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Title: The whys and wherefores of navigation
Author: Gershom Bradford
Release date: May 19, 2024 [eBook #73652]
Language: English
Original publication: New York: D. Van Nostrand Company, 1918
Credits: Carol Brown, Chris Curnow and the Online Distributed Proofreading Team at https://www.pgdp.net (This file was produced from images generously made available by The Internet Archive)
*** START OF THE PROJECT GUTENBERG EBOOK THE WHYS AND WHEREFORES OF NAVIGATION ***
THE
WHYS AND WHEREFORES
OF
NAVIGATION
THE
WHYS AND WHEREFORES
OF
NAVIGATION
BY
GERSHOM BRADFORD 2d
_Navigating Officer and Senior Instructor New York State Nautical
Schoolship “Newport,” Late Nautical Expert,
Hydrographic Office, U. S. N._
_ILLUSTRATED_
[Illustration: colophon]
NEW YORK
D. VAN NOSTRAND COMPANY
25 PARK PLACE
1918
Copyright, 1918, by
D. VAN NOSTRAND COMPANY
PRESS OF
BRAUNWORTH & CO.
BOOK MANUFACTURERS
BROOKLYN, N. Y.
TO THE
AMERICAN SUB-PATROL
WHOSE WAR ZONE RECORD OF SKILL AND DARING
HAS THE AUTHOR’S PROFOUND RESPECT
AND ADMIRATION
PREFACE
When taking into consideration the number of excellent works now
published on navigation there would seem to be a small chance of
finding a different angle from which to discuss the subject. The
purpose of the majority of such books is to give instruction to
beginners. This book, however, is written particularly for the men who
do their work mostly by rote and wish to know more of the reasons; or,
perhaps, for schoolship graduates who may here find a chance to extend
their horizons.
I have not considered it desirable to avoid repetition and in order
to closely follow a line of thought have freely repeated many points
already taken up. This has in a number of cases avoided the distraction
of seeking a page of reference elsewhere.
I have had in mind that it serve as supplementary reading to the
American Practical Navigator, Bowditch, that great bulwark of
navigation which for over a hundred years has protected American ships
through every deed of valor and every commercial adventure. It is
placed beyond criticism by its venerable name and its remarkable record
and, as a reference book for the navigator, it stands without a peer,
but as a text book it founders the student.
If to such mariners as these a little insight is given to the “Whys and
Wherefores” of their work, I shall be well repaid for the work of many
watches below.
These discussions appeared several years ago, in a less extended form,
in the Master, Mate and Pilot, the magazine formerly published by the
American Association of Masters, Mates and Pilots.
I have freely consulted the following standard works: American
Practical Navigator, Bowditch; Wrinkles in Practical Navigation, Lecky;
The Theory and Practice of Navigation, Dunraven; General Astronomy,
Young; Navigation and Compass Adjustment, Muir; Guide to the Marine
Board’s Examinations, Reed.
I have received and am grateful for very valuable help and suggestions
from Mr. George W. Littlehales, Hydrographic Engineer, U.S.N., Mr.
Felix Riesenberg, C.E., Commander New York State Schoolship Newport,
and George A. Collie (deceased), Nautical Expert, Hydrographic Office,
U.S.N.
G. B.
NEW YORK, April 15, 1918.
CONTENTS
PAGE
CHAPTER I
INTRODUCTORY REMARKS 1
CHAPTER II
NAUTICAL ASTRONOMY 5
CHAPTER III
DECLINATION AND RIGHT ASCENSION, INCLUDING PRECESSION 15
CHAPTER IV
TIME 33
CHAPTER V
CORRECTIONS FOR OBSERVED ALTITUDES 68
CHAPTER VI
LATITUDE 78
CHAPTER VII
AZIMUTHS AND AMPLITUDES 93
CHAPTER VIII
LONGITUDE 101
CHAPTER IX
SUMNER METHOD, INCLUDING NEW NAVIGATION 107
CHAPTER X
THE MOON 139
CHAPTER XI
CHARTS 146
INDEX 161
THE WHYS AND WHEREFORES
OF NAVIGATION
CHAPTER I
Introductory Remarks
In embarking upon the study of navigation familiarity with the compass
is the first logical step: the quick mental conversion of a course or
direction given in points to the same direction in degrees expressed
in quadrants as S. 35° E., and again into a system by which direction
is indicated by degrees from 0° to 360°. A mariner will encounter all
three of these systems and will find constant conversion necessary
back and forth for various purposes. The 0° to 360° system is the most
up-to-date and the simplest form of handling direction.
Following the compass may be taken up the use and description of
other nautical instruments with which every mariner is supposed to be
familiar.
Dead reckoning is the first calculation to appear and this involves
the correction of the compass courses back and forth between true,
magnetic and compass directions. This is dealt with under Azimuths
and Amplitudes. In practical navigation a vessel commences her voyage
and attempts to sail in a certain direction, but the well-known
elements of compass error, variation and deviation, current, wind, seas
and poor steering all divert the vessel from the projected course.
In dead reckoning a navigator strives to keep track of his position
by keeping a record of actual courses steered and distances run. He
then is obliged to guess at the amount the vessel has diverted both
in direction and distance until an astronomical observation sets him
straight again. It is here particularly shown that navigation becomes
an art of estimating position and the better the navigator’s bump of
locality, the greater his success. This is a peculiar gift and usually
is born in the man, at least it cannot be learned from books. The
process of finding latitude and longitude by dead reckoning is supposed
to be already well known to the reader and will not be detailed at
length.
However, every course angle is laid off from a meridian (which is true
N. and S.) and terminates in a parallel of latitude. This meridian
and parallel intersect at right angles; hence these with the distance
run (which is the hypothenuse) form a plane right angle triangle,
plane because the curvature of the earth is not considered in short
distances. To solve this triangle, we have the course angle and one
side--the distance run. With these the other two sides are easily
found by computation, but more easily by tables No. 1 and 2, Bowditch.
The side along the meridian is represented by the column headed Lat.
(difference of latitude) and the side lying in the parallel is in the
column headed Dep. (departure). Thus the values of the sides of the
triangle are given in miles and tenths, showing the distance good
made N. or S. and E. or W. It will be noted that at the top of the
pages of these tables are four different courses and at the bottom are
likewise four courses making the same page serve for eight different
courses. This is accomplished by the fact that triangles formed by
these particular eight courses are the same in _shape_. Thus N. 30° E.,
for instance, makes a similar triangle to N. 30° W. (330°); S. 30° E.
(150°); or S. 30° W. (210°). They have identically the same difference
of latitude and departure. If this fact is not clear draw a diagram and
be convinced. In the cases of N. 60° E. (60°); N. 60° W. (300°); S.
60° E. (120°); S. 60° W. (240°), the same shaped triangle as above is
found, but reversed in that what was the difference of latitude side
now has become the departure side. The values of these sides are read
from the bottom of the page and are found in the reverse columns to fit
the reverse triangle. The latitude value read from the top of the page
as 30° becomes a departure value when read from the bottom with 60°.
The subject of Sailings is one of the early problems confronting the
student of navigation and will be considered briefly. The above remarks
on dead reckoning cover the principle of plane sailing, the simple
method where the spherical surface of the earth is ignored and a flat
ocean substituted. This method will not serve for anything but short
distances of a few hundred miles without sufficient error to render it
impracticable. Traverse sailing is a series of plane sailing courses
made, for instance, by a sailing vessel beating to windward.
In parallel sailing the vessel pursues a true E. or W. course and runs
along a parallel of latitude. Thus all her progress is in the terms
of departure with no difference of latitude. As all meridians converge
from the equator towards the poles the length in miles of a degree
of longitude keeps on diminishing as the poles are approached and,
conversely, miles of departure have an increasing value in degrees of
longitude. So in parallel sailing what we desire to know is what is
the value in the particular latitude of our course of our departure
(miles) in ° ´ ´´ of longitude. Having this and applying it to the
longitude left will give the longitude in. Middle latitude sailing is
very similar to parallel sailing in that it is assumed, for the purpose
of getting the difference of longitude, that the whole departure of the
course or courses sailed has been made in the mean or middle latitude,
because the greater value (in the northern hemisphere) in difference of
longitude of a mile northward of the middle latitude is counteracted by
the corresponding lesser value southward of the middle latitude.
Mercator sailing is perhaps the most extensively used, as the Mercator
principle is employed almost universally in the construction of
navigational charts. It is described under Charts in this book.
Also under Charts is a description of the Gnomonic Chart which is also
called the Great Circle Chart and used in Great Circle Sailing, also
referred to in those pages.
The young navigator is counselled never to _know_ where his vessel
is, lest through over confidence he be led into close and dangerous
quarters.
CHAPTER II
Nautical Astronomy
The sun is the center of the solar system, with all the planets,
including the earth, revolving around it, some with orbits greater and
others less than that of the earth. The planets in some instances have
satellites revolving in turn around them, as the moon in the case of
the earth.
The movements of the earth will be, perhaps, more readily understood by
assuming a position at the North Pole; here beneath the observer the
earth is rotating left-handed--against the hands of a watch, once each
day; while at the same time it is speeding onward through space in a
left-handed curve, which in the course of a year resolves itself into a
complete revolution around the sun.
The sun and stars are considered to be stationary for all navigational
purposes, the _apparent_ movements of these bodies being entirely due
to the motions of the earth. The result of our daily rotation from
west to east is, that the heavenly bodies march past our meridian in
a majestic procession for 24 hours, after which the performance is
repeated. The uninitiated are here forewarned against becoming confused
by the assumption used for convenience by navigators that the heavens
revolve around the earth.
The planets and moon join the sun and stars in the daily parade
past our meridian, but their apparent movements are not entirely an
illusion, for they have motions of their own that somewhat affect the
precision of their daily revolution. This is readily observed in the
moon’s hour of rising, which is very perceptibly later each evening.
Her actual revolution around the earth, being from west to east, is
contrary to the apparent diurnal motion and thereby each evening finds
her farther to the eastward and consequently rising later. As a result
of this change in the time of rising, the moon must of necessity rise
in every hour of both day and night in the course of a month--the time
she requires to revolve around the earth.
The onward movement of the earth in its orbit as we face the sun in
latitudes north of the tropics, is toward our right, and this causes
the sun to apparently move slowly eastward or to the left among the
stars, corresponding exactly to our movement westward. This movement is
opposite to his daily course across the heavens. As a year is required
for the earth to accomplish our revolution around the sun, it follows
that this same length of time is consumed by the sun in making its
_apparent_ eastward revolution of the heavens.
The movements of the planets are more complex. They all revolve around
the sun in the same direction as the earth, but as their orbits are
of vastly different sizes, they will be found in various positions
relative to the sun; they overhaul and pass each other, but owing to
their uniform direction of revolution they never meet. The planetary
system is like the horse race at a county fair; the pole horse has the
advantage, but the varying speeds of the contestants soon place them at
various parts of the track.
From the earth the movements of the planets, aside from the diurnal
movement, are composed of their own actual movement around the sun,
combined with an apparent motion, due to the earth’s onward movement
precisely as described above in the case of the sun. The combined
movement of a planet may be noted by reference to the fixed stars
beyond it.
The positions of heavenly bodies are determined by two
measurements--coordinates--the distance north or south of the celestial
equator, called the declination, and the distance east from the prime
celestial meridian taken as a reference, called the right ascension,
each of which will be subsequently treated at length under its
individual heading. The movement of the planets eastward or westward
as described, constantly changes their right ascension; and as their
orbits are inclined at different angles to the celestial equator, they
are always changing their declination.
The planets whose orbits are smaller than that of the earth are
called inferior, while those whose orbits are of greater dimensions
are known as superior planets. Mercury and Venus are inferior planets
and consequently are always nearer the sun; their comparatively close
proximity making them appear to us as morning and evening stars.
In fact, Mercury is so close that it is unavailable, owing to the
brilliancy of the sun, for observation with a sextant; while Venus, on
the other hand, a little more remote, is an excellent body to observe,
and is always found in the east or west, conveniently near the prime
vertical, the most favorable place for a time sight for longitude. The
twilight or dawn which usually prevails at the time of a Venus sight
gives the navigator a good horizon to observe upon. Mars, Jupiter and
Saturn are superior planets and their travels are so extended that they
may be found almost anywhere in the heavens within the limits of their
declinations.
The earth’s orbit is slightly elliptical, with the sun located a little
out of center--a little nearer one end. Should a line or axis be drawn
through the long diameter, its intersection with that part of the orbit
nearest the sun is called the Perihelion while the opposite point is
known as the Aphelion. The former is used as a point of reference
from which the earth’s position can be located in terms of angular
measurement from time to time. This angle, known as the anomaly, is
formed by the line from the sun to the Perihelion and that drawn from
the sun to the earth. The latter distance is called the radius vector
of the earth. We (the earth), are at the Perihelion about January 1,
and consequently this angle at that date is 0°, but from this time on,
the angle increases approximately one degree a day throughout the year.
The plane of the earth’s equator makes at all times an angle of about
23° 28´ with the plane of its orbit. This is a highly important
angle to mankind, for upon it depends the climate of the world. The
axis of the earth, if we can conceive it as represented by a slender
imaginary staff, extends through the unlimited distance to a point
in the heavens--the celestial pole; this point is in the zenith for
a person at our north pole. Since the distance between these points
is mathematically infinite, any number of lines parallel to this
“staff” will appear to penetrate the sky at the single point of the
celestial pole. Thus the parallel positions of the axis corresponding
to the earth’s various positions, even those at opposite sides of the
orbit, converge into this common point. To be clearer, the parallel
lines representing the different positions of the axis during the
year according to our geometry form a group of separate points on the
heavens, but the distance being beyond all reckoning, our limited
conceptions fail to identify the group of points and it resolves into
one point.
By the same line of reasoning the plane of the earth’s equator remains
parallel in all its positions throughout the yearly cruise around the
sun, and its projection marks but one celestial equator upon the sky.
While the direction of the axis and corresponding position of
the equator are constant for all practical purposes, there is,
nevertheless, an extremely slow circular movement of the axis, called
the precession of the equinoxes, a subject which is reserved for
subsequent discussion.
COORDINATES
In nautical astronomy the earth is assumed to be the center of space
with the heavens forming a globular shell around it, known as the
celestial sphere. All fixed stars are assumed to lie on its concave
surface from the earth regardless of their actual distances. The tracks
of all other bodies moving, or appearing to move, across the sky are
considered to be on the surface of this sphere. It is necessary, in
order to conveniently define the position of heavenly bodies to mark
this celestial sphere with imaginary circles to serve as coordinates,
as we mark the earth with meridians of longitude and parallels of
latitude.
Before going into the explanation of these coordinates, it may be well
to consider a few facts concerning circles. A great circle is of course
understood to be one whose plane passes through the center of a sphere,
dividing it into two equal parts. There can be an infinite number of
these circles whose planes cut the sphere at every possible inclination
as long as they pass through its center. A circle may be a great circle
of either the celestial sphere, the earth, or even of a baseball. The
poles of a great circle are the points on the surface of its sphere,
penetrated by the diameter perpendicular to the plane of the great
circle. As for example, the poles of the earth are connected by the
diameter that is perpendicular to the plane of the equator. An angle
at any pole is measured on the great circle which subtends it. For
instance, angles at the poles of the earth are measured on the equator;
angles at the zenith on the horizon. With these facts well in mind we
will proceed, showing the scheme of circles employed in laying off the
surface of the heavens.
There are three systems of circles, each designed to fulfil a different
requirement.
The first system depends upon the position of the observer and changes
its whole imaginary structure to correspond with his movements. The
plumb-line, if extended to the heavens overhead, will determine the
zenith, the point of origin of this system on the celestial sphere.
The corresponding point directly beneath us is known as the nadir.
The great circle of the celestial sphere everywhere equally distant
from both the zenith and the nadir is the horizon. It is plain that
a new zenith and new horizon are created with every movement of the
observer. The facts that man is on the surface and not at the center of
the earth, and that his eye is elevated above its surface, each creates
another horizon.
The rational horizon is marked by a plane, perpendicular to a
plumb-line and passing through the earth’s center; while the sensible
horizon is determined by a plane, also perpendicular to the plumb-line,
but passing through the eye of the observer. It will therefore be
seen that these two parallel horizons are some 4000 miles apart, the
semi-diameter of the earth; but this distance when projected on the
celestial sphere becomes insignificant when compared with the infinite
distance of this sphere from the earth, and the rational and sensible
horizons shrink into a single line so far as we can perceive.
While this statement is true when dealing with the stars, it needs
modifying when dealing with the sun and moon, and in very accurate
observations of planets, as their distances are insufficient to
eliminate the angle formed between the line from the body to the center
of the earth, and that from the body to the observer. This is allowed
for when observing these bodies by applying the correction of parallax
to the observed altitude.
The visible horizon is the boundary seen between the sea and sky. If
the observer’s eye were at the level of the sea, his visible horizon
would coincide with the sensible horizon, defined above; but the
elevation above the surface from which sights are taken causes the line
of vision, tangent to the sea, to be depressed below the plane of the
sensible horizon making an angle with it called the dip of the horizon.
In practice all altitudes of heavenly bodies taken from a vessel are
measured to the visible horizon and corrected for the dip to reduce
them to the sensible horizon, then again corrected for parallax to
obtain the true altitude of the body above the rational horizon; or
what is the same thing, the altitude as observed at the center of the
earth.
From the zenith, an infinite number of great circles, known as
vertical circles, sweep around the celestial sphere, cutting the
horizon at right angles and passing through the nadir. The one which
cuts the north and south points is called the celestial meridian, and
is evidently a projection of the terrestrial meridian. The vertical
circle passing through the east and west points is called the prime
vertical, and has a distinction above other vertical circles by virtue
of its being the most favorable position for a body in observations
for longitude. The heavens are further swept by an infinite number of
parallels of altitude which are, as their name implies, parallel to the
horizon.
The azimuth of a body is its angular distance from the north or south
points of the horizon, determined by the angle formed at the zenith,
or by the arc of the horizon between the meridian of the observer and
the vertical circle passing through the body. Amplitude is the angle
at the zenith formed by the prime vertical and the vertical circle
passing through the body or it is the angular distance from the east or
west points, measured on the horizon, and is measured similarly to the
azimuth.
The system of laying off the heavens just described is well enough for
the momentary locating of a body, which is a very important feature in
navigation, but for some purposes a more stable point than the zenith,
which on shipboard is ever changing, is needed, from which to form a
system that is constant the world over. To meet this demand we take
the point in the sky penetrated by the prolongation of the earth’s
northern axis--the celestial pole--and from this point meridians and
parallels are developed upon the celestial sphere as has been done on
the earth. In fact, it is as though these terrestrial coordinates were
projected to the heavens where they hold the same relative positions
as upon the earth. The northern celestial pole is in the zenith at our
north pole. The same is true of the south pole. The celestial equator,
or equinoctial, is a great circle, which is midway between the poles
and everywhere 90° from them. It marks the termination of the plane of
our equator extended to the celestial sphere, or in other words, it is
always directly over our equator.
The parallels of the celestial sphere evolved by this system,
corresponding to those of latitude upon the earth are called parallels
of declination, while the celestial meridians having as their point of
origin the prolongation of the earth’s axis, are known as hour circles.
The particular hour circle passing through the zenith is one and the
same with the celestial meridian. It will be seen that this circle must
pass through the zenith, nadir and the poles. The angle formed at the
pole by the hour circle passing through a body and a local meridian is
the hour angle of that body, and is measured westward through 24 hours,
although A.M. hour angles of the sun are reckoned eastward through 12
hours.
At the north pole where the zenith is identical with the celestial
pole, the vertical circles, parallels of altitude and rational horizon
are coincident with the hour circles, parallels of declination and the
equator, respectively; but departing from this point they form angles
with each other corresponding to the degrees of latitude from the pole;
at the equator the angle reaches 90°.
The system of circles described above is by far the most extensively
used, and positions determined by its coordinates are comparatively
constant, but there is still a third system of circles which was used
and handed down to us by the ancients. In the place of the celestial
equator, a similar great circle is used, known as the ecliptic. This
circle is determined by the extension of the plane of the earth’s
orbit to the celestial sphere. The poles of the ecliptic everywhere
90° from this circle are the points from which meridians depart as
upon the earth. The prime meridian of this system passes through the
intersection of the celestial equator, and the ecliptic--the vernal
equinox or First Point of Aries. Celestial latitude and longitude are
the coordinates used with this system, but navigators universally
prefer to use the well-known declination and right ascension. Hence the
path of usefulness of the former seldom leads beyond the observatories.
CHAPTER III
Declination and Right Ascension
Owing to the important place that declination holds in nautical
astronomy, a detailed explanation will appropriately follow closely
in the wake of the preceding remarks. It must be made clear, before
getting under way, that declination is the distance, in degrees,
minutes and seconds, of a body north (+) or south (-) of the celestial
equator measured on the hour circle passing through the body. This
distance is identical with the latitude of the place in the zenith of
which the body happens to be. What declination is to a body in the
heavens, latitude is to the place on the earth directly beneath it.
The declination of fixed stars changes very slowly from month to month,
but the planets meander about on the celestial sphere in a way that
is liable to puzzle anyone other than an astronomer. This element,
however, is worked out in the observatory and given in the nautical
almanac in a way that relieves the navigator of worry concerning the
complex movements of these latter bodies. The same may be said of the
moon, but the subject will be treated, somewhat superficially though
sufficiently for the needs and desires of the practical mariner, in
a special talk on the moon. This eliminates all the celestial bodies
except the sun, the most important; and for this reason the facts
relative to its declination will be considered at some length.
As has already been stated, the sun is stationary, but our movements
around it to the right causes it to appear to move to the left;
precisely as you see, when under way, an anchored vessel’s masts move
to the left along the land behind her, while you move on to the right.
We have no landmarks behind the sun by which to observe his apparent
movements, so in lieu of such ranges, we resort to the fixed stars,
which serve as excellent marks to get a bearing on Old Sol and keep tab
on him as he moves eastward among them. This movement must in no way be
confounded with his apparent daily motion westward. As an illustration,
we may see Orion--a familiar friend--swinging high in the western sky
in the early evening; some weeks later he is riding low, and yet a
little later still, he is swallowed up in the brilliancy of the setting
sun. In other words, the sun and Orion have approached and passed each
other. We know Orion does not move, for he is composed of fixed stars,
and this seeming westward movement of his is in reality the apparent
eastward marching of the sun, which is due to the earth’s movement of
revolution. The sun in this apparent movement eastward follows a course
at a rate equal to that of the earth, along a great circle of the
celestial sphere called the ecliptic, a circle that plays an important
part in the explanation of declination, particularly that of the sun.
The ecliptic is marked by the extension of the earth’s orbit to the
celestial sphere.
A few more words concerning great circles will be introduced here,
and the following statements, while they apply to great circles
in general, especially fit the relationship of the equinoctial or
celestial equator to the ecliptic. These two great circles cut each
other at an angle of 23° 28´. Great circles always bisect each other,
and hence any two great circles of the celestial sphere, regardless
of the angle they may take with the celestial equator, must intersect
each other at exactly opposite points, 180° apart. What is true in this
regard of the celestial sphere is equally true of the great circles of
the earth. A vertex of a great circle is the point which departs the
greatest distance from the equator--the highest point of the circle
reached in declination. There are two vertices 180° apart with the two
points of intersection 90° in either direction. The declination or
latitude of either vertex is equal to the angle at which the circles
intersect each other. The intersections are called the equinoxes, and
it may be well to say here that the word equinox has several meanings
in navigation, often rendering it necessary to judge by the text which
is intended. The vernal equinox, for instance, refers to a certain
time of year--March 21st. The sun is that day directly overhead at
the intersection of the equator and the terrestrial ecliptic and this
_point_ is sometimes called the vernal equinox. Again, the sun at the
same time occupies a point on the heavens also known as the vernal
equinox, which is at the intersection of the celestial equator and the
ecliptic. The point in the orbit occupied by the earth at this time is
also spoken of as the vernal equinox.
The reader is now asked to arouse his imagination and if possible
to conceive himself a passenger in an aeroplane equipped with some
remarkable power capable of carrying him to a position in space,
above, yet a little outside, the earth’s orbit, near the Perihelion,
and there to heave to and view awhile an astronomical picture. Spread
out before his unrestricted vision will be the earth, its orbit, and
the sun. It is to be hoped that the imagination of the reader is
still sufficiently supple to suppose the plane of the orbit to be the
surface of an infinite ocean stretching away beyond human conception of
distance and “breaking” against the celestial sphere; the “surfline”
there marks the ecliptic; the “ocean’s” surface representing the great
plane of the ecliptic. The sun will be seen as if at anchor in his
proper place within the orbit. The earth is “underway,” half submerged,
and listed 23° 28´ _toward our point of vantage_. This inclination, or
direction of the axis, is in a general way toward the perihelion, and
within a few degrees of being parallel with the long diameter of the
orbit. The earth maintains this nearly parallel position of its axis
with the long diameter throughout the period of its revolution; a fact
of importance to remember.
It will be readily seen that during the encircling of the sun there
must be one position where the northern axis is inclined directly
toward that body, another opposite where it is headed away from him,
and two positions midway where the bearing of the axis (projected on
the plane of the orbit) is at right angles to the bearing of the sun
from the earth; another feature to be “salted down” in the memory.
If the earth revolved on an even keel, the equator and the “waterline”
would be coincident, but fortunately this is not the case, and owing
to the inclination of the axis another great circle is defined by the
“waterline,” called the terrestrial ecliptic, being directly beneath
its celestial namesake. The inclination of the northern pole being
in a general way toward the perihelion, correspondingly depresses or
“submerges” that half of the equator below the plane of the ecliptic,
represented by the “water surface,” and at the same time the opposite
side rolls the equator above it. At two points (the equinoxes) on
opposite sides of the earth, and at right angles to the direction of
its inclination, the equator and terrestrial ecliptic cross each other
at the “water’s edge.”
The sun is always exactly overhead for that point of the earth which
is nearest to it. This is an essential fact to remember in navigation.
Bearing in mind that the sun is stationary and ignoring for a time the
rotation of the earth, each advance in its orbit brings about a change
of bearing of the sun and a new position becomes the nearest point,
and thereby directly beneath the sun. The constant changing of the
sun’s bearing continues throughout the year, or one revolution, and a
circle of these overhead positions is marked upon the earth, which is
coincident with the terrestrial ecliptic--the visionary “waterline.”
It is obvious that the vertical rays of the sun must apparently follow
this line, for it can only be overhead for places that are in the same
plane, and this again is the level of the “ocean.”
This circle of overhead positions projected on the celestial sphere
marks the ecliptic--the “margin” of the infinite ocean, and the path
that the sun seems to follow eastward among the stars.
The above paragraphs show us that the sun in following this line
around the earth crosses the equator twice, and twice he attains a
distance of 23° 28´ from it, and so must be on the equator twice and
reach a declination of 23° 28´ north and 23° 28´ south in the course of
one year.
Returning to our imaginary illustration, we will now follow the
peregrinations of the earth for a year and note the effect of its
inclination in the different parts of the orbit upon the declination of
the sun.
It will be assumed that it is the 21st of March and from our airy
position we see the earth away on our right nearly 90° from the
Perihelion. As this is the vernal equinox, there are a number of
interesting points to be considered: The direction of the earth’s axis,
projected on the plane of the orbit, is at right angles to the bearing
of the sun from the earth; the sun is directly over the intersection
of the equator and terrestrial ecliptic, and being overhead for
this point on the equator, the declination must be 0°. Moreover, a
line drawn from this intersection, or terrestrial vernal equinox,
through the center of the sun and extended to the celestial sphere
would strike the corresponding intersection of the ecliptic and the
equinoctial or celestial equator--the celestial vernal equinox. The
arrival of the earth at this position is the signal of spring for the
northern hemisphere, likewise it announces the advent of autumn to our
southern neighbors below the “Line.” The sun this day rises in the east
(approximately) and passing through the zenith, sets in the west for
those living on the equator. The explorer at the north pole is cheered
by the first light as the sun appears in the horizon, while the south
pole becomes enshrouded in the long Antarctic night. Without lingering
for ceremonies over the change of seasons, the earth continues steadily
on its way toward the aphelion; the sun’s vertical rays leave the
intersection of the equator and the terrestrial ecliptic, and follow
along the latter, thus widening its distance from the equator as the
earth proceeds. As the ecliptic in this half of the orbit is above,
or north, of the equator the former is in north latitude and the sun,
following along it, is thereby also in north declination. A line from
any place having the vertical rays, through the sun to the celestial
sphere, always terminates on the celestial ecliptic, all being in the
same plane, and shows the corresponding celestial position of the sun
on it. Its declination distance from the celestial equator, in degrees,
minutes and seconds, is identical with that of the place on the earth
directly beneath it relative to our equator. So by showing the course
of the sun’s overhead positions on the earth its celestial positions
are, at the same time, indicated. The overhead position of the sun on
the terrestrial ecliptic gradually departs from the equator culminating
about June 21st, the summer solstice, in a declination of 23° 28´ at
a point near the aphelion in the orbit, 90° (approximately) from the
equinox.
The positions in the orbit of the summer and winter solstices are
reached by the earth several days before the points of the aphelion and
perihelion. These respective positions would be in conjunction were it
not for a slow and remarkable motion of the earth’s axis before spoken
of, and later to be described, called the precession of the equinoxes.
The summer solstice is the great half-way point of the earth’s annual
circumnavigation of the sun; it is a matter of moment all over the
world, and another great change of seasons is at hand. The sun is
overhead for places along the parallel of 23° 28´ N. and bears north
23° 28´ from the zenith at noon from places on the equator.
At the north pole, since its appearance on the horizon on March 21st,
the sun has mounted to an altitude of 23° 28´ and to nearly 67° at
places on the Arctic circle. The earth’s northern axis is, in this
position, inclined 23° 28´ directly toward the sun, which pours its
rays continuously upon the northern regions, uninterrupted even by
the earth’s daily rotation. It is on this day that the whole Arctic
zone enjoys the full glory of the midnight sun. The earth’s continuous
movement of revolution does not allow a delay of this favorable season
in northern latitudes, but continues to make the sun’s vertical rays
follow the terrestrial ecliptic as before on its way toward the
intersection with the equator 90° away. On this leg of the journey, the
sun is traveling on the upper one of two converging lines and thereby
gradually lessening its distance from the other--the equator--or, in
other words, reducing its declination. This continues until September
21st when the autumnal equinox is reached and the sun’s declination
becomes 0°. The sun now being overhead at the intersection of the
equator and the terrestrial ecliptic, is on the opposite side of the
earth from the intersection of March 21st. In fact the conditions are
similar, but now the earth is on the opposite side of the sun, and
the change of seasons is the entrance of spring for the dwellers in
southern latitudes.
The sun has dropped lower and lower in the sky at the north pole since
June, until on this day it is in the horizon and it is time for the
Esquimos to seek their igloos and prepare to hibernate during the long
Arctic night now ushered in.
The sunshine at the time of the equinoxes is equally distributed over
the northern and southern zones, and the zenith distance of the sun
at noon at any place is, theoretically, equal to the latitude of the
place (except a small error due to change of declination accumulated
subsequent, or previous, to the instant of the equinox).
The conditions during the next six months are reversed as the earth
proceeds into that half of the orbit containing the perihelion. Now the
sun following the terrestrial ecliptic enters southern latitudes or
south declination, for in this part of the orbit the equator is above
(or north) the plane of the ecliptic. The sun’s diverging course from
the equator leads it farther and farther southward until on or about
December 21st it arrives at the winter solstice with a culmination of
23° 28´ south declination. At this point the earth is but a few degrees
from the perihelion as it was from the aphelion at the summer solstice.
The earth’s north pole is now inclined directly away from the sun and
its rays have entirely forsaken the Arctic for the Antarctic zone;
notwithstanding the earth’s daily rotation, which brings alternating
light and darkness to the greater part of the world, the northern polar
regions are in a continuous shadow, and no sunlight reaches these
remote parts. At this time of the year the northern hemisphere above
the tropic of Cancer, is in an unfavorable position relative to the
sun, and as a result places situated on parallels less remote than the
Arctic are having long nights and short days in proportion to their
latitude north. On the other hand, in the southern hemisphere the days
are longer and the nights shorter, as the southern latitude increases
until at the Antarctic circle night disappears and the sunshine is
uninterrupted. It is seen that this is an exact reversal of the
conditions at the summer solstice.
The earth enters the last quadrant of the great ellipse of its orbit,
the sun now approaches the equator as the earth nears the vernal
equinox. The south declination diminishes until on March 21st it
becomes 0° and the earth has completed its revolution. We will now
go on another tack and instead of considering only the effects of
declination due to the earth’s revolution, will assume that the earth
has been halted in its onward course of revolution and is making its
daily rotation in the same position. The earth turning from west to
east causes the sun to appear to proceed from east to west in its
diurnal motion. Each rotation, requiring 24 hours, marks upon the
earth a circle of overhead positions parallel to the equator and
hence without change of declination. The result of such a remarkable
condition would be, no change of seasons and no change in the length
of the days and nights. In reality, however, we are saved from such
monotony, for both the motion of rotation and revolution of the earth
are acting together and giving a compound effect on the apparent
movements of the sun. This alters the daily circles just mentioned to
a fine spiral of overhead positions, ever changing in declination. The
daily difference of the sun’s declination shown in the Nautical Almanac
is equivalent to the distance between two threads of this spiral.
The change of declination is most directly seen and felt in the polar
regions, where the activities of the denizens are mostly limited to
the favorable phases of this change. At the north pole, after the sun
has appeared above the horizon, this spiral of declination can be
continuously followed. The sextant will disclose a constant increase
in altitude as the sun circles round and round the sky, winding itself
up and finally culminating at 23° 28´. The process is then immediately
reversed. The stars here make daily circles of _equal_ altitudes as
their change of declination is insignificant; but the circles of the
planets and the moon are converted into spirals, the fineness of which
is in proportion to the rate of their change of declination.
The fact that the sun reaches an altitude of 23° 28´ at the pole at
the summer solstice with its declination of a like amount and that on
March 21st, when the sun is in the horizon with the altitude 0°, it is
directly over the equator with 0° declination, shows that at this place
(the pole) the altitude is equal to the declination. Should an explorer
travel southward 1°, his sextant would show an altitude 1° greater
than at the pole, yet moving about does not affect the declination at
a given time. It follows by taking his altitude at noon the explorer
in the polar regions may readily learn his distance from the pole by
subtracting the declination in the Nautical Almanac from his sextant
reading.
It may not generally be known that the southern summer is shorter
than the summer of the northern hemisphere, but such is the case by
approximately eight days. The reason of this inequality lies in the
fact that the sun is nearer one end of the orbital ellipse, and the
short diameter passing through this body divides the orbit into unequal
parts. The smaller part being that traveled by the earth during the
_southern_ summer. Furthermore the nearer proximity of the sun causes
an accelerated motion which further tends to lessen the time spent by
the earth in this part of the orbit.
RIGHT ASCENSION
Declination and right ascension being used together as coordinates,
we will not separate them. It will be remembered that the equator
and the terrestrial ecliptic cross each other on opposite sides of
the earth; that on or about March 21, the sun is overhead at the
intersection that is the vernal equinox. Now if at this intersection
on this day a plumb-line were carried upward, it would at length reach
the sun, and continued to infinity and projected on the celestial
sphere would locate a point called the celestial vernal equinox, known
by many as the First Point of Aries. This point is one of the most
important celestial “landmarks” used in astronomy and navigation,
but, unfortunately, no heavenly body marks its place. However, as its
relative position among the neighboring stars is well known, its exact
location is easily ascertained.
The hour circle which passes through this point is known as the
equinoctial colure, and may be considered the prime meridian of the
heavens, for from it is measured the right ascension of all bodies.
Right ascension of a body is the angle at the celestial pole between
this meridian of reference and the hour circle passing through the
body. It is always measured eastward through 24 hours of sidereal time
(360°). The angle is measured by the arc intercepted on the celestial
equator. For example, a star 15° east of the equinoctial colure has a
right ascension of 1 hour or 15°, but, if the star is 15° west, its
right ascension is 23 hours or 345°.
The positions of heavenly bodies are defined by right ascension and
declination exactly as positions upon the earth are expressed by
longitude and latitude, right ascension corresponding to longitude and
declination to latitude.
In the discussion of Time, to follow, more facts concerning right
ascension will be found.
PRECESSION OF THE EQUINOXES
A comparison of the present positions of the fixed stars with their
places as recorded in ancient times shows a great discrepancy. The
celestial latitudes, which were reckoned from the ecliptic, show no
appreciable change; but in the declinations and right ascensions
there is a great departure from the old positions. The error of right
ascension was found by the old Greek astronomer, Hipparchus, to appear
as a uniform eastward movement of all the stars, which led him to
reason that, instead of the stars themselves changing, their point of
reference was moving westward, thus lengthening all right ascensions.
The famous astronomer after further reasoning decided that the
position of the celestial pole was changing, in fact that the line of
the earth’s axis was describing a circle on the heavens, which was
left-handed or against the hands of a watch as viewed from the north
pole of the earth. This movement was found to be extremely slow,
requiring 25,800 years to complete the circle which has as its radius
the amount of the inclination of the earth’s axis--23° 28´.
If a match is put through a piece of cardboard about the size of a half
dollar to the distance of ¼ inch, and spun, the motion of the cardboard
just as it staggers through loss of speed, gives some idea, although
exaggerated, of the precession movement of the plane of the equator,
which is of course infinitely slower. The movement of the top of the
match is a semblance of the corresponding motion of the vanishing point
of the axis on the celestial sphere.
The earth, as already explained, points its axis at practically the
same spot in the heavens throughout the year, and if it were not for
this annual precession of 50´´ it would for all intents and purposes
hold a permanent direction. About December 21, the winter solstice,
while the earth is still some degrees from the perihelion, its northern
axis, is inclined directly away from the sun. Each year this distance
from the perihelion is becoming greater, widening this angle between
the direction of the axis, projected on the plane of the orbit, and
the major diameter of the orbit, until in time the north pole will be
headed directly away from the sun in that part of the orbit which the
earth now occupies in September, and so on.
[Illustration: This diagram shows the successive positions of the earth
at the Vernal Equinox (March 21st) due to the revolution of the axis
and the consequent westward movement of the First Point of Aries.
FIG. 1.]
In the year A.D. 1250 the winter solstice occurred at perihelion and in
the year 6400 A.D. the vernal equinox will occur at this point of the
orbit. That is, the axis of the earth was inclined directly away from
the sun at perihelion in the former year but in the latter year the
inclination will have changed about 90° backward against the earth’s
course about the sun, and it will be the beginning of spring (the
vernal equinox) when the earth is at perihelion instead of the first of
winter as in 1250 A.D. Since 1250 A.D. the inclination has changed an
equivalent of about 11 days for now the earth is at perihelion about
January 1st, and the solstice occurring about December 21st, shows the
present relative situation.
In other words, the vernal equinox is slipping back in the orbit
towards the perihelion, and as the solstices maintain their positions
at 90° from the equinoxes they must likewise be “slipping a cog” each
year.
The vernal equinox was situated many centuries ago in the first part of
the constellation of Aries, and was known as the First Point of Aries,
but owing to the movement of precession it has dropped back or westward
(as we face our southern horizon) 50´´ a year until it has left that
constellation entirely and is now about leaving the constellation
of Pisces, some 30° from the position used by Hipparchus in his
calculations. The majority of navigators still call this point of the
celestial vernal equinox the First Point of Aries.
Holding these facts in mind, it may be clear that as the earth
approaches that part of the orbit where the vernal equinox occurs it
has turned its pole, and correspondingly points on the equator, 50´´
to the right or west during the year; thus causing the point of the
terrestrial equinox to meet (or come under) the sun that much sooner.
In other words, referring to the effect as seen on the heavens, the
celestial equinox was advanced to the westward that much to meet
the sun in its eastward movement among the stars and will become the
nearest point to the sun, 50´´ before the position of the equinox
of last year. As the points in the orbit where the vernal equinox
occurs year by year works back toward the perihelion, the range line
through the sun to the heavens beyond must each year correspondingly
edge its way westward along the celestial ecliptic through different
constellations. This is what is known as the precession of the
equinoxes.
The course of the celestial pole in the heavens is shown by a circle
drawn about the pole of the ecliptic using 23° 28´ as a radius. This
path will pass 1¼° from our pole star and this position marks the
present termination of our extended axis; half way around the circle it
passes the first magnitude star Vega close aboard, thus making this the
future pole star some 12,000 years hence. If there be such creatures as
navigators in those far-away days, latitude by Vega will no doubt be a
popular sight among them.
The cause of this remarkable movement of the earth is due to the fact
that the earth is not a true sphere, and the influence of the sun is
not exerted equally upon its mass. Its flattening at the poles is
attended by a corresponding bulging along the equatorial belt. When
the earth is in the vicinity of the perihelion, leaning away from the
sun, the half of this ring of extra matter on the side towards the
sun is above the plane of the ecliptic or orbit. The tendency of the
added attraction exerted upon it, is to draw the earth to an upright
position, or in other words, at this time the sun is pulling stronger
on the northern or upper side than on the lower. Again, when near the
aphelion and summer solstice, leaning towards the sun, that part of the
ring of extra matter on the side towards the sun is below the level of
the orbit, and the attraction is again tending as before to pull the
earth upright. At the equinoxes there is an equal amount of this extra
matter above and below the plane of the orbit evenly distributing the
attraction.
The effect of this influence would in time bring the earth’s equator
and the plane of the ecliptic into coincidence and the earth’s pole
would be directly beneath the pole of the ecliptic, were it not for
its rotation. The two forces acting upon the earth result in the slow
revolution of the axis. The exact effect of these forces is rather
complex but it is a demonstration of the principle of the gyroscope.
The movement of the axis is affected very slightly by other influences
than that of the sun, the most notable of which is the moon, whose
monthly revolutions around the earth produce a similar influence in the
bulging mass within the tropics, but as its revolutions are so rapid,
it has but a slight effect on the precession movement of the earth.
It is sufficient, however, to cause the extended axis to nod slightly
and make a waved circle of precession on the heavens. This is called
“Nutation,” from the Latin word _nuto_, meaning to nod.
CHAPTER IV
Time
A thorough understanding of time, one of the most important elements in
navigation, clears the way to a better idea of the theory of finding
one’s position at sea; there is, in the minds of many, considerable fog
hanging about certain portions of this subject, and it is hoped that
this explanation will clear some of this away.
Worcester’s Dictionary defines time as measured duration. It is the
interval between events. It flows ceaselessly and with uniformity, yet
the mortal mind is unable to conceive its beginning or its end. Man,
in order to measure his activities, has blocked it off into different
denominations convenient for his uses. Of these, the navigator uses the
following in his determinations and reckonings: years, months, days,
hours, minutes and seconds.
Certain astronomical phenomena were naturally enlisted by the ancient
astronomers to furnish standards for time measurements; the value of
a year was determined by the time necessary for a complete revolution
of the earth around the sun, while the length of a day was fixed by
the time of a rotation of the earth on its axis. The precision with
which these evolutions are accomplished gives the required accuracy.
The revolution of the earth governs the change of seasons, while
the rotation is responsible for the alternating periods of day and
night. With the exception of the month, the other measurements of
time mentioned above are denominations of these standards. The month,
one-twelfth of a year, is measured by the revolution of the moon around
the earth.
Solar time, as its name implies, is measured by the apparent diurnal
movement of the sun. It is the variety of time in universal use by
which is regulated the daily activities of life; and this is indeed
quite natural, for of necessity the work and play of the world depend
upon the light and darkness that this body serves out to us.
While we are unconscious of the earth’s rotation, its effect is seen in
the apparent daily course of the sun across the heavens, caused by our
turning past it, yet in common practice the sun is assumed to revolve
around the earth, and is usually thus spoken of for the purpose of
simpler explanation.
The time at each meridian is necessarily different from that of every
other, as only one of them holds the same position relative to the
sun at the same time or putting it in another way, only one meridian
can cross the sun at the same time, determining local noon for those
places located upon it. It is forenoon for that part of the world
westward of the sun and afternoon for that portion eastward of it.
As the earth turns from west to east, the places or meridians to the
eastward are first favored with the sun’s light, and those meridians
cross this body before those to the westward. The sun apparently moves
from the eastward to the westward, crossing each meridian in succession
until in a few hours it is afternoon for places to the eastward and
noon with us. The sun is now in our meridian, and it is forenoon for
people to the westward of us. For example, at 7 A.M., 75th meridian
time, it is noon in England and dead of night in our Pacific Coast; at
our noon (75th), it is late afternoon in England and breakfast time in
California.
It requires 24 hours, solar time, for the sun to make its apparent
revolution around the earth, this course being a circle; it contains
360° of arc. It follows that in one hour it passes over 15° of arc,
while 4 minutes are required for 1° to be traveled. Thus it is evident
that any arc of the circumference of the earth, or difference of
longitude, which is the same thing, has an equivalent time value
and vice versa. That is, the arc comprised between the meridian of
Greenwich and the 60th meridian west, for instance, besides being
measured as 60° W., is equal to 4 hours of time. Again 4 hours of
Greenwich time indicates that the sun has crossed the Greenwich
meridian 4 hours ago and is at that particular instant crossing the
meridian 60° west of Greenwich. If the arc were between Greenwich and
a place 60° E., the equivalent time interval would also be 4 hours,
because 60° of arc is everywhere equal to 4 hours of time; but the time
at Greenwich, with sun on the 60th meridian east, is 20 hours of the
previous day, or 8 A.M. of the present day. Thus: May 14, 20 hours, or
May 15, 8 A.M.
The meridians extend from pole to pole, and it matters not what
parallel you may be on, whether north or south latitude, your distance
can always be measured to the Greenwich meridian in arc or time
precisely as well as though you were on the exact parallel of Greenwich
itself. If the time at Greenwich is carried, and the local time of
any other meridian is desired, turn the difference of longitude into
time and apply it with regard to signs: - if west of Greenwich and +
if east. The local time at any place can thus be calculated; or to go
farther, if the time of any meridian is at hand, the time of any other
place can be readily found.
Every meridian carries a time of its own, and the instant of the click
of a telegraph key may be recorded all over the world in the local time
of each locality, yet the interval between this and a subsequent click
has an absolute value which is the same at every place, regardless
of whether it is expressed in solar, sidereal or lunar time, and its
actual value is invariable.
For convenience, on land, our country is blocked off into belts
of standard time, 15° wide, each carrying the time of its central
meridian. For instance, 75th meridian time is used by the eastern
states, while just westward the clocks’ faces show an hour earlier
time, that of the 90° belt, and so on.
It is a good rule to remember in reckoning all kinds of time that the
clock’s _face_ shows earlier time to the westward, and from this it is
easy to deduce the proper application of a correction.
There are two kinds of solar time used in navigation; the first to
be considered is apparent time, the kind shown by the sun dial, or
measured by the sun as we see it. It is noon of the apparent day
when the sun is seen with the sextant to dip while taking a meridian
altitude. It is at the moment of dipping that the navigator announces
12 o’clock, and with the striking of eight bells begins a new apparent
day on shipboard.
=The Day Lost and the Day Gained.=--The fact that the sun seems to
travel from east to west, determining the local time for successive
meridians or places along the way, causes an interesting condition in
reckoning time aboard ship. A vessel steaming westward on a parallel
sails with the sun; in the forenoon she is sailing away from it, at
noon the sun overhauls the vessel and they race together, but it
becomes a hopeless chase for the steamer during the afternoon. In
consequence of their similar course, however, the vessel will hold the
sun longer, and the length of daylight will be increased over that
time allotted a stationary position in proportion to the speed of
the vessel. On the other hand, a vessel steaming eastward each hour
advances to meet the sun; at noon the effect is as if they pass each
other, and during the remainder of the day they are moving in opposite
directions, hence this vessel has a shorter term of light and is
deprived of its full share of sunshine.
In practice these facts require the continuous setting back of the
ship’s clock, keeping apparent time on a westbound vessel. Take a
concrete case for illustration: to-day assume we are at sea on the
45th meridian west and set the clock at the dipping of the sun,
apparent noon; the vessel is westbound, steaming along the equator,
and rolls along at a good 15-knot clip. In 24 hours by the clock we
will cover 360 miles, or 360´ of arc on the equator, which is equal
to 6° difference of longitude. (Should the easting or westing be made
in higher latitudes, the difference of longitude will be increased
proportionately.) So when the ship’s clock shows noon we will be
6° farther west than at the preceding noon, or in 51° west. The
navigator, should he observe the sun, would find it had not reached
its highest altitude (the meridian), and he would be obliged to wait
(approximately) 24 minutes, the equivalent in time of 6°, before the
sun would dip. The clock is carrying 45th meridian time, and we are now
determining noon for the 51st meridian. He sings out 8 bells, but the
clock shows 12.24 P.M. The ship has gained 24 minutes by sailing with
the sun, and the clock is set back and a fresh start is made.
A vessel sailing east has the opposite experience. The navigator, if
guided by the ship’s clock, would find that the sun had dipped some 24
minutes before noon if a run similar to the above mentioned was made
eastward. In this case the apparent time of the 51st meridian is shown
by the clock, while the ship has moved on to the 45th, and the time of
noon is correspondingly approximately 24 minutes earlier than the clock
admits.
In the above example, the clock in the first instance is 24 minutes
fast and is set back that amount to correct it for the time of the
51° meridian W.; but this time cannot be thus arbitrarily thrown away
without some subsequent reckoning. There is just so much time all over
the world, and there are no gaps or extra intervals; it is absolute in
its uniform flow. Therefore, there must be a way of squaring ourselves
with Old Father Time.
But let us follow the voyage farther and see what transpires:
Continuing the course westward and ignoring for convenience all
intervening land, each day it becomes necessary to set the clock back
24 minutes until we have circumnavigated the earth. Suppose we took
our departure from the Greenwich meridian and kept our log throughout
the voyage with great care, expecting, according to our reckoning, to
arrive on a Saturday, we would indeed be mystified on arrival to hear
the ringing of church bells and find that it was Sunday. We have lost
a whole day according to our log, by throwing away 24 minutes at a
time. The time of the world goes on just the same, regardless of how we
juggle the hands of the clock. Now, if we try a similar voyage eastward
around the earth, we will be setting the clock ahead 24 minutes each
day, and when the anchor is dropped on our return, we will discover
that it is Friday instead of Saturday. The ship’s clock has skipped
this 24 minutes each day, and our log is a day ahead of what it should
be.
In order to prevent this difference of date, it was decided years
ago to establish an international date line, which should correspond
approximately, with the 180th meridian. The logs of vessels going west
around the earth will be a day behind the calendar when they reach
Greenwich, so a day is dropped from the reckoning when crossing the
180th meridian; that is, if it is Monday, the next day in the log will
be Wednesday. On the contrary vessels bound eastward will be a day
ahead when they reach their destination of Greenwich, so the date of
crossing the date line is entered twice in the log, as for instance,
there will appear two Mondays. By this method the accumulated errors
of chasing local time, are in a measure straightened out, and ship’s
logs are kept in agreement with the calendar of those at home. Thus it
will be seen that it is the accumulation of time thus gained or lost
that obliges navigators to add or drop a day to or from their logs when
crossing the 180th meridian.
In slow cargo steamers and sailing vessels, particularly when the
course creates but little departure, the change of time due to
difference of longitude is not sufficiently large to cause much
inconvenience and can be taken care of by setting the clock back
or ahead at noon. But with the development of the modern steamer,
speed has increased to such an extent that the easting or westing of
certain day’s runs correspond to a considerable amount of time, and
to correct the clock to local time, all at once, would be a source
of inconvenience and a bother. This is especially true where a fast
steamer covers much easting and westing in high latitudes where the
convergence of the meridians has shortened the degrees of longitude,
thereby increasing the difference of longitude over a similar day’s
run in lower latitudes. Hence, in order to more equally distribute
its error, the longitude at noon is anticipated by the navigator and
the clocks set at 8 A.M. for the local time of the approaching noon
meridian.
When the clocks are set at noon, they are correct only for the moment
and then start an accumulating error, depending in amount upon the
rapidity of the easting or westing made. But by anticipating the
longitude at noon, the forenoon watch will experience a decreasing
error instead of one accumulated for twenty hours, and still
increasing. It serves to keep the time of day more nearly correct.
In the transatlantic service, where high speed is maintained and the
courses result in a large amount of easting and westing, another method
is used for convenience. The navigator estimates the noon position
of the next day and accordingly divides the error into thirds. The
amount of the first third is applied at 11 P.M., the second at 3 A.M.
and the last third at 5 A.M. By this method the error is distributed
between the “first,” “mid” and “morning” watches. It is a matter of
considerable moment, and no joke, to the hard-working stokers to have
the clock set back on them the full amount of the day’s run all at one
time; and likewise going east, it would be giving an unfair advantage
to those on duty to set the clock ahead nearly an hour during the
morning watch.
The apparent or real sun is not a very accurate timekeeper and its
days are unequal in length. The aborigines and even our ancestors were
content with the time of day indicated by the sun dial, but as the
generations have passed, each bringing increased development, time has
become valuable; the crude timepieces have been forced aside by more
reliable instruments until to-day we figure down at times to a one
hundredth part of a second.
It is impossible to construct a clock that will follow the
irregularities of the apparent sun, so an imaginary sun has been
devised which is assumed to make its revolution at a uniform speed
along the celestial equator with exactly 24 hours between its transits
of the same meridian. This interval is the average of all apparent days
in one year.
The varying rate of the sun’s apparent motion is due to several causes
which will be subsequently discussed under the Equation of Time.
The time measured by the transit and progress of the mean sun is called
mean time; if at Greenwich, it is Greenwich mean noon and Greenwich
mean time (G. M. T.); if it represents the time of the observer’s place
or meridian, it is local mean time (L. M. T.). It is mean time that is
shown by all clocks and chronometers used in every day life.
The distance between the apparent and mean suns, expressed in time,
is known as the Equation of Time, and the application of this
correction depends upon which sun is ahead. It is tabulated in the
nautical almanac for every two hours of Greenwich mean time, with
hourly differences, so it can be reduced for longitude in time to
any meridian, or corrected to any intermediate Greenwich time. It is
applied according to the sign accompanying it, and can be used to
change apparent into mean, or mean into apparent time.
The progress of the mean sun across the sky with reference to the
meridian is measured by the angle at the pole (expressed in time),
between the meridian and the hour circle passing through the mean sun.
This is the hour angle of the mean sun as well as the local mean time.
Civil Time is a variety of mean time, and is reckoned through 12 hours
from midnight to noon, and again 12 hours from noon to midnight,
dividing the day into the well-known periods of A.M. and P.M. With
this kind of time, the day begins at midnight and the hour angle until
noon is measured eastward through 180° of the revolution and westward
through the remaining half from noon to midnight. In other words, 4:00
P.M. signifies that the sun has a westerly hour angle of 4 hours, while
8:00 A.M. indicates that the sun is 4 hours eastward of the meridian.
Astronomical time is reckoned westward through the whole 24 hours of
the day, 0 hours being noon. From noon to noon is an astronomic day.
Thus 5 P.M. civil is the same as 5 hours astronomical time, while 5
A.M., May 14th is the same as May 13th, 17 hours.
In every solar observation for time the real or apparent sun is
observed and hence the time derived from the sight must be local
apparent; to which the equation of time must be applied to convert it
into local mean time. It has already been made clear that the longitude
is equal to the difference between the local mean time and Greenwich
mean time, or between local apparent time and Greenwich apparent time.
SIDEREAL TIME
Sidereal is derived from the Latin word sidus, meaning of or belonging
to the stars. Sidereal time is measured by the apparent diurnal
revolution of the stars, resulting from the rotation of the earth.
By their use the conditions which render the sun inaccurate as a
timekeeper are eliminated; for the period of rotation of the earth is
so regular that the passages of the stars across the meridian occur
with great precision. This exactness enables the astronomer to keep the
observatory clock checked to a remarkable degree of accuracy. These
observatory clocks carry sidereal time, and for convenience it is
customary to divide their faces into 24 instead of 12 hours.
Sidereal time is the bedrock of all time; for it is by converting it
into solar time and sending it throughout the country by telegraph and
radio that the people of the world get the standard by which to set
their clocks and chronometers. Sidereal time is not practicable for
every day use as its noon occurs, without regard to light or darkness,
at every hour of day or night during a year. In March, at the time of
the vernal equinox, it agrees with the solar clocks, but in September
at the autumnal equinox, its noon occurs at the solar midnight.
While the sun is employed as the object of reference in solar time, it
may appear strange that no particular star is thus used in sidereal;
but in lieu of a definite stellar object by which to measure the
sidereal movement of the heavens, we refer to the celestial vernal
equinox.
This point was located in the constellation of Aries centuries ago, and
hence its popular name--The First Point of Aries; but this has become
a misnomer, for the point has long ago moved westward into another
constellation, as discussed under the Precession of the Equinoxes.
Navigators still cling to the name, however, and the equinox continues
to serve its purpose, regardless of its slow drift westward.
This imaginary point of reference crosses the observer’s meridian
much as the stars do, with the difference that it is always on the
celestial equator and acquires no declination. The value of this point
becomes further enhanced by the fact that it always lies in the same
direction regardless of our position in the orbit. In other words, the
distance of the equinox being infinite, lines drawn from perihelion and
aphelion, respectively, to it, fail to produce an appreciable angle.
In explanation of this statement, it must be understood that for all
uses on the earth the terrestrial system of direction (that is, using
the bearing of the north pole as a standard, with east to the left
and west to the right when the back is toward the pole) is entirely
adequate, but when dealing with the direction of celestial bodies,
a broader standard must be considered. North and South both have a
definite place in the heavens, being the points of the extended axis
of the earth, but east and west are only relative expressions. To
demonstrate this: it is possible for a man, traveling westward on the
Trans Siberian Railroad, to see from the rear platform, in the evening,
a certain star bearing eastward. At the same moment it is possible
for an officer of a transpacific liner in the early morning, to be
taking a sight of this same star bearing westward. In terms of absolute
direction that star bore the same from both sides of the world.
On the 21st of March the earth, sun, and celestial vernal equinox are
in range, with the sun between the earth and the equinox. For a place
in north latitude on the meridian of the terrestrial equinox, the sun
as usual bears south at noon this day, and hence the range mentioned
above bears south at that time.
This coincidence of bearing is only momentary, for the earth with its
onward motion immediately moves out of range and forms an angle between
the sun and the celestial equinox. At noon on the day succeeding the
equinox the sun bears to the left of the so-called First Point of Aries
(celestial vernal equinox). The sun according to terrestrial direction
always bears south by true compass at noon, yet the First Point of
Aries being at an infinite distance always bears the same by absolute
direction. If this point could be seen and a bearing of it taken by
compass simultaneously with the sun, it would be, perhaps, S. 1° W. and
so on widening the angle, roughly speaking, a degree each day.
[Illustration: The acceleration of Sidereal over Mean Time
FIG. 2.]
The interval between two successive transits of the sun across the
meridian constitutes a solar day, and likewise the period required
for a certain star to return to the same meridian is a sidereal day,
but these two days are not of the same length. The solar day we know
is 24 hours long, but its sidereal contemporary has a length of only
23 hours, 56 minutes (approximately) solar time. The sidereal clocks,
however, are geared to show 24 hours, sidereal time, in 23 hours, 56
minutes, solar time. By this it will be seen that in any given period
the face of the sidereal clock will show more hours than the solar
clock.
Both the solar and sidereal clocks start even at the vernal equinox,
about March 21st, but from then on, the sidereal clock gains on the
solar time clock about 4 minutes a day until in a year it is a full 24
hours ahead, showing that there is one more day in a sidereal year than
in a solar year. The approximate relation of the times shown by these
clocks is readily calculated by allowing a gain in the sidereal clock
of one hour for each 15 days after March 21st, or two hours for each
month.
In order to aid in a simpler explanation, let us again follow the earth
around its orbit and note the conditions that distinguish sidereal from
solar time. Let us once more assume it to be the time of the vernal
equinox, the clocks, both sidereal and solar, now show 0 hours, and the
sun, the earth and the First Point of Aries are in range. The earth
immediately moves out of line by virtue of its onward motion, and the
sun correspondingly appears to move eastward; this is imperceptible at
first, however, and not noticeable without a careful measurement, as it
seems to be swallowed up in the contrary (westward) diurnal movement.
After 24 hours of rotation from the instant of the equinox the earth
turns the meridian until it causes the First Point of Aries to transit,
marking sidereal noon of the first day. The sidereal clock at this
moment reads 24 hours, but a glance at the solar clock shows 11 hours
56 minutes A.M., about 4 minutes short of (solar) noon. An observation
will show that the sun has apparently moved about a degree eastward
of the hour circle passing through the First Point of Aries since the
preceding noon, and the earth must turn this extra degree before the
sun will be brought to the meridian, thus occupying the 4 minutes
mentioned above. In other words, the earth turns 360° in a sidereal
day but must turn about 361° in a solar day.
Three months after the vernal equinox, the angle between the First
Point of Aries and the sun becomes, in round numbers, 90°, and it
requires 6 hours for the earth to bring the sun to the meridian after
the passage of the First Point of Aries. In plainer language, when the
First Point of Aries crosses the meridian (sidereal noon) the sun is
about 90° to the left--about rising in the eastern sky; the earth must
make a quarter turn, or 6 hours, before it will be solar noon. Thus it
will be seen that at this point sidereal time is 6 hours ahead of solar
time.
In six months, when the First Point of Aries is on the meridian the
noonday sun is shining on the antipodes, and it lacks 12 hours of solar
noon. The difference between the sidereal and solar clocks has now
reached 12 hours and through a continuation of the same process the
interval between their readings, widens throughout the remainder of the
year.
When the 21st of March comes around again, and the meridian presents
itself to the sun and the First Point of Aries in range, a careful
count of the number of times this latter point has crossed the meridian
during the year, discloses 366¼ transits. That is, the earth has
actually turned about its axis 366¼ times. The sun is found to have
passed the meridian only 365¼ times. Counting the rotations of the
earth by the number of the sun’s transits while we are revolving around
him, causes the apparent loss of a day due to the earth unwinding
itself once, so to speak, during the year. The accumulated difference
amounts to one sidereal day. Hence it will be seen that a year
contains 366¼ sidereal days of 23 hours, 56 minutes each, and 365¼
solar days of 24 hours each.
Now for a recapitulation of the subject of time.
The rotation of the earth is the real standard of measuring time
intervals; the period required for this rotation does not vary. It
has been suggested that the tide waves have a minute effect on the
regularity of this movement, but the construction of our clocks is
such, that if any variation exists we are unable to detect it. Whether
we use the passages of the stars, or the transits of the sun to reckon
our time, it falls back in either case upon the diurnal rotation of the
earth.
Apparent time is measured by the seeming progress of the actual sun.
The time of its transit of the meridian is irregular, but is always
shown by its “dip,” culmination, with the sextant.
Mean time is reckoned by the revolutions of a fictitious sun, called
the mean sun, and the length of one of these revolutions is the average
of a year of apparent days. This, owing to its uniformity, is the time
used for the everyday purposes of life. The difference between apparent
and mean time is called the equation of time, and, by applying it
according to signs given in the Nautical Almanac, one can be converted
to the other as desired.
Sidereal time is indicated by the position of the First Point of Aries
relative to the meridian; it is star time. The stars make a complete
revolution of the heavens in 4 minutes less time than is required by
the mean sun. Therefore the sidereal day is that amount shorter than
the solar day.
The point of the celestial vernal equinox or First Point of Aries is
a sort of celestial “bench mark;” besides indicating sidereal time,
it serves as a point from which right ascension is measured eastward.
This subject has been discussed previously, but as it is intimately
associated with sidereal time, perhaps it may be made clearer since the
latter has been so fully explained.
Right ascension is measured on the celestial equator, precisely the
same as longitude on the earth, excepting that it is always measured
eastward through the full 24 sidereal hours, contrary to the diurnal
movement of heavenly bodies. Moreover, the meridian passing through the
vernal equinox is called the celestial prime meridian, and sometimes
the Greenwich of the heavens. There is another point of distinction,
however, between this prime meridian of the sky and our meridian of
Greenwich, which, while it does not effect practical navigation, has
to receive consideration in the long run; our longitude values on the
earth remain at all times constant, but owing to the precession of the
equinoxes the celestial prime meridian is slowly moving westward, thus
causing the right ascensions measured from it to become very slowly in
error (50´´ yearly).
The hour angle of a body is the angle formed at the pole between the
meridian and the hour circle passing through the body measured westward.
With all these important facts well in mind we will go ahead under a
slow bell, through a few more statements which may be found a little
perplexing. However, a careful study of the Time Diagram will, no
doubt, drive away the haziness so often surrounding the subject of
time.
[Illustration: This diagram represents the plane of the equator
looking down upon the North Pole. The 75° W meridian is chosen as that
of the observer and local time reckoned therefrom. The arrows on the
outer circumference indicate the directions of the earth’s rotation;
the other smaller arrows indicate the direction in which each element
is reckoned.
FIG. 3.]
The hour angle of the mean sun is the local mean time, and the hour
angle of the First Point of Aries is the local sidereal time. The local
mean time and longitude in time accelerated by Table III Nautical
Almanac plus right ascension of the mean sun is equal to the local
sidereal time. The right ascension of the meridian is the same thing,
exactly, as the hour angle of the First Point of Aries, and both of
these are identical with the local sidereal time. The sidereal time of
Greenwich mean noon is the same as the right ascension of the mean sun
at that time. The hour angle of a star plus the right ascension of the
same star is equal to the local sidereal time.
Difference of longitude can be represented by an interval of sidereal
time or by a difference of right ascension, precisely the same as by a
difference of solar time. Thus with the local sidereal time calculated
from an observation of a star, and the corresponding Greenwich sidereal
time taken from the Nautical Almanac, the longitude is at hand, by
turning their difference into arc. The fact that the actual time
interval is longer in the case of solar time than in an interval of
the same number of hours of sidereal time, has no influence on the
resulting difference of longitude. The number of degrees in any arc
can be the same, yet vary in linear measurement, but the same number
of hours of solar and sidereal time represent the same proportionate
part of a circle. It was just stated that the hour angle of a star plus
its right ascension is the same as the local sidereal time; now this
is also true of the sun. The hour angle of the mean sun plus the right
ascension of the mean sun is equal to the local sidereal time; by means
of this equality we are able to find the Greenwich sidereal time on any
occasion. It is necessary to have this element in order to compare it
with the local sidereal time, which we find by observation of a star,
to obtain difference of longitude (in time). In page 2 of the Nautical
Almanac will be found the Right Ascension of the Mean Sun at Greenwich
Mean Noon; this must always be taken out for the preceding noon. We now
have a measure of sidereal time to be added to a measure of mean time,
but it will be remembered in early arithmetic that an apple and a peach
can not be added together any more than ½ can be added to ⅓. The only
course to steer is to reduce the quantities to a common denominator,
or like quantities. So in handling these two varieties of time, solar
time must be accelerated by adding a correction to it, or sidereal
time retarded by subtracting an amount necessary to make it equal to
a corresponding value of solar time. The tables for the conversion of
one of these varieties of time to the other are found in the American
Ephemeris Tables II and III, Bowditch Tables 8 and 9, and at the foot
of pages 2 and 3, Nautical Almanac.
A practical illustration of this may make a clearer impression. In the
early evening of April 8th, secured a sight of Regulus; the chronometer
showed 8 days, 9 hours, 56 minutes, and 0 seconds (corrected), with
other necessary elements given, the sight is worked as usual to find
the star’s hour angle, which proves to be:
h. m. s.
H A Regulus 1 59 47 bearing west.
R A 10 3 02 RAMS 1 h. 6 m. 7 s.
-- -- -- GMT 9 h. 56 m. 0 s.
L. S. T. 12 02 49 Accel. 9 h. 1 m. 29 s.
G. S. T. 11 03 45 Accel. 56 m. 9 s.
-- -- -- ----- ----- -----
Long. in time 59 04 G. S. T. 11 h. 3 m. 45 s.
This being a star sight we obtain from it the sidereal time at place of
observation and as the chronometer carries Greenwich mean time we seek
the corresponding sidereal time by adding this and its acceleration for
a sidereal interval to the right ascension of the mean sun taken from
Nautical Almanac. The result is the Greenwich Sidereal Time.
It is occasionally required to find the sidereal time at the ship in
which case it is only necessary to apply the longitude in time to the
Greenwich Sidereal Time.
As Greenwich mean time is the most used and is the best understood it
is a very convenient practice to carry G. M. T. on the navigator’s
watch. It is readily converted into any other time with ease but serves
more purposes as it is without conversion. A stop watch is an excellent
instrument for taking time sights where great accuracy is essential. By
setting it at 0 minutes a man can observe alone starting the watch as
he makes contact with the horizon and when subsequently comparing with
the chronometer subtract the reading of the watch to get the G. T. of
observation.
The most expeditious way to convert time into arc is to multiply the
hours by 15 and add the number of minutes divided by 4 to get the
degrees; multiply the remaining minutes by 15 and add the seconds
divided by 4 to get the minutes; multiply the remaining seconds by 15
to get the seconds in arc:
Thus:
Long. in time 2 hrs. 42 m. 23 s.
30 30 45
10 5
--------------------
Long. in arc 40° 35´ 45´´
To change arc into time divide the degrees by 15 to get the hours;
multiply the remainder by 4 and divide the minutes by 15 and add to get
the number of minutes (m.); multiply the remainder of minutes (´) by 4
and divide the seconds (´´) by 15 carrying the division of tenths if
desired, adding the result to get the seconds (s.):
Thus:
Long. in arc 40° 35´ 45´´
2 hrs. 40 m. 20 s.
2 3
-------------------
Long. in time 2 hrs. 42 m. 23 s.
This may appear complicated at first but is much the quickest way of
conversion. However, Table No. 7, Bowditch is always available if
desired.
In getting an understanding of any time problem, that is such as
changing mean time into sidereal time; obtaining the hour angle of a
star or planet; in seeking the local time from the chronometer, or any
time values that are found perplexing, always draw a diagram. Make the
circle on the plane of the equator, with the pole as the center, and
meridians radiating from it towards the circumference of the circle.
Now imagine for the moment that you are at the north pole, and the
date is the 21st of September. The sun is traveling in the horizon,
and if the direction of the Greenwich meridian is known, this body
serves as a time piece, for the angle between this meridian (direction)
and the sun is the Greenwich time. This angle corresponds with twice
the angle between XII hours on the watch and the hour hand; or would
(disregarding equation of time) coincide with it if the watch’s face
was divided into 24 hours. Likewise when the vernal equinox or First
Point of Aries lies in the direction of Greenwich it is Greenwich
sidereal noon, and the subsequent angle that appears through the
rotation of the earth, shows the Greenwich sidereal time.
EQUATION OF TIME
It is necessary in considering this subject to reiterate some of the
statements made in the preceding talk on time, but, as they are very
important, no time is wasted by further impressing them on the mind.
Let it be understood that the apparent orbit of the sun is actually due
to the earth’s revolution around him, yet for simpler explanation it is
considered to be the sun’s own revolution.
The apparent movement of the real sun is not of uniform speed and, in
consequence, it has become necessary to devise a fictitious sun whose
assumed revolutions around the earth are at all times regular in their
rate.
The equation of time is the difference between these two suns and, as
they are at times in conjunction and at other times attain a distance
from each other of 16 minutes 20 seconds, and, moreover, as the real
sun is sometimes ahead and again in the wake of the mean sun, it
becomes evident that the equation of time is an ever varying quantity.
The irregularity of the sun’s apparent movement as compared with the
uniformity of the mean sun, is subject to two causes: First, the earth
travels in an ellipse, and, as the length of a degree varies in the
different parts of the circumference, the motion would appear to be
irregular, that is, if the sun actually traveled at a uniform rate, it
would, from the above fact, appear to us to be variable in its motion;
furthermore, the laws of forces only allow a body traveling in a circle
the privilege of a uniform speed so the earth, owing to its varying
distance from the sun, experiences a corresponding change in the amount
of attraction exerted upon it by the sun and its velocity, actually
becomes variable. Thus, during the winter, December and January, when
they are nearest each other, the attraction is strongest and the earth
increases its speed in revolution; while in June and July the earth
is at its greatest distance from the sun and the attraction is less,
resulting in a slowing down in the rate of the onward movement. As the
sun appears to us to take on movements corresponding to those of the
earth, these variable movements of the latter are seen in the apparent
motion of the sun. Second, the plane of the earth’s orbit is inclined
at an angle to that of the equator, which makes the sun appear to be
traveling at a variable speed along the ecliptic.
With these two errors combined influencing the apparent sun, he
becomes unreliable for regulating timepieces. The mean sun, which was
originated to obviate these irregularities, is assumed to travel in
a circle with the earth located in the centre, which disposes of the
first reason for an apparent variable motion; and again, the mean sun
revolves in the plane of the equator, thus eliminating the second
obstacle in the way of uniform time.
Now we will continue a little farther into the explanation of the
reasons for the irregular movement of the real sun. A law discovered by
Kepler, and named for him, provides that a radius from the sun to the
earth covers sectors of equal areas in equal times; a sector equal in
area to any sector covered in the same time. That is, when the earth is
in that part of the orbit near the vernal equinox, the radius of the
orbit will in a given time, say a week, sweep over a certain area; the
earth proceeds toward aphelion and when in the vicinity of that point,
the radius becomes greatly increased in length. Now in a week with
this longer radius, a far greater area would be covered if the earth
maintained the same rate of speed as at the equinox, but Kepler’s law
says, “equal areas in equal times,” so in order to conform with the
law, the earth’s speed of revolution must be reduced. The earth does
not slow down just for the sake of obeying Kepler, but at this part of
the orbit it is at its greatest distance from the sun and hence the
reduced attraction causes the earth to lag a little.
At the time of the autumnal equinox, an area similar to that of the
vernal equinox is covered. As the earth approaches perihelion, the
radius is gradually shortened by the eccentricity of the sun in this
part of the orbit and the increased attraction causes the earth to
speed up correspondingly. At the increased speed, the shorter radius
sweeps the same area in a week as at other parts of the orbit and
Kepler’s law still holds good.
As the apparent motion of the real sun corresponds exactly with the
real motion of the earth, it is evident from the above that the real
sun apparently moves at different rates of speed along the ecliptic,
faster in winter and slower in summer than the mean sun.
The value of a mean solar day is the average of a year of apparent
days, or in other words, there is the same number of mean solar days in
a year as there are apparent days.
In considering the effect of the variable motion of the earth in its
orbit, we will recall the conditions used when defining sidereal and
solar days. The former comprises the interval between two successive
passages of a certain star across the meridian, or perhaps better,
between two successive passages of the meridian over a star. This is
the true length of the earth’s rotation and is the standard to which we
may refer the length of the mean or apparent solar days.
Now it requires about 3 minutes 56 seconds longer for the meridian to
sweep around from sun to sun than from star to star, owing to the fact
that the mean sun moves uniformly eastward that amount daily, thereby
requiring the meridian, after reaching its position of yesterday
noon, to overhaul the mean sun this 3 minutes 56 seconds of eastward
movement. The mean sun maintains this uniform difference between its
days and the length of the sidereal day. Without this daily easting of
the sun, the sidereal and solar day would be the same.
But, in considering the apparent sun, we find the length of its days
continually varies from both that of the sidereal and mean day. This is
explained by the fact that the eastward movement of the apparent sun
is due to the movement of the earth in its orbit, and as this movement
becomes faster or slower the eastward movement of the sun becomes
correspondingly faster or slower. Thus, we readily see that with the
apparent sun moving eastward faster or slower at times, the length
of the apparent day must vary accordingly and we cannot establish a
uniform difference between it and the sidereal day, as in the case of
the latter and the mean day. The apparent days exceed the mean days
in length, between September and March, while the earth is traveling
fastest in its orbit. Beginning at the autumnal equinox with the
apparent sun eight minutes behind the mean sun, the former gains slowly
at first but with increasing rapidity. About the end of December, at
perihelion, it overhauls the mean sun and they are coincident _as
regards this correction only_. Leaving perihelion, the apparent sun
rapidly takes the lead but with a gradually decreasing amount until
at the equinox in March, reaches its maximum lead of 8 minutes.
Entering that portion of the year March to September, we find the earth
traveling slower and the mean sun gaining on the apparent sun; between
the vernal equinox and aphelion, the mean sun gains until both are
together at the summer solstice and then forging ahead the mean sun
attains a lead of 8 minutes in September.
It must be borne in mind that this error is caused only by the
eccentricity of the orbit and is but a component part of the whole
correction of the equation of time. The other portion is due to the
obliquity of the orbit, or its inclination to the equator.
This error is introduced through the fact that the apparent sun
moves in the ecliptic and the mean sun is assumed to proceed along
the celestial equator. In considering this phase of the question, we
will ignore entirely, for the time being, the error of eccentricity
described above.
The error of equation of time due to the obliquity of the orbit is a
simple one to see, but like many simple things it is easier to show it
by a diagram than to explain in words, so the reader is referred to the
accompanying figure, that a study of it may be made before proceeding.
[Illustration: FIG. 4.]
At the equinoxes, where the ecliptic and equator cross and the
solstices--the vertices of the ecliptic--that is, four times a year,
the true and mean sun are together, but departing from these points
they do not travel with the same right ascension, remembering that
right ascension is measured on the equator. Taking, for example, the
earth in that quadrant of the orbit comprised between the vernal
equinox and summer solstice, the apparent sun in the heavens would
be by cause of obliquity alone, to the right or to the westward of
the mean sun, and thus it will be seen that with the earth rotating
from right to left the apparent sun will cross the meridian first;
consequently between March 21st and June 21st that part of the equation
of time due to the obliquity of the orbit bears a minus sign when mean
time is desired from the apparent time. This correction reaches its
maximum half-way or 45° from the equinox, amounting at that point to
nearly 10 minutes.
[Illustration: FIG. 5.]
Now the reason for this difference between the
mean and apparent sun when each (so far as this problem is concerned)
moves along its respective path--the equator and the ecliptic--at the
same rate, is this: suppose the equator between the equinox and the
solstice is divided into an equal number of parts and an hour circle
drawn through each point of division. Beginning from the equinox (the
common apex of the triangle) the arc of each hour circle between
the equator and the ecliptic, forms the altitude of a right-angled
triangle, while the equator and the ecliptic are base and hypothenuse
respectively. Thus it will be seen that each portion of the equator
(base) is shorter than the corresponding part of the ecliptic as
defined the hour circle, to the extent of the ratio of the base to the
hypothenuse.
This amount increases with the increasing size of the triangle, but
a new element enters to counteract its effect. With the increasing
divergence of the ecliptic and equator, the divergence of the hour
circles becomes a factor and as the solstice is approached the
divisions on the equator, are represented on the ecliptic by gradually
decreasing spaces between the hour circles.
The combination of these effects produces the error due to the
obliquity of the orbit. The error has the opposite effect in the next
quadrant, that is, from June to September; and in opposite quadrants it
is the same.
So it will be seen that error due to the eccentricity causes the
apparent sun to lead the mean sun from December 31st to June 30th,
reaching a maximum of 8 minutes about April 2d. This sun then falls
astern until December, again attaining a maximum of -8 minutes about
October 1st. The error due to the obliquity of the orbit accumulates
between the equinoxes and solstices for at these points the two suns
are together and there is no error, but about the 6th of May, August,
November, and February, it reaches a maximum of 10 minutes; in August
and February, the mean sun takes the lead.
These two errors of equation of time combined algebraically will result
in the plain line of the diagram.
CALENDAR
The ancients, in order to keep track of past events and to anticipate
the future, devised a calendar, which, while not identical with the
one now in use, was of itself a remarkable production. They chose the
revolutions of the moon as their basis of measuring the passage of
time, and, as a lunar month contains only 27⅓ days, the 12 months used
in this early calendar comprised a year of about 354 days.
It became apparent as the world progressed that use of the moon
was very unsatisfactory for this purpose, as the calendar became
complicated and confusion resulted, owing to the difference between
the lunar and solar years. This condition remained until the reign of
Caesar, when that monarch determined to establish a more satisfactory
method of reckoning time. With the aid of an eminent astronomer,
he completely revised the calendar, using the revolution of the
earth around the sun as the standard for the length of a year. The
time required for this is 365¼ days, approximately, and as it was
inconvenient to include the ¼ day in the year, it was allowed to
accumulate for 4 years, when as a whole day, it was added to the end of
February.
Caesar, evidently proud of this accomplishment, honored himself by
naming his astronomer’s invention the Julian Calendar and in order to
further immortalize himself he changed the name of the seventh month
to July. Augustus, his successor, apparently envious of the honor his
predecessor derived from this source, and determined not to be thus
outdone in perpetuating his name, changed the month Sextilis to August.
The commonly accepted 365¼ days as the length of year is only an
approximation, however, and the small difference between this and the
actual length of a year began to accumulate until this weak point in
the Julian Calendar became a matter of moment. The exact length of
a year from vernal equinox to vernal equinox is 365 days 5 hours 48
minutes and 46 seconds, which lacks just 11 minutes 14 seconds of 365¼
days. This caused the dates of the equinoxes and solstices to keep
slipping back 11 minutes each year and when considerable time had
passed the difference became large enough to cause inconvenience; the
date of the vernal equinox having dropped back to March 11th in 1582.
In this year Pope Gregory, acting under the advice of an astronomer by
the name of Clavius, modified the Julian Calendar. He first added 10
days to restore the date, and then to forestall a further retrogressing
of the calendar, provided that only those century years divisible by
400 should be considered leap years. In this way the 11 minutes 14
seconds was prevented from causing further mischief. This calendar
known as the Gregorian Calendar is now in almost universal use, though
at first it was adopted only by Catholic countries.
It is interesting to note that the time consumed by the sun in making
his apparent yearly revolution from a certain star back to that star
again is a sidereal year of 365 days 6 hours 9 minutes 9 seconds.
The tropical year--the one in common use--is shorter, being the time
required for the sun to leave and return to the vernal equinox, or
First Point of Aries. This point, it will be remembered, is moving
westward about 50´´ annually, and it will be seen that while the sun
starts its eastward revolution among the stars, the equinox is very
slowly moving westward to meet him, thus making the tropical year about
20 minutes shorter than the sidereal year.
While discussing the calendar it is an opportune time to explain a
matter concerning the dates of the equinoxes and solstices. It has of
course been noticed by everyone that the vernal equinox occurs one year
on March 20th and another on March 21st, or the summer solstice on June
22d and yet another year on June 21st and so on.
Aside from the slight change due to the dropping back of the seasons in
the orbit by the precession of the equinoxes, the actual time of the
equinoxes and solstices may be considered as constant, yet the dates
vary a few hours.
The year in common use--the tropical year--contains approximately 365¼
days, yet we take account of only 365 days, the extra ¼ day being laid
aside for future reckoning. During the next year this 6 hours will be
augmented by 6 more; the next by another 6, making 18 hours ahead of
the calendar. The fourth year this amount reaches 24 hours, and a full
day, the 29th of February, is added to the calendar for that year, and
we are square again. But the different equinoxes and solstices occur
at just 364¼ days; taking for example the vernal equinox, it occurs on
the following (approximate) dates, which it will be noticed are 6 hours
later each year:
1908--March 20, 12 hours (leap year).
1909--March 20, 18 hours.
1910--March 20, 24 hours or March 21st noon.
1911--March 21, 6 hours.
1912--March 20, 12 hours (leap year).
It is evident by the above that the insertion of the extra day just
previous to the equinox in the leap years, sets the date of the equinox
back a day by the calendar. Juggling the ¼ day as shown above causes
the change in the calendar dates of these phenomena.
CHAPTER V
Corrections for Observed Altitudes
The observed altitude of a heavenly body, as measured with the sextant,
requires treatment for numerous errors to reduce it to the true
altitude and make it ready for use in working any of the navigational
observations. The amount of error varies in different bodies, the moon
requiring the maximum and the fixed star the minimum correction. All
the errors are not common to all bodies, that is, with some, certain
errors become so insignificant that they are cast aside.
These errors comprise the index error of the sextant, refraction, dip,
semi-diameter, and parallax. In Table 46, Bowditch, will be found
the combined corrections (index error excepted) to be applied to an
observed altitude of a star or planet and to that of the sun’s lower
limb. A supplementary table furnishes an additional correction to be
applied to the semi-diameter of the sun when accuracy is desired. These
corrections will now be discussed in the order named:
INDEX CORRECTION
The index and horizon glasses of a sextant are supposed to be parallel
when the zero of the vernier and the zero of the limb are in one, and
with this the case, the true and reflected images seen in the horizon
glass should exactly coincide. Any difference between them is the index
error.
It is seldom that a sextant is so well adjusted that no index error
exists, but it is not desirable to keep tampering with the instrument
with an attempt to eliminate this error, for it will in time injure its
accuracy.
By testing the sextant at each sight, the error can be closely watched
and allowance made for it in correcting altitude. The easiest and most
accurate method of ascertaining this error is by using a star in the
following manner: Set the zero of the vernier a little to either side
of the zero of the limb, and observe a 2d or 3d magnitude star--move
the reflected image past the real and note if they pass directly over
one another. If not, the horizon glass is not perpendicular and needs
adjustment. Bring the reflected star in exact conjunction with the real
star, and read off the index correction--if the zero of the vernier
is to the left of the zero of the limb--on the arc--the difference is
minus (-) and subtracted from the observed altitude; and if to the
right--off the arc--it is plus (+) and added. A well-known rule of
thumb expresses it thus: if it’s _on_ it’s _off_, and if it’s _off_
it’s _on_. The sea horizon is also available for determining this
correction and serves the purpose with fairly accurate results.
SEMI-DIAMETER
In measuring the altitude of certain bodies for navigational purposes,
it is necessary to determine the distance of the center of the body
above the horizon. To accomplish this in an accurate manner the lower
edge or limb is brought down to the horizon and the semi-diameter
applied to this measured altitude. When the lower limb is used, as is
the usual practice, the correction for semi-diameter is obviously plus
(+). The upper limb can be resorted to, however, should the lower side
of the body become veiled by cloud, and in this case the correction is
minus (-).
Semi-diameter of the sun is obtained readily from the Nautical Almanac
for each ten-day period, for it must be remembered that the sun is
continually changing its distance from the earth, and consequently
the diameter of the former is increased and lessened slightly at
different times of the year. For instance: On January 2d, when the
earth is near perihelion and we are at our nearest point to the sun,
the semi-diameter is 16´ 17´´.90, while on July 2d, when we are in
the remote parts of the orbit, the semi-diameter is only 15´ 45´´.69,
making a difference of over 32´´.
The moon being such a near neighbor of ours gives more trouble in
determining her diameter. Besides being greatly affected by her rapidly
changing distance from the earth, a further correction is occasioned
by the fact that our position on the surface is nearer the moon at
times than is the center of the earth. That is, when the moon is in
the zenith we are 4000 miles, the earth’s radius, nearer that body
than when she is in our horizon. It is evident that the direction of
the moon in our sensible horizon is at right angles to a perpendicular
erected at our place of observation and passing through the earth’s
center, and this again makes it evident that the moon is about equally
distant from the earth’s center and our position on the surface; but
as she ascends the heavens she comes nearer our position until in the
zenith the distance has been reduced by 4000 miles and the diameter
appears correspondingly larger. Draw a diagram and see for yourself.
This Augmentation of the Semi-diameter, due to the altitude, is found
tabulated in Bowditch, Table 18.
The semi-diameter becomes too small to consider in ordinary navigation
when observing any of the planets, and of course fixed stars are beyond
its scope.
REFRACTION
Everyone knows that the blade of an oar when dipped in the water
appears to be bent in a remarkable manner at the surface. This is
a clear case of refraction. Should the oar, however, be held under
everywhere at an equal depth, a square look downward at it would fail
to show any refraction. So it becomes evident that refraction is caused
by the rays of light passing obliquely from a rarer to a denser medium
or vice versa. A ray of light coming from a heavenly body to the earth
passes through a medium of gradually increasing density, from the thin
outer air to the denser atmosphere at the surface of the earth. The ray
of light consequently becomes curved downward and reaches the earth at
a point nearer the heavenly body than would be the case if the light
ray traveled in a straight line. The effect of this to the observer is
that the body appears higher than it really is. The difference between
the actual direction of the ray of light unaffected by the air, and our
line of vision as we see the body, is the refraction.
The amount of refraction ordinarily affecting an observed altitude
depends upon the distance of the body above the horizon. At the
zenith, the rays of light, entering our atmosphere perpendicularly, are
not deflected and refraction is nil. But, on the other hand, when the
body is near the horizon, the rays of light pass through the atmosphere
at a sharp angle and are consequently subject to the greatest bending,
thus giving us our maximum refraction. In fact, this element becomes so
unreliable in low altitudes that it is not advisable to observe a body
when less than 10° or 13° above the horizon. This in no way concerns
bearings taken of bodies in the horizon for amplitude, as refraction
affects the altitude and not the azimuth of a body.
DIP
It is a well-known fact to every seaman that by going aloft, he can
pick up a light sooner than on deck; that is, the higher his elevation
the wider his horizon becomes. The horizon of a man in a small boat is
only about 3 miles away, but, if he climbs to the bridge of a steamer
some 60 feet above the water, he finds that the horizon has receded
until he has a range of about 9 miles.
The fact that the horizon can be altered by changing the altitude
should appeal to every navigator as a possible means of getting a
horizon in foggy weather, by going aloft or getting as low as possible,
provided the fog bank is lying above or close to the water.
The altitude of a body is measured to the visible horizon, yet the
measurement must be adjusted to the sensible horizon before the true
altitude can be obtained. This correction is accomplished by applying
to the observed altitude the amount of the angle formed at the
observers eye by the planes of the sensible and visible horizons. The
angle is known as the dip of the horizon. It is readily seen that
this angle always makes the observed altitude too large, for the eye
if located at the exact surface of the sea, theoretically sees the
sensible and visible horizons in one, while at every elevation above
the surface it depresses the visible horizon correspondingly. It is,
therefore, always necessary to apply the dip as found in Table 14,
Bowditch, with a minus (-) sign.
An inspection of the table of dip will show that the rate of
increase of this error becomes more rapid as the height of the eye
is diminished. To illustrate: The reader will note that between an
elevation of 4 feet and one of 9 feet there is a difference of 1
minute in the dip, while higher up, say between 26 feet and 38 feet,
a difference of 1 minute is likewise found, yet in the first instance
there was a range of 5 feet and in the second a range of 12 feet. This
fact in itself is an argument in favor of observing altitudes at a good
height above the water.
In calculating a meridian altitude, an error in the dip directly
affects the result by a corresponding amount, so extra care should be
exercised in this respect. In this instance, we endeavor to locate the
body relative to our zenith and anything that affects its altitude
directly affects the latitude. In a time sight, a different principle
is involved. Here the position of the body as defined by the latitude
locates the apex of one angle of the astronomical triangle and hence a
small error in the altitude will very likely cause a greater effect on
the longitude.
An allusion was made under the caption of Refraction to the
displacement of the visible horizon by terrestrial refraction to detect
which requires watchfulness on the navigator’s part. The familiar
“loom” seen along the coast is an example of the workings of variable
refraction. Now imagine this distortion less aggravated with no land to
show its existence and you have a good illustration of this error.
Refraction of this nature is usually found during light airs and calms
when the different layers of air arrange themselves according to their
temperatures. The heated air over land below the horizon in hot weather
will displace the intervening horizon; moreover, when the air is warmer
than the sea, the horizon is elevated above the normal and, when the
conditions are reversed, the horizon is unduly depressed. Thus lights
become visible a little sooner after a hot day ashore. The Red Sea,
Gulf Stream, mouth of the Amazon, and other large rivers are places
where the horizon should be especially distrusted. Capt. Lecky, in his
famous Wrinkles in Practical Navigation, refers to an experience he
once had with this error. The latitude had been found “by an excellent
meridian altitude of the sun to be as much as 14´ in error. The time
was mid-winter--the day a clear cloudless one--the sea smooth, and
the horizon clean-cut. Five observers at noon agreed within the usual
minute or half minute of arc, nevertheless, on making Long Island (U.
S. A.) in less than two hours afterwards, the latitude was found wrong
to the amount stated. Many such cases have come under the writer’s
notice, but this one alone is cited on account of the magnitude of the
phenomenon.”
What Captain Lecky said in his work on navigation is reliable and
this should serve to make an impression as to the dangers of such
occurrences.
In clear weather this displacement of the horizon may be lessened
somewhat by observing from aloft. By extending the horizon, such
disturbing influences as the motion of the vessel and an irregular
horizon caused by rough sea are minimized. In hazy weather, however,
it is recommended to observe low, bringing the horizon as close as
possible.
PARALLAX
In calculating the true altitude of a body the distance of its center
above the horizon is supposed to be measured from the center of the
earth, or what is the same thing, the altitude above the rational
horizon.
The application of semi-diameter adjusts the measured angle with the
center of the body, while parallax corrects the error due to our
observing from the surface of the earth to the sensible horizon,
instead of from the center to the rational horizon.
Parallax, in other words, is the angle formed at the body by the lines
drawn from the observer’s position, and from the center of the earth,
respectively. This angle is subtended by the radius of the earth, and
it is obvious that the farther away a body is, the smaller the angle,
and consequently the less the parallax. So when dealing with planets or
fixed stars, it becomes insignificant and no parallax is considered.
The moon, on the contrary, is so close aboard that the angle of
parallax reaches a value of nearly 1´; as a minute of altitude means a
minute of latitude and in turn a mile, so with this body the error due
to parallax must be carefully determined.
In the case of the sun, however, it is somewhat of a waste of time
to bother with parallax, for it never exceeds 8´´ or 9´´ and such
fine calculation is uncalled for in ordinary navigation where so
many greater errors must be kept in sight. However, we desire to
eliminate every known element of error without undue figures, so it
is recommended that Table 20B, Bowditch, be used, where without extra
trouble the parallax may be found conveniently combined with the
refraction.
When a body is in the sensible horizon, the parallax is greatest. The
angle of parallax subtended by the radius of the earth is then an acute
angle of a right-angled triangle and is as large as it can possibly be
with the body at the same distance. As the body obtains altitude above
the horizon, the right angle of the triangle (at the observer) becomes
obtuse and our acute angle of parallax becomes smaller and smaller
until the body reaches an altitude of 90°--in our zenith, when the
obtuse-angled triangle has resolved itself into the perpendicular line
that passes through our position and the earth’s center. The angle of
parallax here disappears.
When a body is in the horizon, its parallax is known as Horizontal
Parallax in contra-distinction to Parallax of Altitude. The latter has
become generally known among navigators merely as parallax.
Our position on the surface causes a body to appear lower than if
viewed from the center of the earth, so the error of parallax is
added to the observed altitude; when, however, it is combined with
refraction it is subtracted in an observation of the sun, but added
when the moon is used.
The parallax of the moon is excessive because the radius of the earth
becomes a considerable amount when compared with the close proximity
of the body, and causes a considerable angle at the body between the
lines drawn from the observer and that drawn from the center of the
earth. The change in parallax is so great that it becomes necessary in
order to preserve accuracy to correct the observed altitude for index
correction, dip and semi-diameter, to secure an approximate corrected
altitude before attempting to correct for parallax. The horizontal
parallax, which is the angle subtended by the earth’s radius when
the moon is in the horizon, is taken from the Nautical Almanac, and
with this and the approximate altitude as arguments, enter Table 18,
Bowditch, and pick out, having regard for correction tables at the
right, the parallax and refraction combined.
The usual corrections to the observed altitude of the sun or stars
can be picked out at once from Table 46, Bowditch, where they are all
combined for a quick correction.
CHAPTER VI
Latitude
MERIDIAN ALTITUDE
It is surprising to us, in these advanced days of nautical science, to
read of our adventurous ancestors of a century ago navigating their
ships to all parts of the known and unknown world with nothing to guide
them but their dead reckoning and the latitude crudely obtained by the
method of meridian altitude. Many of our finest ships, as late as the
first decade of the nineteenth century, sailed to China and back with
no knowledge of their longitude save what the master guessed it to be.
Even in later days much navigating has been done in the less lucrative
trades by mariners who had no knowledge of the method of finding
longitude. It required more time and distance to navigate by latitude
and dead reckoning only, as it was not always safe to lay a course from
an indefinite position directly for the coast. It was the custom in the
old days to keep off soundings until on the latitude of the port of
destination, then steer due west, and whatever the longitude might turn
out to have been the master would sooner or later make the land in the
vicinity of his port.
The first step in obtaining the latitude by meridian altitude is the
measurement with the sextant of the sun’s altitude. This is done when
it reaches its highest point in its course across the sky; this occurs
when it bears due N. or S. true and this moment is local apparent noon.
A few minutes before this time the image of the sun should be brought
to the horizon, and by swinging the lower part of the instrument the
image will be made to swing likewise in an arc; the lowest point of its
lower edge (limb) should then be brought in contact with the horizon as
closely as the circumstances will permit. The image will keep rising
from the horizon, but by using the tangent screw it can be continually
brought back to contact. At noon it will hang, and dip below; the
reading of the sextant at this moment is the meridian altitude.
In working the problem three quantities are used and the navigator must
be familiar with them:
The first is the zenith distance (_z_), which as its name implies is
the sun’s (or stars) distance from the zenith. Zenith is 90° from the
horizon, so the true altitude of the body subtracted from 90° is _z_
the quantity desired.
The second element is the declination (_d_), which is the distance in
degrees, minutes, seconds, of the body either north or south of the
equator. This is taken from the Nautical Almanac.
The third and resulting quantity is the latitude, which is the distance
in degrees, minutes, seconds, of the ship either north or south of the
Equator.
The altitude observed taken with the sextant at noon is corrected for
semi-diameter, parallax, dip, refraction and instrument error (if any
exists). These corrections are explained in detail in _Corrections for
Observed Altitudes_.
The declination of the sun is constantly changing between 23½° N. and
23½° S. This is given in the Nautical Almanac for each two hours of
Greenwich mean time with the difference for each hour given for each
day. So it becomes necessary to ascertain the declination at the moment
of observation, namely, at local noon. This anywhere in the Atlantic
will occur subsequent to Greenwich noon, as the sun (apparently) passes
around the world from the eastward to the westward once a day--24
hours--which corresponds to 360° of longitude. The rate of travel
is therefore equivalent to 15° in an hour. Hence if the sun crosses
Greenwich meridian and five hours later crosses the meridian of the
ship, say in 75° W., the interval is 75 divided by 15, or 5 hours.
During this interval the sun has changed in declination northward or
southward and should be picked out of the Almanac for 5 hours Greenwich
mean time.
When the zenith distance and declination are at hand the latitude is
obtained by a mere algebraic addition, which is, _z_ + _d_ = latitude;
where, if the body bears south the _z_ is marked +, if north it is
marked -; if the declination is south it is marked - and if north it is
+. The result of the addition if - indicates south latitude, if + north
latitude. The meridian altitude of a star, planet or moon is found in a
similar manner. The formula of _z_ + _d_ = latitude, having regard to
signs named as above, is applicable to each. The declination and the
correction of the observed altitude are picked out of the Almanac and
Bowditch tables in a somewhat different manner peculiar to each body.
It is found by many navigators to be more convenient to observe a
body for meridian altitude by time than in waiting for the “dip.” The
altitude is taken at exactly local apparent noon in case of the sun
and the time of meridian passage in the cases of other bodies. This
expedient is especially desirable in observing stars, as the horizon is
not as distinct and the “dip” not so easily detected as with the sun.
In order to secure the local mean time (L. M. T.) of a star’s transit,
the G. M. T. of the star’s transit over the Greenwich meridian is found
in the Nautical Almanac (p. 96) for the first day of the month and
correct for the day by table on next page (N. A.). The ship’s mean time
of transit will be the same, as both sun and star hold their relative
positions as the star moves from Greenwich to the ship’s meridian
except for the small retardation of the sun’s movement over the star’s
movement. This is best found at the foot of page 2, Nautical Almanac,
where the longitude in time gives the correction to be subtracted from
(G. M. T.) of transit which will give the local mean time of transit
at ship--the time to observe the star. An observation of a planet is
similarly handled. The moon is somewhat unreliable owing to its rapid
changes in position and the large corrections necessary to correct the
altitude, and is consequently rather an unpopular body to observe.
However, there are times when she might prove valuable in giving
position when much needed.
In the case of the sun the time of transit is local apparent noon, by
applying the longitude in time gives Greenwich apparent time of local
noon, and corrected for equation of time gives Greenwich mean time of
transit.
It is often necessary to report the latitude at noon very quickly to
the master. This can be accomplished by calculating the problem to a
point where the addition or subtraction of the observed altitude is
all that is necessary to give the latitude. The corrections are applied
in advance by the estimated altitude, and declination corrected by the
estimated longitude. Art. 325, Bowditch, gives the constants to be used
in four different situations.
CIRCUM- OR EX-MERIDIAN ALTITUDE
It frequently happens, especially in the higher latitudes, that an
aggravating mass of cloud drives over the sun or other objects that
you are chasing, with the tangent screw, and it is lost from view
together with all hope of a meridian altitude. But such an unfortunate
occurrence as the loss of the mid-day latitude may be averted by
employing the Circum-meridian sight or Ex-meridian, as our English
cousins call it.
The mariner accustomed to its use “shoots” the sun and notes the time
by chronometer or watch. Or on cloudy days, he would be standing by,
near apparent noon watching for a chance to catch a glimpse of the
object through a rift of cloud, and thereby forestall the loss of his
latitude.
The theory of this observation is extremely simple, being merely to add
to the observed altitude, taken before or after apparent noon when the
sun is being considered, the amount of rise or fall between the time
of sight and the time of culmination, and proceed with this amended
altitude as in an ordinary meridian altitude sight.
The use of this method of obtaining the latitude is restricted to
certain limits. Those who use Bowditch Tables will find themselves
restricted to 26 minutes from the time of transit and a declination of
63°, while Brent’s Ex-meridian Tables allow a greater scope and their
limit of 70° of declination includes many stars that would be otherwise
unavailable. A good guide is to never allow the number of degrees in
the zenith distance to be exceeded by the number of minutes from noon.
In very high altitudes circum-meridians are not to be recommended, and
the higher the altitude, the more accurate must be the time used. This
is plain when it is realized that the lower the sun’s altitude at noon,
the more nearly its diurnal path approaches the line of the horizon;
with the lessening curve of its course, comes a lessening rise near
noon, hence less accuracy is needed in the exact time of sight from
that of transit. In the tropics, however, where high altitudes of the
sun prevail, the clouds do not offer such an element of bother as they
do farther north or south, and there this problem as applied to the sun
loses its popularity.
In practice the use of the tables of Bowditch makes this problem an
exceedingly simple one, requiring but few figures. Table 27 contains
the value of rise of the body for one minute, but as this rise varies
as the square of the interval from noon, it becomes necessary to resort
to another table (26) of constants for a multiplier, in lieu of the
number of minutes from noon. That is, if we should multiply the amount
of rise or fall for 1 minute by the number of minutes from noon, we
would not be taking into account the decreasing rapidity of rise or
the increasing rapidity of fall as the body approaches or leaves the
meridian. But Table 26 provides a multiplier which reconciles this
inequality and gives the proper correction to apply to the observed
altitude.
This quantity is added in every case where the upper transit is
observed but subtracted when a sight is taken below the pole where the
conditions are reversed.
There are several pamphlets and books on the market from which the
correction to the observed altitude may be obtained. All are simple in
form and with their explanations are readily understood. Notable among
these Ex-meridian Tables are those by Capt. Armisted Rust, U. S. N.
The circum-meridian is a reliable method of finding the latitude, but
the time used should be accurate to produce satisfactory results. If,
however, the conditions be favorable, it is not necessary to discard
this observation even if the correctness of the time is somewhat in
doubt, for in Towson’s Ex-meridian Tables is found this note:
“If equal altitudes be taken before and after the meridian passage,
half the elapsed time may be employed as the hour angle for determining
the reduction. Or, when the altitudes before and after noon differ by
only a few minutes, the mean of the two may be reduced by employing
half the elapsed time as the hour angle for reducing the mean altitude.”
In practicing this suggestion it is necessary, in order to preserve
accuracy, to put the vessel on the nearest east or west course during
the run between these equal altitude observations. This is imperative
in a swift vessel.
The stars and planets offer themselves for use in this problem as in
all others, and here they possess special advantages of which the
mariner may well avail himself. Indeed, it may be said in truth that
when a horizon can be obtained the latitude is always available
through this problem.
And right here should be impressed upon the navigator the great
advantage of becoming familiar with the stars, not merely those of
greatest brilliancy, but the “lesser lights” that can be observed.
Among the latter, especial acquaintance should be sought with those
whose right ascensions place them in the gaps between the larger stars,
thus almost the entire heavens are included in the scope of operations,
making the latitude and longitude practically always available,
provided again there is a horizon.
Star charts, planispheres, and globes are for sale everywhere and no
study is more interesting than that of the ways of these celestial
travelers. They appear and pass each day, year after year, until
you consider them as old friends, and, as you come on deck for the
mid-watch, you look for Orion, for instance, the same as you look for
the members of your watch at their proper stations.
But we are off our course. The increasing popularity of the
circum-meridian and its undoubted accuracy when used with time obtained
from a carefully rated chronometer, is breaking the hold of the
time-honored meridian altitude. There is no waiting with cold fingers,
perhaps, for the body to dip for this sight, just shoot the star, note
the time and duck for the chart room to work it up.
The most favorable position of a body for a circum-meridian altitude
is one in which the rise and fall near the meridian are slow. In the
case of the sun, it was explained that a low altitude proved the
best, but, in the case of the stars, we find another condition; those
near the pole, or in other words, of large declinations, describe
such small diurnal circles that here also the change in altitude is
correspondingly small, thus fulfilling a desired condition for the
successful working of this problem. To illustrate this point the reader
is referred to Polaris. Now this star has an extremely small diurnal
circle and it will be remembered that the altitude is for all practical
purposes the same for a half hour either side of the meridian, showing
the extreme slowness of its movement of revolution.
The stars are used in the same way as the sun except, of course, that
the distance from the meridian becomes the star’s hour angle instead of
local apparent time. This is readily obtained as follows: Adding to the
Greenwich mean time the sidereal time of the preceding Greenwich mean
noon (Nautical Almanac), together with the acceleration of Greenwich
mean time (Table 9 Bowditch), gives the right ascension of the
meridian. Taking the difference between this latter quantity and the
right ascension of the star, we have the star’s hour angle, west, if
the right ascension of the meridian is greater than that of the star,
and east, if contrary conditions exist.
The circum-meridian, as well as the straight meridian altitude, is
available for use of stars near the meridian below the pole, and, as
one proceeds into higher latitudes, the pole becomes more and more
elevated, offering thereby more opportunities for practicing this phase
of the problem. The only feature to be remembered in this case is
that the body is higher at the time of a circum-meridian than when it
transits, so the correction to be applied to the observed altitude must
be subtracted (-) in order to obtain the meridian altitude.
The planets, too, are used by the ex-meridian altitude method, but
being wanderers in the heavens their right ascensions and declinations
must be determined for the Greenwich date from the Nautical Almanac.
The amended altitude of any body is assumed to be the meridian altitude
and is used in the familiar formula _z_ + _d_ = latitude (see Latitude
by meridian altitude); but it must be borne in mind that the result is
not the latitude at noon but at the time of sight. If the observation
was made say 9 minutes before noon and the latitude considered to
be the position at local apparent noon as in an ordinary meridian
altitude, there would be an error of 3 miles from the correct position
for a 20-knot steamer.
Another point to be guarded against is that when taking several
altitudes and their corresponding times their mean cannot be obtained
in the ordinary way, but each altitude must be separately reduced and
the mean taken of the results.
It is again necessary to diverge from the subject to impress on the
mariner an urgent warning against anything but the most untiring
vigilance in the care of his chronometer, and the keeping of accurate
time. If this element cannot be depended upon there will be many
hours of anxiety coming to him and probably sooner or later downright
disaster. The almost universal establishment of time signals in all
good-sized sea ports of the world together with radio time signals sent
broadcast allows but little excuse for not obtaining a good rate by the
time a vessel is ready for sea. Every well-known work on navigation
deals with the subject of rating chronometers and so no space will
here be given to it. After reading this talk on one of the most
important and up-to-date observations where so much depends upon the
accuracy of the time, the reader cannot fail to appreciate this earnest
admonishment.
POLARIS
The process of finding the latitude by means of Polaris is valuable,
comparatively short and the result, if the conditions are favorable, is
accurate. We will consider it first in a general way.
The imaginary line representing the earth’s axis, if extended
indefinitely, is presumed to pierce the celestial sphere at the
celestial pole, therefore for an observer standing at our north pole
this imaginary point would be exactly in the zenith and hence 90° from
the horizon just as the pole is 90° from the equator, these amounts
evidently bear a relation to one another. Should the person at the
pole leave his frigid surroundings and proceed toward the equator, he
would note that the pole had dropped lower and lower in the heavens,
precisely in proportion to his progress southward, until at length,
when the equator (latitude 0°) was reached, the pole would be observed
to be exactly in the horizon (altitude 0°). From this it is easy to
deduce the statement that the altitude of the celestial pole is equal
to the latitude of the place of observation.
The object of this problem then is to obtain the altitude of the
celestial pole. This point, unfortunately, is marked by no star of
which a direct altitude may be observed to aid the navigator in
reaching this desired result. There is, however, a star of the 2d
magnitude, called Polaris (because of its proximity to the pole) with
a polar distance of only 1¼°. As all fixed stars are apparently
revolving in circles around the celestial pole, this star joins the
grand procession with its little radius of 1¼°.
It is plain that at no time can this star be more than the amount of
this radius (1¼°) from the pole, and when on the meridian either above
or below the pole the full amount of the radius is subtracted from or
added to the corrected altitude of the star to obtain the true altitude
of the pole. When the star is on a line passing through the pole
and parallel to the horizon at its elongations as it is called, the
altitude is then equal to the latitude, for its elevation is the same
as that of the pole.
It requires 24 hours for this star to complete the small circle of
revolution, the same time required by every star; its movement is
necessarily very slow. By computing its hour angle, we can locate its
position on this circle, and hence by applying a correction to its
altitude, subtracting or adding according to the position of the star
above or below the pole, we will obtain the altitude of the pole.
A rough estimate of the position of the pole may be secured by noting
the position of the Big Dipper, the second star in the handle, called
Mizar, is approximately in line with Polaris and the pole.
We will now proceed to show the method by which the hour angle is
obtained:
In the talk on Time, it was stated that the local (astronomical) mean
time plus the right ascension of the mean sun is equal to the local
sidereal time; and again, that the right ascension of a star plus its
hour angle equals local sidereal time. With these facts as a basis, the
formula for latitude by Polaris given in the Nautical Almanac will be
followed in explanation.
[Illustration: FIG. 6.]
The time of observation must be noted by chronometer and converted
into local (astronomical) mean time; this must be corrected by Table
III (Nautical Almanac) in order to change this solar interval into
a sidereal time interval; to this converted time must be added the
Greenwich sidereal time of mean noon (page 2); that is, the hour angle
of the First Point of Aries, or what is the same thing, the right
ascension of the mean sun; to this sum must be applied a correction for
longitude, in time, taken from the foot of page 2, N. A. The sum is the
local sidereal time.
The reason for the correction of longitude is this: The difference
between the right ascension of the mean sun at noon on two successive
days is 3 m. 56 s., the same as the accumulated difference between
solar and sidereal time in 1 day. Now we take from the Nautical Almanac
this element for Greenwich mean noon, yet the sun has since covered
the distance equal to the longitude, and during the interval required
to do this, the sidereal time has accelerated over the solar an amount
which bears the same ratio to the 3 m. 56 s., that the longitude in
time bears to 24 hours. The Nautical Almanac handles the terms of this
proportion in tabular form at the foot of page 2. It is stated that the
sun has traveled from the meridian of Greenwich to the local meridian,
and it might be suggested that at the time of observation the sun has
covered this amount plus the local hour angle or the local astronomical
mean time. This is true but the amount of local hour angle has been
previously accelerated to sidereal time by the correction to local
astronomical mean time.
With the local sidereal time enter Table I (Nautical Almanac) and pick
out the correction to be applied according to sign to the altitude.
It is probably needless to say that the observed altitude must be
corrected for index error, dip and refraction before applying this
latter correction, which converts it into latitude.
This is called the Nautical Almanac method and is sufficiently accurate
for navigational purposes, but should a greater refinement be desired
there are tables of further corrections given in the American Ephemeris
and Nautical Almanac.
It is always advantageous to get an observation of a star near twilight
or dawn, in order that a well-defined horizon may be available; but,
in taking a sight of Polaris, another important feature is to be
considered. When the star’s hour angle is at or near 6 or 18 hours,
that is, near that part of its orbit cut by a line passing through the
pole and parallel to the horizon, it is rising or falling most rapidly,
with the result that a small error in time will produce a considerable
error in the hour angle, an error of 3 minutes introducing a difference
of 1´ in the latitude.
It is quite worth while, therefore, to select a time for the
observation of Polaris when this star is near either of its
culminations, its highest or lowest positions, where the time need not
be especially accurate; but by carefully noting the time it is possible
to get good results at other times when the horizon is defined. By
using the position of the star Mizar, as suggested above, however, the
navigator will be greatly aided in selecting the most propitious time
for observing Polaris.
CHAPTER VII
Azimuths and Amplitudes
Of all the navigational instruments now in practical use, there are
few, if any, that exceed the mariner’s compass in usefulness to
mankind. The part it has played in the development of the world has
been most important, and its utility is no less to-day than in the
past, for the intercourse of nations is still guided by the compass
needle. With such a responsibility depending on this instrument, it
would naturally be supposed that its indications must be very accurate,
but, on the contrary, the needle is swayed by the slightest magnetic
influence and points North only on rare occasions; and in steel vessels
only by mere chance.
The needle is drawn from true north first by the direction of the
earth’s magnetic force which is not coincident with the meridian owing
to the position of the magnetic poles. The north magnetic pole being
in the extreme northern part of Canada, all the lines of force in the
northern hemisphere converge toward this locality. The needle when
otherwise undisturbed lies in the direction of these lines of force and
takes an angle with the meridian depending on the locality.
The amount of divergence from the true north, or variation, as it is
called, differs in different localities but is readily obtained by
a glance at the chart where each compass rose shows the amount of
variation at that place. From a magnetic course, or bearing, the true
course, or bearing, is readily found by the proper application of
this variation, which may be either easterly or westerly. The _true_
course is to the _right_ of the magnetic course, when considered
from the center of the compass, in _easterly_ variation; T. R.
E.--True-Right-Easterly. Remember these three words and the whole
lesson is learned, for if true is to the right in easterly variation
it must be to the left in westerly; and if true is to the right in
easterly, the magnetic course must be to the left of true course
in easterly and to the right in westerly. In this way the true and
magnetic courses are converted one to the other at will.
If we were to sail always in an entirely wooden ship, our compass
troubles would be very few, for the above would include every
phase of the situation. As wood is non-magnetic the compass would
be uninfluenced by outside disturbances. Wood, as a ship-building
material, having been so much displaced by iron and steel, the use of
these metals has brought many problems to solve in connection with the
deflection of the compass needle.
The effect on the compass needle of the magnetism in a vessel and her
cargo is known as deviation and is very complicated, owing to many
influences which are at work at all times giving an ever-varying value
to this element of error.
The causes of deviation and its treatment in the way of compensating
the compass are subjects much too extensive for this little book;
furthermore, they are carefully dealt with in a half dozen of the
well-known works on navigation, so we will touch only on the every-day
side of compass work.
The deviation changes with every alteration of the vessel’s head, owing
to the change in position of prominent parts of the vessel’s hull
relative to each other, to the compass, and to the terrestrial lines of
force (magnetism).
As a result of these influences on the compass needle, the mariner
has three courses to deal with. The first is the true course, which
is based on a compass whose needle points true north. The second, the
magnetic course, is taken from a compass affected by variation alone
and therefore pointing to the magnetic pole. The third is the compass
course, or that course actually shown by an ordinary standard compass
in a steel ship, affected by the error of variation combined with the
error of deviation.
The combination of the deviation and the variation is the compass
error and is obtained by adding the deviation and variation if both
are of the same name, the compass error taking that name; for instance
suppose we have a variation of 2° W. and deviation of 10° W., the
combined error is 12° W. If, however, the variation and deviation are
of different names, it becomes necessary to find the difference between
the two and name the result after the greater quantity; thus, with a
deviation of 4° E. and a variation of 10° W., the error is 6° W.
The compass error is applied to compass course to obtain true course
and vice versa by the same rule as for variation.
The navigator in planning his course between two positions lays the
parallel rulers on these positions on the chart and carries this
direction to the nearest compass rose. This may be a true rose, in
which event he remembers his T. R. E. rule, reversing it in this
case, and with the variation given on the chart secures the magnetic
course. In an iron or steel vessel, the deviation for that course
must be ascertained from the deviation card by trial or from a Napier
Diagram direct and applied to the magnetic course in order to obtain
the compass course. This is accomplished precisely as in finding the
magnetic from the true course (to the left if deviation is easterly and
to the right if westerly). The course by standard compass is now at
hand by which we can steam from one selected point to the other.
The deviation as has been said is an ever-varying error, and
consequently it is quite impracticable to depend wholly on a fixed
deviation card. We may take aboard some magnetic cargo or change our
latitude to a great extent, the vessel may be pounded excessively by
heavy seas, a stroke of lightning or by stranding; all these are causes
liable to affect the deviation more or less.
In order to forestall the serious consequences that are liable to
attend such a derangement of the normal and expected deviation, the
careful navigator takes azimuths or amplitudes on every course when
practicable. Azimuths and amplitudes are nothing more nor less than
astronomical bearings of heavenly bodies; they indicate the true
bearing of the body, and the difference between this bearing and the
bearing taken simultaneously by standard compass is the compass error.
The azimuth of a body is the angle at the zenith between the meridian
and the vertical circle passing through the body. It is customary,
however, to consider the azimuth as measured by the arc of the horizon
rather than by the angle at the zenith. It is measured from the north
or south point according to the latitude, toward either the east or
west point, through 180°.
An amplitude, unlike an azimuth, is restricted as to time of
observation, for the body must be on the horizon either rising or
setting; and should be observed when the sun is about its own diameter
above the horizon and with a not excessive height of eye. The amplitude
is measured from the east or west point through 90° to the north or
south point. If the body observed has a south declination and is
rising, the amplitude will be East so much South; if declination is
north, East so much North, for a body rises in the East point when its
declination is 0°--on the equator.
The principle of the amplitude lies in the solution of a right-angled
spherical triangle, whose sides are the body’s polar distance, the
co-latitude, and the zenith distance which is 90°. We desire the
complement of the angle at the zenith. It is unnecessary to compute
an amplitude, for in Table 39, Bowditch, will be found the desired
bearings for different latitudes and declinations. The sun will be
found the most satisfactory of the heavenly bodies to utilize for
amplitudes.
There are two methods of calculating an azimuth, one known as the time
azimuth and the other as the altitude azimuth. The former is the most
popular owing to the tables that have been compiled, an inspection of
which facilitates the navigator in quickly obtaining the true azimuth
of a body. Before entering the tables, it is necessary to have as
arguments the latitude and declination, and, if using the sun, the
local apparent time, or for stellar work the hour angle. Should the
star’s hour angle exceed 12 hours, 12 hours should be subtracted from
it, and the remainder used as P.M. time. A planet may be employed
precisely in the same manner as a star.
One of the simplest and most expeditious methods of securing the
azimuth is by means of a diagram. Upon this convenient invention
the bearing of a body can very quickly be taken off with a pair of
rulers. Weir’s Azimuth Diagram is sold by the Hydrographic Office for
a very small sum. The only argument that can be used against its use
is that it requires a small table to lay it upon. Simple and complete
directions are printed on the diagram.
The altitude azimuth is often computed at the same time as the ordinary
A.M. and P.M. time sights, utilizing the altitude of the body for
both operations. The principle involved in computing both an altitude
azimuth and a time azimuth is the solution of the same astronomical
triangle for the same angle, but in the case of the altitude azimuth
three sides are given (the co-latitude, the zenith distance, and polar
distance) to find the angle at the zenith. In the time azimuth, two
sides and the included angle are given (the polar distance, co-latitude
and local apparent time or hour angle) to find also, the angle at the
zenith.
The azimuth found by computation should be named North if in north
latitude, or South if in south latitude.
It has been customary to add up the logs, divide by 2 and the cosine
will be half the azimuth named from the elevated pole, but a more
expeditious way is after adding the logs seek in the log haversines
and find the azimuth directly but named from the opposite pole to the
latitude. With the correct bearing of the sun, and its simultaneous
bearing by standard compass at hand, the compass error is found by
merely taking their difference. Now this error, as said before, is
composed of the sum or difference of the deviation and the variation,
so, if either is subtracted from their sum, or added to their
difference the remainder is the other quantity. The variation being
always known is subtracted from or added to the compass error to obtain
the deviation, thus checking the deviation card for that particular
course the vessel was steering at the time of observation.
With the compass error at hand, many students become perplexed as to
the proper manner of dealing with this error, and finding from it the
deviation. The compass error is first named, by considering the two
bearings (compass and true) from the center of the compass; if the true
is to the right of the compass bearing, the error is easterly, if to
the left, westerly.
Now should the variation happen to be identical with the compass error,
both in amount and in name, there is no deviation; if the variation is
0°, then the whole error is deviation. If by chance the compass error
is 0°, it indicates that the variation and deviation are equal in
amount and opposed to each other in their influence on the needle. The
deviation, in such a case, naturally takes the opposite name from the
variation.
In separating the variation from the compass error, it is necessary to
exercise a little thought and to consider what deviation applied to the
given variation will produce that compass error. This will be readily
seen after a little practice. There are, however, some rules which are
here given, by which the deviation can be obtained mechanically.
The deviation is the difference between the variation and the compass
error if they are of the same name or adding them if of different
names. It is given the same name as the compass error unless the
compass error is subtracted from the variation, when the deviation
takes the opposite name.
Or a diagram in which the error is shown by its particular number
of degrees east or west of the true north line may be drawn and the
variation likewise properly shown east or west of true north. If the
error is to the left of the variation the deviation is west and if to
the right the deviation is east.
CHAPTER VIII
Longitude
The longitude of any position on the earth is its distance east or west
from the meridian of Greenwich, which has been chosen as the meridian
of origin. Longitude is measured on the equator eastward and westward
through 180°, completing in this way the whole circumference of the
earth.
The circumference of every circle comprises 360°, whether it is a great
circle of the earth or any of the parallels which range in size from a
point at the poles to a great circle at the equator. There are always
360° but the length of each degree is determined by the size of the
circle. Thus a degree of longitude on the equator is 60 miles, while
on the 50th parallel of latitude it is only about 39 miles, owing to
the decreasing size of the parallels of latitude. A minute of longitude
on the equator, like a minute of latitude, is equal to one mile, but
the difference between the meridians in actual distance decreases
toward the poles gradually lessening the linear value of a degree of
longitude. Thus it will be seen that when it is desired to represent
a difference of longitude in distance, it must be done in terms of
departure (miles) corresponding to the particular parallel of latitude
of the position.
The sun apparently moves around the earth in its diurnal motion,
covering 360° in 24 hours, whether the declination is north or south,
and a little simple division shows that in one hour he passes over
15° of longitude, whatever the latitude. This reduced shows that 1°
is passed over every 4 minutes. As the standard time, the world over,
is reckoned by the movements of the sun, it is plainly seen that when
considering longitude, a definite relation exists between time and arc
(°-´-´´). Owing to this relation, time and arc become interchangeable
by a simple process of conversion.
So it follows that if we have the time at Greenwich by a chronometer,
and through a trigonometrical calculation we determine the local mean
time at the ship, the difference in time between Greenwich and the
ship’s meridian represents the longitude in time, which is readily
converted into arc.
The calculation involved in determining the local mean time is the
solution of the astronomical triangle, or in other words it is a
problem in spherical trigonometry. This triangle has its apex at the
pole with one side as the polar distance (90° - declination of the
observed body), another side the co-latitude (90° - dead reckoning
latitude) and the third side the zenith distance (90° - the corrected
altitude of the body).
It is one of the principles of trigonometry that with any three
elements given in a triangle any of the remaining elements may be
computed; that is, any angle or side is obtainable. The solution of
the astronomical triangle for various elements includes the finding of
the zenith distance and from this the altitude, which forms the main
feature of the problem involved in the New Navigation. It also provides
us with the angle between co-latitude and the zenith distance, which
is the azimuth of the body, by which the mariner is able to ascertain
the error of his compass.
The most important feature of the astronomical triangle is the angle
at the pole, known as the hour angle, which when found secures for
the navigator his local time. The problem presents itself in the form
of three sides being given to find one angle. It is found by the time
sight formula, which is too well-known to need any discussion here.
The shape of the triangle is determined by the declination of the
body, its altitude and the latitude of the vessel, and the polar or
hour angle; and it stands to reason that a formula will not produce
the same accuracy in the hour angle with every shape of the triangle.
For instance, in high latitudes or when the body has a declination
approaching 90°, the accuracy of the time sight formula becomes
effected.
Another very important point to bear in mind when observing a body with
the view of computing its hour angle, is its azimuth. When the bearing
is nearly east or west, on the prime vertical, the body is rising or
falling faster than at any other time, and an error in altitude or
latitude will produce the least error in the resulting longitude. The
necessity for close attention to this point is increased with the
latitude. Observations for time taken when the body has an azimuth of
less than 45° or over 135° are wholly unreliable.
The sun does not always cross the prime vertical in his daily track
across the heavens, for under certain conditions, say during the
northern winter, he will rise southward of east and set southward of
west. Under these adverse conditions, the calculation of longitude is
not dependable, and the best a navigator can do when using the sun is
to observe as soon as he has an altitude sufficient to clear the excess
refraction existing near the horizon.
It is under such circumstances that star sights are of incalculable
value, for a star can always be found in a suitable position with but
little waiting, or we may employ the New Navigation method, where the
azimuth of the sun is as good one place as another.
In order that a body will cross the prime vertical, the latitude must
be of the same name and greater than the declination. In conditions
cited above the declination of the sun is south and the latitude is
north, hence the body will never be on the prime vertical. If the
latitude is less than the declination, the sun’s diurnal track is
tilted toward the zenith, instead of away from it as when the latitude
is greater, and the result is that the sun, while never on the prime
vertical, approaches it for a time after rising, then recedes again. It
should be observed when at its nearest point to the bearing of east of
west.
The bearing of various bodies can be readily found by an inspection of
Hydrographic Office Azimuth Tables Nos. 71 and 120, the declination and
latitude being used as arguments.
There is a method of finding the longitude known as the equal
altitude method, but it is not valuable. The conditions are exacting
where accurate results are required and when these conditions exist
the ordinary time sight is available and at its best advantage, so
longitude by equal altitudes is not popular. To secure good results,
the body must have an altitude above 70° and near the prime vertical;
and, furthermore, the ship must be kept on an east or west course or
remain stationary. The theory of the problem is simplicity itself, and
for this reason is very alluring, but the best use that equal altitudes
can be put to is the determination of chronometer error ashore, and
in these days of radio time signals even this use is almost obsolete.
The rule is as follows: Observe the sun’s altitude, simultaneously
noticing the time by chronometer and clamp the sextant to prevent any
chance of the altitude becoming disturbed. When the sun has fallen to
the same altitude as of the forenoon sight, note the time again by
the chronometer. The mean of the two times, corrected for chronometer
error, equation of time, and the equation of equal altitudes due to
change in declination, in case of the sun, is the Greenwich apparent
time corresponding to our local noon or our longitude in time, which
should then be converted into arc.
The stars and planets are available as well as the sun for the finding
of longitude and when there is a distinct horizon, stellar sights
have many advantages. The problem depends upon the solution of the
astronomical triangle by the same formula as with the sun.
There are a few points of difference between a time sight of the sun
and one of a star or planet needing explanation. In the case of the
former body, we naturally compare the solar time of Greenwich with
the solar time of the local meridian, but in stellar work we employ
for this comparison stellar time, or, as it is more popularly called,
sidereal time. So it becomes necessary to turn the Greenwich mean time
of the chronometer into Greenwich sidereal time and compare it with
local sidereal time. The difference, as in mean time, is the longitude
in time, which is converted into arc in precisely the same way.
The Greenwich mean time is turned into sidereal time by adding to it
the right ascension of the mean sun, taken from the Nautical Almanac
and the acceleration for the Greenwich mean time (Table 9, Bowditch).
The local sidereal time is the result of an addition of the star’s
right ascension and the star’s hour angle, the right ascension is
taken from the Nautical Almanac without correction if a fixed star is
being considered and the computation of a time sight gives the star’s
hour angle. The local sidereal time being the right ascension of the
meridian, it follows that the angle from the vernal equinox to the star
plus the angle from the star to the meridian is what we desire; hence
the above rule for obtaining the local sidereal time.
Should the star bear east of the meridian, the local sidereal time may
be found by subtracting the (easterly) hour angle from the star’s right
ascension or adding them as above and subtracting 24 hours. Reference
to the Time Diagram, Fig. 3, will make these points clear also.
It is customary to add up the familiar logs of time sight--sec. lat.,
cosec. p. d., cos ½ sum, sin, remainder--divide by 2 and seek the H. A.
(hour angle) in the A.M. or P.M. column of Table 44, Bowditch, using
the log as a sin; but a more expeditious way is to use the sum of the
logs as the log haversine in Table 45 and pick out the hour angle
directly.
CHAPTER IX
Sumner Method
Every mariner who has reached a position in the profession where he
is intrusted with the responsibilities of navigating a vessel is
under obligation to the late Capt. Thomas H. Sumner, of Boston. This
shipmaster discovered and developed the principle of the so-called
Sumner or Position Lines, a principle which has proved of inestimable
value and which, with its subsequent improvements, has well-nigh
revolutionized the methods of navigation.
The discovery was purely accidental and for that reason is interesting.
Here, in Capt. Sumner’s own words, is how it occurred: “Having sailed
from Charleston, S. C., 25th November, 1837, bound for Greenock, a
series of heavy gales from the westward promised a quick passage. After
passing the Azores, the wind prevailed from the southward, with thick
weather, after passing longitude 21° W., no observation was had until
near the land, but soundings were had not far, as was supposed, from
the edge of the bank. The weather was now more boisterous, and very
thick, and the wind still southerly.
“Arriving about midnight, 17th December, within 40´ by dead reckoning,
of Tuskar light, the wind hauled S.E. (true), making the Irish coast a
lee shore. The ship was then kept close to the wind and several tacks
made to preserve her position as nearly as possible until daylight,
when, nothing being in sight, she was kept on E.N.E. under short
sail, with heavy gales. At about 10 A.M. an altitude of the sun was
observed, and chronometer time noted; but having run so far without any
observation, it was evident that the latitude by dead reckoning was
liable to error and could not be entirely relied upon.
“However, the longitude by chronometer was determined, using the
uncertain D. R. latitude, and the ship’s position fixed in accordance.
A second latitude was then assumed 10´ to the north of the last and
working with this latitude a second position of the ship was obtained
and again a third position by means of a third latitude still 10´
further north.
“On picking off these three positions on the chart it was discovered
that the three points were all disposed in a straight line lying E.N.E.
and W.S.W., and that when this line was produced on the first-named
direction it also passed through the Smalls Light. The conclusion
arrived at was that the observed altitude must have happened at all
three points, at the Smalls Light, and at the ship at the same instant
of time. The deduction followed that, though the absolute position of
the ship was doubtful, yet the true bearing of the Smalls Light was
certain, provided the chronometer was correct. The ship was therefore
kept on her course, E.N.E. and in less than an hour the Smalls Light
was made bearing E. by N. ½N. and close aboard. The latitude by D. R.
turned out to be 8´ in error.”
If the captain had worked more time sights using different latitudes,
he would have added new positions on the line to which he refers,
each placed upon it according to the latitude used. Had he cared to
pursue his experiments farther, and used latitudes very wide of his
dead reckoning position, he would have discovered that the resulting
positions instead of lying in a straight line, were in a curve and an
arc of a circle.
The principle involved is very crudely illustrated in the following
experiment: Let the reader consider himself aboard ship lying at
anchor--say a full-rigged ship, so as to insure a foremast of good
height. Lower the dinghy and take along a sextant.
We start with a series of measurements to determine the angle, as read
from the sextant in the dinghy, between the truck and the waterline
about the vessel. As a result of these measurements, we discover that
this angle becomes smaller as the distance from the vessel increases.
Carrying our tests farther, suppose when the sextant shows the altitude
of the fore truck above the waterline to be 70°, that the distance
to the vessel be determined. With this distance as a radius and the
foremast as the center, we row in a circle around the vessel, the
sextant will continue to read 70° all around the circle.
It is thus demonstrated that a circle surrounds that foremast upon
which the altitude of its truck is everywhere 70°--a circle of equal
altitudes.
Not being quite sure of this interesting fact, perhaps, another angle
is selected by moving a little farther from the ship. The sextant shows
the fore truck to have an altitude of 50°; the distance to the vessel
is established, whereupon the dinghy is rowed around the vessel with
this distance as a radius. Again the sextant reveals no change from
50° and it is clearly shown that we have moved about on a circle of
50° elevation of the truck.
We can continue experimenting in this way until the distance from the
ship becomes so great that some physical condition prevents our reading
the angle of the truck’s altitude.
These investigations show that there is a system of concentric
circles of equal altitude about every elevated object like the little
undulations we have seen so many times produced by the splash of a
stone thrown into a pool.
These circles of equal altitudes surround not only elevated terrestrial
objects but also celestial bodies, as will now be shown. As the sun
is the most convenient body for this illustration, let us substitute
it for the fore truck of the foregoing experiments, while for the
waterline of the vessel we will use the point on the earth touched by a
plumb-bob suspended from the center of the sun.
This point will fall on the equator on the 21st of March or
thereabouts, as the sun coming up from his southern declination
crosses the equator into north declination at this time. The instant
of the transit is the vernal equinox. Now this point will be found an
excellent one from which to study this problem, but, as this takes some
time and the sun is ever on the move, we will imagine ourselves endowed
with the power of Joshua to command the sun and moon, which will enable
us to study this phenomenon while free from the restlessness of the
Universe.
First of all, it must be understood that the sun shines on one
hemisphere of the earth at all times; it matters not how the earth is
tipped in relation to him, one half of the world is always enjoying
sunshine. The center of the lighted area is the spot directly beneath
the sun where the plumb-bob touches and about this point lies the
system of concentric circles of equal altitudes of the sun.
Under the conditions shown above, the sun is in the zenith of the
terrestrial vernal equinox, shining on the earth for a distance of 90°
in every direction; but its altitude diminishes in direct proportion
with the distance of the observer from the point of the equinox. On the
great circle everywhere 90° from the equinox the sun is in the horizon
with an altitude of 0° (provided we disregard dip and refraction).
Suppose the members of some intrepid expedition have reached the
northern or southern pole; they would, at the time being considered,
see the sun in the horizon and in the direction of the meridian passing
through the vernal equinox.
Eastward along the equator 90° of longitude from the vernal equinox,
the inhabitants are just resting from the toils of the day, for with
them the sun is setting in their western horizon, while away to the
westward 90° the people are showing signs of activity, for it is just
sun-up in their eastern horizon.
So all around the world just 90° from this selected position and at
this appointed time is a circle of equal altitudes, namely 0°, for is
not the sun seen in the horizon at all points on this circle?
The altitude of the sun is 90° at the point of observation and 0° on
its outer circle of altitude; these are the two extremes and between
them lies an infinite number of concentric circles of equal altitude
for navigators to utilize. The zenith distance, derived by subtracting
the altitude from 90°, indicates the distance of each circle from the
center or sun’s position. Thus if an observation was taken by some
bewildered mariner in which the altitude was found to be 80°, the
corresponding zenith distance of 10° multiplied by 60 would indicate
that the altitude was taken 600 miles from the sun’s position, or to
put it in another way, the circle of equal altitudes upon which the
observer was located in this case had a radius of 600 miles.
What is true of the sun on the equator regarding the principle of
the circles of equal altitudes holds good throughout its range
of declination, the whole system moving north and south with the
continuous change of declination and from east to west with its
apparent diurnal motion.
In the quoted article, Capt. Sumner shows a method by which the
position of a vessel may be established on some particular circle of
equal altitude; it matters not where the observed body happens to be
at the time, for with the Nautical Almanac and chronometer it can be
located should we care to know. The navigator, however, cares to deal
ordinarily only with a very small arc of the circle embraced within
his immediate whereabouts. Should he be somewhat uncertain of these
he would simply require the use of a longer line to extend beyond the
limits of his possible position.
Except when in a latitude that differs but little from the declination
of the observed body, the circle of equal altitude will be sufficiently
large to allow the mariner to represent its arc in his vicinity by
a straight line. Thus the lines of position used to plot a vessel’s
position on the chart are in reality chords or tangents of the circle
of equal altitude. In geometry it will be remembered that we used to
study about circumscribed and inscribed polygons and here we have a
practical application of their use. If we consider the line of position
to be a tangent, it is one side of a great polygon with a vast number
of sides circumscribed about the circle of equal altitude; and if we
consider it to be a chord, it is likewise a side of a great polygon
inscribed within the circle of equal altitude. It matters not, however,
if the line or curve of position is considered a straight line, except
in the ill-chosen condition of the body near the zenith when the radius
of the circle will be proportionately small. If exactly in the zenith
there will be no circle of equal altitude at all and the sextant will
measure an altitude of 90°. It is comparatively rare, however, that
such a condition will embarrass the use of this method.
Another point to be remembered in connection with the inscribed and
circumscribed polygon propositions and one which has a practical
application in the use of position lines, is that the tangent or chord
of a circle is at right angles to the radius passing through the point
of tangency or center of the chord. It follows that the sub-celestial
or terrestrial position of the observed body, being at the center of
the circle, is always at right angles to a line of position.
This important fact gives the navigator an opportunity to check his
compass error each time he establishes a position line, by comparing a
compass bearing of the body taken simultaneously with the measurement
of the altitude, with the true bearing.
To establish a position line as Capt. Sumner did and as it was done
for years afterwards, by assuming two latitudes usually 10´ each side
of the dead reckoning latitude, and drawing the line through the two
resulting longitudes, is known as the chord method. The two longitudes
being positions on the circle a line drawn between them is a chord of
the circle.
The work of computing a time sight is more or less laborious to
everyone and with some seafarers forms their most arduous mental
exercise. At any rate no one wants to work any more than is necessary
to insure accurate results. So when establishing a position line it
will often be found convenient to use the short cut known as the
tangent method.
With the latitude by account work the observed altitude as in the
ordinary time sight, instead of assuming two latitudes. Seek the true
azimuth in the tables or on diagram, using the latitude and declination
employed in the time sight and the local apparent time gained from it,
as the arguments. The true azimuth, it will be remembered, always bears
at right angles to the position line. Hence if the azimuth is laid down
through the position furnished by the time sight, the position line may
also be readily plotted at right angles to the line of azimuth at the
time sight position.
The navigator now-a-days is expected to think in position lines when he
is clear of the land, as a pilot thinks in shore bearings and marks.
That is, he must see these imaginary lines of the different visible
bodies, and keep track of their availibility for his particular use. It
is easy to get into the habit of this, for they are simply astronomical
bearings instead of bearings on distant terrestrial objects, with the
distinction that the celestial bearing allows of a 90° correction to
produce a position line.
The morning sun on the prime vertical with a sufficient altitude to
avoid any dangerous refraction, will produce a north and south line of
position. During the forenoon as the sun passes toward the meridian,
the northern end of the position line will move in direct proportion
with the body’s change in azimuth to the eastward and the southern end
to the westward, until at noon with the sun on the meridian we have an
east and west position line.
It will be seen that at one moment of the day it is a very easy matter
to establish a line of position; the mere working of a meridian
altitude does this. This simple expedient of finding a position line
was utilized a great deal as a means of making a landfall in the days
before chronometers were perfected. In those good old days, before
the clipper ship era, time was not held at such a premium as in the
present hustling period, and a few days more or less at sea mattered
but little. The shrewd shipmasters then would keep well offshore until
in the latitude of Boston or the Virginia capes, as the case might be,
when they would haul due west and let her go, making, no doubt, first
rate landfalls, if the old pig yoke was in good working order.
The value of a position line was demonstrated to the writer some years
ago when bound in from the eastward and running into a heavy and very
extensive fog bank somewhere southeastward of Halifax. During a break
in the prevailing conditions the navigator succeeded in securing
an ex-meridian sight and fortunately got a fairly good idea of the
latitude. The vessel was under sail and making but slow progress, and
as a result of the protracted period of overcast sky the longitude
became considerably a matter of guess work. The vessel, however, was
kept on a _west course_ with a careful allowance made for the set.
“Sir William Thompson” was kept going at regular intervals and it was
surprising to see the soundings check up with the chart as the vessel
approached, crossed, and left astern the Roseway Bank, southward of
Cape Sable. One felt as sure of the position as did the old Nantucket
sailor in crossing “Marm Hackett’s garden.”
In cases where the soundings do not check so precisely as in this
instance, it will sometimes be found a great help to lay off to scale
the depths obtained on the edge of a piece of draftman’s transparent
linen. Place it on the chart in the line of the course, and, should the
soundings fail to agree, move the scale forward and back or to either
side, always preserving the direction of the course, until a position
is found where the soundings on the scale agree with the depths given
by the chart.
Progress has been made in the science of navigation as in all other
sciences, and the modern shipmaster is not obliged to hold aloof from
Nantucket Shoals and Georges Bank under ordinary conditions as our
ancestors were compelled to do, for with a correct chronometer and a
knowledge of the position line such outlying dangers have been robbed
of many of their anxiety-producing elements. Before showing the method
of working around such places another point of value of the position
line is called to the reader’s attention.
A line of position extended until it reaches the land or some danger
will indicate to the mariner the bearing of that particular point of
the coast or danger. If it so happens that this point is not the place
of destination, the navigator, not being able to lay a course direct
for his objective port through inability to determine the vessel’s
distance offshore, overcomes the difficulty by sailing a sufficient
distance at right angles, then hauling on to a new position line
parallel to the original one. This is similar to what our ancestors
did in the simple way cited above. If the line lies in the direction
of an off-lying or isolated shoal that is dangerously near the course,
an offset like that shown above will allow a course parallel to the
position line to be sailed in safety. Here is an example to show its
useful application:
A steamer sailing from St. John, N. B., for New York proceeded but
about 10 hours on her voyage, when she ran into a terrific gale. The
master was soon forced to heave his vessel to and ride it out as best
he could. The driving snow and mountainous seas occupied the attention
of the officers in their efforts to save the steamer and in this way
the dead reckoning position became a matter of mere guesswork. The wind
after some 20 hours in the northeast quadrant hauled to the northward,
at length blowing out in the northwest with clearing weather.
It was the master’s intention to pass through the South Channel,
between Georges Bank and Nantucket Shoals, but as he had lost his
reckoning to such an extent he hesitated about laying a course through
such a danger-strewn locality.
In the late twilight immediately following the clearing sky, the
master succeeded in catching the altitude of a star bearing 300° and
established a line, the direction of which led close westward of
Cultivator Shoal (a 6-foot spot on Georges Bank). So to be on the side
of prudence and give this shoal a good berth, the master steamed 8
miles at right angles to this position line. The course or direction of
the new position parallel to the first was found to lead directly into
the range of visibility of Nantucket Lightship. So the master’s mind
was put at rest as he laid his course along the second position line,
knowing he would at length make the lightship.
It often happens that a distant mountain peak is visible and the sun
is in a suitable position to establish a set of cross bearings, using
the mountain for one object and the sun for the other. Now with what
has previously been stated, it is hardly necessary to remind the reader
that a “line of position” obtained from observations of the sun will
be at right angles to the sun’s true bearing; therefore, in order
to judge whether these objects are properly placed to give a good
intersection, due consideration must be given to the relative bearings
of the objects. It is evident that the sun must bear by compass nearly
in the direction of the mountain or in the opposite direction to have
the position line and the line of bearing of the mountain cut at nearly
right angles. Of course, as with any set of cross bearings the angle of
intersection may still be effectual if the lines cut at 50° to 60°, but
the nearer a 90° cut the more accurate the resulting position.
A position line is liable to displacement through a variety of causes
among which is an inaccurate altitude and through incorrect Greenwich
mean time. In the former instance, an error of 1´ will displace the
position one mile; if the altitude is 1´ too large, the correct
position of the line will be 1 mile directly away from the bearing of
the body and vice versa. The effect of an error in time upon a position
line is to displace it bodily eastward or westward the amount of arc
corresponding to the error in the chronometer; the direction of the
line is, however, unaltered. The sun carries his system of circles of
equal altitude with him from east to west as he travels along a certain
parallel of latitude corresponding to his declination (neglecting the
slight change in declination). It is quite evident that any arc of a
selected circle, will, if its position is plotted on a small scale
chart--say every 20 minutes--be found continuously parallel with
itself. And the intervals between each two plotted positions of the arc
will be 5° (of arc) the corresponding value of 20 minutes. Thus the
displacement of the position line due to an error of time is explained.
If the time was slow, the line was too far to the eastward, if fast, it
was too far to the westward.
The value of a position line has been demonstrated, yet with all it
does not positively establish the position of a vessel. The mariner
in locating his vessel in a harbor does not usually stop after he
has taken one bearing, but proceeds to find another object whose
bearing will make a favorable “cut” with the first, and thus at their
intersection determines his position. As a further check against
possible error a third object may be chosen and, if the three bearings
plot without forming a triangle at their intersection, a very reliable
fix will be obtained.
What applies to terrestrial objects thus employed may be used as an
illustration to be followed in taking celestial bearings. If the
mariner establishes a position line and knows his vessel is located
at a point somewhere along it, let him look about for another body
so placed that the position line derived from it will make a good
intersection with the first line; if all data are correct this point
will indicate the position of the vessel.
When the sun is used this is seldom possible but in lieu of another
body the sun can again be employed to establish the second position
line after it has moved sufficiently in azimuth to make a good cut. The
thought no doubt immediately arises as to the effect of the vessel’s
change in position during the interval. This is easily taken care of by
means of the course and distance run during the interval between the
sights.
The first position line must be considered carried bodily by the vessel
without change of bearing from its first position to the position of
the second observation. That is, if at 9 A.M. a position line was
established bearing in a 15°-195° direction, and the vessel then
steamed and made good a 40°-course for 6 hours and 10 knots an hour,
when another position line was established, the 15°-195° line of 9
A.M. would be moved bodily in a 40° direction 60 miles; where its
intersection with the second line would indicate the position of the
vessel at 3 P.M. The determination of position at sea by employing two
position lines of a body with the run between sights is called Sumner’s
double altitude problem.
It has already been shown that one body, notably the sun, can be used
to get an intersection of two of its lines of position by waiting a
sufficient time between observations for the body to change its bearing
at least 30°, the nearer 90° the better. The relationship between the
interval of time and the amount of change of bearing varies greatly,
depending upon the latitude of the observer and declination of the
body. For example, let us consider the two extreme cases: Suppose a
mariner to be observing the sun on the equator on March 21st, he will
note practically no change in azimuth during the whole forenoon. Yet
another mariner in the Polar sea, whose latitude differs about 90° from
that of the former, will have the sun encircling his horizon making the
whole amount of the sun’s movement a corresponding change in azimuth.
Therefore it will be seen that with a low-riding sun (or other body)
the change of azimuth is greater in a given time, and for this reason
the position lines derived from the sun are more advantageously
practiced in higher latitudes, especially in winter. This is a point of
great value in view of the fact that the sun’s diurnal course is such
that it is never on the prime vertical in northern latitudes during the
winter months, making longitudes derived from chronometer sights very
unreliable.
But to go back to the mariner on the equator whose latitude and sun’s
declination so nearly agree. He is in a predicament should he persist
in the plan to determine his whereabouts by position lines of the sun.
In such an unusual case, it would be well to resort to some other
method or wait until evening and determine the ship’s position by
establishing the position line of some star or stars. It will be but
a few days before the ship’s progress will cause the sun to leave his
right course across the sky and take the hour circles at an angle. Take
a case when the sun at noon has a zenith distance of 10°, the change
of azimuth during the forenoon is still small, but suppose the bearing
was noted 1 hour, or even less, before noon and again in similar amount
after noon, a change will be found of perhaps 90°, the difference
of moving from the southeast quadrant (if declination is south of
latitude) to the southwest quadrant. In this way, a remarkably good
cut may be had within a comparatively short time. The foregoing will
convince the reader that he must be governed by the change of bearing
and not by time elapsed, in predicting the value of the cut of his
position lines.
In the use of position lines, it is necessary to bear in mind, that,
when the body’s altitude begins to approach the zenith, or, what is the
same thing, when the ship is getting close to the body’s sub-celestial
position, the circle is getting proportionately smaller. Under such
conditions the arcs of the circle of equal altitudes can no longer
be shown as a straight line. The double altitude as it is ordinarily
practiced is here impracticable. And even outside this impracticable
area, discretion must be shown. The dead reckoning position must be
proportionately accurate, and the assumed latitudes must be brought
correspondingly close together, in order to have a shorter line of
position, because the curvature of the circle is getting sharper as the
sub-celestial point is approached. To put it in another way, a smaller
arc must be used in order to avoid the error due to excessive curvature.
Very good results can be obtained by noting the time of observation
by chronometer (G. M. T.) and correcting it for equation of time in
order to get Greenwich apparent time. This, if converted into arc, is
the longitude of the sub-solar position. By using the Greenwich mean
time to correct the declination taken from the Nautical Almanac for
that day, the latitude of the sub-solar position may be obtained. Plot
this position on the chart and use it as the center of a circle; then
with the zenith distance (90° - altitude) as a radius, draw an arc in
the probable position of the vessel. Somewhere along this arc is the
ship’s position. The bearing of the sun (rather hard to get so nearly
overhead) corrected for compass error, reversed, and laid off from the
sub-solar position will give a fair idea of the position of the vessel.
Now by waiting a sufficient time for the sun to change its azimuth
enough to make a good cut and using its new sub-solar position as a
center with the zenith distance of a second observation as a radius,
an arc may be drawn which will intersect the first arc at the position
of the vessel. The run between the sights will, of course, require
the first arc to be carried forward as the first position line in the
ordinary double altitude problem.
JOHNSON’S METHOD
It is not always found convenient to plot the position lines of a set
of observations on a chart; perhaps for lack of a chart of proper
scale or possibly for want of the chart itself. Again many navigators
do not take kindly to the graphic method, but prefer to solve their
latitudes or longitudes by computation. In any event Johnson’s Method
comes as a relief to such persons, saving them from the arduous duty of
establishing a set of position lines by the chord method of assuming
two latitudes to get two longitudes.
Johnson’s Method can be practiced in both the double altitude problem
of the sun, where the first sight, or position line, is brought forward
to the second sight by correcting it for the intervening run, or where
stars are used simultaneously.
Chief among its merits is the saving of figures. It is only necessary
to compute two (instead of four) chronometer sights in order to find
the ship’s position, thus obtaining a mathematically accurate result
by a short cut. But also a great advantage in the Johnson Method is
that the resulting longitude is obtained by calculation and it is not
necessary to plot the lines upon the chart to secure the position.
In using Johnson’s method it is not absolutely necessary to observe two
stars simultaneously as the quick work of a good man is sufficiently
close for the practical purposes of navigation.
It becomes evident to anyone reading the foregoing pages that every
ordinary time sight places the vessel on a circle of equal altitude,
the longitude resulting from the computation, depending on the
latitude, by dead reckoning, used. Now rather than work two sights
employing two assumed latitudes on either side of the supposed
position, make the calculation only once, using the latitude by account.
Suppose by way of explanation that the altitude of a star bearing S.
55° E. is observed simultaneously with that of another star bearing
S. 25° E. The longitudes derived by working the time sight of each
should be identical, provided the altitudes are the true altitudes, the
Greenwich time is without error, and the latitude used is correct. A
combination of accuracy, indeed, and one not likely to be experienced
often in actual practice. However, a skillful navigator should find no
great difficulty these days in always having the correct Greenwich time
at hand. There is always, of course, an opportunity for the display
of skill in measuring altitudes, refraction particularly being an
illusive element and not always easy to detect. But if care has been
taken to eliminate the errors as much as possible from the time and the
altitude, it is safe to consider any discrepancy between the resulting
longitudes as accountable to an error in the dead reckoning latitude.
The method of obtaining the ship’s position from the difference in
the longitudes, derived from double or simultaneous observations, was
originated by A. C. Johnson, R. N., and its many advantages have for
years made it the most popular form among progressive shipmasters.
The working of this problem involves the application of a correction
to each calculated longitude in such a way as to bring them into
agreement. The tables (Bowditch Tables 47 and 48) furnish this
correction, which is known as the longitude factor and is symbolized by
the letter F. It constitutes the change in longitude due to a change
of 1´ in latitude. This quantity changes directly with the change of
azimuth of the body; for example, the change in longitude is nil if the
change in latitude is made on a due north or south line, and change in
longitude increases as the change in latitude is made on lines bearing
more and more eastward or westward. So it is necessary in order to
obtain these corrections to have the true azimuths of the bodies at the
moment of observation to use as an argument in the table of longitude
factors. These are readily taken from the Azimuth Tables or diagram
using the data furnished by the time sight.
The two longitudes obtained from time sights in which the same dead
reckoning latitude is used, lie on the parallel of this latitude,
but (unless the two longitudes happen to be coincident) the ship’s
position is either north or south of this parallel according to the
error existing in the dead reckoning latitude. If the observed azimuth
of the body (or bodies) fall within the same quadrant or in opposite
quadrants, the correct longitude will be found to the eastward or the
westward of both calculated longitudes. This is clearly shown in Fig.
7; both azimuths are between south and east. If the observed azimuth of
the body (or bodies) fall in adjacent quadrants say, one between south
and east and the other between south and west, the ship’s position
will be found between the two calculated or erroneous longitudes.
The position of this true longitude is determined by means of the
before-mentioned factors. The factor of a longitude is the distance of
the true longitude east or west of the meridian passing through the
calculated or erroneous longitude, assuming the latitude to be in error
1´. The moment of this factor, it will be seen, depends on the azimuth
of the body, which in turn determines the direction of the position
line.
[Illustration: FIG. 7.]
The combination of the two factors, by adding if the bodies are in
the same or opposite quadrants or vice versa, is the combined error
in difference of longitude due to 1´ of error in latitude. It now
becomes a matter of proportion by which to obtain the error in
the dead-reckoning latitude. As the combined error in difference
of longitude for 1´ of latitude, is to 1´ of latitude, so is the
difference between the two calculated longitudes, to the error in
latitude.
The longitude factors are based upon an error of 1´, so if the error
is more than 1´ it becomes necessary to multiply the factor by the
error in order to obtain the correction to the calculated or erroneous
longitude.
[Illustration: FIG. 8.]
An altitude may be taken of any body and after a suitable change in
bearing has taken place (not less than 30°) a second altitude may be
taken and the first longitude advanced for the run during the interval
to the parallel of the latitude by dead reckoning at the time of second
sight.
In the usual event of a disagreement in the calculated longitudes the
rule of procedure is as follows: With the body’s true azimuth at each
observation, the difference between the longitudes and the latitude by
dead reckoning, used at second sight, enter Table 47, Bowditch, and
take out the corresponding numbers. If the azimuths are in adjacent
quadrants, these quantities should be added, but if in the same or
opposite quadrants, they must be subtracted. The result in each case
gives the combined error in difference of longitude for an error of 1´
in latitude.
It is now only necessary to divide the difference between the two
longitudes by this combined error and we have the error between the
correct latitude and the latitude by dead reckoning. Now multiply the
error in latitude by the number taken from Table 47 corresponding to
the first longitude, to obtain the correction to that longitude, and
by multiplying the same error in latitude by the number corresponding
to the second longitude we have the correction for that longitude.
The application of these corrections should bring the two calculated
longitudes into agreement at the position of the true longitude.
Some difficulty may be experienced in learning how to apply these
corrections to the calculated longitudes, but it is always easy to make
a rough diagram if at all in doubt. A horizontal line representing
the parallel of latitude by dead reckoning at second sight may be
drawn with the two longitudes plotted upon it; establish the position
lines through these longitudes by drawing them at right angles to
the sun’s (or star’s) azimuth. The intersection of the two position
lines indicates the true longitude and a glance shows how to apply the
corrections to each calculated longitude to get the true. In Fig. 8 the
westerly longitude requires the correction to be applied to the east
and the easterly longitude correction to the west in order to arrive at
the true longitude.
Without the use of a diagram a rule easy to remember in deciding
whether to apply the correction in longitude to the eastward or
westward is here given: If the error in latitude is of the same name as
the first letter of the bearing, the change in longitude is contrary in
name to that of the second letter and vice versa. For example take the
case just cited.
When a body’s azimuth is less than 45°, it is wiser, and insures
more accurate results to work by New Navigation, or, if sufficiently
close to the meridian, as an ex-meridian. In the case of the latter
the corrections in such a case are taken from the table of latitude
factors (Bowditch, Table 48), the problem being the same in principle
and in solution as that described above. Good results are even obtained
by using two ex-meridians, one on each side of the meridian. The
corrections in the latitude may be applied according to the following
rule, if it is preferred to the rough diagram method: if the error of
longitude is of the same name as the second letter of the bearing, the
change in latitude is of the contrary name to the first letter, and
vice versa.
THE NEW NAVIGATION
In every branch of science and industry since time immemorial a
continuous process of simplification and increased accuracy has been
taking place, and amid this general evolution of working systems the
science of navigation will not be found an exception. Even now there
is a tendency to displace the time-honored chronometer sight, together
with a long list of more or less bewildering ways of obtaining latitude
and longitude.
The advance method is popularly known as the New Navigation, yet its
principles were originally brought forward by Marcq St. Hilaire, a
French admiral, nearly 40 years ago. It is not a new method of finding
position, but rather an improved way of establishing a Sumner line.
Like many innovations it has taken all these years for navigators to
become reconciled to the change and break away from the more familiar
forms.
In order to facilitate a simple explanation of New Navigation it will
be brought to mind that every heavenly body has a corresponding point
on the earth directly beneath it, which bears the same relation in
latitude and longitude to the earth, that the body does in declination
and right ascension to the celestial sphere. To an observer at such a
sub-celestial point the body is in the zenith with an altitude of 90°;
and about him lies a system of concentric circles of equal altitude,
which extends over a hemisphere of the earth 90° in every direction
from the point of origin. This point, through the apparent diurnal
revolution of the body, carries this whole system of circles around
the earth each day and northward and southward with the body’s change
in declination. On the outer limit of this system of circles, the
altitude of the body is 0°. Thus it is seen that the altitude of the
body decreases and its zenith distance (90° - altitude) correspondingly
increases in direct proportion as the observer departs from the
sub-celestial point, and vice versa. If, for instance, an observer is
100 miles (nautical) from this point, the zenith distance is 100´ or 1°
40´ and the altitude of the body is 88° 20´; at 2700 miles 2700´/60 =
45° of zenith distance, and 90° - 45° of altitude.
A feature is now introduced that has a close bearing upon the
principle under discussion, to serve as an opening view of the subject:
A navigator fortunate enough to have a body reasonably near his zenith,
say 5°, has at hand an extremely simple way of graphically finding
his ship’s position. This situation has previously been described,
but is repeated to make clear the principle of New Navigation. The
sub-celestial position of the body at the moment of observation is
readily ascertained by noting the time by chronometer and recourse
to the Nautical Almanac for its declination. With the point thus
established as a center, and the zenith distance derived from the
observed altitude as a radius, swing a circle upon the chart. The
ship’s position is somewhere on the circumference of this circle of
equal altitudes. This circle is now carried forward the amount and
direction of the run of the vessel between this observation and a
subsequent one similarly taken. During this interval the bearing of
the body should have changed sufficiently to make a good intersection
of the circles. The ship being on both circles must be at one of the
two intersections, between which the mariner can readily decide. The
conditions cited are comparatively unusual but show the practical use
of a circle of equal altitude in its simplest form.
The zenith distance is ordinarily too large to become a radius for
such use on the chart. The circles of equal altitude are in practice
so large that 10 to 40 mile arcs in the vicinity of the vessel are
treated by her navigator as straight lines, known as Sumner or position
lines. These lines are, theoretically, chords or tangents according
to the method employed in establishing the line, but in practice the
divergence from the circle is negligible, excepting always when the
body is too close to the zenith. The establishment of the position line
has been done in several ways for many years until the advent of this
new and more expeditious method.
The altitude of a body at any selected time for an assumed position can
be readily calculated. If this altitude does not agree (and it seldom
does) with the altitude measured simultaneously with the sextant,
corrected for the usual errors, the assumed position is not coincident
with the actual position of the vessel. The navigator now proceeds to
lay off from the assumed position, the line of azimuth of the body
taken from the azimuth tables, Weir’s Azimuth Diagram, or determined
by observation. On this line the distance between the observed and
computed altitude, expressed in minutes of arc, is measured, towards
the body if the observed altitude is greater, and away from it if less,
than the computed altitude. The point thus indicated is a position on
a circle of equal altitudes, the arc in the immediate vicinity of the
computed point being, approximately, the position line. This line is at
right angles to the azimuth for the reason that a tangent is at right
angles to the radius of a circle at a given point.
It is now known that the ship is somewhere on this line of position,
and it is necessary to cut it with another such line to determine
definitely her position. If the sun is the body being observed, it
becomes necessary, in order to provide a good angle of intersection,
to wait until the azimuth changes at least 30°, when the observation
is repeated, a second line established, and the first line brought
forward in exact accordance with the ship’s run. The interval required
naturally depends upon the latitude of the ship and the declination of
the sun. The intersection of the lines will be the position of the ship
at the time of the second sight.
The use of stars has a decided advantage in that there are always some
of these bodies available for observation lying in various azimuths; it
is practicable, with a well-defined horizon, to observe simultaneously
two or more of these bodies whose bearings show that they would produce
desirable position lines. From the resulting intersections the position
of the ship is secured. This obviates waiting for the second line, a
feature that is always inconvenient and sometimes, perhaps, dangerous.
The calculation of the altitude is accomplished by the solution of
the spherical triangle in which we have given the co-latitude (90°
- assumed latitude), the polar distance and the hour angle of the
meridian of the assumed position. Thus with two sides and the included
angle, the third side or the zenith distance (90° - altitude) is easily
determined by either of several formulas.
With the use of this method all the formulas that formerly, and still,
often puzzle the navigator to remember can be reduced to this one
sight. One of the most important features it possesses is that it can
be utilized regardless of the altitude of the body (except when very
high), its azimuth, or its hour angle, all of which are elements that
have to be used under certain favorable circumstances in order to get
accurate results from the older forms. The navigator is now given a
greater freedom in choosing bodies to observe than is found in any
other method.
The mariner to-day has been almost entirely relieved from the labor
of computing position at sea, should he care to avail himself of a set
of altitude tables, several excellent ones have made their appearance
on the market, among them Hydrographic Office Publication No. 200.
From them the altitude can be selected corresponding to the conditions
of any particular observation. With a set of these tables a navigator
is no longer required to be a mathematician or to remember the forms
of a half dozen sights. Thus in this wonderful age the mariner’s
utopian dream of obtaining position at sea by inspection, is, in a way,
realized.
* * * * *
In order to illustrate the practical working of a problem by this
method, the following example is taken up point by point:
Early on the morning of May 21, 1899, while in the assumed position of
latitude 55° 00´ N., longitude 112° 08´ E. observed the true altitude
of the star Arcturus to be 37° 14´ 50´´, bearing west of the meridian.
The chronometer carrying Greenwich mean time read 20 d. 6 h. 20 m. 03
s. The observer desired his position.
The problem by the St. Hilaire method resolves itself into the solution
of the spherical triangle shown in Fig. 10, where two sides and an
included angle are given:
Polar distance = 90° - declination (Nautical Almanac).
Co-latitude = 90° - latitude (by dead reckoning).
Hour angle of star. See figure and solution below.
The hour angle of the sun is more readily found than that of a star.
It is accomplished by applying the longitude (in time) of the assumed
position to the Greenwich time shown by the chronometer at the time
of sight. This hour angle of the mean sun must be corrected by the
equation of time to obtain the hour angle of the actual sun.
[Illustration: FIG. 9.]
[Illustration: FIG. 10.]
The cosine-haversine formula serves the purposes of this problem very
satisfactorily:
Hav _z_ = hav (_L_ ~ _d_) + cos _L_ cos _d_ hav _h_
which is derived from the well-known expression:
Cos _z_ = sin _L_ sin _d_ + cos _L_ cos _d_ cos _h_
where _z_ = zenith distance; _L_ = the latitude; and _h_ = the hour
angle.
SOLUTION
Dec. Arcturus 19° 42´ 29´´.
Lat. 55° 00 N. (Assumed). G. M. T. 20 d. 6 h. 20 m. 03 s.
R. A. M. ⊙︎ 3 51 42
Acceleration 1 02
--------------------------------------
Lat. 55° 00´ 00´´. G. S. T. 10 12 47
Dec. 19 42 29 Long. 7 28 32
------------------------ --------------------------------------
_L_ ~ _d_ 35 17 31 L. S. T. 17 41 19
R. A. ⁜ 14 11 03
--------------------------------------
H. A. ⁜ 3 30 16 W.
(Observer) 52 34 00
Lat. 55° 00´ 00´´ = cos. 9.75859
Dec. 19 42 29 = cos. 9.97378
H. A. ⁜ 52 34 00 = hav. 9.29244
--------------------
9.02481 = nat. hav. .10588
nat. hav. 35° 17´ 31´´ .09189
--------
_z_ = 52° 48´ 35´´ = nat. hav. .19777
90 00 00
------------
Computed altitude 37 11 25
Observed altitude 37 14 50
------------
Altitude difference = 3´ 25´´.
[Illustration: FIG. 11.]
A ship’s position is usually obtained by plotting the lines of azimuth
and the position lines much in the manner shown in the chartlet. The
azimuth of the body at the moment of observation is readily taken by
inspection from the azimuth tables or better still from Weir’s Azimuth
Diagram, both published by the U. S. Hydrographic Office.
In order to get an intersection of two lines of position and thereby
ascertain the latitude and longitude at once it is assumed that the
observer took an observation of another star bearing S. 45° E.,
simultaneously with Arcturus.
When ordinary A.M. time sights are taken the resulting longitude
establishes a north and south sumner line but the latitude is by D.
R.; at noon the latitude by meridian altitude establishes an east and
west line but the longitude is by D. R. so it is with a sumner line
a position is established upon it but the position along it is by D.
R. the latitude and longitude, however, can be obtained by a slight
calculation without drawing the lines on the chart; that is, the most
probable position. the altitude difference having been determined enter
table 2, bowditch, using the azimuth, or its reciprocal as the case may
be, as the course, and with the altitude difference as the distance,
pick out the difference of latitude and the departure and apply them to
the dead reckoning latitude and longitude as is the usual practice. the
result is the most probable position (according to the D. R.) on the
sumner line.
CHAPTER X
THE MOON
The moon is the most interesting of the heavenly bodies not only from a
romantic viewpoint, but from the astronomical as well. Looking at the
practical side, it is due mostly to the moon’s influence of attraction
on the waters of the earth that we have the highly important phenomena
of the tides. The moon is our nearest neighbor in the heavens; in fact,
she is a satellite, that is, revolves around the earth. This movement
is from west to east at an average rate of 51 minutes each day. The
moon’s orbit is elliptical with the earth lying a little out of center,
not unlike the situation of the sun in the earth’s orbital ellipse but
more pronounced. When the moon is at the nearest point to the earth she
is said to be at “perigee” and the point where she is most remote is
called the “apogee.”
The moon is a non-luminous body and gives off nothing but reflected
sunlight. The lunar hemisphere facing the sun is therefore the only
illuminated portion of the body, and as she turns on her axis precisely
as the earth does, the same side is always towards us. The astronomers
have seen but one side of our satellite. This solar illumination
accounts for the various interesting phases of the moon which we see
each month. When this body in her monthly revolution around the earth
passes between us and the sun, the illuminated side is towards the sun
and the dark side towards us. We see no moon at this time and call it
new moon. Two weeks later, she has completed one-half of her revolution
and is now on the other side of the earth and we are between the moon
and the sun. The illuminated face of the moon is now directly towards
us and we call it full moon. At the time of new moon, the eastward
movement quickly brings her out of range with the sun and in a couple
of days we are able to see a fine crescent in the western sky. This is
the very edge of the illuminated face--we can see around the corner
just that much. Day after day the moon’s lighted surface becomes larger
and larger until in about a week she is near our meridian at sunset and
therefore at, roughly (depending on the time of the year), 90° from
the bearing of the sun. The moon now presents to us a face one-half
dark and one-half light. This is called the quadrature. This term also
applies to the similar condition occurring a week after full moon when
she is again bearing at right angles to the sun. These occasions are
also called the first and last quarter, respectively.
The movements of the moon are very rapid. She makes her revolution
around the earth in 27⅓ days, making a change in right ascension of
360° or 24 hours in this interval, a change of over two minutes each
hour. The declination passes through its whole cycle of change from
north to south and return also in 27⅓ days; the sun requires a year
to pass through its extremes of declination and return. The change of
the moon’s declination averages about 9´ per hour. These facts demand
careful attention when employing the moon in navigation.
It is a very curious and happy circumstance that in the higher
latitudes when the short days of the winter sun occur, the moon at
full rides its highest declinations, and consequently gives extra
long nights of moonlight; and that in the summer, when the sun is in
higher declination and the days are long, the moon at full is in low
declination and there is less moonlight when it is least needed. The
reason of these conditions is that the full moon occurs when on the
opposite side of the earth from the sun and at the winter solstice
when the earth’s north pole is inclined away from the sun she must be
inclined towards the moon passing that body in high declination. The
reverse conditions exist at the summer solstice.
Another fortunate provision for lovers of moonlight nights is the
fact that the plane of the moon’s orbit is not in the same plane as
that of the earth’s orbit, for if such were the case each time the
three bodies, the earth, sun and moon, came in range there would be an
eclipse. The new moon coming in between the earth and the sun would
cause an eclipse of the sun, and at full moon when the earth is between
the sun and moon, there would be an eclipse of the moon. Therefore,
there would be an eclipse twice a month. This fortunately is avoided by
the angle of 5° that the plane of the moon’s orbit takes with that of
the earth. As a result they only come in exact range occasionally when
the moon at new and full happens to be on the ecliptic--the earth’s
orbit. If, to repeat, this occurs at full there is an eclipse of the
moon, if it occurs at new, there is an eclipse of the sun. the moon
moves eastward through the heavens on her monthly course of revolution;
it then becomes apparent that she must return to the meridian later and
later each day the amount of “retardation,” as it is called. This
retardation is a variable quantity dependent upon the moon’s irregular
change in right ascension. It is caused by the moon’s motion in her
elliptical orbit and at the inclination which her orbit takes with the
celestial equator. These causes are precisely the same in character as
those producing the equation of time in the conditions relative to the
sun and the earth’s orbit, but those of the moon are much greater. The
errors causing a variation in the right ascension of the sun requiring
a year where the similar conditions in the moon are brought about in a
month, which accounts for the marked changes in the moon’s rate of
eastward motion. The _average_ daily retardation, or average later
time in arriving at the meridian, is very close to 51 minutes. Yet the
extremes of retardation range from 38 to 66 minutes. The average of 51
minutes daily retardation is also noticed in the later rising and
setting of the moon. The extreme times between successive risings or
settings during the year, while they average 51 minutes like the
crossing of the meridian, they do not maintain the same extremes,
changing on account of the latitude of the observer as well as upon
her own motions. At 41° north the retardation on successive risings
and settings ranges between 23 minutes and 1 hour and 17 minutes. As
the vessel proceeds farther north the range is greater until near 66°
north when the moon is in her average greatest declination north she
does not set at all becoming circumpolar for a certain time each
month. In the duration of a month the moon changes her right ascension
24 hours, where the sun takes a year to accomplish this amount as it
(apparently for navigational purposes) moves eastward around the
earth. This shows the much more rapidly increasing change in right
ascension in the case of our satellite. Thus again the moon’s rapid
motions are accounted for.
The moon’s orbit around the earth is not coincident--does not lie
in the same plane as the earth’s orbit (the ecliptic) but takes an
angle of about 5° 8´ with it. The point of intersection between the
moon’s orbit and the ecliptic are called nodes (corresponding with the
equinoxes). The point crossed by the moon as it passes from southern
to the northern side of the ecliptic is called the ascending and the
other the descending node. The moon’s axis is very slowly describing a
circle in the heavens similar to that of the earth; and in consequence
the nodes are slowly moving westward along the ecliptic year by year.
Just as is the equinox by the movement of precession, but at a much
greater rate (see remarks on precession elsewhere). The moon’s axis
completes its revolution in about 19 years, while the earth requires
26,000 years. This is called the lunar cycle. At the time in the lunar
cycle when the ascending node of the moon’s orbit is in range with the
vernal equinox the moon has her greatest range of declination--about
57° from extreme north to extreme south. She is then 23° north, the
amount that the ecliptic is from the equator and 5° more, the amount
that the moon’s orbit is above the ecliptic. About 9½ years later when
the moon’s axis has listed in the opposite direction and the descending
node coincides with the vernal equinox, the moon’s maximum declination
equals 23° minus 5° or 18° north or south, a range of only about 26°.
[Illustration: This figure is viewed from the sun looking towards the
earth on march 21st--the vernal equinox.
FIG. 12.]
In the autumn, there occurs an interesting phenomenon regarding the
moon called the Harvest Moon. This is the time of unusually fine
moonlight nights in which the moon rises for three or four evenings at
about the same time instead of the usual rapid retardation. The time
the sun or moon is above our horizon depends upon its declination and
our latitude. As the sun moves northward in declination from March to
June, our days lengthen by the sun rising earlier and farther in the
northeast, and setting later and farther to the northwest. Similarly
the moon in September is moving northward in declination very rapidly
and would be rising earlier each evening were it not for its own
eastward movement of revolution which causes her to slip eastward an
average of 51 minutes daily and causes her later rising at night. The
result is that these two influences at work almost counteract each
other and cause the moon to rise at about the same time for several
days giving us three or four glorious moonlight nights called the
Harvest Moon.
It will be seen by the foregoing that great care must be exercised in
having the time of observation accurately determined owing to those
rapid movements of the moon. It is also a matter of great difficulty
to correct the observed altitude of this body on account of numerous
errors that become considerable in amounts due to her proximity to the
earth. And for these reasons this body is not popular for observations
with the general run of navigators. In the case of the semi-diameter,
considerable error is apparent and is fully described, with parallax,
which is excessive, under “Corrections for Observed Altitudes.”
CHAPTER XI
Charts
A difficulty was encountered when the early cartographers attempted to
represent the earth’s spherical surface on a flat sheet. It can not be
done, of course, without distortion being introduced in some manner.
There are various methods of taking care of this error and one is
adopted for one certain purpose while another scheme is used in some
other work. These methods of caring for the error or distortion are
known as projections, the principal being the Mercator, the gnomonic
and polyconic. The Mercator projection is almost universally used for
navigational purposes; the gnomonic projection facilitates the use of
great circle sailing, and the polyconic is used for surveying sheets.
The Mercator chart represents the earth as though it were a cylinder
instead of a sphere.
If we take the skin of one-half an orange, and assume it to represent
the northern hemisphere of the earth, an attempt to forcibly bring it
flat upon a table will result in the tearing or stretching of the skin.
It can, however, be brought flat to the table in a regular uniform way
by cutting it in a saw tooth fashion from the stem (pole) to the edge
(equator), as shown in the diagram.
[Illustration: FIG. 13.]
The shaded portions represent actual earth’s surface and the blank
parts show the error introduced by using this method. In this form it
is useless as a chart, so the real parts are stretched or extended each
way to the dotted lines making a complete chart. It is now, however,
without a vestige of accuracy in representing the bodies of land and
water as they really exist. The result would be that if a round island
should be in the latitude of the top of the chart it would be stretched
into an elongated island lying east and west giving a very erroneous
inaccurate idea of it, as shown by the east and west shading. If,
however, the island had been on the equator where no east and west
stretching would have occurred the island would appear in its natural
shape, but the farther north or south it lies just in proportion to
the latitude will it be stretched in an east and west direction. Such
a condition will not serve the purposes of navigation and it becomes
necessary to extend the degrees of latitude, making them appear
longer and longer as the equator is departed from. This stretches the
elongated east and west island in a north and south direction and
brings it back approximately to its actual shape of a round island. If
there was a round island in 10° N. and a similar-sized similar-shaped
island in 50° N. the Mercator chart would show the northern one to be
almost twice as large owing to this artificial distortion. But its
relative shape would remain practically correct. It will be seen that
the latitude scale on the sides of the chart carries an increasing
value towards the north--on a chart where a degree is about ¼´´ long at
the equator it would be about ½´´ long in 60° N. or S.
A minute of latitude is equal to a mile on this scale, but it becomes
necessary to use it in the latitude in which the measurement is taken.
If a course runs N. 60° E. from latitude 30° N. to 40° N. and the
distance is desired, take at the middle latitude at the side of the
chart a convenient multiple of distance, say 30´, on the dividers and
step off the distance. Or the whole course can be taken off at once and
with the points of the dividers at equal distances north and south of
the middle latitude read off the number of minutes of latitude lying
between them.
In very high latitudes the Mercator chart is not reliable. The
distortion becomes excessive and bearings taken will not plot correctly.
All the meridians on a Mercator chart are parallel and cut the equator
at right angles. They all lie in a true north and south direction. The
parallels of latitude all lie east and west and are parallel to each
other and at right angles to the meridians. The degrees of longitude
on the globe grow smaller and smaller as the pole is approached due
to the actual convergence of the meridians, but as all meridians are
parallel on the Mercator chart the length of a degree must be shown
the same length at the top as well as at the bottom of the chart. In
just the proportion that the degrees of longitude have been lengthened
artificially beyond their true length must the degrees of latitude be
lengthened in each latitude. This amount is shown in Table 3, Bowditch,
reckoned as the distance in miles each parallel is from the equator by
the Mercator projection. Thus in latitude 40° N. the distance is 2400´
or miles, the table shows that in the construction of a Mercator chart
this parallel should be increased artificially to 2607.6. These are
called the meridional parts.
On a Mercator chart the ship’s course is represented by a straight
line and cuts each meridian at the same angle and is called a rhumb
line. For all practical purposes on short runs this rhumb line is the
best to use, but it is not the shortest distance between two points.
Should you be able in a course a thousand miles long, to see your port
of destination your rhumb line course at the outset would not head
your ship for it, but (in northern latitude) to the southward of it.
However, as you proceeded the ship’s head would gradually draw towards
the port and you would eventually arrive. What appears to be a straight
line on this chart is really a curve on the sphere of the earth. Your
line of actual vision is a great circle, and in order to follow such a
bee line you must constantly change your compass course (on a long run)
and describe a curve on the Mercator chart unless the ship is headed
north or south or east or west along the equator, in which cases she
is sailing on a great circle. The well-known Hydrographic Office Pilot
Charts are on the Mercator projection and show all steamship tracks as
curves, for they are great circles.
The gnomonic chart is based on a projection of the earth’s surface
upon a plane tangent to any chosen point which is to be the center of
the chart. The eye is assumed to be at the center of the earth looking
outward to the point of tangency. It will be seen that the surface of
the earth adjacent to the point of tangency will be very accurately
shown on the chart, but becomes distorted gradually from the center,
the sides of which show the land in such an unnatural shape that it is
hardly recognizable.
With a chart on this projection great circle sailing is much
simplified. The straight line between two points indicates the great
circle to follow, and the course and distance is obtained by following
the directions and illustrated example given on each chart. They
are constructed for the different oceans and are for sale by the
Hydrographic Office.
The course can be transferred to a Mercator chart by taking successive
positions from the gnomonic chart and plotting them according to
latitude and longitude, and joining by straight or curved lines.
In setting out on a voyage the port of destination could it be seen
ahead would indicate the great circle course, and in order to continue
to head directly for it, the course must be continually changed. While
in the North Atlantic bound for Europe the course must be changed
constantly to the east (right) in order to remain on the great
circle--the straight and shortest distance.
The course and distance can be computed by the form given in Bowditch,
in which two sides and an included angle are given to find the other
side and the (course) angle at the point of departure. The co-latitudes
of the points of departure and destination and the angle between
them at the pole, are respectively the sides and the included angle.
However, the gnomonic chart gives the course and distance graphically.
* * * * *
When a chart is purchased or received from the Government offices the
date of issue stamped upon it should be carefully noted. It can safely
be taken for granted that the chart has been corrected up to that date
and it is incumbent upon the navigator or master to seek in all Notices
to Mariners subsequent to this date for any that affect the chart. If
a Notice contains information requiring a correction the number of the
chart appears in boldface type. The alterations should be made neatly
with India waterproof ink, and if by the nature of the information
it is impracticable to make the changes a note should be made in a
conspicuous place.
Charts are printed from copper, zinc or aluminum plates and small
changes easily made by hand are not changed on the plate until an
accumulation of errata make it necessary, or sweeping changes of a more
extensive nature takes place such as a new survey, dredged channels,
etc. A chart under extensive correction is brought up to date in every
particular, including the latest geographic spelling, new docks and
public works. The date is noted on the right of the center margin and
the dates of smaller hand corrections are indicated at the lower left
corner; the figures denote the number of the weekly Notice to Mariners,
in which the information is found, and the year.
The different scales of charts range from those of the world to a
harbor plan. There are charts of oceans; general coasts, such as from
the St. Lawrence to below New York; intermediate coasts, as from
Eastport, Maine to Cape Ann; and approaches to ports, say from Cape Ann
to Cape Cod for the port of Boston; and lastly there are harbor plans.
Those covering large areas are known as small-scale charts while harbor
charts are called large-scale charts.
A chart depends on the surveys that furnished its data, and its
accuracy and reliability rests upon that survey. Even with the most
careful surveys, where the lead is used to ascertain depths, there are
many instances where pinnacle rocks have escaped detection by coming
between the casts of the lead taken by the surveying party. These
isolated rocks become points of great danger to vessels of deep draft,
and it becomes a measure of safety to avoid rocky coasts and offshore
patches by giving them a wide berth. Spaces devoid of soundings may
well be viewed with suspicion if in reasonably shallow water, for it
would appear to indicate a lack of thoroughness in the survey, at least
a lack of soundings. The wire drag, a device used to sweep important
areas to a certain depth, is the only sure way of discovering all the
dangers of the bottom.
The aids to navigation shown on the charts are described by symbols and
abbreviations as fully as possible with the limited space. All symbols
are placed in the location of the aid, but in some cases the actual
position may be in doubt by the nature of the symbol, for instance a
buoy’s location is denoted by the ring that accompanies the symbol and
not the triangle; a light vessel by the position of the dot of the
light, or between them if there are two dots (lights). Buoys and light
vessels often drag their moorings or go adrift entirely, especially in
the winter season. It is therefore the part of wisdom to check a ship’s
position by shore marks when possible and be prepared to find buoys out
of position. The mechanism of a light buoy is often disarranged through
various causes.
The characteristics of all lights are briefly given with the visibility
and height above the sea. The charted visibility is the distance they
should be seen from a vessel’s deck on which the height of the eye is
fifteen feet above the sea, so, from the deck of an ordinary power boat
a light will not be seen until well within the range of visibility
as published, while from the deck of a large steamer the light will
be seen outside its charted visibility. This refers to high-powered
lights where the curvature of the earth has to be given consideration.
A flashing light is one in which the flash is of less duration than
the eclipse, while an occulting light has an eclipse equal to or less
than the period of light. Flashes and eclipses are often grouped and
receive the name of group flashing or group occulting. An alternating
light is one in which two colors are shown each for an equal interval
with no intervening eclipse, but if an eclipse separates the color
flash from a white flash, for instance, it becomes a flashing white
light varied by a red flash. It is a very common practice to insert
sectors of different colors into the arc of visibility of a light in
order to cover a dangerous shoal or to indicate a channel. Bearings
defining these sectors are taken from seaward and not from the light.
The term luminous range will be met with, and indicates the distance
the power of the light can carry the visibility irrespective of an
intervening horizon. A light may have a luminous range much in excess
of its visibility which is limited by the horizon but in a haze or fog
its penetrating power will greatly exceed that of a light of similar
visibility but less luminous range. The power of a light is more
commonly shown by units of a thousand candle power, thus, 5.6 indicates
a power of fifty-six hundred candles. The catoptric (C.) light employs
the reflecting, and the dioptric (D.) the refracting principle.
There is a large amount of useful information given on every chart that
the average mariner allows to escape his notice. This failure on his
part is mostly due to familiarity, or reliance on pilots with local
knowledge.
The first important feature of a chart to be considered is the
shoreline, which is shown as a continuous line representing the
high-water mark. This, it must be borne in mind, is much changed at
low water, and where the range of tide is large the shoreline is
proportionately in error. Again, where the water is shallow the change
is more marked than where the shores are steep-to. If account is not
taken of the stage of the tide it is easy to be very much deceived.
In approaching a strange harbor the chart should be scrutinized for
prominent marks, and these identified as soon as possible, then the
lesser objects can be picked up by their relative positions with the
already identified landmarks. Among the lesser marks may be found
cliffs, boulders, sandy beaches, vegetation, buildings (particularly
church spires and houses with cupolas). Prominent elevations of land
always serve to identify a locality. The chart shows these elevations
clearly by contour lines.
A twenty-foot contour line, for instance, shows the line of the cut,
should the hill be sawed off twenty feet above the sea. When contour
lines are wide apart the land has a gradual slope, and as the grade
becomes steeper the contour lines come closer proportionately.
The chart, wherever possible, represents the earth as seen from
overhead, but in the case of vertical objects, they are of necessity
shown horizontally. One of the notable cases of this is in the
representation of cliffs, which in order to show their height are drawn
with the side view as seen from the water.
Numbers seen on the land show the height above the high water.
From the topographical features of a chart we turn to those of
hydrography. All the depths indicated on a chart are those existing at
mean low water, on the Atlantic Coast, and mean lower low water on the
Pacific Coast.
All the British Admiralty and most of the Hydrographic Office charts
are reduced to the level of low water at ordinary spring tides. While
there is usually more water to be expected than shown, it must be
remembered that when the plane of mean low water is used, the low
waters that, roughly speaking, come between the moon’s first and last
quarters, will fall below the soundings on the charts.
[Illustration]
[Illustration]
The effect of an abnormal barometer and high winds must at times be
borne in mind, for a continuous northwest wind will make a vital
difference in the depths along the Atlantic seaboard, especially in
the Delaware and Chesapeake Bays.
Very often there is information that can not be symbolized on the chart
and is placed in italics in the form of a _note_. These are always
important and should be read carefully.
Sailing directions are written to supplement the charts and preserve
for the mariner a mass of information which otherwise would not reach
him.
It is always well to pay attention to the current arrows, as they are
a means by which the strength and direction of the tidal stream may be
ascertained. The symbol of a tide rip should not be ignored by one in a
small boat, as the conditions might be right to make them dangerous.
On the water areas of the chart we find contours of depths as on the
land are contours of height. It is a good scheme, if one wanted to take
the trouble to run a contour line indicating a depth a few feet greater
than the draft of his boat, and tint the shallow water with a brush.
Large vessels of deep draft upon approaching the coast are guided by
the ten-fathom curve unless the shore is very steep-to and the water
very deep. The masters of such vessels would remain outside that curve
until their position was well established. Lighter draft vessels are
guided by the five-fathom curve in a similar manner.
When approaching the land and the landmarks are not available, the
character of the bottom further assists the mariner as he sounds slowly
towards the land. The kind of bottom is indicated by abbreviations
which are obvious on almost every portion of the charts.
The three-fathom curve is the most important to the greatest number
of navigators and for this reason is made the most pronounced. In the
majority of charts it is shown by a “sanded” area within it, but in
many new charts it is heavily tinted.
In changing from one chart to another while working in an unfamiliar
locality, take especial note whether the soundings are in feet or
fathoms.
It is an excellent practice when a vessel is brought to anchor and
cross bearings taken, to estimate the radius of her swinging circle by
adding the amount of chain out to the length of the vessel and with
this describe a circle on the chart and note if there is any danger of
tailing into shoal water at any quarter.
* * * * *
This has been a long voyage and I am glad to tie up and let the
printers take charge. If any of my readers see places where I have
stood into the shallow water of inaccuracy, I will be grateful for a
passing hail that I may shift helm and get out with as little damage as
possible.
INDEX
A
Acceleration for longitude, 91
Amplitude, 12, 97
Aphelion, 8
Apogee, 139
Astronomical triangle, 102
Azimuth, 12, 96
altitude, 98
Weir’s diagram of, 98
Azimuths and amplitudes, 93
C
Calendar, 64
Celestial latitude, 14
longitude, 14
Chart, contours, 155
corrections, 151
scales, 152
Charts, 146
datum, 155
diagrams of symbols, 156
fathom curves, 158
information on, 154
symbols, 152
Chronometer time, necessity of accuracy in, 87
Circles of equal altitudes, 110
Circum-meridian altitude, 82
Compass, 93
deviation of, 94
error of, 94
naming of, 99
Coordinates, 9
Corrections for observed altitudes, 68
D
Day lost and day gained, 37
Dead reckoning, 1
Declination, 15
daily change of, 24
Deviation, naming of, 100
Dip, 72
Double altitude problem, 120
E
Earth’s orbit, 8
revolution around the sun, 20
Eclipses, 141
Ecliptic, 14
Equal altitude circle, 109
Equal altitude method, 104
Equation of Time, 56
diagram of, 62
error due to obliquity, 60
Equinoctial colure, 26
Ex-meridian altitude, 82
bodies below the pole, 86
planets, 86
stars, 86
F
First point of Aries, 14, 26
G
Gnomonic chart, 150
Great circles, 10, 17
Great-circle sailing, 150
Greenwich mean time on navigator’s watch, 54
H
Horizon, rational, 11, 75
sensible, 12, 73
visible, 11, 73
Hour circles, 13
I
Index correction, 68
J
Johnson’s method, 123
diagrams, 126
K
Kepler’s Law, 56
L
Latitude, 78
by Polaris, 88
factor, 129
formula for finding, 80
Laying a course, 96
Lights, characteristics, etc., 153
Longitude, 101
and time, 102
by equal altitudes, 104
factor, 125
prime vertical, 103
stars and planets, 105
Lunar cycle, 143
M
Mean time into sidereal, 106
Mercator Projection, 146
diagram, 147
Meridian altitude, 78
time to observe a star, 81
to report at noon, 81
Meridional parts, 149
Middle latitude sailing, 4
Mizar, 89
Moon, 139
full, 140
harvest, 143
new, 140
Moon’s declination, 143
diagram, 144
nodes, 143
retardation, 142
revolution, 140
right ascension, 142
N
Nadir, 11
Napier’s diagram, 96
Nautical astronomy, 5
New navigation, 102, 129
advantages of, 133
diagrams of, 135, 137
finding position without plotting on chart, 138
illustrated examples, 134
Notices to mariners, 151
Nutation, 32
P
Parallax, 75
horizontal, 76
moon, 77
Parallel sailing, 3
Perigee, 139
Perihelion, 8
Plane sailing, 2
Planets, 6-7
Polaris, 88
diagram, 90
time to take sight, 92
Precession of the equinoxes, 27
causes of, 31
Prime vertical, 12
observations on, 103
Q
Quadrature, 140
R
Refraction, 71
Rhumb line, 149
Right ascension, 26, 50
S
Sailings, 3
Seasons, the, 20
Semi-diameter, 69
Ship’s time, 40
Sidereal time, 43
(local) how to obtain, 52
longitude from, 53
year, 66
Solar system, 5
Summer, length of northern and southern, 25
lines, as chords and tangents, 112
when under the sun, 121
Sumner, Capt., his experience, 107
line, displacement of, 118
tangent method, 114
value of, 115
method, 107
Sun’s change in azimuth, 120
eastward movement, 16
T
Time diagram, 51
Tropical year, 66
V
Vega as a pole star, 31
Vernal equinox, 20
dates of, 67
Vertical circles, 12
W
Weir’s azimuth diagram, 98
Z
Zenith, 10
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