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l for: the founding of a College Oi-.tnis Colony'

Bought with the income
of the
Class of 1872 Fund

MATHEMATICAL GEOGRAPHY

BY
WILLIS E. JOHNSON, Ph.B.
VICE PRESIDENT AND PROFESSOR OF GEOGRAPHY AND
SOCIAL SCIENCES, NORTHERN NORMAL AND
INDUSTRIAL SCHOOL, ABERDEEN,
SOUTH DAKOTA

NEW YORK v CINCINNATI -.- CHICAGO
AMERICAN BOOK COMPANY

Copyright, 1907,
BY
WILLIS^ E. JOHNSON
red. at Stationers* Hall, Li
JOHNSON MATH. GEO.
E-P 4

PREFACE

In the greatly awakened interest in the common-school
subjects during recent years, geography has received a
large share. The establishment of chairs of geography in
some of our greatest universities, the giving of college
courses in physiography, meteorology, and commerce,
and the general extension of geography courses in normal
schools, academies, and high schools, may be cited as
evidence of this growing appreciation of the importance of
the subject.
While physiographic processes and resulting land forms
occupy a large place in geographical control, the earth in
its simple mathematical aspects should be better under
stood than it generally is, and mathematical geography
deserves a larger place in the literature of the subject than
the few pages generally given to it in our physical geog
raphies and elementary astronomies. It is generally
conceded that the mathematical portion of geography
is the most difficult, the most poorly taught and least
understood, and that students require the most help in
understanding it. The subject-matter of mathematical
geography is scattered about in many works, and no one
book treats the subject with any degree of thoroughness,
or even makes a pretense at doing so. It is with the
view of meeting the need for such a volume that this
work has been undertaken.
Although designed for -use in secondary schools and for
teachers' preparation, much material herein organized
3

4 PREFACE
may be used in the upper grades of the elementary school.
The subject has not been presented from the point of
view of a little child, but an attempt has been made to
keep its scope within the attainments of a student in a
normal school, academy, or high school. If a very short
course in mathematical geography is given, or if students
are relatively advanced, much of the subject-matter may
be omitted or given as special reports.
To the student or teacher who finds some portions too
difficult, it is suggested that the discussions which seem
obscure at first reading are often made clear by additional
explanation given farther on in the book. Usually the
second study of a topic which seems too difficult should be
deferred until the entire chapter has been read over care
fully. The experimental work which is suggested is given for
the purpose of making the principles studied concrete and
vivid. The measure of the educational value of a labora
tory exercise in a school of secondary grade is not found
in the academic results obtained, but in the attainment of
a conception of a process. The student's determination
of latitude, for example, may not be of much value if its
worth is estimated in terms of facts obtained, but the
forming of the conception of the process is a result of
inestimable educational value. Much time may be wasted,
however, if the student is required to rediscover the facts
and laws of nature which are often so simple that to see
is to accept and understand.
Acknowledgments are due to many eminent scholars
for suggestions, verification of data, and other valuable
assistance in the preparation of this book.
To President George W. Nash of the Northern Normal
and Industrial School, who carefully read the entire manu-

PREFACE 5
script and proof, and to whose thorough training, clear
insight, and kindly interest the author is under deep
obligations, especial credit is gratefully accorded. While
the author has not availed himself of the direct assistance
of his sometime teacher, Professor Frank E. Mitchell, now
head of the department of Geography and Geology of the
State Normal School at Oshkosh, Wisconsin, he wishes
formally to acknowledge his obligation to him for an
abiding interest in the subject. For the critical exami
nation of portions of the manuscript bearing upon fields
in which they are acknowledged authorities, grateful
acknowledgment is extended to Professor Francis P.
Leavenworth, head of the department of Astronomy of
the University of Minnesota; to Lieutenant-Commander
E. E. Hayden, head of the department of Chronometers
and Time Service of the United States Naval Observatory,
Washington; to President F. W. McNair of the Michigan
College of Mines; to Professor Cleveland Abbe of the
United States Weather Bureau; to President Robert S.
Woodward of the Carnegie Institution of Washington; to
Professor T. C. Chamberlin, head of the department of
Geology of the University of Chicago; and to Professor
Charles R. Dryer, head of the department of Geography
of the State Normal School at Terre Haute, Indiana. For
any errors or defects in the book, the author alone is
responsible.

CONTENTS

CHAPTER I page
Introductory  9
CHAPTER II
The Form op the Earth  24
CHAPTER III
The Rotation op the Earth  45
CHAPTER IV
Longitude and Time  62
CHAPTER V
Circumnavigation and Time  92
CHAPTER VT
The Earth's Revolution  104
CHAPTER VII
Time and the Calendar  132
CHAPTER VIII
Seasons  146
CHAFfER IX
Tides  176
CHAPTER X
Map Projections  190
7

8 CONTENTS
CHAPTER XI page
The United States Government Land Survey  226
CHAPTER XII
Triangulation in Measurement and Survey .  237
CHAPTER XIII
The Earth in Space  246
CHAPTER XIV
Historical Sketch  268

Appendix  279
Glossary  314
Index  323

MATHEMATICAL GEOGRAPHY

CHAPTER I

INTRODUCTORY

Observations and Experiments

Observations of the Stars. On the first clear evening,
observe the " Big Dipper " * and the polestar. In Septem
ber and in December, early in the evening, they will be
nearly in the positions represented in Figure 1. Where
is the Big Dipper
later in the evening ?
Find out by obser
vations. Learn readily to
pick out Cassiopeia's
Chair and the Little
Dipper. Observe
their apparent mo
tions also. Notice
the positions of stars
in different portions of the sky and observe where they are
later in the evening. Do the stars around the polestar
remain in the same position in relation to each other, —
the Big Dipper always like a dipper, Cassiopeia's Chair
* In Ursa Major, commonly called the "Plow," "The Great
Wagon," or "Charles's Wagon" in England, Norway, Germany, and
other countries. 9

CD 03 .
%
o c

' ' ¦ 3K*
<¦  :--... cf/
^jttW Oippet. ¦» North Sur •

» *. <>-
? *••-:-
Cr *
A ;
"" *-¦.
September

/
•*\*
¦/<*

Fig. i

10

INTRODUCTORY

always like a chair, and both always on opposite sides of
the polestar? In what sense may they be called " fixed "
stars (see pp. 108, 265)?
Make a sketch of the Big Dipper and the polestar,
recording the date and time of observation. Preserve
your sketch for future reference, marking it Exhibit 1.
A month or so later, sketch again at the same time of
night, using the same sheet of paper with a common
polestar for both sketches. In making your sketches
be careful to get the angle formed by a line through
the " pointers " and the polestar with a perpendicular to
the horizon. This angle can be formed by observing the
side of a building and the pointer line. It can be
measured more accurately in the fall months with a pair
of dividers having straight edges, by placing
one outer edge next to the perpendicular
side of a north window and opening the
dividers until the other outside edge is
parallel to the pointer line (see Fig. 2).
Now lay the dividers on a sheet of paper
and mark the angle thus formed, repre
senting the positions of stars with asterisks.
Two penny rulers pinned through the ends
will serve for a pair of dividers.
Phases of the Moon. Note the position
of the moon in the sky on successive nights at the same
hour. Where does the moon rise? Does it rise at the
same time from day to day? When the full moon may
be observed at sunset, where is it? At sunrise? When
there is a full moon at midnight, where is it? Assume
it is sunset and the moon is high in the sky, how much of
the lighted part can be seen?
Answers to the foregoing questions should be based upon.

Fig. .

THE NOON SHADOW 11
first-hand observations. If the questions cannot easily
be answered, begin observations at the first opportunity.
Perhaps the best time to begin is when both sun and moon
may be seen above the horizon. At each observation
notice the position of the sun and of the moon, the portion
of the lighted part which is turned toward the earth, and
bear in mind the simple fact that tlie moon always shows a
lighted half to the sun. If the moon is rising when the sun
is setting, or the sun is rising when the moon is setting,
the observer must be standing directly between them, or
approximately so. With the sun at your back in the east
and facing the moon in the west, you see the moon as
though you were at the sun. How much of the lighted
part of the moon is then seen? By far the best apparatus
for illustrating the phases of the moon is the sun and
moon themselves, especially when both are observed above
the horizon.
The Noon Shadow. Some time early in the term from
a convenient south window, measure upon the floor the
length of the shadow when it is shortest during the
day. Record the measurement and the date and time of
day. Repeat the measurement each week. Mark this
Exhibit 2.
On a south-facing window sill, strike a north-south line
(methods for doing this are discussed on pp. 61, 130).
Erect at the south end of this line a perpendicular board,
say six inches wide and two feet long, with the edge next
the north-south line. True it with a plumb line; one
made with a bullet and a thread will do. This should
be so placed that the shadow from the edge of the board
may be recorded on the window sill from 11 o'clock, a.m.,
until 1 o'clock, p.m. (see Fig. 3).
Carefully cut from cardboard a semicircle and mark the

12

INTRODUCTORY

degrees, beginning with the
middle radius as zero. Fasten
this upon the window sill
with the zero meridian coin
ciding with the north-south
line. Note accurately the
clock time when the shadow
from the perpendicular board
crosses the line, also where
the shadow is at twelve
o'clock. Record these facts
with the date and preserve as
Noon shadow Exhibit 3. Continue the ob-
F;g- 3 servations every few days.
The Sun's Meridian Altitude. When the shadow
is due north, carefully
measure the angle formed
by the shadow and a level
line. The simplest way is
to draw the window shade
down to the top of a sheet
of cardboard placed very
nearly north and south with
the bottom level and then
draw the shadow line, the
lower acute angle being the
one sought (see Fig. 4).
Another way is to drive a
pin in the side of the window
casing, or in the edge of the
vertical board (Fig. 3) ; fasten a thread to it and connect
the other end of the thread to a point on the sill where the
shadow falls. A still better method is shown on p. 172.

^

k Altitude of Sun at noon"

Fig. 4

CENTRIFUGAL FORCE

13

Since the shadow is north, the sun is as high in the sky
as it will get during the day, and the angle thus measured
gives the highest altitude of the sun for the day. Record
the measurement of the angle with the date as Exhibit 4.
Continue these records from week to week, especially
noting the angle on one of the following dates: March 21,
June 22, September 23, December 22. This angle on
March 21 or September 23, if subtracted from 90°, will
equal the latitude* of the observer.
A Few Terms Explained

Centrifugal Force. The literal meaning of the word
suggests its current meaning. It comes from the Latin
centrum, center; -and
fugere, to flee. A cen
trifugal force is one
directed away from a
center. When a stone
is whirled at the end of
a string, the pull which
the stone gives the
string is called centri
fugal force. Because
of the inertia of the
stone, the whirling
motion given to it
by the arm tends to
make it fly off in a straight line (Fig. 5), — and this
it will do if the string breaks. The measure of the
centrifugal force is the tension on the string. If the
string be fastened at the end of a spring scale and the
* This is explained on pp. 170, 171.

Tends to fly off
Fig. 5

14 INTRODUCTORY
stone whirled, the scale will show the amount of the centri
fugal force which is given the stone by the arm that
whirls it. The amount of this force * (C) varies with the
mass of the body (m), its velocity (v), and the radius
of the circle (r) in which it moves, in the following ratio:

0 =

The instant that the speed becomes such that the avail
able strength of the string is less than the value of

r
however slightly, the stone will cease to follow the curve
and will immediately take a motion at a uniform speed
in the straight line with which its motion happened to
coincide at that instant (a tangent to the circle at the point
reached at that moment).
Centrifugal Force on the Surface of the Earth. The
rotating earth imparts to every portion of it, save along
the axis, a centrifugal force which varies according to the
foregoing formula, r being the distance to the axis, or the
radius of the parallel. It is obvious that on the surface
of the earth the centrifugal force due to its rotation is
greatest at the equator and zero at the poles.
At the equator centrifugal force (C) amounts to about
jf 9 that of the earth's attraction (g), and thus an object
there which weighs 288 pounds is lightened just one pound
by centrifugal force, that is, it would weigh 289 pounds
were the earth at rest. At latitude 30°, C = 7^7— (that is,
385
* On the use of symbols, such as C for centrifugal force, <j> for latitude,
etc., see Appendix, p. 307.

CENTRIPETAL FORCE 15
centrifugal force is ¦gfa the force of the earth's attraction) ;
at W.C -£;*<»>, C = _JL_.
For any latitude the " lightening effect " of centrifugal
force due to the earth's rotation equals -£- times the
289
square of the cosine of the latitude (C = -g— X cos2 <f>).
289
By referring to the table of cosines in the Appendix, the
student can easily calculate the " lightening " influence
of centrifugal force at his own latitude. For example,
say the latitude of the observer is 40°.

Cosine 40° = .7660. ¦£- X .76602 = -g— •
• 289 492
Thus the earth's attraction for an object on its surface
at latitude 40° is 492 times as great as centrifugal force
there, and an object weighing 491 pounds at that latitude
would weigh one pound more were the earth at rest.*
Centripetal Force. A centripetal {centrum, center; petere,
to seek) force is one directed toward a center, that is, at
right angles to the direction of motion of a body. To
.distinguish between centrifugal force and centripetal
force, the student should realize that forces never occur
singly but only in pairs and that in any force action there
are always two bodies concerned. Name them A and B.
Suppose A pushes or pulls B with a certain strength.
This cannot occur except B pushes or pulls A by the same
amount and in the opposite direction. This is only a
simple way of stating Newton's third law that to every
* These calculations are based upon a spherical earth and make no
allowances for the oblateness.

16 INTRODUCTORY
action (A on B) there corresponds an equal and opposite
reaction (B on A).
Centrifugal force is the reaction of the body against the
centripetal force which holds it in a curved path, and it
must always exactly equal the centripetal force. In the
case of a stone whirled at the end of a string, the necessary
force which the string exerts on the stone to keep it in a
curved path is centripetal force, and the reaction of the
stone upon the string is centrifugal force.
The formulas for centripetal force are exactly the same
as those for centrifugal force. Owing to the rotation of
the earth, a body at the equator describes a circle with
uniform speed.- The attraction of the earth supplies the
centripetal force required to hold it in the circle. The
earth's attraction is greatly in excess of that which is
required, being, in fact, 289 times the amount needed.
The centripetal force in this case is that portion of the attrac
tion which is used to hold the object in the circular course.
The excess is what we call the weight of the body or the
force of gravity.
If, therefore, a spring balance suspending a body at the
equator shows 288 pounds, we infer that the earth really
pulls it with a force of 289 pounds, but one pound of this
pull is expended in changing the direction of the motion
of the body, that is, the value of centripetal force is one
pound. The body pulls the earth to the same extent,
that is, the centrifugal force is also one pound. At the
poles neither centripetal nor centrifugal force is exerted
upon bodies and hence the weight of a body there is the
full measure of the attraction of the earth.
Gravitation. Gravitation is the all-pervasive force by
virtue of which every particle of matter in the universe
is constantly drawing toward itself every other particle

GRAVITATION

17

of matter, however distant. The amount of this attrac
tive force existing between two bodies depends upon
(1) the amount of matter in them, and (2) the distance
they are apart.
There are thus two laws of gravitation. The first law,
the greater the mass, or amount of matter, the greater the
attraction, is due to the fact that each particle of matter
has its own independent attractive force, and the more
there are of the par
ticles, the greater
is the combined
attraction. The Second Law
Explained. In gen
eral terms the law
is that the nearer
an object is, the
greater is its at
tractive force. Just
as the heat and
light of a flame are
greater the nearer one gets to it (Fig. 6), because more rays
are intercepted, so the nearer an object is, the greater is its
H

More rays intercepted when near the flame"
Fig. 6

^^^

F^~-^

G

^^^~

B
C
D
Fig. 7
attraction. The ratio of the increase of the power of gravita
tion as distance decreases, may be seen from Figures 7 and 8.
JO. MATH. GEO.-
18

INTRODUCTORY

Two lines, AD and AH (Fig. 7), are twice as far apart
at C as at B because twice as far away; three times as far
apart at D as at B because three times as far away, etc.
Now light radiates out in every direction, so that light
coming from point A' (Fig. 8), when it reaches B' will be

H

-""i i *»*
 ' ^-vl
c
^t-_— p  "^a
«4
B'
C
D'
Fig. 8
spread over the square of B' F' at C", on the square C G' ;
at D' on the square D' H', etc. C being twice as far
away from A' as B' , the side C G' is twice that of B' F' ,
as we observed in Fig. 7, and its square is four times as
great. Line D' H' is three times as far away, is three
times as long, and its square is nine times as great. The
light being spread over more space in the more distant
objects, it will light up a. given area less. The square
at B' receives all the light within the four radii, the
same square at C receives one fourth of it, at D' one
ninth, etc. The amount of light decreases as the square of
the distance increases.- The force of gravitation is exerted
in every direction and varies in exactly the same way.
Thus the second law of gravitation is that the force varies
inversely as the square of the distance.
Gravity. The earth's attractive influence is called
gravity. The weight of an object is simply the measure of
GRAVITY 19
the force of gravity. An object on or above the surface
of the earth weighs less as it is moved away from the
center of gravity.* It is difficult to realize that what we
call the weight of an object is simply the excess of attrac
tion which the earth possesses for it as compared with
other forces acting upon it, and that it is a purely relative
affair, the same object having a different weight in different
places in the solar system. Thus the same pound-weight
taken from the earth to the sun's surface would weigh 27
pounds there, only one sixth of a pound at the surface of
the moon, over 2 \ pounds on Jupiter, etc. If the earth
were more dense, objects would weigh more on the surface.
Thus if the earth retained its present size but contained as
much matter as the sun has, the strongest man in the world
could not lift a silver half dollar, for it would then weigh
over five tons. A pendulum clock would then tick 575
times as fast. On the other hand, if the earth were no
denser than the sun, a half dollar would weigh only a
trifle more than a dime now weighs, and a pendulum clock
would tick only half as fast.
From the table on p. 266 giving the masses and dis
tances of the sun, moon, and principal planets, many
interesting problems involving the laws of gravitation
may be suggested. To illustrate, let us take the problem
" What would you weigh if you were on the moon ? "
Weight on the Moon. The mass of the moon, that is,
the amount of matter in it, is -j-j that of the earth.
Were it the same size as the earth and had this mass, one
pound on the earth would weigh a little less than one
eightieth of a pound there. According to the first law of
gravitation we have this proportion:
1. Earth's attraction : Moon's attraction : : 1: ¥*j.
* For a more accurate and detailed discussion of gravity, see p. 279.

20 INTRODUCTORY
But the radius of the moon is 1081 miles, only a little
more than one fourth that of the earth. Since a person
on the moon would be so much nearer the center of gravity
than he is on the earth, he would weigh much more there
than here if the moon had the same mass as the earth.
According to the second law of gravitation we have this
proportion :
2. Earth's attraction : Moon's attraction : :  ; : — — — ; •
40002 10812
We have then the two proportions:
1. Att. Earth : Att. Moon : : 1 : ,V. 1 1

2. Att. Earth : Att. Moon

40002 ' 10812
Combining these by multiplying, we get
Att. Earth : Att. Moon : : 6 : 1.
Therefore six pounds on the earth would weigh only
one pound on the moon. Your weight, then, divided by
six, represents what it would be on the moon. There
you could jump six times as high — if you could live to
jump at all on that cold and almost airless satellite (see
pp. 263, 264).
The Sphere, Circle, and Ellipse. A sphere is a solid
bounded by a curved surface all points of which are equally
distant from a point within called the center.
A circle is a plane figure bounded by a curved line all
points of which are equally distant from a point within
called the center. In geography what we commonly call
circles such as the equator, parallels, and meridians, are
really only the circumferences of cireles. Wherever used

THE SPHERE, CIRCLE, AND ELLIPSE 21
in this book, unless otherwise stated, the term circle
refers to the circumference.
Every circle is conceived to be divided into 360 equal
parts called degrees. The greater the size of the circle,
the greater is the length of each degree. A radius of a
circle or of a sphere is a straight line from the boundary
to the center. Two radii forming a straight line con
stitute a diameter.
Circles on a sphere dividing it into two hemispheres are
called great circles. Circles on a sphere dividing it into
unequal parts are called small circles.
All great circles on the same sphere bisect each other,
regardless of the angle at which they cross one another.
That this may be clearly seen, with a globe before you test
these two propositions:
a. A point 180° in any direction from one point in a
great circle must lie in the same circle.
b. Two great circles on the same sphere must cross
somewhere, and the point 180° from the one where they
cross, lies in both of the circles, thus each great circle
divides the other into two equal parts.
An angle is the difference in direction of two lines which,
if extended, would meet. Angles are measured by using
the meeting point as the center of a circle and finding the
fraction of the circle, or number of degrees of the circle,
included between the lines. It is well to practice esti
mating different angles and then to test the accuracy of
the estimates by reference to a graduated quadrant or
circle having the degrees marked.
An ellipse is a closed plane curve such that the sum of
the distances from one point in it to two fixed points within,
called foci, is equal to the sum of the distances from any
other point in it to the foci. The ellipse is a conic section

22

INTRODUCTORY

formed by cutting a right cone by a plane passing obliquely
through its opposite sides (see Ellipse in Glossary).
To construct an ellipse,
drive two pins at points
for foci, say three inches
apart. With a loop of
non-elastic cord, say ten
inches long, mark the
boundary line as repre
sented in Figure 10.
Orbit of the Earth.
The orbit of the earth
is an ellipse. To lay off
an ellipse which shall
quite correctly represent
the shape of the earth's
orbit, place pins one

Ellipse.
A&A', Foci. C D, Minor Axis
XY, Major Axis. A toA'.Fpcal
Distance. A M+AM=AN+AN
Fig. 9

tenth of an inch apart and make a loop of string 12.2 inches
long. This loop
can easily be made
by driving two pins
6.1 inches apart
and tying a string
looped around
them. Shape of the
Earth. The earth
is a spheroid, or a
solid approaching
a sphere (see Sphe
roid in Glossary).
The diameter upon which it rotates is called the axis.
The ends of the axis are its poles. Imaginary lines on the

To Construct
An Ellipse

Fig.

SHAPE OF THE EARTH 23
surface of the earth extending from pole to pole are called
meridians* While any number of meridians may be
conceived of, we usually think of them as one degree apart.
We say, for example, the ninetieth meridian, meaning the
meridian ninety degrees from the prime or initial meridian.
What kind of a circle is a meridian circle? Is it a true
circle? Why?
The equator is a great circle midway between the poles.
Parallels are small circles parallel to the equator.
It is well for the student to bear in mind the fact that
it is the earth's rotation on its axis that determines most of
the foregoing facts. A sphere at rest would not have
equator, meridians, etc.
* The term meridian is frequently used to designate a great circle
passing through the poles. In this book such a circle is designated a
meridian circle, since each meridian is numbered regardless of its oppo
site meridian.

CHAPTER II
the form of the earth
The Earth a Sphere
Circumnavigation. The statements commonly given as
proofs of the spherical form of the earth would often apply
as well to a cylinder or an egg-shaped or a disk-shaped
body. " People have sailed around it," " The shadow of
the earth as seen in the eclipse of the moon is always cir
cular," etc., do not in themselves prove that the earth is a
sphere. They might be true if the earth were a cylinder
or had the shape of an egg. " But men have sailed around
it in different directions." So might they a lemon-shaped
body. To make a complete proof, we must show that men
have sailed around it in practically every direction and
have found no appreciable difference in the distances in
the different directions.
Earth's Shadow always Circular. The shadow of the
earth as seen in the lunar eclipse is always circular. But
a dollar, a lemon, an egg, or a cylinder may be so placed
as always to cast a circular shadow. When in addition
to this statement it is shown that the earth presents many
different sides toward the sun during different eclipses of
the moon and the shadow is always circular, we have a
proof positive, for nothing but a sphere casts a circular
shadow when in many different positions. The fact that
eclipses of the moon are seen in different seasons and at
different times of day is abundant proof that practically
24

TELESCOPIC OBSERVATIONS

25

Fig. u. Ship's rigging distinct.
Water hazy.

all sides of the earth are turned toward the sun during
different eclipses.
Almost Uniform Surface Gravity. An object has almost
exactly the same weight in
different parts of the earth
(that is, on the surface),
showing a practically common
distance from different points
on the earth's surface to the
center of gravity. This is
ascertained, not by carrying
an object all over the earth
and weighing it with a pair
of spring scales (why not
balances?) ; but by noting the
time of the swing of the
pendulum, for the rate of its swing varies according
to the force of gravity. Telescopic Observations. If
we look through a telescope
at a distant object over a
level surface, such as a body
of water, the lower part is
hidden by the intervening
curved surface. (Figs. 11, 12.)
This has been observed in
many different places, and the
rate of curvature seems uni
form everywhere and in every
direction. Persons ascending
in balloons or living on high
elevations note the appreciably earlier time of sunrise
or later time of sunset at the higher elevation.

Fig. 12.

Water distinct.
ill-defined.

Rigging

26 THE FORM OF THE EARTH
Shifting of Stars and Difference in Time. The proof
which first demonstrated the curvature of the earth, and
one which the student should clearly understand, is the
disappearance of stars from the southern horizon and the
rising higher of stars from the northern horizon to persons
traveling north, and the sinking of northern stars and the
rising of southern stars to south-bound travelers. After
people had traveled far enough north and south to make
an appreciable difference in the position of stars, they
observed this apparent rising and sinking of the sky. Now
two. travelers, one going north and the other going south,
will see the sky apparently elevated and depressed at the
same time; that is, the portion of the sky that is rising for
one will be sinking for the other. Since it is impossible
that the stars be both rising and sinking at the same
time, only one conclusion can follow, — the movement of
the stars is apparent, and the path traveled north and
south must be curved.
Owing to the rotation of the earth one sees the same
stars in different positions in the sky east and west, so the
proof just given simply shows that the earth is curved in
a north and south direction. Only when timepieces were
invented which could carry the time of one place to differ
ent portions of the earth could the apparent positions
of the stars prove. the curvature of the earth east and
west. By means of the telegraph and telephone we
have most excellent proof that the earth is curved east
and west.
If the earth were flat, when it is sunrise at Philadelphia
it would be sunrise also at St. Louis and Denver. Sun
rays extending to these places which are so near together
as compared with the tremendous distance of the sun, over
ninety millions of miles away, would be almost parallel

ACTUAL MEASUREMENT 27
on the earth and would strike these points at about the
same angle. But we know from the many daily messages
between these cities that sun time in Philadelphia is an
hour later than it is in St. Louis and two hours later than
in Denver.
When we know that the curvature of the earth north
and south as observed by the general and practically
uniform rising and sinking of the stars to north-bound and
south-bound travelers is the same as the curvature east
and west as shown by the difference in time of places
east and west, we have an excellent proof that the earth is
a sphere. Actual Measurement. Actual measurement in many
different places and in nearly every direction shows a prac
tically uniform curvature in the different directions. In
digging canals and laying watermains, an allowance must
always be made for the curvature of the earth; also in
surveying, as we shall notice more explicitly farther on.
A simple rule for finding the amount of curvature for
any given distance is the following:
Square the number of miles representing the distance, and
two thirds of this number represents in feet the departure
from a straight line.
Suppose the distance is 1 mile. That number squared
is 1, and two thirds of that number of feet is 8 inches.
Thus an allowance of 8 inches must be made for 1 mile.
If the distance is 2 miles, that number squared is 4, and
two thirds of 4 feet is 2 feet, 8 inches. An object, then,
1 mile away sinks 8 inches below the level line, and at 2
miles it is below 2 feet, 8 inches.
To find the distance, the height from a level line being
given, we have the converse of the foregoing rule:
Multiply the number representing the height in feet by IJ,

28

THE FORM OF THE EARTH

and the square root of this product represents the number of
miles distant the object is situated.
The following table is based upon the more accurate
formula:

Distance

(miles) =

- 1.317 Vh

eight (feet).

Ht. ft.

Dist. miles

Ht. ft.

Dist. miles

Ht. ft.

Dist. miles

1

1.32

50

9.31

170

17.17

2

1.86

55

9.77

180

17.67

3

2.28

60

10.20

190

18.15

4

2.63

65

10.62

200

18.63

5

2.94

70

11.02

300

22.81

6

3.23

75

11.40

400

26.34

7

3.48

80

11.78

500

29.45

8

3.73

85

12.14

600

32.26

9

3.95

90

12.49

700

34.84

10

4.16

95

12.84

800

37.25

15

5.10

100

13.17

900

39.51

20

5.89

110

13.81

1000

41.65

25

6.59

120

14.43

2000

58.90

30

7.21

130

15.02

3000

72.13

35

7.79

140

15.58

4000

83.50

40

8.33

150

16.13

5000

93.10

45

8.83

160

16.66

Mile

95.70

The Earth an Oblate Spheroid
Richer's Discovery. In the year 1672 John Richer, the
astronomer to the Royal Academy of Sciences of Paris,
was sent by Louis XIV to the island of Cayenne to make
certain astronomical observations. His Parisian clock had
its pendulum, slightly over 39 inches long, regulated to
beat seconds. Shortly after his arrival at Cayenne, he
noticed that the clock was losing time, about two and a
half minutes a day. Gravity, evidently, did not act with
so much force near the equator as it did at Paris. The
astronomer found it necessary to shorten the pendulum
nearly a quarter of an inch to get it to swing fast enough.

AMOUNT OF OBLATENESS 29'
Richer reported these interesting facts to his colleagues
at Paris, and it aroused much discussion. At first it was
thought that greater centrifugal force at the equator,
counteracting the earth's attraction more there than else
where, was the explanation. The difference in the force
of gravity, however, was soon discovered to be too great
to be thus accounted for. The only other conclusion was
that Cayenne must be farther from the center of gravity
than Paris (see the discussion of Gravity, Appendix,
p. 279; also Historical Sketch, pp. 273-275).
Repeated experiments show it to be a general fact that
pendulums swing faster on the surface of the earth as one
approaches the poles. Careful measurements of arcs of
meridians prove beyond question that the earth is flattened
toward the poles, somewhat like an oblate spheroid.
Further evidence is found in the fact that the sun and
planets, so far as ascertained, show this same flattening.
Cause of Oblateness. The cause of the oblateness is
the rotation of the body, its flattening effects being more
marked in earlier plastic stages, as the earth and other
planets are generally believed to have been at one time.
The reason why rotation causes an equatorial bulging
is not difficult to understand. Centrifugal force increases
away from the poles toward the equator and gives a lifting
or lightening influence to portions on the surface. If the
earth were a sphere, an object weighting 289 pounds at
the poles would be lightened just one pound if carried to
the swiftly rotating equator (see p. 280). The form given
the earth by its rotation is called an oblate spheroid or an
ellipsoid of rotation.
Amount of Oblateness. To represent a meridian circle
accurately, we should represent the polar diameter about
gg-g- part shorter than the equatorial diameter. That this

30 THE FORM OF THE EARTH
difference is not perceptible to the unaided eye will be
apparent if the construction of such a figure is attempted,
say ten inches in diameter in one direction and ^ of an
inch less in the opposite direction. The oblateness of
Saturn is easily perceptible, being thirty times as great as
that of the earth, or one tenth (see p. 257). Thus an
ellipsoid nine inches in polar diameter (minor axis) and
ten inches in equatorial diameter (major axis) would rep
resent the form of that planet.
Although the oblateness of the earth seems slight when
represented on a small scale and for most purposes may be
ignored, it is nevertheless of vast importance in many
problems in surveying, astronomy, and other subjects.
Under the discussion of latitude it will be shown how this
oblateness makes a difference in the lengths of degrees of
latitude, and in the Appendix it is shown how this equa
torial bulging shortens the length of the year and changes
the inclination of the earth's axis (see Precession of the
Equinoxes and Motions of the Earth's Axis).
Dimensions of the Spheroid. It is of very great impor
tance in many ways that astronomers and surveyors know
as exactly as possible the dimensions of the spheroid.
Many men have made estimates based upon astronomical
facts, pendulum experiments and careful surveys, as to the
equatorial and polar diameters of the earth. Perhaps the
most widely used is the one made by A. R. Clarke, for many
years at the head of the English Ordnance Survey, known
as the Clarke Spheroid of 1866.
Clarke Spheroid ov 1866.
A. Equatorial diameter  7,926.614 miles
 B. Polar diameter  7,899.742 miles
A-B 1
Oblateness-^--  —

DIMENSIONS OF THE SPHEROID 31
It is upon this spheroid of reference that all of the
work of the United States Geological Survey and of the
United States Coast and Geodetic Survey is based, and
upon which most of the dimensions given in this book are
determined. In 1878 Mr. Clarke made a recalculation, based upon
additional information, and gave the following dimensions,
though it is doubtful whether these approximations are
any more nearly correct than those of 1866.
Clarke Spheroid op 1878.
A. Equatorial diameter  7,926.592 miles
B. Polar diameter  7,899.580 miles
A-B 1

Oblateness

293.46

Another standard spheroid of reference often referred
to, and one used by the United States Governmental
Surveys before 1880, when the Clarke spheroid was
adopted, was calculated by the distinguished Prussian
astronomer, F. H. Bessel, and is called the
Bessel Spheroid op 1841.
A. Equatorial diameter  7,925.446 miles
B. Polar diameter  7,898.954 miles
Oblateness ±^  ^
Many careful pendulum tests and a great amount of
scientific triangulation surveys of long arcs of parallels
and meridians within recent years have made available
considerable data from which to determine the true
dimensions of the spheroid. In 1900, the United States
Coast and Geodetic Survey completed the measurement
of an arc across the United States along the 39th parallel

32 THE FORM OF THE EARTH
from Cape May, New Jersey, to Point Arena, California,
through 48° 46' of longitude, or a distance of about 2,625
miles. This is the most extensive piece of geodetic sur
veying ever undertaken by any nation and was so carefully
done that the total amount of probable error does not
amount to more than about eighty-five feet. A long arc
has been surveyed diagonally from Calais, Maine, to
New Orleans, Louisiana, through 15° V of latitude and
22° 47' of longitude, a distance of 1,623 miles. Another
long arc will soon be completed along the 98th meridian
across the United States. Many shorter arcs have also
been surveyed in this country.
The English government undertook in 1899 the gigantic
task of measuring the arc of a meridian extending the
entire length of Africa, from Cape Town to Alexandria.
This will be, when completed, 65° long, about half on
each side of the equator, and will be of great value in
determining the oblateness. Russia and Sweden have
lately completed the measurement of an arc of 4° 30' on
the island of Spitzbergen, which from its high latitude,
76° to 80° 30' N., makes it peculiarly valuable. Large
arcs have been measured in India, Russia, France, and
other countries, so that there are now available many times
as much data from which the form and dimensions of the
earth may be determined as Clarke or Bessel had.
The late Mr. Charles A. Schott, of the United States
Coast and Geodetic Survey, in discussing the survey of the
39th parallel, with which he was closely identified, said: *
" Abundant additional means for improving the existing
deductions concerning the earth's figure are now at hand,
and it is perhaps not too much to expect that the Interna-
* In his Transcontinental Triangulation and the American Arc oi
the Parallel.

EQUATOR ELLIPTICAL 33
tional Geodetic Association may find it desirable in the
near future to attempt a new combination of all available
arc measures, especially since the two large arcs of the
parallel, that between Ireland and Poland and that of the
United States of America, cannot fail to have a paramount
influence in a new general discussion."
A spheroid is a solid nearly spherical. An oblate sphe
roid is one flattened toward the poles of its axis of rotation.
The earth is commonly spoken of as a sphere. It would
be more nearly correct to say it is an oblate spheroid.
This, however, is not strictly accurate, as is shown in the
succeeding discussion.
The Earth a Geoid
Conditions Producing Irregularities. If the earth had
been made up of the same kinds of material uniformly
distributed throughout its mass, it would probably have
assumed, because of its rotation, the form of a regular
oblate spheroid. But the earth has various materials
unevenly distributed in it, and this has led to many slight
variations from regularity in form.
Equator Elliptical. Pendulum experiments and measure
ments indicate not only that meridians are elliptical but
that the equator itself may be slightly elliptical, its longest
axis passing through the earth from 15° E. to 165° W. and
its shortest axis from 105° E. to 75° W. The amount of this
oblateness of the equator is estimated at about 5,^ or a
difference of two miles in the lengths of these two diameters
of the equator. Thusthe meridian circle passing through
central Africa and central Europe (15° E.) and around
near Behring Strait (165° W.) may be slightly more oblate
than the other meridian circles, the one which is most
JO. MATH. GEO. — 3

34

THE FORM OF THE EARTH

nearly circular passing through central Asia (105° E.),
eastern North America, and western South America
(75° W.).
United States Curved Unequally. It is interesting to
note that the dimensions of the degrees of the long arc
of the 39th parallel surveyed in the United States bear out

Fig. 13. Gravimetric lines showing variation in force of gravity

to a remarkable extent the theory that the earth is slightly
flattened longitudinally, making it even more than that
just given, which was calculated by Sir John Herschel and
A. R. Clarke. The average length of degrees of longitude
from the Atlantic coast for the first 1,500 miles corresponds
closely to the Clarke table, and thus those degrees are
longer, and the rest of the arc corresponds closely to the
Bessel table and shows shorter degrees.

GEOID DEFINED

35

Cape May to Wallace (Kansas)
Wallace to Uriah (Calif.) .

Diff. in
long.

26.661° 21.618°

Length of 1°

53.829 mi.
53.822 mi

Clarke

53.828 mi.

Bessel

53.821 mi.

Earth not an Ellipsoid of Three Unequal Axes. This
oblateness of the meridians and oblateness of the equa
tor led some to treat the earth as an ellipsoid of three
unequal axes: (1) the longest equatorial axis, (2) the
shortest equatorial axis, and (3) the polar axis. It has
been shown, however, that meridians are not true ellipses,
for the amount of flattening northward is not quite the
same as the amount southward, and the mathematical
center of the earth is not exactly in the plane of the equator.
Geoid Denned. The term geoid, which means " like the
earth," is now applied to that figure which most nearly cor
responds to the true shape of the earth. Mountains, valleys,
and other slight deviations from evenness of surfaces are
treated as departures from the geoid of reference. The
following definition by Robert S. Woodward, President of
the Carnegie Institution of Washington, very clearly
explains what is meant by the geoid.*
" Imagine the mean sea level, or the surface of the sea
freed from the undulations due to winds and to tides.
This mean sea surface, which may be conceived to extend
through the continents, is called the geoid. It does not
coincide exactly with the earth's spheroid, but is a slightly
wavy surface lymg partly above and partly below the
spheroidal surface, by small but as yet not definitely known
amounts. The determination of the geoid is now one of
the most important problems of geophysics."
* Encyclopaedia Americana.

36 THE FORM OF THE EARTH
An investigation is now in progress in the United States
for determining a new geoid of reference upon a plan never
followed hitherto. The following is a lucid description *
of the plan by John F. Hayford, Inspector of Geodetic
Work, United States Coast and Geodetic Survey.
Area Method of Determining Form of the Earth. " The
arc method of deducing the figure of the earth may be
illustrated by supposing that a skilled workman to whom is
given several stiff wires, each representing a geodetic arc,
either of a parallel or a meridian, each bent to the radius
deduced from the astronomic observations of that arc, is
told in what latitude each is located on the geoid and then
requested to construct the ellipsoid of revolution which
will conform most closely to the bent wires. Similarly,
the area method is illustrated by supposing that the work
man is given a piece of sheet metal cut to the outline of
the continuous triangulation which is supplied with neces
sary astronomic observations, and accurately molded to fix
the curvature of the geoid, as shown by the astronomic
observations, and that the workman is then requested to
contruct the ellipsoid of revolution which will conform
most accurately to the bent sheet. Such a bent sheet
essentially includes within itself the bent wires referred to
in the first illustration, and, moreover, the wires are now
held rigidly in their proper relative positions. The sheet is
much more, however, than this rigid system of bent lines,
for each arc usually treated as a line is really a belt of
considerable width which is now utilized fully. It is ob
vious that the workman would succeed much better in con
structing accurately the required ellipsoid of revolution
from the one bent sheet than from the several bent wires.
When this proposition is examined analytically it will be
* Given at the International Geographic Congress, 1904.

ON A MERIDIAN CIRCLE 37
seen to be true to a much greater extent than appears
from this crude illustration."
" The area of irregular shape which is being treated as a
single unit extends from Maine to California and from
Lake Superior to the Gulf of Mexico. It covers a range of
57° in longitude and 19° in latitude, and contains 477
astronomic stations. This triangulation with its numerous
accompanying astronomical observations will, even with
out combination with similar work in other countries,
furnish a remarkably strong determination of the figure
and size of the earth."
It is possible that at some distant time in the future the
dimensions and form of the geoid will be so accurately
known that instead of using an oblate spheroid of reference
(that is, a spheroid of such dimensions as most closely
correspond to the earth, treated as an oblate spheroid such
as the Clarke Spheroid of 1866), as is now done, it will be
possible to treat any particular area of the earth as having
its own peculiar curvature and dimensions.
Conclusion. What is the form of the earth? We went
to considerable pains to prove that the earth is a sphere.
That may be said to be its general form, and in very many
calculations it is always so treated. For more exact cal
culations, the earth's departures from a sphere must be
borne in mind. The regular geometric solid which the
earth most nearly resembles is an oblate spheroid. Strictly
speaking, however, the form of the earth (not considering
such irregularities as mountains and valleys) must be
called a geoid.
Directions on the Earth
On a Meridian Circle. Think of yourself as standing
on a great circle of the earth passing through the poles.

38

THE FORM OF THE EARTH

Pointing from the northern horizon by way of your feet
to the southern horizon, you have pointed to all parts
of the meridian circle beneath you. Your arm has
swung through an angle of 180°, but you have pointed
through all points of the meridian circle, or 360° of it.
Drop your arm 90°, or from the horizon to the nadir,
and you have pointed through half of the meridian
circle, for 180° of latitude. It is apparent, then, that for
every degree you drop your arm, you point through
a space of two degrees of latitude upon the earth
beneath. The north pole is, let us say, 45° from you. Drop
your arm 22|° from the northern horizon, and you will
point directly toward the north pole (Fig. 14). What
ever your latitude, drop your arm half as many degrees from
the northern horizon as you are degrees from the pole, and you
will point directly toward that pole*
You may be so accustomed to thinking of the north
pole as northward in a
horizontal line from you
that it does not seem
real to think of it as
below the horizon. This
is because one is liable
to forget that he is
living on a ball. To
point to the horizon is
to point away from the earth.
A Pointing Exercise. It may not be easy or even
essential to learn exactly to locate many places in rela
tion to the home region, but the ability to locate readily
* The angle included between a tangent and a chord is measured by
one half the intercepted arc.

S Horizon

N. Horizon

HPala

Fig. 14

A POINTING EXERCISE

39

Horizon Lino

Fig. 15

some salient points greatly clarifies one's sense of loca
tion and conception of the earth as a ball.
The following exer
cise is designed for
students living not far
from the 45th parallel.
Since it is impossible
to point the arm or
pencil with accuracy
at any given angle, it
is roughly adapted for
the north temperate
latitudes (Fig. 15).
Persons living in the
southern states may
use Figure 16, based
on the 30th parallel.
The student should make the necessary readjustment for
his own latitude.
Drop the arm from
the northern horizon
quarter way down, or
22J°, and you are
pointing toward the
north pole (Fig. 15).
Drop it half way
down, or 45° from the
horizon, and you are
pointing 45° the other
side of the north pole,
or half way to the
equator, on the same parallel but on the opposite side of
the earth, in opposite longitude. Were you to travel half

Horizon Line

Fig. 16

40 THE FORM OF THE EARTH
way around the earth in a due easterly or westerly direc
tion, you would be at that point. Drop the arm 22J°
more, or 67J° from the horizon, and you are pointing 45°
farther south or to the equator on the opposite side of the
earth. Drop the arm 22J° more, or 90° from the horizon,
toward your feet, and you are pointing toward our anti
podes, 45° south of the equator on the meridian opposite
ours. Find where on the earth this point is. Is the
familiar statement, " digging through the earth to China,"
based upon a correct idea of positions and directions on
the earth?
From the southern horizon drop the arm 22J°, and you
are pointing to a place having the same longitude but on
the equator. Drop the arm 22J° more, and you point to
a place having the same longitude as ours but opposite
latitude, being 45° south of the equator on our meridian.
Drop the arm 22£° more, and you point toward the south
pole. Practice until you can point directly toward any of
these seven points without reference to the diagram.
Latitude and Longitude
Origin of Terms. Students often have difficulty in
remembering whether it is latitude that is measured east
and west, or longitude. When we recall the fact that
to the people who first used these terms the earth was
believed to be longer east and west than north and south,
and now. we know that owing to the oblateness of the
earth this is actually the case, we can easily remember that
longitude (from the Latin longus, long) is measured east
and west. The word latitude is from the Latin latitudo,
which is from latus, wide, and was originally used to
designate measurement of the " width of the earth," or
north and south.

LONGITUDE

41

Antipodal Areas. From a globe one can readily ascer
tain the point which is exactly opposite any given one on
the earth. The map showing antipodal areas indicates
at a glance what portions of the earth are opposite each
other; thus Australia lies directly through the earth from

Fig. 17. Map of Antipodal Areas
mid-Atlantic, the point antipodal to Cape Horn is in
central Asia, etc.
Longitude is measured on parallels and is reckoned from
some meridian selected as standard, called the prime
meridian. The meridian which passes through the Royal
Observatory at Greenwich, near London, has long been the
prime meridian most used. In many countries the
meridian passing through the capital is taken as the prime
meridian. Thus, the Portuguese use the meridian of the
Naval Observatory in the Royal Park at Lisbon, the

42 THE FORM OF THE EARTH
French that of the Paris Observatory, the Greeks that of
the Athens Observatory, the Russians that of the Royal
Observatory at Pulkowa, near St. Petersburg.
In the maps of the United States the longitude is often
reckoned both from Greenwich and Washington. The
latter city being a trifle more than 77° west of Greenwich,
a meridian numbered at the top of the map as 90° west
from Greenwich, is numbered at the bottom as 13° west
from Washington. Since the United States Naval Obser
vatory, the point in Washington reckoned from, is 77° 3'
81" west from Greenwich, this is slightly inaccurate.
Among all English speaking people and in most nations of
the world, unless otherwise designated, the longitude of a
place is understood to be reckoned from Greenwich.
The longitude of a place is the arc of the parallel inter
cepted between it and the prime meridian. Longitude
may also be defined as the arc of the equator intercepted
between the prime meridian and the meridian of the
place whose longitude is sought.
Since longitude is measured on parallels, and parallels
grow smaller toward the poles, degrees of longitude are
shorter toward the poles, being degrees of smaller circles.
Latitude is measured on a meridian and is reckoned
from the equator. The number of degrees in the arc of
a meridian circle, from the place whose latitude is sought
to the equator, is its latitude. Stated more formally, the
latitude of a place is the arc of the meridian intercepted
between the equator and that place. (See Latitude in
Glossary.) What is the greatest number of degrees of
latitude any place may have? What places have no
latitude? Comparative Lengths of Degrees of Latitude. If the earth
were a perfect sphere, meridian circles would be true mathe-

LATITUDE

43

Pole

matical circles. Since the earth is an oblate spheroid,
meridian circles, so called, curve less rapidly toward the
poles. Since the curvature is greatest near the equator,
one would have to travel less distance on a meridian there
to cover a degree of curvature, and a degree of latitude is
thus shorter near the equator. Conversely, the meridian
being slightly flat
tened toward the
poles, one would
travel farther there
to cover a degree
of latitude, hence
degrees of latitude
are longer toward
the poles. Perhaps
this may be seen
more clearly from
Figure 18.
While all circles
have 360°, the de
grees of a small
circle are, of course,
shorter than the degrees of a greater circle. Now an
arc of a meridian near the equator is obviously a part
of a smaller circle than an arc taken near the poles and,
consequently, the degrees are shorter. Near the poles,
because of the flatness of a meridian there, an arc of a
meridian is a part of a larger circle and the degrees are
longer. As we travel northward, the North star (polestar)
rises from the horizon. In traveling from the equator on
a meridian, one would go 68.7 miles to see the polestar
rise one degree, or, in other words, to cover one degree of
curvature of the meridian. Near the pole, where the earth

Fig. iB

44

THE FORM OF THE EARTH

is flattest, one would have to travel 69.4 miles to cover one
degree of curvature of the meridian. The average length
of a degree of latitude throughout the United States is
almost exactly 69 miles.
Table of Lengths of Degrees. The following table
shows the length of each degree of the parallel and of the
meridian at every degree of latitude. It is based upon
the Clarke spheroid of 1866.

Deg.

Deg.

Deg.

Deg.

Deg.

Deg.

Lat.

Par.

Mer.

Lat.

Par.

Mer.

Lat.

Par.

Mer.

Miles
Miles
Miles
Miles
Miles
Miles
0°
69.172
68.704
31°
59.345
68.890
61°
33.623
69.241
1
69.162
68.704
32
58.716
68.901
62
32.560
69.251
2
69.130
68.705
33
58.071
68.912
63
31.488
69.261
3
69.078
68.706
34
57.407
68.923
64
30.406
69.271
4
69.005
68.708
35
56.725
68.935
65
29.315
69.281
5
68.911
68.710
36
56.027
68.946
66
28.215
69.290
6
68.795
68.712
37
55.311
68.958
67
27.106
69.299
7
68.660
68.715
38
54.579
68.969
68
25.988
69.308
8
68.504
68.718
39
53.829
68.981
69
24.862
69.316
9
68.326
68.721
40
53.063
68.993
70
23.729
69.324
10
68.129
68.725
41
52.281
69.006
71
22.589
69.332
11
67.910
68.730
42
51.483
69.018
72
21.441
69.340
12
67.670
68.734
43
50.669
69.030
73
20.287
69.347
13
67.410
68.739
44
49.840
69.042
74
19.127
69.354
14
67.131
68.744
45
48.995
69.054
75
17.960
69.360
15
66.830
68.751
46
48.136
69.066
76
16.788
69.366
16
66.510
68.757
47
47.261
69.079
77
15.611
•69.372
17
66.169
68.764
48
46.372
69.091
78
14.428
69.377
18
65.808
68.771
49
45.469
69.103
79
13.242
69.382
19
65.427
68.778
50
44.552
69.115
80
12.051
69.386
20
65.026
68.786
51
43.621
69.127
81
10.857
69.390
21
64.606
68.794
52
42.676
69.139
82
9.659
69.394
22
64.166
68.802
53
41.719
69.151
83
8.458
69.397
23
63.706
68.811
54
40.749
69.163
84
7.255
69.400
24
63.228
68.820
55
39.766
69.175
85
6.049
69.402
26
62.729
68.829
56
38.771
69.186
86
4.842
69.404
26
62.212
68.839
57
37.764
69.197
87
3.632
69.405
27
61.676
68.848
58
36.745
69.209
88
2.422
69.407
28
61.122
68.858
59
35.716
69.220
89
1.211
69.407
29
60.548
68.869
60
34.674
69.230
90
0.000
69.407
30
59.956
68.879
CHAPTER III
THE ROTATION OF THE EARTH
The Celestial Sphere
Apparent Dome of the Sky. On a clear night the stars
twinkling all over the sky seem to be fixed in a dark dome
fitting down around the horizon. This apparent concavity,
studded with heavenly bodies, is called the celestial sphere.
Where the horizon is free from obstructions, one can see
half * of the celestial sphere at a given time from the same
place. A line from one side of the horizon over the zenith point
to the opposite side of the horizon is half of a great circle
of the celestial sphere. The horizon line extended to the
celestial sphere is a great circle. Owing to its immense

If these lines met at a point 50,000 miles
distant, the difference in their direction
could not be measured. Such is the ratio
of the diameter of the earth and the dis
tance to the very nearest of the stars.

Fig. 19

distance, a line from an observer at A (Fig. 19), pointing
to a star, will make a line apparently parallel to one from
B to the same star. The most refined measurements at
* No allowance is here made for the refraction of rays of light or the
slight curvature of the globe in the locality.
45

46 THE ROTATION OF THE EARTH
present possible fail to show any angle whatever between
them. We may note the following in reference to the celestial
sphere. (1) The earth seems to be a mere point in the
center of this immense hollow sphere. (2) The stars,
however distant, are apparently fixed in this sphere.
(3) Any plane from the observer, if extended, will divide
the celestial sphere into two equal parts. (4) Circles
may be projected on this sphere and positions on it indi
cated by degrees in distance from established circles or
points. Celestial Sphere seems to Rotate. The earth rotates on
its axis (the term rotation applied to the earth refers to
its daily or axial motion). To us, however, the earth
seems stationary and the celestial sphere seems to rotate.
Standing in the center of a room and turning one's body
around, the objects in the room seem to rotate around in
the opposite direction. The point overhead will be the
only one that is stationary. Imagine a fly on a rotating
sphere. If it were on one of the poles, that is, at the end
of the axis of rotation, the object directly above it would
constantly remain above it while every other fixed object
would seem to swing around in circles. Were the fly to
walk to the equator, the point directly away from the
globe would cut the largest circle around him and the
stationary points would be along the horizon.
Celestial Pole. The point in the celestial sphere directly
above the pole and in line with the axis has no motion.
It is called the celestial pole. The star nearest the pole
of the celestial sphere and directly above the north pole
of the earth is called the North star, and the star nearest
the southern celestial pole the South star. It may be of
interest to note that as we located the North star by refer-

AT THE NORTH POLE-

47

ence to the Big Dipper, the South star is located by refer
ence to a group of stars known as the Southern Cross.
Celestial Equator. A great circle is conceived to extend
around the celestial sphere 90°
from the poles (Fig. 20). This
is called the celestial equator.
The axis of the earth, if pro
longed, would pierce the celes
tial poles, almost pierce the
North and South stars, and
the equator of the earth if ex
tended would coincide with the
celestial equator.
At the North Pole. An Fig- M
observer at the north pole
will see the North star almost exactly overhead, and
as the earth turns around under his feet it will remain
constantly overhead (Fig. 21). Half way, or 90° from
the North star, is the celestial equator around the
horizon. As the earth
rotates, — though it
seems to us per
fectly still, — the stars
around the sky seem
TfiftrtSL— / ^*~ \ *° SWU1§ m circles in
\mT^^^^^^- I ^ne °PPosite direction.
' '' '' OfciL ^----' The planes of the star
paths are parallel to
Fig-" the horizon. The
same half of the celestial sphere can be seen all of the
time, and stars below the horizon always remain so.
All stars south of the celestial equator being forever
invisible at the north pole, Sirius, the brightest of the

48 THE ROTATION OF THE EARTH
stars, and many of the beautiful constellations, can never
be seen from that place. How peculiar the view of the
heavens must be from the pole, the Big Dipper, the
Pleiades, the Square of Pegasus, and other star groups
swinging eternally around in courses parallel to the
horizon. When the sun, moon, and planets are in the
portion of their courses north of the celestial equator,
they, of course, will be seen throughout continued rotations
of the earth until they pass below the celestial equator,
when they will remain invisible again for long periods.
The direction of the daily apparent rotation of the stars
is from left to right (westward), the direction of the
hands of a clock looked at from above. Lest the direction
of rotation at the North pole be a matter of memory
rather than of insight, we may notice that in the United
States and Canada when we face southward we see the
sun's daily course in the direction left to right (westward),
and going poleward the direction remains the same though
the sun approaches the horizon more and more as we
approach the North pole.
At the South Pole. An observer at the South pole, at
the other end of the axis, will see the South star directly
overhead, the celestial equator on the horizon, and the
plane of the star circles parallel with the horizon. The
direction of the apparent rotation of the celestial sphere
is from right to left, counter-clockwise. If a star is seen
at one's right on the horizon at six o'clock in the morning,
at noon it will be in front, at about six o'clock at night at
his left, at midnight behind him, and at about six o'clock
in the morning at his right again.
At the Equator. An observer at the equator sees the
stars in the celestial sphere to be very different in their
positions in relation to himself. Remembering that he is

BETWEEN EQUATOR AND POLES

49

Fig. 22

standing with the line of his body at right angles to the
axis of the earth, it is easy to understand why all the stars
of the celestial sphere seem to be shifted around 90° from
where they were at the poles. The celestial equator is a
great circle extending
from east to west di
rectly overhead. The
North star is seen
on the northern hori
zon and the South
star on the southern
horizon. The planes
of the circles followed
by stars in their daily
orbits cut the horizon
at right angles, the horizon being parallel to the axis. At
the equator one can see the entire celestial sphere, half at
one time and the other half about twelve hours later.
Between Equator
and Poles. At places
between the equator
and the poles, the ob
server is liable to feel
that a star rising due
east ought to pass
the zenith about six
hours later instead
of swinging slantingly
around as it actually
seems tc do. This is because one forgets that the axis
is not squarely under his feet excepting when at the
equator. There, and there only, is the axis at right
angles to the line of one's body when erect. The
Jo. Math. Geo, — 4

Fig. 2;

50 THE ROTATION OF THE EARTH
.apparent rotation of the celestial sphere is at right angles
to the axis.
Photographing the Celestial Sphere. Because of the
earth's rotation, the entire celestial sphere seems to rotate.
Thus we see stars daily circling around, the polestar
always stationary. When stars are photographed, long
exposures are necessary that their faint light may affect
the sensitive plate of the camera, and the photographic
instruments must be constructed so that they will move
at the same rate and in the same direction as the stars,
otherwise the stars will leave trails on the plate. When
the photographic instrument thus follows the stars in
their courses, each is shown as a speck on the plate and
comets, meteors, planets, or asteroids, moving at different
rates and in different directions, show as traces.
Rotation of Celestial Sphere is Only Apparent. For a long
time it was believed that the heavenly bodies rotated
around the stationary earth as the center. It was only
about five hundred years ago that the astronomer Coper
nicus established the fact that the motion of the sun and
stars around the earth is only apparent, the earth rotating.
We may be interested in some proofs that this is the case.
It seems hard to believe at first that this big earth, 25,000
miles in circumference can turn around once in a day.
" Why, that would give us a whirling motion of over a
thousand miles an hour at the equator." " Who could
stick to a merry-go-round going at the rate of a thousand
miles an hour? " When we see, however, that the sun
93,000,000 miles away, would have to swing around in a
course of over 580,000,000 miles per day, and the stars, at
their tremendous distances, would have to move at unthink
able rates of speed, we see that it is far easier to believe
that it is the earth and not the celestial sphere that rotates

EASTWARD DEFLECTION OF FALLING BODIES 51
daily. We know by direct observation that other planets,
the sun and the moon, rotate upon their axis, and may
reasonably infer that the earth does too.
So far as the whirling motion at the equator is concerned,
it does give bodies a slight tendency to fly off, but the
amount of this force is only g^g as great as the attractive
influence of the earth; that is, an object which would
weigh 289 pounds at the equator, were the earth at rest,
weighs a pound less because of the centrifugal force of
rotation (see p. 14).
Proofs of the Earth's Rotation
Eastward Deflection of Falling Bodies. Perhaps the
simplest proof of the rotation of the earth is one pointed
out by Newton, although he had no means of demon
strating it. With his clear vision be said that if the earth
rotates and an object were dropped from a considerable
height, instead of falling directly toward the center of
the earth in the direction of the plumb line,* it would be
deflected toward the east. Experiments have been made
in the shafts of mines where air currents have been shut off
and a slight but unmistakable eastward tendency has been
observed. During the summer of 1906, a number of newspapers
and magazines in the United States gave accounts of the
eastward falling of objects dropped in the deep mines of
northern Michigan, one of which (Shaft No. 3 of the Tam
arack mine) is the deepest in the world, having a vertical
depth of over one mile (and still digging!). It was stated
that objects dropped into such a shaft never reached the
* The slight geocentric deviations of the plumb line are explained
on pp. 28.1-282.

52 THE ROTATION OF THE EARTH
bottom but always lodged among timbers on the east
side. Some papers added a touch of the grewsome by
implying that among the objects found clinging to the east
side are " pieces of a dismembered human body " which
were not permitted to fall to the bottom because of the rota
tion of the earth. Following is a portion of an account *
by F. W. McNair, President of the Michigan College of
Mines. Mc Nair's Experiment. " Objects dropping into the shaft
under ordinary conditions nearly always start with some
horizontal velocity, indeed it is usually due to such initial
velocity in the horizontal that they get into the shaft
at all. Almost all common objects are irregular in shape,
and, drop one of them ever so carefully, contact with the
air through which it is passing soon deviates it from
the vertical, giving it a horizontal velocity, and this when
the air is quite still. The object slides one way or another
on the air it compresses in front of it. Even if the body
is a sphere, the air will cause it to deviate, if it is rotating
about an axis out of the vertical. Again, the air in the
shaft is in ceaseless motion, and any obliquity of the
currents would obviously deviate the falling body from
the vertical, no matter what its shape. If the falling
object is of steel, the magnetic influence of the air mains
and steam mains which pass down the shaft, and which
invariably become strongly magnetic, may cause it to
swerve from a vertical course . . .
" A steel sphere, chosen because it was the only con
venient object at hand, was suspended about one foot
from the timbers near the western corner of the compart
ment. The compartment stands diagonally with refer
ence to the cardinal points. Forty-two hundred feet below
* In the Mining and Scientific Press, July 14, 1906.

McNAIR'S EXPERIMENT 53
a clay bed was placed, having its eastern edge some five
feet east of the point of suspension of the ball. When
the ball appeared to be still the suspending thread was
burned, and the instant of the dropping of the ball was
indicated by a prearranged signal transmitted by tele
phone to the observers below, who, watch in hand, waited
for the sphere to strike the bed of clay. It failed to
appear at all. Another like sphere was hung in the center
of the compartment and the trial was repeated with the
same result. The shaft had to be cleared and no more
trials were feasible. Some months later, one of the spheres,
presumably the latter one, was found by a timberman
where it had lodged in the timbers 800 feet from the
surface. " It is not probable, however, that these balls lodged
because of the earth's rotation alone. . . . The matter is
really more complicated than the foregoing discussion
implies. It has received mathematical treatment from
the great Gauss. According to his results, the deviation
to the east for a fall of 5,000 feet at the Tamarack mine
should be a little under three feet. Both spheres had that
much to spare before striking the timbers. It is almost
certain, therefore, that others of the causes mentioned in
the beginning acted to prevent a vertical fall. At any
rate, these trials serve to emphasize the unlikelihood that
an object which falls into a deep vertical shaft, like those
at the Tamarack mine, will reach the bottom, even when
some care is taken in selecting it and also to start it verti
cally. " If the timbering permits lodgment, as is the case in
most shafts, it may truthfully be said that if a shaft is
deep in proportion to its cross section few indeed will be
the objects falling into it which will reach the bottom,

54

the Rotation OF The earth

and such objects are more likely to lodge on the easterly
side than on any other."
The Foucault Experiment. Another simple demonstra
tion of the earth's rotation is by the celebrated Foucault
experiment. In 1851, the French physicist, M. Leon Fou
cault, suspended from the dome of the Pantheon, in Paris,
a heavy iron ball by wire two hundred feet long. A pin
was fastened to the lowest
side of the ball so that when
swinging it traced a slight
mark in a layer of sand placed
beneath it. Carefully the
long pendulum was set swing
ing. It was found that the
path gradually moved around
toward the right. Now either
the pedulum changed its
plane or the building was
gradually turned around. By
experimenting with a ball
suspended from a ruler one
can readily see that gradually
turning the ruler will not
Fig. 24 change the plane of the
swinging pendulum. If the
pendulum swings back and forth in a north and south direc
tion, the ruler can be entirely turned around without chang
ing the direction of the pendulum's swing. If at the north
pole a pendulum was set swinging toward a fixed star, say
Arcturus, it would continue swinging toward the same
star and the earth would thus be seen to turn around in
a day. The earth would not seem to turn but the pendu
lum would seem to deviate toward the right or clockwise.

THE FOUCAULT EXPERIMENT

55

Conditions for Success. The Foucault experiment has
been made in many places at different times. To be suc
cessful there should be a long slender wire, say forty feet
or more in length, down the well of a stairway. The weight
suspended should be heavy and spherical so that the
impact against the air may not cause it to slide to one
side, and there SHOWING THE EARTH'S MOTION

Interesting Experiment in the Dome
of the Pantheon.
We«c Tor fc Sim Special Service
Paris, Oct, 23.— An interesting experi
ment under the auspices of the; astro
nomical society of France took place yea- I
terday afternoon when ocular proof of the
revolution. of the earth was given by I

should be protec
tion against drafts
of air. A good sized
circle, marked off
in degrees, should
be placed under it,
with the center
exactly under the
ball when at rest.
From the rate of
the deviation the
latitude may be
easily determined
or, knowing the
latitude, the devia
tion may be cal
culated. To Calculate Amount of Deviation. At first thought
it might seem as though the floor would turn completely
around under the pendulum in a day, regardless of the
latitude. It will be readily seen, however, that it is only
at the pole that the earth would make one complete rota
tion under the pendulum in one day * or show a deviation
of 15° in an hour. At the equator the pendulum will
show no deviation, and at intermediate latitudes the rate
* Strictly speaking, in one sidereal day.

means of a pendulum, consisting of a ball
weighing 60 pounds attached by a wire ,70
yards in length to .the Interior of the dome
of the Pantheon. Mr. Chauraio, minister
of public instruction, - -. who presided,
burned a string that tied: the weight to a
Pillar aria: the Immense pendulum began
Its Jqurney.r^Sand had been. placed on the
floor and ' each- time the pendulum passed
over It a. new track was., marked In regu
lar deviation, though the plane of the
pendulum's swrng remained unchanged.
The .experiment was completely success-
I £u]- - IV© 2.
Fig. 25

56 THE ROTATION OF THE EARTH
of deviation varies. Now the ratio of variation from the
pole considered as one and the equator as zero is shown
in the table of " natural sines " (p. 311). It can be
demonstrated that the number of degrees the plane of the
pendulum will deviate in one hour at any latitude is found
by multiplying 15° by the sine of the latitude.
d = deviation
<f> = latitude
.'. d = sine <f> X 15°.
Whether or not the student has a very clear conception of
what is meant by " the sine of the latitude " he may easily
calculate the deviation or the latitude where such a pen
dulum experiment is made.
Example. Suppose the latitude is 40°. Sine 40° =
.6428. The hourly deviation at that latitude, then, is
.6428 X 15° or 9.64°. Since the pendulum deviates 9.64°
in one hour, for the entire circuit it will take as many
hours as that number of degrees is contained in 360° or
about 37J. hours. It is just as simple to calculate one's
latitude if the amount of deviation for one hour is known.
Suppose the plane of the pendulum is observed to deviate
9° in an hour. Sine of the latitude X 15° = 9°.
.*. Sine of the latitude = ^ or .6000.
From the table of sines we find that this sine, .6000, corre
sponds more nearly to that of 37° (.6018) than to the sine
of any other whole degree, and hence 37° is the latitude
where the hourly deviation is 9°. At that latitude it would
take forty hours (360 ¦*¦ 9 = 40) for the pendulum to
make the entire circuit.

OTHER EVIDENCE

57

Table of Variations. The following table shows the
deviation of the plane of the pendulum for one hour and
the time required to make one entire rotation.

Latitude.

Hourly

Circuit of

Latitude.

Hourly

• Circuit of

Deviation.

Pendulum.

Deviation.

Pendulum.

5°

1.31°

275' hrs.

50°

11.49°

31 hrs.

10

2.60

138

55

12.28

29

15

3.88

93

60

12.99

28

20

5.13

70

65

13.59

26

25

6.34

57

70

14.09

25.5

30

7.50

48

75

14.48

24.8

35

8.60

42

80

14.77

24.5

40

9.64

37

85

14.94

24.1

45

10.61

34

90

15.00

24.0

Other Evidence. Other positive evidence of the rotation
of the earth we have in the fact that the equatorial winds
north of the equator veer toward the east and polar winds
toward the west — • south of the equator exactly opposite —
and this is precisely the result which would follow from the
earth's rotation. Cyclonic winds in the northern hemi
sphere in going toward the area of low pressure veer toward
the right and anti-cyclonic winds also veer toward the
right in leaving areas of high pressure, and in the southern
hemisphere their rotation is the opposite. No explanation
of these well-known facts has been satisfactorily advanced
other than the eastward rotation of the earth, which easily
accounts for them.
Perhaps the best of modern proofs of the rotation of
the earth is demonstrated by means of the spectroscope.
A discussion of this is reserved until the principles are
explained (pp. 107, 108) in connection with the proofs of
the earth's revolution.

58

THE ROTATION OF THE EARTH

Velocity of Rotation
The velocity of the rotation at the surface, in miles per
hour, in different latitudes, is as follows:

Latitude.

Velocity.

Latitude.

Velocity.

Latitude.

Velocity.

0

1038

44

748

64

456

5

1034

45

735

66

423

10

1022

46

722

68

390

15

1002

47

709

70

356

20

975

48

696

72

322

25

941

49

682

74

287

30

899

50

668

76

252

32

881

51

654

78

216

34

861

52

640

80

181

36

840

53

626

82

145

38

819

54

611

84

109

39

807

55

596

86

73

40

796

56

582

88

36

41

784

58

551

89

18

42

772

60

520

89J

9

43

760

62

488

90

0

Uniform Rate of Rotation. There are theoretical grounds
for believing that the rate of the earth's rotation is getting
gradually slower. As yet, however, not the slightest
variation has been discovered. Before attacking the
somewhat complex problem of time, the student should
clearly bear in mind the fact that the earth rotates on its
axis with such unerring regularity that this is the most
perfect standard for any time calculations known to
science. Determination of Latitude
Altitude of Celestial Pole Equals Latitude. Let us return,
in imagination, to the equator. Here we may see the North
star on the horizon due north of us, the South star on the

TO FIND YOUR LATITUDE 59
horizon due south, and halfway between these two points,
extending from due east through the zenith to due west,
is the celestial equator. If we travel northward we shall
be able to see objects which were heretofore hidden from
view by the curvature of the earth. We shall find that
the South star becomes hidden from sight for the same
reason and the North star seems to rise in the sky. The
celestial equator no longer extends through the point
directly overhead but is somewhat to the south of the
zenith, though it still intersects the horizon at the east
and west points. As we go farther north this rising of
the northern sky and sinking of the southern sky becomes
greater. If we go halfway to the north pole we shall find
the North star halfway between the zenith and the northern
horizon, or at an altitude of 45° above the horizon. For
every degree of curvature of the earth we pass over,
going northward, the North star rises one degree from
the horizon. At New Orleans the North star is 30° from
the horizon, for the city is 30° from the equator. At
Philadelphia, 40° north latitude, the North star is 40°
from the horizon. South of the equator the converse of
this is true. The North star sinks from the horizon and
the South star rises as one travels southward from the
equator. The altitude of the North star is the latitude of
a place north of the equator and the altitude of tlie South
star is the latitude of a place south of the equator. It is
obvious, then, that the problem of determining latitude is
the problem of determining the altitude of the celestial
pole. To Find Your Latitude.* By means of the compasses
and scale, ascertain the altitude of the North star. This
can be done by placing one side of the compasses on
a level window sill and sighting the other side toward

60 THE ROTATION OF THE EARTH
the North star, then measuring the angle thus formed.
Another simple process for ascertaining latitude is to
determine the altitude of a star not far from the North
star when it is highest and when it is lowest ; the average
of these altitudes is the altitude of the pole, or the latitude.
This may easily be done in latitudes north of 38° during
the winter, observing, say, at 6 o'clock in the morning
and at 6 o'clock in the evening. This is simple in that
it requires no tables. Of course such measurements are
very crude with simple instruments, but with a little
care one will usually be surprised at the accuracy of his
results. Owing to the fact that the North star is not exactly at
the north pole of the celestial sphere, it has a slight rotary
motion. It will be more accurate, therefore, if the obser
vation is made when the Big Dipper and Cassiopeia are

A

True Pole

True Pole
 &• 

C Q True Pole ^ _ -. ..
TruePaleQD

Fig. 26
in one of the positions (A or B) represented by Figure 26.
In case of these positions the altitude of the North star
will give the true latitude, it then being the same altitude
as the pole of the celestial sphere. In case of position
D, the North star is about 1J° below the true pole, hence

TO FIND YOUR LATITUDE 61
1£° must be added to the altitude of the star. In case of
position C, the North star is l\° above the true pole, and
that amount must be subtracted from its altitude. It is
obvious from the diagrams that a true north and south
line can be struck when the stars are in positions C and D,
by hanging two plumb lines so that they lie in the same
plane as the zenith meridian line through Mizar and Delta
Cassiopeise. Methods of determining latitude will be further
discussed on pp. 172-174. The instrument commonly
used in observations for determining latitude is the meri
dian circle, or, on shipboard, the sextant. Read the de
scription of these instruments in any text on astronomy.
Queries
In looking at the heavenly bodies at night do the stars,
moon, and planets all look as though they were equally
distant, or do some appear nearer than others? The fact
that people of ancient times believed the celestial sphere
to be made of metal and all the heavenly bodies fixed or
moving therein, would indicate that to the observer who is
not biased by preconceptions, all seem equally distant.
If they did not seem equally distant they would not
assume the apparently spherical arrangement.
The declination, or distance from the celestial equator,
of the star (Benetnasch) at the end of the handle of the
Big Dipper is 50°. How far is it from the celestial pole?
At what latitude will it touch the horizon in its swing
under the North star? How far south of the equator could
one travel and still see that star at some time?

CHAPTER IV
longitude and time
Solar Time
Sun Time Varies. The sun is the world's great time
keeper. He is, however, a slightly erratic one. At the
equator the length of day equals the length of night the
year through. At the poles there are six months day and
six months night, and at intermediate latitudes the time
of sunrise and of sunset varies with the season. Not only
does the time of sunrise vary, but the time it takes the sun
apparently to swing once around the earth also varies.
Thus from noon by the sun until noon by the sun again is
sometimes more than twenty-four hours and sometimes
less than twenty-four hours. The reasons for this varia
tion will be taken up in the chapter on the earth's revolution.
Mean Solar Day. By a mean solar day is meant the
average interval of time from sun noon to sun noon.
While the apparent solar day varies, the mean solar day
is exactly twenty-four hours long. A sundial does not
record the same time as a clock, as a usual thing, for the
sundial records apparent solar time while the clock records
mean solar time.
Relation of Longitude to Time. The sun's apparent daily
journey around the earth with the other bodies of the
celestial sphere gives rise to day and night.* It takes the
sun, on the average, twenty-four hours apparently to swing
* Many thoughtlessly assume that the fact of day and night is a
proof of the earth's rotation. 62

HOW LONGITUDE IS DETERMINED 63
once around the earth. In this daily journey it crosses
360° of longitude, or 15° for each hour. It thus takes
four minutes for the sun's rays to sweep over one degree
of longitude. Suppose it is noon by the sun at the 90th
meridian, in four minutes the sun will be over the 91st
meridian, in four more minutes it will be noon by the sun
on the 92d meridian, and so on around the globe.
Students are sometimes confused as to the time of day
in places east of a given meridian as compared with the
time in places west of it. When the sun is rising here, it
has already risen for places east, of us, hence their time is
after sunrise or later than ours. If it is noon by the
sun here, at places east of us, having already been noon
there, it must be past noon or later in the day. Places
to the east have later time because the sun reaches them first.
To the westward the converse of this is true. If the sun
is rising here, it has not yet risen for places west of us and
their time is before sunrise or earlier. When it is noon by
the sun in Chicago, the shadow north, it- is past noon by
che sun in Detroit and other places eastward and before
noon by the sun in Minneapolis and other places westward.
How Longitude is Determined. A man when in London,
longitude 0°, set his watch according to mean solar time
there. When he arrived at home he found the mean solar
time to be six hours earlier (or slower) than his watch,
which he had not changed. Since his watch indicated
later time, London must be east of his home, and since the
sun appeared six hours earlier at London, his home must
be 6 X 15°, or 90°, west of London. While on shipboard
at a certain place he noticed that the sun's shadow was due
north when his watch indicated 2:30 o'clock, p.m. Assum
ing that both the watch and the sun were " on time " we
readily see that since London time was two and one half

64 LONGITUDE AND TIME
hours later than the time at that place, he must have been
west of London 2| X 15°, or 37° 30'.
Ship's Chronometer. Every ocean vessel carries a very
accurate watch called a chronometer. This is regulated
to run as perfectly as possible and is set according to the
mean solar time of some well known meridian. Vessels
from English speaking nations all have their chronometers
set with Greenwich time. By observing the time accord
ing to the sun at the place whose longitude is sought and
comparing that time with the time of the prime meridian
as indicated by the chronometer, the longitude is reck
oned. For example, suppose the time according to the
sun is found by observation to be 9:30 o'clock, a.m., and
the chronometer indicates 11:20 o'clock, a.m. The prime
meridian, then, must be east as it has later time. Since
the difference in time is one hour and fifty minutes and
there are 15° difference in longitude for an hour's differ
ence in time, the difference in longitude must be If X 15°,
or 27° 30'.
The relation of longitude and time should be thoroughly
mastered. From the table at the close of this chapter,
giving the longitude, of the principal cities of the world,
one can determine the time it is in those places when it is
noon at home. Many other problems may also be sug
gested. It should be borne in mind that it is the mean
solar time that is thus considered, which in most cities is
not the time indicated by the watches and clocks there.
People all over Great Britain set their timepieces to agree
with Greenwich time, in Ireland with Dublin, in France
with Paris, etc. (see " Time used in Various Countries " at
the end of this chapter).
Local Time. The mean solar time of any place is often
called its local time. This is the average time indicated

ORIGIN OF PRESENT SYSTEM 65
by the sundial. All places on the same meridian have the
same local time. Places on different meridians must of
necessity have different local time, the difference in time
being four minutes for every degree's difference in longitude.
Standard Time
Origin of Present System. Before the year 1883, the
people of different cities in the United States commonly
used the local time of the meridian passing through the city.
Prior to the advent of the railroad, telegraph, and telephone,
little inconvenience was occasioned by the prevalence of so
many time systems. But as transportation and communi
cation became rapid and complex it became very difficult to
adjust one's time and calculations according to so many
standards as came to prevail. Each railroad had its own
arbitrary system of time, and where there were several
railroads in a city there were usually as many species of
" railroad time " besides the local time according to
longitude. " Before the adoption of standard time there were some
times as many as five different kinds of time in use in a
single town. The railroads of the United States followed
fifty-three different standards, whereas they now use five.
The times were very much out of joint." *
His inability to make some meteorological calculations
in 1874 because of the diverse and doubtful character of
the time of the available weather reports, led Professor
Cleveland Abbe, for so many years connected with the
United States Weather service, to suggest that a system of
standard time^ should be adopted. At about the same time
several others made similar suggestions and the subject was
soon taken up in an official way by the railroads of the
* The Scrap Book, May, 1906,
JO. MATH. GEO. — 5

Fig. 27. Standard Time Belts

MOUNTAIN STANDARD TIME 67
country under the leadership of William F. Allen, then sec
retary of the General Time Convention of Railroad Officials.
As a result of his untiring efforts the railway associations
endorsed his plan and at noon of Sunday, November 18,
1883, the present system was inaugurated.
Eastern Standard Time. According to the system all
cities approximately within 7^-° of the 75th meridian use
the mean solar time of that meridian, the clocks and
watches being thus just five hours earlier than those of
Greenwich. This belt, about 15° wide, is called the eastern
standard time belt. The 75th meridian passes through the
eastern portion of Philadelphia, so the time used through
out the eastern portion of the United States corresponds
to Philadelphia local mean solar time.
Central Standard Time. The time of the next belt is the
mean solar time of the 90th meridian or one hour slower
than eastern standard time. This meridian passes through
or very near Madison, Wisconsin, St. Louis, and New
Orleans, where mean local time is the same as standard
time. When it is noon at Washington, D. C, it is 11
o'clock, a.m., at Chicago, because the people of the former
city use eastern standard time and those at the latter use
central standard time.
Mountain Standard Time. To the west of the central
standard time belt lies the mountain region where the
time used is the mean solar time of the 105th meridian.
This meridian passes through Denver, Colorado, and its
clocks as a consequence indicate the same time that the
mean sun does there. As the standard time map shows,
all the belts are bounded by irregular lines, due to the
fact that the people of a city usually use the same time
that their principal railroads do, and where trains change
their time depends in a large measure upon the conven-

68 LONGITUDE AND TIME
ience to be served. This belt shows the anomaly of being
bounded on the east by the central time belt, on the west
by the Pacific time belt, and on the south by the same belts.
The reasons why the mountain standard time belt tapers
to a point at the south and the peculiar conditions which
consequently result, are discussed under the topic " Four
Kinds of Time around El Paso " (p. 75).
Pacific Standard Time. People living in the states bor
dering or near the Pacific Ocean use the mean solar time
of the 120th meridian and thus have three hours earlier
time than the people of the Atlantic coast states. This
meridian forms a portion of the eastern boundary of
California. In these great time belts * all the clocks and other time
pieces differ in time by whole hours. In addition to astro
nomical observatory clocks, which are regulated according
to the mean local time of the meridian passing through the
observatory, there are a few cities in Michigan, Georgia,
New Mexico, and elsewhere in the United States, where
mean local time is still used.
Standard Time in Europe. In many European countries
standard time based upon Greenwich time, or whole hour
changes from it, is in general use, although there are many
more cities which use mean local time than in the United
States. Western European time, or that of the meridian
of Greenwich, is used in Great Britain, Spain, Belgium, and
Holland. Central European time, one hour later than that
of Greenwich, is used in Norway, Sweden, Denmark, Luxem7
burg, Germany, Switzerland, Austria-Hungary, Servia, and
Italy. Eastern European time, two hours later than that
of Greenwich, is used in Turkey, Bulgaria, and Roumania.
* For a discussion of the time used in other portions of North
America and elsewhere in the world see pp. 81-87.

GETTING THE TIME 69
Telegraphic Time Signals
Getting the Time. An admirable system of sending time
signals all over the country and even to Alaska, Cuba, and
Panama, is in vogue in the United States, having been
established in August, 1865. The Naval Observatories at
Washington, D. C, and Mare Island, California, send out
the signals during the five minutes preceding noon each
day. It is a common custom for astronomical observatories to
correct their own clocks by careful observations of the stars.
The Washington Observatory sends telegraphic signals to
all the cities east of the Rocky Mountains and the Mare
Island Observatory to Pacific cities and Alaska. A few
railroads receive their time corrections from other observ
atories. Goodsell Observatory, Carleton College, North-
field, Minnesota, has for many years furnished time to the
Great Northern, the Northern Pacific, the Great Western,
and the Sault Ste. Marie railway systems. Allegheny
Observatory sends out time to the Pennsylvania system
and the Lick Observatory to the Southern Pacific system.
How Time is Determined at the United States Naval
Observatory. The general plan of correcting clocks at the
United States Naval Observatories by stellar observations
is as follows: A telescope called a meridian transit is situ
ated in a true north-south direction mounted on an east-
west axis so that it can be rotated in the plane of the
meridian but not in the slightest degree to the east or
west. Other instruments used are the chronograph and
the sidereal clock. The chronograph is an instrument
which may be electrically connected with the clock and
which automatically makes a mark for each second on a
sheet of paper fastened to a cylinder. The sidereal clock

70 LONGITUDE AND TIME
is regulated to keep time with the stars — not with the
sun, as are other clocks. The reason for this is because
the stars make an apparent circuit with each rotation of
the earth and this, we have observed, is unerring while
the sun's apparent motion is quite irregular.
To correct the clock, an equatorial or high zenith star
is selected. A well known one is chosen since the exact
time it will cross the meridian of the observer (that is, be
at its highest point in its apparent daily rotation) must be
calculated. The chronograph is then started, its pen and
ink adjusted, and its electrical wires connected with the
clock. The observer now sights the telescope to the point
where the expected star will cross his meridian and, with
his hand on the key, he awaits the appearance of the star.
As the star crosses each of the eleven hair lines in the field
of the telescope, the observer presses the key which auto
matically marks upon the chronographic .cylinder. Then
by examining the sheet he can tell at what time, by the
clock, the star crossed the center line. He then calculates
just what. time the clock should indicate and the difference
is the error of the clock. By this means an error of one
tenth of a second can be discovered.
The Sidereal Clock. The following facts concerning the
sidereal clock may be of interest. It is marked with
twenty-four hour spaces instead of twelve. Only one
moment in the year does it indicate the same time as
ordinary timepieces, which are adjusted to the average sun.
When the error of the clock is discovered the clock is
not at once reset because any tampering with the clock
would involve a slight error. The correction is simply
noted and the rate of the clock's gaining or losing time is
calculated so that the true time can be ascertained very
precisely at any time by referring to the data showing the

SENDING TIME SIGNALS 71
clock error when last corrected and the rate at which it
varies. A while before noon each day the exact sidereal time is
calculated ; this is converted into local mean solar time and
this into standard time. The Washington Naval Observ
atory converts this into the standard time of the 75th
meridian or Eastern time and the Mare Island Observatory
into that of the 120th meridian or Pacific time.
Sending Time Signals. By the cooperation of telegraph
companies, the time signals which are sent out daily from
the United States Naval Observatories reach practically
every telegraph station in the country. They are sent
out at noon, 75th meridian time, from Washington, which
is 11 o'clock, a.m., in cities using Central time and 10
o'clock, a.m., where Mountain time prevails; and at noon,
120th meridian time, they are sent to Pacific coast cities
from the Mare Island Observatory — three hours after
Washington has flashed the signal which makes correct
time accessible to sixty millions of our population living
east of the Rockies.
Not only are the time signals sent to the telegraph
stations and thence to railway offices, clock makers and
repairers, schools, court houses, etc., but the same tele
graphic signal that marks noon also actually sets many
thousands of clocks, their hands whether fast or slow auto
matically flying to the true mark in response to the electric
current. In a number of cities of the United States,
nineteen at present, huge balls are placed upon towers or
buildings and are automatically dropped by the electric
noon signal. The time ball in Washington is conspicu
ously placed on the top of the State, War, and Navy build
ing and may be seen at considerable distances from many
parts of the city.

72

LONGITUDE AND TIME

A few minutes before noon each day, one wire at each
telegraphic office is cleared of all business and " thousands
of telegraph operators sit in silence, waiting for the click
of the key which shall tell them that the ' master clock '
in Washington has begun to speak." * At five minutes
before twelve the instrument begins to click off the seconds.
Figure 28 (adapted from a cut appearing in Vol. IV, Appen
dix IV, United States Naval Observatory Publications)

0

10

20 30

4C

50 60
55 «. Minute before noon
 i  i 
5 Sec omitted
56 «* Minute before noon
5 Sec omitted
 I  1
57th Minute before noon
5 Sec omitted
 1... 1
SB"1 Minute before noon
5 Sec omitted
 I Noont
59 "• Minute before noon
Fig. 28
10 Sec omitted
f graphically shows which second beats are sent along the
wires during each of the five minutes before noon by the
transmitting clock at the Naval Observatory.
Explanation of the Second Beats. It will be noticed
that the twenty-ninth second of each minute is omitted.
This is for the purpose of permitting the observer to
distinguish, without counting the beats, which is the one
denoting the middle of each minute; the five seconds at
the end of each of the first four minutes are omitted to
mark the beginning of a new minute and the last ten
seconds of the fifty-ninth minute are omitted to mark
conspicuously the moment of noon. The omission of the
* From " What's the Time," Youth's Companion, May 17 and June
14, 1906.
SENDING TIME SIGNALS 73
last ten seconds also enables the operator to connect
his wire with the clock to be automatically set or the
time ball to be dropped. The contact marking noon is
prolonged a full second, not only to make prominent this
important moment but also to afford sufficient current
to do the other work which this electric contact must
perform. Long Distance Signals. Several times in recent years
special telegraphic signals have been sent out to such dis
tant points as Madras, Mauritius, Cape Town, Pulkowa
(near St. Petersburg), Rome, Lisbon, Madrid, Sitka,
Buenos Ayres, Wellington, Sydney, and Guam. Upon
these occasions " our standard clock may fairly be said to
be heard in ' the remotest ends of the earth,' thus antici
pating the day when wireless telegraphy will perhaps allow
of a daily international time signal that will reach every
continent and ocean in a small fraction of a second." *
These reports have been received at widely separated
stations within a few seconds, being received at the Lick
Observatory in 0.05s, Manila in 0.11s, Greenwich in
1.33s, and Sydney, Australia, in 2.25s.
Confusion from Various Standards
Where different time systems are used in the same
community, confusion must of necessity result. The
following editorial comment in the Official Railway Guide
for November, 1900, very succinctly sets forth the con
dition which prevailed in Detroit as regards standard and
local time.
" The city of Detroit is now passing through an agitation
which is a reminiscence of those which took place through-
* "The Present Status of the Use of Standard Time," by Lieut.
Commander E. E. Hayden, U. S. Navy.

74 LONGITUDE AND TIME
out the country about seventeen years ago, when stand
ard time was first adopted. For some reason, which it
is difficult to explain, the city fathers of Detroit have
refused to change from the old local time to the standard,
notwithstanding the fact that all of the neighboring cities
— Cleveland, Toledo, Columbus, Cincinnati, etc., — in
practically the same longitude, had made the change years
ago and realized the benefits of so doing. The business
men of Detroit and visitors to that city have been for a
long time laboring under many disadvantages owing to
the confusion of standards, and they have at last taken
the matter into their own hands and a lively campaign,
with the cooperation of the newspapers, has been
organized during the past two months. Many of the
hotels have adopted standard time, regardless of the
city, and the authorities of Wayne County, in which
Detroit is situated, have also decided to hold court on
Central Standard time, as that is the official standard .of
the state of Michigan. The authorities of the city have
so far not taken action. It is announced in the news
papers that they probably will do so after the election,
and by that time, if progress continues to be made, the
only clock in town keeping the local time will be on the
town hall. All other matters will be regulated by standard
time, and the hours of work will have- been altered
accordingly in factories, stores, and schools. Some
opposition has been encountered, but this, as has been
the case in every city where the change has been made,
comes from people who evidently do not comprehend the
effects of the change. One individual, for instance, writes
to a newspaper that he will decline to pay pew rent in
any church whose clock tower shows standard time; he
refuses to have his hours of rest curtailed. How these will

FOUR KINDS OF TIME AROUND EL PASO

75

be affected by the change he does not explain. Every
visitor to Detroit who has encountered the difficulties
which the confusion of standards there gives rise to, will
rejoice when the complete change is effected."
The longitude of Detroit being 83° W., it is seven degrees
east of the 90th meridian, hence the local time used in the
city was twenty-eight minutes faster than Central time
and thirty-two minutes slower than Eastern time. In
Gainesville, Georgia, mean local sun time is used in the city,

Silver City

x x x x X Mexican Time
E X I C

Fig. 29
while the Southern railway passing through the city uses
Eastern time and the Georgia railway uses Central time.
Four Kinds of Time Around El Paso. Another place of
peculiar interest in connection with this subject is El Paso,
Texas, from the fact that four different systems are em
ployed. The city, the Atchison, Topeka, and Santa Fe,
and the El Paso and Southwestern railways use Mountain
time. The Galveston, Harrisburg, and San Antonio, and
the Texas and Pacific railways use Central time. The
Southern Pacific railway uses Pacific time. The Mexican
Central railway uses Mexican standard time.

76 LONGITUDE AND TIME
From this it will be seen that when clocks in Strauss,
N. M., a few miles from El Paso, are striking twelve, the
clocks in El Paso are striking one; in Ysleta, a, few miles
east, they are striking two; while across the river in Juarez,
Mexico, the clocks indicate 12:24.
Time Confusion for Travelers. The confusion which
prevails where several different standards of time obtain
is well illustrated in the following extract from " The
Impressions of a Careless Traveler " by Lyman Abbott,
in the Outlook, Feb. 28, 1903.
" The changes in time are almost as interesting and
quite as perplexing as the changes in currency. Of course
our steamer time changes every day; a sharp blast on the
whistle notifies us when it is twelve o'clock, and certain
of the passengers set their watches accordingly every
day. I have too much respect for my faithful friend to
meddle with him to this extent, and I keep my watch
unchanged and make my calculations by a mental com
parison of my watch with the ship's time. But when we
are in port we generally have three times — ship's time,
local time, and railroad time, to which I must in my own
case add my own time, which is quite frequently neither.
In fact, I kept New York time till we reached Genoa;
since then I have kept central Europe railroad time.
Without changing my watch, I am getting back to that
standard again, and expect to find myself quite accurate
when we land in Naples."
The Legal Aspect of Standard Time
The legal aspect of standard time presents many
interesting features. Laws have been enacted in many
different countries and several of the states of this country
legalizing some standard of time. Thus in Michigan,

THE LEGAL ASPECT OF STANDARD TIME 77
Minnesota, and other central states the legal time is the
mean solar time of longitude 90° west of Greenwich.
When no other standard is explicitly referred to, the
time of the central belt is the legal time in force. Similarly,
legal time in Germany was declared by an imperial decree
dated March 12, 1903, as follows:*
" We, Wilhelm, by the grace of God German Emperor, King of
Prussia, decree in the name of the Empire, the Bundesrath and Reich
stag concurring, as follows:
" The legal time in Germany is the mean solar time of longitude 15°
east from Greenwich."
Greenwich time for Great Britain, and Dublin time for
Ireland, were legalized by an act of Parliament as follows :
A Bill to remove doubts as to the meaning of expressions relative
to time occurring in acts of ParUament, deeds, and other legal instru
ments. Whereas it is expedient to remove certain doubts as to whether
expressions of time occurring in acts of Parliament, deeds, and other
legal instruments relate in England and Scotland to Greenwich time,
and in Ireland to Dublin time, or to the mean astronomical time in
each locality:
Be it therefore enacted by the Queen's most Excellent Majesty,
by and with the advice and consent of the Lords, spiritual and tem
poral, and Commons in the present Parliament assembled, and by
the authority of the same, as follows (that is to say):
1. That whenever any expression of time occurs in any act of
Parliament, deed, or other legal instrument, the time referred to
shall, unless it is otherwise specifically stated, be held in the case of
Great Britain to be Greenwich mean time and in the case of Ireland,
Dublin mean time.
2. This act may be cited as the statutes (definition of time) act, 1880.
Seventy-fifth meridian time was legalized in the District
of Columbia by the following act of Congress :
An Act to establish a standard of time in the District of Columbia.
Be it enacted by the Senate and House of Representatives of the
* Several of the following quotations are taken from the " Present
Status of the Use of Standard Time," by E. E. Hayden.

78 LONGITUDE AND TIME
United States of America in Congress assembled, That the legal
standard of time in the District of Columbia shall hereafter be the
mean time of the seventy-fifth meridian of longitude west from
Greenwich. Section 2. That this act shall not be so construed as to affect
existing contracts.
Approved, March 13, 1884.
In New York eastern standard time is legalized in
section 28 of the Statutory Construction Law as follows:
The standard time throughout this State is that of the 75th meridian
of longitude west from Greenwich, and all courts and public offices, and
legal and official proceedings, shall be regulated thereby. Any act
required by or in pursuance of law to be performed at or within a pre
scribed time, shall be performed according to such standard time.
A New Jersey statute provides that the time of the
same meridian shall be that recognized in all the courts and
public offices of the State, and also that " the time named
in any notice, advertisement, or contract shall be deemed
and taken to be the said standard time, unless it be other
wise expressed." In Pennsylvania also it is provided
that " on and after July 1, 1887, the mean solar time of
the seventy-fifth meridian of longitude west of Greenwich,
commonly called eastern standard time," shall be the
standard in all public matters; it is further provided that
the time " in any and all contracts, deeds, wills, and
notices, and in the transaction of all matters of business,
public, legal, commercial, or otherwise, shall be construed
with reference to and in accordance with the said standard
hereby adopted, unless a different standard is therein
expressly provided for."
Where there is no standard adopted by legal authority,
difficulties may arise, as the following clipping from the
New York Sun, November 25, 1902, illustrates:

THE LEGAL ASPECT OF STANDARD TIME

79

WHAT'S NOON IN A FIRE
POLICY ?

Solar Noon or Standard Time
Noon — Courts Asked to Say.

Fire in Louisville at 11:45 a.m., Stand
ard Time, Which Was 12: 02 1-2 p.m.
Solar Time — Policies Expired at
Noon and 13 Insurance Companies
Wont Pay.
Whether the word "noon,"
which marks the beginning and
expiration of all fire insurance
policies, means noon by standard
time, or noon by solar time, is a
question which is soon to be
fought out in the courts of Ken
tucky, in thirteen suits which
have attracted the attention of
fire insurance people all over the
world. The suits are being
brought by the Peaslee-Gaulbert
Company and the Louisville Lead
and Color Company of Louisville,
and $19,940.70 of insurance
money depends on the result.
Now, although the policies in
these companies all state that
they were in force from noon of
April 1, 1901, to noon of April 1,
1902, not one of them says what
kind of time that period of the
day is to be reckoned in. In
Louisville the solar noon is 17£
minutes earlier than the stand
ard noon, so that a fire occurring
in the neighborhood of noon on
the day of a policy's expiration,
may easily be open to attack.
The records of the Louisville
fire department show that the fire
that destroyed the buildings of
the two companies was discov
ered at 11:45 o'clock Louisville
standard time in the forenoon of
April 1, last. The fire began in
the engine room of the main fac
tory and spread to the two other
buildings which were used mainly
as warehouses. When the fire
department recorded the time of

the fire's discovery it figured, of
course, by standard time. Solar
time would make it just two and (
a half minutes after noon. If
noon in the policies means noon
by solar time, of course the thir
teen companies are absolved from
any responsibility for the loss.
If it means noon by standard
time, of course they must pay.
When the insurance people got
the claims of the companies they
declined to pay, and when asked
for an explanation declared that
noon in the policies meant noon
by solar time. The burned-out
companies immediately began suit,
and in their affidavits they say
that not only is standard time
the official time of the State of
Kentucky and the city of Louis
ville, but it is also the time upon
which all business engagements
and all domestic and social en
gagements are reckoned. They
state further that they are pre
pared to show that in 1890 the
city of Louisville passed an ordi
nance making standard time the
official time of the city, that all
legislation is dated according to
standard time, and that the gov
ernor of the state is inaugurated
at noon according to the same
measurement of time.
Solar time, state the companies,
can be found in use in Louisville
by only a few banking institu
tions which got charters many
years ago that compel them to
use solar time to this day. Most
banks, they say, operate on stand
ard time, although they keep
clocks going at solar time so as to
do business on that basis if
requested. Judging by standard
time the plaintiffs allege their fire
took place fifteen minutes before
their policies expired.
The suits will soon come to trial,
and, of course, will be watched with
great interest by insurance people.

80 LONGITUDE AND TIME
Iowa Case. An almost precisely similar case occurred
at Creston, Iowa, September 19, 1897. In this instance
the insurance policies expired " at 12 o'clock at noon,"
and the fire broke out at two and a half minutes past
noon according to standard time, but at fifteen and one-
half minutes before local mean solar noon. In each of
these cases the question of whether standard time or local
mean solar time was the accepted meaning of the term
was submitted to a jury, and in the first instance the ver
dict was in favor of standard time, in the Iowa case the
verdict was in favor of local time.
Early Decision in England. In 1858 and thus prior to
the formal adoption of standard time in Great Britain, it
was held that the time appointed for the sitting of a court
must be understood as the mean solar time of the place
where the court is held and not Greenwich time, unless it be
so expressed, and a new trial was granted to a defendant
who had arrived at the local time appointed by the court
but found the court had met by Greenwich time and the
case had been decided against him.
Court Decision in Georgia. In a similar manner a court
in the state of Georgia rendered the following opinion:
"The only standard of time in computation of a day, or hours of a
day, recognized by the laws of Georgia is the meridian of the sun ; and
a legal day begins and ends at midnight, the mean time between meri
dian and meridian, or 12 o'clock post meridiem. An arbitrary and arti
ficial standard of time, fixed by persons in a certain line of business,
cannot be substituted at will in a certain locality for the standard
recognized by the law."
Need for Legal Time Adoption on a Scientific Basis. There
is nothing in the foregoing decisions to determine whether
mean local time, or the time as actually indicated by the
sun at a particular day, is meant. Since the latter some-

TIME USED IN VARIOUS COUNTRIES 81
times varies as much as fifteen minutes faster or slower
than the average, opportunities for controversies are mul
tiplied when no scientifically accurate standard time is
adopted by law.
Even though statutes are explicit in the definition of
time, they are still subject to the official interpretation
of the courts, as the following extracts show:
Thomas Mier took out a fire insurance policy on his saloon at
11: 30 standard time, the policy being dated noon of that day. At
the very minute that he was getting the policy the saloon caught fire
and was burned. Ohio law makes standard time legal time, and the
company refused to pay the $2,000 insurance on Mier's saloon. The
case was fought through to the Supreme Court, which decided that
" noon " meant the time the sun passed the meridian at Akron, which is
at 11: 27 standard time. The court ordered the insurance company to
pay. — Law Notes, June, 1902.
In the 28th Nebraska Reports a ease is cited in which judgment by
default was entered against a defendant in a magistrate's court who
failed to make an appearance at the stipulated hour by standard time,
but arrived within the limit by solar time. He contested the ruling
of the court, and the supreme judiciary of the state upheld him in the
contest, although there was a Nebraska statute making standard time
the legal time. The court held that "at noon" must necessarily
mean when the sun is over the meridian, and that no construction
could reasonably interpret it as indicating 12 o'clock standard time.
Time Used in Various Countries
The following table is taken, by permission, largely
from the abstracts of official reports given in Vol. IV,
Appendix IV of the Publications of the United States
Naval Observatory, 1905. The time given is fast or slow
as compared with Greenwich mean solar time.
Argentina, 4 h. 16 m. 48.2 s. slow. Official time is referred to the merid
ian of Cordoba. At 11 o'clock, a.m., a daily signal is telegraphed
from the Cordoba Observatory.
JO. MATH. GEO. — 6

82 LONGITUDE AND TIME
Austria-Hungary, 1 h. fast. Standard time does not exist except for
the service of railroads where it is in force, not by law, but by order
of the proper authorities.
Belgium. Official time is calculated from 0 to 24 hours, zero corre
sponding to midnight at Greenwich. The Royal Observatory at
Brussels communicates daily the precise hour by telegraph.
British Empire.
Great Britain. The meridian of Greenwich is the standard time
meridian for England, Isle of Man, Orkneys, Shetland Islands,
and Scotland.
Ireland, 0 h. 25 m. 21.1 s. slow. The meridian of Dublin is the
standard time meridian.
Africa (English Colonies), 2 h. fast. Standard time for Cape Colony,
Natal, Orange River Colony, Rhodesia and Transvaal.
Australia. New South Wales, Queensland, Tasmania and Victoria, 10 h. fast.
South Australia and Northern Territory, 9 h. 30 m. fast.
Canada. Alberta and Saskatchewan, 7 h. slow.
British Columbia, 8 h. slow.
Keewatin and Manitoba, 6 h. slow.
Ontario and Quebec, 5 h. slow.
New Brunswick, Nova Scotia, and Prince Edward Island, 4 h. slow.
Chatham Island, 11 h. 30 m. fast.
Gibraltar, Greenwich time.
Hongkong, 8 h. fast.
Malta, 1 h. fast.
New Zealand, 11 h. 30 m. fast.
India. Local mean time of the Madras Observatory, 5 h. 20 m. 59.1 s.,
is practically used as standard time for India and Ceylon, being
telegraphed daily all over the country; but for strictly local use it
is generally converted into local mean time. It is proposed soon
to adopt the standard time of 5 h. 30 m. fast of Greenwich for India
and Ceylon, and 6 h. 30 m. fast of Greenwich for Burmah.
Newfoundland, 3 h. 30 m. 43.6 s. slow. (Local mean time of
St. John's.)
Chile, 4 h. 42 m. 46.1 s. slow. The official railroad time is furnished by
the Santiago Observatory. It is telegraphed over the country daily
at 7 o'clock, a.m. The city of Valparaiso uses the local time, 4 h.
46 h. 34.1 m. slow, of the observatory at the Naval School located
there.

TIME USED IN VARIOUS COUNTRIES 83
China. An observatory is maintained by the Jesuit mission at
Zikawei near Shanghai, and a time ball suspended from a mast on
the French Bund in Shanghai is dropped electrically precisely at
noon each day. This furnishes the local time at the port of
Shanghai 8 h. 5 m. 43.3 s. fast, which is adopted by the railway and
telegraph companies represented there, as well as by the coastwise
shipping. From Shanghai the time is telegraphed to other ports.
The Imperial Railways of North China use the same time, taking
it from the British gun at Tientsin and passing it on to the stations
of the railway twice each day, at 8 o'clock a.m. and at 8 o'clock p.m.
Standard time, 7 h. and 8 h. fast, is coming into use all along the
east coast of China from Newchwang to Hongkong.
Colombia. Local mean time is used at Bogota, 4 h. 56 m. 54.2 s. slow,
taken every day at noon in the observatory. The lack of effective
telegraphic service makes it impossible to communicate the time
as corrected at Bogota to other parts of the country, it frequently
taking four and five days to send messages a distance of from 50 to
100 miles.
Costa Rica, 5 h. 36 m. 16.9 s. slow. This is the local mean time of the
Government Observatory at San Jose.
Cuba, 5 h. 29 m. 26 s. slow. The official time of the Republic is the civil
mean time of the meridian of Havana and is used by the railroads
and telegraph lines of the government. The Central Meteorological
Station gives the time daily to the port and city of Havana as well
as to all the telegraph offices of the Republic.
Denmark, 1 h. fast. In Iceland, the Faroe Islands and the Danish
West Indies, local mean time is used.
Egypt, 2 h. fast. Standard time is sent out electrically by the standard
clock of the observatory to the citadel at Cairo, to Alexandria, Port
Said and Wady-Halfa.
Equador, 5 h. 14 m. 6.7 s. slow. The official time is that of the meridian
of Quito, corrected daily from the National Observatory.
France, 0 h. 9 m. 20.9 s. fast. Legal time in France, Algeria and Tunis is
local mean time of the Paris Observatory. Local mean time is
considered legal in other French colonies.
German Empire.
Germany, 1 h. fast.
Kiaochau, 8 h. fast.
Southwest Africa, 1 h. fast.
It is proposed to adopt standard time for the following:
Bismarck Archipelago, Carolines, Mariane Islands and New Guinea,
10 fast.

84 LONGITUDE AND TIME
German East Africa, 2 h. fast or 2 h. 30 m. fast.
Kamerun, 1 h. fast.
Samoa (after an understanding with the U. S.), 12 h. fast.
Toga, Greenwich time.
Greece, 1 h. 34 m. 52.9 s. fast. By royal decree of September 14, 1895, the
time in common use is that of the mean time of Athens, which is
transmitted from the observatory by telegraph to the towns of
the kingdom.
Holland. The local time of Amsterdam, 0 h. 19 m. 32.3 s. fast, is
generally used, but Greenwich time is used by the post and telegraph
administration and the railways and other transportation com
panies. The observatory at Leyden communicates the time twice
a week to Amsterdam, The Hague, Rotterdam and other cities,
and the telegraph bureau at Amsterdam signals the time to all
the other telegraph bureaus every morning.
Honduras. In Honduras the half hour nearest to the meridian of
Tegucigalpa, longitude 87° 12' west from Greenwich, is generally
used. Said hour, 6 h. slow, is frequently determined at the National
Institute by means of a solar chronometer and communicated by
telephone to the Industrial School, where in turn it is indicated to
the public by a steam whistle. The central telegraph office com
municates it to the various sub-offices of the Republic, whose
clocks generally serve as a basis for the time of the villages, and in
this manner an approximately uniform time is established through
out the Republic.
Italy, 1 h. fast. Adopted by royal decree of August 10, 1893. This
time is kept in all government establishments, ships of the Italian
Navy in the ports of Italy, railroads, telegraph offices, and Italian
coasting steamers. The hours are numbered from 0 to 24,
beginning with midnight.
Japan. Imperial ordinance No. 51, of 1886: "The meridian that
passes through the observatory at Greenwich, England, shall be
the zero (0) meridian. Longitude shall be counted from the above
meridian east and west up to 180 degrees, the east being positive
and the west negative. From January 1, 1888, the time of the
135th degree east longitude shall be the standard time of Japan."
This is 9 h. fast.
Imperial ordinance No. 167, of 1895: "The standard time hitherto
used in Japan shall henceforth be called central standard time. The
time of the 120th degree east longitude shall be the standard time
of Formosa, the Pescadores, tho Yaeyama, and the Miyako groups,
and shall be called western standard time. This ordinance shall
take effect from the first of January, 18S6." This is 8 h. fast.

TIME USED IN VARIOUS COUNTRIES 85
Korea, 8 h. 30 m. fast. Central standard time of Japan is telegraphed
daily to the Imperial Japanese Post and Telegraph Office at Seoul.
Before December, 1904, this was corrected by subtracting 30 m.,
which nearly represents the difference in longitude, and was then
used by the railroads, street railways, and post and telegraph offices,
and most of the better classes. Since December 1, 1904, the Jap
anese post-offices and railways in Korea have begun to use central
standard time of Japan. In the country districts the people use
sundials to some extent.
Luxemburg, 1 h. fast, the legal and uniform time.
Mexico, 6 h. 36 m. 26.7 s. slow. The National Astronomical Observatory
of Tacubaya regulates a clock twice a day which marks the local
mean time of the City of Mexico, and a signal is raised twice a week
at noon upon the roof of the national palace, such signal being
used to regulate the city's public clocks. This signal, the clock at
the central telegraph office, and the public clock on the cathedral,
serve as a basis for the time used commonly by the people. The
general telegraph office transmits this time daily to all of its branch
offices. Not every city in the country uses this time, however,
since a local time, very imperfectly determined, is more commonly
observed. The following railroad companies use standard City of
Mexico time corrected daily by telegraph: Central, Hidalgo, Xico
and San Rafael, National and Mexican. The Central and National
railroads correct their clocks to City of Mexico time daily by means
of the noon signal sent out from the Naval Observatory at Wash
ington (see page 71) and by a similar signal from the observatory
at St. Louis, Missouri. The Nacozari, and the Cananea, Yaqui River
and Pacific railroads use Mountain time, 7 h. slow, and the Sonora
railroad uses the local time of Guaymas, 7 h. 24 m. slow.
Nicaragua, 5 h. 45 m. 10 s. slow. Managua time is issued to all public
offices, railways, telegraph offices and churches in a zone that
extends from San Juan del Sur, latitude 11° 15' 44" N., to El Ocotal,
latitude 12° 46' N., and from El Castillo, longitude 84° 22' 37" W.,
to Corinto, longitude 87° 12' 31" W. The time of the Atlantic
ports is usually obtained from the captains of ships.
Norway, 1 h. fast. Central European time is used everywhere through
out the country. Telegraphic time signals are sent -out once a
week to the telegraph stations throughout the country from the
observatory of the Christiania University.
Panama. Both the local mean time of Colon, 5 h. 19 m. 39 s. slow, and
eastern standard time of the United States, 5 h. slow, are used. The
latter time is cabled daily by the Central and South American Cable

86 LONGITUDE AND TIME
Company from the Naval Observatory at Washington, and will
probably soon be adopted as standard.
Peru, 5 h. 9 m. 3 s. slow. There is no official time. The railroad from
Callao to Oroya takes its time by telegraph from the noon signal at
the naval school at Callao, which may be said to be the standard
time for Callao, Lima, and the whole of central Peru. The railroad
from Mollendo to Lake Titicaca, in southern Peru, takes its time
from ships in the Bay of Mollendo.
Portugal, 0 h. 36 m. 44.7 s. slow. Standard time is in use throughout
Portugal and is based upon the meridian of Lisbon. It is estab
lished by the Royal Observatory in the Royal Park at Lisbon,
and from there sent by telegraph to every railway station through
out Portugal having telegraphic communication. Clocks on railway
station platforms are five minutes behind and clocks outside of
stations are true.
Russia, 2 h. 1 m. 18.6 s. fast. All telegraph stations use the time of the
Royal Observatory at Pulkowa, near St. Petersburg. At railroad
stations both local and Pulkowa time are given, from which it is
to be inferred that for all local purposes local time is used.
Salvador, 5 h. 56 m. 32 s. slow. The government has established a national
observatory at San Salvador which issues time on Wednesdays and
Saturdays, at noon, to all public offices, telegraph offices, railways,
etc., throughout the Republic.
Santo Domingo, 4 h. 39 m. 32 s. slow. Local mean time is used, but there
is no central observatory and no means of correcting the time. The
time differs from that of the naval vessels in these waters by about
30 minutes.
Servia, 1 h. fast. Central European time is used by the railroad, tele
graph companies, and people generally. Clocks are regulated by
telegraph from Budapest every day at noon.
Spain, Greenwich time. This is the official time for use in govern
mental offices in Spain and the Balearic Islands, railroad and
telegraph offices. The hours are numbered from 0 to 24, beginning
with midnight. In some portions local time is still used for private
matters.
Sweden, 1 h. fast. Central European time is the standard in general
use. It is sent out every week by telegraph from the Stockholm
Observatory.
Switzerland, 1 h. fast. Central European time is the only legal time.
It is sent out daily by telegraph from the Cantonal Observatory at
Neuchatel.

TIME USED IN VARIOUS COUNTRIES 87
Turkey. Two kinds of time are used, Turkish and Eastern European
time, the former for the natives and the latter for Europeans. The
railroads generally use both, the latter for the actual running of
trains and Turkish time-tables for the benefit of the natives.
Standard Turkish time is used generally by the people, sunset being
the base, and twelve hours being added for a theoretical sunrise.
The official clocks are set daily so as to read 12 o'clock at the theo
retical sunrise, from tables showing the times of sunset, but the
tower clocks are set only two or three times a week. The govern
ment telegraph lines use Turkish time throughout the empire, and
St. Sophia time, 1 h. 56 m. 53 s. fast, for telegrams sent out of the
country.
United States. Standard time based upon the meridian of Greenwich,
varying by whole hours from Greenwich time, is almost universally
used, and is sent out daily by telegraph to most of the country, and
to Havana and Panama from the Naval Observatory at Washington,
and to the Pacific coast from the observatory at Mare Island Navy
Yard, California. For further discussions of standard time belts in
the United States, see pp. 66-68 and the U. S. standard time belt
map. Insular possessions have time as follows :
Porto Rico, 4 h. slow, Atlantic standard time.
Alaska, 9 h. slow, Alaska standard time.
Hawaiian Islands, 10 h. 30 m. slow, Hawaiian standard time.
Guam, 9 h. 30 m. fast, Guam standard time.
Philippine Islands, 8 h. fast, Philippine standard time.
Tutuila, Samoa, 11 h. 30 m. slow, Samoan standard time.
Uruguay, 3 h. 44 m. 48.9 s. slow. The time in common use is the mean
time of the meridian of the dome of the Metropolitan Church of
Montevideo. The correct time is indicated by a striking clock in
the tower of that church. An astronomical geodetic observatory,
with meridian telescope and chronometers, has now been estab
lished and will in the future furnish the time. It is proposed to
install a time ball for the benefit of navigators at the port of Monte
video. An electric time service will be extended throughout the
country, using at first the meridian of the church and afterwards
that of the national observatory, when constructed.
Venezuela, 4 h. 27 m. 43.6 s. The time is computed daily at the Caracas
Observatory from observations of the sun and is occasionally tele
graphed to other parts of Venezuela. The cathedral clock at Caracas
is corrected by means of these observations. Railway time is at
least five minutes later than that indicated by the cathedral clock,
which is accepted as standard by the people. Some people take
time from the observatory flag, which always falls at noon.

88

LONGITUDE AND TIME

Latitude and Longitude of Cities
The latitude and longitude of cities in the following
table was compiled from various sources. Where possible,
the exact place is given, the abbreviation " 0 " standing
for observatory, " C " for cathedral, etc.

Adelaide, S. Australia, Snap
per Point 
Aden, Arabia, Tel. Station .
Alexandria, Egypt, Eunos Pt.
Amsterdam, Holland, Ch.
Antwerp, Belgium, O. . . .
Apia, Samoa, Ruge's Wharf
Athens, Greece, O 
Bangkok, Siam, Old Br. Fact.
Barcelona, Spain, Old Mole
Light 
Batavia, Java, O 
Bergen, Norway, C 
Berlin, Germany, O 
Bombay, India, O. .
Bordeaux, France, O. . . .
Brussels, Belgium, O . . . .
Buenos Aires, Custom House
Cadiz, Spain, O 
Cairo, Egypt, O. .....
Calcutta, Ft. Wm. Semaphore
Canton, China, Dutch Light
Cape Horn, South Summit .
Cape Town, S. Africa, O.
Cayenne, Fr. Guiana, Landing
Christiania, Norway, O. . .
Constantinople, Turkey, C. .
Copenhagen, Denmark, New O.
Dublin, Ireland, O 
Edinburgh, Scotland, O. . .
Florence, Italy, O 
Gibraltar, Spain, Dock Flag
Glasgow, Scotland, O. . . .
Hague, The, Holland, Ch. .
Hamburg, Germany, O. . .
Havana, Cuba, Morro Lt. H.
Hongkong, China, C. . . .

Latitude

Longitude from
Greenwich

34°

46'

50" S

138°

30'

39" E

12°

46'

40" N

44°

58'

58" E

31°

11'

43" N

29°

51'

40" E

52°

22'

30" N

4°

53'

04" E

51°

12'

28" N

4°

24'

44" E

13°

48'

56" S

171°

44'

56" W

37°

58'

21" N

23°

43'

55" E

13°

44'

20" N

100°

28'

42" E

41°

22'

10" N

2°

10'

55" E

6°

07'

40" N

106°

48'

25" E

60°

23'

37" N

5°

20'

15" E

52°

30'

17" N

13°

23'

44" E

18°

53'

45" N

72°

48'

58" E

44°

50'

07" N

00°

31'

23" W

50°

51'

11" N

4°

22'

18" E

34°

36'

30" S

58°

22'

14" W

36°

27'

40" N

6°

12'

20" W

30°

04'

38" N

31°

17'

14" E

22°

33'

25" N

88°

20'

11" E

23°

06'

35" N

113°

16'

34" E

55°

58'

41" S

67°

16'

15" W

33°

56'

03" S

18°

28'

40" E

4°

56'

20" N

52°

20'

25" W

59°

54'

44" N

10°

43'

35" E

41°

00'

16" N

28°

58'

59" E

55°

41'

14" N

12°

34'

47" E

53°

23'

13" N

6°

20'

30" W

55°

57'

23" N

3°

10'

54" W

43°

46'

04" N

11°

15'

22" E

36°

07'

10" N

5°

21'

17" W

55°

52'

43" N

4°

17'

39" W

52°

04'

40" N

4°

18'

30" E

53°

33'

07" N

9°

58'

25" E

23°

09'

21" N

82°

21'

30" W

21°

16'

52" N

114°

09'

31" E

LATITUDE AND LONGITUDE OF CITIES

89

Latitude

Longitude from
Greenwich

Jerusalem, Palestine, Ch.

31°

46'

45" N

35°

13'

25" E

Leipzig, Germany, 0. . . .
Lisbon, Portugal, 0. (Royal)

51°

20'

06" N

12°

23'

30" E

3S°

42'

31" N

9°

11'

10" W

Liverpool, England, 0.

53°

24"

04" N

3°

04'

16" W

Madras, India, 0.

13°

04'

06" N

80°

14'

51" E

Marseilles, France, New 0. .

43°

18'

22" N

5°

23'

43" E

Melbourne, Victoria, 0.

37°

49'

53" S

144°

58'

32" E

Mexico, Mexico, 0. . . .

19°

26'

01" N

99°

06'

39" W

Montevideo, Uruguay, C.

34°

54'

33" S

56°

12'

15" W

Moscow, Russia, 0.

55°

45'

20" N

37°

32'

36" E

Munich, Germany, 0. . . .

48°

08'

45" N

11°

36'

32" E

Naples, Italy, 0 

40°

51'

46" N

14°

14'

44" E

Panama, Cent. Am., C. . . .

8°

57'

06" N

79°

32'

12" W

Para, Brazil, Custom H.

1°

26'

59" S

48°

30'

01" W

Paris, France, 0.

48°

50'

11" N

2°

20'

14" E

Peking, China. . . .

39°

56'

00" N

116°

28'

54" E

Pulkowa, Russia, 0 . .

59°

46'

19" N

30°

19'

40" E

Rio de Janeiro, Brazil, 0. .

22°

54'

24" S

43°

10'

21" W

Rome, Italy, 0. . .

41°

53'

54" N

12°

28'

40" E

Rotterdam, HolL, Time Ball

51°

54'

30" N

4°

28'

50" E

St. Petersburg, Russia, see

Pulkowa
Stockholm, Sweden, 0..
59°
20'
35" N
18°
03'
30" E
Sydney, N. S. Wales, 0.
33°
51'
41" S
151°
12'
23" E
Tokyo, Japan, 0 
35°
39'
17" N
139°
44'
30" E
Valparaiso, Chile, Light House
33°
01'
30" S
71°
39'
22" W
United States
Aberdeen, S. D., N. N. & I. S.
45°
27'
50" N
98°
28'
45" W
Albany, N. Y., New O.
42°
39'
13" N
73°
46'
42" W
Ann Arbor, Mich., O.
42°
16'
48" N
83°
43'
48" W
Annapolis, Md., O.
38°
58'
53" N
76°
29'
08" W
Atlanta, Ga., Capitol . . .
33°
45'
19" N
84°
23'
29" W
Attu Island, Alaska, Chi-
chagoff Harbor 
52°
56'
01" N
173°
12'
24" E
Augusta, Me., Baptist Ch.
44°
18'
52" N
69°
46'
37" W
32°
00'
40" N
100°
27'
35" W
Baltimore, Md., Wash. Mt. .
39°
17'
48" N
76°
36'
59" W
Bangor, Me., Thomas Hill
44°
48'
23" N
68°
46'
59" W
Beloit, Wis., College . .
42°
30'
13" N
89°
1'
46" W
37°
52'
24" N
122°
15'
41" W
Bismarck, N. D.
46°
49'
12" N
100°
45'
08" W
Boise, Idaho, Ast. Pier
43°
35'
58" N
116°
13'
04" W
Boston, Mass., State House
42°
21'
28" N
71°
03'
50" W
90

LONGITUDE AND TIME

Latitude

Longitude from
Greenwich

Buffalo, N. Y 
Charleston, S. O, Lt. House
Cheyenne, Wyo., Ast. Sta. .
Chicago, 111., O. . .
Cincinnati, Ohio
Cleveland, Ohio, Lt. H.
Columbia, S. C. . .
Columbus, Ohio . .
Concord, N. H. . .
Deadwood, S. D., P. O.
Denver, Col., 0. . .
Des Moines, Iowa . .
Detroit, Mich. . .
Duluth, Minn. . . .
Erie, Pa., Waterworks
Fargo, N. D., Agri. College .
Galveston, Tex., C. . .
Guthrie, Okla 
JJartford, Conn 
Helena, Mont 
Honolulu, Sandwich Islands
Indianapolis, Ind 
Jackson, Miss 
Jacksonville, Fla., M. E. Ch.
Kansas City, Mo 
Key West, Fla., Light House
Lansing, Mich., Capitol . .
Lexington, Ky., Univ. .
Lincoln, Neb 
Little Rock, Ark 
Los Angeles, Cal., Ct. House
Louisville, Ky 
Lowell, Mass 
Madison, Wis., O 
Manila, Luzon, C 
Memphis, Tenn 
Milwaukee, Wis., Ct. House.
Minneapolis, Minn., O. . . .
Mitchell, S. D 
Mobile, Ala., Epis. Church .
Montgomery, Ala. . . .
Nashville, Tenn., 0 
Newark, N. J., M. E. Ch. . .
New Haven, Conn., Yale .
New Orleans, La., Mint. . .
New York, N. Y., City Hall.
Northfield, Minn., O. . . .

42°

53'

03" N

78°

52'

42" W

32°

41'

44" N

79°

52'

58" W

41°

07'

47" N

104°

48'

52« W

41°

50'

01" N

87°

36'

36" W

39°

08'

19" N

84°

26'

00" W

41°

30'

02" N

81°

42'

10" W

33°

59'

12" N

81°

00'

12" W

39°

57'

40" N

82°

59'

40" W

43°

11'

48" N

71°

32'

30" W

44°

22'

34" N

103°

43'

19" W

39°

40'

36" N

104°

59'

23" W

41°

35'

08" N

93°

37'

30" W

42°

20'

00" N

83°

02'

54" W

46°

48'

00" N

92°

06'

10" W

42°

07'

53" N

80°

05'

51" W

46°

52'

04" N

96°

47'

11" W

29°

18'

17" N

94°

47'

26" W

35°

51'

48" N

100°

26'

24" W

41°

45'

59" N

72°

40'

45" W

46°

35'

36" N

111°

52'

45" W

21°

18'

12" N

157°

51'

34" W

39°

47'

00" N

86°

05'

00" W

31°

16'

00" N

91°

36'

18" W

30°

19'

43" N

81°

39'

14" W

39°

06'

08" N

94°

35'

19" W

24°

32'

58" N

81°

48'

04" W

42°

43'

56" N

84°

33'

23" W

38°

02'

25" N

84°

30'

21" W

40°

55'

00" N

96°

52'

00" W

34°

40'

00" N

92°

12'

00" W

34°

03'

05" N

118°

14'

32" W

38°

15'

08" N

85°

45'

29" W

42°

22'

00" N

71°

04'

00" W

43°

04'

37" N

89°

24'

27" W

14°

35'

31" N

120°

58'

03" E

35°

08'

38" N

90°

03'

00" W

43°

02'

32" N

87°

54'

18" W

44°

58'

38" N

93°

14'

02" W

43°

49'

00" N

98°

00'

14" W

30°

41'

26" N

88°

02'

28" W

32°

22'

46" N

86°

17'

57" W

36°

08'

54" N

86°

48'

00" W

40°

44'

06" N

74°

10'

12" W

41°

18'

28" N

72°

55'

45" W

29°

57'

46" N

90°

03'

28" W

40°

42'

44" N

74°

00'

24" W

44°

27'

42" N

93°

08'

57" W

LATITUDE AND LONGITUDE OF CITIES

91

Ogden, Utah, O 
Olympia, Wash 
Omaha, Neb 
Pago Pago, Samoa ....
Philadelphia, Pa., State House
Pierre, S. D., Capitol. . . .
Pittsburg, Pa 
Point Barrow (highest lati
tude in the United States)
Portland, Ore 
Princeton, N. J., O 
Providence, R. I., Unit. Ch.
Raleigh, N. C 
Richmond, Va., Capitol . .
Rochester, N. Y., O 
Sacramento, Cal 
St. Louis, Mo 
St. Paul, Minn 
San Francisco, Cal., C. S. Sta.
San Juan, Porto Rico, Morro
Light House . .
Santa Fe, N. M 
Savannah, Ga., Exchange .
Seattle, Wash., C. S. Ast. Sta.
Sitka, Alaska, Parade Ground
Tallahassee, Fla 
Trenton, N. J. Capitol . . .
Virginia City, Nev. . .
Washington, D. C, O. . . .
Wheeling, W. Va 
Wilmington, Del., Town Hall
Winona, Minn 

Latitude

Longitude from
Greenwich

41°

13'

08" N

111°

59'

45" W

47°

03'

00" N

122°

57'

00" W

41°

16'

50" N

95°

57'

33" W

14°

IS'

06" S

170°

42'

31" W

39°

56'

53" N

75°

09'

03" W

44°

22'

50" N

100°

20'

26" W

40°

26'

34" N

80°

02'

38" W

71°

27'

00" N

156°

15'

00" W

45°

30'

00" N

122°

40'

30" W

40°

20'

58" N

74°

39'

24" W

41°

49'

28" N

71°

24'

20" W

35°

47'

00" N

78°

40'

00" W

37°

32'

19" N

77°

27'

02" W

43°

09'

17" N

77°

35'

27" W

38°

33'

38" N

121°

26'

00" W

38°

38'

04" N

90°

12'

16" W

44°

52'

56" N

93°

05'

00" W

37°

47'

55" N

122°

24'

32" W

18°

28'

56" N

66°

07'

28" W

35°

41'

19" N

105°

56'

45" W

32°

04'

52" N

81°

05'

26" W

47°

35'

54" N

122°

19'

59" W

57°

02'

52" N

135°

19'

31" W

30°

25'

00" N

84°

18'

00" W

40°

13'

14" N

74°

46'

13" W

39°

17'

36" N

119°

39'

06" W

38°

53'

39" N

77°

03'

06" W

40°

05'

16" N

80°

44'

30" W

39°

44'

27" N

75°

33'

03" W

44°

04'

00" N

91°

30'

00" W

CHAPTER V
CIRCUMNAVIGATION AND TIME
Magellan's Fleet. When the sole surviving ship of
Magellan's fleet returned to Spain in 1522 after having
circumnavigated the globe, it is said that the crew were
greatly astonished that their calendar and that of the
Spaniards did not correspond. They landed, according to
their own reckoning, on September 6, but were told it was
September 7. At first they thought they had made a mis
take, and some time elapsed before they realized that they
had lost a day by going around the world with the sun.
Had they traveled toward the east, they would have
gained a day, and would have recorded the same date as
September 8. "My pilot is dead of scurvy: may
I ask the longitude, time and day? "
The first two given and compared;
The third, — the comrhandante stared!
"The first of June? I make it second,"
Said the stranger, "Then you've wrongly reckoned!"
— Bret Harte, in The Lost Galleon.
The explanation of this phenomenon is simple. In
traveling westward, in the same way with the sun, one's
days are lengthened as compared with the day at any
fixed place. When one has traveled 15° westward, at what
ever rate of speed, he finds his watch is one hour behind
the time at his starting point, if he changes it according
to the sun. He has thus lost an hour as compared with
92

WESTWARD TRAVEL— DAYS ARE LENGTHENED 93
the time at his starting point. After he has traveled 15°
farther, he will set his watch back two hours and thus
record a loss of two hours. And so it continues through
out the twenty-four belts of 15° each, losing one hour in
each belt; by the time he arrives at his starting point

Fig. 30
again, he has set his hour hand back twenty-four hours
and has lost a day.
Westward Travel — Days are Lengthened. To make this
clearer, let us suppose a traveler starts from London Mon
day noon, January 1st, traveling westward 15° each day.
On Tuesday, when he finds he is 15° west of London, he
sets his watch back an hour. It is then noon by the sun
where he is. He says, " I left Monday noon, it is now

94 CIRCUMNAVIGATION AND TIME
Tuesday noon; therefore I have been out one day." The
tower clock at London and his chronometer set with it,
however, indicate a different view. They say it is Tuesday,
1 o'clock, p.m., and he has been out a day and an hour.
The next day the process is repeated. The traveler hav
ing covered another space of 15° westward, sets his watch
back a second hour and says, " It is Wednesday noon and
I have been out just two days." The London clock, how
ever, says Wednesday, 2 o'clock, p.m. — two days and
two hours since he left. The third day this occurs again,
the traveler losing a third hour; and what to him seems
three days, Monday noon to Thursday noon, is in reality
by London time three days and three hours. Each of his
days is really a little more than twenty-four hours long,
for he is going with the sun. By the time he arrives at
London again he finds what to him was twenty-four days
is, in reality, twenty-five days, for he has set his watch
back an hour each day for twenty-four days, or an entire
day. To have his calendar correct, he must omit a day,
that is, move the date ahead one day to make up the date
lost from his reckoning. It is obvious that this will be
true whatever the rate of travel, and the day can be omitted
from his calendar anywhere in the journey and the error
corrected. Eastward Travel — Days are Shortened. Had our trav
eler gone eastward, when he had covered 15° of longitude
he would set his watch ahead one hour and then say,
" It is now Tuesday noon. I have been out one day."
The London clock would indicate 11 o'clock, a.m., of
Tuesday, and thus say his day had but twenty-three
hours in it, the traveler having moved the hour hand
ahead one space. He has gained one hour. The second
day he would gain another hour, and by the time he arrived

THE INTERNATIONAL DATE LINE

95

at London again, he would
have set his hour hand ahead
twenty-four hours or one
full day. To correct his cal
endar, somewhere on his voy
age he would have to repeat
a day. The International Date
Line. It is obvious from
the foregoing explanation
that somewhere and some
time in circumnavigation, a
day must be omitted in trav
eling westward and a day
repeated in traveling east
ward. Where and when the
change is made is a mere
matter of convenience . The
theoretical location of the
date line commonly used is
the 180th meridian. This
line where a traveler's cal
endar needs changing varies
as do the boundaries of the
standard time belts and for
the same reason. While
the change could be made
at any particular point on
a parallel, it wpuld make a
serious inconvenience were
the change made in some
places. Imagine, for ex
ample, the 90th meridian,

Fig. 31

96 CIRCUMNAVIGATION AND TIME
west of Greenwich, to be the fine used. When it was
Sunday in Chicago, New York, and other eastern points,
it would be Monday in St. Paul, Kansas City, and western
points. A traveler leaving Minneapolis on Sunday night
would arrive in Chicago on Sunday morning and thus
have two Sundays on successive days. Our national holi
days and elections would then occur on different days
in different parts of the country. To reduce to the
minimum such inconveniences as necessarily attend chang
ing one's calendar, the change is made where there is a
relatively small amount of travel, away out in the Pacific
Ocean. Going westward across this fine one must set his
calendar ahead a day; going eastward, back a day.
As shown in Figures 31 and 32, this fine begins on the
180th meridian far to the north, sweeps to the eastward
around Cape Deshnef, Russia, then westward beyond the
180th meridian seven degrees that the Aleutian islands
may be to the east of it and have the same day as
continental United States; then the fine extends to the
180th meridian which it follows southward, sweeping
somewhat eastward to give the Fiji and Chatham islands
the same day as Australia and New Zealand. The follow
ing is a letter, by C. B. T. Moore, commander, U. S. N.,
Governor of Tutuila, relative to the accuracy of the map
in this book: Pago-Pago, Samoa, December 1, 1906.
Dear Sir: — The map of your Mathematical Geography is
correct in placing Samoa to the east of the international date lines
The older geographies were also right in placing these islands west
of the international date line, because they used to keep the same date
as Australia and New Zealand, which are west of the international
date line.
The reason for this mistake is that when the London Missionary
Society sent its missionaries to Samoa they were not acquainted with

THE INTERNATIONAL DATE LINE 97
the trick of changing the date at the 180th meridian, and so carried
into Samoa, which was east of the date line, the date they brought
with them, which was, of course, one day ahead.
This false date was in force at the time of my first visit to Samoa,
in 1889. While I have no record to show when the date was
corrected, I believe that it was corrected at the time of the annexa
tion of the Samoan Islands by the United States and by Germany.
The date in Samoa is, therefore, the same date as in the United
State-}, and is one day behind what it is in Australia and New Zealand ;
Example: To-day is the 2d; day of December in Auckland, and the
1st day of December in Tutuila.
Very respectfully, C. B. T. Moore,
Commander, U. S. Navy,
Governor.
Mr. Willis E. Johnson,
Vice President Northern Normal and Industrial School,
Aberdeen, Soulh Dakota.
" It is fortunate that the 180th meridian falls where
it does. From Siberia to the Antarctic continent this
imaginary line traverses nothing but water. The only
land which it passes at all near is one of the archipelagoes
of the south Pacific; and there it divides but a handful of
volcanoes and coral reefs from the main group. These
islands are even more unimportant to the world than
insignificant in size. Those who tenant them are few,
and those who are bound to these few still fewer. . . .•
There, though time flows ceaselessly on, occurs that
unnatural yet unavoidable jump of twenty-four hours;
and no one is there to be startled by the fact, — no one to
be perplexed in trying to reconcile the two incongruities,
continuous time and discontinuous day. There is nothing
but the ocean, and that is tenantless. . . . Most fortunate
was it, indeed, that opposite the spot where man was most
destined to think there should have been placed so little
to think about." *
* From Chosbn, by Percival Lowell.
JO. MATH. GEO. — 7

98 CIRCUMNAVIGATION AND TIME
Where Days Begin. When it is 11:30 o'clock, p.m., on
Saturday at Denver, it is 1:30 o'clock, a.m., Sunday, at
New York. It is thus evident that parts of two days
exist at the same time on the earth. Were one to travel
around the earth with the sun and as rapidly it would be
perpetually noon. When he has gone around once, one
day has passed. Where did that day begin? Or, sup
pose we wished to be the first on earth to hail the new
year, where could we go to do so? The midnight line,
just opposite the sun, is constantly bringing a new day
somewhere. Midnight ushers in the new year at Chicago.
Previous to this it was begun at New York. Still east
of this, New Year's Day began some time before. If we
keep going around eastward we must surely come to
some place where New Year's Day was first counted, or we
shall get entirely around to New York and find that the
New Year's Day began the day before, and this midnight
would commence it again. As previously stated, the date
line commonly accepted nearly coincides with the 180th
meridian. Here it is that New Year's Day first dawns
and each new day begins.
The Total Duration of a Day. While a day at any
particular place is twenty-four hours long, each day lasts
on earth at least forty-eight hours. Any given day, say
Christmas, is "first counted as that day just west of the
date line. An hour later Christmas begins 15° west of
that line, two hours later it begins 30° west of it, and so on
around the globe. The people just west of the date line
who first hailed Christmas have enjoyed twelve hours of
it .when it begins in England, eighteen hours of it when
it begins in central United States, and twenty-four hours
of it, or the whole day, when it begins in western Alaska,
just east of the date line. Christmas, then, has existed

CONFUSION OF TRAVELERS 99
twenty-four hours on the globe, but having just begun in
western Alaska, it will tarry twenty-four hours longer
among mankind, making forty-eight hours that the day
blesses the earth.
If the date line followed the meridian 180° without
any variation, the total duration of a day would be exactly
forty-eight hours as just explained. But that line is quite
irregular, as previously described and as shown on the
map. Because of this irregularity of the date line the
same day lasts somewhere on earth over forty-nine hours.
Suppose we start at Cape Deshnef, Siberia, longitude 169°
West, a moment after midnight of the 3d of July. The
4th of July has begun, and, as midnight sweeps around
westward, successive places see the beginning of this day.
When it is the 4th in London it has been the 4th at
Cape Deshnef twelve hours and forty-four minutes. When
the glorious day arrives at New York, it has been seven
teen hours and forty-four minutes since it began at Cape
Deshnef. When it reaches our most western point on this
continent, Attu Island, 173° E., it has been twenty-five
hours and twelve minutes since it began at Cape Deshnef.
Since it will last twenty-four hours at Attu Island, forty-
nine hours and twelve minutes will have elapsed since the
beginning of the day until the moment when all places
on earth cease to count it that day.
When Three Days Coexist. Portions of three days
exist at the same time between 11:30 o'clock, a.m., and
12:30 o'clock, p.m., London time. When it is Monday
noon at London, Tuesday has begun at Cape Deshnef,
but Monday morning has not yet. dawned at Attu Island;
nearly half an hour of Sunday still remains there.
Confusion of Travelers. Many stories are told of the
confusion to travelers who pass from places reckoning

100

CIRCUMNAVIGATION AND TIME

one day across this line, to places having a different day.
" If it is such a deadly sin to work on Sunday, one or the
other of Mr. A and Mr. B coming one from the east,
the other from the west of the 180th meridian, must, if
he continues his daily vocations, be in a bad way. Some
of our people in the Fiji are in this unenviable position,
as the line 180° passes through Loma-Loma. I went
from Fiji to Tonga in Her Majesty's ship Nymph and
arrived at our destination on Sunday, according to our
reckoning from Fiji, but on Saturday, according to the
proper computation west from Greenwich. We, how-

Fig. 32

ever, found the natives all keeping Sunday. On my ask
ing the missionaries about it they told me that the mis
sionaries to that group and Samoa having come from

CONFUSION OF TRAVELERS 101
the westward, had determined to observe their Sabbath
day, as usual, so as not to subject the natives to any
puzzle, and agreed to put the dividing line farther off,
between them and Hawaii, somewhere in the broad ocean
where no metaphysical natives or ' intelligent ' Zulus
could cross-question them." *
" A party of missionaries bound from China, sailing
west, and nearing the line without their knowledge, on
Saturday posted a notice in the cabin announcing that
' To-morrow being Sunday there will be services in this
cabin at 10 a.m.' The following morning at 9, the captain
tacked up a notice declaring that ' This being Monday
there will be no services in this cabin this morning.' "
It should be remembered that this line, called " inter
national," has not been adopted by all nations as a hard
and fast line, making it absolutely necessary to change
the date the moment it is crossed. A ship sailing, say,
from Honolulu, which has the same day as North America
and Europe, to Manila or Hongkong, having a day later,
may make the change in date at any time between these
distant points; and since several days elapse in the pas
sage, the change is usually made so as to have neither
two Sundays in one week nor a week without a Sunday.
Just as the traveler in the United States going from a
place having one time standard to a place having a dif
ferent one would find it necessary to change his watch
but could make the change at any time, so one passing
from a place having one day to one reckoning another,
could suit his convenience as to the precise spot where
he make the change. This statement needs only the
* Mr. E. L. Layard, at the British Consulate, Noumea, New Cale
donia, as quoted in a pamphlet on the International Date Line by
Henry Collins.

102 CIRCUMNAVIGATION AND TIME
modification that as all events on a ship must be regu
lated by a common timepiece, changed according to
longitude, so the community on board in order to adjust
to a common calendar must accept the change when made
by the captain.
Origin and Change of Date Line. The origin of this
line is of considerable interest. The day adopted in any
region depended upon the direction from which the
people came who settled the country. For example,
people who went to Australia, Hongkong, and other Eng
lish possessions in the Orient traveled around Africa or
across the Mediterranean. They thus set their watches
ahead an hour for every 15°. " For two centuries after
the Spanish settlement the trade of Manila with the
western world was carried on via Acapulco and Mexico "
(Ency. Brit.). Thus the time which obtained in the
Philippines was found by setting watches backwards an
hour for every 15°, and so it came about that the calendar
of the Philippines was a day earlier than that of Australia,
Hongkong, etc. The date line at that time was very indefi
nite and irregular. In 1845 by a decree of the Bishop of
Manila, who was also Governor-General, Tuesday, Decem
ber 31, was stricken from the calendar; the day after
Monday, December 30, was Wednesday, January 1, 1846.
This cutting the year to 364 days and the week to 6 days
gave the Philippines the same day as other Asiatic places,
and shifted the date line to the east of that archipelago.
Had this change never been made, all of the possessions
of the United States would have the same day.
For some time after the acquisition of Alaska the people
living there, formerly citizens of Russia, used the day later
than ours, and also used the Russian or Julian calendar,
twelve days later than ours. As people moved there from

PROBLEM 103
the United States, our system gradually was extended, but
for a time both systems were in vogue. This made affairs
confusing, some keeping Sunday when others reckoned the
same day as Saturday and counted it as twelve days later
in the calendar, New Year's Day, Christmas, etc., coming
at different times. Soon, however, the American system
prevailed to the entire exclusion of the Russian, the
inhabitants repeating a day, and thus having eight days
in one week. While the Russians in their churches in
Alaska are celebrating the Holy Mass on our Sunday,
their brethren in Siberia, not far- away, and in other parts
of Russia, are busy with Monday's duties.
Date Line East of Fiji Islands. Fiji, No XIV, 1879: An ordi
nance enacted by the governor of the colony of Fiji, with the advice
and consent of the legislative council thereof, to provide for a univer
sal day throughout the colony.
Whereas, according to the ordinary rule of noting time, any given
time would in that part of the colony lying to the east of the meridian
of 180° from Greenwich be noted as of a day of the week and month
different from the day by which the same time would be noted in the
part of the colony lying to the west of such meridian; and
Whereas, by custom the ordinary rule has been set aside and time
has been noted throughout the colony as though the whole were situ
ated to the west of such meridian; and
Whereas, in order to preclude uncertainty for the future it is expe
dient that the above custom should be legalized ; therefore
Be it enacted by the governor, with the advice and consent of the
legislative council, as follows:
Time in this colony shall be noted as if the whole colony were
situated to the west of the meridian of 180° from Greenwich.
{Exempli gratia — To-day, which according to the ordinary rule
for noting time is on the island of Ovalau the 5th day of June, and
on the island of Vanua-Balevu the 4th day of June, would by this
ordinance be deemed as the 5th day of June, 1879, in the whole
colony.) Problem. Assuming it was 5 a.m., Sunday, May 1, 1898, when the
naval battle of Manila began, what time was it in Milwaukee, the city
using standard time and Dewey using the local time of 120° east?

CHAPTER VI
the earth's revolution
Proofs of Revolution
For at least 2400 years the theory of the revolution
of the earth around the sun has been advocated, but only
in modern times has the fact been demonstrated beyond
successful contradiction. The proofs rest upon three sets
of astronomical observations, all of which are of a delicate
and abstruse character, although the underlying principles
are easily understood.
Aberration of Light. When rain is falling on a calm day
the drops will strike the top of one's head if he is stand
ing still in the rain; but if one moves, the direction of the
drops will seem to have changed, striking one in the face
more and more as the speed is increased (Fig. 33). Now
light rays from the sun, a star, or other heavenly body,
strike the earth somewhat slantingly, because the earth is
moving around the sun at the rate of over a thousand
miles per minute. Because of this fact the astronomer
must tip his telescope slightly to the east of a star in order
to see it when the earth is in one side of its orbit, and to
the west of it when in the opposite side of the orbit. The
necessity of this tipping of the telescope will be apparent
if we imagine the rays passing through the telescope are
like raindrops falling through a tube. If the tube is car
ried forward swiftly enough the drops will strike the sides
of the tube, and in order that they may pass directly
through it, the tube must be tilted forward somewhat,
104

ABERRATION OF LIGHT

105

the amount varying with (a) the rate of its onward motion,
and (6) the rate at which the raindrops are falling.
Since the telescope must at One time be tilted one way
to see a star and at another season tilted an equal amount
in the opposite direction, each star thus seems to move

Fig. 33

about in a tiny orbit, varying from a circle to a straight
line, depending upon the position of the star, but in every
case the major axis is 41", or twice the greatest angle at
which the telescope must be tilted forward.
Each of the millions of stars has its own apparent aber
rational orbit, no two being exactly alike in form, unless
the two chance to be exactly the same distance from the
plane of the earth's orbit. Assuming that the earth

106 THE EARTH'S REVOLUTION
revolves around the sun, the precise form of this aberra
tional orbit of any star can be calculated, and observation
invariably confirms the calculation. Rational minds can
not conceive that the millions of stars, at varying dis
tances, can all actually have these peculiar annual motions,
six months toward the earth and six months from it, in
addition to the other motions which many of them (and
probably all of them, see pp. 265-267) have. The dis
covery and explanation of these facts in 1727 by James
Bradley (see p. 278), the English Astronomer Royal, forever
put at rest all disputes as to the revolution of the earth.
Motion in the Line of Sight. If you have stood near by
when a swiftly moving train passed with its bell ringing,
you may have noticed a sudden change in the tone of the
bell; it rings a lower note immediately upon passing. The
pitch of a note depends upon the rate at which the sound
waves strike the ear; the more rapid they are, the higher
is the pitch. Imagine a boy throwing chips* into a river
at a uniform rate while walking down stream toward a
bridge and then while walking upstream away from the
bridge. The chips will be closer together as they pass
under the bridge when the boy is walking toward it than
when he is walking away from it. In a similar way the
sound waves from the bell of the rapidly approaching
locomotive accumulate upon the ear of the listener, and
the pitch is higher than it would be if the train were sta
tionary, and after the train passes the sound waves will be
farther apart, as observed by the same person, who will
hear a lower note in consequence.
Color varies with Rate of Vibration. Now in a precisely
similar manner the colors in a ray of light vary in the rate
*This illustration is adapted from Todd's New Astronomy, p.
432.

MOTION IN THE LINE OF SIGHT 107
of vibration. The violet is the most rapid,* indigo about
one tenth part slower, blue slightly slower still, then green,
yellow, orange, and red. The spectroscope is an astro
nomical instrument which spreads out the fine of light
from a celestial body into a band and breaks it up into
its several colors. If a ringing bell rapidly approaches us,
or if we approach it, the tone of the bell sounds higher
than if it recedes from us or if we recede from it. If we
rapidly approach a star, or a star approaches us, its color
shifts toward the violet end of the spectroscope ; and if we
rapidly recede from it, or it recedes from us, its color shifts
toward the red end. Now year after year the thousands
of stars in the vicinity of the plane of the earth's pathway
show in the spectroscope this change toward violet at one
season and toward red at the opposite season. The far
ther from the plane of the earth's orbit a star is located,
the less is this annual change in color, since, the earth
neither approaches nor recedes from stars toward the
poles. Either the stars near the plane of the earth's
orbit move rapidly toward the earth at one season, gradu
ally stop, and six months later as rapidly recede, and stars
away from this plane approach and recede at rates dimin
ishing exactly in proportion to their distance from this
plane, or the earth itself swiftly moves about the sun.
Proof of the Rotation of the Earth. The same set of
* The rate of vibration per second for each of the colors in a ray
of light is as follows :
Violet .... 756.0 x IO12 YeUow  508.8 x 10"
Indigo .... 698.8 x IO12 Orange .... 457.1 x 1012
Blue  617.1 x 10'2 Red  393.6 x 10"
Green .... 569.2 x IO12
Thus the violet color has 756 . 0 millions of millions of vibrations each
second; indigo, 698.8 millions of millions, etc.

108 THE EARTH'S REVOLUTION
facts and reasoning applies to the rotation of the earth.
In the evening a star in the east shows a color approaching
the violet side of the spectroscope, and this gradually shifts
toward the red during the night as the star is seen higher
in the sky, then nearly overhead, then in the west. Now
either the star swiftly approaches the earth early in the
evening, then gradually pauses, and at midnight begins to
go away from the earth faster and faster as it approaches
the western horizon, or the earth rotates on its axis,
toward a star seen in the east, neither toward nor from it
when nearly overhead, and away from it when seen near
the west. Since the same star rises at different hours
throughout the year it would have to fly back and forth
toward and from the earth, two trips every day, varying
its periods according to the time of its rising and setting.
Besides this, when a star is rising at Calcutta it shows the
violet tendency to observers there (Calcutta is rotating
toward the star when the star is rising), and at the same
moment the same star is setting at New Orleans and thus
shows a shift toward the red to observers there. Now the
distant star cannot possibly be actually rapidly approach
ing Calcutta and at the same time be as rapidly receding
from New Orleans. The spectroscope, that wonderful
instrument which has multiplied astronomical knowledge
during the last half century, demonstrates, with mathe
matical certainty, the rotation of the earth, and multiplies
millionfold the certainty of the earth's revolution.
Actual Motions of Stars. Before leaving this topic we
should notice that other changes in the colors of stars show
that some are actually approaching the earth at a uniform
rate, and some are receding from it. Careful observa
tions at long intervals show other changes in the posi
tions of stars. The latter motion of a star is called its

THE PARALLAX OF STARS 109
proper motion to distinguish it from the apparent motion
it has in common with other stars due to the motions of
the earth. The spectroscope also assists in the demon
stration that the sun with the earth and the rest of the
planets and their attendant satellites is moving rapidly
toward the constellation Hercules.
Elements of Orbit Determined by the Spectroscope. As
an instance of the use of the spectroscope in determining
motions of celestial bodies, we may cite the recent calcu
lations of Professor Kustner, Director of the Bonn Observa
tory. Extending from June 24, 1904, to January 15, 1905,
he made careful observations and photographs of the
spectrographs lines shown by Arcturus. He then made
calculations based upon a microscopic examination of
the photographic plates, and was able to determine (a) the
size of the earth's orbit, (6) its form, (c) the rate of the
earth's motion, and (d) the rate at which the solar system
and Arcturus are approaching each other (10,849 miles per
hour, though not in a direct line).
The Parallax of Stars. Since the days of Copernicus
(1473-1543) the theory of the revolution of the earth
around the sun has been very generally accepted. Tycho
Brahe (1546-1601), however, and some other astrono
mers, rejected this theory because they argued that if the
earth had a motion across the great distance claimed for its
orbit, stars would change their positions in relation to the
earth, and they could detect no such change. Little did
they realize the tremendous distances of the stars. It was
not until 1838 that an astronomer succeeded in getting
the orbital or heliocentric parallax of a star. The German
astronomer Bessel then discovered that the faint star 61
Cygni is annually displaced to the extent of 0.4". Since
then about forty stars have been found to have measurable

110

THE EARTH'S REVOLUTION

parallaxes, thus multiplying the proofs of the motion of
the earth around the sun.
Displacement of a Star Varies with its Distance. Figure 34
_^ shows that the amount of
the displacement of a star
in the background of the
heavens owing to a change
in the position of the earth,
varies with the distance of
the star. The nearer the
star, the greater the displace
ment; in every instance, how
ever, this apparent shifting of
a star is exceedingly minute,
owing to the great distance
(see pp. 45, 246) of the very
nearest of the stars.
Since students often con
fuse the apparent orbit of a
star described under aberra
tion of light with that due to the parallax, we may make
the following comparisons:

Fig. 34

Aberrational Orbit
1. The earth's rapid motion
causes the rays of light to slant
(apparently) into the telescope so
that, as the earth changes its
direction in going around the sun,
the star seems to shift slightly
about. 2. This orbit has the same
maximum width for all stars,
however near or distant.

Parallactic Orbit
1. As the earth moves about
in its orbit the stars seem to
move about upon the back
ground of the celestial sphere.

2. This orbit varies in width
with the distance of the star; the
nearer the star, the greater the
width.

WINTER CONSTELLATIONS INVISIBLE IN SUMMER 111

Effects of Earth's Revolution
Winter Constellations Invisible in Summer. You have
doubtless observed that some constellations which are
visible on a winter's night cannot be seen on a summer's
night. In January, the beautiful constellation Orion may
be seen early in the evening and the whole night through;
in July, not at all. That this is due to the revolution of
the earth around the sun may readily be made apparent.
In the daytime we cannot easily see the stars around the

*N.Star

sun, because of its great light and the peculiar properties
of the atmosphere; six months from now the earth will
have moved halfway around the sun, and we shall be
between the sun and the stars he now hides from view,
and at night the stars now invisible will be visible.
If you have made a record of the observations suggested
in Chapter I, you will now find that Exhibit I (Fig. 35),
shows that the Big Dipper and other star groups have
slightly changed their relative positions for the same time
of night, making a little more than one complete rotation
during each twenty-four hours. In other words, the stars

112

THE EARTH'S REVOLUTION

have been gaming a little on the sun in the apparent daily
swing of the celestial sphere around the earth.
The reasons for this may be understood from a careful

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Fig. 36
study of Figure 36. The outer circle, which should be
indefinitely great, represents the celestial sphere; the inner
ellipse, the path of the earth around the sun. Now the sun
does not seem to be, as it really is, relatively near the
earth, but is projected into the celestial sphere among the
TWO APPARENT MOTIONS OF THE SUN 113
stars. When the earth is at point A the sun is seen among
the stars at o; when the earth has moved to B the sun
seems to have moved to b, and so on throughout the annual
orbit. The sun, therefore, seems to creep around the celestial
sphere among the stars at the same rate and in the same
direction as the earth moves in its orbit. If you walk around
a room with someone standing in the center, you will see
that his image may be projected upon the wall opposite,
and as you walk around, his image on the wall will move
around in the same direction. Thus the sun seems to move
in the celestial sphere in the same direction and at the same
rate as the earth moves around the sun.
Two Apparent Motions of the Sun: Daily Westward,
Annual Eastward. The sun, then, has two apparent
motions, — a daily swing around the earth with the celestial
sphere, and this annual motion in the celestial sphere among
the stars. The first motion is in a direction opposite to that
of the earth's rotation and is from east to west, the second
is in the same direction as the earth's revolution and is
from west to east. If this is not readily seen from the
foregoing statements and the diagram, think again of the
rotation of the earth making an apparent rotation of
the celestial sphere in the opposite direction, the reasons
why the sun and moon seem to rise in the east and set in
the west ; then think of the motion of the earth around the
sun by which the sun is projected among certain stars
and then among other stars, seeming to creep among
them from west to east.
After seeing this clearly, think of yourself as facing the
rising sun and a star which is also rising. Now imagine
the earth to have rotated once, a day to have elapsed,
and the earth to have gone a day's journey in its orbit in
the direction corresponding to upward. The sun would
JO. MATH. GEO.— 8

114 THE EARTH'S REVOLUTION
not then be on the horizon, but, the earth having moved
" upward," it would be somewhat below the horizon.
The same star, however, would be on the horizon, for the
earth does not change its position in relation to the stars.
After another rotation the earth would be, relative to the
stars viewed in that direction, higher up in its orbit and
the sun farther below the horizon when the star was just
rising. In three months when the star rose the sun would
be nearly beneath one's feet, or it would be midnight; in
six months we should be on the other side of the sun,
and it would be setting when the star was rising; in nine
months the earth would have covered the " downward "
quadrant of its journey around the sun, and the star
would rise at noon; twelve months later the sun and
star would rise together again. If the sun and a star
set together one evening, on the next evening the star
would set a little before the sun, the next night earlier
still. Since the sun passes around its orbit, 360°, in a year,
365 days, it passes over a space of nearly one degree each
day. The diameter of the sun as seen from the earth
covers about half a degree of the celestial sphere. During
one rotation of the earth, then, the sun creeps eastward
among the stars about twice its own width. A star rising
with the sun. will gain on the sun nearly s^ of a day
during each rotation, or a little less than four minutes.
The sun sets nearly four minutes later than the star with
which it set the day before.
Sidereal Day. Solar Day. The time from star-rise to
star-rise, or an exact rotation of the earth, is called a
sidereal day. Its exact length is 23 h. 56 m. 4.09 s. The
time between two successive passages of the sun over a
given meridian, or from noon by the sun until the next

CAUSES OF APPARENT MOTIONS OF THE SUN 115
noon by the sun, is called a solar day* Its length varies
somewhat, for reasons to be explained later, but aver
ages twenty-four hours. When we say "day," if it is not
otherwise qualified, we usually mean an average solar
day divided into twenty-four hours, from midnight to
midnight. The term "hour, " too, when not otherwise
qualified, refers to one twenty-fourth of a mean solar
day. Causes of Apparent Motions of the Sun. The apparent
motions of the sun are due to the real motions of the

Ec/'BUS-

Aog. 23

Ai/g.22

Aug. 2/

Fig. 37

earth. If the earth moved slowly around the sun, the
sun would appear to move slowly among the stars. Just
as we know the direction and rate of the earth's rotation
by observing the direction and rate of the apparent rota-
* A solar day is sometimes defined as the interval from sunrise to
sunrise again. This is true only at the equator. The length of the
solar day corresponding to February 12, May 15, July 27, or November
3, is almost exactly twenty-four hours. The time intervening between
sunrise and sunrise again varies greatly with the latitude and season.
On the dates named a solar day at the pole is twenty-four hours long,
as it is everywhere else on earth. The time from sunrise to sunrise
again, however, is almost six months at either pole.

116

THE EARTH'S REVOLUTION

tion of the celestial sphere, we know the direction and
rate of the earth's revolution by observing the direction
and rate of the sun's apparent annual motion.
The Ecliptic. The path which the center of the sun
seems to trace around the celestial sphere in its annual

Fig. 38. Celestial sphere, showing zodiac

orbit is called the ecliptic* The line traced by the center
of the earth in its revolution about the sun is its orbit.
Since the sun's apparent annual revolution around the sky
is due to the earth's actual motion about the sun, the path
of the sun, the ecliptic, must lie in the same plane with the
* So called because eclipses can occur only when the moon crosses
the plaqe of the ecliptic.

THE ZODIAC 117
earth's orbit. The earth's equator and parallels, if ex
tended, would coincide with the celestial equator and
parallels; similarly, the earth's orbit, if expanded in the
same plane, would coincide with the ecliptic. We often
use interchangeably the expressions " plane of the earth's
orbit " and " plane of the ecliptic."
The Zodiac. The orbits of the different planets and of
the moon are inclined somewhat to the plane of the eclip
tic, but, excepting some of the minor planets, not more
than eight degrees. The moon and principal planets,
therefore, are never more than eight degrees from the
pathway of the sun. This belt sixteen degrees wide, with
the ecliptic as the center, is called the zodiac (more fully
discussed in the Appendix, p. 293). Since the sun appears
to pass around the center of the zodiac once each year,
the ancients, who observed these facts, divided it into
twelve parts, one for each month, naming each part from
some constellation in it. It is probably more nearly cor
rect historically, to say that these twelve constellations
got their names originally from the position of the sun in
the zodiac. Libra, the Balance, probably got its name
from the fact that in ancient days the sun was among the
group of stars thus named about September 23, when the
days and nights are equal, thus balancing. In some such
way these parts came to be called the " twelve signs of
the zodiac," one for each month.
The facts in this chapter concerning the apparent
annual motion of the sun were well known to the ancients,
possibly even more generally than they are to-day. The
reason for this is because there were few calendars and
almanacs in the earlier days of mankind, and people had
to reckon their days by noting the position of the sun.
Thus, instead of saying that the date of his famous

118 THE EARTH'S REVOLUTION
journey to Canterbury was about the middle of April,
Chaucer says it was
When Zephirus eek with his sweete breeth
Enspired hath in every holt and heath
The tendre croppes, and the younge sonne
Hath in the Ram his halfe course yronne.
Even if clothed in modern English such a description
would be unintelligible to a large proportion of the stu
dents of to-day, and would need some such translation as
the following.
" When the west wind of spring with its sweet breath
hath inspired or given new life in every field and heath
to the tender crops, and the young sun (young because it
had got only half way through the sign Aries, the Ram,
which marked the beginning of the new year in Chaucer's
day) hath run half his course through the sign the Ram."
Obliquity of the Ecliptic. The orbit of the earth is
not at right angles to the axis. If it were, the ecliptic
would coincide with the celestial equator. The plane of
the ecliptic and the plane of the celestial equator form an
angle of nearly* 23|°. This is called the obliquity of the
ecliptic. We sometimes speak of this as the inclination of
the earth's axis from a perpendicular to the plane of its
orbit. Since the plane of the ecliptic forms an angle of 23|° with
the plane of the equator, the sun in its apparent annual
course around in the ecliptic crosses the celestial equator
* The exact amount varies slightly from year to year. The follow
ing table is taken from the Nautical Almanac, Newcomb's Calculations:
1903 .... 23° 27' 6.86" 1906  23° 27' 5.45"
1904 .... 23° 27' 6.39" 1907  23° 27' 4.98"
1905  23° 27' 5.92" 1908 . .... 23° 27' 4.51"

VARYING SPEED OF THE EARTH

119

Fig. 39

twice each year, and at one season gets 23£° north of it,
and at the opposite season 23£° south of it. The sun thus
never gets nearer the pole of the celestial sphere than 66J°.
On March 21 and September 23 the sun is on the celes
tial equator. On June 21
and December 22 the sun
is 23|° from the celestial
equator. Earth's Orbit. We have
learned that the earth's orbit
is an ellipse, and the sun is
at a focus of it. While the
eccentricity is not great, and
when reduced in scale the
orbit does not differ materi
ally from a circle, the differ
ence is sufficient to make an appreciable difference in the
rate of the earth's motion in different parts of its orbit.
Figure 113, p. 285, represents the orbit of the earth, greatly
exaggerating the ellipticity. The point in the orbit nearest
the sun is called perihelion (from peri, around or near, and
helios, the sun). This point is about 91 J million miles from
the sun, and the earth reaches it about December 31st.
The point in the earth's orbit farthest from the sun is
called aphelion (from a, away from, and helios, sun). Its
distance is about 94| million miles, and the earth reaches
it about July 1st.
Varying Speed of the Earth. According to the law of
gravitation, the earth moves faster in its orbit when near
perihelion, and slower when near aphelion. In December
and January the earth moves fastest in its orbit, and
during that period the sun moves fastest in the ecliptic
and falls farther behind the stars in their rotation in the

120

THE EARTH'S REVOLUTION

celestial sphere. Solar days are thus longer then than
they are in midsummer when the earth moves more slowly
in its orbit and more nearly keeps up with the stars.
Imagine the sun and a star are rising together January
1st. After one exact rotation of the earth, a sidereal day,
the star will be rising again, but since the earth has moved
rapidly in its course around the sun, the sun is somewhat
farther behind the star than it would be in summer when
the earth moved more slowly around the sun. At star-
rise January 3d, the sun is behind still farther, and in
the course of a few weeks the sun will be several minutes
behind the point where it would be if the earth's orbital
motion were uniform. The sun is then said to be slow of
the average sun. In July the sun creeps back less rapidly
in the ecliptic, and thus a solar day is more nearly the
same length as a sidereal day, and hence shorter than the
average. Another factor modifies the foregoing statements. The
daily courses of the stars swinging around with the celes
tial sphere are parallel and are at right angles to the axis.
The sun in its annual
26°
25°24°
'23°
22 o
21°

path creeps diagonally
across their courses.
When farthest from the
celestial equator, in June
and in December, the
sun's movement in the
ecliptic is nearly parallel
Kg- 40 to the courses of the
stars (Fig. 40); as it
gets nearer the celestial equator, in March and in Sep
tember, the course is more oblique. Hence in the latter
part of June and of December, the sun, creeping back in

—  e

SIDEREAL DAY SHORTER THAN SOLAR DAY 121

the ecliptic, falls farther behind the stars and becomes
slower than the average. In the latter part of March and
of September the sun creeps in a more diagonal course and
hence does not fall so far behind the stars in going the
same distance, and thus becomes faster than the average
(Fig. 41).
Some solar days being longer than others, and the
sun being sometimes
3° /°
•r°
'2'°

^:

^3

Fig. 41

slow and sometimes
fast, together with
standard time adop
tions whereby most
places have their
watches set by mean
solar time at some
given meridian, make
it unsafe to set one's
watch by the sun without making many corrections.
The shortest day in the northern hemisphere h about
December 22d; about that time the sun is neither fast nor
slow, but it then begins to get slow. So as the days get
longer the sun does not rise any earlier until about the
second week of January. After Christmas one may notice
the later and later time of sunsets. In schools in the
northern states beginning work at 8 o'clock in the morning,
it is noticed that the mornings are actually darker for a
while after the Christmas holidays than before, though the
shortest day of the year has passed.
Sidereal Day Shorter than Solar Day. If one wanted
to set his watch by the stars, he would be obliged to
remember that sidereal days are shorter than solar days;
if the star observed is in a certain position at a given time
of night, it will be there nearly four minutes earlier the

122 THE EARTH'S REVOLUTION
next evening. The Greek dramatist Euripides (480-407
B.C.), in his tragedy " Rhesus," makes the Chorus say:
Whose is the guard? Who takes my turn? The first signs
are setting, and the seven Pleiades are in the sky, and the Eagle
glides midway through the sky. Awake! See ye not the brilliancy
of the moon? Morn, morn, indeed is approaching, and hither is one
of the forewarning stars.

Summary.
Note carefully these propositions:
1. The earth's orbit is an ellipse.
2. The earth's orbital direction is the same as the direction of its
axial motion.
3. The rate of the earth's rotation is uniform, hence sidereal days
are of equal length.
4. The orbit of the earth is in nearly the same plane as that of the
equator.
5. The earth's revolution around the sun makes the sun seem to
creep backward among the stars from west to east, falling
behind them about a degree a day. The stars seem to swing
around the earth, daily gaining about four minutes upon the
sun.
6. The rate of the earth's orbital motion determines the rate of the
sun's apparent annual backward motion among the stars.
7 The rate of the earth's orbital motion varies, being fastest when
the earth is nearest the sun or in perihelion, and slowest when
farthest from the sun or in aphelion.
8. The sun's apparent annual motion, backward or eastward among
the stars, is greater when in or near perihelion (December 31)
than at any other time.
9. The length of solar days varies, averaging 24 hours in length.
There are two reasons for this variation.
a. Because the earth's orbital motion is not uniform, it being faster
when nearer the sun, and slower when farther from it.
b. Because when near the equinoxes the apparent annual motion
of the sun in the celestial sphere is more diagonal than when
near the tropics.

SUN FAST OR SUN SLOW 123
10. Because of these two sets of causes, solar days are more than 24
hours in length from December 25 to April 15 and from June
15 to September 1, and less than 24 hours in length from April
15 to June 15 and from September 1 to December 25.
Equation of Time
Sun Fast or Sun Slow. The relation of the apparent
solar time to mean solar time is called the equation of
time. As just shown, the apparent eastward motion of
the sun in the ecliptic is faster than the average twice a
year, and slower than the average twice a year. A ficti
tious sun is imagined to move at a uniform rate eastward
in the celestial equator, starting with the apparent sun at
the vernal equinox (see Equinox in Glossary) and com
pleting its annual course around the celestial sphere in
the same time in which the sun apparently makes its cir
cuit of the ecliptic. While, excepting four times a year,
the apparent sun is fast or slow as compared with this
fictitious sun which indicates mean solar time, their differ
ence at any moment, or the equation of time, may be
accurately calculated.
The equation of time is indicated in various ways. The
usual method is to indicate the time by which the appar
ent sun is faster than the average by a minus sign, and the
time by which it is slower than the average by a plus sign.
The apparent time and the equation of time thus indicated,
when combined, will give the mean time. Thus, if the sun
indicates noon (apparent time), and we know the equation
to be — 7 m. (sun fast, 7 m.), we know it is 11 h. 53 m., a.m.
by mean solar time.
Any almanac shows the equation of time for any day of
the year. It is indicated in a variety of ways.
a. In the World Almanac it is given under the title

124 THE EARTH'S REVOLUTION
" Sun on Meridian." The local mean solar time of the
sun's crossing a meridian is given to the nearest second.
Thus Jan. 1, 1908, it is given as 12 h. 3 m. 16 s. We know
from this that the apparent sun is 3 m. 16 s. slow of the
average on that date.
b. In the Old Farmer's Almanac the equation of time is
given in a column headed " Sun Fast," or " Sun Slow."
c. In some places the equation of time is indicated by
the words, " clock ahead of sun," and " clock behind sun."
Of course the student knows from this that if the clock is
ahead of the sun, the sun is slower than the average, and,
conversely, if the clock is behind the sun, the latter must
be faster than the average.
d. Most almanacs give times of sunrise and of sunset.
Now half way between sunrise and sunset it is apparent
noon. Suppose the sun rises at 7:24 o'clock, a.m., and
sets at 4 : 43 o'clock, p.m. Half way between those times
is 12: 03 J o'clock, the time when the sun is on the
meridian, and thus the sun is 3£ minutes slow (Jan. 1, at
New York).
e. The Nautical Almanac* has the most detailed and
accurate date obtainable. Table II for each month gives
in the column " Equation of Time " the number of min
utes and seconds to be added to or subtracted from 12
o'clock noon at Greenwich for the apparent sun time. The
adjoining column gives the difference for one hour to be
added when the sun is gaining, or subtracted when the sun
is losing, for places east of Greenwich, and vice versa for
places west.
Whether or not the student has access to a copy of the
* Prepared annually three years in advance, by the Professor of
Mathematics, United States Navy, Washington, D. C. It is sold by
the Bureau of Equipment at actual cost of publication, one dollar.

SUN FAST OR SUN SLOW

125

Nautical Almanac it may be of interest to notice the use
of this table.

AT GREENWICH MEAN NOON.

i ® ©
•M o>>
a

4 ao
S o
s 0

THE

SUN'S

Equation of Time
to be
Sub
tracted from
Mean
Time

Diff. for 1
Hour

Sidereal Time,
or Right
Ascension of
Mean Sun

Apparent Right
Ascension

Diff. for
lHour

Apparent
Declination

Diff. for
lHour

Wed.Thur. Frid.

1
2
3

h m s
18 42 9.88
18 46 35.09
18 50 59.99

11 .057
11.04411.030

O 1 It
S. 23 5 47.3
23 1 6.3
22 55 57.7

+ 11.13
12.28 13.42

m s
3 10.29
3 38.93
4 7.28

s
1.2001.1881.174

li m s
18 38 59.60
18 42 56.16
18 46 52.71

Sat.SUN. Mon.

45 6

18 55 24.54
18 59 48.70
19 4 12.45

11.01510.998 10.979

22 50 21.8
22 44 18.6
22 37 48.2

+ 14.5615.7016.82

4 35.27
5 2.87
5 30.06

1.158 1.141 1.123

18 50 49.27
18 54 45.83
18 58 42.39

Tues.Wed.Thur.

789

19 8 35.74
19 12 50.56
19 17 : ).85

10.959 10.93910.918

22 30 51.0
22 23 27.1
22 15 36.8

+ 17.9419.04
20.14

5 56.80
6 23.06
6 48.79

1.104 1.083
1.061

19 2 38.94
19 6 35.50
19 10 32.08

Frid. Sat.
SUN.

101112

19 21 :2.61
19 26 3.79
19 30 24.39

10.89510.87110.846

22 7 20.2
21 58 37.7
21 49 29.5

+ 21.2322.3023.37

7 13.99
7 38.62
8 2.66

1.0381.014
0.989

19 14 28.62
19 18 25.17
19 22 21.73

Part of a page from The American EphemeHs and Nautical A Imanac, Jan. 1908.
This table indicates that at 12 o'clock noon, on the
meridian of Greenwich on Jan. 1, 1908, the sun is slow 3 ni.
10.29 s., and is losing 1.200 s. each hour from that moment.
Wc know it is losing, for we find that on January 2 the
sun is slow 3 m. 38.93 s., and by that time its rate of loss
is slightly less, being 1.188 s. each hour.
Suppose you are at Hamburg on Jan. 1, 1908, when
it is noon according to standard time of Germany, one
hour before Greenwich mean noon. The equation of time
will be the same as at Greenwich less 1.200 s. for the hour's
difference, or (3 m. 10.29 s. — 1.200 s.) 3 m. 9.09 s. If you
are at New York on that date and it is noon, Eastern stand-

126 THE EARTH'S REVOLUTION
ard time, five hours after Greenwich noon, it is obvious
that the sun is 5 X 1.200 s. or 6 s. slower than it was at
Greenwich mean noon. The equation of time at New
York would then be 3 m. 10.29 s. + 6 s. or 3 m. 16.29 s.
/. The Analemma graphically indicates the approximate
equation of time for any day of the year, and also indicates
the declination of the sun (or its distance from the celestial
equator). Since our year has 365| days, the equation of
time for a given date of one year will not be quite the
same as that of the same date in a succeeding year. That
for 1910 will be approximately one fourth of a day or six
hours later in each day than for 1909; that is, the table
for Greenwich in 1910 will be very nearly correct for Cen
tral United States in 1909. Since for the ordinary pur
poses of the student using this book an error of a few
seconds is inappreciable, the analemma will .answer for
most of his calculations.
The vertical lines of the analemma represent the num
ber of minutes the apparent sun is slow or fast as com
pared with the mean sun. For example, the dot repre
senting February 25 is a little over half way between the
lines representing sun slow 12 m. and 14 m. The sun is then
slow about 13 m. 18 s. It will be observed that April 15,
June 15, September 1, and December 25 are on the central
line. The equation of time is then zero, and the sun may
be said to be " on time." Persons living in the United
States on the 90th meridian will see the shadow due north
at 12 o'clock on those days; if west of a standard time
meridian one will note the north shadow when it is past
12 o'clock, four minutes for every degree; and, if east of
a standard time meridian, before 12 o'clock four minutes
for each degree. Since the analemma shows how fast or
slow the sun is each day, it is obvious that, knowing one's

SUN FAST OR SUN SLOW

127

THE ANALEMMA
Showing position of apparent Sun and
its declination euery day of the year.

- mt-10

128 THE EARTH'S REVOLUTION
longitude, one can set his watch by the sun by reference
to this diagram, or, having correct clock time, one can
ascertain his longitude.
Uses of the Analemma
To Ascertain Your Longitude. To do this your watch
must show correct standard time. You must also have a
true north-south line.
1. Carefully observe the time when the shadow is north.
Ascertain from the analemma the number of minutes and
seconds the sun is fast or slow.
2. If fast, add that amount to the time by your watch;
if slow, subtract. This gives your mean local time.
3. Divide the minutes and seconds past or before twelve
by four. This gives you the number of degrees and
minutes you are from the standard time meridian. If
the corrected time is before twelve, you are east of it; if
after, you are west of it.
4. Subtract (or add) the number of degrees you are
east (or west) of the standard time meridian, and this is
your longitude.
For example, say the date is October 5th. 1. Your
watch says 12 h. 10 m. 30 s., p.m., when the shadow is north.
The analemma shows the sun to be 1 1 m. 30 s. fast. 2. The
sun being fast, you add these and get 12:22 o'clock, p.m.
This is the mean local time of your place. 3. Dividing
the minutes past twelve by four, you get 5 m. 30 s. This is
the number of degrees and minutes you are west from the
standard meridian. If you live in the Central standard
time belt of the United States, your longitude is 90° plus
5° 30', or 95° 30'. If you are in the Eastern time belt,
it is 75° plus 5° 30'. If you are in Spain, it is 0° plus
5° 30', and so on.

TO SET YOUR WATCH 129
To Set Your Watch. To do this you must know your
longitude and have a true north-south line.
1. Find the difference between your longitude and that
of the standard time meridian in accordance with which
you wish to set your watch. In Eastern United States the
standard time meridian is the 75th, in Central United
States the 90th, etc.
2. Multiply the number of degrees and seconds of
difference by four. This gives you the number of minutes
and seconds your time is faster or slower than local time.
If you are east of the standard meridian, your watch must
be set slower than local time; if west, faster.
3. From the analemma observe the position of the sun
whether fast or slow and how much. If fast, subtract
that time from the time obtained in step two ; if slow, add.
This gives you the time before or after twelve when the
shadow will be north; before twelve if you are east of the
standard time meridian, after twelve if you are west.
4. Carefully set your watch at the time indicated in step
three when the sun's shadow crosses the north-south line.
For example, suppose your longitude is 87° 37' W.
(Chicago). 1. The difference between your longitude and
your standard time meridian, 90°, is 2° 23'. 2. Multi
plying this difference by four we get 9° 32', the minutes
and seconds your time is slower than the sun's average
time. That is, the sun on the average casts a north
shadow at 11 h. 50 m. 28 s. at your longitude. 3. From the
analemma we see the sun is 14 m. 15 s. slow on February 6.
The time being slow, we add this to 11 h. 50 m. 28 s. and
get 12 h. 4 m. 43 s., or 4 m. 43 s. past twelve when the
shadow will be north. 4. Just before the shadow is north
get your watch ready, and the moment the shadow is north
set it 4 m. 43 s. past twelve.
JO. HATH. GEO.— 9

130 THE EARTH'S REVOLUTION
To Strike a North-South Line. To do this you must
know your longitude and have correct time.
Steps 1, 2, and 3 are exactly as in the foregoing explana
tion how to set your watch by the sun. At the time you
obtain in step 3 you know the shadow is north; then draw
the line of the shadow, or, if out of doors, drive stakes or
otherwise indicate the line of the shadow.
To Ascertain Your Latitude. This use of the analemma
is reserved for later discussion.
Civil and Astronomical Days. The mean solar day of
twenty-four hours reckoned from midnight is called a civil
day, and among all. Christian nations has the sanction of
law and usage. Since astronomers work at night they
reckon a day from noon. Thus the civil forenoon is
dated a day ahead of the astronomical day, the afternoon
being the last half of the civil day but the beginning of
the astronomical day. Before the invention of clocks and
watches, the sundial was the common standard for the
time during each day, and this, as we have seen, is a con
stantly varying one. When clocks were invented it was
found impossible to have them so adjusted as to gain or
lose with the sun. Until 1815 a civil day in France was a
day according to the actual position of the sun, and hence
was a very uncertain affair.
A Few Facts: Do You Understand Them ?
1. A day of twenty-four hours as we commonly use the
term, is not one rotation of the earth. A solar day is a
little more than one complete rotation and averages
exactly twenty-four hours in length. This is a civil or
legal day.
2. A sidereal day is the time of one rotation of the
earth on its axis.

CIVIL AND ASTRONOMICAL DAYS 131
3. There are 366 rotations of the earth (sidereal days)
in one year of 365 days (solar days).
4. A sundial records apparent or actual sun time, which
is the same as mean sun time only four times a year.
5. A clock records mean sun time, and thus corresponds
to sundial time only four times a year.
6. In many cities using standard time the shadow of the
sun is never in a north-south line when the clock strikes
twelve. This is true of all cities more than 4° east or
west of the meridian on which their standard time is based.
7. Any city within 4° of its standard time meridian will
have north-south shadow lines at twelve o'clock no more
than four times a year at the most. Strictly speaking,
practically no city ever has a shadow exactly north-south
at twelve o'clock.

CHAPTER VII
TIME AND THE CALENDAR
" In the early days of mankind, it is not probable that
there was any concern at all about dates, or seasons, or
years. Herodotus is called the father of history, and his
history does not contain a single date. Substantially
the same may be said of Thucydides, who wrote only a
little later — somewhat over 400 B.C. If Geography and
Chronology are the two eyes of history, then some histories
are blind of the one eye and can see but little out of the
other." *
Sidereal Year. Tropical Year. As there are two kinds
of days, solar and sidereal, there are two kinds of years,
solar or tropical years, and sidereal years, but for very
different reasons. The sidereal year is the time elapsing
between the passage of the earth's center over a given
point in its orbit until it crosses it again. For reasons
not properly discussed here (see Precession of the Equi
noxes, p. 286), the point in the orbit where the earth is
when the vertical ray is on the equator shifts slightly
westward so that we reach the point of the vernal equinox
a second time a few minutes before a sidereal year has
elapsed. The time elapsing from the sun's crossing of the
celestial equator in the spring until the crossing the next
spring is a tropical year and is what we mean when we
say " a year." f Since it is the tropical year that we
* R. W. Farland in Popular Astronomy for February, 1895.
f A third kind of year is considered in astronomy, the anomalistic
year, the time occupied by the earth in traveling from perihelion to
perihelion again. Its length is 365 d. 6 h. 13 m. 48.09 s. The lunar year,
132

THE MOON THE MEASURER 133
attempt to fit into an annual calendar and which marks
the year of seasons, it is well to remember its length:
365 d. 5 h. 48 m. 45.51 s. (365.2422 d.). The adjust
ment of the days, weeks, and months into a calendar
that does not change from year to year but brings
the annual holidays around in the proper seasons, has
been a difficult task for the human race to accomplish.
If the length of the year were an even number of days
and that number was exactly divisible by twelve,
seven, and four, we could easily have seven days in a week,
four weeks in a month, and twelve months in a year and
have no time to carry over into another year or month.
The Moon the Measurer. Among the ancients the
moon was the great measurer of time, our word month
comes from the word moon, and in connection with its
changing phases religious feasts and celebrations were
observed. Even to-day we reckon Easter and some other
holy days by reference to the moon. Now the natural
units of time are the. solar day, the lunar month (about
29£ days), and the tropical year. But their lengths are
prime to each other. For some reasons not clearly known,
but believed to be in accordance with the four phases of
the moon, the ancient Egyptians and Chaldeans divided
the month into four weeks of seven days each. The
addition of the week as a unit of time which is naturally
related only to the day, made confusion worse confounded.
Various devices have been used at different times to make
the same date come around regularly in the same season
year after year, but changes made by priests who were
ignorant as to the astronomical data and by more igno
rant kings often resulted in great confusion. The very
twelve new moons, is about eleven days shorter than the tropical
year. The length of a sidereal year is 365 d. 6 h. 9 m. 8.97 s.

134 TIME AND THE CALENDAR
exact length of the solar year in the possession of the
ancient Egyptians seems to have been little regarded.
Early Roman Calendar. Since our calendar is the same
as that worked out by the Romans, a brief sketch of their
system may be helpful. The ancient Romans seem to
have had ten months, the first being March. We can see
that this was the case from the fact that September means
seventh; October, eighth; November, ninth; and Decem
ber, tenth. It was possibly during the reign of Numa
that two months were added, January and February.
There are about 29£ days in a lunar month, or from one
new moon to the next, so to have their months conform
to the moons they were given 29 and 30 days alter
nately, beginning with January. This gave them twelve
lunar months in a year of 354 days. It was thought
unlucky to have the number even, so a day was added for
luck. This year, having but 355 days, was over ten days too
short, so festivals that came in the summer season would
appear ten days earlier each year, until those dedicated
to Bacchus, the god of wine, came when the grapes were
still green, and those of Ceres, the goddess of the harvest,
before the heads of the wheat had appeared. To correct
this an extra month was added, called Mercedonius, every
second year. Since the length of this month was not fixed
by law but was determined by the pontiffs, it gave rise
to serious corruption and fraud, interfering with the collec
tion of debts by the dropping out of certain expected
dates, lengthening the terms of office of favorites, etc.
The Julian and the Augustan Calendars. In the year
46 B.C., Julius Caesar, aided by the Egyptian astronomer,
Sosigenes, reformed the calendar. He decreed that begin
ning with January the months should have alternately 31

Jan.

31

31

Feb.

29-30

28-29

Mar.

31

31

Apr.

30

30

May

31

31

June

30

30

July

31

31

Aug.

30

31

Sept.

31

30

Oct.

30

31

Nov.

31

30

Dec.

30

31

THE GREGORIAN CALENDAR 135
and 30 days, save February, to which was assigned 29
days, and every fourth year an additional day. This
made a year of exactly 365^ days. Since the true year has
365 days, 5 hours, 48 min., 45.51 sec, and the Julian year
had 365 days, 6 hours, it was 11
min., 14.49 sec. too long.
During the reign of Augustus
another day was taken from Feb
ruary and added to August in order
that that month, the name of which
had been changed from Sextilis to
August in his honor, might have as
many days in it as the month
Quintilis, whose name had been
changed to July in honor of Julius Caesar. To prevent the
three months, July, August, and September, from having 31
days each, such an arrangement being considered unlucky,
Augustus ordered that one day be taken from September
and added to October, one from November and added to
December. Thus we find the easy plan of remembering
the months having 31 days, every other one, was dis
arranged, and we must now count our knuckles or learn:
" Thirty days hath September, April, June, and November.
All the rest have thirty-one, save the second one alone,
Which has four and twenty-four, till leap year gives it one day more."
The Gregorian Calendar. This Julian calendar, as it
is called, was adopted by European countries just as they
adopted other Roman customs. Its length was 365.25
days, whereas the true length of the year is 365.2422
days. While the error was only .0078 of a day, in the
course of centuries this addition to the true year began
to amount to days. By 1582 the difference had amounted
to about 13 days, so that the time of the spring equinox,

136 TIME AND THE CALENDAR
when the sun crosses the celestial equator, occurred the
11th of March. In that year Pope Gregory XIII reformed
the calendar so that the March equinox might occur on
March 21st, the same date as it did in the year 325 a.d.,
when the great Council of Nicaa was held which finally
decided the method of reckoning Easter. One thousand
two hundred and fifty-seven years had elapsed, each being
11 min. 14 sec. too long. The error of 10 days was
corrected by having the date following October 4th of that
year recorded as October 15. To prevent a recurrence of
the error, the Pope further decreed that -thereafter the
centurial years not divisible by 400 should not be counted
as leap years. Thus the years 1600, 2000, 2400, etc., are
leap years, but the years 1700, 1900, 2100, etc., are not
leap years. This calculation reduces the error to a very
low point, as according to the Gregorian calendar nearly
4000 years must elapse before the error amounts to a
single day.
The Gregorian calendar was soon adopted in all Roman
Catholic countries, France recording the date after Decem
ber 9th as December 20th. It was adopted by Poland in
1786, and by Hungary in 1787. Protestant Germany,
Denmark, and Holland adopted it in 1700 and Protestant
Switzerland in 1701. The Greek Catholic countries have
not yet adopted this calendar and are now thirteen days
behind our dates. Non-Christian countries have calendars
of their own.
In England and her colonies the change to the Gre
gorian system was effected in 1752 by having the date
following September 2d read September 14. The change
was violently opposed by some who seemed to think that
changing the number assigned to a particular day modi
fied time itself, and the members of the Government are

OLD STYLE AND NEW STYLE

137

SEPTEMBER. IX Month.

Shall Fruits, which none, but brutal Eyej tytvey.
Untouch'd grow npe. uniattcd drop away,'
Shall here ih* irrational, the falvage Kind V,
Lord it o'er Stores by Heavn for Man def.gn*d>
And trample whan mild Sam benignly raife, _
While Man mud lofe the Ufe. and Heav*n the fitiji'
Shall it then be.'" (Indignant here (he ro'e.
Indignant, yet humane, her fiofom glows) No'

Remark.days.{3lr.|o';''lOrf'II)pl.| Afpefls

55 47
5 49
5 S°
S 5
S 53
5 5*
i S
5 57
o" 6
6

'3
6 it
6 to

said to have heen mobbed in London by laborers who
cried " give us back our eleven days."
Old Style and New Style. Dates of events occurring
before this change
are usually kept as
they were then writ
ten, the letters o.s.
sometimes being
written after the
date to signify the
old style of dating.
To translate a date
into the Gregorian or
new style, one must
note the century in
which it occurred.
For example, Colum
bus discovered land
Oct. 12, 1492, o.s.
According to the
Gregorian calendar
a change of 10 days
was necessary in
1582. In 1500, leap
year was counted by
the old style but
should not have
been counted by the
new style. Hence,
in the century end
ing 1500, only 9 days
difference had been made. So the discovery of America
occurred October 12, o.s. or October 21, n.s. English

j| Wind 1 1 666
London burnt, t
and thudi luilb
Day break 4 tt.
c pall Trin.
ram,
Nat'Y V .Mas v
then cltat
St Matthiw1^
Days deer, a *6
and
6 pad Trin.
Holy Rood, Ivind,.'
Ember Week,
Plta/anl
St. Michael, 'weather.

o
59
5 57
5 5'
5 55
5 53
5 i>
S 5'
5 49

rt\ 1 gYTbe too obliging
t 17 *» 'if* 8 40
,1 3| I *"h Ij Itm-
i( Ijfet 10 ao fit
Unfe 11 ci
it evermort
di [obliging
iOi t'/ttf.
. Hold /due
Countil
O'nsS -i f 5
before Dinner;
7*s-'nfe 8 0
the full Bell j
batei Tbtntittg
6 S 5 o: it/ell

yt 7
_>9 '3
»5
X 7

r 3
»9
8 IJ
*7
11 1 1
25

leaood Di, ot Seftrmhr, tbi hid Count of Stflion and E.cheauei
.no.iii.tn Milken, F.ifi, and Mam otafoitfeld, and all Coum
nc.arni 01 oilonj,n[ iheielo, than be holden and kept upon, 01 ac-
roed.nglo 11, lame IMalor.l Dly>, upoo, or acceding 10 »hirh the
tanK Ihould Have b„n lo ken 01 holden, in cafe in,. Act had nor
been made , met ,i to fa., Eleven Day, Hie, than the fame would
ban happened, leeoidjn. to ,ha Nominal Dayi of the faid Ne*
Soppul.,.<>n olTin.e. b, »»,,>, the Commencement ofeatr, Month,
lei \,°m?i, D*" '""»'. ""anneipated .. btooet) for.itd,
ronii.i, ibneof ,n to, »,f, nolwiihrlinding.
Jlni ¦u.t,..., Hawaii., ,o d.vea, Oarlomt. Pe.fce,plloni. tnd U-
['•"••  » PL"" -'""» •*!» Klntdom, c.ilLTl not and
Gnwoa.., on,„;,tvl.a Nomlotl D.,t ,lt Tlmatin fhJ'HvJe'l.

Fig. 43. Page from Franklin's Almanac Showing
Omission of Eleven Days, 1752.

138 TIME AND THE CALENDAR 12
historians often write such dates October — , the upper date
referring to old style and the lower to new style.
A historian usually follows the dates in the calendar used
by his country at the time of the event. If, however, the
event refers to two nations having different calendars, both
dates are given. Thus, throughout Macaulay's " History
of England " one sees such dates as the following: Avaux,
r^2-?, 1689. (Vol. III.) A few dates in American his-
Aug. 6
tory prior to September, 1752, have been changed to agree
with the new style. Thus Washington was born Feb. 11,
1731, o.s, but we always write it Feb. 22, 1732. The
reason why all such dates are not translated into new
style is because great confusion would result, and, besides,
some incongruities would obtain. Thus the principal ship
of Columbus was wrecked Dec. 25, 1492, and Sir Isaac
Newton was born Dec. 25, 1642, and since in each case
this was Christmas, it -would hardly do to record them as
Christmas, Jan. 3, 1493, in the former instance, or as
Christmas, Jan. 4, 1643, in the latter case, as we should
have to do to write them in new style.
The Beginning of the Year. With the ancient Romans
the year had commenced with the March equinox, as
we notice in the names of the last months, September,
October, November, December, meaning 7th, 8th, 9th,
10th, which could only have those names by counting
back to March as the first month. By the time of Julius
Caesar the December solstice was commonly regarded as
the beginning of the year, and he confirmed the change,
making his new year begin January first. The later
Teutonic nations for a long time continued counting the
beginning of the year from March 25th. In 1563, by an

OLD STYLE IS STILL USED IN ENGLAND 139
edict of Charles IX, France changed the time of the begin
ning of the year to January first. In 1600 Scotland made
the same change and England did the same in 1752 when
the Gregorian system was adopted there. Dates between
the first of January and the twenty-fifth of March from
' 1600 to 1752 are in one year in Scotland and another year
in England. In Macaulay's " History of England " (Vol.
Ill, p. 258), he gives the following reference : "Act. Pari.
Scot., Mar. 19, 1689-90." The date being between Jan
uary 1st and March 25th in the interval between 1600 and
1752, it was recorded as the year 1689 in England, and a
year later, or 1690, in Scotland — Scotland dating the new
year from January 1st, England from March 25th. This
explains also why Washington's birthday was in 1731, o.s.,
and 1732, n.s., since English colonies used the same
system of dating as the mother country.
Old Style is still used in England's Treasury Depart
ment. " The old style is still retained in the accounts
of Her Majesty's Treasury. This is why the Christmas
dividends are not considered due until Twelfth Day, and
the midsummer dividends not till the 5th of July, and in
just the same way it is not until the 5th of April that
Lady Day is supposed to arrive. There is another piece
of antiquity in the public accounts. In the old times, the
year was held to begin on the 25th of March, and this
change is also still observed in the computations over
which the Chancellor of the Exchequer presides. The
consequence is, that the first day of the financial year is
the 5th of April, being old Lady Day, and with that day
the reckonings of our annual budgets begin and end." —
London Times* Feb. 16, 1861.
* Under the date of September 10, 1906, the same authority says
that the facts above quoted obtain in England at the present time.

140 TIME AND THE CALENDAR
Greek Catholic Countries Use Old Style. The Greek
Catholic countries, Russia, some of the Balkan states and
Greece, still employ the old Julian calendar which now,
with their counting 1900 as a leap year and our not
counting it so, makes their dates 13 days behind ours.
Dates in these countries recorded by Protestants or Roman
Catholics or written for general circulation are commonly
recorded in both styles by placing the Gregorian date
under the Julian date. For example, the date we cele
brate as our national holiday would be written by an
American in Russia as T , - . The day we commem-
July 4 J 12
orate as the anniversary of the birth of Christ, Dec. — ; the
day they commemorate, — :— : — ^t^ttt^- It should be
J Jan. 7, 1907
remembered that if the date is before 1900 the differ
ence will be less than thirteen days. Steps are being
taken in Russia looking' to an early revision of the
calendar. Mohammedan and Jewish Calendars. The old system
employed before the time of the Caesars is still used by
the Mohammedans and the Jews. The year of the former
is the lunar year of 354J,j days, and being about .03 of
a year too short to correspond with the solar year, the
same date passes through all seasons of the year in the
course of 33 years. Their calendar dates from the year
of the Hegira, or the flight of Mohammed, which occurred
July, 622 a.d. If their year was a full solar year, their
date corresponding to 1900 would be €22 years less than
that number, or 1278, but being shorter in length there are
more of them, and they write the date 1318, that year
beginning with what to us was May 1. That is to say,

CHALDEAN CALENDAR 141
what we called May 1, 1900, they called the first day of
their first month, Muharram, 1318.
Chinese Calendar. The Chinese also use a lunar calen
dar; that is, with months based upon the phases of the
moon, each month beginning with a new moon. Their
months consequently have 29 and 30 days alternately.
To correct the error due to so short a year, seven out of
every nineteen years have thirteen months each. This
still leaves the average year too short, so in every cycle of
sixty years, twenty-two extra months are intercalated.
Ancient Mexican Calendar. The ancient Mexicans had a
calendar of 18 months of 20 days each and five additional
days, with every fourth year a leap year. Their year began
with the vernal equinox.
Chaldean Calendar. Perhaps the most ancient calendar
of which we have record, and the one which with modifi
cations became the basis of the Roman calendar which we
have seen was handed down through successive genera
tions to us, was the calendar of the Chaldeans. Long
before Abraham left Ur of the Chaldees (see Genesis xi,
31; Nehemiah ix, 7, etc.) that city had a royal observatory,
and Chaldeans had made subdivisions of the celestial sphere
and worked out the calendar upon which ours is based.
Few of us can fail to recall how hard fractions were
when we first studied them, and how we avoided them in
our calculations as much as possible. For exactly the
same reason these ancient Chaldeans used the number
60 as their unit wherever possible, because that number
being divisible by more numbers than any other less than
100, its use and the use of any six or a multiple of six
avoided fractions. Thus they divided circles into 360
degrees (6 X 60), each degree into 60 minutes, and each
minute into 60 seconds. They divided the zodiac into

142

TIME AND THE CALENDAR

spaces of 30° each, giving us the plan of twelve months
in the year. Their divisions of the day led to our 24
hours, each having 60 minutes, with 60 seconds each.
They used the week of seven days, one for each of the
heavenly bodies that were seen to move in the zodiac.
This origin is suggested in the names of the days of the
week.
Days of the Week

ModernEnglish

Celestial Origin

Roman

Modern French.

Ancient Saxon

ModernGerman

1. Sunday

Sun

Dies Solis

Dimanche

Sunnan-daeg

Sonntag

2. Monday

Moon

Dies LunsB

Lundi

Monan-daeg

Montag

3. Tuesday

Mars

Dies Martis

Mardi

Mythical God
Tiew or Tuesco
Tues-daeg

Dienstag

4. Wednesday

Mercury

Dies Mercurii

Mercredi

"Woden
Woden' s-daeg

(Mid-week) Mittwoche

5. Thursday

Jupiter

Dies Jovis

Jeudi

Thor (thunderer)
Thors-daeg

Doimerstag

6. Friday

Venus

Dies Veneris

VeDdredi

Friga
Frigedaeg

Freitag

7. Saturday

Saturn

Dies Saturni

Samedi

Saeter-daeg

Sams tag or
Sonnabend

Complex Calendar Conditions in Turkey. " But it is in
Turkey that the time problem becomes really complicated,
very irritating to him who takes it seriously, very funny
to him who enjoys a joke. To begin with, there are four
years in Turkey — a Mohammedan civil year, a Moham
medan religious year, a Greek or Eastern year, and a Euro
pean or Western year. Then in the year there are both
lunar months depending on the changes of the moon, and
months which, like ours, are certain artificial proportions
of the solar year. Then the varieties of language in

COMPLEX CALENDAR CONDITIONS IN TURKEY 143

Turkey still further complicate the calendars in custom
ary use. I brought away with me a page from the diary
which stood on my friend's library table, and which is
customarily sold in Turkish shops to serve the purpose of
a calendar; and I got from my friend the meaning of the
hieroglyphics, which I record here as well as I can remem
ber them. This page represents one day. Numbering the
compartments in it from left to right, it reads as follows:

1. March, 1318 (Civil Year).
2. March, 1320 (Religious
Year).
3. Thirty-one days (Civil
Year).
4. Wednesday.
5. Thirty days (Religious
Year).
6. 27 (March: Civil Year).
7. (March: Religious Year.)
8. March, Wednesday (Ar
menian).
9. April, Wednesday(French)
10. March, Wednesday(Greek)
11. Ecclesiastical Day (French
R. C. Church).
12. March, Wednesday (Rus
sian).
13. Month Day (Hebrew).
14. Month Day (Old Style).
15. Month Day (New Style).
16. Ecclesiastical Day (Ar
menian).
17. Ecclesiastical Day(Greek)
18. Midday, 5:35, 1902; Mid
day, 5:21.

'T'A^rU j££irr-

: i *

rvn

prs-anrt*tUFi»h

MAPTIOi- TETAP.

MAPTT, CPtiAA

279
IJ3.. tap JW ajieeXeaJe

AVRIL - MERCR

S. Hugues-

IP'S £ cbipyr

MaVrpiiyT); trtt&rj T7J{ Jv 6l(OxXo>(x1)

>atu or 5. 35 am 1902 IMidi 5 21 Jr"30
Fig. 44

" I am not quite clear in my mind now as to the meaning
of the last section, but I think it is that noon according to
European reckoning, is twenty-one minutes past five accord-

144 TIME AND THE CALENDAR
ing to Turkish reckoning. For there is in Turkey, added
to the complication of year, month, and day, a further
complication as to hours. The Turks reckon, not from an
artificial or conventional hour, but from sunrise, and their
reckoning runs for twenty-four hours. Thus, when the
sun rises at 6 : 30 our noon will be 5 : 30, Turkish time. The
Turkish hours, therefore, change every day.' The steam
ers on the Bosphorus run according to Turkish time, and
one must first look in the time-table to see the hour, and
then calculate from sunrise of the day what time by his
European clock the boat will start. My friends in Turkey
had apparently gotten used to this complicated calendar,
with its variable years and months and the constantly
changing hours, and took it as a matter of course." *
Modern Jewish Calendar. The modern Jewish calendar
employs also a lunar year, but has alternate years length
ened by adding extra days to make up the difference
between such year and the solar year. Thus one year
will have 354 days, and another 22 or 23 days more.
Sept. 23, 1900, according to our calendar, was the begin
ning of their year 5661.
Many remedies have been suggested for readjusting
our calendar so that the same date shall always recur on
the same day of the week. While it is interesting for the
student to speculate on the problem and devise ways of
meeting the difficulties, none can be suggested that does
not involve so many changes from our present system that
it will be impossible for a long, long time to overcome
social inertia sufficiently to accomplish a reform.
If the student becomes impatient with the complexity
of the problem, he may recall with profit these words of
* The Impressions of a Careless Traveler, by Lyman Abbott. — The
Outlook, Feb. 28, 1903.

TOPICS FOR SPECIAL REPORTS 145
John Fiske: "It is well to simplify things as much as
possible, but this world was not so put together as to save
us the trouble of using our wits."
Three Christmases in One Year. " Bethlehem, the
home of Christmases, is that happy Utopia of which every
American child dreams — it has more than one Christmas.
In fact, it has three big ones, and, strangely enough, the
one falling on December 25th of our calendar is not the
greatest of the three. It is, at least, the first. Thirteen
days after the Latin has burned his Christmas incense in
the sacred shrine, the Greek Church patriarch, observing
that it is Christmas-time by his slower calendar, catches up
the Gloria, and bows in the Grotto of the Nativity for the
devout in Greece, the Balkan states, and all the Russias.
After another period of twelve days the great Armenian
Church of the East takes up the anthem of peace and
good-will, and its patriarch visits the shrine." *
Topics for Special Reports. The gnomon. The clep
sydra. Other ancient devices for reckoning time. The
week. The Metonic cycle and the Golden Number. The
calculation of Easter. The Roman calendar. Names of
the months and days of the week. Calendar reforms.
The calendar of the French Revolution. The Jewish
calendars. The Turkish calendar.
* Ernest I. Lewis in Woman's Home Companion, December, 1903.

.TO. MATH. ORO.- 10

CHAPTER VIII
SEASONS
Vertical and Slanting Rays of the Sun. He would be
unobservant, indeed, who did not know from first-hand
experience that the morning and evening rays of the sun
do not feel so warm as those of midday, and, if living out
side the torrid zone, that rays from the low winter sun in
some way lack the heating power of those from the high
summer sun. The reason for this difference may not be
so apparent. The vertical rays are not warmer than the
slanting ones, but the more nearly vertical the sun, the
more heat rays are intercepted by a given surface. If
you place a tub in the rain and tip it so that the rain falls
in slantingly, it is obvious that less water will be caught
than if the tub stood at right angles to the course of the
raindrops. But before we take up in detail the effects of
the shifting rays of the sun, let us carefully examine the
conditions and causes of the shifting.
Motions of the Earth. The direction and rate of the
earth's rotation are ascertained from the direction and
rate of the apparent rotation of the celestial sphere. The
direction and rate of the earth's revolution are ascertained
from the apparent revolution of the sun among the stars
of the celestial sphere. Just as any change in the rotation
of the earth would produce a corresponding change in the
apparent rotation of the celestial sphere, so any change
in the revolution of the earth would produce a correspond
ing change in the apparent revolution of the sun.
Were the sun to pass among the stars at right angles to
146

EQUINOXES 147
the celestial equator, passing through the celestial poles,
we should know that the earth went around the sun in a
path whose plane was perpendicular to the plane of the
equator and was in the plane of the axis. In such an
event the sun at some time during the year would shine
vertically on each point on the earth's surface. Seasons
would be nearly the same in one portion of the earth
as in another. The sun would sometimes cast a north
shadow at any given place and sometimes a south shadow.
Were the sun always in the celestial equator, the ecliptic
coinciding with it, we should know that the earth traveled
around the sun at right angles to the axis. The vertical
ray of the sun would then always be overhead at noon on
the equator, and no change in season would occur. Were
the plane of the earth's orbit at an angle of 45° from the
equator the ecliptic would extend half way between the
poles and the equator, and the sun would at one time get
within 45° of the North star and six months later 45°
from the South star. The vertical ray on the earth would
then travel from 45° south latitude to 45° north latitude,
and the torrid zone would be 90° wide.
Obliquity of the Ecliptic. But we know that the vertical
ray never gets farther north or south of the equator than
about 23^°, or nearer the poles than about 66J°. The
plane of the ecliptic or of the earth's orbit is, then, inclined
at an angle of 66J° to the axis, or at an angle of 23J° to
the plane of the equator. This obliquity of the ecliptic
varies slightly from year to year, as is shown on
pp. 118, 288.
Equinoxes. The sun crosses the celestial equator twice
a year, March 20 or 21, and September 22 or 23,* varying
* The reason why the date shifts lies in the construction of our
calendar, which must fit a year of 365 days, 5 h. 48 m. 45.51 s. The time

148 SEASONS
from year to year, the exact date for any year being easily
found by referring to any almanac. These dates are
called equinoxes (equinox; cequus, equal; nox, night), for
the reason that the days and- nights are then twelve
hours long everywhere on earth. March 21 is called
the vernal (spring) equinox, and September 23 is called
the autumnal equinox, for reasons obvious to those
who live in the northern hemisphere (see Equinox in
Glossary). Solstices. About the time when the sun reaches its most
distant point from the celestial equator, for several days it
seems neither to recede from it nor to approach it. The
dates when the sun is at these two points are called the
solstices (from sol, sun; and stare, to stand). June 21 is
the summer solstice, and December 22 is the winter solstice;
vice versa for the southern hemisphere. The same terms
are also applied to the two points in the ecliptic farthest
from the equator ; that is, the position of the sun on those
dates. At the Equator. March 21. Imagine you are at the
equator March 21. Bear in mind the fact that the North
star (strictly speaking, the north pole of the celestial
sphere) is on the northern horizon, the South star on the
southern horizon, and the celestial equator extends from
due east, through the zenith, to due west. It is sunrise of
the vernal equinox. The sun is seen on the eastern hori
zon ; the shadow it casts is due west and remains due west
until noon, getting shorter and shorter as the sun rises
higher.of the vernal equinox in 1906 was March 21, 7: 46 a.m., Eastern
standard time. In 1907 it occurred 365 days, 5 h. 48 m. 45.51 s. later, or
at 1 : 35 p.m., March 21. In 1908, being leap year, it will occur 366 days,
5 h. 48 m. 45.51 s. later, or at about 7: 24 p.m., March 20. The same facts
are true of the solstices; they occur June 21-22 and December 22-23.

AT THE EQUATOR 149
Shadows. At noon the sun, being on the celestial
equator, is directly overhead and casts no shadow, or the
shadow is directly underneath. In the afternoon the

Fig. 45. Illumination of the earth in twelve positions, corresponding to months.
The north pole is turned toward us.
shadow is due east, lengthening as the sun approaches
the due west point in the horizon. At this time the sun's
rays extend from pole to pole. The circle of illumination,
that great circle separating the lighted half of the earth
from the half which is turned away from the sun, since it

150 SEASONS
extends at this time from pole to pole, coincides with a
meridian circle and bisects each parallel. Half of each
parallel being in the light and half in the dark, during
one rotation every point will be in the light half a day and
away from the sun the other half, and day and night are
equal everywhere on the globe.
After March 21 the sun creeps back in its orbit, gradu
ally, away from the celestial equator toward the North
star. At the equator the sun thus rises more and more
toward the north of the due east point on the horizon,
and at noon casts a shadow toward the south. As the
sun gets farther from the celestial equator, the south noon
shadow lengthens, and the sun rises and sets farther
toward the north of east and west.
On June 21 the sun has reached the point in the eclip
tic farthest from the celestial equator, about 23J° north.
The vertical ray on the earth is at a corresponding distance
from the equator. The sun is near the constellation
Cancer, and the parallel marking the turning of the sun
from his course toward the polestar is called the Tropic
(from a Greek word meaning turning) of Cancer. Our
terrestrial parallel marking the southward turning of the
vertical ray is also called the Tropic of Cancer. At this
date the circle of illumination extends 23J° beyond the
north pole, and all of the parallels north of 66J° from the
equator are entirely within this circle of illumination and
have daylight during the entire rotation of the earth. At
this time the circle of illumination cuts unequally parallels
north of the equator so that more than half of them are in
the lighted portion, and hence days are longer than nights
in the northern hemisphere. South of the equator the
conditions are reversed. The circle of illumination does
not extend so far south as the south pole, but falls short

AT THE EQUATOR 151
of it 23£°, and consequently all parallels south of 66|° are
entirely in the dark portion of the earth, and it is con
tinual night. Other circles south of the equator are so
intersected by the circle of illumination that less than
half of them are in the lighted side of the earth, and the
days are shorter than the nights. It is midwinter there.
After June 21 gradually the sun creeps along in its
orbit away from this northern point in the celestial sphere
toward the celestial equator. The circle of illumination
again draws toward the poles, the days are more nearly
of the same length as the nights, the noon sun is more
nearly overhead at the equator again, until by September
23, the autumnal equinox, the sun is again on the celestial
equator, and conditions are exactly as they were at the
March equinox.
After September 23 the sun, passing toward the South
star from the celestial equator, rises to the south of a due
east line on the equator, and at noon is to the south of the
zenith, casting a north shadow. The circle of illumina
tion withdraws from the north pole, leaving it in darkness,
and extends beyond the south pole, spreading there the
glad sunshine. Days grow shorter north of the equator, less
than half of their parallels being in the lighted half, and
south of the equator the days lengthen and summer comes.
On December 22 the sun has reached the most distant
point in the ecliptic from the celestial equator toward the
South star, 23J° from the celestial equator and 66J° from
the South star, the vertical ray on the earth being at corre
sponding distances from the equator and the south pole.
The sun is now near the constellation Capricorn, and every
where within the tropics the shadow is toward the north;
on the tropic of Capricorn the sun is overhead at noon,
and south of it the shadow is toward the south. Here

152 SEASONS
the vertical ray turns toward the equator again as the sun
creeps in the ecliptic toward the celestial equator.
Just as the tropics are the parallels which mark the
farthest limit of the vertical ray from the equator, the
polar circles are the parallels marking the farthest extent
of the circle of illumination beyond, the poles, and are the
same distance from the poles that the tropics are from the
equator. The Width of the Zones is thus determined by the dis
tance the vertical ray travels on the earth, and with the
moving of the vertical ray, the shifting of the day circle.
This distance is in turn determined by the angle which
the earth's orbit forms with the plane of the equator. The
planes of the equator and the orbit forming an angle of
23J°, the vertical ray travels that many degrees each side
of the equator, and the torrid zone is 47° wide. The circle
of illumination never extends more than 23J° beyond each
pole, and the frigid zones are thus 23^° wide. The remain
ing or temperate zones between the torrid and the frigid
zones must each be 43° wide.
At the North Pole. Imagine you are at the north pole.
Bear in mind the fact that the North star is always almost
exactly overhead and the celestial equator always on the
horizon. On March 21 the sun is on the celestial equa
tor and hence on the horizon.* The sun now swings
around the horizon once each rotation of the earth, casting
long shadows in every direction, though, being at the
north pole, they are always toward the south, f After the
* Speaking exactly, the sun is seen there before the spring equinox
and after the autumnal equinox, owing to refraction and the dip of the
horizon. See p. 160.
t The student should bear in mind the fact that directions on the
earth are determined solely by reference to the true geographical
pole, not the magnetic pole of the mariner's compass. At the north

AT THE NORTH POLE 153
spring equinox, the sun gi'adually rises higher and higher
in a gently rising spiral until at the summer solstice, June
21, it is 23£° above the horizon. After this date it gradu
ally approaches the horizon again until, September 23,
the autumnal equinox, it is exactly on the horizon, and
after this date is seen no more for six months. Now the
stars come out and may be seen perpetually tracing their
circular courses around the polestar. Because of the reflec
tion and refraction of the rays of light in the air, twilight
prevails when the sun is not more than about 18° below
the horizon, so that for only a small portion of the six
months' winter is it dark, and even then the long journeys
of the moon above the celestial equator, the bright stars
that never set, and the auroras, prevent total darkness
(see p. 164). On December 22 the sun is 23£° below the
horizon, after which it gradually approaches the horizon
again, twilight soon setting in until March 21 again shows
the welcome face of the sun.
At the South Pole the conditions are exactly reversed.
There the sun swings around the horizon in the opposite
direction; that is, in the direction opposite the hands of a
watch when looked at from above. The other half of the
celestial sphere from that seen at the north pole is always
above one, and no stars seen at one pole are visible at the
other pole, excepting the few in a very narrow belt around
the celestial equator, lifted by refraction of light.
pole the compass points due south, and at points between the magnetic
pole and the geographical pole it may point in any direction excepting
toward the north. Thus Admiral A. H. Markham says, in the Youth's
Companion for June 22, 1902:
" When, in 1876, I was sledging over the frozen sea in my endeavor
to reach the north pole, and therefore traveling in a due north direc
tion, I was actually steering by compass E. S. E., the variation of the
compass in that locality varying from ninety-eight degrees to one
hundred and two degrees westerly."

154 SEASONS
Parallelism of the Earth's Axis. Another condition of
the earth in its revolution should be borne in mind in
explaining change of seasons. The earth might rotate on
an axis and revolve around the sun with the axis inclined
23J° and still give us no change in seasons. This can
easily be demonstrated by carrying a globe around a cen
tral object representing the sun, and by rotating the axis
one can maintain the same inclination but keep the verti
cal ray continually at the equator or at any other circle
within the tropics. In order to get the shifting of the
vertical ray and change of seasons which now obtain, the
axis must constantly point in the same direction, and its
position at one time be parallel to its position at any other
time. This is called the parallelism of the earth's axis.
That the earth's axis has a very slow rotary motion, a
slight periodic " nodding " which varies its inclination
toward the plane of the ecliptic, and also irregular motions
of diverse character, need not confuse us here, as they are
either so minute as to require very delicate observations
to determine them, or so slow as to require many, years to
show a change. These three motions of the axis are dis
cussed in the Appendix under " Precession of the Equi
noxes," " Nutation of the Poles," and " Wandering of the
Poles" (p. 286).
Experiments with the Gyroscope. The gyroscope, prob
ably familiar to most persons, admirably illustrates the
causes of the parallelism of the earth's axis. A disk, sup
ported in a ring, is rapidly whirled, and the rotation tends
to keep the axis of the disk always pointing in the same
direction. If the ring be held in the hands and carried
about, the disk rapidly rotating, it will be discovered that
any attempt to change the direction of the axis will meet
with resistance. This is shown in the simple fact that a

DAY'S LENGTH AT THE EQUINOXES 155
rapidly rotating top remains upright and is not easily
tipped over; and, similarly, a bicycle running at a rapid
rate remains erect, the rapid motion of the wheel (or
top) giving the axis a tendency to remain in the same
plane. The gyroscope shown in Figure 46 * is one used by
Professor R. S. Holway of the University of California.
It was made by mounting a six-inch sewing machine wheel
on ball bearings in the fork of an old bicycle. Its advan-

Fig. 46

tages over those commonly used are its simplicity, the
ball bearings, and its greater weight.
Foucault Experiment. In 1852, the year after his
famous pendulum experiment, demonstrating the rotation
of the earth, M. Leon Foucault demonstrated the same
facts by means of a gyroscope so mounted that, although
the earth turned, the axis of the rotating wheel remained
constantly in the same direction.
Comparative Length of Day and Night
Day's Length at the Equinoxes. One half of the earth
being always in the sunlight, the circle of illumination is a
great circle. The vertical ray marks the center of the
lighted half of the surface of the earth. At the equinoxes
* Taken, by permission, from the Journal of Geography for February,
1904.

156 SEASONS
the vertical ray is at the equator, and the circle of illumi
nation extends from pole to pole bisecting every parallel.
Since at this time any given parallel is cut into two equal
parts by the circle of illumination, one half of it is in the
sunlight,. and one half of it is in darkness, and during one
rotation a point on a parallel will have had twelve hours
day and twelve hours night. (No allowance is made for
refraction or twilight.)
Day's Length after the Equinoxes. After the vernal
equinox the vertical ray moves northward, and the circle
of illumination extends beyond the north pole but falls
short of the south pole. Then all parallels, save the
equator, are unequally divided by the circle of illumination,
for more than half of each parallel north of the equator
is in the light, and- more than half of each parallel south
of the equator is in darkness. Consequently, while the
vertical ray is north of the equator, or from March 21 to
September 23, the days are longer than the nights north
of the equator, but are shorter than the nights south of
the equator.
During the other half of the year, when the vertical ray
is south of the equator, these conditions are exactly
reversed. The farther the vertical ray is from the equator,
the farther is the circle of illumination extended beyond
one pole and away from the other pole, and the more
unevenly are the parallels divided by it; hence the days
are proportionally longer in the hemisphere where the
vertical ray is, and the nights longer in the opposite hemi
sphere. The farther from the equator, too, the greater
is the difference, as may be observed from Figure 50,
page 162. Parallels near the equator are always nearly
bisected by the circle of illumination, and hence day
nearly equals night there the year around.

LONGEST DAYS AT DIFFERENT LATITUDES 157
Day's Length at the Equator. How does the length of
day at the equator compare with the length of night?
When days are shorter south of the equator, if they are
longer north of it. and vice versa, at the equator they must
be of the same length. The equator is always bisected
by the circle of illumination, consequently half of it is
always in the sunlight. This proposition, simple though
it is, often needs further demonstration to be seen clearly.
It will be obvious if one sees:
(a) A point on a sphere 180° in any direction from a
point in a great circle lies in the same circle.
(b) Two great circles on the same sphere must cross
each other at least once.
(c) A point 180° from this point of intersection common
to both great circles, will lie in each of them, and hence
must be a point common to both and a point of inter
section. Hence two great circles, extending in any
direction, intersect each other a second time 180° from
the first point of crossing, or half way around. The circle
of illumination and equator are both great circles and
hence bisect each other. If the equator is always bisected
by the circle of illumination, half of it must always be in
the light and half in the dark.
Day's Length at the Poles. The length of day at the
north pole is a little more than six months, since it extends
from March 21 until September 23, or 186 days. At the
north pole night extends from September 23 until March
21, and is thus 179 days in length. It is just opposite at
the south pole, 179 days of sunshine and 186 days of
twilight and darkness. This is only roughly stated in full
days, and makes no allowance for refraction of light or
twilight. Longest Days at Different Latitudes. The length of the

158

SEASONS

longest day, that is, from sunrise to sunset, in different
latitudes is as follows:

Lat.

Day

Lat.

Day

Lat.

Day

Lat.

Day

0°

12 h.

25°

13 h. 34 m.

50°

16 h.

9 m.

70°

65 days

5U

12 h. 17 m.

30°

13 h. 56 m.

55°

17 h.

7 m.

75°

103 "

10°

12 h. 35 m.

35°

14 h. 22 m.

60°

18 h.

30 m.

80°

134 "

15°

12 h. 53 m.

40°

14 h. 51m.

65°

21 h.

09 m.

85°

161 "

20°

13 h. 13 m.

45°

15 h. 26 m.

66° 33'

24 h.

00 m.

90°

6 mos.

The foregoing table makes no allowance for the fact
that the vertical ray is north of the equator for a longer
time than it is south of the equator, owing to the fact that
we are farther from the sun then, and consequently the
earth revolves more slowly in its orbit. No allowance is
made for refraction, which lifts up the rays of the sun
when it is near the horizon, thus lengthening days every
where.
Refraction of Light
The rays of light on entering the atmosphere are bent

Fig. 47

out of straight courses. Whenever a ray of light enters
obliquely a medium of greater or of less density, the ray
is bent out of its course (Fig. 47). Such a change in

AMOUNT OF REFRACTION VARIES

159

direction is called refraction. When a ray of light enters
obliquely a medium of greater density, as in passing
through from the upper rarer atmosphere to the lower
denser layers, or from air into water, the rays are bent in
the direction toward a perpendicular to the surface or less
obliquely. This is called the first law of refraction. The
second law of refraction is the converse of this; that is, on
entering a rarer medium the ray is bent more obliquely
or away from a perpendicular to the surface. When a

BSEgatsi!8SMWi«j »*-<<»¦«¦

|i»«iiwv#«%'.

Fig. 48

ray of light from an object strikes the eye, we see the
object in the direction taken by the ray as it enters the
eye, and if the ray is refracted this will not be the real
position of the object. Thus a fish in the water (Fig. 48)
would see the adjacent boy as though the boy were nearly
above it, for the ray from the boy to the fish is bent
downwards, and the ray as it enters the eye of the fish
seems to be coming from a place higher up.
Amount of Refraction Varies. The amount of refrac
tion depends upon the difference in the density of the

160

SEASONS

media and the obliqueness with which the rays enter.
Rays entering perpendicularly are not refracted at all.
The atmosphere differs very greatly in density at different
altitudes owing to its weight and elasticity. About one
half of it is compressed within three miles of the surface
of the earth, and at a height of ten miles it is so rare that
sound can scarcely be transmitted through it. A ray of
light entering the atmosphere obliquely is thus obliged to
traverse layers of air of increasing density, and is refracted
more and more as it approaches the earth.
Effect of Refraction on Celestial Altitudes. Thus, refrac
tion increases the apparent altitudes of all celestial objects
excepting those at the
zenith (Fig. 49). The
amount of refraction at
the horizon is ordinarily
36' 29"; that is to say,
a star seen on the hori
zon is in reality over
one half a degree below
the horizon. The ac
tual amount of refrac
tion varies with the temperature, humidity, and pressure
of the air, all of which affect its density and which must
be taken into consideration in accurate calculations.
Since the width of the sun as seen from the earth is about
32', when the sun is seen just above the horizon it actually
is just below it, and since the sun passes one degree in
about four minutes, the day is thus lengthened about four
minutes in the latitudes of the United States and more in
higher latitudes. This accounts for the statement in alma
nacs as to the exact length of the day at the equinoxes.
Theoretically the day is twelve hours long then, but prac-

Fig. 49

REFRACTION ON CELESTIAL ALTITUDES

161

tically it is a few minutes longer. Occasionally there is
an eclipse of the moon observed just before the sun has
gone down. Tho earth is exactly between the sun and the
moon, but because of refraction, both sun and moon are
seen above the horizon.
The sun and moon often appear flattened when near
the horizon, especially when seen through a haze. This
apparent flattening is due to the fact that rays from the
lower portion are more oblique than those from the upper
portion, and hence it is apparently lifted up more than
the upper portion. Mean Refraction Table
(For Temperature 50° Fahr., barometric pressure 30 in.)

Apparent

Mean Re

Apparent

Mean Re

Apparent

Mean Re

Altitude.

fraction.

Altitude.

fraction.

Altitude.

fraction. .
•4

0°

36' 29.4"

8°

6' 33.3"

26°

V 58.9"

1

24 53.6

9

5 52.6

30

1 40.6

2

18 25.5

10

5 19.2

40

1 9.4

3

14 25.1

12

4 27.5

50

0 48\9

4

11 44.4

14

3 49.5

60

0 33.6

5

9 52.0

16

3 20.5

70

0 21.2

6

8 28.0

18

2 57.5

80

0 10.3

7

7 23.8

22

2 23.3

90

0 00.0

Twilight
The atmosphere has the peculiar property of reflecting
and scattering the rays of light in every direction. Were
not this the case, no object would be visible out of the
direct sunshine, shadows would be perfectly black, our
houses, excepting where the sun shone, would be perfectly
dark, the blue sky would disappear and we could see the
stars in the day time just as well as at night. Because
of this diffusion of light, darkness does not immediately
set in after sunset, for the rays shining in the upper air
JO. MATH. GEO. — 11

162

SEASONS

are broken up and reflected to the lower air. This, in
brief, is the explanation of twilight. There being practi
cally no atmosphere on. the moon there is no twilight
there. These and other consequences resulting from the
lack of an atmospheric envelope on the moon are described
on pp. 263, 264.
Length of Twilight. Twilight is considered to last while
the sun is less than about 18° below the horizon, though
the exact distance varies somewhat with the condition of
the atmosphere, the latitude, and the season of the year.
There is thus a
twilight zone im
mediately beyond
the circle of illumi
nation, and outside
of this zone is the
true night. Figure
50 represents these
three portions :(1)
the hemisphere re
ceiving direct rays (slightly more than a hemisphere owing
to refraction), (2) the belt 18° from the circle of illumina--
tion, and (3) the segment in darkness — total save for
starlight or moonlight. The height of the atmosphere is,
of course, greatly exaggerated. The atmosphere above
the line AB receives direct rays of light and reflects
and diffuses them to the lower layers of atmosphere.
Twilight Period Varies with Season. It will be seen from
Figure 50 that the fraction of a parallel in the twilight
zone varies greatly with the latitude and the season. At
the equator the sun drops down at right angles to the
18
horizon, hence covers the 18° twilight zone in — —
oOU

Fig. 50

of a

TWILIGHT NEAR THE EQUATOR 163
day or one hour and twelve minutes. This remains prac
tically the same the year around there. In latitudes of
the United States, the twilight averages one and one-half
hours long, being greater in midsummer. At the poles,
twilight lasts about two and one-half months.
Twilight Long in High Latitudes. The reason why the
twilight lasts so long in high latitudes in the summer will
be apparent if we remember that the sun, rising north of
east, swinging slantingly around and setting to the north
of west, passes through the twilight zone at the same
oblique angle. At latitude 48° 33' the sun passes around
so obliquely at the summer solstice that it does not sink 18°
below the horizon at midnight, and stays within the twi
light zone from sunset to sunrise. At higher latitudes on
that date the sun sinks even less distance below the
horizon. For example, at St. Petersburg, latitude 59° 56'
30", the sun is only 6° 36' 25" below the horizon at mid
night June 21 and it is light enough to read without
artificial light. From 66° to the pole the sun stays
entirely above the horizon throughout the entire summer
solstice, that being the boundary of the " land of the mid
night sun."
Twilight Near the Equator. " Here comes science now
taking from us another of our cherished beliefs — the wide
superstition that in the tropics there is almost no twilight,
and that the ' sun goes down like thunder out o' China
'crosst the bay.' Every boy's book of adventure tells of
travelers overtaken by the sudden descent of night, and
men of science used to bear out these tales. Young, in
his ' General Astronomy,' points out that 'at Quito the
twilight is said to be at best only twenty minutes.' In
a monograph upon ' The Duration of Twilight in the
Tropics,' S. I. Bailey, points out, by carefully verified

164 SEASONS
observation and experiments, that the tropics have their
fair share of twilights. He says: 'Twilight may be said
to last until the last bit of illuminated sky disappears from
the western horizon. In general it has been found that
this occurs when the sun has sunk about eighteen degrees
below the horizon. . . . Arequipa, Peru, lies within the
tropics, and has an elevation of 8,000 feet, and the air is
especially pure and dry, and conditions appear to be
exceptionally favorable for an extremely short twilight.
On Sunday, June 25, 1899, the following observations
were made at the Harvard Astronomical Station, which is
situated here: The sun disappeared at 5:30 p.m., local
mean time. At 6 p.m., thirty minutes after sunset, I
could read ordinary print with perfect ease. At 6 : 30 p.m.
I could see the time readily by an ordinary watch. At
6:40 p.m., seventy minutes after sunset, the illuminated
western sky was still bright enough to cast a faint shadow
of an opaque body on a white surface. At 6:50 p.m.,
one hour and twenty minutes after sunset, it had dis
appeared. On August 27, 1899, the following observa
tions were made at Vincocaya. The latitude of this place
is about sixteen degrees south and the altitude 14,360
feet. Here it was possible to read coarse print forty-
seven minutes after sunset, and twilight could be seen for
an hour and twelve minutes after the sun's disappearance.'
So the common superstition about no twilight in the
tropics goes to join the William Tell myth." — • Harper's
Weeekly, April 5, 1902.
Twilight Near the Pole. " It may be interesting to re
late the exact amount of light and darkness experienced
during a winter passed by me in the Arctic regions within
four hundred and sixty miles of the Pole.
" From the time of crossing the Arctic circle until we

VERTICAL RAYS AND INSOLATION 165
established ourselves in winter quarters on the 3d of
September, we rejoiced in one long, continuous day. On
that date the sun set below the northern horizon at mid
night, and the daylight hours gradually decreased until
the sun disappeared at noon below the southern horizon
on the 13th of October.
" From this date until the 1st of March, a period of one
hundred and forty days, we never saw the sun; but it
must not be supposed that because the sun was absent we
were living in total darkness, for such was not the case.
During the month following the disappearance of the sun,
and for a month prior to" its return, we enjoyed for an
hour, more or less, on either side of noon, a glorious twi
light; but for three months it may be said we lived in
total darkness, although of course on fine days the stars
shone out bright and clear, rendered all the more brilliant
by the reflection from the snow and ice by which we were
surrounded, while we also enjoyed the light from the moon
in its regular lunations.
" On the 21st of December, the shortest day in the year,
the sun at our winter quarters was at noon twenty degrees
below the horizon. I mention this because the twilight
circle, or, to use its scientific name, the crepusculum, when
dawn begins and twilight ends, is determined when the
sun is eighteen degrees below the horizon.
" On our darkest day it was not possible at noon to read
even the largest-sized type." — Admiral A. H. Markham,
R. N., in the Youth's Companion, June 22, 1899.
Effect of the Shifting Rays of the Sun.
Vertical Rays and Insolation. The more nearly vertical
the rays of the sun are the greater is the amount of heat
imparted to the earth at a given place, not because a verr

166

SEASONS

tical ray is any warmer, but because more rays fall over
a given area. In Figure 51 we notice that more perpen
dicular rays extend over a given area than slanting ones.

z
1 Sd ¦(¦¦>,

Fig. 51
We observe the morning and evening rays of the sun,
even when falling perpendicularly upon an object, say
through a convex lens or burning glass, are not so
warm as those at midday. The reason is apparent
from Figure 52, the
slanting rays tra
verse through more
of the atmosphere.
At the summer
solstice the sun's
rays are more nearly
vertical over Europe
and the United K 52
States than at other
times. In addition to the greater amount of heat received
because of the less oblique rays, the days are longer than

MAXIMUM HEAT FOLLOWS SUMMER SOLSTICE 167
nights and consequently more heat is received during the
day than is radiated off at night. This increasing length
of day time greatly modifies the climate of regions far to
the north. Here the long summer days accumulate enough
heat to mature grain crops and forage plants. It is inter
esting to note that in many northern cities of the United
States the maximum temperatures are as great as in some
southern cities.
How the Atmosphere is Heated. To understand how
the atmosphere gets its heat we may use as an illustration
the peculiar heat-receiving and heat-transmitting proper
ties of glass. We all know that glass permits heat rays
from the sun to pass readily through it, and that the dark
rays of heat from the stove or radiator do not readily pass
through the glass. Were it not for this fact it would be
no warmer in a room in the sunshine than in the shade,
and if glass permitted heat to escape from a room as
readily as it lets the sunshine in we should have to dis
pense with windows in cold weather. Stating this in
more technical language, transparent glass is diatherma-
nous to luminous heat rays but athermanous to dark rays.
Dry air possesses this same peculiar property and permits
the luminous rays from the sun to pass readily through to
the earth, only about one fourth being absorbed as they
pass through. About three fourths of the heat the atmos
phere receives is that which is radiated back as dark rays
from the earth. Being athermanous to these rays the heat
is retained a considerable length of time before it at length
escapes into space. It is for this reason that high alti
tudes are cold, the atmosphere being heated from the
bottom upwards.
Maximum Heat Follows Summer Solstice. Because of
these conditions and of the convecting currents of air, and,

168 SEASONS
to a very limited extent, of water, the heat is so distrib
uted and accumulated that the hottest weather is in the
month following the summer solstice (July in the northern
hemisphere, and January in the southern) ; conversely, the
coldest month is the one following the winter solstice.
This seasonal variation is precisely parallel to the diurnal
change. At noon the sun is highest in the sky and pours
in heat most rapidly, but the point of maximum heat is
not usually reached until the middle of the afternoon, when
the accumulated heat in the atmosphere begins gradually
to disappear.
Astronomical and Climatic Seasons. Astronomically
there are four seasons each year: spring, from the vernal
equinox to the summer solstice; summer, from the sum
mer solstice to the autumnal equinox; autumn, from the
autumnal equinox to the winter solstice; winter, from the
winter solstice to the spring equinox. As treated in phy
sical geography, seasons vary greatly in number and
length with differing conditions of topography and posi
tion in relation to winds, mountains, and bodies of water.
In most parts of continental United States and Europe
there are four fairly marked seasons: March, April, and
May are called spring months; June, July, and August,
summer months; September, October, and November,
autumn months; and December, January, and February,
winter months. In the southern states and in western
Europe the seasons just named begin earlier. In Cali
fornia and in most tropical regions, there are two seasons,
one wet and one dry. In northern South America there
are four seasons, — two wet and two dry.
From the point of view of mathematical geography
there are four seasons having the following lengths in the
northern hemisphere:

HEMISPHERES UNEQUALLY HEATED

169

Spring: Vernal equinox
Summer solstice .

Summer: Summer solstice
Autumnal equinox

Autumn: Autumnal equinox
Winter solstice

Winter: Winter solstice
Vernal equinox

March 21
June 21

I 92 days "

June 21 | „ . ,
Sept. 23f94daysJ

Summer half
* 186 days.

. Dm!' I}90 days] 1 Winter half 179
_ [ days.
Dec. 22 ) oa ,
. March2l}89daysJ

Hemispheres Unequally Heated. For the southern
hemisphere, spring should be substituted for autumn, and
summer for winter. From the foregoing it will be seen
that the northern hemisphere has longer summers and
shorter winters than the southern hemisphere. Since the
earth is in perihelion, nearest the sun, December 31, the
earth as a whole then receives more heat than in the north
ern summer when the earth is farther from the sun.
Though the earth as a whole must receive more heat in
December than in July, the northern hemisphere is then
turned away from the sun and has its winter, which is thus
warmer than it would otherwise be. The converse is true
of the northern summer. The earth then being in aphelion
receives less heat each day, but the northern hemisphere
being turned toward the sun then has its summer, cooler
than it would be were this to occur when the earth is in
perihelion. It is well to remember, however, that while
the earth as a whole receives more heat in the half
year of perihelion, there are only 179 days in that
portion, and in the cooler portion there are 186 days,
so that the total amount of heat received in each
portion is exactly the same. (See Kepler's Second Law,
p. 284.)

170

SEASONS

Determination of Latitude from Sun's Meridian
Altitude.
In Chapter II we learned how latitude is determined by
ascertaining the altitude of the celestial pole. We are now
in a position to see how this is commonly determined by
reference to the noon sun.
Relative Positions of Celestial Equator and Celestial Pole.
The meridian altitude of the celestial equator at a given
place and the altitude of the celestial pole at that place are
complementary angles, that is, together they equal 90°.
Though when understood this proposition is exceedingly
simple, students sometimes only partially comprehend it,
and the later conclusions are consequently hazy.
1. The celestial equator is always 90° from the celestial
pole. 2. An arc of the celestial sphere from the northern hori-

Zenith

Horizon Line at Latitude 40°N
Fig. 53

zon through the zenith to the southern horizon comprises
180°. 3. Since there are 90° from the pole to the equator,
from the northern horizon to the pole and from the
southern horizon to the equator must together equal 90°.

DECLINATION OF THE SUN 171
One of the following statements is incorrect. Find
which one it is.
a. In latitude 30° the altitude of the celestial pole is
30° and that of the celestial equator is 60°.
b. In latitude 36° the altitude of the celestial equator
is 54°.
c. In latitude 48° 20' the altitude of the celestial equa
tor is 41° 40'.
d. If the celestial equator is 51° above the southern
horizon, the celestial pole is 39° above the northern horizon.
e. If the altitude of the celestial equator is 49° 31', the
latitude must be 40° 29'.
/. If the altitude of the celestial equator is 21° 24', the
latitude is 69° 36'.
On March 21 the sun is on the celestial equator.* If on
this day the sun's noon shadow indicates an altitude of
40°, we know that is the altitude of the celestial equator,
and this subtracted from 90° equals 50°, the latitude of
the place. On September 23 the sun is again on the celes
tial equator, and its noon altitude subtracted from 90°
equals the latitude of the place where the observation is
made. Declination of the Sun. The declination of the sun or of
any other heavenly body is its distance north or south of
the celestial equator. The analemma, shown on page 127,
gives the approximate declination of the sun for every day
in the year. The Nautical Almanac, Table 1, for any
* Of course, the center of the sun is not on the celestial equator all
day, it is there but the moment of its crossing. The vernal equinox
is the point of crossing, but we commonly apply the term to the day
when the passage of the sun's center across the celestial equator
occurs. During this day the sun travels northward less than 24',
and since its diameter is somewhat more than 33' some portion of the
sun's disk is on the celestial equator the entire day.

172

SEASONS

month gives the declination very exactly (to the tenth of
a second) at apparent sun noon at the meridian of Green
wich, and the difference in declination for every hour, so
the student can get the declination at his own longitude
for any given day very exactly from this table. With
out good instruments, however, the proportion of error of
observation is so great that the analemma will answer
ordinary purposes.
How to Determine the Latitude of Any Place. By ascer
taining the noon altitude of the sun, and referring to the
analemma or a declination
table, one can easily compute
the latitude of a place.
1. First determine when
the sun will be on your me
ridian and its shadow strike
a north-south line. This is
discussed on pp. 128, 129.
2. By some device meas
ure the altitude of the sun at
apparent noon; i.e., when the
shadow is north. A card
board placed level under a
window shade, as illustrated
in Figure 54, will give sur
prisingly accurate results; a
carefully mounted quadrant
(see Fig. 55), however, will
give more uniformly successful measurements. Angle A
(Fig. 54), the shadow on the quadrant, is the altitude of
the sun. This is apparent from Figure 56, since xy is the
line to the sun, and angle B = angle A.
3. Consult the analemma and ascertain the declination

Fig. 54

EXAMPLE

17.3

of the sun. Add this to the sun's altitude if south declina
tion, and subtract it if north declination. If you are south
of the equator you
must subtract dec
lination south and
add declination
north. (If the ad
dition makes the
altitude of the sun
more than 90°, sub
tract 90° from it,
as under such cir
cumstances you are
north of the equa
tor if it is a south
shadow, or south of
the equator if it
is a north shadow.
This will occur only
within the tropics.)
4. Subtract the result of step three from 90°, and the
remainder is your latitude.
Example. For example,
say you are at San Fran
cisco, October 23, and wish
to ascertain your latitude.
1. Assume you have a
north-south line. (The sun's
shadow will cross it on that
date at 11 h. 54 m. 33 s.,
a.m., Pacific time.)
Fig'56 2. The altitude of the
sun when the shadow is north is found to be 41°.

Fig. 55

174

SEASONS

3. The declination is 11° S. Adding we get 52°, the
altitude of the celestial equator.
4. 90°— 52° equals 38°, latitude of place of observer.
Conversely, knowing the latitude of a place, one can
ascertain the noon altitude of the sun at any given day.
From the analemma and the table of latitudes many inter
esting problems will suggest themselves, as the following
examples illustrate.
Problem. 1. How high above the horizon does the sun
get at St. Petersburg on December 22?
Solution. The latitude of St. Petersburg is 59° 56' N.,
hence the altitude of the celestial equator is 30° 4'. The
declination of the sun December 22 is 23° 27' S. Since
south is below the celestial equator
at St. Petersburg, the altitude of the
sun is 30° 4' less 23° 27', or 6° 37'.
Problem. 2. At which city is the
noon sun higher on June 21, Chicago
or Quito?
Solution. The latitude of Chicago
is 41° 50', and the altitude of the
celestial equator, 48° 10'. The dec
lination of the sun June 21 is 23°
27' N. North being higher than the
celestial equator at Chicago, the
noon altitude of the sun is 48° 10'
plus 23° 27', or 71° 37'.
The latitude of Quito being 0°, the
altitude of the celestial equator is
90°. The declination of the sun being
23° 27' from this, the sun's noon alti
tude must be 90° less 23° 27', or 66° 33'. The sun is thus
5° 4' higher at Chicago than at the equator on June 21.

^

Fig- 57- Taking the altitude
of the sun at sea

LATITUDE FROM MOON OR STARS 175
Latitude from Moon or Stars. With a more extended
knowledge of astronomy and mathematics and with suitable
instruments, we might ascertain the position of the celes
tial equator in the morning or evening from the moon,
planets, or stars as well as from the sun. At sea the
latitude is commonly ascertained by making measurements
of the altitudes of the sun at apparent noon with the sex
tant. The declination tables are used, and allowances
are made for refraction and for the " dip " of the horizon,
and the resultant calculation usually gives the latitude
within about half a mile. At observatories, where the
latitude must be ascertained with the minutest precision
possible, it is usually ascertained from star observations
with a zenith telescope or a " meridian circle " telescope,
and is verified in many ways.

CHAPTER IX
TIDES
Tides and the Moon. The regular rise and fall of the
level of the sea and the accompanying inflows and out
flows of streams, bays, and channels, are called tides. Since
very ancient times this action of the water has been asso
ciated with the moon because of the regular interval
elapsing between a tide and the passage of the moon over
the meridian of the place, and a somewhat uniform increase
in the height, of the tide when the moon in its orbit around
the earth is nearest the sun or is farthest from it. This
unquestioned lunar influence on the ocean has doubtless
been responsible as the basis for thousands of unwarranted
associations of cause and effect of weather, vegetable
growth, and even human temperament and disease with
phases of the moon or planetary or astral conditions.
Other Periodic Ebbs and Flows. Since there are other
periodical ebbs and flows due to various causes, it may be
well to remember that the term tide properly applies only
to the periodic rise and fall of water due to unbalanced
forces in the attraction of the sun and moon. Other con
ditions which give rise to more or less periodical ebbs and
flows of the oceans, seas, and great lakes are:
a. Variation in atmospheric pressure; low barometer
gives an uplift to water and high barometer a depression.
b. Variability in evaporation, rainfall and melting snows
produces changes in level of adjacent estuaries.
c. Variability in wind direction, especially strong and
continuous seasonal winds like monsoons, lowers the
176

MOON'S ORBIT 177
level on the leeward of coasts and piles it up on the wind
ward side.
d. Earthquakes sometimes cause huge waves.
A few preliminary facts to bear in mind when consider
ing the causes of tides :
The Moon
Sidereal Month. The moon revolves around the earth
in the same direction that the earth revolves about the
sun, from west to east. If the moon is observed near a
given star on one night, twenty-four hours later it will be
found, on the average, about 13.2° to the eastward. To
reach the same star a second time it will require as many
days as that distance is contained times in 360° or about
27.3 days. This is the sidereal month, the time required
for one complete .revolution of the moon.
Synodic Month. Suppose the moon is near the sun at a
given time, that is, in the same part of the celestial sphere.
During the twenty-fours hours following, the moon will
creep eastward 13.2° and the sun 1°. The moon thus
gains on the sun each day about 12.2°, and to get in con
junction with it a second time it will take as many days
as 12.2° is contained in 360° or about 29.5 days. This is
called a synodic (from a Greek word meaning " meeting ")
month, the time from conjunction with the sun — new
moon — until the next conjunction or new moon. The
term is also applied to the time from opposition or full
moon until the next opposition or full moon. If the
phases of the moon are not clearly understood it would
be well to follow the suggestions on this subject in the
first chapter.
Moon's Orbit. The moon's orbit is an ellipse, its
JO. MATH. GEO.  12

178 TIDES
nearest point to the earth is called perigee (from peri,
around or near; and ge, the earth) and is about 221,617
miles. Its most distant point is called apogee (from apo,
from; and ge, earth) and is about 252,972 miles. The
average distance of the moon from the earth is 238,840
miles. The moon's orbit is inclined to the ecliptic 5° 8'
and thus may be that distance farther north or south than
the sun ever gets.
The new moon is said to be in conjunction with the sun,
both being on the same side of the earth. If both are
then in the plane of the ecliptic an eclipse of the sun must
take place. The moon being so small, relatively (diam
eter 2,163 miles), its shadow on the earth is small and
thus the eclipse is visible along a relatively narrow path.
The full moon is said to be in opposition to the sun,
it being on the opposite side of the earth. If, when in
opposition, the moon is in the plane of the ecliptic it will
be eclipsed by the shadow of the earth. When the moon
is in conjunction or in opposition it is said to be in syzygy.
Gravitation
Laws Restated. This force was discussed in the first
chapter where the two laws of gravitation were explained
and illustrated. The term gravity is applied to the force
of gravitation exerted by the earth (see Appendix, p. 279).
Since the explanation of tides is simply the application of
the laws of gravitation to the earth, sun, and moon, we
may repeat the two laws:
First law: The force of gravitation varies directly as
the mass of the' object.
Second law: The force of gravitation varies inversely
a= the square of the distance of the object.

SUN'S ATTRACTION 179
Sun's Attraction Greater, but Moon's Tide-Producing
Influence Greater. There is a widely current notion that
since the moon causes greater tides than the sun, in the
ratio of 5 to 2, the moon must have greater attractive
influence for the earth than the sun has. Now this cannot
be true, else the earth would swing around the moon as
certainly as it does around the sun. Applying the laws of
gravitation to the problem, we see that the sun's attrac
tion for the earth is approximately 176 times that of the
moon.* The reasoning which often leads to the erroneous con
clusion just referred to, is probably something like this:
Major premise: Lunar and solar attraction causes tides.
Minor premise: Lunar tides are higher than solar tides.
Conclusion: Lunar attraction is greater than solar
attraction. We have just seen that the conclusion is in error. One
or both of the premises must be in error also. A study
of the causes of tides will set this matter right.
Causes of Tides -f
It is sometimes erroneously stated that wind is caused
by heat. It would be more nearly correct to say that
wind is caused by the unequal heating of the atmosphere.
Similarly, it is not the attraction of the sun and moon
for the earth that causes tides, it is the unequal attraction
for different portions of the earth that gives rise to un
balanced forces which produce tides.
* For the method of demonstration, see p. 19. The following data
are necessary: Earth's mass, 1; sun's mass, 330,000; moon's mass, g>T ;
distance of earth to sun, 93,000,000 miles ; distance of earth to moon,
239,000 miles.
| A mathematical treatment will be found in the Appendix.

180

TIDES

Portions of the earth toward the moon or sun are 8,000
miles nearer than portions on the side of the earth opposite
the attracting body, hence the force of gravitation is
slightly different at those points as compared with other
points on the earth's surface. It is obvious, then, that at
A and B (Fig. 58) there are two unbalanced forces, that
is, forces not having counterparts elsewhere to balance
them. At these two sides, then, tides are produced,

To Sun or Moon
)»> >

Fig. 58

since the water of the oceans yields to the influence of
these forces. That this may be made clear, let us examine
these tides separately.
The Tide on the Side of the Earth Toward the Moon. If A
is 239,000 miles from the moon, B is 247,000 miles away
from it, the diameter of the earth being AB (Fig. 58).
Now the attraction of the moon at A, C, and D, is away
from the center of the earth and thus lessens the force of
gravity at those points, lessening more at A since A is
nearer and the moon's attraction is exerted in a line

SOLAR TIDES COMPARED WITH LUNAR TIDES 181
directly opposite to that of gravity. The water, being
fluid and easily moved, yields to this lightening of its
weight and tends to " pile up under the moon." We thus
have a tide on the side of the earth toward the moon.
Tidal Wave Sweeps Westward. As the earth turns
on its axis it brings successive portions of the earth toward
the moon and this wave sweeps around the globe as nearly
as possible under the moon. The tide is retarded some
what by shallow water and the configuration of the coast
and is not found at a given place when the moon is at
meridian height but lags somewhat behind. The time
between the passage of the moon and high tide is called
the establishment of the port. This time varies greatly at
different places and varies somewhat at different times of
the year for the same place.
Solar Tides Compared with Lunar Tides. Solar tides
are produced on the side of the earth toward the sun for
exactly the same reason, but because the sun is so far
away its attraction is more uniform upon different parts
of the earth. If A is 93,000,000 miles from the sun, B is
93,008,000 miles from the sun. The ratio of the squares of
these two numbers is much nearer unity than the ratio
of the numbers representing the squares of the distances
of A and B from the moon. If the sun were as near as
the moon, the attraction for A would be greater by an
enormous amount as compared with its attraction for B.
Imagine a ball made of dough with lines connected to
every particle. If we pull these lines uniformly the ball
will not be pulled out of shape, however hard we pull. If,
however, we pull some lines harder than others, although
we pull gently, will not the ball be pulled out of shape?
Now the pull of the sun, while greater than that of the
moon, is exerted quite evenly throughout the earth and

182 TIDES
has but a slight tide-producing power. The attraction of
the moon, while less than that of the sun, is exerted less
evenly than that of the sun and hence produces greater
tides. It has been demonstrated that the tide-producing force
of a body varies inversely as the cube of its distance and
directly as its mass. Applying this to the moon and sun
we get: Let T = sun's tide-producing power,
and t = moon's tide-producing power.
The sun's mass is 26,500,000 times the moon's mass,
.-. T : t : : 26,500,000 : 1.
But the sun's distance from the earth is 390 times the
moon's distance, • T • t • • • 1
..1 .t.. 39Q3 . 1.
Combining the two proportions, we get,
T : t : : 2 : 5.
It has been shown that, owing to the very nearly equal
attraction of the sun for different parts of the earth, a
body's weight is decreased when the sun is overhead, as
compared with the weight six hours from then, by only
— -r-— — ¦ ; that is, an object weighing a ton varies in
Ji\j , UUU j uuu
weight | of a grain from sunrise to noon. In case of the
moon this difference is about 2\ times as great, or nearly
2 grains.

THE TIDE ON SIDE OF EARTH OPPOSITE MOON 183
Tides on the Moon. It may be of interest to note that
the effect of the earth's attraction on different sides of the
moon must be twenty times as great as this, so it is
thought that when the moon was warmer and had oceans *
the tremendous tidal waves swinging around in the oppo
site direction to its rotation caused a gradual retardation
of its rotation until, cs ages passed, it came to keep the
same face toward the earth. The planets nearest the sun,
Mercury and Venus, probably keep the same side toward
the sun for a similar reason. Applying the same reasoning
to the earth, it is believed that the period of rotation must
be gradually shortening, though the rate seems to be
entirely inappreciable.
The Tide on the Side of the Earth Opposite the Moon. A
planet revolving around the sun, or a moon about a planet,
takes a rate which varies in a definite mathematical ratio
to its distance (see p. 285). The sun pulls the earth to
ward itself about one ninth of an inch every second. If
the earth were nearer, its revolutionary motion would be
faster. In case of planets having several satellites it is
observed that the nearer ones revolve about the planet
faster than the outer ones (see p. 255). Now if the
earth were divided into three separate portions, as in
Figure 59, the ocean nearest the sun, the earth proper,
and the ocean opposite the sun would have three separate
motions somewhat as the dotted lines show. Ocean A
would revolve faster than earth C or ocean B. If these
three portions were connected by weak bands their stretch
ing apart would cause them to separate entirely. The
* The presence of oceans or an atmosphere is not essential to the
theory, indeed, is not usually taken into account. It seems most cer
tain that the earth is not perfectly rigid, and the theory assumes that
the planets and the moon have sufficient viscosity to produce body
tides.

184

TIDES

tide-producing power at B is this tendency it has to fall
away, or more strictly speaking, to fall toward the sun less
rapidly than the rest of the earth.
Moon and Earth Revolve About a Common Center of
Gravity. What has been said of the earth's annual rev
olution around the sun applies equally to the earth's

Fig. 59

monthly swing around the center of gravity common to
the earth and the moon. We commonly say the earth
revolves about the sun and the moon revolves about the
earth. Now the earth attracts the sun, in its measure,
just as truly as the sun attracts the earth; and the moon
attracts the earth, in the ratio of its mass, as the earth
attracts the moon. Strictly speaking, the earth and sun
revolve around their common center of gravity and the
moon and earth revolve around their center of gravity.

COURSE OF THE TIDAL WAVE 185
It is as if the earth were connected with the moon by
a rigid bar of steel (that had no weight) and the two,
thus firmly bound at the ends of this rod 239,000 miles
long, were set spinning. If both were of the same weight,
they would revolve about a point equidistant from each.
The weight of the moon being somewhat less than g1^ that
of the earth, this center of gravity, or point of balance,
is only about 1,000 miles from the earth's center.
Spring Tides. When the sun and moon are in con
junction, both on the same side of the earth, the unequal
attraction of both for the side toward them produces an
unusually high tide there, and the increased centrifugal
force at the side opposite them also produces an unusual
high tide there. Both solar tides and both lunar tides
are also combined when the sun and moon are in opposi
tion. Since the sun and moon are in syzygy (opposition
or conjunction) twice a month, high tides, called spring
tides, occur at every new moon, and at every full moon.
If the moon should be in perigee, nearest the earth, at the
same time it was new or full moon, spring tides would be
unusually high.
Neap Tides. When the moon is at first or last quarter
— moon, earth, and sun forming a right angle — the solar
tides occur in the trough of the lunar tides and they are
not as low as usual, and lunar tides occurring in the trough
of the solar tides are not so high as usual.
Course of the Tidal "Wave. While the tidal wave is gen
erated at any point under or opposite the sun or moon, it
is out in the southern Pacific Ocean that the absence of
shallow water and land areas offers least obstruction to its
movement. Here a general lifting of the ocean occurs,
and as the earth rotates the lifting progresses under or
opposite the moon or sun from east to west. Thus a huge

186

TIDES

wave with crest extending north and south starts twice a
day off the western coast of South America. The general
position of this crest is shown on the co-tidal map, one line
for every hour's difference in time. The tidal wave is
retarded along its northern extremity, and as it sweeps
along the coast of northern South America and North
America, the wave assumes a northwesterly direction and
sweeps down the coast of Asia at the rate of about 850

Fig. 60. Co-tidal lines
miles per hour. The southern portion passes across the
Indian Ocean, being retarded in the north so that the
southern portion is south of Africa when the northern por
tion has just reached southern India. The time it has
taken the crest to pass from South America to south Africa
is about 30 hours. Being retarded by the African coast,
the northern portion of the wave assumes an almost north
erly direction, sweeping up the Atlantic at the rate of about
700 miles an hour. It moves so much faster northward
in the central Atlantic than along the coasts that the crest

TIME BETWEEN SUCCESSIVE TIDES 187
bends rapidly northward in the center and strikes all points
of the coast of the United States within two or three hours
of the same time. To reach France the wave must swing
around Scotland and then southward across the North Sea,
reaching the mouth of the Seine about 60 hours after
starting from South America. A new wave being formed
about every 12 hours, there are thus several of these tidal
waves following one another across the oceans, each
slightly different from the others.
While the term " wave " is correctly applied to this tidal
movement it is very liable to leave a wrong impression
upon the minds of those who have never seen the sea.
When thinking of this tidal wave sweeping across the
ocean at the rate of several hundred miles per hour, we
should also bear in mind its height and length (by height
is meant the vertical distance from the trough to the crest,
and by length the distance from crest to crest). Out in
midocean the height is only a foot or two and the length
is hundreds of miles. Since the wave requires about three
hours to pass from trough to crest, it is evident that a ship
at sea is lifted up a foot or so during six hours and then
as slowly lowered again, a motion not easily detected. On
the shore the height is greater and the wave-length shorter,
for about six hours the water gradually rises and then for
about six hours it ebbs away again. Breakers, bores, and
unusual tide phenomena are discussed on p. 189.
Time Between Successive Tides. The time elapsing
from the passage of the moon across a meridian until it
crosses the same meridian again is 24 hours 51 min.* This,
* More precisely, 24 h. 50 m. 51 s. This is the mean lunar day, or
interval between successive passages of the moon over a given meridian.
The apparent lunar day varies in length from 24 h. 38 m. to 25 h. 5 m.
for causes somewhat similar to those producing a variation in the length
of the apparent solar day.

188

TIDES

in contradistinction to the solar and sidereal day, may
be termed a lunar day. It takes the moon 27.3 solar days
to revolve around the earth, a sidereal month. In one
day it journeys sW of a day or 51 minutes. So if the
moon was on a given meridian at 10 a.m., on one day,
by 10 a.m. the next day the moon would have moved
12.2° eastward, and to direct the same meridian a second
time toward the moon it takes on the average 51 minutes
longer than 24 hours, the actual time varying from 38 m. to

Fig. 6i. Low tide

lh. 6 m. for various reasons. The tides of one day, then,
are later than the tides of the preceding day by an average
interval of 51 minutes.
In studying the movement of the tidal wave, we observed
that it is retarded by shallow water. The spring tides
being higher and more powerful move faster than the
neap tides, the interval on successive days averaging only
38 minutes. Neap tides move slower, averaging somewhat
over an hour later from day to day. The establishment
of a port, as previously explained, is the average time

BORE OF THE AMAZON 189
elapsing between the passage of the moon and the high
tide following it. The establishment for Boston is 11 hours,
27 minutes, although this varies half an hour at different
times of the year.
Height of Tides. The height of the tide varies greatly
in different places, being scarcely discernible out in mid-
ocean, averaging only IJ feet in the somewhat sheltered
Gulf of Mexico, but averaging 37 feet in the Bay of Fundy.
The shape and situation of some bays and mouths of
rivers is such that as the tidal wave enters, the front part
of the wave becomes so steep that huge breakers form
and roll up the bay or river with great speed. These
bores, as they are called, occur in the Bay of Fundy, in
the Hoogly estuary of the Ganges, in that of the Dor-
dogne, the Severn, the Elbe, the Weser, the Yangtze, the
Amazon, etc.
Bore of the Amazon. A description of the bore of the
Amazon, given by La Condamine in the eighteenth cen
tury, gives a good idea of this phenomenon. ' ' During three
days before the new and full moons, the period of the
highest tides, the sea, instead of occupying six hours to
reach its flood, swells to its highest limit in one or two
minutes. The noise of this terrible flood is heard five or
six miles, and increases as it approaches. Presently you
see a liquid promontory, 12 or 15 feet high, followed by
another, and another, and sometimes by a fourth. These
watery mountains spread across the whole channel, and
advance with a prodigious rapidity, rending and crush
ing everything in their way. Immense trees are instantly
uprooted by it, and sometimes whole tracts of land are
swept away."

CHAPTER X

MAP PROJECTIONS

To represent the curved surface of the earth, or any
portion of it, on the plane surface of a map, involves
serious mathematical difficulties. Indeed, it is impossible
to do so with perfect ac
curacy. The term pro
jection, as applied to the
representation on a plane
of points corresponding
to points on a globe, is
not always used in
geography in its strictly
mathematical sense, but
denotes any representa
tion on a plane of paral
lels and meridians of the
earth. The Orthographic
Projection
This is, perhaps, the
most readily understood
projection, and is one of
the oldest known, having
been used by the ancient
Greeks for celestial representation. The globe truly repre
sents the relative positions of points on the earth's surface.
190

Fig. fa

PARALLELS AND MERIDIANS

191

It might seem that a photograph of a globe would
correctly represent
on a flat surface
the curved surface
of the earth. A
glance at Figure 62,
from a photograph
of a globe, shows
the parallels near
the equator to be v
farther apart than
those near the
poles. This is not
the way they are
on the globe. The
orthographic pro

se
Fig. 63. Equatorial orthographic projection

jection is the representation of the globe as a photo
graph would show it from a great distance.

Fig. 64

Parallels and Meridians Farther Apart in Center of Map.
Viewing a globe from a distance, we observe that par-

192

MAP PROJECTIONS

allels and meridians appear somewhat as represented in
Figure 63, being farther apart toward the center and in
creasingly nearer toward the outer portion. Now it is
obvious from Figure 64 that the farther the eye is placed
from the globe, the less will be the distortion, although a
removal to an infinite distance will not obviate all distor
tion. Thus the eye at x sees lines to E and F much
nearer together than lines to A and B, but the eye at the
greater distance sees less difference.
When the rays are perpendicular to the axis, as in Figure
65, the parallels at A, B, C, D, and E will be seen on the
tangent plane XY
at A', B', C, D',
and E'. While the
distance from A to
B on the globe
is practically the
same- as the dis
tance from D to E,
to the distant eye
A' and B' will ap
pear much nearer
together than D'
and E'. Since A
(or A') represents a pole and E (or E') the equator, line
XY is equivalent to a central meridian and points A', B',
etc., are where the parallels cross it.
How to Lay off an Equatorial Orthographic Projection.
If parallels and meridians are desired for every 15°, divide
the circle into twenty-four equal parts ; any desired number
of parallels and meridians, of course, may be drawn. Now
connect opposite points with straight lines for parallels
(as in Fig. 65). The reason why parallels are straight lines

Fig. 65.

TO LAY OFF AN EQUATORIAL PROJECTION 193

in the equatorial orthographic projection is apparent if
one remembers that if the eye is in the plane of the equator
and is at an infinite distance, the parallels will lie in practi
cally the same plane as the eye.
To lay off the meridians, mark on the equator points
exactly as on the central meridian where parallels intersect
it. The meridians may now be made as arcs of circles
passing through the poles and these points. With one
foot of the compasses in the equator, or equator extended,
place the other so that
it will pass through the
poles and one of these
points. After a little
trial it will be easy to
lay off each of the meri
dians in this way.
To be strictly correct
the meridians should not
be arcs of circles as just
suggested but should be
semi-ellipses with the
central meridian as
major axis as shown in
Figure 66. While somewhat more difficult, the student
should learn how thus to lay them off. To construct the
ellipse, one must first locate the foci. This is done by taking
half the major axis (central meridian) as radius and with
the point on the equator through which the meridian is
to be constructed as center, describe an arc cutting the
center meridian on each side of the equator. These points
of intersection on the central meridian are the foci of the
ellipse, one half of which is a meridian. By placing a pin
at each of the foci and also at the point in the equator
JO. MATH. GEO.  13

Kg. 66.

Western hemisphere, in equatorial
orthographic projection

194

MAP PROJECTIONS

where the meridian must cross and tying a string as a loop
around these three pins, then withdrawing the one at the
equator, the ellipse may be made as described in the first
chapter. How to Lay off a Polar Orthographic Projection. This
is laid off more easily than the former projection. Here
the eye is conceived to be directly above a pole and the
equator is the boundary
of the hemisphere seen.
It is apparent -that
from this position the
equator and parallels
will appear as circles
-and, since the planes
of the meridians pass
through the eye, each
meridian must appear
as a straight line.
Lay off for the equa
tor a circle the same
size as the preceding
one (Fig. 65), sub
dividing it into twenty-four parts, if meridians are desired
for every 15°. Connect these points with the center,
which represents the pole. On any diameter mark off
distances as on the center meridian of the equatorial
orthographic projection (Fig. 65). Through these points
draw circles to represent parallels. You will then have
the complete projection as in Figure 67.
Projections may be made with any point on the globe as
center, though the limits of this book will not permit the
rather difficult explanation as to how it is done for lati
tudes other than 0° or 90°. With the parallels and

Fig. 67. Polar orthographic projection

TO LAY OFF A POLAR ORTHOGRAPHIC PROJECTION 195
meridians projected, the map may be drawn. The student
should remember that all maps which make any claim to
accuracy or correctness are made by locating points of
an area to be represented according to their latitudes and
longitudes; that is, in reference to parallels and meridians.
It will be observed that the orthographic system of projec
tion crowds together areas toward the outside of the map
and the scale of miles suitable for the central portion will
not be correct for the outer portions. For this reason a
scale of miles never appears upon a hemisphere made on
this projection. SUMMARY
In the orthographic projection:
1. The eye is conceived to be at an infinite distance.
2. Meridians and parallels are farther apart toward the center of
the map.
3. When a point in the equator is the center, parallels are straight
lines.
4. When a pole is at the center, meridians are straight lines. If the
northern hemisphere is represented, north is not toward the top
of the map but toward the center.

Stereographic Projection
In the stereographic projection the eye is conceived to be
upon the surface of the globe, viewing the opposite hemi
sphere. Points on the opposite hemisphere are projected
upon a plane tangent to it. Thus in Figure 68 the eye is
at E and sees A at A', B at B', C at C", etc. If the earth
were transparent, we should see objects on the opposite
half of the globe from the view point of this projection.

196

MAP PROJECTIONS

How to Lay off an Equatorial Stereographic Projection.
In Figure 68, E represents the eye at the equator, A and N
are the poles and A'N' is the corresponding meridian of

Fig. 68

the projection with B', C, etc., as the points where the
parallels cross the meridian. Taking the line A'N' of Figure
68 as diameter, construct upon it a circle (see Fig. 69).

THE POLAR STEREOGRAPHIC PROJECTION 197

Divide the circumference into twenty-four equal parts
and draw parallels as
arcs of circles. Lay
off the equator and
subdivide it the same
as the central meridian,
that is, the same as
A'N' of Figure 68.
Through the points in
the equator, draw me
ridians as arcs of circles
and the projection is
complete. The Polar Stereo
graphic Projection is
made on the same
plan as the polar orthographic projection, excepting that
the parallels have the
distances from the pole

Fig. 69. Equatorial stereographic projection

Fig. 70. Polar stereographic
projection

Fig. 71. Northern hemisphere in polar
stereographic projection

that are represented by the points in A'N' of Figure 68
(see Figs. 70, 71).

198

MAP PROJECTIONS

Areas are crowded together toward the center of the
map when made on the stereographic projection and a
scale of miles suitable for the central portion would be too
small for the outer portion. This projection is often used,
however, because it it so easily laid off.

Fig. 72 . Hemispheres in equatorial stereographic projection
SUMMARY

/

In the stereographic projection:
1. The eye is conceived to be on the surface of the globe.
2. Meridians and parallels are nearer together toward the center of
the map.
3. When a point in the equator is the center of the map, parallels and
meridians are represented as arcs of circles.
4. When a pole is the center, meridians are straight lines.
Globular Projection.
1
With the eye at an infinite distance (as in the ortho
graphic projection), parallels and meridians are nearer
together toward the outside of the map; with the eye on
the surface (as in the stereographic projection), they are
nearer together toward the center of the map. It would
seem reasonable to expect that with the eye at some point

THE POLAR STEREOGRAPHIC PROJECTION 199

Fig- 73

intermediate between an infinite distance from the surface
and the surface itself that the parallels and meridians
would be equidistant at dif
ferent portions of the map.
That point is the sine of an
angle of 45°, or a little less
than the length of a radius
away from the surface. To
find this distance at which
the eye is conceived to be
placed in the globular projec
tion, make a circle of the same
size as the one which is the
basis of the map to be made,
draw two jadii at an angle of 45° (one eighth of the circle)
and draw a line,
AB, from the ex
tremity of one ra
dius perpendicular
to the other radius.
The length of this
perpendicular is the
distance sought (A B,
Fig. 73).
Thus with the eye
at E (Fig, 74) the
pole A is projected
to the tangent plane
at A', B at B', etc.,
and the distances
A'B', B'C, etc., are
practically equal so that they are constructed as though
they were equal in projecting parallels and meridians.

Fig. 74

200

MAP PROJECTIONS

Fig 75-

Hemispheres in equatorial globular
projection

How to Lay off an Equatorial Globular Projection. As in
the orthographic or stereographic projections, a circle is
divided into equal parts,
according to the number
of parallels desired, the
¦ central meridian and
equator being subdi
vided into half as many
equal parts. Parallels
and meridians may be
drawn as arcs of circles, being sufficiently accurate for
ordinary purposes (see Fig. 75).
The polar globular projection is laid off precisely
like the orthographic and the stereographic projections
having the pole as the center, excepting that the con
centric circles representing
parallels are equidistant (see
Fig. 76).
By means of starlike addi
tions to the polar globular pro
jection (see Fig. 77), the entire
globe may be represented. If
folded back, the rays of the
star would meet at the south
pole. It should be noticed
that " south " in this pro
jection is in a line directly
away from the center ; that is, the top of the map is south,
the bottom south, and the sides are also south. While
portions of the southern hemisphere are thus spread out,
proportional areas are well represented, South America
and Africa being shown with little distortion of area and
outline.

Fig. 76. Polar globular projection

THE POLAR GLOBULAR PROJECTION

201

The globular projection is much used to represent
hemispheres, or with the
star map to represent
the entire globe, because
the parallels on a me
ridian or meridians on a
parallel are equidistant
and show little exagger
ation of areas. For this
reason it is sometimes
called an equidistant
projection, although
there are other equi
distant projections. It
is also called the De la Hire projection from its discoverer
(1640-1718). SUMMARY
In the globular projection:
1. The eye is conceived to be at a certain distance from the globe
(sine 45°).
2. Meridians are divided equidistantly by parallels, and parallels are
divided equidistantly by meridians.
3. When a pole is the center of the map, meridians are straight lines.
4. There is little distortion of areas.

Fig. 77. World in polar globular projection

The Gnomonic Projection
When we look up at the sky we see what appears to be
a great dome in which the sun, moon, planets, and stars
are located. We seem to be at the center of this celestial
sphere, and were we to imagine stars and other heavenly
bodies to be projected beyond the dome to an imaginary
plane we should have a gnomonic projection. Because
of its obvious convenience in thus showing the position

202

MAP PROJECTIONS

of celestial bodies, this projection is a very old one, having
often been used by the ancients for celestial maps.
Since the eye is at the center for mapping the celestial
sphere, it is conceived to be at the center of the earth in
projecting parallels and meridians of the earth. As will
be seen from Figure 78,
the distortion is very
great away from the
center of the map and
an entire hemisphere
cannot be shown.
All great circles on
this projection are repre
sented as straight lines.
This will be apparent if
one imagines himself at
the center of a trans
parent globe having par
allels and meridians
traced upon it. Since
the plane of every great
circle passes through the
center of the globe, the
eye at that point will
see every portion of a
great circle as in one
plane and will project
it as a straight line. As will be shown later, it is be
cause of this fact that sailors frequently use maps made
on this projection.
To Lay off a Polar Gnomonic Projection. Owing to the
fact that parallels get so much farther apart away from
the center of the map, the gnomonic projection is almost

Fig. 78

GREAT CIRCLE SAILING

203

never made with any other point than the pole for center,
and then only for latitudes about forty degrees from the
pole. The polar gno
monic projection is
made like the polar
projections previously
described, excepting
that parallels intersect
the meridians at the
distances represented
in Figure 78. The
meridians, being great
circles, are represented
as straight lines and
the parallels as concen
tric circles.
Great Circle Sailing.

Fig. 79. Polar gnomonic projection

Bear-

It would seem at first thought
that a ship sailing to a
point due eastward,
say from New York to
Oporto, would follow
the course of a paral
lel, that is, would sail
due eastward. This,
however, would not be
its shortest course.
The solution of the
following little catch
problem in mathe
matical geography will
make clear the reason
for this. " A man was forty rods due east of a bear,
his gun would carry only thirty rods, yet with no change

Fig. 8o

204 MAP PROJECTIONS
of position he shot and killed the bear. Where on earth
were they?" Solution: This could occur only near the
pole where parallels are very small circles. The bear was
westward from the man and westward is along the course
of a parallel. The bear was thus distant forty rods in a
curved line from the man but the bullet flew in a straight
line (see Fig. 80).
The shortest distance between two points on the earth
is along the arc of a great circle. A great circle passing
through New York and Oporto passes a little to the north
of the parallel on which both cities are located. Thus it
is that the course of vessels plying between the United
States and Europe curves somewhat to the northward of
parallels. This following of a great circle by navigators
is called great circle sailing. The equator is a great circle
and parallels near it are almost of the same length. In
sailing within the tropics, therefore, there is little advan
tage in departing from the course of a parallel. Besides
this, the trade winds and doldrums control the choice of
routes in that region and the Mercator projection is always
used in sailing there. In higher latitudes the gnomonic
projection is commonly used.
Although the gnomonic projection is rarely used ex
cepting by sailors, it is important that students understand
the principles underlying its construction since the most
important projections yet to be discussed are based upon it.
SUMMARY
In the gnomonic projection:
1. The eye is conceived to be at the center of the earth.
2. There is great distortion of distances away from the centerof the map.
3. A hemisphere cannot be shown.
4. All great circles are shown as straight lines.
a. Therefore it is used largely for great circle sailing.
5. The pole is usually the center of the map.

HOMOLOGRAPHIC PROJECTION
The Homolographic Projection

205

The projections thus far discussed will not permit the
representation of the entire globe on one map, with the
exception of the starlike extension of the polar globular
projection. The homolographic projection is a most
ingenious device which is used quite extensively to repre-

Fig. 81. Homolographic projection
sent the entire globe without distortion of areas. It is a
modification of the globular projection.
How to Lay off a Homolographic Projection. First lay
off an equatorial globular projection, omitting the parallels.
The meridians are semi-ellipses, although those which are
no more than 90° from the center meridian may be drawn
as arcs of circles.
Having laid off the meridians as in the equatorial
globular projection, double the length of the equator,
extending it equally in both directions, and subdivide
these extensions as the equator was subdivided. Through

206

MAP PROJECTIONS

these points of subdivision and the poles, draw ellipses for
meridians. To draw the outer elliptical meridians. Set the points of
the compasses at the distance from the point through
which the meridian is to be drawn to the central meridian.
Place one point of the compasses thus set at a pole and
mark off points on the equator for foci of the ellipse.
Drive pins in these foci and also one in a pole. Around
these three pins form a loop with a string. Withdraw
the pin at the pole and draw the ellipse as described on

Fig. 82. World In homolographic projection
page 22. This process must be repeated for each pair of
meridians. The parallels are straight lines, as in the orthographic
projection, somewhat nearer together toward the poles.
If nine parallels are drawn on each side of the equator,
they may be drawn in the following ratio of distances,
beginning at the equator: 2, 1|, 1|, If, 1|, 1|, If, 1|, 1|.
This will give an approximately correct representation.
One of the recent books to make frequent use of this
projection is the " Commercial Geography " by Gannett,
Garrison, and Houston (see Fig. 82).

EQUATORIAL DISTANCES OF PARALLELS

207

Equatorial Distances -of Parallels. The following table
gives the exact relative distances of parallels from the
equator. Thus if a map twenty inches wide is to be
drawn, ten inches from equator to pole, the first parallel
will be .69 of an inch from the equator, the second 1.37
inches, etc.

<p

Dis
tance

0
20°2530

Dis
tance

35°4045

Dis
tance

<t>

Dis
tance

<t>

Dis
tance

<t>

Dis
tance

5°
10 15

.069.137
.205

.272 .339.404

.468.531.592

50° 55
60

.651 .708.762

65°70
75

.814.862 .906

80°85 90

.945 .978
1.000

The homolographic projection is sometimes called the
Mollweide projection from its inventor (1805), and the
Babinet, or Babinet-homolographic projection from a
noted cartographer who used it in an atlas (1857) . From
the fact that within any given section bounded by paral
lels and meridians, the area of the surface of the map is
equal to the area within similar meridians and parallels
of the globe, is it sometimes called the equal-surface pro
jection.
SUMMARY
In the homolographic projection:
1. The meridians are semi-ellipses, drawn as in the globular projec
tion, 360° of the equator being represented.
2. The parallels are straight lines as in the orthographic projection.
3. Areas of the map represent equal areas of the globe.
4. There is no distortion of area and not a very serious distortion
of form of continents.
5. The globe is represented as though its surface covered half of
an exceedingly oblate spheroid.

208

MAP PROJECTIONS

The Van der Grinten Projection
The homolographic projection was invented early in the
nineteenth century. At the close of the century Mr.
Alphons Van der Grinten of Chicago invented another pro
jection by which the entire surface of the earth may be
represented. This ingenious system reduces greatly the

Fig. 83. World in Van der Grinten projection
angular distortion incident to the homolographic projec
tion and for the inhabitable portions of the globe there is
very little exaggeration of areas.
In the Van der Grinten projection the outer boundary
is a meridian circle, the central meridian and equator are
straight lines, and other parallels and meridians are arcs
of circles. The area of the circle is equal to the surface
of a globe of one half the diameter of this circle. The
equator is divided into 360°, but the meridians are, of
course, divided into 180°.

GNOMONIC CYLINDRICAL PROJECTION

209

A modification of
this projection is
shown in Figure
83. In this the
central meridian is
only one half the
length of the equa
tor, and parallels are
at uniform distances
along this meridian.
Cylindrical Pro
jections

Gnomonic Cylin
drical Proiection In Fig. 84. World in Van der Grinten projection
this projection the sheet on which the map is to be made
is conceived to be
wrapped as a cyl
inder around the
globe, touching the
equator. The eye
is conceived to be
at the center of the
globe, projecting the
parallels and meri
dians upon the tan
gent cylinder. Fig
ure 85 shows the
cylinder partly un
wrapped with meri
dians as parallel
straight lines and
As in the gnomonic

Fig. 85
parallels also as parallel straight lines.

JO. MATH. GEO.

-14

210

MAP PROJECTIONS

projection, the parallels are increasingly farther apart away
from the equator.
An examination of Figure 86 will show the necessity
for the increasing distances of parallels in higher latitudes.
The eye at the center (E) sees A at A', B at B', etc. Be
yond 45° from the equator the distance between parallels
becomes very great. A'B' represents the same distance
(15° of latitude) as G'H', but is over twice as long on the
map. At A' (60°
north latitude) the
meridians of the
globe are only half
as far apart as they
are at the equator,
but they are repre
sented on the map
as though they were
just as far apart
there as at the equa
tor. Because of the
rapidly increasing
distances of paral
lels, to represent
higher latitude than
60° would require a
very large sheet, so
the projection is usu
ally modified for a map of the earth as a whole, sometimes
arbitrarily. G' H' is the distance from the equator to the first par
allel, and since a degree of latitude is about equal to a
degree of longitude there, this distance may be taken
between meridians.

60°

N

/
1
1
1
—-Ay
a!
45°
N
a'
30°
N
f'
15°
N
ft'
q'
Eq
uat
or
H'
15°
S
* v "v/
r
30°
S
 v. \
j'
40°
S
\ \ \ \ \ \ \
K'
c
60°
S
Fig. 86
THE MERCATOR (CYLINDRICAL) PROJECTION

211

Stereographic Cylindrical Projection. For reasons just
given, the gnomonic or central cylindrical projection needs
reduction to show the poles at all or any high latitudes
without great distortion. Such a reduction is well shown
in the stereographic projection. In this the eye is con
ceived to be on the equator, projecting each meridian from
the view point of the meridian opposite to it. Figure 87
shows the plan on which it is laid off, meridians being par
allel straight lines
and equidistant and
parallels being paral
lel straight lines at
increasing distances
away from the equa
tor. The Mercator (Cyl
indrical) Projection.
In the orthographic, '
stereographic, globu
lar, gnomonic, hom
olographic, and Van
der Grinten projec
tions, parallels or
meridians, or both,
are represented as
curved lines. It
should be borne in
mind that directions on the earth are determined from
parallels and meridians. North and south are along a
meridian and when a meridian is represented as a curved
line, north and south are along that curved line. Thus the
two arrows shown at the top of Figure 81, are pointing in
almost exactly opposite directions and yet each is point-

/
/
/

A

75°

' s'
/ s

B'

60°
'' ^^"'C\
C
45°
/ ' ''
30°
( '^'
a
15°
/-aEfX"'-'- " " '
  r
r
Cq
uat
or
l^V-:
c
c
\ *N>^
\ s ^ ^
^y v
^-— ^ \
\ X
\ \ \ \
Fig. 87
'212 MAP PROJECTIONS
ing due north. The arrows at the bottom point opposite
each other, yet both point due south. The arrows point
ing to the right point the same way, yet one points north
and the other points south. A line pointing toward the
top of a map may or may not point north. Similarly,
parallels lie in a due east-west direction and to the right
on a map may or may not be to the east.
It should be obvious by this time that the map projec
tions studied thus far represent directions in a most unsat
isfactory manner, however well they may represent areas.
Now to the sailor the principal value of a chart is to show
directions to steer his course by and if the direction is rep
resented by a curved line it is a slow and difficult process
for him to determine his course. We have seen that the
gnomonic projection employs straight lines to represent
arcs of great circles, and, consequently, this projection is
used in great circle sailing. The Mercator projection shows
all parallels and meridians as straight lines at proportional
distances, hence directions as straight lines, and is another,
and the only other, kind of map used by sailors in plotting
their courses.
Maps in Ancient Times. Before the middle of the fif
teenth century, sailors did not cover very great portions
of the earth's surface in continuous journeys out of sight
of land where they had to be guided almost wholly by the
stars. Mathematical accuracy in maps was not of very
great importance in navigation until long journeys had to
be made with no opportunity for verification of calcula
tions. Various roughly accurate map projections were
made. The map sent to Columbus about the year 1474
by the Italian astronomer Toscanelli, with which he sug
gested sailing directions across the " Sea of Darkness," is
an interesting illustration of a common type of his day.'

THE MERCATOR (CYLINDRICAL) PROJECTION 2l3

214 MAP PROJECTIONS
The long journeys of the Portuguese along the coast of
Africa and around to Asia and the many voyages across
the Atlantic early in the sixteenth century, made accurate
map projection necessary. About the middle of that cen
tury, Emperor Charles V of Spain employed a Flemish
mathematician named Gerhard Kramer to make maps for
the use of his sailors. The word Kramer means, in Ger
man, " retail merchant," and this translated into Latin,
then the universal language of science, becomes Mercator,
and his invention of a very valuable and now widely used
map projection acquired his Latinized name.
Plan of Mercator Chart. The Mercator projection is
made on the same plan as the other cylindrical projec
tions, excepting as to the distances between parallels.
The meridians are represented as parallel lines, whereas
on the globe they converge. There is thus a distortion of
longitudes, greater and greater, away from the equator.
Now the Mercator projection makes the parallels farther
apart away from the equator, exactly proportional to the
meridional error. Thus at latitude 60° the meridians on
the earth are almost exactly half as far apart as at the
equator, but being equidistant on the map, they are rep
resented as twice as far apart as they should be. The
parallels in that portion of the Mercator map are accord
ingly made twice as far apart as they are near the equator.
Since the distortion in latitude exactly equals the distor
tion in longitude and parallels and meridians are straight
lines, all directions are represented as straight lines. A
navigator has simply to draw upon the map a line from
the point where he is to the point to which he wishes to
sail in a direct course, measure the angle which this line
forms with a parallel or meridian, and steer his ship accord
ing to the bearings thus obtained.

THE MERCATOR (CYLINDRICAL) PROJECTION 215

To Lay off a Mercator Projection. Figure 89 shows the
simplest method of laying off this projection. From the
extremity of each radius drop a line to the nearest radius,
parallel to the tangent A'L. The lengths of these lines,
respectively, represent the distances * between parallels.
Thus N'M equals CP, K'N' equals BN, A'K' equals A K.
The meridians are
equidistant and are
the same distance
apart as the first
parallel is from the
equator. The table of me
ridional parts on
page 217 gives the
relative distances of
parallels from the
equator. By means
of this table a more
exact projection
may be laid off than
by the method just suggested. To illustrate: Suppose we
wish a map about twenty inches wide to include the 70th
parallels. We find in the table that 5944.3 is the distance
to the equator. Then, since the map is to extend 10 inches
on each side of the equator, r_. . » is the scale to be used
H 5944.3
in making the map; that is, 1 inch on the map will be
represented by 10 inches h- 5944.3. Suppose we wish to

A

/s'
60°

N
N

a
MuAD
n
45°
N
N
30°
N
M
15°
N
i l^r^-yXT
Equ
aior
o f c n
r
15°
S
30°
S
45°
s
60°
s
Fig. 8o
L
75°
s
* Technically speaking, the distance is the tangent of the angle of
latitude and any table of natural tangents will answer nearly as well
as the table of meridional parts, although the latter is more accurate,
being corrected for the oblateness of the meridian.
216

MAP PROJECTIONS

lay off parallels ten degrees apart. The first parallel to be
drawn north of the equator has, according to the table,
599.1 for its meridional distance. This multiplied by
cqtj-q equals slightly more than 1. Hence the parallel
10° should be laid off 1 inch from the equator. The 20th
parallel has for its meridional distance 1217.3. This mul-
10

tiplied by the scale

gives 2.03 inches from the

5944.3
equator. The 30th parallel has a meridional distance

-?

-*

k

!

C

(

(.
a

E *

.1

\vL

if

^

fi

» %

&

OC, 'fdH

(
r1
iw\a
ff^
•S 0
J
<
'it
„
1
'ffljfl ILL
e
P
*':'&
iS-
XL,
'ib
H-
-+¦-**
V
_
,
,
\i
iiAfijiyi
W
•4
"
Xf\ 1
M°±k
»
—
~n'
r
o
0
E j.
A
»>*-
K-E-
w
*h *
1
#
iC
mU
1
t>
0
R
C-T-
10-
0
CE
^
1°
:
j
^
1
1
11
Fig. go. World in mercator projection
1876.9, this multiplied by the scale gives 3.15 inches. In
like manner the other parallels are laid off. The meridians
'  " ' 5944.3 for every de-
will be
X 60 or 600 inches
5944.3
gree, or for ten degrees 6000 inches -h 5944.3, which equals
1.01 inches. This makes the map 36.36 inches long (1.01
inches X 36 = 36.36 inches).
We see, then, that the same scale of miles cannot be used
for different parts of the map, though within 30° of the
equator representations of areas will be in very nearly true
proportions. The parallels in a map not wider than this,
say for Africa, may be drawn equidistant and the same
THE MERCATOR (CYLINDRICAL) PROJECTION 217

distance apart as the meridians, the inaccuracy not being
very great. Table of Meridional Parts *

1°

59.6

18°

1091 . 1

35°

2231 . 1

52°

3647.1

69°

2°

119.2

19°

1154.0

36°

2304.5

53°

3745.4

70°

3°

178.9

20°

1217.3

37°

2378.8

54°

3846.1

71°

4°

238.6

21°

1281.0

38°

2454.1

55°

3949.1

72°

5°

298.4

22°

1345.1

39°

2530.5

56°

4054.9

73°

6°

358.3

23°

1409.7

40°

2607.9

57°

4163.4

74°

7°

418.3

24°

1474.7

41°

2686.5

58°

4274.8

75°

8°

478.4

25°

1540.3

42°

2766.3

59°

4389.4

76°

9°

538.6

26°

1606.4

43°

2847.4

60°

4507.5

77°

10°

599.1

27°

1673.1

44°

2929.9

61°

4628.1

78°

11°

659.7

28°

1740.4

45°

3013.7

62°

4754.7

79°

12°

720.6

29°

1808.3

46°

3099.0

63°

4884.5

80°

13°

781.6

30°

1876.9

47°

3185.9

64°

5018.8

81°

14°

842.9

31°

1946.2

48°

3274.5

65°

5158.0

82°

15°

904.5

32°

2016.2

49°

3364.7

66°

5302.5

83°

16°

966.4

33°

2087.0

50°

3456.9

67°

5452.8

84°

17°

1028.6

34°

2158.6

51°

3551.0

68°

5609.5

85°

5773.1
5944.3 6124.0
6313.06512.46723.66948.1
7187.87444.8
7722.1 8023.18352.6
8716.4 9122.79583.0
10114.010741.7

SUMMARY
In the cylindrical projection:
1. A cylinder is conceived to be wrapped around the globe, tangent
to the equator.
2. All parallels and meridians are represented as straight lines, the
former intersecting the latter at right angles.
3. The parallels are made at increasing distances away from the
equator: a. In the gnomonic projection, as though projected from
the center of the earth to the tangent cylinder.
b. In the stereographic projection, as projected from the
equator upon an opposite meridian, the projection point
varying for each meridian.
c. In the Mercator projection, at distances proportional to
the meridional excess.
Directions are better represented in this projection than in
any other. Here northward is directly toward the top
of the map, eastward directly toward the right, etc.
For this reason it is the projection most commonly
employed for navigators' charts.
* From Bowditch's Practical Navigator.

218

MAP PROJECTIONS

4. There ir: great distortion of areas and outlines of continents in
hifffc [latitudes; Greenland appears larger than South America.
5. The entire globe may be represented in one continuous map.
6. Ij c same scale of miles cannot be used for high latitudes that is
used neai the equator. Conic Projection
The portion of a sphere between the
planes of two parallels which are near
together is very similar to the zone of a
cone (see Fig. 91). Hence, if we imagine
a paper in the form of a cone placed
upon the globe and parallels and
meridians projected upon this cone from
the center of the globe, then this coni
cal map unrolled, we can understand
this system.
Along the parallel tangent to the
cone, points on the map will correspond
exactly to points upon the globe. Par
allels which are near the line of tangency will be repre
sented very much in the relative
positions they occupy on the
globe. In a narrow zone, there
fore, near the tangent parallel,
there will be very little distor
tion in latitudes and longitudes
and an area mapped within the
zone will be very similar in form
and area to the form and area
as it appears upon the globe
itself. For this reason the conic
projection, or some modification of it, is almost always
employed in representing small areas of the earth's surface.

Fig. 91

Fig. 93

TO LAY OFF A CONIC PROJECTION

219

If the forty-fifth parallel

To Lay off a Conic Projection.
is the center of the
area to be mapped,
draw two straight
lines tangent to the
forty-fifth parallel
of a circle (see Fig.
93). Project upon
these lines points
for parallels as in
the gnomonic pro
jection. With the
apex as center,
draw arcs of circles
through these points for parallels. Meridians are straight
lines meeting at the
apex and are equidis
tant along any parallel.
It will be observed
that parallels are farther
apart away from the
tangent parallel (45°,
in this case) as in
the Mercator projection
they are farther apart
away from the equator,
which is tangent to the
globe in that projection.
There is also an exag
geration of longitudes
away from the tangent
Fig. 94. North America in conic projection parallel. Because of this
lengthening of parallels, meridians are sometimes curved

220

MAP PROJECTIONS

inwardly to prevent too much distortion of areas. The
need for this will be apparent if one draws parallels be-

Fig. 95. The world in conic projection
yond the equator, for he will find they are longer than
the equator itself unless meridians curve inwardly there.
By taking the tan
gent parallel ten degrees
north of the equator and
reducing distances of
parallels, a fan-shaped
map of the world may
be shown. In this map
of the world on the
conic projection, there
is even greater distor
tion of parallels south of
the equator, but since
meridians converge somewhat north of the equator there
is less distortion in northern latitudes. Since most of the
land area of the globe is in the northern hemisphere, this

/ /yLJ H-4JI \ \

-)

i \ \x\y\!x\K

(\ \ L-V*\*\ \^A:
¦/y*\f- Jt0~i*~°~£u. ^hcv
^^^v
Fig. 96. Europe in conic projection
INTERSECTING CONIC PROJECTION

221

projection is much better suited to represent the entire
world than the Mercator projection.
Bonne's (Conic) Projection. This is a modification of
the conic projection as previously described to prevent
exaggeration of areas away from the parallel which is con
ceived to be touching the globe. The central meridian is
a straight line and parallels are concentric equidistant
circles. The distance between parallels is the length
of the arc of the circle which is used as a basis for the
projection. For ordinary purposes, the distance AB
(Fig. 93) may be taken for each of the distances between
parallels. Having laid off the central meridian and marked off the
arcs for parallels, the true distance of the meridian on each
parallel is laid off and the meri
dian is drawn through these
points. This gives a gentle
inward curve for meridians
toward the outside of the map
of continents. Instead of fol
lowing Bonne's system with
strict accuracy, the map maker
sometimes makes the curve a
little less in lower latitudes,
allowing a slight exaggeration
of areas to permit the putting
in of more details where they
are needed.
Intersecting Conic Projection. Where a considerable
extent in latitude is to be represented, the cone is some
times conceived to cut into the sphere. In this case,
each meridian intersects the sphere at two parallels (see
Fig. 97) and since along and near the tangent parallels

Fig. 97

222 MAP PROJECTIONS
(A and B) there is little distortion, this plan is better
adapted for a map showing greater width north and south
than is the conic projection.
The map of Europe well illustrates this difference.
Europe lies between 35° and 75° north latitude. On a
conic projection the tangent parallel would be 55°. Near
this parallel there would be no exaggeration of areas
but at the extreme north and south, 20° away from this
parallel, there would be considerable distortion. If, instead,
we make an intersecting conic projection, we should have
the cone pass through parallels 45° and 65° and along
these parallels there would be no distortion and no part
of the map being more than 10° away from these lines,
there would be very little exaggeration anywhere.
It should be noticed that the region between the inter
sections of the meridians must be projected back toward
the center of the sphere and thus be made smaller in the
map than it appears on the globe. The central parallel
would be too short in proportion to the rest. Since this
area of Europe (between 45° and 65°) is the most impor
tant portion and should show most details, it would be a
serious defect, from the practical map maker's point of
view, to minify it.
Polyconic Projection. This differs from the conic pro
jection in that it is readjusted at each parallel which is
drawn, so that each one is tangent to the sphere. This
makes the circumscribing cone bent at each parallel, a
series of conic sections. The word polyconic means
" many cones." The map constructed on this projection
is thus accurate along each parallel, instead of along but
one as in the conic projection or along two as in the inter
secting conic projection. For representing small areas
this is decidedly the most accurate projection known.

POLYCONIC PROJECTION

223

Since the zone along each parallel is projected on an
independent cone, the point
which is the apex for one cone
will not be the same for any
other (unless both north and
south latitudes are shown' in
the same map). In the conic
projection the parallels are all
made from the apex of the
cone as the center. In the
polyconic projection each par
allel has its own conical apex
and hence its own center. This
may easily be observed by a
comparison of the parallels in
Figure 94 (conic projection,
all made from one center) and
those in Figure 98 (polyconic projection, each made from
a different center)

MM

Fig. 98. Africa ana Europe in
polyconic projection

SUMMARY

In the conic projection :
1. A cone is conceived to be fitted about a portion of the globe,
tangent to some parallel.
2. The tangent parallel shows no distortion and portions near it have
but Uttle. This projection is therefore used extensively for
mapping small areas.
a. In the conic projection on the gnomonic or central plan,
the eye is conceived to be at the center of the globe,
parallels are crowded closer together toward the central
parallel, and distant areas are exaggerated.
The cone may be conceived to intersect the globe at two
parallels, between which there is a diminution of areas and
beyond which there is an exaggeration of areas.

224 MAP PROJECTIONS
6. In the Bonne projection parallels are drawn at equidistant
intervals from a common center and meridians are slightly
curved to prevent distortion in longitudes.
u. In the polyconic projection many short conic sections are
conceived to be placed about the globe, one for each parallel
represented. Parallels are drawn from the apexes of the
cones.
The Scale
The area of any map bears some proportion to the actual
area represented. If the map is so drawn that each mile
shall be represented by one inch on the map, since one
mile equals 63,360 inches, the scale is said to be 1 : 63,360.
This is often written fractionally, 7777-^7: ¦ A scale of two
63,360
inches to the mile is 1:31,680. These, of course, can be
used only when small areas are mapped. The following
scales with their equivalents are most commonly used in
the United States Geological Survey, the first being the
scale employed in the valuable geological folios covering
a large portion of the United States.
Scale 1:125,000, 1 mile = 0.50688 inches.
Scale 1 :90,000, 1 mile = 0.70400 inches.
Scale 1:62,500, 1 mile = 1.01376 inches.
Scale 1:45,000, 1 mile = 1.40800 inches.
Some Conclusions
The following generalizations from the discussion of
map projections seem appropriate.
1. In all maps north and south lie along meridians and
east and west along parallels. The top of the map may or
may not be north; indeed, the cylindrical projection is the
only one that represents meridians by perpendicular lines.

POLYCONIC PROJECTIONS 225
2. Maps of the same country on different projections
may show different shapes and yet each may be correct.
To make maps based upon some arbitrary system of
triangles or lines is not scientific and often is not even
helpful. 3. Owing to necessary distortions in projecting the
parallels and meridians, a scale of miles can rarely be used
with accuracy on a map showing a large area.
4. Straight lines on maps are not always the shortest
distances between two points This will be clear if we
remember that the shortest distance between two points
on the globe is along the arc of a great circle. Now great
circles, such as meridians and the equator, are very often
represented as curved lines on a map, yet along such a
curved line is the shortest distance between any two places
in the line on the globe which the map represents.
5. To ascertain the scale of miles per inch used on any
map, or verify the scale if given, measure the space along
a meridian for one inch and ascertain as correctly as possible
the number of degrees of latitude contained in the inch.
Multiply this by the number of miles in one degree of
latitude, 69, and you have the number of miles on the
earth represented by one inch on the map.

CHAPTER XI
THE UNITED STATES GOVERNMENT LAND SURVEY
Allowance for Curvature. One of the best proofs that
the earth is a sphere is the fact that in all careful measure
ments over any considerable area, allowance must be made
for the curvature of the surface. If two lines be drawn
due northward for one mile in the northern part of the
.United States or in central Europe, say from the 48th
parallel, they will be found nearer together at the north
ern extremities than they are at the southern ends.
Origin of Geometry. One of the greatest of the prac
tical problems of mathematics and astronomy has been
the systematic location of lines and points and the measure
ment of surfaces of the earth by something more definite,
more easily described and relocated than metes and bounds.
Indeed, geometry is believed to have had its origin in the
need of the ancient Egyptians for surveying and reloca
ting the boundaries of their lands after the Nile floods.
Locating by Metes and Bounds. The system of locating
lands by metes and bounds prevails extensively over the
world and, naturally enough, was followed in this country
by the early settlers from Europe. To locate an area by
landmarks, some point of beginning is established and the
boundary lines are described by means of natural objects
such as streams, trees, well established highways, and
stakes, piles of stone, etc., are placed for the purpose. The
directions are usually indicated by reference to the magnetic
compass and distances as ascertained by surveyors' chains.
But landmarks decay and change, and rivers change their
226

LOCATING BY METES AND BOUNDS 227
courses.* The magnetic needle of the compass does not
point due north (excepting along two or three isogonal
lines, called agones), and varies from year to year. This
gives rise to endless confusion, uncertainty, and litigation.
Variation almost without limit occurs in such descrip
tions, and farms assume innumerable forms, sometimes
having a score of angles. The transitory character of such
platting of land is illustrated in the following excerpt from
a deed to a piece of property in Massachusetts Bay Colony,
bearing the date: " Anno Domini one thousand seven hun
dred and thirty-six and in the tenth year of the reign of
our sovereign Lord George the Second, King." In this,
Emma Blowers deeds to William Stanley, " A certain par
cel of Upland and Swamp Ground Situate and lying in the
Township of Manchester being the thirty-first lot into the
Westerly Division of Common Rights made in said Man
chester by the proprietors thereof in the year of our Lord
one thousand six hundred ninety-nine, Said lot containing
Ten Acres, more or less, being cutted and bounded as
followeth Viz : At the Northeast Corner with a maple tree
between Sowest and Abraham Master's, from that South-
* Where a, meandering river constitutes the boundary of a nation
or state, changes in the course of the stream give rise to problems in
civil government, as the following incident illustrates. A minister in
the southern part of South Dakota was called upon to officiate at a
wedding in a home in a bend of the Missouri River. During the high
water of the preceding spring, the river had burst over the narrow
neck at the bend and at the time of the wedding it was flowing on
both sides of the cut-off so that there was a doubt as to whether the
main channel of the stream, the interstate boundary line, was north
of them and they were in Nebraska, or south and they were still in
South Dakota. To be assured of the legality of the marriage rite,
the bridal couple, minister, and witnesses rowed to the north bank,
and up on the South Dakota bluff the marriage service was per
formed, the bridal party returning — they cared not to which state,
for the festivities.

228 THE UNITED STATES GOVERNMENT LAND SURVEY

easterly thirty-nine poles to Morgan's Stump, so called,
from that Southeasterly fourty-four poles upon said west
Farm Line to a black Oak tree, from that Sixty-six poles
Northward to the first bounds, or however Otherwise the
Said Lot is or ought to have been bounded."
Survey of Northwest Territory. When, in 1785, practi
cally all of the territory north and west of the Ohio River
had been ceded to the United States by the withdrawal of
state claims, Congress
provided for its sur
vey, profiting from the
experiences resulting
from hastily marked
boundaries. Thomas
Hutchins was ap
pointed Geographer of
the United States, and
after the selection of
thirteen assistants, he
was instructed to begin
its survey. Starting in
1786 from the south
west corner of Pennsylvania, he laid off a line due north
to a point on the north bank of the Ohio River. From
this point he started a line westward. According to the
directions of Congress, every six miles along this east-
west " geographer's line," meridians were to be laid off
and parallels to it at intervals of six miles, each of the
six miles square to be divided into thirty-six square
miles and these divided into " quarters," thus spreading a
huge " gridiron " over the land. The larger squares were
called "townships," an adaptation of the New Eng
land " town." They are commonly called " Congressional

36

30

24-

18

12

6

35

29

23

17

II

5

34-

28

22

16

IO

4-

33

27

21

15

9

3

32

26

20

14-

8

Z

3/

25

19

13

7

1

Fig. 99

SURVEY OF NORTHWEST TERRITORY

229

6

5

4

3

2

1

7

8

9

IO

II

12

18

17

16

IS

14-

13

19

20

21

22

23

24

30

29

28

27

26

25

31

32

33

34

35

36

Fig. ioo

townships " in most parts of the United States, to distin
guish them from the political subdivision of the county
called the " civil town
ship " or the " muni
cipal township."
Jefferson is believed
to have suggested this
general plan which with
some variations has
been continued over
the major portion of
the United States and
the western portion of
Canada. Hutchins and
his crew laid off the
" geographer's line "
only forty-two miles, making seven ranges of townships
west of the Pennsyl
vania state boundary,
when they were fright
ened away by the
Indians. The work
was continued, how
ever, on the same gen
eral plan one exception
being the method of
numbering the sec
tions. In these first
" seven ranges " the
sections are numbered
as in Figure 99, else
where in the United States they are numbered as in
Figure 100, and in western Canada as in Figure 101.

31

32

33

34

35

36

30

29

28

27

26

25

19

20

21

22

23

24

18

17

16

15

14

13

7

8

9

IO

II

12

6

5

4

3

2

1

Fig. ioi

230 THE UNITED STATES GOVERNMENT LAND SURVEY
Each of the square miles is commonly called a " sec
tion." The law passed by Congress May 20, 1785, provided that,
" The surveyors . . . shall proceed to divide the said ter
ritory into townships of six miles square, by lines running
due north and south, and others crossing these at right
angles, as near as may be." Owing to the convergence of
the meridians this, of course, was a mathematical impos
sibility; " as near as may be," however, has been broadly
interpreted. According to the provisions of this act and
the acts of May 18, 1796, May 10, 1800, and Feb. 11, 1805,
and to rules of commissioners of the general land office, a
complete system has been evolved, the main features of
which are as follows:
Principal Meridians. These are run due north, south,
or north and south from some initial point selected
with great care and located in latitude and longitude by
astronomical means. Thirty-two or more of these prin
cipal meridians have been surveyed at irregular intervals
and of varying lengths. Some of these are known by
numbers and some by names. The first principal meri
dian is the boundary line between Indiana and Ohio;
the second is west of the center of Indiana, extending the
entire length of the state; the third is in the center of
Illinois, extending the entire length of the state ; the Talla
hassee principal meridian passes directly through that
city and is only about twenty-three miles long; other
principal meridians are named Black Hills, New Mexico,
Indian, Louisiana, Mount Diablo, San Bernardino,* etc.
* The entire platting of the portions of the United States to which
this discussion refers is clearly shown on the large and excellent maps
of the United States, published by the Government and obtainable
at the actual cost, eighty cents, from the Commissioner of the General
Land office, Washington, D. C.

BASE LINES 231
To the east, west, or east and west of principal meridians,
north and south rows of townships called ranges are laid
off. Each principal meridian, together with the system of
townships based upon it, is independent of every other
principal meridian and where two systems come together.
irregularities are found.
Base Lines. Through the initial point selected from
which to run the principal meridian, an east-west base line
is run, at right angles to it, and corresponds to a true geo
graphic parallel. As in case of the principal meridian,
this line is laid off with great care since the accuracy of
these controlling lines determines the accuracy of the
measurements based upon them.
Tiers of townships are laid off and numbered north and
south of these base lines. In locating a township the word
tier is usually omitted; township number 4 north, range 2
west of the Michigan principal meridian, means the town
ship in tier 4 north of the base line and in the second range
west of the Michigan principal meridian. This is the
township in which Lansing, Michigan, is located.
The fourth principal meridian in western Illinois and
Wisconsin has two base lines, one at its southern extremity
extending westward to the Mississippi River and the other
constituting the interstate boundary line between Wiscon
sin and Illinois. The townships of western Illinois are
numbered from the southern base line, and all of those in
Wisconsin and northeastern Minnesota are numbered from
the northern base line. The fourth principal meridian is
in three sections, being divided by an eastern bend of the
Mississippi River and by the western portion of Lake
Superior. The largest area embraced within one system is that
based upon the fifth principal meridian. This meridian

232 THE UNITED STATES GOVERNMENT LAND SURVEY

STANDARD PARALLELS 233
extends northward from the mouth of the Arkansas River
until it again intersects the Mississippi River in north
eastern Missouri and then again it appears in the big east
ern bend of the Mississippi River in eastern Iowa. Its
base line passes a few miles south of Little Rock, Arkan
sas, from which fact it is sometimes called the Little Rock
base line. From this meridian and base line all of Arkan
sas, Missouri, Iowa, North Dakota, and the major portions
of Minnesota and South Dakota have been surveyed, an
area considerably larger than that of Germany and Great
Britain and Ireland combined. The most northern tier
from this base lies about a mile south of the forty-ninth
parallel, the boundary line between the United States and
Canada, and is numbered 163. The southern row of sec
tions of tier 164 with odd Jottings lies between tier 163 and
Canada. Its most northern township is in the extreme
northern portion of Minnesota, west of the Lake of the
Woods, and is numbered 168. It thus lies somewhat
more than a thousand miles north of the base from which
it was surveyed. There are nineteen tiers south of the
base line in Arkansas, making the extreme length of this
area about 1122 miles. The most eastern range from the
fifth principal meridian is numbered 17 and its most
western, 104, making an extent in longitude of 726
miles. Standard Parallels. The eastern and western boundaries
of townships are, as nearly as may be, true meridians, and
when they have been extended northward through several
tiers, their convergence becomes considerable. At latitude
40° the convergence is about 6.7 feet per mile or somewhat
more than 40 feet to each township. To prevent this
diminution in size of townships to the north of the base
line, standard parallels are run, along which six-mile

234 THE UNITED STATES GOVERNMENT LAND SURVEY

measurements are made for a new set of townships. These
lines are also called correction lines for obvious reasons.
Division of Dakotas. When Dakota Territory was
divided and permitted to enter the Union as two states,
the dividing line agreed upomwas the seventh standard par
allel from the base line of the fifth principal meridian.
This line is about four miles south of the parallel 46° from
the equator and was chosen in preference to the geo
graphic parallel because it was the boundary line between
farms, sections, town
ships, and, to a consid
erable extent, counties.
The boundary line be
tween Minnesota and
Iowa is what is called
a secondary base line
and corresponds to a
standard parallel be
tween tiers 100 and 101
north of the base line
of the fifth principal
meridian. The standard paral
lels have been run at varying intervals, the present dis
tance being 24 miles. None at all were used in the earlier
surveys. Since public roads are usually built on section
and quarter section lines, wherever a north-south road
crosses a correction line there is a " jog " in the road, as a
glance at Figure 103 will show.
Townships Surveyed Northward and Westward. The
practice in surveying is to begin at the southeast corner of
a township and measure off to the north and west. Thus
the sections in the north and west are liable to be larger

Fig. 103

LEGAL SUBDIVISIONS OF A SECTION

235

or smaller than 640 acres, depending upon the accuracy of
the survey. In case of a fractional township, made by the
intervention of large bodies of water or the meeting of

^T~
* 3
% 1
* 3
"^T7!
* 13
TV* — r-'
7 Is
ST -> |j_
e'.i
r*~TZ
2 }/
J*^l
5 !6-,
7 |e
s|e^
r|a
s !e.
-j Sivy*
se/*
0
SWA
SE/4
SW/*
SE1/*
O
SW/*
SE'/*
SW/*
SE'/*
/
SW'/*.
SE'/*
2!
NE/*
O
O
10
II
12
-! sw/*
tL-
SE&
O
if
11
V Lot 4, Section 7
XLot /.Section I.
Fig. 104
.another system of survey or a state line, the sections bear
the same numbers they would have if the township were
full. Irregular surveys and other causes sometimes make
the townships or sec
tions considerably
larger than the desired
area. In such cases 40
acre lots, or as near
that size as possible,
appear in the northern
row of sections, the
other half section re
maining as it would
otherwise be. These
lots may also appear
in the western part of
a township, and the
discrepancy should appear in the western half of each
section. This is illustrated in Figure 104.
Legal Subdivisions of a Section. The legal subdivisions
N.'/s
320
Sec. 4
Acres
N'/z Sivy*
80 A.
SE'/+
160 Acres
N'/iSWASW/s SO A.
SE'A-SW'/*
40A.
5W/+SW/4
IOA.
Fig. 105
236 THE UNITED STATES GOVERNMENT LAND SURVEY
of a section are by halves, quarters, and half quarters. The
designation of the portions of a section is marked in Figure
105. The abbreviations look more unintelligible than they
really are. Thus N. E. 1 of S. E. | of Set. 24, T. 123 N. R.
64 W. 5 p.m. means the northeast quarter of the southeast
quarter of section 24, in tier of townships number 123 north,
and in range 64 west of the fifth principal meridian. Any
such description can easily be located on the United States
map issued by the General Land Office.

CHAPTER XII

TRIANGULATION IN MEASUREMENT AND SURVEY

The ability to measure the distance and size of objects
without so much as touching them seems to the child or
uneducated person to be a great mystery, if not an impos
sibility. Uninformed persons sometimes contend that
astronomers only guess at the distances and dimensions
of the sun, moon, or a planet. The principle of such meas
urement is very simple and may easily be applied.
To Measure the Width of a Stream. Suppose we wish to
measure the width of a river,
yard, or field without actually
crossing it. First make a
triangle having two equal
sides and one right angle
(Fig. 106) . Select some easily
distinguished point on the
farther side, as X (Fig. 107), and find a convenient point
opposite it, as B. Now carry the triangle to the right or
left of B until by sight-
 ing you see that the
long side is in line with
B when the short side
 is in line with X. You
^c will then form the tri
angle BAX or BCX.
It is apparent (by simi
lar triangles) that AB or CB equals BX. Measure off AB
or BO and you will have BX, the distance sought. If
237

Fig. 106

"7TT

River, or

Fie{d

Fig. 107

238 TRIANGULATION IN MEASUREMENT AND SURVEY
you measure both to the right and to the left and take the
.average of the two you will get a more nearly correct
result. To Measure the Height of an Object. In a similar man
ner one may measure the height of a flagstaff or building.
Let X represent the highest point in the flagstaff (Fig. 108)
and place the triangle on or near the ground, with the
short side toward X and long side level. The distance to
the foot of the pole is its height. It is easy to see from
this that if we did not have a triangle just as described,
say the angle at the
p£ point of sighting was
less, by measuring
that angle and look
ing up the value of
itstangent in a trigo
nometrical table, one
could as easily cal-
'^h et^^c- culate the height or
distance. The angle
of the triangle from
which sighting was done is 45°, its tangent is 1. OOOO, that is,
XB equals 1.0000 times BC. If the angle used were 20°, in
stead of 45°, its tangent would be .3640; that is, XB would
equal .3640 times BC. If the angle were 60°, the tangent
would be 1.7321, that is, XB would equal that number
times BC. A complete list of tangents for whole degrees
is given in the Appendix. With the graduated quadrant
the student can get the noon altitude of the sun (though
for this purpose it need not be noon), and by getting the
length of shadow and multiplying this by its natural tan
gent get the height of the object. If it is a building that
is thus measured, the distance should be measured from

>\

Fig. 108

TO MEASURE THE HEIGHT OF AN OBJECT 239
the end of the shadow to the place directly under the point
casting the longest shadow measured.
Two examples may suffice to illustrate how this may
be done.
1. Say an object casts a shadow 100 feet from its base
when the altitude of the sun is observed to be 58°. The
table shows the tangent of 58° to be 1.6003. The height
of the object, then, must be 1.6003 times 100 feet or 160.03
feet. 2. Suppose an object casts a shadow 100 feet when the
sun's height is observed to be 68° 12'. Now the table does
not give the tangent for fractions of degrees, so we must
add to tan 68° \ of the difference between the values of
tan 68° and tan 69° (12' = \°).
The table shows that
tan 69° = 2.6051, and
tan 68° = 2.4751, hence the
difference = 0.1300.
\ of .1300 = 0.0260, and since
tan 68° = 2.4751, and we have found that
tan 12' = 0.0260, it follows that
tan 68° 12'= 2.5011.
Multiplying 100 feet by this number representing the
value of tan 68° 12'
100 feet X 2.5011 = 250.11 feet, answer.
By simple proportion one may also measure the height
of an object by the length of the shadow it casts. Let XB
represent a flagstaff and BC its shadow on the ground
(Fig. 108). Place a ten-foot pole (any other length will
do) perpendicularly and measure the length of the shadow

240 TRIANGULATION IN MEASUREMENT AND SURVEY
it casts and immediately mark the limit of the shadow
of the flagstaff and measure its length in a level line.
Now the length of the flagstaff will bear the same ratio to
the length of the pole that the length of the shadow of
the flagstaff bears to the length of the shadow of the
pole. If the length of the flagstaff's shadow is 60 feet and
that of the pole is 6 feet, it is obvious that the former is
ten times as high as the latter, or 100 feet high. In formal
proportion
BX : B'X' ::BC : B'C.
To Measure the Width of the Moon. To measure the
width of the moon if its distance is known. Cut from a
piece of paper a circle one inch in diameter and paste it

Moon

Fig. 109
high up on a window in view of the full moon. Find how
far the eye must be placed from the disk that the face of
the moon may be just covered by the disk. To get this
distance it is well to have one person hold the end of a
tapeline against the window near the disk and the observer

HOW ASTRONOMERS MEASURE SIZES AND DISTANCES 241
hold the line even with his eye. You then have three ele
ments of the following proportion:
Dist. to disk: dist. to moon:: width of disk: width of moon.
From these elements, multiplying extremes and means
and dividing, it is not difficult to get the unknown element,
the diameter of the moon. If the student is careful in his
measurement and does not forget to reduce all dimensions
to the same denomination, either feet or inches, he will be
surprised at the accuracy of his measurement, crude though
it is.
How Astronomers Measure Sizes and Distances. It is by
the aid of these principles and the use of powerful and
accurate instruments that the distances and dimensions of
celestial bodies are determined, more accurately, in some
instances, than would be likely to be done with rod and
chain, were such measurement possible.
In measuring the distance of the moon from the earth
two observations may be made at the same moment from
widely distant points on the earth. Thus a triangle is
formed from station A and station B to the moon. The
base and included angles being known, the distance can
be calculated to the apex of the triangle, the moon. There
are several other methods based upon the same general
principles, such as two observations from the same point
twelve hours apart. Since the calculations are based upon
lines conceived to extend to the center of the earth, this is
called the geocentric parallax (see Parallax in Glossary).
It is impossible to get the geocentric parallax of other
stars than the sun because they are so far away that lines
sighted to one from opposite sides of the earth are appar
ently parallel. It is only by making observations six
months apart, the diameter of the earth's orbit forming the
JO. MATH. GEO. — 16

242 TRIANGULATION IN MEASUREMENT AND SURVEY
base of the triangle, that the parallaxes of about forty stars
have been determined and even then the departure from
the parallel is so exceedingly slight that the distance can
be given only approximately. The parallax of stars is
called heliocentric, since the base passes through the cen
ter of the sun.

Survey by Triangulation
A method very extensively employed for exact measure
ment of land surfaces is by laying off imaginary triangles
across the surface, and by measuring the length of one side
and the included angles all other dimensions may be accu
rately computed. Immense areas in India, Russia, and
North America have been thus surveyed. The triangu
lation surveys of the United States comprise nearly a

87140'

87

30'

87'

20'

TRIANGULATION

IN MICHIGAN
.Triloba
SCALE OF MILES
0 12 3 16 10
Baldwin^
„ Dexter^--""^
18° a>- —
Morgan
^Mesnard
46°
80' "^-~-^__^
F'
Carp
30'
tr
40'
87'
30'
87"
20'
Fig. no
million square miles extending from the Atlantic to the
Pacific. This work has been carried on by the United*
States Geological Survey for the purpose of mapping the
topography and making geological maps, and by the United
States Coast and Geodetic Survey.
DETERMINATION OF BASE LINE 243
Determination of Base Line. The surveyor selects two
points a few miles apart where the intervening surface is
level. The distance between these points is ascertained,
great care being used to make it as correct as possible, for
this is the base line and all calculations rest for their accu
racy upon this distance as it is the only line measured.
The following extracts from the Bulletin of the United
States Geological Survey on Triangulation, No. 122, illus
trate the methods employed. " The Albany base line ("in
central Texas) is about nine miles in length and was meas
ured twice with a 300-foot steel tape stretched under a
tension of 20 pounds. The tape was supported by stakes
at intervals of 50 feet, which were aligned and brought to
the grade established by more substantial supports, the
latter having been previously set in the ground 300 feet
apart, and upon which markings of the extremities of the
tape were made. The two direct measurements differed
by 0.167 foot, but when temperature corrections were
applied the resulting discrepancy was somewhat greater,
owing possibly to difficulty experienced at the time of
measurements in obtaining the true temperature of the
tape. The adopted length of the line after applying the
corrections for temperature, length of tape, difference on
posts, inclination, sag, and sea level, was 45,793.652 feet."
" The base line (near Rapid City, South Dakota) was
measured three times with a 300-foot steel tape; temper
ature was taken at each tape length; the line was supported
at each 50 feet and was under a uniform tension of 20
pounds. The adopted length of the line after making cor
rections for slope, temperature, reduction to sea level, etc.,
is 25,796.115 feet (nearly 5 miles), and the probable error
of the three measurements is 0.84 inch." " The Gunni
son line (Utah) was measured under the direction of Prof.

244 TRIANGULATION IN MEASUREMENT AND SURVEY
A. H. Thompson, in 1875, the measurement being made
by wooden rods carried in a trussed wooden case. These
rods were oiled and varnished to prevent absorption of
moisture, and their length was carefully determined by
comparisons with standard steel rods furnished by the
United States Coast and Geodetic Surveys."
Completion of Triangle. From each extremity of the
base line a third point is sighted and with an instrument
the angle this line forms with the base line is determined.
Thus suppose AB (Fig. Ill) represents the base line. At
A the angle CAB is determined and at B the angle CBA
is determined. Then by trig
onometrical tables the lengths
of lines CA and BC are ex
actly determined. Any one
of these lines may now be
used as a base for another
triangle as with base AB.
If the first base line is cor-
Fig. m rect, and the angles are de
termined accurately, and
proper allowances are made for elevations and the curva
ture of the earth, the measurement is very accurate and
easily obtained, whatever the intervening obstacles between
the points. In some places in the western part of the
United States, long lines, sometimes many miles in length,
are laid off from one high elevation to another. The
longest side thus laid off in the Rocky Mountain region
is 183 miles long.
" On the recent primary triangulation much of the
observing has been done at night upon acetylene lamps;
directions to the distant light keepers have been sent by
the telegraphic alphabet and flashes of light, and the

SURVEY OF INDIAN TERRITORY 245
necessary observing towers have been built by a party
expert in that kind of work in advance of the observing
party." *
Survey of Indian Territory. In March, 1895, Congress
provided for the survey of the lands of Indian Territory
and the work was placed in charge of the Director of the
Geological Survey instead of being let out on contract as
had been previously done. The system of running prin
cipal and guide , meridians, base and correction parallels,
and township and section lines was adopted as usual and
since the topographic map was made under the same direc
tion, a survey by triangulation was made at the same time.
The generally level character of the country made it possi
ble to make triangles wherever desired, so the " checker
board " system of townships has superimposed upon it
triangles diagonally across the townships. In this way the
accurate system of triangulation was used to correct the
errors incident to a survey by the chain. Since so many
lines were thus laid off and all were made with extreme
accuracy, the work of making the contour map was rendered
comparatively simple.
* John F. Hayford, Inspector of Geodetic Work, United States
Coast and Geodetic Survey, in a paper relating to Primary Triangu
lation before the Eighth International Geographic Congress, 1904.

CHAPTER XIII
THE EARTH IN SPACE
The Solar System. The group of heavenly bodies to
which the earth belongs is called, after its great central
sun, the solar system. The members of the solar system
are the sun; eight large planets, some having attendant
satellites or moons; several hundred smaller planets called
asteroids, or planetoids ; and occasional comets and meteors.
The planets with their satellites, and the asteroids all
revolve around the sun in the same direction in elliptical
orbits not far from a common plane. Those visible to the
naked eye may be seen not far from the ecliptic, the path
of the sun in its apparent revolution. The comets and
swarms of meteors also revolve around the sun in greatly
elongated orbits.
The solar system is widely separated from any of the
stars, with which the planets should not be confused. If
one could fly from the earth to the sun, 93,000,000 miles,
in a single day, it would take him only a month to reach
the orbit of the most distant planet, Neptune, but at that
same terrific rate, it would take over seven hundred years
to reach the very nearest of the distant stars. If a circle
three feet in diameter be made to represent the orbit of
the earth, an object over seventy miles away would repre
sent the nearest of the distant stars.
The earth's orbit as seen from the nearest star is as a
circle a trifle over half an inch in diameter seen at a dis
tance of a mile. Do not imagine that the brightest stars
are nearest. 246

NEBULAR HYPOTHESIS 247
From the foregoing one should not fail to appreciate the
immensity of the earth's orbit. It is small only in a rela
tive sense. The earth's orbit is so large that in traveling
eighteen and one half miles the earth departs from a
perfectly straight line only about one ninth of an inch; it
is nearly 584,000,000 miles in length and the average
orbital velocity of the earth is 66,600 miles per hour.
Sun's Onward Motion. It has been demonstrated that
many of the so-called fixed stars are not fixed in relation
to each other but have " proper " motions of their own.
It is altogether probable that each star has its own motion
in the universe. Now the sun is simply one of the stars
(see p. 265), and it has been demonstrated i^hat with its
system of planets it is moving rapidly, perhaps 40,000
miles per hour, toward the constellation Hercules. Many
speculations are current as to whether our sun is controlled
by some other sun somewhat as it controls the planets,
and also as to general star systems. Any statement of
such conditions with present knowledge is little, if any,
more than a guess.
Nebular Hypothesis. Time was when it was considered
impious to endeavour to ascertain the processes by which
God works " in His mysterious way, His wonders to per
form;" and to assign to natural causes and conditions
what had been attributed to God's fiat was thought sacri
legious. It is hoped that day has forever passed.
This great theory as to the successive stages and con
ditions in the development of the solar system, while
doubtless faulty in some details, is at present almost the
only working hypothesis advanced and " forms the foun
dation of all the current speculations on the subject." It
gives the facts of the solar system a unity and significance
scarcely otherwise obtainable.

248 THE EARTH IN SPACE
A theory or a hypothesis, if worthy of serious attention,
is always based upon facts. Some of the facts upon
which the nebular theory is based are as follows :
1. All of the planets are not far from a common
plane. 2. They all revolve around the sun in the same direc
tion. 3. Planetary rotation and revolution are in the same
direction, excepting, perhaps, in case of Uranus and
Neptune. 4. The satellites revolve around their respective planets
in the direction of their rotation and not far from the
plane of revolution.
5. All the members seem to be made up of the same
kinds of material.
6. Analogy.
a. The nebulae we see in the heavens have the same
general appearances this theory assumes the solar system
to have had.
b. The swarms of meteorites making the rings of Saturn
are startlingly suggestive of the theory.
c. The gaseous condition of the sun with its corona
suggests possible earlier extensions of it. The fact that
the sun rotates faster at its equator than at other parts
also points toward the nebular theory. The contraction
theory of the source of the sun's heat, so generally accepted,
is a corollary of the nebular theory.
d. The heated interior of the earth and the charac
teristics of the geological periods suggest this theory as
the explanation.
The Theory. These facts reveal a system intimately
related and pointing to a common physical cause. Accord
ing to the theory, at one time, countless ages ago, all

NEBULAR HYPOTHESIS 249
the matter now making up the solar system was in one
great cloudlike mass extending beyond the orbit of the
most distant planet. This matter was not distributed with
uniform density. The greater attraction of the denser
portions gave rise to the collection of more matter around
them, and just as meteors striking our atmosphere gener
ate by friction the flash of light, sometimes called falling
or shooting stars, so the clashing of particles in this nebu
lous mass generated intense heat.
Rotary Motion. Gradually the whole mass balanced
about its center of gravity and a well-defined rotary
motion developed. As the great nebulous mass con
densed and contracted, it rotated faster and faster. The
centrifugal force at the axis of rotation was, of course,
zero and increased rapidly toward the equator. The force
of gravitation thus being partially counteracted by cen^
trifugal force at the equator, and less and less so at other
points toward the axis, the mass flattened at the poles.
The matter being so extremely thin and tenuous and acted
upon by intense heat, also a centrifugal force, it flattened
out more and more into a disklike form.
As the heat escaped, the mass contracted and rotated
faster than ever, the centrifugal force in the outer portion
thus increased at a greater rate than did the power of
gravitation due to its lessening diameter. Hence, a time
came when the centrifugal force of the outer portions
exactly balanced the attractive power of gravitation and
the rim or outer fragments ceased to contract toward the
central mass; and the rest, being nearer the center of
gravity, shrank away from these outer portions. The
outer ring or ringlike series of fragments, thus left off,
continued a rotary motion around the central mass,
remaining in essentially the same plane.

250 THE EARTH IN SPACE
Planets Formed from Outlying Portions. Since the
matter in the outlying portions, as in the whole mass, was
somewhat unevenly distributed, the parts of it consoli
dated. The greater masses in the outer series hastened
by their attraction the lesser particles back of them,
retarded those ahead of them, and thus one mass was
formed which revolved around the parent mass and
rotated on its axis. If this body was not too dense it
might collect into the satellites or moons revolving around
it. This process continued until nine such rings or lumps
had been thrown off, or, rather, left off. The many small
planets around the sun between the orbit of Mars and
that of Jupiter were probably formed from one whose parts
were so nearly of the same mass that no one by its pre
ponderating attraction could gather up all into a planet.
The explanation of the rings of Saturn is essentially the
same. Conclusion as to the Nebular Hypothesis. This theory,
with modifications in detail, forms the basis for much of
scientific speculation in subjects having to do with the
earth. That it is the ultimate explanation, few will be so
hardy as to affirm. Many questions and doubts have been
thrown on certain phases recently but it is, in a sense,
the point of departure for other theories which may dis
place it. Perhaps even the best of recent theories to
receive the thoughtful attention of the scientific world,
the " planetesimal hypothesis," can best be understood
in general outline, in terms of the nebular theory.
The Planetesimal Hypothesis. This is a new explana
tion of the genesis of our solar system which has been
worked out by Professors Chamberlin and Moulton of
the University of Chicago, and is based upon a very
careful study of astronomical facts in the light of mathe-

NEBULAR HYPOTHESIS 251
matics and astrophysics. It assumes the systems to have
been evolved from a spiral nebula, similar to the most
common form of nebula observed in the heavens. It is
supposed that the nebulous condition may have been
caused by our sun passing so near a star that the tremen
dous tidal strain caused the eruptive prominences (which
the sun shoots out at frequent intervals) to be much
larger and more vigorous than usual, and that these, when
projected far out, were pulled forward by the passing star
and given a revolutionary course about the sun. The arms
of spiral nebula have knots of denser matter at intervals
which are supposed to be due to special explosive impulses
and to become the centers of accretion later. The mate
rial thus shot out was very hot at first, but soon cooled
into discrete bodies or particles which moved independ
ently about the sun like planets (hence the term planetesi
mal). When their orbits crossed or approached each other,
the smaller particles were gathered into the knots, and
these ultimately grew into planets. Less than one seven-
hundredth of the sun was necessary to form the planets
and satellites.
This hypothesis differs from the nebular hypothesis in a
number of important particulars. The latter assumes the
earth to have been originally in a highly heated condition,
while under the planetesimal hypothesis the earth may
have been measurably cool at the surface at all times, the
interior heat being due to the compression caused by
gravity. The nebular hypothesis views the atmosphere as
the thin remnant of voluminous original gases, whereas
the new hypothesis conceives the atmosphere to have been,
gathered gradually about as fast as consumed, and to have
come in part from the heated interior, chiefly by volcanic
action, and in part from outer space. The oceans, accord-

252 THE EARTH IN SPACE
ing to the old theory, were condensed from the great
masses of original aqueous vapors surrounding the earth;
according to the new theory the water was derived from
the same sources as the atmosphere. According to the
planetesimal hypothesis the earth, as a whole, has been
solid throughout its history, and never in the molten state
assumed in the nebular hypothesis.
Solar System not Eternal. Of one thing we may be
reasonably certain, the solar system is not an eternal one.
When we endeavor to extend our thought and imagina
tion backward toward " the beginning," it is only toward
creation; when forward, it is only toward eternity.
"Thy kingdom is an everlasting kingdom,
And thy dominion endureth throughout all generations."
— Psalms, 145, 13.
The Mathematical Geography of the Planets, Moon,
and Sun
The following brief sketches of the mathematical geog
raphy of the planets give their conditions in terms corre
sponding to those applied to the earth. The data and
comparisons with the earth are only approximate. The
more exact figures are found in the table at the end of
the chapter.
Striving for vividness of description occasionally results
in language which implies the possibility of human inhab
itancy on other celestial bodies than the earth, or suggests
interplanetary locomotion (see p. 305). Such conditions
exist only in the imagination. An attempt to exclude
astronomical facts not bearing upon the topic in hand and
not consistent with the purpose of the study, makes nec
essary the omission of some of the most interesting facts.

FORM AND DIMENSIONS 253
For such information the student should consult an astron
omy. The beginner should learn the names of the planets
in the order of their nearness to the sun. Three minutes
repetition, with an occasional review, will fix the order:
Mercury, Venus, Earth, Mars, Asteroids,
Jupiter, Saturn, Uranus, Neptune.
There are obvious advantages in the following discussion
in not observing this sequence, taking Mars first, then
Venus, etc. Mars
Form and Dimensions. In form Mars is very similar
to the earth, being slightly more flattened toward the
poles. Its mean diameter is 4,200 miles, a little more
than half the earth's. A degree of latitude near the
equator is 36.6 miles long, getting somewhat longer toward
the poles as in case of terrestrial latitudes.
Mars has a little less than one third the surface of the
earth, has one seventh the volume, weighs but one ninth
as much, is three fourths as dense, and an object on its
surface weighs about two fifths as much as it would here.
A man weighing one hundred and fifty pounds on the
earth would weigh only fifty-seven pounds on Mars, could
jump two and one half times as high or far, and could
throw a stone two and one half times the distance he could
here.* A pendulum clock taken from the earth to Mars
would lose nearly nine hours in a day as the pendulum
would tick only about seven elevenths as fast there. A
* He could not throw the stone any swifter on Mars than he could
on the earth ; gravity there being so much weaker, the stone would move
farther before falling to the surface.

254 THE EARTH IN SPACE
watch, however, would run essentially the same there as
here. As we shall see presently, either instrument would
have to be adjusted in order 'to keep Martian time as the
day there is longer than ours.
Rotation. Because of its well-marked surface it has
been possible to ascertain the period of rotation of Mars
with very great precision. Its sidereal day is 24 h. 37 m.
22.7 s. The solar day is 39 minutes longer than our solar
day and owing to the greater ellipticity of its orbit the
solar days vary more in length than do ours.
Revolution and Seasons. A year on Mars has 668
Martian days,* and is nearly twice as long as ours. The
orbit is much more elliptical than that of the earth, peri
helion being 26,000,000 miles nearer the sun than aphelion.
For this reason there is a marked change in the amount of
heat received when Mars is at those two points, being
almost one and one half times as much when in perihelion
as when in aphelion. The northern summers occur when
Mars is in aphelion, so that hemisphere has longer, cooler
summers and shorter and warmer winters than the southern
hemisphere. Northern Hemisphere Southern Hemisphere
Spring  191 days Spring  149 days
Summer  181 days Summer  147 days
Autumn  149 days Autumn  191 days
Winter  147 days Winter  181 days
Zones. The equator makes an angle of 24° 50' with the
planet's ecliptic (instead of 23° 27' as with us) so the change
in seasons and zones is very similar to ours, the climate,
of course, being vastly different, probably very cold because
of the rarity of the atmosphere (about the same as on our
* Mars, by Percival Lowell.

FORM AND DIMENSIONS 255
highest mountains) and absence of oceans. The distance
from the sun, too, makes a great difference in climate.
Being about one and one half times as far from the earth,
the sun has an apparent diameter only two thirds as
great and only four ninths as much heat is received over
a similar area.
Satellites. Mars has two satellites or moons. Since
Mars was the god of war of the Greeks these two satellites
have been given the Greek names of Deimos and Phobos,
meaning "dread" and "terror," appropriate for "dogs
of war." They are very small, only six or seven miles in
diameter. Phobos is so near to Mars (3,750 miles from the
surface) that it looks almost as large to a Martian as our
moon does to us, although not nearly so bright. Phobos,
being so near to Mars, has a very swift motion around the
planet, making more than three revolutions around it
during a single Martian day. Now our moon travels
around the earth from west to east, but only about 13° in
a day, so because of the earth's rotation the moon rises
in the east and sets in the west. In case of Phobos, it
revolves faster than the planet rotates and thus rises in
the west and sets in the east. Thus if Phobos rose in the
west at sunset in less than three hours it would be at
meridian height and show first quarter, in five and one
half hours it would set in the east somewhat past the
full, and before sunrise would rise again in the west almost
at the full again. Deimos has a sidereal period of 30.3
hours and thus rises in the east and sets in the west, the
period from rising to setting being 61 hours.
Venus
Form and Dimensions. Venus is very nearly spherical
and has a diameter of 7,700 miles, very nearly that of the

256 THE EARTH IN SPACE
earth, so its latitude and longitude are very similar to ours.
Its surface gravity is about T9T that of the earth. A man
weighing 150 pounds here would weigh 135 pounds there.
Revolution. Venus revolves around the sun in a period
of 225 of our days, probably rotating once on the journey,
thus keeping essentially the same face toward the sun.
The day, therefore, is practically the same as the year, and
the zones are two, one of perpetual sunshine and heat and
the other of perpetual darkness and cold. Its atmosphere
is of nearly the same density as that of the earth. Being
a little more than seven tenths the distance of the earth
from the sun, that blazing orb seems to have a diameter
nearly one and one half times as great and pours nearly
twice as much light and heat over a similar area. Its
orbit is more nearly circular than that of any other planet.
Jupiter
Form and Dimensions. After Venus, this is the bright
est of the heavenly bodies, being immensely large and
having very high reflecting power. Jupiter is decidedly
oblate. Its equatorial diameter is 90,000 miles and its
polar diameter is 84,200 miles. Degrees of latitude near
the equator are thus nearly 785 miles long, increasing to
over 800 miles near the pole. The area of the surface is
122 times that of the earth, its volume 1,355, its mass or
weight 317, and its density about one fourth.
Surface Gravity. The weight of an object on the surface
of Jupiter is about two and two thirds times its weight
here. A man weighing 150 pounds here would weigh 400
pounds there but would find he weighed nearly 80 pounds
more near the pole than at the equator, gravity being so
much more powerful there. A pendulum clock taken from

FORM AND DIMENSIONS 257
the earth to Jupiter would gain over nine hours in a day
and would gain or lose appreciably in changing a single
degree of latitude because of the oblateness of the planet.
Rotation. The rotation of this planet is very rapid,
occupying a little less than ten hours, and some portions
seem to rotate faster than others. It seems to be in a
molten or liquid state with an extensive envelope of gases,
eddies and currents of which move with terrific speed.
The day there is very short as compared with ours and a
difference of one hour in time makes a difference of over
36° in longitude, instead of 15° as with us. Their year
being about 10,484 of their days, their solar day is only a
few seconds longer than their sidereal day.
Revolution. The orbit of Jupiter is elliptical, perihelion
being about 42,000,000 miles nearer the sun than aphelion.
Its mean distance from the sun is 483,000,000 miles, about
five times that of the earth. The angle its equator forms
with its ecliptic is only 3°, so there is little change in
seasons. The vertical ray of the sun never gets more
than 3° from the equator, and the torrid zone is 6° wide.
The circle of illumination is never more than 3° from or
beyond a pole so the frigid zone is only 3° wide. The
temperate * zone is 84° wide.
Jupiter has seven moons. Saturn
Form and Dimensions. The oblateness of this planet
is even greater than that of Jupiter, being the greatest of
* These terms are purely relative, meaning, simply, the zone on
Jupiter corresponding in position to the temperate zone on the earth.
The inappropriateness of the term may be seen in the fact that Jupiter
is intensely heated, so that its surface beneath the massive hot vapors
surrounding it is probably molten.
JO. MATH. GEO.  17

258 THE EARTH IN SPACE
the planets. Its mean diameter is about 73,000 miles. It,
therefore, has 768 times the volume of the earth and 84
times the surface. Its density is the lowest of the planets,
only about one eighth as dense as the earth. Its surface
gravity is only slightly more than that of the earth, vary
ing, however, 25 per cent from pole to equator.
Rotation. Its sidereal period of rotation is about 10 h.
14 m., varying slightly for different portions as in case
of Jupiter. The solar day is only a few seconds longer
than the sidereal day.
Revolution. Its average distance from the sun is
866,000,000 miles, varying considerably because of its
ellipticity. It revolves about the sun in 29.46 of our
years, thus the annual calendar must comprise 322,777
of the planet's days.
The inclination of Saturn's axis makes an angle of 27°
between the planes of its equator and its ecliptic. Thus
the vertical ray sweeps over 54° giving that width to its
torrid zone, 27° to the frigid, and 36° to the temperate.
Its ecliptic and our ecliptic form an angle of 2.5°, so we
always see the planet very near the sun's apparent path.
Saturn has surrounding its equator immense disks, of
thin, gauzelike rings, extending out nearly 50,000 miles
from the surface. These are swarms of meteors or tiny
moons, swinging around the planet in very nearly the
same plane, the inner ones moving faster than the outer
ones and being so very minute that they exert no appre
ciable attractive influence upon the planet.
In addition to the rings, Saturn has ten moons.
Uranus
Form and Dimensions. This planet, which is barely
visible to the unaided eye, is also decidedly oblate, nearly

FORM AND DIMENSIONS 259
as -much so as Saturn. Its mean diameter is given as
from 34,900 miles to 28,500 miles. Its volume, on basis
of the latter (and latest) figures, is 47 times that of the
earth. Its density is very low, about three tenths that of
the earth, and its surface gravity is about the same as
ours at the equator, increasing somewhat toward the
pole. Nothing certain is known concerning its rotation as it
has no distinct markings upon its surface. Consequently
we know nothing as to the axis, equator, days, calendar,
or seasons.
Its mean distance from the sun is 19.2 times that of the
earth and its sidereal year 84.02 of our years.
Uranus has four satellites swinging around the planet
in very nearly the same plane at an angle of 82.2° to the
plane of the orbit. They move from west to east around
the planet, not for the same reason Phobos does about
Mars, but probably because the axis of the planet, the
plane of its equator, and the plane of these moons has
been tipped 97.8° from the plane of the orbit and the
north pole has been tipped down below or south of the
ecliptic, becoming the south pole, and giving a backward
rotation to the planet and to its moons.
Neptune
Neptune is the most distant planet from the sun, is
probably somewhat larger than Uranus, and has about
the same density and slightly greater surface gravity.
Owing to the absence of definite markings nothing is
known as to its rotation. Its one moon, like those of
Uranus, moves about the planet from west to east in a
plane at an angle of 34° 48' to its ecliptic, and its back
ward motion suggests a similar explanation, the inclina-

260 THE EARTH IN SPACE
tion of its axis is more than 90° from the plane of its
ecliptic.
Mercury
This is the nearest of the planets to the sun, and as it
never gets away from the sun more than about the width
of forty suns (as seen from the earth), it is rarely visible
and then only after sunset in March and April or before
sunrise in September and October.
Form and Dimensions. Mercury has about three
eighths the diameter of the earth, one seventh of the sur
face, and one eighteenth of the volume. It probably has
one twentieth of the mass, nine tenths of the density,
and a little less than one third of the surface gravity.
Rotation and Revolution. It is believed that Mercury
rotates once on its axis during one revolution. Owing to
its elliptical orbit it moves much more rapidly when near
perihelion than when near aphelion, and thus the sun
loses as compared with the average position, just as it
does in the case of the earth, and sweeps eastward about
23J° from its average position. When in aphelion it gains
and sweeps westward a similar amount. This shifting
eastward making the sun " slow " and westward making
the sun " fast " is called libration.
Thus there are four zones on Mercury, vastly different
from ours, indeed, they are not zones (belts) in a terres
trial sense.
a. An elliptical central zone of perpetual sunshine,
extending from pole to pole and 133° in longitude. In
this zone the vertical ray shifts eastward 23£° and back
again in the short summer of about 3d days, and westward
a similar extent during the longer winter of about 58
days. Two and one half times as much heat is received

ROTATION 261
in the summer, when in perihelion, as is received in the
winter, when in aphelion. Thus the eastward half of
this zone has hotter summers and cooler winters than
does the western half. Places along the eastern and
western margin of this zone of perpetual sunshine see the
sun on the horizon in winter and only 23^° high in the
summer. b. An elliptical zone of perpetual darkness, extending
from pole to pole and 133° wide from east to west.
c. Two elliptical zones of alternating sunshine and
darkness (there being practically no atmosphere on Mer
cury, there is no twilight there), each extending from
pole to pole and 47° wide. The eastern of these zones
has hotter summers and cooler winters than the western
one has.
The Moon
Form and Dimensions. The moon is very nearly
spherical and has a diameter of 2,163 miles, a little over
one fourth that of the earth, its volume one forty-fourth,
its density three fifths, its mass jYt- an(l ^s surface
gravity one sixth that upon the earth. A pendulum
clock taken there from the earth would tick so slowly
that it would require about sixty hours to register one of
our days. A degree of latitude (or longitude at its
equator) is a little less than nineteen miles long.
Rotation. The moon rotates exactly once in one revo
lution around the earth, that is, keeps the same face
toward the earth, but turns different sides toward the
sun once each month.
Thus what we call a sidereal month is for the moon
itself a sidereal day, and a synodic month is its solar
day. The latter is 29.5306 of our days, which makes the

262 THE EARTH IN SPACE
moon's solar day have 708 h. 44 m. 3.8 s. If its day were
divided into twenty-four parts as is ours, each one would
be longer than a whole day with us.
Revolution and Seasons. The moon's orbit around the
- sun has essentially the same characteristics as to peri
helion, aphelion, longer and shorter days, etc., as that of
the earth. The fact that the moon goes around the earth
does not materially affect it from the sun's view point.
To illustrate the moon's orbit about the sun, draw a circle
78 inches in diameter. Make 26 equidistant dots in
this circle to represent the earth for each new and full
moon of the year. Now for each new moon make a dot
one twentieth of an inch toward the center (sun) from
every other dot representing the earth, and for every full
moon make a dot one twentieth of an inch beyond the
alternate ones. These dots representing the moon, if
connected, being never more than about one twentieth
of an inch from the circle, will not vary materially from
the circle representing the orbit of the earth, and the
moon's orbit around the sun will be seen to have in every
part a concave side toward the sun.
The solar day of the moon being 29.53 of our days, its
tropical year must contain as many of those days as that
number is contained times in 365.25 days or about 12.4
days. The calendar for the moon does not have any
thing corresponding to our month, unless each day be
treated as a month, but has a year of 12.4 long days of
nearly 709 hours each. The exact length of the moon's
solar year being 12.3689 d., its calendar would have the
peculiarity of having one leap year in every three, that is,
two years of 12 days each and then one of 13 days,
with an extra leap year every 28 years.
The earth as seen from the moon is much like the moon

ABSENCE OF ATMOSPHERE 263
as seen from the earth, though very much larger, about
four times as broad. Because the moon keeps the same
face constantly toward the earth, the latter is visible to
only a little over half of the moon. On this earthward
side our planet would be always visible, passing through
precisely the same phases as the moon does for us, though
in the opposite order, the time of our new moon being
" full earth " for the moon. So brightly does our earth
then illuminate the moon that when only the faint cres
cent of the sunshine is visible to us on the rim of the
moon, we can plainly see the " earth shine " on the rest
of the moon's surface which is toward us.
Zones. The inclination of the plane of the moon's
equator to the plane of the ecliptic is 1° 32' (instead of
23° 27' as in the case of the earth). Thus its zone corre
sponding to our torrid* zone is 3° 4' wide, the frigid zone
1° 32', and the temperate zones 86° 56'.
Absence of Atmosphere. The absence of an atmosphere
on the moon makes conditions there vastly different from
those to which we are accustomed. Sunrise and sunset
show no crimson tints nor beautiful coloring and there
is no twilight. Owing to the very slow rotation of the
moon, 709 hours from sun-noon to sun-noon, it takes
nearly an hour for the disk of the sun to get entirely above
the horizon on the equator, from the time the first glint
of light appears, and the time of sunset is equally pro
longed; as on the earth, the time occupied in rising or
setting is longer toward the poles of the moon. The stars
* Again we remind the reader that these terms are not appropriate
in case of other celestial bodies than the earth. The moon has almost
no atmosphere to retain the sun's heat during its long night of nearly
354 hours and its dark surface must get exceedingly cold, probably
several hundred degrees below zero.

264 THE EARTH IN SPACE
do not twinkle, but shine with a clear, penetrating light.
They may be seen as easily in the daytime as at night,
even those very near the sun. Mercury is thus visible
the most of the time during the long daytime of 354
hours, and Venus as well. Out of the direct rays of the
sun, pitch darkness prevails. Thus craters of the vol
canoes are very dark and also cold. In the tropical portion
the temperature probably varies from two or three hundred
degrees below zero at night to exceedingly high tempera
tures in the middle of the day. During what is to the
moon an eclipse of the sun, which occurs whenever we see
the moon eclipsed, the sun's light shining through our
atmosphere makes the most beautiful of coloring as
viewed from the moon. The moon's atmosphere is so
rare that it is incapable of. transmitting sound, so that a
deathlike silence prevails there. Oral conversation is
utterly impossible and the telephone and telegraph as we
have them would be of no use whatever. Not a drop of
water exists on that cold and cheerless satellite.
Perhaps it is worth noting, in conclusion, that it is
believed that our own atmosphere is but the thin remnant
of dense gases, and that in ages to come it will get more
and more rarified, until at length the earth will have the
same conditions as to temperature, silence, etc., which
now prevail on the moon. The Sun
Dimensions. The diameter is 866,500 miles, nearly
four times the distance of the moon from the earth. Its
surface area is about 12,000 times that of the earth, and
its volume over a million times. Its density is about
one fourth that of the earth, its mass 332,000 times, and
its surface gravity is 27.6 times our earth's. A man

THE SUN A STAR 265
weighing 150 pounds here would weigh over two tons
there, his arm would be so heavy he could not raise it
and his bony framework could not possibly support his
body. A pendulum clock there would gain over a hun
dred hours in a day, so fast would the attraction of the
sun draw the pendulum.
Rotation. The sun rotates on its axis in about 25| of
our days, showing the same portion to the earth every
27J days. This rate varies for different portions of the
sun, its equator rotating considerably faster than higher
latitudes. The direction of its rotation is from west to
east from the sun's point of view, though as viewed from
the earth the direction is from our east to our west. The
plane of the equator forms an angle of about 26° with the
plane of our equator, though only about 7£° with the
plane of the ecliptic.
When we realize that the earth, as viewed from the sun,
is so tiny that it receives not more than one billionth of
its light and heat, we may form some idea of the immense
flood of energy it constantly pours forth.
The Sun a Star. " The word ' star ' should be omitted
from astronomical literature. It has no astronomic mean
ing. Every star visible in the most penetrating telescope
is a hot sun. They are at all degrees of heat, from dull
red to the most terrific white heat to which matter can be
subjected. Leaves in a forest, from swelling bud to the
' sere and yellow,' do not present more stages of evolution.
A few suns that have been weighed, contain less matter
than our own; some of equal mass; others are from ten
to twenty and thirty times more massive, while a few are
so immensely more massive that all hopes of comparison
fail. " Every sun is in motion at great speed, due to the attrac-

266

THE EARTH IN SPACE

tion and counter attraction of all the others. They go in
every direction. Imagine the space occupied by a swarm
of bees to be magnified so that the distance between each
bee and its neighbor should equal one hundred miles. The
insects would fly in every possible direction of their own
Solar System Tablb

Object

.a a
02

Mean Diameter (miles)

Sidereal Day

As compared with the earth*

a
s* CD
CD u
CD (3

>>0
A

Mass

CD t>a
as
3>®

"3 .
a) H
Fh c$
CD CD
to

0
ifA*

Mercury

,»

3,000

88 days

6.800

0.85

0.04C

0.330

0.24

0.4

Venus

?

7,700

223 days

1.900

0.94

0.820

0.900

C.62

0.71.0

Earth

©

7,918

*

1.000

1.00

1.000

1.000

1.00

Mars

$

4,230

24h 37m 22.7s

0.440

0.73

0.110

0.380

1.88

1.5

Jupiter

%

88,000

9h55m

0.040

0.23

317.000

2.650

11.86

5.2

Saturn

\

73,000

lOh 14m

0.010

0.13

95.000

1.180

29.46

9.5

Uranus

t
0

31,700

?

0.003

0.31

14.600

1.110

84.02

19.2

Neptune

V

32,000

1

0.001

0.34

17.000

1.250

164.78

30.1

Sun

0

866,400

25d 7h 48m

0.25

332,000.000

27.650

Moon

c

2,163

27d 7h 43m

0.61
0.012
0.166
* The dimensions of the earth and other data are given in the
table of geographical constants p. 310.
THE SUN A STAR 267
volition. Suns move in every conceivable direction, not
as they will, but in abject servitude to gravitation. They
must obey the omnipresent force, and do so with mathe
matical accuracy." Fr>m "New Conceptions in Astron
omy," by Edgar L. Larkin, in Scientific American, February
3, 1906.

CHAPTER XIV
HISTORICAL SKETCH
The Form of the Earth
While various views have been held regarding the form
of the earth, those worthy of attention* may be grouped
under four general divisions.
I. The Earth Flat. Doubtless the universal belief of
primitive man was that, save for the irregularities of moun
tain, hill, and valley, the surface of the earth is flat. In all
the earliest literature that condition seems to be assumed.
The ancient navigators could hardly have failed to observe
the apparent convex surface of the sea and very ancient
literature as that of Homer alludes to the bended sea.
This, however, does not necessarily indicate a belief in the
spherical form of the earth.
Although previous to his time the doctrine of the spher
ical form of the earth had been advanced, Herodotus
(born about 484 b.c, died about 425 b.c.) did not believe
in it and scouted whatever evidence was advanced in its
favor. Thus in giving the history of the Ptolemys, kings
of Egypt, he relates the incident of Ptolemy Necho (about
610-595 b.c.) sending Phoenician sailors on a voyage
around Africa, and after giving the sailors' report that
they saw the sun to the northward of them, he says, " I,
* As for modern, not to say recent, pseudo-scientists and alleged
divine revealers who contend for earths of divers forms, the reader
is referred to the entertaining chapter entitled " Some Cranks and
their Crotchets" in John Fiske's A Century of Science, also the footnote
on pp. 267-268, Vol. I, of his Discovery of America.
268

THE EARTH A SPHERE 269
for my part, do not believe them." Now seeing the sun
to the northward is the most logical result if the earth be
a sphere and the sailors went south of the equator or south
of the tropic of Cancer in the northern summer.
Ancient travelers often remarked the apparent sinking
of southern stars and rising of northern stars as they
traveled northward, and the opposite shifting of the heav
ens as they traveled southward again. In traveling east
ward or westward there was no displacement of the heav
ens and travel was so slow that the difference in time of
sunrise or star-rise could not be observed. To infer that
the earth is curved, at least in a north-south direction, was
most simple and logical. It is not strange that some began
to teach that the earth is a cylinder. Anaximander (about
611-547 b.c), indeed, did teach that it is a cylinder* and
thus prepared the way for the more nearly correct theory.
II. The Earth a Sphere. The fact that the Chaldeans
had determined the length of the tropical year within less
than a minute of its actual value, had discovered the pre
cession of the equinoxes, and could predict eclipses over
two thousand years before the Christian era and that in
China similar facts were known, possibly at an earlier
period, would indicate that doubtless many of the astron
omers of those very ancient times had correct theories as
to the form and motions of the earth. So far as history
has left any positive record, however, Pythagoras (about
582-507 b.c), a Greek f philosopher, seems to have been
the first to advance the idea that the earth is a sphere.
His theory being based largely upon philosophy, nothing
* According to some authorities he taught that the earth is a
sphere and made terrestrial and celestial globes. See Ball's History of
Mathematics, p. 18.
t Sometimes called a Phoenician.

270 HISTORICAL SKETCH
but a perfect sphere would have answered for his con
ception. He was also the first to teach that the earth
rotates * on its axis and revolves about the sun.
Before the time of Pythagoras, Thales (about 640-546
b.c), and other Greek philosophers had divided the earth
into five zones, the torrid zone being usually considered
so fiery hot that it could not be crossed, much less inhab
ited. Thales is quoted by Plutarch as believing that the
earth is a sphere, but it seems to have been proved that
Plutarch was in error. Many of the ancient philosophers
did not dare to teach publicly doctrines not commonly
accepted, for fear of punishment for impiety. It is
possible that his private teaching was different from his
public utterances, and that after all Plutarch was right.
Heraclitus, Plato, Eudoxus, Aristotle and many others
in the next two centuries taught the spherical form of the
earth, and, perhaps, some of them its rotation. Most of
them, however, thought it not in harmony with a perfect
universe, or that it was impious, to consider the sun as
predominant and so taught the geocentric theory.
The first really scientific attempt to calculate the size
of the earth was by Eratosthenes (about 275-195 b.c).
He was the keeper of the royal library at Alexandria, and
made many astronomical measurements and calculations
of very great value, not only for his own day but for ours
as well. Syene, the most southerly city of the Egypt of
his day, was situated where the sundial cast no shadow
at the summer solstice. Measuring carefully at Alexan-
* Strictly speaking, Pythagoras seems to have taught that both
sun and earth revolved about a central fire and an opposite earth
revolved about the earth as a shield from the central fire. This rather
complicated machinery offered so many difficulties that his followers
abandoned the idea of the central fire and "opposite earth" and had
the earth rotate on its own axis.

THE EARTH A SPPIERE 271
dria, he found the noon sun to be one fiftieth of the cir
cumference to the south of overhead. He then multiplied
the distance between Syene and Alexandria, 5,000 stadia
by 50 and got the whole circumference of the earth to be
250,000 stadia. The distance between the cities was not
known very accurately and his calculation probably con
tained a large margin of error, but the exact length of the
Greek stadium of his day is not known * and we cannot
tell how near the truth he came.
Any sketch of ancient geography would be incomplete
without, mention of Strabo (about 54 b.c — 21 a.d.) who is
sometimes called the " father of geography." He believed
the earth to be a sphere at the center of the universe.
He continued the idea of the five zones, used such circles
as had commonly been employed by astronomers and
geographers before him, such as the equator, tropics, and
polar circles. His work was a standard authority for
many centuries.
About a century after the time of Eratosthenes, Posi-
donius, a contemporary of Strabo, made another measure
ment, basing his calculations upon observations of a star
instead of the sun, and getting a smaller circumference,
though that of Eratosthenes was probably too small.
Strabo, Hipparchus, Ptolemy and many others made esti
mates as to the size of the earth, but we have no record of
any further measurements with a view to exact calculation
until about 814 a.d. when the Arabian caliph Al-Mamoum
sent astronomers and surveyors northward and southward,
carefully measuring the distance until each party found a
star to have shifted to the south or north one degree.

* The most reliable data seem to indicate the length of the stadium
was 606| feet.

272 HISTORICAL SKETCH
This distance of two degrees was then multiplied by 180
and the whole circumference obtained.
The period of the dark ages was marked by a decline in
learning and to some extent a reversion to primitive con
ceptions concerning the size, form, or mathematical prop
erties of the earth. Almost no additional knowledge
was acquired until early in the seventeenth century.
Perhaps this statement may appear strange to some
readers, for this was long after the discovery of America
by Columbus. It should be borne in mind that his voyage
and the resulting discoveries and explorations contributed
nothing directly to the knowledge of the form or size of
the earth. That the earth is a sphere was generally
believed by practically all educated people for centuries
before the days of Columbus. The Greek astronomer
Cleomecles, writing over a thousand years before Colum
bus was born, said that all competent persons excepting
the Epicureans accepted the doctrine of the spherical
form of the earth.
In 1615 Willebrord Snell, professor of mathematics at
the University of Leyden, made a careful triangular sur
vey of the level surfaces about Leyden and calculated
the length of a degree of latitude to be 66.73 miles. A
recalculation of his data with corrections which he sug
gested gives the much more accurate measurement of 69.07
miles. About twenty years later, an Englishman named
Richard Norwood made measurements and calculations
in southern England and gave 69.5 as the length of a
degree of latitude, the most accurate measurement up to
that time.
It was about 1660 when Isaac Newton (1642-1727)
discovered the laws of gravitation, but when he applied
the laws to the motions of the moon his calculations did

THE EARTH AN OBLATE SPHEROID 273
not harmonize with what he assumed to be the size
of the earth. About 1671 the French astronomer, Jean
Picard, by the use of the telescope, made very careful
measurements of a little over a degree of longitude and
obtained a close approximation to its length. Newton,
learning of the measurement of Picard, recalculated the
mass of the earth and motions of the moon and found his
law of gravitation as the satisfactory explanation of all the
conditions. Then, in 1682, after having patiently waited
over twenty years for this confirmation, he announced
the laws of gravitation, one of the greatest discoveries
in the history of mankind. We find in this an excellent
instance of the interdependence of the sciences. The
careful measurement of the size of the earth has contrib
uted immensely to the sciences of astronomy and physics.
III. The Earth an Oblate Spheroid. From the many
calculations which Newton's fertile brain could now make,
he soon was enabled to announce that the earth must be,
not a true sphere, but an oblate spheroid. Christian
Huygens, a celebrated contemporary of Newton, also con
tended for the oblate form of the earth, although not on
the same grounds as those advanced by Newton.
In about 1672 the trip of the astronomer Richer to
French Guiana, South America, and his discovery that
pendulums swing more slowly there (see the discussion
under the topic The Earth an Oblate Spheroid, p. 28),
and the resulting conclusion that the earth is not a true
sphere, but is flattened toward the poles, gave a new
impetus to the study of the size of the earth and other
mathematical properties of it.
Over half a century had to pass, however, before the
true significance of Richer's discovery was apparent to
all or generally accepted. An instance of a commonly
JO. MATH. GEO. — 18

274 HISTORICAL SKETCH
accepted reason assigned for the shorter equatorial pen
dulum is the following explanation which was given to
James II of England when he made a visit to the Paris
Observatory in 1697. " While Jupiter at times appears
to be not perfectly spherical, we may bear in mind the
fact that the theory of the earth being flattened is suffi
ciently disproven by the circular shadow which the earth
throws on the moon. The apparent necessary shortening
of the pendulum toward the south is really only a correc
tion for the expansion of the pendulum in consequence of
the higher temperature." It is interesting to note that if
this explanation were the true one, the average tempera
ture at Cayenne would have to be 43° above the boiling
point. Early in the eighteenth century Giovanni Cassini, the
astronomer in charge of the Paris Observatory, assisted by
his son, continued the measurement begun by Picard and
came to the conclusion that the earth is a prolate spheroid.
A warm discussion arose and the Paris Academy of
Sciences decided to settle the matter by careful measure
ments in polar and equatorial regions.
In 1735 two expeditions were sent out, one into Lap
land and the other into Peru. Their measurements, while
not without appreciable errors, showed the decided differ
ence of over half a mile for one degree and demonstrated
conclusively the oblateness of a meridian and, as Voltaire
wittily remarked at the time, " flattened the poles and
the Cassinis."
The calculation of the oblateness of the earth has occu
pied the attention of many since the time of Newton. His
calculation was g-^j-; that is, the polar diameter was ^i^
shorter than the equatorial. Huygens estimated the flat
tening to be about ^^. The most commonly accepted

THE EARTH A GEOID 275
spheroid representing the earth is the one calculated in
1866 by A. R. Clarke, for a long time at the head of the
English Ordnance Survey (see p. 30). Purely astronomical
calculations, based upon the effect of the bulging of the
equator upon the motion of the moon, seem to indicate
slightly less oblateness than that of General Clarke. Pro
fessor William Harkness, formerly astronomical director
of the United States Naval Observatory, calculated it to
be very nearly TJTy.
IV. The Earth a Geoid. During recent years many
careful measurements have been made on various portions
of the globe and extensive pendulum tests given to ascer
tain the force of gravity. These measurements demon
strate that the earth is not a true sphere ; is not an oblate
spheroid; indeed, its figure does not correspond to that of
any regular or symmetrical geometric form. As explained
in Chapter II, the equator, parallels, and meridians are
not true circles, but are more or less elliptical and wavy
in outline. The extensive triangulation surveys and the
application of astrophysics to astronomy and geodesy
make possible, and at the same time make imperative, a
careful determination of the exact form of the geoid.
The Motions of the Earth
The Pythagoreans maintained as a principle in their
philosophy that the earth rotates on its axis and revolves
about the sun. Basing their theory upon a priori reason
ing, they had little better grounds for their belief than
those who thought otherwise. Aristarchus (about 310-
250 b.c), a Greek astronomer, seems to have been the
first to advance the heliocentric theory in a systematic
manner and one based upon careful observations and cal
culations. From this time, however, until the time of

276 HISTORICAL SKETCH
Copernicus, the geocentric theory was almost universally
adopted. The geocentric theory is often called the Ptolemaic sys
tem from Claudius Ptolemy (not to be confused with
ancient Egyptian kings of the same name), an Alexandrian
astronomer and mathematician, who seems to have done
most of his work about the middle of the second century,
a.d. He seems to have adopted, in general, the valuable
astronomical calculations of Hipparchus (about 180-110
b.c). The system is called after him because he com
piled so much of the observations of other astronomers
who had preceded him and invented a most ingenious
system of " cycles," " epicycles," "deferents," " Gentries,"
and " eccentrics " (now happily swept away by the Coper
nican system) by which practically all of the known facts
of the celestial bodies and their movements could be
accounted for and yet assume the earth to be at the center
of the universe.
Among Ptolemy's contributions to mathematical geog
raphy were his employment of the latitude and longitude
of places to represent their positions on the globe (a scheme
probably invented by Hipparchus), and he was the first
to use the terms " meridians of longitude " and " parallels
of latitude." It is from the Latin translation of his sub
divisions of degrees that we get the terms " minutes " and
" seconds " (for centuries the division had been followed,
originating with the Chaldeans. See p. 141). The sixty
subdivisions he called first small parts; in Latin, " minu-
tce prima," whence our term " minute." The sixty sub
divisions of the minute he called second small parts ; in
Latin, " minutce secundce," whence our term " second."
The Copernican theory of the solar system, which has
universally displaced all others, gets its name from the

THE EARTH A GEOID 277
Polish astronomer Nicolas Copernicus (1473—1543). He
revived the theory of Aristarchus, and contended that the
earth is not at the center of the solar system, but that the
sun is, and planets all revolve around the sun. He had
no more reasons for this conception than for the geocentric
theory, excepting that it violated no laws or principles,
was in harmony with the known facts, and was simpler.
Contemporaries and successors of Copernicus were far
from unanimous in accepting the heliocentric theory. One
of the dissenters of the succeeding generation is worthy of
note for his logical though erroneous argument against it.
Tycho Brahe * contended that the Copernican theory was
impossible, because if the earth revolved around the sun,
and at one season was at one side of its orbit, and at
another was on the opposite side, the stars would appar
ently change their positions in relation to the earth (tech
nically, there would be an annual parallax), and he could
detect no such change. His reasoning was perfectly sound,
but was based upon an erroneous conception of the dis
tances of the stars. The powerful instruments of the past
fifty years have made these parallactic motions of many of
the stars a determinable, though a very minute, angle, and
constitute an excellent proof of the heliocentric theory
(see p. 109).
Nine years after the death of Brahe, Galileo Galilei
(1564—1642) by the use of his recently invented telescope
discovered that there were moons revolving about Jupiter,
indicating by analogy the truth of the Copernican theory.
Following upon the heels of this came his discovery that
Venus in its swing back and forth near the sun plainly
* Tycho Brahe (1546-1601) a famous Swedish astronomer, was born
at Knudstrup, near Lund, in the south of Sweden, but spent most of
his life in Denmark,

278 HISTORICAL SKETCH
shows phases just as our moon does, and appears larger
when in the crescent than when in the full. The only
logical conclusion was that it revolves around the sun,
again confirming by analogy the Copernican theory. Gali
lei was a thorough-going Copernican in private belief, but
was not permitted to teach the doctrine, as it was con
sidered unscriptural.
As an illustration of the humiliating subterfuges to
which he was compelled to resort in order to present an
argument based upon the heretical -theory, the following
is a quotation from an argument he entered into con
cerning three comets which appeared in 1618. He based
his argument as to their motions upon the Copernican
system, professing to repudiate that theory at the same
time. " Since the motion attributed to the earth, which I as
a pious and Christian person consider most false, and not
to exist, accommodates itself so well to explain so many and
such different phenomena, I shall not feel sure that, false
as it is, it may not just as deludingly correspond with the
phenomena of comets."
One of the best supporters of this theory in the next
generation was Kepler (1571—1630), the German astrono
mer, and friend and successor of Brahe. His laws of
planetary motion (see p. 284) were, of course, based upon
the Copernican theory, and led to Newton's discovery of
the laws of gravitation.
James Bradley (1693-1762) discovered in 1727 the
aberration of light (see p. 104), and the supporters of the
Ptolemaic system were routed, logically, though more
than a century had to pass before the heliocentric theory
became universally accepted.

APPENDIX GRAVITY
Gravity is frequently defined as the earth's attractive
influence for an object. Since the attractive influence
of the mass of the earth for an object on or near its sur
face is lessened by centrifugal force (see p. 14) and in
other ways (see p. 183), it is more accurate to say that
the force of gravity is the resultant of
a. The attractive force mutually existing between the
earth and the object, and
b. The lessening influence of centrifugal force due to
the earth's rotation.
Let us consider these two factors separately, bearing
in mind the laws of gravitation (see p. 17).
a. Every particle of matter attracts every other
particle. (1) Hence the point of gravity for any given object
on the surface of the earth is determined by the mass of
the object itself as well as the mass of the earth. The
object pulls the earth as truly and as much as the earth
attracts the object. The common center of gravity of the
earth and this object lies somewhere between the center
of the earth's mass and the center of the mass of the
object. Each object on the earth's surface, then, must
have its own independent common center of gravity be
tween it and the center of the earth's mass. The position
of this common center will vary —
(a) As the object varies in amount of matter (first
law), and 279

280 APPENDIX
(b) As the distance of the object from the center of
the earth's mass varies (inversely as the square of the
distance). (2) Because of this principle, the position of the sun
or moon slightly modifies the exact position of the center
of gravity just explained. It was shown in the dis
cussion of tides that, although the tidal lessening of
the weight of an object is as yet an immeasurable
quantity, it is a calculable one and produces tides (see
p. 183).
b. The rotation of the earth gives a centrifugal force
to every object on its surface, save at the poles.
(1) Centrifugal force thus exerts a slight lifting influ
ence on objects, increasing toward the equator. This
lightening influence is sufficient to decrease the weight
of an object at the equator by g-^-g of the whole. That
is to say, an object which weighs 288 pounds at the
equator would weigh a pound more if the earth did not
rotate. Do not infer from this that the centrifugal force
at the pole being zero, a body weighing 288 pounds at the
equator would weigh 289 pounds at the pole, not being
lightened by centrifugal force. This would be true if the
earth were a sphere. The bulging at the equator decreases
a body's weight there by ^gg as compared with the weight
at the poles. Thus a body at the equator has its weight
lessened by 2|9 because of rotation and by ^-9 because
of greater distance from the center, or a total of -jg-g of
its weight as compared with its weight at the pole. A
body weighing 195 pounds at the pole, therefore, weighs
but 194 pounds at the equator. Manifestly the rate of
the earth's rotation determines the amount of this cen
trifugal force. If the earth rotated seventeen times as
fast, this force at the equator would exactly equal the

GRAVITY

281

earth's attraction,* objects there would have no weight;
that is, gravity would be zero. In such a case the plumb
line at all latitudes would point directly toward the nearest
celestial pole. A clock at the 45th parallel with a pendu
lum beating seconds would gain one beat every 19£ minutes
if the earth were at rest, but would lose three beats in the
same time if the earth rotated twice as fast.
(2) Centrifugal force due to the rotation of the earth
not only affects the amount of gravity, but modifies the
direction in which it is exerted. Centrifugal force acts
in a direction at right angles to the axis, not directly
opposite the earth's attraction excepting at the equator.
Thus plumb lines, excepting at the equator and poles,
are slightly tilted toward the poles.
If the earth were at rest a plumb line at latitude 45°
would be in the direction toward the center of the mass
of the earth at C (Fig.
112). The plumb line
would then be PC. But
centrifugal force is ex
erted toward CF, and the
resultant of the attrac
tion toward C and cen
trifugal force toward CF
makes the line deviate
to a point between those
directions, as CG, the
true center of gravity, and Fis- Iia
the plumb line becomes P'GG. The amount of the cen-

»CF

* Other things equal, centrifugal force varies with the square of
the velocity (see p. 14), and since centrifugal force at the equator equals
289 times gravity, if the velocity of rotation were increased 17 times,
centrifugal force would equal gravity (172 = 289).

282 APPENDIX
trifugal force is so small as compared with the earth's
attraction that this deviation is not great. It is greatest
at the 45th parallel where it amounts to 5' 57",. or nearly
one tenth of a degree. There is an almost equal devia
tion due to the oblateness of the earth. At latitude 45°
the total deviation of the plumb line from a line drawn
to the center of the earth is 11' 30.65."
LATITUDE
In Chapter II the latitude of a place was simply defined
as the arc of a meridian intercepted between that place
and the equator. This is true geographical latitude, but
the discussion of gravity places us in a position to under
stand astronomical and geocentric latitude, and how geo
graphic latitude is determined from astronomical latitude.
Owing to the elliptical form of a meridian " circle," the
vertex of the angle constituting the latitude of a place is
not at the center of the globe. A portion of a meridian
circle near the equator is an arc of a smaller circle than
a portion of the same meridan near the pole (see p. 43
and Fig. 18).
Geocentric Latitude. It is sometimes of value to speak
of the angle formed at the center of the earth by two
lines, one drawn to the place whose latitude is sought,
and the other to the equator on the same meridian. This
is called the geocentric latitude of the place.
Astronomical Latitude. The astronomer asceitains lati
tude from celestial measurements by reteience to a level
line or a plumb line. Astronomical latitude, then, is the
angle formed between the plumb line and the plane of
the equator.
In the discussion of gravity, the last effect of centri-

LATITUDE 283
fugal force noted was on the direction of the plumb line.
It was shown that this line, excepting at the equator and
poles, is deviated slightly toward the pole. The effect of
this is to increase correspondingly the astronomical lati
tude of a place. Thus at latitude 45°, astronomical lati
tude is increased by 5' 57", the amount of this deviation.
If there were no rotation of the earth, there would be no
deviation of the plumb line, and what we call latitude 60°
would become 59° 54' 51". Were the earth to rotate twice
as fast, this latitude, as determined by the same astronom
ical instruments, would become 60° 15' 27".
If adjacent to a mountain, the plumb line deviates
toward the mountain because of its attractive influence on
the plumb bob; and other deviations are also observed,
such as with the ebb and flow of a near by tidal wave.
These deviations are called " station errors," and allowance
must be made for them in making all calculations based
upon the plumb line.
Geographical latitude is simply the astronomical latitude,
corrected for the deviation of the plumb line. Were it not
for these deviations the latitude of a place would be deter
mined within a few feet of perfect accuracy. As it is,
errors of a few hundred feet sometimes may occur (see
p. 289).
Celestial Latitude. In the discussion of the celestial
sphere many circles of the celestial sphere were described
in the same terms as circles of the earth. The celestial
equator, Tropic of Cancer, etc., are imaginary circles which
correspond to the terrestrial equator, Tropic of Cancer, etc.
Now as terrestrial latitude is distance in degrees of a meri
dian north or south of the equator of the earth, one would
infer that celestial latitude is the corresponding distance
along a celestial meridian from the celestial equator, but

284 APPENDIX
this is not the case. Astronomers reckon celestial latitude
from the ecliptic instead of from the celestial equator. As
previously explained, the distance in degrees from the
celestial equator is called declination.
Celestial Longitude is measured in degrees along the
ecliptic from the vernal equinox as the initial point, meas
ured always eastward the 360° of the ecliptic.
In addition to the celestial pole 90° from the celestial
equator, there is a pole of the ecliptic, 90° from the ecliptic.
A celestial body is thus located by reference to two sets of
circles and two poles.
(a) Its declination from the celestial equator and posi
tion in relation to hour circles, as celestial meridians are
commonly called (see Glossary).
(6) Its celestial latitude from the ecliptic and celestial
longitude from " ecliptic meridians."
KEPLER'S LAWS
These three laws find their explanation in the laws of
gravitation, although Kepler discovered them before New
ton made the discovery which has immortalized his name.
First Law. The orbit of each planet is an ellipse, having
the sun as a focus.
Second Law. The planet moves about the sun at such
rates that the straight line connecting the center of the
sun with the center of the planet (this line is called the
planet's radius vector) sweeps over equal areas in equal
times (see Fig. 113).
The distance of the earth's journey for each of the
twelve months is such that the ellipse is divided into
twelve equal areas. In the discussion of seasons we
observed (p. 169) that when in perihelion, in January, the

KEPLER'S LAWS 285
earth receives more heat each day than it does when in
aphelion, in July. The northern hemisphere, being turned
away from the sun in January, thus has warmer winters
than it would otherwise have, and being toward the sun
in July, has cooler summers. This is true only for corre
sponding days, not for the seasons as a whole. According
to Kepler's second law the earth must receive exactly the
same total amount of heat from the vernal equinox (March

A — - - r y^j»  IP
Aphelion ^y^-^y?W\ Perihelion

Fig. 113 t-
21) to the autumnal equinox (Sept. 23), when farther from
the sun, as from the autumnal to the vernal equinox, when
nearer the sun. During the former period, the northern
summer, the earth receives less heat day by day, but
there are more days.
Third Law. The squares of the lengths of the times (side
real years) of planets are proportional to the cubes of their
distances from the sun. Thus,
(Earth's year)2: (Mars' year)2: : (Earth's distance)3 :
(Mars' distance)3. Knowing the distance of the earth to

286 APPENDIX
the sun and the distance of a planet to the sun, we have
three of the quantities for our proportion, calling the
earth's year 1, and can find the year of the planet; or,
knowing the time of the planet, we can find its distance.

MOTIONS OF THE EARTH'S AXIS
In the chapter on seasons it was stated that excepting
for exceedingly slow or minute changes the earth's axis
at one time is parallel to itself at other times. There are
three such motions of the axis.
Precession of the Equinoxes. Since the earth is slightly
oblate and the bulging equator is tipped at an angle
of 23J° to the ecliptic, the sun's attraction on this rim
tends to draw the axis over at right angles to the equator.
The rotation of the earth, however, tends to keep the
axis parallel to itself, and the effect of the additional accel
eration of the equator is to cause the axis to rotate slowly,
keeping the same angle to the ecliptic, however.
At the time of Hipparchus -(see p. 276), who discovered
this rotation of the axis, the present North star, Alpha
Ursa Minoris, was about 12° from the true pole of the
celestial sphere, toward which the axis points. The course
which the pole is taking is bringing it somewhat nearer
the polestar; it is now about 1° 15' away, but a hundred
years hence will be only half a degree from it. The period
of this rotation is very long, about 25,000 years, or 50.2"
each year. Ninety degrees from the ecliptic is the pole
of the ecliptic about which the pole of the celestial equator
rotates, and from which it is distant 23J°.
As the axis rotates about the pole of the ecliptic, the
point where the plane of the equator intersects the plane

MOTIONS OF THE EARTH'S AXIS 287
of the ecliptic, that is, the equinox, gradually shifts around
westward. Since the vernal equinox is at a given point
in the earth's orbit one year, and the next year is reached
a little ahead of where it was the year before, the term
precession of the equinoxes is appropriate. The sidereal
year (see p. 132) is the time required for the earth to
make a complete revolution in its orbit. A solar or tropi
cal year is the interval from one vernal equinox to the
next vernal equinox, and since the equinoxes " precede,"
a tropical year ends about twenty minutes before the
earth reaches the same point in its orbit a second time.
As is shown in the discussion of the earth's revolution
(p. 169), the earth is in perihelion December 31, making
the northern summer longer and cooler, day by day, than
it would otherwise be, and the winter shorter and warmer.
The traveling of the vernal equinox around the orbit,
however, is gradually shifting the date of perihelion, so
that in ages yet to come perihelion will be reached in July,
and thus terrestrial climate is gradually changing. This
perihelion point (and with it, aphelion) has a slight west
ward motion of its own of 11.25" each year, making, with
the addition of the precession of the equinoxes of 50.2", a
total shifting of the perihelion point (see " Apsides " in
the Glossary) of 1' 1.45". At this slow rate, 10,545 years
must pass before perihelion will be reached July 1. The
amount of the ellipticity of the earth's orbit is gradually
decreasing, so that by the time this shifting has taken
place the orbit will be so nearly circular that there may be
but slight climatic effects of this shift of perihelion. It
may be of interest to note that some have reasoned that
ages ago the earth's orbit was so elliptical that the
northern winter, occurring in aphelion, was so long and
cold that great glaciers were formed in northern North

288 APPENDIX
America and Europe which the short, hot summers could
not melt. The fact of the glacial age cannot be disputed,
but this explanation is not generally accepted as satis
factory. Nutation of the Poles. Several sets of gravitative influ
ences cause a slight periodic motion of the earth's axis
toward and from the pole of the ecliptic. Instead of
" preceding" around the circle 47° in diameter, the axis
makes a slight wavelike motion, a " nodding," as it is
called. The principal nutatory motion of the axis is
due to the fact that the moon's orbit about the earth
(inclined 5° 8' to the ecliptic) glides about the ecliptic in
18 years, 220 days, just as the earth's equator glides about
the ecliptic once in 25,800 years. Thus through periods
of nearly nineteen years each the obliquity of the ecliptic
(see pp. 118, 147) gradually increases and decreases again.
The rate of this nutation varies somewhat and is always
very slight; at present it is 0.47" in a year.
Wandering of the Poles. In the discussion of gravity
(p. 279), it was shown that any change in the position of
particles of matter effects a change in the point of gravity
common to them. Slight changes in the crust of the earth
are constantly taking place, not simply the gradational
changes of wearing down mountains and building up of
depositional features, but great diastrophic changes in
mountain structure and continental changes of level.
Besides these physiographic changes, meteorological con
ditions must be factors in displacement of masses, the
accumulation of snow, the fluctuation in the level of great
rivers, etc. For these reasons minute changes in the
position of the axis of rotation must take place within the
earth. Since 1890 such changes in the position of the axis
within the globe have been observed and recorded. The

MOTIONS OF THE EARTH'S AXIS 289
" wandering of the poles," as this slight shifting of the
axis is called, has been demonstrated by the variation in
the latitudes of places. A slight increase in the latitude
of an observatory is noticed, and at the same time a cor
responding decrease is observed in the latitude of an
observatory on the opposite side of the globe. " So
definite are the processes of practical astronomy that the
position of the north pole can be located with no greater
uncertainty than the area of a large Eskimo hut." *
In 1899 the International Geodetic Association took
steps looking to systematic and careful observations and
records of this wandering of the poles. Four stations
not far from the thirty-ninth parallel but widely separated
in longitude were selected, two in the United States, one
in Sicily, and the other in Japan.
All of the variations since 1889 have been within an
area less than sixty feet in diameter.
Seven Motions of the Earth. Seven of the well-defined
motions of the earth have been described in this book:
1. Diurnal Rotation.
2. Annual Revolution in relation to the sun.
3. Monthly Revolution in relation to the moon (see
p. 184).
4. Precessional Rotation of Axis about the pole of the
ecliptic. 5. Nutation of the poles, an elliptical or wavelike motion
in the precessional orbit of the axis.
6. Shifting on one axis of rotation, then on another,
leading to a " wandering of the poles."
7. Onward motion with the whole solar system (see
"Sun's Onward Motion," p. 247).
* Todd's New Astronomy, p. 95.
JO. MATH. GEO. — 19

290 APPENDIX
MATHEMATICAL TREATMENT OF TIDES
The explanation of the cause of tides in the chapter
on that subject may be relied upon in every particular,
although mathematical details are omitted. The mathe
matical treatment is difficult to make plain to those who
have not studied higher mathematics and physics. Sim
plified as much as possible, it is as follows:
Let it be borne in mind that to find the cause of tides
we must find unbalanced forces which change their positions.
Surface gravity over the globe varies slightly in different
places, being less at the equator and greater toward the
poles. As shown elsewhere, the force of gravity at the
equator is less for two reasons:
a. Because of greater centrifugal force.
b. Because of the oblateness of the earth.
(a) Centrifugal force being greater at the equator than
elsewhere, there is an unbalanced force which must cause
the waters to pile up to some extent in the equatorial
region. If centrifugal force were sometimes greater at the
equator and sometimes at the poles, there would be a cor
responding shifting of the accumulated waters and we
should have a tide — and it would be an immense one.
But we know that this unbalanced force does not change
its position, and hence it cannot produce a tide.
(b) Exactly the same course of reasoning applies to the
unbalanced force of gravity at the equator due to its
greater distance from the center of gravity. The position
of this unbalanced force does not shift, and no tide results.
Since the earth turns on its axis under the sun and
moon, any unbalanced forces they may produce will neces
sarily shift as different portions of the earth are succes
sively turned toward or from them. Our problem, then,

MATHEMATICAL TREATMENT OF TIDES

291

is to find the cause anil direction of the unbalanced forces
produced by the moon or sun.
In Figure 114, let CA be the acceleration toward the
moon at C, due to the moon's attraction. Let BD be

Fig. 114
the acceleration at B. Now B is nearer the moon than
C, so BD will be greater than CA, since the attraction
varies inversely as the square of the distance.
From B construct BE equal to CA. Comparing forces
BE and BD, the latter is greater. Completing the paral
lelogram, we have BFDE. Now it is a simple demonstra
tion in physics that if two forces act upon B, one to F and
the other to E, the resultant of the two forces will be the
diagonal BD. Since BE and BF combined result in BD,
it follows that BF represents the unbalanced force at B.
At B, then, there is an unbalanced force as compared
with C as represented by BF. At B' the unbalanced
force is represented by B'F'. Note the pulling direction
in which these unbalanced forces are exerted.
Note. — For purposes of illustration the distance of the moon
represented in the figures is greatly diminished. The distance CA is
taken arbitrarily, likewise the distance BD. If CA were longer,
however, BD would be still longer; and while giving CA a different
length, would modify the form of the diagram, the mathematical rela
tions would remain unchanged. Because of the short distance given
CM in the figures, the difference between the BF in Figure 114 and BF
in Figure 115 is greatly exaggerated. The difference between the
unbalanced or tide-producing force on the side toward the moon and
that on the opposite side is approximately .0467 BF (Fig. 114).

292

APPENDIX

In Figure 115, B is farther from the moon than C, hence
BE (equal to CA) is greater than BD, and the unbalanced
force at B is BF, directed away from the moon. A study
of Figures 114 and 115 will show that the unbalanced force
on the side towards the moon (BF in Fig. 114) is slightly
greater than the unbalanced force on the side opposite the
moon (BF in Fig. 115). The difference, however, is ex-

Fig. 115
ceedingly slight, and the tide on the opposite side is prac
tically equal to the tide on the side toward the attracting
body. Combining the arrows showing the directions of the unbal
anced forces in the two figures, we have the arrows shown

-M

Fig. n6

in Figure 116. The distribution and direction of the un
balanced forces may be thus summarized: " The disturbing
force produces a pull along A A' and a squeeze along BB'."*
* Mathematical Astronomy, Barlow and Bryan, p. 377.

THE ZODIAC 293
THE ZODIAC
This belt in the celestial sphere is 16° wide with the
ecliptic as the center. The width is purely arbitrary. It
could have been wider or narrower just as well, but was
adopted by the ancients because the sun, moon, and plan
ets known to them were always seen within 8° of the path
way of the sun. We know now that several asteroids, as
truly planets as the earth, are considerably farther from
the ecliptic than 8°; indeed, Pallas is sometimes 34° from
the ecliptic — to the north of overhead to people of north
ern United States or central Europe.
Signs. As the sun " creeps backward " in the center of
the zodiac, one revolution each year, the ancients divided
its pathway into twelve parts, one for each month. To
each of these sections of thirty degrees (360° -^12 = 30°)
names were assigned, all but one after animals, each one
being considered appropriate as a " sign " of an annual
recurrence (see p. 117). Aries seems commonly to have
been taken as the first in the series, the beginning of
spring. Even yet the astronomer counts the tropical year
from the " First point of Aries," the moment the center
of the sun crosses the celestial equator on its journey
northward. As explained in the discussion of the precession of the
equinoxes (p. 286), the point in the celestial equator where
the center of the sun crosses it shifts westward one degree
in about seventy years. In ancient days the First point of
Aries was in the constellation of that name but now it is
in the constellation to the west, Pisces. The sign Aries
begins with the First point of Aries, and thus with the west
ward travel of this point all the signs have moved back
into a constellation of a different name. Another differ-

294

APPENDIX

ence between the signs and the constellations of the zodiac
is that the star clusters are of unequal length, some more
than 30° and some less, whereas the signs are of uniform
length. The positions and widths of the signs and con-

Fig. 117
stellations with the date when the sun enters each are
shown in Figure 117.
Aries, the first sign, was named after the ram, probably
because to the ancient Chaldeans, where the name seems
to have originated, this was the month of sacrifice. The
sun is in Aries from March 21 until April 20. It is repre-

THE ZODIAC 295
sented by a small picture of a ram ( mtf ) or by a hiero
glyphic (r).
Taurus, the second sign ( ps$ ), was dedicated to the
bull. In ancient times this was the first of the signs,
the vernal equinox being at the beginning of this sign.
According to very ancient mythology it was the bull that
drew the sun along its " furrow " in the sky. There
are, however, many other theories as to the origin of the
designation. The sun is in Taurus from April 20 until
May 21.
Gemini, the third sign, signifies twins ( f$> ) and gets its
name from two bright stars, Castor and Pollux, which used
to be in this sign, but are now in the sign Cancer. The
sun is in Gemini from May 21 until June 22.
Cancer, the fourth sign ( m|g ), was named after the
crab, probably from the fact that when in this sign the
sun retreats back again, crablike, toward the south. The
sun is in Cancer from June 22 until July 23.
Leo, signifying lion, is the fifth sign ( $^$* ) and seems
to have been adopted because the lion usually was used
as a symbol for fire, and when the sun was in Leo the
hottest weather occurred. The sun is in this sign from
July 23 until August 23.
Virgo, the virgin ( %¦ ), refers to the Chaldean myth of
the descent of Ishtar into hades in search of her husband.
The sun is in Virgo from August 23 until September 23.
The foregoing are the summer signs and, consequently,
the corresponding constellations are our winter constel
lations. It must be remembered that the sign is always
about 30° (the extreme length of the " Dipper ") to the
west of the constellation of the same name.
Libra, the balances ( j*j ), appropriately got its name
from the fact that the autumnal equinox, or equal balanc-

296 APPENDIX
ing of day and night, occurred when the sun was in the
constellation thus named the Balances. The sun is now
in Libra from September 23 until October 24,
Scorpio is the eighth sign ( «|K )¦ The scorpion was a
symbol of darkness, and was probably used to represent
the shortening of days and lengthening of nights. The
sun is now in Scorpio from October 24 until November 23.
Sagittarius, meaning an archer or bowman, is
sometimes represented as a Centaur with a
bow and arrow. The, sun is in this sign from November
23 until December 22.
Capricorn, signifying goat, is often represented as hav
ing the tail of a fish (ia3^ )• It probably has its origin
as the mythological nurse of the young solar god. The
sun is in Capricorn from December 22 until January 20.
Aquarius, the water-bearer ( $§, ), is the eleventh sign
and probably has a meteorological origin, being associated
as the cause of the winter rains of Mediterranean coun
tries. The sun is in this sign from January 20 until
February 19.
Pisces is the last of the twelve signs. In accordance
with the meaning of the term, it is represented as two
fishes ( Jg; ). Its significance was probably the same as
the water-bearer. The sun is in this sign from February
19 until the vernal equinox, March 21, when it has com
pleted the " labors " of its circuit, only to begin over
again. The twelve signs of the ancient Chinese zodiac were
dedicated to a quite different set of animals; being, in
order, the Rat, the Ox, the Tiger, the Hare, the Dragon,
the Serpent, the Horse, the Sheep, the Monkey, the Hen,
the Dog, and the Pig. The Egyptians adopted with a few
changes the signs of the Greeks.

THE ZODIAC 297
Myths and Supekstitions as to the Relation of
the Zodiac to the Earth
When one looks at the wonders of the heavens it does
not seem at all strange that in the early dawn of history,
ignorance and superstition should clothe the mysterious
luminaries of the sky with occult influences upon the
earth, the weather, and upon human affairs. The ancients,
observing the apparent fixity of all the stars excepting the
seven changing ones of the zodiac — the sun, moon, and
five planets known to them — endowed this belt and its
seven presiding deities with special guardianship of the
earth, giving us seasons, with varying length of day and
change of weather; bringing forth at its will the sprouting
of plants and fruitage and harvest in their season; count
ing off inevitably the years that span human life ; bringing
days of prosperity to some and of adversity to others;
and marking the wars and struggles, the growth and
decay of nations. With such a background of belief, at
once their science and their religion, it is not strange that
when a child was born the parents hastened to the astrol
oger to learn what planet or star was in the ascendancy,
that is, most prominent during the night, and thus learn
in advance what his destiny would be as determined from
the character of the star that would rule his life.
The moon in its monthly path around the earth must
pass through the twelve signs of the zodiac in 29J days or
spend about 2\ days to each sign. During the blight of
intelligence of the dark ages, some mediaeval astrologer
conceived the simple method of subdividing the human
body into twelve parts to correspond to the twelve con
stellations of the zodiac. Beginning with the sign Aries,
he dedicated that to the head, the neck he assigned to

298 APPENDIX
Taurus, the arms were given over to Gemini, the stars of
Cancer were to rule the breast, the heart was presided
over by Leo, and so on down to Pisces which was to rule
the feet. Now anyone who was born when the moon was in
Aries would be strong in the head, intellectual ; if in Taurus,
he would be strong in the neck and self-willed, etc. More
over, since the moon makes a circuit of the signs of the
zodiac in a month, according to his simple scheme when
the moon is in Aries the head is especially affected; then
diseases of the head rage (or is it then that the head
is stronger to resist disease?), and during the next few days
when the moon is in Taurus, beware of affections of the
neck, and so on down the list. The very simplicity of this
scheme and ease by which it could be remembered led to
its speedy adoption by the masses who from time imme
morial have sought explanations of various phenomena
by reference to celestial bodies.
Now there is no astronomical or geographical necessity
for considering Aries as the first sign of the zodiac. Our
year begins practically with the advent of the sun into
Capricorn — the beginning of the year was made January
1-for this very purpose. The moon is not in any peculiar
position in relation to the earth March 21 any more than
it is December 23. If when the calendar was revised the
numbering of the signs of the zodiac had been changed
also, then Capricorn, the divinities of which now rule the
knees, would have been made to rule the head, and the
whole artificial scheme would have been changed ! Besides,
the sign Capricorn does not include the constellation Capri
corn, so with the precession of the equinoxes the subtle
influences once assigned to the heavenly bodies of one
constellation have been shifted to an entirely different set
of stars! The association of storms with the sun's cross-

THE ZODIAC 299
ing the equinox and with the angle the cusps of the moon
show to the observer (a purely geometric position varying
with the position of the observer) is in the same class as
bad luck attending the taking up of the ashes after the sun
has gone down or the wearing of charms against rheuma
tism or the " evil " eye.
" The fault, dear Brutus, is not in our stars,
But in ourselves, that we are underlings." — Shakespeare.

300

APPENDIX

PRACTICAL WORK IN MATHEMATICAL
GEOGRAPHY
Concrete work in this subject has been suggested directly,
by implication, or by suggestive queries and problems
throughout the book. No instruments of specific char
acter have been suggested for use excepting such as are
easily provided, as a graduated quadrant, compasses, an
isosceles right triangle, etc. Interest in the subject will
be greatly augmented if the following simple instruments,
or similar devices,
are made or pur
chased and used.
To Make a Sundial
This is not espe
cially difficult and
may be accom
plished in several
Fie- II8 ways. A simple
plan is shown in Figure 118. Angle BAC should be the
co-latitude of the place, that is, the latitude subtracted
from 90°, though this is not at all essential. The hour
lines may be marked off according to two systems, for
standard time or for local time.
Standard Time Dial. If you wish your dial to indicate
clock time as correctly as possible, it will be necessary to
consult the analemma or an almanac to ascertain the equa
tion of time when the hour lines are drawn. Since the sun
is neither fast nor slow April 14, June 15, September 1, or
December 25, those are the easiest days on which to lay
off the hours. On one of those dates you can lay them off
according to a reliable timepiece.

PRACTICAL WORK 301
If you mark the hour lines at any other date, ascertain
the equation of time (see p. 127) and make allowances
accordingly. Suppose the date is October 27. The ana
lemma shows the sun to be 16 minutes fast. You should
mark the hour lines that many minutes before the hour as
indicated by your timepiece, that is, the noon line when
your watch says 11: 44 o'clock, the 1 o'clock line when
the watch indicates 12:44, etc. If the equation is slow,
say five minutes, add that time to your clock time, mark
ing the noon line when your watch indicates 12 : 05, the
next hour line at 1 : 05, etc. It is well to begin at the hour
for solar noon, at that time placing the board so that the
sun's shadow is on the XII mark and after marking off the
afternoon hours measure from the XII mark westward
corresponding distances for the forenoon. Unless you
chance to live upon the meridian which gives standard
time to the belt in which you are, the noon line will be
somewhat to the east or west of north.
This sundial will record the apparent solar time of the
meridian upon which the clock time is based. The differ
ence in the time indicated by the sundial and your watch
at any time is the equation of time. Test the accuracy
of your sundial by noticing the time by your watch when
the sundial indicates noon and comparing this difference
with the equation of time for that day. If your sundial
is accurate, you can set your watch any clear day by look
ing up the equation of time and making allowances accord
ingly. Thus the analemma shows that on May 28 the sun
is three minutes fast. When the sundial indicates noon
you know it is three minutes before twelve by the clock.
Local Time Dial. To mark the hour lines which show
the local mean solar time (see p. 64), set the XII hour
line due north. Note accurately the clock time when the

302 APPENDIX
shadow is north. One hour later mark the shadow line
for the I hour line, two hours later mark the II hour line,
etc. This dial will indicate the apparent solar time of
your meridian. You can set your watch by it by first
converting it into mean solar time and then into standard
time. (This is explained on pp. 128, 129.)
It should be noted that these two sundials are exactly
the same for persons who use local time, or, living on the
standard time meridian, use standard time.
The Sun Board
The uses of the mounted quadrant in determining lat
itude were shown in the chapter on seasons (see p. 173).

Fig. 119

Dr. J. Paul Goode, of the University of Chicago, has
designed a very convenient little instrument which an
swers well for this and other purposes.
A vertically placed quadrant enables one to ascertain

PRACTICAL WORK

303

the altitude of the sun for determining latitude and cal
culating the heights of objects.
By means of a graduated circle
placed horizontally the azimuth
of the sun (see Glossary) may
be ascertained. A simple vernier
gives the azimuth readings to
quarter degrees. It also has a
device for showing the area cov
ered by a sunbeam of a given
size, and hence its heating power.
The Heliodon
This appliance was designed by
Mr. J. F. Morse, of the Medill
High School, Chicago. It vividly
illustrates the apparent path of
the sun at the equinoxes and sol
stices at any latitude. The points Fie- I2°
of sunrise and sunset can also be shown and hence the
length of the longest day or night can be calculated.

304

APPENDIX

WHAT KEEPS THE MEMBERS OF THE SOLAR
SYSTEM IN THEIR ORBITS?
When a body is thrown in a direction parallel to the
horizon, as the bullet from a level gun, it is acted upon by
two forces:
(a) The projectile force of the gun, AB. (Fig. 121.)
(b) The attractive force of the earth, AC.
The course it will actually take from point A is the
diagonal AA'. When it reaches A' the force AB still

rg.

Fig. 121
acts (not considering the friction of the air), impelling it
in the line A'B'. Gravity continues to pull it in the line
A'C, and the' projectile takes the diagonal direction A' A"
and makes the curve (not a broken line as in the figure)
A A' A". It is obvious from this diagram that if the im
pelling force be sufficiently great, line AB will be so long in
relation to line AC that the bullet will be drawn to the
earth just enough to keep it at the same distance from
the surface as that of its starting point.
The amount of such a projectile force near the surface
of the earth at the equator as would thus keep an object

PATHS OF PRO.IECTILES

305

at an unvarying distance from the earth is 26,100 feet per
second. Fired in a horizontal direction from a tower (not
allowing for the friction of the air) such a bullet would
forever circle around the earth. Dividing the circumfer
ence of the earth (in feet) by this number we find that
such a bullet would return to its starting point in about
5,000 seconds, or 1 h. 23 m., making many revolutions
around the earth during one day. Since our greatest guns,
throwing a ton of steel a distance of twenty-one miles,
give their projectiles a speed of only about 2,600 feet
per second, it will be seen that the rate we have given
is a terrific one. If this speed were increased to 37,000 feet

37,000 feet a second.

Flies offindefin itely.

Return of the bullet of 26,100 ft

Fig. 12a. Paths of Projectiles ol Different Velocities (Scientific American
Supplement, Sept. 32, 1906. Reproduced by permission)

per second, the bullet would never return to the earth.
One is tempted here to digress and demonstrate the utter
impossibility of human beings even " making a trip to the
moon," to say nothing of one to a much more distant
planet. The terrific force with which we should have to
be hurled to get away from the earth, fourteen times the
speed of the swiftest cannon ball, is in itself an insuperable
difficulty. Besides this, there would have to be the most
exact calculation of the force and direction, allowing for
(a) the curve given a projectile by gravity, (6) the cen-

JO. MATH. GEO.

306 APPENDIX
trifugal force of rotation, (c) the revolution of the earth,
(d) the revolution of the moon, (e) the friction of the air,
a variable quantity, impossible of calculation with abso
lute accuracy, (/) the inevitable swerving in the air by
reason of its currents and varying density, and (g) the
influence on the course by the attraction of the sun and
planets. In addition to these mathematical calculations
as to direction and projectile force, there would be the
problem of (h) supply of air, (i) air pressure, to which our
bodies through the evolution of ages have become adapted,
(j) the momentum with which we would strike into the
moon if we did " aim " right, etc.
Returning to our original problem, we may notice that
if the bullet were fired horizontally at a distance of 4,000
miles from the surface of the earth, the pull of gravity would
be only. one fourth as great (second law of gravitation),
and the projectile would not need to take so terrific a speed
to revolve around the earth. As we noticed in the discus
sion of Mars (see p. 255), the satellite Phobos is so near
its primary, 1,600 miles from the surface, that it revolves
at just about the rate of a .cannon ball, making about three
revolutions while the planet rotates once.
While allusion has been made only to a bullet or a moon,
in noticing the application of the law of projectiles, the
principle applies equally to the planets. Governed by the
law here illustrated, a planet will revolve about its primary
in an orbit varying from a circle to an elongated ellipse.
Hence we conclude that a combination of projectile and
attractive forces keeps the members of the solar system
in their orbits.

FORMULAS AND TABLES 307
FORMULAS AND TABLES
Symbols Commonly Employed
There are several symbols which are generally used in
works dealing with the earth, its orbit or some of its other
properties. To the following brief list of these are added
a few mathematical symbols employed in this book,
which may not be familiar to many who will use it. The
general plan of using arbitrary symbols is shown on page
14, where 0 represents universal gravitation and g repre
sents gravity; C represents centrifugal force and c centri
fugal force due to the rotation of the earth.
4> (Phi), latitude.
£ (Epsilon) , obliquity of the ecliptic, also eccentricity of
an ellipse.
tz (Pi), the number which when multiplied by the diam
eter of a circle equals the circumference; it is
3.14159265, nearly 3.1416, nearly 3f tt2= 9.8696044.
d (Delta), declination, or distance in degrees from the
celestial equator.
« , " varies as; " x <x y means x varies as y.
< , "is less than; " x < y means x is less than y.
> , "is greater than; " x > y means x is greater than y.

Formulas

The Circle and Sphere
r ¦¦
= radius.
c = circumference
d
= diameter.
a = area.
308 APPENDIX

red

= c.

c n

¦d.

nr2

= area.

4ct2 = surface of sphere.
£ot-3 = volume of sphere = 4.1888r3 (nearly).
The Ellipse
a = \ major axis. o = oblateness.
b = \ minor axis. e = eccentricity.
¦k ab = area of ellipse.

a — b
o = 

r a

-_6_2 2

The Earth Compared with Other Bodies
P = the radius of the body as compared with the
radius of the earth. Thus in case of the moon, the
moon's radius = 1081, the earth's radius = 3959,
and P = mi
P2 = surface of body as compared with that of the
earth.
P3 = volume of body as compared with that of the
earth.
JYJQgg -™- = surface gravity as compared with that of the
earth.

FORMULAS AND TABLES 309

Centrifugal Force
c = centrifugal force, r = radius.
v = velocity. m = mass. mv2
c =  ¦
r
Lessening of surface gravity at any latitude by reason
of the centrifugal force due to rotation.
g = surface gravity.
c at any latitude = 5^- X cos2 <f>-
Deviation of the plumb line from true vertical by reason
of centrifugal force due to rotation.
d = deviation.
d = 357" X sin 2 <f>. Miscellaneous
Rate of swing of pendulum varies inversely as the
square root of the surface gravity, r = — — •
Density of a body = — j— •
Hourly deviation of the plane of a pendulum due to
the rotation of the earth = sin latitude X 15° (d = sin </>
X 15°).
Weight of bodies above the surface of the earth.
w = weight,
d = distance from the center of the earth.
wtt7
Weight of bodies below the surface of the earth.
w <x d.

310 APPENDIX

GEOGRAPHICAL CONSTANTS*
Equatorial semi-axis:
in feet  20,926,962.
in meters  6,378,206.4
in miles  3,963.307
Polar semi-axis :
in feet  20,855,121.
in meters  6,356,583.8
in miles  3,949.871
Oblateness of earth  1-^294.9784
Circumference of equator (in miles)  24,901 . 96
Circumference through poles (in miles)  24,859.76
Area of earth's surface, square miles  196,971,984.
Volume of earth, cubic miles  259,944,035,515.
Mean density (Harkness)  5.576
Surface density (Harkness)  2 . 56
Obliquity of ecliptic (see page 118)  23° 27' 4.98 s.
Sidereal year  365 d. 6 h. 9 m. 8 . 97 s. or 365 . 25636 d.
Tropical year . . . . 365 d. 5 h. 48 m. 45.51 s. or 365.24219 d.
Sidereal day 23 h. 56 m. 4 . 09 s. of mean solar time.
Distance of earth to sun, mean (in miles)  92,800,000.
Distance of earth to moon, mean (in miles)  238,840.
MEASURES OF LENGTH
Statute mile  5,280. 00 feet
Nautical mile,t or knot  6,080. 27
German sea mile  6,076. 22
Prussian mile, law of 1868  24,604. 80
Norwegian and Swedish mile  36,000. 00
Danish mile  24,712.51
Russian werst, or versta  3,500. 00
Meter.  3.28
Fathom  6.00
Link of surveyor's chain  0. 66
* Dimensions of the earth are based upon the Clarke spheroid of
1866. f As defined by the United States Coast and Geodetic Survey.

FORMULAS AND TABLES

311

TABLE OF NATURAL SINES AND COSINES

Sin

Cos

Sin

Cos

Sin

Cos

0°

.0000

90°

31°

.5150

59°

61°

.8746

29°

1

.0175

89

32

.5299

58

62

.8829

28

2

.0349

88

33

.5446

57

63

.8910

27

3

.0523

87

34

.5592

56

64

.8988

26

4

.0698

86

35

.5736

55

65

.9063

25

5

.0872

85

36

.5878

54

66

.9135

24

6

.1045

84

37

.6018

53

67

.9205

23

7

.1219

83

38

.6157

52

68

.9272

22

8

.1392

82

39

.6293

51

69

.9336

21

9

.1564

81

40

.6424

50

70

.9397

20

10

.1736

80

41

.6561

49

71

.9455

19

11

.1908

79

42

.6691

48

72

.9511

18

12

.2079

78

43

.6820

47

73

.9563

17

13

.2250

77

44

.6947

46

74

.9613

16

14

.2419

76

45

.7071

45

75

.9659

15

15

.2588

75

46

.7193

44

76

.9703

14

16

.2756

74

47

.7314

43

77

.9744

13

17

.2924

73

48

.7431

42

78

.9781

12

18

.3090

72

49

.7547

41

79

.9816

11

19

.3256

71

50

.7660

40

80

.9848

10

20

.3420

70

51

.7771

39

81

.9877

9

21

.3584

69

52

.7880

38

82

.9903

8

22

.3746

68

53

.7986

37

83

.9925

7

23

.3907

67

54

.8090

36

84

.9945

6

24

.4067

66

55

.8192

35

85

.9962

5

25

.4226

65

56

.8290

34

86

.9976

4

26

.4384

64

57

.8387

33

87

.9986

3

27

.4540

63

58

.8480

32

88

.9994

2

28

.4695

62

59

.8572

31

89

.9998

1

29

.4848

61

60

.8660

30

90

1.0000

0

30

.5000

60

312

APPENDIX

TABLE OF NATURAL TANGENTS AND COTANGENTS

Tan

Cot

Tan

Cot

Tan

Cot

0°

.0000

90°

31°

.6009

59°

61°

1.8040

29°

1

.0175

89

32

.6249

58

62

1.8807

28

2

.0349

88

33

.6494

57

63

1.9626

27

3

.0524

87

34

.6745

56

64

2.0503

26

4

.0699

86

35

.7002

55

65

2.1445

25

5

.0875

85

36

.7265

54

66

2.2460

24

6

.1051

84

37

.7536

53

67

2.3559

23

7

.1228

83

38

.7813

52

68

2.4751

22

8

.1405

82

39

.8098

51

69

2.6051

21

9

.1584

81

40

.8391

50

70

2.7475

20

10

.1763

80

41

.8693

49

71

2.9042

19

11

.1944

79

42

.9004

48

72

3.0777

18

12

.2126

78

43

.9325

47

73

3.2709

17

13

.2309

77

44

.9657

40

74

3.4874

16

14

.2493

76

45

1.0000

45

75

3.7321

15

15

.2679

75

46

1.0355

44

76

4.0108

14

16

.2867

74

47

1.0724

43

77

4.3315

13

17

.3057-

73

48

1.1106

42

78

4.7046

12

18

.3249

72

49

1.1504

41

79

5.1446

11

19

.3443

71

50

1.1918

40

80

5.6713

10

20

.3640

70

51

1.2349

39

81

6.1338

9

21

.3839

69

52

1.2794

38

82

7.1154

8

22

.4040

68

53

1.3270

37

83

8.1443

7

23

.4245

67

54

1.3764

36

84

9.5144

6

24

.4452

66

55

1.4281

35

85

11.43

5

25

.4663

65

56

1.4826

34

86

14.30

4

26

.4877

64

57

1.5399

33

87

19.08

3

27

.5095

63

58

1.6003

32

88

24.64

2

28

.5317

62

59

1.6643

31

89

57.29

1

29

.5543

61

60

1.7321

30

90

0.0000

0

30

.5774

60

FORMULAS AND TABLES 313

LIST OF TABLES GIVEN IN THIS BOOK PAGE
Curvature of earth's surface  ... .28
Cosines. . . . . . .... 311
Cotangents   312
Day, length of longest day at different latitudes . 158
Declination of the sun, see analemma  127
Deviation of freely swinging pendulum due to earth's rotation 57
Distances, etc., of planets . . 266
Equation of time, see analemma.  127
Earth's dimensions, etc. . . ... .... 310
Latitudes, lengths of degrees  44
of principal cities of the world . ... .88
Longitudes, lengths of degrees . .  ; 44
of principal cities of the world  88
Measures of length   310
Meridional parts . .  217
Sines, natural ...  311
Solar system table . . .  266
Standard time adoptions  81
Tangents, natural  312
Time used in various countries  .... 81
Velocity of earth's rotation at different latitudes 58
Vertical ray of sun, position on earth, see analemma  127

GLOSSARY
Aberration, the apparent displacement of sun, moon, planet, or star pro
duced as a resultant of (a) the orbital velocity of the earth, and (6)
the velocity of light from the heavenly body.
Acceleration, increase or excess of mean motion or velocity.
Altitude, elevation in degrees (or angle of elevation) of an object above
the horizon.
Analemma, a scale showing (a) the mean equation of time and (6)
the mean declination of the sun for each day of the year.
Aphelion (3, fe' li on), the point in a planet's orbit which is farthest
from the sun.
Apogee (ap' o je), the point farthest from the earth in any orbit;
usually applied to the point in the moon's orbit farthest from the
earth.
Apparent solar day, see Day.
Apparent (solar) time, see Time.
Apsides (ap' si dez), line of, a, line connecting perihelion and aphelion
of a planet's orbit, or perigee and apogee of a moon's orbit.
Apsides is plural for apsis, which means the point in an orbit
nearest to the primary or farthest from it.
Arc, part of a circle; in geography, part of the circumference of a
circle.
Asteroids, very small planets. A large number of asteroids revolve
around the sun between the orbits of Mars and Jupiter.
Autumnal equinox, see Equinox.
Axis, the line about which an object rotates.
Azimuth (5z' I miith) the angular distance of an object from the -celestial
meridian of the place of the observer to the celestial meridian of
the object. The azimuth of the sun is the distance in degrees from
its point of rising or setting to a south point on the horizon.
Celestial sphere, the apparent hollow sphere in which the sun, moon,
planets, comets, and stars seem to be located.
Center of gravity the point about which a body (or group of
bodies) balances.
Centrifugal force (sen trif u gal), a force tending away from a center.
Centripetal force (sen trip' e tai), a force tending toward a center.
Colures (ko liirz'), the four principal meridians of the celestial sphere,
two passing through the equinoxes and two through the solstices.
Conjunction, see Syzygy. 314

GLOSSARY 315
Copernican system (k5 per' ni can), the theory of the solar system
advanced by Copernicus (1473-1543) that the sun is the center of
the solar system, the planets rotating on their axes and revolving
around the sun. See Heliocentric theory.
Cotidal lines, lines passing through places that have high tide at the
same time.
Day. Astronomical day, a period equal to a mean solar day, reckoned
from noon and divided into twenty-four hours, usually numbered
from one to twenty-four.
Civil day, the same as an astronomical day excepting that it is
reckoned from midnight. It is also divided into twenty-four
hours, usually numbered in two series, from one to twelve.
Sidereal day, the interval between two successive passages of a
celestial meridian over a given terrestrial meridian. The zero
meridian from which the sidereal day is reckoned is the one
passing through the First point of Aries. The length of the
sidereal day is 23 h. 56 m. 4.09 s. The sidereal day is divided
into twenty-four hours, each shorter than those of the civil or
astronomical day; they are numbered from one to twenty-four.
Solar day.
Apparent solar day, the interval between two successive passages
of the sun's center over the meridian of a place; that is, from
sun noon to the next sun noon; this varies in length from 23 h.
59 m. 38.8 s. to 24 h. 0 m. 30 s.
Mean solar day, the average interval between successive passages
of the sun's center over the meridian of a place; that is, the aver
age of the lengths of all the solar days of the year; this average
is 24 h. as we commonly reckon civil or clock time.
Declination is the distance in degrees of a celestial body from the
celestial equator. Declination in the celestial sphere corresponds
to latitude on the earth.
Eccentricity (gk sen tris' l ty), see Ellipse.
Ecliptic (6 klip' tik), the path of the center of the sun in its apparent
orbit in the celestial sphere. A great circle of the celestial sphere
whose plane forms an angle of 23° 27' with the plane of the equator.
This inclination of the plane of the ecliptic to the plane of the
equator is called the obliquity of the ecliptic. The points 90°
from the ecliptic are called the poles of the ecliptic. Celestial
latitude is measured from the ecliptic.
Ellipse, a plane figure bounded by a curved line, every point of which
is at such distances from two points within called the foci (pro
nounced fo' si; singular, focus) that the sum of the distances is
constant.

316 GLOSSARY
Eccentricity (Sk sSn tris' I ty) is the fraction obtained by dividing
the distance of a focus to the center of the major axis by one half
the major axis.
Oblateness or ellipticity is the deviation of an ellipse from a circle
and is the fraction obtained by dividing the difference between
the major and minor axes by the major axis.
Ellipticity (61 lip tis' I ty), see Ellipse.
Equation of time (e kwa' shun), the difference between apparent
solar time, or time as actually indicated by the sun, and the mean
solar time, or the average time indicated by the sun. It is usually
indicated by the minus sign when the apparent sun is faster than
the mean sun and with the plus sign when the apparent time is
slow. The apparent sun time combined with the equation of time
gives the mean time; e.g., by the apparent sun it is 10 h. 30 m., the
equation is — 2 m. (sun fast 2 m.), combined we get 10 h. 28 m.,
the mean sun time. See Day.
Equator (g kwa' ter), when not otherwise qualified means terrestrial
equator.Celestial equator, the great circle of the celestial sphere in the
plane of the earth's equator. Declination is measured from the
celestial equator.
Terrestrial equator, the great circle of the earth 90° from the
poles or ends of the axis of rotation. Latitude is measured from
the equator.
Equinox, one of the two points where the ecliptic intersects the celes
tial equator. Also the time when the sun is at this point.
Autumnal equinox, the equinox which the sun reaches in autumn.
Also the time when the sun is at that point, September 23.
Vernal equinox, the equinox which the sun reaches in spring.
This point is called the First point of Aries, since that sign of
the zodiac begins with this point, the sign extending eastward
from it 30°. The celestial meridian (see colure) passing through
this point is the zero meridian of the celestial sphere, from which
celestial longitude is reckoned. The vernal equinox is also the
time when the sun is at this point, about March 21, the beginning
of the astronomical year. See Year.
Geocentric (je o sSn' trik ; from ge, earth; centrum, center),
Theory of the solar system assumes the earth to be at the center
of the solar system ; see Ptolemaic system.
Latitude, see Latitude.
Parallax, see Parallax.
Geodesy (je 5d' S sy), a branch of mathematics or surveying which
is applied to the determination, measuring, and mapping of lines or
areas on the surface of the earth.

GLOSSARY 317
Gravitation, the attractive force by which all particles of matter tend
to approach one another.
Gravity, the resultant of (a) the earth's attraction for any portion of
matter rotating with the earth and (6) the centrifugal force due to
its rotation. The latter force (6) is so small that it is usually
ignored and we commonly speak of gravity as the earth's attrac
tion for an object. Gravity is still more accurately defined in the
Appendix.
Heliocentric (he 11 5 sBn'trik ; from helios, sun; centrum, center).
Theory of the solar system assumes the sun to be at the center of ,
the solar system; also called the Copernican system (see Coper
nican system).
Parallax, see Parallax.
Horizon (ho ri' zon), the great circle of the celestial sphere cut by a
plane passing through the eye of the observer at right angles to
the plumb line.
Dip of horizon. If the eye is above the surface, the curvature of
the earth makes it possible to see beyond the true horizon. The
angle formed, because of the curvature of the earth, between the
true horizon and the visible horizon is called the dip of the
horizon.
Visible horizon, the place where the earth and sky seem to meet.
At sea if the eye is near the surface of the water the true horizon
and the visible horizon are the same, since water levels and forms
a right angle to the plumb line.
Hour- circles, great circles of the celestial sphere extending from pole
to pole, so called because they are usually drawn every 15° or one
for each of the twenty-four hours of the day. While hour-circles
correspond to meridians on the earth, celestial longitude (see
Longitude) is not reckoned from them as they change with the
rotation of the earth.
Latitude, when not otherwise qualified, geographical latitude is
meant.Astronomical latitude, the distance in degrees between the
plumb line at a given point on the earth and the plane of the
equator.
Celestial latitude, the distance in degrees between a celestial
body and the ecliptic.
Geocentric latitude, the angle formed by a line from a given
point on the earth to the center of the earth (nearly the same
as the plumb line) and the plane of the equator.
Geographical latitude, the distance in degrees of a given point
on the earth from the equator. Astronomical, geocentric, and

318 GLOSSARY
geographical latitude are nearly the same (see discussion of Lati
tude in Appendix).
Local time, see Time.
Longitude. Celestial longitude, the distance in degrees of a celestial body
from lines passing through the poles of the ecliptic (see Ecliptic),
called ecliptic meridians; the zero meridian, from which celestial
longitude is reckoned, is the one passing through the First point
of Aries (see Equinox).
Terrestrial longitude, the distance in degrees of a point on the
earth from some meridian, called the prime meridian.
Mass, the amount of matter in a body, regardless of its volume or
size.
Mean solar time, see Time.
Meridian. Celestial meridian, a great circle of the celestial sphere passing
through the celestial poles and the zenith of the observer. The
celestial meridian passing through the zenith of a given place
constantly changes with the rotation of the earth.
Terrestrial meridian, an imaginary line on the earth passing
from pole to pole. A meridian circle is a great circle passing
through the poles.
Month. Calendar month, the time elapsing from a given day of one month
to the same numbered day of the next month; e.g., January 3
to February 3. This is the civil or legal month.
Sidereal month, the time it takes the moon to revolve about the
earth in relation to the stars; one exact revolution of the moon
about the earth; it varies about three hours in length but aver
ages 27.32166 d.
Synodic month, the time between two successive new moons or
full moons. This is what is commonly meant by the lunar
month, reckoned from new moon to new moon; its length varies
about thirteen hours but averages 29.53059 d. There are sev
eral other kinds of lunar months important in astronomical
calculations.
Solar month, the time occupied by the sun in passing through a
sign of the zodiac; mean length, 30.4368 d.
Nadir (na' der), the point of the celestial sphere directly under the
place on which one stands; the point 180° from the zenith.
Neap tides, see Tides.
Nutation, a small periodic elliptical motion of the earth's axis, due
principally to the fact that the plane of the moon's orbit is not the
same as the plane of the ecliptic, so that when the moon is on one

GLOSSARY 319
side of the plane of the ecliptio there is a tilting tendency given the
bulging equatorial region. The inclination of the earth's axis, or
tho obliquity of the ecliptic, is thus slightly changed through a
period of 18.6 years, varying each year from 0" to 2" (See
Motions of the Axis in the Appendix.)
Oblateness, the same as ellipticity; see Ellipse.
Oblate spheroid, see Spheroid.
Obliquity (5b Ilk' wl ty), of the ecliptic, see Ecliptic.
Opposition, see Syzygy.
Orbit, the path described by a heavenly body in its revolution about
another heavenly body.
Parallax, the apparent displacement, or difference of position, of an
object as seen from two different stations or points of view.
Annual or heliocentric parallax of a star is the difference in
the star's direction as seen from the earth and from the sun.
The base of the triangle thus formed is based upon half the major
axis of the earth's orbit.
Diurnal or geocentric parallax of the sun, moon, or a planet is
the difference in its direction as seen from the observers' station
and the center of the earth. The base of the triangle thus formed
is half the diameter of the equator.
Perigee (p6r' I je), the point in the orbit of the moon which is nearest
to the earth. The term is sometimes applied to the nearest point
of a planet's orbit.
Perihelion (per I he' II on) , the point in a planet's orbit which is nearest
to the sun.
Poles. Celestial, the two points of the celestial sphere which coincide
with the earth's axis produced, and about which the celestial
sphere appears to rotate.
Op the ecliptic, the two points of the celestial sphere which are 90°
from the ecliptic.
Terrestrial, the ends of the earth's axis.
Ptolemaic system (tol e ma' ik), the theory of the solar system advanced
by Claudius Ptolemy (100-170 a.d.) that the earth is the center of
the universe, the heavenly bodies daily circling around it at different
rates. Called also the geocentric theory (see Geocentric).
Radius (plural, radii, ra' di I) , half of a diameter.
Radius Vector, a line from the focus of an ellipse to a point in the
boundary Une. Thus a line from the sun to any planet is a radius
vector of the planet's orbit.
Refraction of light, in general, the change in direction of a ray of
light when it enters obliquely a medium of different density. As

320 GLOSSARY
used in astronomy and in this work, refraction is the change in
direction of a ray of light from a celestial body as it enters the
atmosphere and passes to the eye of the observer. The effect is to
cause it to seem higher than it really is, the amount varying with the
altitude, being zero at the zenith and about 36' at the horizon.
Revolution, the motion of a planet in its orbit about the sun, or of a
satellite about its planet.
Rotation the motion of a body on its axis.
Satellite, a moon.
Sidereal day, see Day.
Sidereal year, see Year.
Sidereal month, see Month.
Sidereal time, see Time.
Signs of the zodiac, its division of 30° each, beginning with the vernal
equinox or First point of Aries.
Solar times, see Time.
Solstices (s51' stis es; sol, sun; stare, to stand), the points in the
ecliptic farthest from the celestial equator, also the dates when the
sun is at these points; June 21, the summer solstice; December 22,
the winter solstice.
Spheroid (sfg' roid), a body nearly spherical in form, usually referring
to the mathematical form produced by rotating an ellipse about
one of its axes; called also an ellipsoid or spheroid of revolution
(in this book, a spheroid of rotation).
Oblate spheroid, a mathematical solid produced by rotating an
ellipse on its minor axis (see Ellipse).
Prolate spheroid, a mathematical solid produced by rotating an
ellipse on its major axis (see Ellipse).
Syzygy (siz' I jy; plural, syzygies), the point of the orbit of the moon
(planet or comet) nearest to the earth or farthest from it. When
in the syzygy nearest the earth, the moon (planet or comet) is said
to be in conjunction ; when in the syzygy farthest from the earth it
is said to be in opposition.
Time. Apparent solar time, the time according to the actual position
of the sun, so that twelve o'clock is the moment when the
sun's center passes the meridian of the place (see Day, apparent
solar).
Astronomical time, the mean solar time reckoned by hours num
bered up to twenty-four, beginning with mean solar noon (see
Day, astronomical).
Civil time, legally accepted time; usually the same as astronomical
time except that it is reckoned from midnight. It is commonly
numbered in two series of twelve hours each day, from midnight

GLOSSARY 321
and from noon, and is based upon a meridian prescribed by law
or accepted as legal (see Day, civil).
Equation op time, see Equation of time.
Sidereal time, the time as determined from the apparent rotation
of the celestial sphere and reckoned from the passage of the
vernal equinox over a given place. It is reckoned in sidereal
days (see Day, sidereal).
Solar time is either apparent solar time or mean solar time,
reckoned from the mean or average position of the sun (see
Day, solar day).
Standard time, the civil time that is adopted, either by law or
usage, in any given region; thus practically all of the people of
the United States use time which is five, six, seven, or eight hours
earlier than mean Greenwich time, being based upon the mean
solar time of 75°, 90°, 105°, or 120° west of Greenwich.
Tropical year, see Year.
Tropics. Astronomical, the two small circles of the celestial sphere parallel
to the celestial equator and 23° 27' from it, marking the north
ward and southward limits of the sun's center in its annual
(apparent) journey in the ecliptic; the northern one is called the
tropic of Cancer and the southern one the tropic of Capricorn,
from the signs of the zodiac in which the sun is when it reaches
the tropics.
Geographical, the two parallels corresponding to the astronomical
tropics, and called by the same names.
Vernal equinox, see Equinox, vernal.
Year. Anomalistic tear (a nom a lis' tik), the time of the earth's revo
lution from perihelion to perihelion again; length 365 d., 6 h.,
13 m., 48 s.
Civil year, the year adopted by law, reckoned by all Christian
countries to begin January 1st. The civil year adopted by
Protestants and Roman Catholics is almost exactly the true
length of the tropical year, 365 . 2422 d., and that adopted by Greek
Catholics is 365 . 25 d. The civil year of non-Christian countries
varies as to time of beginning and length, thus the Turkish civil
year has 354 d.
Lunar year, the period of twelve lunar synodical months (twelve
new moons); length, 354 d.
Sidereal year, the time of the earth's revolution around the sun
in relation to a star; one exact revolution about the sun; length,
365.2564 d.
JO. MATH. OEO. — 21

322 GLOSSARY
Tropical year, tne period occupied by the sun in passing from one
tropic or one equinox to the same again, having a mean length of
365 d. 5 h. 48 m. 45.51 s. or 365.2422 d. A tropical year is shorter
than a sidereal year because of the precession of the equinoxes.
Zenith (ze' nith), the point of the celestial sphere directly overhead;
180° from the nadir.
Zodiac (zo' di ak), an imaginary belt of the celestial sphere extending
about eight degrees on each side of the ecliptic. It is divided into
twelve equal parts (30° each) called signs, each sign being somewhat
to the west of a constellation of the same name. The ecliptic being
the central line of the zodiac, the sun is always in the center of it,
apparently traveling eastward through it, about a month in each
sign. The moon being only about 5° from the ecliptic is always in
the zodiac, traveling eastward through its signs about 13° a day.

INDEX

Abbe, Cleveland, 5, 65.
Abbott, Lyman, 76, 142-144.
Aberdeen, S. D., 89.
Aberration of light (see Glos
sary, p. 314), 104-106, 110,
278.
Abraham, 141.
Acapulco, Mexico, 102.
Acceleration (see Glossary, p.
314), 290-292.
Adelaide, Australia, 88.
Aden (a' den), Arabia, 88.
Africa, 102, 186, 200, 214, 216,
223, 268.
Akron, O., 81.
Alabama, 90.
Alaska, 69, 87, 89, 91, 98, 102,
103.
Albany, N. Y., 89.
Albany, Tex., 243.
Alberta, Canada, 82.
Aleutian Is., 96.
Alexandria, Egypt, 32, 83, 88,
270, 271.
Algeria, 83.
Allegheny Observatory, Alle
gheny, Pa., 69.
Allen, W. F., 67.
Allowance for curvature of
earth's surface, 226, 230,
233.
Al-Mamoum, 271.
Almanac, 118, 123-125, 171.
Altitude, of noon sun, 12, 13,
170-174, 302, 303.
of polestar or celestial pole,
58-61, 170-174.

Amazon, bores of, 189.
American Practical Navigator,
217.
Amsterdam, Holland, 82, 88.
Analemma (see Glossary, p.
314), description of, 126.
representation of, 127.
uses of, 128-130, 171-174,
301.
Anaximander (an ax I man'der),
269.
Annapolis, Md., 89.
Ann Arbor, Mich., 89.
Antipodal (fin tip'6 dal) areas,
map showing, 41.
Antwerp, Belgium, 88.
Aphelion (see Glossary, p. 314),
119, 285, 287.
Apia (a pe'a), Samoa, 88.
Apogee (see Glossary, p. 314),
178.
Apsides (see Glossary, p. 314),
287.
Aquarius (a kwfi'ri u.s) , 296.
Arabia, 88.
Arcturus (ark tu'rus) , 54, 109.
Area method of determining
geoid, 36, 37.
Area of earth's surface, 310.
Arequipa (a ra, ke'pa), Peru,
164.
Argentina (ar jen te'na), 81.
Aries (a'rf ez) , constellation, 294.
first point of, 293, 316.
sign of zodiac, 118, 297, 298.
Aristarchus (fir Istar'kus), 275,
277.
Aristotle (ar'istotl), 270.
Arkansas (ar'kan sa) , 90, 233.

323

324

INDEX

Arkansas River, 233.
Armenian Church, 145.
Asia, 34, 41, 186, 214.
Asteroids, 50, 246, 253, 293, 314.
Astronomical day, 130, 315.
Athens, Greece, 42, 84, 88.
Atlanta, Ga., 89.
Atlantic Ocean, 141, 186, 242.
Atmosphere, 161-167.
absence of, on moon, 263, 264.
how heated, 167.
on Jupiter, 257.
on Mars, 254.
on Mercury, 261.
on Venus, 256.
origin of, 251, 252.
Attu Island, 89, 99.
Auckland, New Zealand, 97.
Augusta, Me., 89.
Augustan calendar, 135.
Austin, Tex., 89.
Australia, 41, 96, 97, 102.
Austria-Hungary, 68, 82.
Axis, changes in position of, 288,
289.
defined, 22, 314.
inclination of, see Obliquity of
ecliptic.
parallelism of, 154.
Azimuth, 303, 314. B
Babinet, 207.
Bacchus, 134.
Bailey, S. I., 163.
Balearic Is., 86.
Balkan States, 140, 145.
Ball's History of Mathematics,
269.
Baltimore, Md., 89.
Bangor, Me., 89.
Bankok, Siam, 88.
Barcelona, Spain, 88.
Barlow and Bryan's Mathe
matical Astronomy, 292.
Barometer, 161, 176.

Base line, 231-234, 243, 244.
Batavia, Java, 88.
Behring Strait, 33.
Belgium, 68, 82, 88.
Beloit, Wis., 89.
Benetnasch, 61.
Bergen, Norway, 88.
Berkeley, Calif., 89.
Berlin, Germany, 88.
Bessel, F. W., 31, 32, 35, 109.
Bethlehem, 145.
Big Dipper, 9,48, 60, 61, 111, 295.
Bismarck Archipelago, 83.
Bismarck, N. D., 89.
Black Hills Meridian, 230.
Bogota (bo go ta'), Columbia, 83.
Boise (boi'za), Ida., 89.
Bombay (bSm ba'), India, 88.
Bonne's projection, 221, 224.
Bonn Observatory, 109.
Bordeaux (bdrdo'.,, France, 88.
Bores, tidal, 187, 189.
Bosphorus, 144.
Boston, Mass., 67, 89.
Bowditch, Nathaniel, 217.
Brahe (bra), Tycho, 109, 277,
278.
Bradley, James, 106, 278.
Brazil, 89.
British Columbia, 82.
British Empire, 82.
Brussels, Belgium, 82, 88.
Budapest (boo' da pest), Hun
gary, 86.
Buenos Aires (bo'nus a'rfz),
Argentina, 73, 88.
Buffalo, N. Y., 90.
Bulgaria, 68.
Bulletin, TJ. S. G. S., 243.
Burmah, 82.

Cadiz (ka'diz), Spain, 88.
Caesar, Augustus, 135, 136.
Julius, 134, 138.
Cairo (ki'ro), Egypt, 83, 88.

INDEX

325

Calais, Me., 32.
Calcutta, India, 88, 108.
Calendar, 132-145.
ancient Mexican, 141.
Augustan, 135.
Chaldean, 141.
Chinese, 141.
early Roman, 134, 138.
Gregorian, 136.
Jewish, 144, 190.
Julian, 135, 140.
Mohammedan, 140, 142-144.
on moon, 262.
Turkish, 142.
California, 32, 35, 37, 68, 89, 90,
91, 155, 168.
Callao (kalla'6), Peru, 86.
Canada, 48, 229, 233.
Cancer, constellation of, 150.
sign of zodiac, 295.
tropic of, 160, 269.
Canterbury Tales, quoted, 118.
Canton, China, 88.
Cape Colony, Africa, 82.
Cape Deshnef, Siberia, 96, 99.
Cape Horn, 41.
Cape May, N. J., 32, 35.
Cape Town, Africa, 32, 73.
Capricorn, constellation of, 150.
sign of zodiac, 296, 298.
tropb of, 151. '
Caracas (kara'kas), Venezuela,
87.
Carleton College, Northfield,
Minn., 69.
Carnegie Institution of Wash
ington, 35.
Carolines, the, 83.
Cassini (kasse'ne), G. D., and
J., 274.
Cassiopeia (kas si 6 pe'ya), 9, 60,
61.
Cayenne (ki fin'), French Guiana,
28, 29, 88.
Celestial equator, 47, 150, 170-
174, 283, 284, 294, 316.

Celestial latitude, 283, 284.
longitude, 284, 318.
meridians, 283, 284.
pole, 46, 47, 58, 61, 170-174.
sphere, 45-51, 62, 314.
tropics, 150, 151, 321.
Central time, in Europe, 68, 76,
85, 86.
in the United States, 67, 71,
74, 75, 77, 128.
Centrifugal force (see Glossary,
p. 314), 13-16, 29, 51, 279-
283, 290, 309.
Centripetal force (see Glossary,
p. 314), 15, 16.
Ceres, 134.
Ceylon (sel6n'), 82.
Chaldeans, 133, 141, 269, 294,
295.
Chamberlin, T. C., 5, 250.
Charles IX., King of France,
139.
Charleston, S. C, 90.
Charles V., Emperor of Spain,
214.
Chatham Islands, 82, 96.
Chaucer, quoted, 118.
Cheyenne (shl en'), Wyo., 90.
Chicago, III, 63, 67, 90, 96, 98,
129, 174, 208.
Chile (che'la), 82, 89.
China, 83, 88, 89, 101, 163, 269.
Chinese calendar, 141.
zodiac, 296.
Choson, Land of the Morning
Calm (Korea), 97.
Christiania (kris te a'ne a) , Nor
way, 88.
Christmases, three in one year,
145.
Chronograph (krSn'o graph), 69.
Chronometer (kro n8m'e ter) , 64.
Cincinnati, O., 74, 90.
Circumference of earth, 30, 31,
310.
Circle defined, 20, 21.

326

INDEX

Circle of illumination, or day cir
cle, 149, 151, 152, 155-157.
Ciudad Juarez, see Juarez.
Civil day, 130, 315.
Clarke, A. R., 30-32, 34, 35, 37,
44, 275, 310.
Cleomedes (kle Sm'edez), 272.
Cleveland, O., 74, 90.
Clock, sidereal, 69, 70.
Collins, Henry, 100.
Colombia, 83.
Colon (ko Ion') , Panama, 85.
Colorado, 67, 90.
Color vibrations, 106, 107.
Columbia, S. C, 90.
Columbus, Christopher, 137, 138,
212, 272.
Columbus, O., 74, 90.
Comets, 50, 246, 278.
Compass, magnetic, or mariner's,
152, 153, 226, 227.
Concord, N. H., 90.
Congressional township, 228.
Conic projection, 218-224.
Conjunction, 178, 185, 320.
Connecticut, 90.
Constantinople, Turkey, 88.
Convergence of meridians, 230,
233.
Copenhagen, Denmark, 88.
Copernican system (see Glos
sary, p. 315), 276-278.
Copernicus (ko per'nikus), 50,
109, 276-278.
Cordoba (kor'do ba), Argentina,
81.
Corinto (ko ren'to) , Nicaragua,
85._
Correction line, 234.
Cosines, natural, table of, 311.
Costa Rica, 83.
Cotangents, natural, table of,
312.
Cotidal lines, 186, 315.
Crepusculum, the, 165.
Creston, Iowa, 80.

Cuba, 69, 83, 88.
Curvature of surface of earth,
rate of, 27, 28, 43, 44, 226,
233.
Cygnus (sig'nus; plural and pos
sessive singular, cygni), 109.
Cylindrical projection, 209-218,
224. D
Dakotas, division of, 234.
Danish West Indies, 83.
Date line, see International
date line.
Day (see Glossary, p. 315), as
tronomical, 130.
circle, see Circle of illumina
tion.
civil, 130.
length of, 155-158.
lunar, 188.
origin of names of days of
week, 142.
sidereal, 114, 130.
solar, 62, 114, 130.
total duration of a, 98."
Deadwood, S. D., 90.
Deimos (di'mus), 255.
Declination (see Glossary, p.
315), 125, 127, 171-175,
284.
De la Hire, Phillippe (fe lap' de
laer'),201.
Denmark, 68, 83, 88, 136.
Density, formula for, 309.
Density of earth, 310.
Denver, Col., 26, 27, 67, 90, 98.
Des Moines (de moin'), Iowa, 90.
Deshnef, Cape, 96, 99.
Detroit, Mich., 63, 73, 90.
Deviation, of pendulum, 54-57.
of plumb line, 281-283.
Dewey, George, 103.
Diameter of earth, 29-31, 45,
310.
Dimensions of earth, 310.

INDEX

327

Dip of horizon (see Glossary, p.
317).
Distances, of planets, 266, 310.
of stars, 45, 246.
District of Columbia, 77, 91, 124.
Diurnal (diur'nal), motion of
earth, see Rotation.
Division of Dakotas, 234.
Dryer, Charles R., 5.
Dublin, Ireland, 64, 82, 88.
Duluth, Minn., 90.
E
Earth in Space, 246-267.
Earth's dimensions, 310.
Eastern time, in Europe, 68, 87.
in the United States, 66, 67,
71, 75, 78.
Eastward deflection of falling
objects, 51-54.
Eclipse, 24, 116, 161, 178.
Ecliptic (see Glossary, p. 315),
116, 119, 284, 286, 287.
obliquity of, 118, 147, 288,
310.
Edinburgh (Sd'in burro), Scot
land, 8S.
Egypt, 83, 88, 133, 268.
El Castillo (61 kas tel'yo), Nica
ragua, 85.
Ellipse (see Glossary, p. 315),
20-22, 193, 206, 308.
Ellipsoid of rotation (see Glos
sary, p. 320), 36.
El Ocotal (61 6k 6 tai'), Nica
ragua, 85.
El Paso, Tex., 68, 75, 76.
Encyclopaedia Britannica, 102.
England, 9, 80, 82, 89, 98, 101,
136, 139, 274.
Ephemeris (e fem'e ris), see
Nautical almanac.
Epicureans, 272.
Equador, 83.
Equation of time (see Glossary,
p. 316), 123-127.

Equator (see Glossary, p. 316),
celestial, 47, 150, 170-174,
283, 284, 294.
length of day at, 157.
terrestrial, 23, 33, 48, 49, 118,
148-152, 272, 280, 310.
Equinox (see Glossary, p. 316),
119, 147, 148, 155, 156, 168,
169, 285.
precession of, 286-288, 303.
Eratosthenes (era tSs' the nez) '
270, 271.
Erie, Pa., 90.
Establishment, the, of a port,
181.
Eudoxu?, 270.
Euripides (Q rip'I dez), 122.
Europe, 101, 166, 168, 220, 222,
223, 226, 288, 293.

Fargo, N. D., 90.
Farland, R. W., 132.
Faroe (fa'ro), Islands, 83.
Fathom, length of, 310.
Fiji Islands, 96, 100, 103.
Fiske, John, 145, 268.
Fixed stars, 10, 108, 109, 265,
266.
Florence, Italy, 88.
Florida, 90, 91.
Form of the earth, 24-44.
Formosa, 84.
Formulas, 307-309.
Foucault (foo ko'), experiment,
with gyroscope, 155.
with pendulum, 54-57.
France, 32, 64, 83, 88, 89, 130,
136, 139, 187.
Franklin's almanac, 137.
Fundy, Bay of, 189.
G
Gainesville, Ga., 75.
Galilei, Galileo (gal I le'o
gal I We), 277, 278.

328

INDEX

Galveston, Tex., 75, 90.
Gannett, Garrison and Hous
ton 'sCommercial Geography,
206.
Gauss, 52.
Gemini, 295, 298.
Genesis, 141.
Genoa, Italy, 76.
Geocentric, latitude, see Lati
tude.
theory (see Glossary, p. 316),
277, 278.
Geodesy (see Glossary, p. 316),
275.
Geodetic Association, Interna
tional, 289.
George IL, King of England,
227.
Geographical constants, 310.
Geoid (je'oid), 33-37, 275.
Geometry, origin of, 226.
Georgia, 68, 75, 80, 89, 91.
German East Africa, 84.
Germany, 68, 77, 83, 84, 88, 89,
97, 136, 233.
Gibraltar, Spain, 82, 88.
Glasgow, Scotland, 88.
Globular projection, 198-201,
211.
Glossary, 314-322.
Gnomonic (no m5n'ik) , cylin
drical projection, 209-211.
Gnomonic projection, 201-204,
211.
Goode, J. Paul, 302.
Goodsell Observatory, North-
field, Minn., 69.
Gravimetric lines, map show
ing, 34.
Gravitation, 16-18, 178, 179,
272 317.
Gravity' 18," 25, 28', 29, 183-
185, 279-282, 290, 304, 305,
308, 309, 317.
on Jupiter, 19, 256.
on Mars, 253.

Gravity on Mercury, 260.
on moon, 19, 20, 261.
on Neptune, 259.
on Saturn, 258.
on sun, 19, 264, 265.
on Uranus, 259.
on Venus, 256.
Great Britain, 64, 68, 77, 80,
182, 233.
Great circle sailing, 203, 204,
212.
Greece, 84, 88.
Greenland, 218.
Greenwich (Am. pron., gren'
wich; Eng. pron., grin'Ij or
gren'ij), England, 41, 42,
64, 67, 68, 73, 77, 78, 80,
82, 84, 86-91, 96, 100, 124-
126, 172.
Gregorian calendar, 135-138.
Guam, 73, 87.
Guaymas, Mexico, 85.
Guiana, French, 88, 273.
Gulf of Mexico, 37, 189.
Gunnison, Utah, 243.
Guthrie, Okla, 90.
Gyroscope (ji'roskop), 154,155.
H
Hague, The, Holland, 84, 88.
Hamburg, Germany, 88, 125.
Harkness, William, 275, 310.
Harper's Weekly, 163, 164.
Harte, Bret, 92.
Hartford, Conn., 90.
Harvard Astronomical Station
(Peru), 164.
Havana, Cuba, 83, 87, 88.
Hawaiian (Sandwich) Islands,
87, 101.
Hayden, E. E., 5, 73, 77.
Hayford, J. F., 36, 37, 245.
Hegira, 140.
Heliocentric theory (see Glos
sary, p. 317), 277, 278.

INDEX

329

Heliodon (he' K o don), 303.
Helena, Mont., 90.
Hemispheres unequally heated,
169, 284.
Heraclitus (her a kli'tus), 270.
Hercules (her'cu lez), constella
tion, 247.
Herodotus (he r6d' o tus), 132,
268.
Herschel, John, 34.
Hidalgo, Mexico, 85.
Hipparchus (hip ar'kus), 271,
286.
Historical sketch, 268-278.
Holland, 68, 84, 88, 89, 136.
Holway, R. S., 155.
Homer, 268.
Homolographic projection, 205-
207, 211.
Honduras, 84.
Hongkong, 82, S3, 88, 1C2.
Honolulu, Hawaiian Islands,
90, 100.
Horizon (see Glossary, p. 317),
38-40, 47, 152, 153, 158,
170, 175.
Hungary, 137.
Hutchins, Thomas, 228, 229.
Huygens (hl'gens), Christian,
273, 274.

Iceland, 83.
Idaho, 89.
Illinois, 90, 230, 231.
Impressions of a Careless Trav
eler, quoted, 76, 142-144.
India, 32, 82, 88, 89, 186, 242.
Indian principal meridian, 230.
Indiana, SO, 230.
Indianapolis, Ind., £0.
Indian Ocean, 186.
Indian Territory, survey of, 245.
Insolation> 165-169.
International date line, 95, 101.

International Geodetic Associa
tion, 289.
Intersecting conic projection,
221, 222.
Iowa, 80, 90.
Ireland, 33, 64, 77, 82, 88, 233.
Isle of Man, 82.
Isogonal (i sog'on al) line, 227.
Italy, 68, 84, 88, 89.

Jackson, Miss., 90.
Jacksonville, Fla., 90.
James II. , King of England, 274.
Japan, 84, 85, 89, 289.
Java, 88.
Jefferson, Thomas, 229.
Jerusalem, 89.
Journal of Geography, 155.
Juarez (hoo a'reth), Mexico, 76.
Julian calendar, 134, 135, 140.
Jupiter, 19, 250, 253, 256, 257,
266, 274, 277. K
Kamerun, Africa, 84.
Kansas, 35.
Kansas City, Mo., 90, 96.
Keewatin, Canada, 82.
Kentucky, 79, 90.
Kepler, Johann, 278.
laws of, 284-286.
Key West, Fja., 90.
Kiaochau (ke a o chow'), China,
83.
Korea (ko re'a), 85.
Kramer, Gerhard, 214.
Kustner, Professor, 109.

La Condamine (lak6n'damen),
189.
Lake of the Woods, 233.
Lake Superior, 231.
Landmarks, use of, in surveys,
226-228.

330

INDEX

Lansing, Mich., 90, 231.
Lapland, 274.
Larkin, E. L., 265-267.
Latitude (see Glossary, p. 317),
astronomical, 282.
celestial, 283, 284.
geocentric, 282.
geographical, 42.
determined by altitude of
circumpolar star, 58-61.
determined by Foucault
experiment, 55, 56.
determined by altitude of
noon sun, 170-175.
lengths of degrees, 42-44.
of principal cities, 88-91.
origin of term, 40.
Law Notes, quoted, 81.
Layard, E. L., 101.
Leavenworth, Francis P., 5.
Leo, 295, 298.
Legal aspect of standard time,
76-81.
Leipzig, Germany, 89.
Length of day, 155-158.
Lewis, Ernest I., 145:
Lexington, Ky., 90.
Leyden, Holland, 84, 272.
Libra (li'bra), 117, 295, 296.
Lick Observatory, 69, 73.
Lima (le'ma), Peru, 86.
Lincoln, Neb., 90.
Link of surveyor's chain, 310.
Lisbon, Portugal, 41, 73, 86, 89.
Little Dipper, 9.
Little Rock, Ark., 90, 233.
Liverpool, England, 89.
London, England, 41, 63, 64,
93-96, 99, 137.
London Times, 139.
Longitude (see Glossary, p. 318),
and time, 62-91.
celestial, 284.
how determined, 63-65, 128.
lengths of degrees, 44.
of principal cities, 88-91.

Longitude, origin of term, 40.
Los Angeles, Calif., 90.
Louisiana, 90, 230.
Louisville, Ky., 79, 90.
Louis XIV., King of France, 28.
Lowell, Mass., 90.
Lowell, Percival, 97, 254.
Luxemburg, 68, 85.
Luzon, 90. M
Macaulay's History of England,
138, 139.
McNair, F. W., 5, 52.
Madison, Wis., 67, 90.
Madras (madras'), India, 73,
82, 89.
Madrid, Spain, 73.
Magnetic compass, 152, 153, 226,
227.
Magnetic pole, 152, 153.
Magellan's fleet, 92.
Maine, 32, 37.
Malta, 82.
Managua (ma na'gua), Nica
ragua, 85.
Manitoba, Canada, 82.
Manila, Philippine Is., 73, 90,
101-103.
change of date at, 101.
Map, 41, 42, 230, 236.
Map projections, 190-225.
Mare Island Naval Observatory,
69, 71, 87.
Mariane Islands, 83.
Markham, A. H, 153, 164, 165.
Mars, 253-255, 266; 285, 306.
Marseilles, France, 89.
Maryland, 89.
Massachusetts, 89, 90, 227.
Mauritius (ma rish' I us) Island,
73.
Mean solar day (see Day).
Measuring diameter of moon,
240,241,

INDEX

331

Measuring distances of objects,
237, 241.
heights of objects, 238.
Measures of length, 310.
Mediterranean, 102.
Melbourne, Australia, 89.
Memphis, Tenn., GO.
Mercator projection, 204, 211—
219, 221.
Mercedonius, 134.
Mercury, 183, 253, 260, 261, 266.
Meridian, 23, 29, 32, 37, 38, 95-
100, 187, 188, 190-225, 283,
318.
celestial, 283, 284, 318.
circle, 23.
length of degrees of, 44.
prime, 41, 42.
principal, for surveys, 230-
236.
rate for convergence, 233.
standard time, 66-68, 71, 75,
77,78,81-87,302.
Meridional parts, table of, 217.
Meteors, 50, 248, 249.
Meter, length of, 310.
Metes and bounds, 226, 228.
Mexico, 85, 89, 102.
Gulf of, 189.
Michigan, 51, 68, 74, 70, 89, 90.
231.
College of Mines, 52.
Midnight sun, 163.
Mile, in various countries, 310.
Milwaukee, Wis., 90, 103.
Mining and Scientific Press,
52-54.
Minneapolis, Minn., 63, 90, 96.
Minnesota, 90, 91, 231, 233.
Mississippi, 90.
River, 231.
Missouri, 90, 91.
River, 227.
Mitchell, Frank E., 5.
Mitchell, S. D., 90.
Miyako (me ya'ko) Islands, S4.

Mobile, Ala., 90.
Mohammedan calendar, 140, 142-
144.
Mollendo, Peru, 86.
Mollweide projection, 207.
Montana, 90.
Montevideo, Uruguay, 87, 89.
Montgomery, Ala., 90.
Month (see Glossary, p. 318),
133, 134.
sidereal, 177, 188.
synodic, 177.
Moon or satellite, 10, 11, 19, 161,
176-185,240,241, 246, 255,
257-259, 261-264, 266, 278,
288, 297, 298, 308.
Moore, C. B. T., 96.
Morse, J. F., 303.
Moscow, Russia, 89.
Motion in the line of sight, 106-
109.
Motions of the earth, 289.
Motions of the earth's axis, 286-
289.
Moulton, F. R., 250.
Mountain time belt, 67, 68, 75.
Mount Diablo meridian, 230.
Munich (mu'nik), Germany, 89.
Myths and superstitions of the
zodiac, 297, 298.
N
Nadir (see Glossary, p. 318), 38.
Naples, Italy, 89.
Nash, George W., 4.
Nashville, Tenn., 90.
Natal, Africa, 82.
Nautical almanac, 118, 124, 171.
Neap tides, 185, 188.
Nebraska, 81, 90, 91, 227.
Nebulae, 248, 251.
Nebular hypothesis, 247-252.
Nehemiah, 141.
Neptune, 253, 259, 260, 266.
Neuchatel, Switzerland, 86.

332

INDEX

Nevada, 91.
Newark, N. J., 90, 91.
New Brunswick, Canada, 82.
New Caledonia, 101.
Newcomb, Simon, 118.
Newchwang, China, 83.
Newfoundland, 82.
New Guinea, 83.
New Hampshire, 90.
New Haven, Conn., 90.
New Jersey, 32, 78, 90, 91.
New Mexico, 68, 76, 91, 230.
New Orleans, La., 59, 67, 90, 108.
New South Wales, 82, 89.
New Style, 137-140, 143.
Newton, Isaac, 15, 51, 138, 272.
New York, 76, 78, 89-91, 96,
98, 99, 204.
New York Sun, 55.
New Zsaland, 82, 96, 97.
Nicaea, Council of, 136.
Nicaragua, 85.
Nile, 226.
North America, 101, 185, 213,
219, 242, 287.
North Carolina, 91.
North, line, 11, 61, 130.
on map, 211, 212, 217, 224.
pole, 22, 47, 152, 153, 289.
star, 43, 46, 47, 49, 58, 148, 286.
North Dakota, 89, 90, 234.
Northfield, Minn., 69, 90.
North Sea, 187.
Northwest Territory, survey of,
228-230.
Norway, 68, 85, 88.
Norwood, Richard, 272.
Nova Scotia, Canada, 82.
Noumea, New Caledonia, 101.
Numa, 134.
Nutation of poles, 288, 318.
0
Oblateness of earth, 28-33, 37,
43, 273-275.

Obliquity of the ecliptic, 118,
147, 288, 310, 315.
Observations of stars, 9.
Official Railway Guide, 73-75.
Ogden, Utah, 91.
Ohio, 90, 230.
River, 228.
Oklahoma, 90.
Old Farmer's Almanac, 124.
Old Style, 137-140, 143.
Olympia, Wash., 91.
Omaha, Neb., 91.
Ontario, Canada, 82.
Oporto, Portugal, 204.
Opposition, 177, 178, 185.
Orange River Colony, 82.
Orbit, of earth, 22, 113, 114, 116-
119, 122, 132, 147, 152, 246,
251, 285, 304-306.
of moon, 177, 178, 262, 288.
Oregon, 91.
Origin of geometry, 226.
Orion (o ri'on), 111.
Oroya, Peru, 86.
Orkneys, The, 82.
Orthographic projection, 190-
195, 198, 200, 211.
Outlook, The, 76, 142-144.

Pacific Ocean, 68, 96, 97, 185,
242.
Pacific time belt, 68.
Pago Pago (pron. pango, pango),
Samoa, 91, 96.
Palestine, 89.
Pallas, 293.
Panama, 69, 85, 87, 89.
Para, Brazil, 89.
Parallax, 109, 241, 277, 319.
Parallelism of earth's axis, 154.
Parallels, 23, 190-225.
Paris, France, 28, 29, 41, 55, 64,
89, 274.
Parliament, 77.

INDEX

333

Pegasus (peg'a siis), Square of,
48.
Peking, China, 89.
Pendulum clock, 28, 54, 309.
Pennsylvania, 78, 90, 228, 229.
Perigee, 178, 319.
Perihelion, 119, 284, 319.
Peru, 86, 274.
Pescadores (peskador'ez)Is.,84.
Phases of the moon, 10, 263.
Philadelphia, Pa., 26, 27, 59, 67,
91.
Philippine Is., 87, 101, 102.
Phobos (fo'bus), 255, 306.
Phoenicians, 268.
Photographing, 50.
Picard (pe kar'), Jean, 273, 274.
Pierre, S. D., 91.
Pittsburg, Pa., 67, 97.
Pisces (pis'sez), 293, 296.
Planetesimal hypothesis, 250.
Planets, 19, 50, 246, 285, 306.
Plato, 270.
Pleiades (ple'yadez), 48, 122.
Plumb line, 11, 51, 281, 282.
Plutarch, 270.
Point Arena, Calif., 32.
Point Barrow, 91.
Pointing exercise, 38-40.
Poland, 33, 136.
Polar diameter of earth, 310.
Polaris, see polestar.
Pole, celestial, 46, 170, 171, 284,
286, 319.
magnetic, 153.
nutation of, 288.
of the ecliptic, 286, 288.
terrestrial, 22, 37, 38, 47, 54-
56, 60, 61, 152-154, 157,
193-212, 280, 286, 288, 290.
Polyconic projection, 222-224.
Polestar (see North star), 9, 10,
286.
Popular Astronomy, 132.
Portland, Ore., 91.
Porto Rico, 87, 91.

Port Said (sa ed'), Egypt, 83.
Portugal, 41, 86, 89.
Posidonius (pSs'I do ni us), 271.
Practical Navigator, 217.
Practical work, 300-303.
Precession of equinoxes, 286-
288.
Prince Edward Island, Can., 82.
Princeton, N. J., 91.
Principal meridian, 230-236.
Projectiles, 304-306.
Projections, map, 190-225.
Proofs, form of earth, 24-29,
33-35, 274.
revolution cf earth, 104-111,
277, 278.
rotation of earth, 51-57, 62,
107, 155.
Proper motion of stars, 109.
Providence, R. I., 91.
Psalms, 252.
Ptolemaic system, 276, 319.
Ptolemy, Claudius, 271.
Ptolemy Necho, of Egypt, 268.
Pulkowa, Russia, 73, 86, 89.
Pythagoras (pi thag'6 ras), 269,
270, 275. Q
Quebec, Canada; 82.
Queensland, Australia, 82.
Quito (ke'to), Equador 83, 163,
174. R
Radius vector, 284, 319.
Raleigh, N. C, 91..
Ranges of townships, 231-233,
236.
Rapid City, S. D., 243.
Rate of curvature of earth's sur
face, 27, 28, 43, 44.
Refraction of light, 45, 158, 319.
Revolution (see Glossary, p.
320), 104-131, 146, 154,
184, 246, 248, 251, 254, 257-
262, 277, 278, 285-289.

334

INDEX

Richer (re shay'), John, 28, 29,
273.
Richmond, Va., 91.
Rio de Janeiro, Brazil, 89.
Rhode Island, 91.
Rhodesia, Africa, 82.
Rochester, N. Y., 91.
Roman calendar, 134, 138.
Rome, Italy, 73, 89.
Rotation of earth, 23, 320.
Rotation, proofs of, 51, 155.
Rotterdam, Holland, 89.
Roumania, 68.
Russia, 32, 86, 89, 96, 102, 103,
140, 242.

S

Sacramento, Calif., 91.
Sagittarius (sag it ta'ri us), 296.
St. John's, Newfoundland, 82.
St. Louis, Mo., 26, 27, 67, 85, 91.
St. Paul, Minn., 91, 96.
St. Petersburg, Russia, 73, 86,
89, 169, 174.
Salvador (Salvador'), 86.
Samoa, 84, 88, 91, 96, 97, 100.
San Bernardino, Calif., 230.
San Francisco, Calif., 93, 173.
San Jose (ho sa'), Costa Rica, 83.
San Juan del Sur, Nicaragua, 85.
San Juan, Porto Rico, 91.
San Rafael (rafagl'), Mexico,
85.
San Salvador, Salvador, 86.
Santiago (san ts a'go), Chile, 82.
Santa Fe, N. M., 91.
Santo Domingo, 86.
Saskatchewan, Canada, 82.
Satellite, see Moon.
Saturn, 30, 248, 253, 257, 258, 266.
Savannah, Ga., 91.
Scale of miles, 195, 216, 224.
Schott, C. A., 32, 33.
Scotland, 82, 88, 139, 187.
Scrap Book, 65.

Scientific American, 267, 305.
Scorpio, 296.
Seasons, 146-175, 168, 169.
Seattle, Wash., 91.
Section, 235, 236.
Seoul (sa ooi'), Korea, 85.
Servia, 68, 86.
Seven motions of earth, 289.
Seven ranges of Ohio, 229.
Sextant, 61.
Shakespeare, 299.
Shanghai (shang'hl), China, 83.
Shetland Is., 82.
Siam, 88.
Siberia, 97, 99, 103.
Sicily, 289.
Sidereal, clock, 69, 70.
day, 55, 315.
month, 177, 318.
year, 132, 287, 310, 321.
Signals, time, 71-73, 81-87.
Signs of zodiac, 116, 294, 320.
Sines, natural, table of, 311.
Sirius (sir'i us), 47.
Sitka, Alaska, 73, 91. :
Snell, Willebrord, 272.
Solar day, see Day.
Solar system, 246-267.
table, 266.
Solstices, 148, 167, 303, 320.
Sosigenes (so sig'e nez), 134.
South America, 34, 168, 186,
187, 200, 213, 218, 273.
South Australia, 82.
Southern Cross, 47.
South Carolina, 90.
South Dakota, 89-91, 227, 233,
234, 243.
South, on map, 200, 211, 212, 224.
pole, 153.
star, 46-49, 58, 59, 148, 151.
Spain, 68, 86, 88, 92, 128, 214.
Spectrograph, 109.
Spectroscope, 57, 107, 109.
Sphere, denned, 20.
Spheroid, 22, 28-33, 320,

INDEX

335

Spitzbergen, 32.
Spring tides, 185, 188.
Square of Pegasus, 48.
Stadium (sta' di um), 271.
Standard parallel, 233, 234.
Standard time, 65-88.
Star, distance of a, 45, 246.
motions of, 108, 109, 265, 266.
sun a, 265-267.
Stereographic projection, 195-
198, 200, 211.
Stockholm, Sweden, 86, 89.
Strabo (stra'bo), 271.
Strauss, N. M., 76.
Sun, 10-12, 19, 161, 246-248,
250, 251, 264-267.
apparent motions of, 113, 294.
a star, 265-267.
declination of, 127, 171-174.
fast or slow, 62, 123-130.
Sun Board, 302, 303.
Sundial, 62, 65, 131, 300-302.
Survey, 31, 32, 36, 226, 272.
Surveyor's chain, 226, 310.
Sweden, 32, 68, 86, 89, 277.
Switzerland, 68, 86, 136.
Sydney, Australia, 89.
Syene, Egypt, 270.
Symbols, 307.
Syzygy, 178, 185, 320.

Tables, list of, 313.
Tacubaya (ta koo ba'ya) , Mex
ico, 85.
Tallahassee, Fla., 91, 230.
Tamarack mine, 51, 53.
Tangents, natural, table of, 312.
Tasmania, 82.
Taurus, 295, 298.
Tegucigalpa, Honduras, 84.
Telegraphic time signals, 69, 81.
Tennessee, 90.
Texas, 89, 90, 243.
Thales (tha'lez), 270.

Thompson, A. H., 244.
Thucydides (thu sld'I dez), 132.
Tidal wave, bore, etc., 185, 189.
Tides, 176-189, 280, 290-292.
Tientsin (te Sn'tsen) , China, 83.
Tiers of townships, 231, 234, 236.
Time (see Glossary, p. 320),
apparent solar, 62.
ball, 71, 83.
confusion, 65, 73-76, 144.
in various countries, 81-87.
local, 64.
how determined, 69.
signals, 69-73, 81-87.
standard, 65-88, 128, 129.
Times, London, 139.
Titicaca, Lake, Peru, 86.
Todd, David, 106, 289.
Toga Is., 84.
Tokyo, Japan, 89.
Toledo, O., 73.
Tonga Is., 100.
Toscanelli, 212, 213.
Township, 228-236.
Transit instrument, 69.
Transvaal, 82.
Trenton, N. J., 91.
Triangulation, 31, 32, 237, 275.
Tropics, 150, 151, 173, 269, 321.
Tunis, 83.
Turkey, 87, 88, 142-144.
Turkish calendar, 142-144.
Tutuila (too twe'la), Samoa, 87,
96, 97.
Twilight, 161-165. U
Unequal heating, 169, 284.
United States, 31-34, 36, 42,
48, 51, 65, 69, 71, 78, 87,
89, 97, 98, 101-103, 126,
128, 129, 160, 168> 187, 224,
226-236, 289, 293.
United States Coast and Geo
detic Survey, 31, 32, 36, 242,
244, 245, 310.

336

INDEX

United States Geological Sur
vey, 31, 242, 243.
United States Government Land
Survey, 226-236.
United States Naval Observa
tory, 69-73, 81.
University of California, 155.
University of Chicago, 250, 302.
Ur, ancient Chaldean city, 141.
Uruguay, 87, 89.
Uranus (u'ranus), 253, 258,
259,266.
Ursa Major, 9.
Utah, 91, 243.

Valparaiso (val pa ri'so), Chile,
82, 89.
Van der Grinten, Alphons, 208.
Vibrations, color, 106, 107.
Victoria, Australia, 82, 89.
Virginia City, Nev., 91.
Virgo, 295.
Velocity of rotation, 58.
Venezuela, 87.
Venus, 183, 253, 255, 266, 277.
Vernal equinox, see Equinox.
Vertical ray of sun, 146, 147,
152, 155, 156, 165,166,313.
Vincocaya (vin ko ka'ya), Peru,
164.
Virginia, 91.
Voltaire, 274.
Volume of earth, 310.
W
Wady-Halfa (wa'de" hal'fa),
Egypt, 83.
Wallace, Kan., 35.
Wandering of the poles, 288.

Washington, 91.
Washington, D. G, 35, 42, 67,
71, 72, 85, 86, 91, 124, 230.
Washington, George, 138, 139.
Watch, to set by sun, 129.
Weight, see Gravity.
Wellington, New Zealand, 73.
Western European time, 68.
West Virginia, 91.
What keeps the members of the
solar system in their or
bits? E04-S06.
Wheeling, W. Va., 91.
Wilhelm II. , Emperor, 77. "
Wilmington, Del., 91.
Winona, Minn., 91.
Winter constellations, 111.
Wisconsin, 67, 89, 90, 103, 231.
Woodward, R. S., 5, 35.
World Almanac, 123.
Wyoming, 90. X
Xico, Mexico, 85. Y
Yaeyama (ye ya'ma) Is., 84.
Yaqui River, Mexico, 85.
Year, 132, 133, 287, 310, 321.
Young, C. A., 163.
Youth's Companion, 72, 153,
165.
Ysleta (Isla'ta), Tex., 76.

Zikawei (zlka'we), China, 83.
Zodiac, 116, 117, 141, 293, 322.
Zones, 152, 254-258, 260, 263.

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