever go straight to its mark? Can you fancy the
shape of the evolute of an ellipse? If a hypocycloid of four arches
be used as an evolute, but the string is taken on two opposite sides
long enough to wrap round the whole arch, and on the other sides of
no length, what sort of a curve would it produce? (An oval, but not an
ellipse.)
CHAPTER XXXV.—What shape must a mirror be to act as a burning mirror
by bringing the sun-light to a point? If a piece of a hollow sphere is
used, at what distance from it will the imperfect point of light be
formed? If you stand in the centre of a field bounded by a circular
fence, where will the echo of your voice sound loudest? If the walls of
a room are in the form of an ellipse, and a man stands in one focus and
speaks, where will the echo sound loudest? If a paraboloid reflector
were placed behind the whistle of a locomotive, what effect would it
have on the sound?
CHAPTER XXXVI.—A man is said to have borrowed a heap of peat-mud,
which was stacked in a cubical form, four feet on a side, and to have
returned two heaps, each a cube of three feet on a side. Did he make a
just return? What is the proportion between the length of a hogshead
holding one hundred and twenty-five gallons, and a keg holding one
gallon, if they are of the same shape? If the smallest of the three men
mentioned on page 141, weighs one hundred and fifty pounds, what do the
others weigh? If a man five and a half feet high weighs one hundred
and sixty pounds, and a man three inches taller weighs one hundred and
eighty, which is stouter in proportion to their height?
CHAPTER XXXVII.—Suppose a pole fastened at the end of a horizontal
revolving arm. If the pole lies horizontal, it keeps in a horizontal
plane; if it is vertical, it describes the surface of a vertical
cylinder. But if it inclines towards the centre-post about which the
arm revolves? If it inclines, but not directly toward the centre-post?
Cut such a hyperboloid surface by a horizontal plane, and what will the
section be? Cut it by a vertical plane, what will the section be? What
changes as you move the vertical plane to and from the centre of the
figure?
CHAPTER XXXVIII.—How many cubic feet of gas will fill a round balloon
seven yards in diameter? How many yards of silk three quarters of a
yard wide will it take to make such a balloon? If the earth were eight
thousand miles in diameter, and a perfect sphere, what would be the
number of square miles of its surface? Of solid miles in its contents?
To how many balls thirteen inches in diameter would it be equivalent?
How many tons would it weigh, if it were all water, one thousand ounces
to a cubic foot? How much if three and one half times that weight?
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