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The Project Gutenberg EBook of Mathematical Geography, by Willis E. Johnson 

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Title: Mathematical Geography 

Author: Willis E. Johnson 

Release Date: February 21, 2010 [EBook #31344] 

Language: English 

Character set encoding: ISO-8859-1 












Copyright, 1907, 



Entered at Stationers ' Hall, London 


Produced by Peter Vachuska, Chris Curnow, Nigel Blower and the 
Online Distributed Proofreading Team at 

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In the greatly awakened interest in the common-school sub- 
jects during recent years, geography has received a large share. 
The establishment of chairs of geography in some of our great- 
est universities, the giving of college courses in physiography, 
meteorology, and commerce, and the general extension of geog- 
raphy 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 oc- 
cupy a large place in geographical control, the earth in its simple 
mathematical aspects should be better understood than it gen- 
erally 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 geographies and elementary astronomies. 
It is generally conceded that the mathematical portion of ge- 
ography is the most difficult, the most poorly taught and least 
understood, and that students require the most help in under- 
standing it. The subject-matter of mathematical geography is 
scattered about in many works, and no one book treats the sub- 
ject 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 teach- 
ers' preparation, much material herein organized 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 attain- 
ments 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 dif- 
ficult, 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 en- 
tire chapter has been read over carefully. 

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 laboratory 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 sug- 
gestions, 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 manuscript 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 him- 
self 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, Wiscon- 
sin, he wishes formally to acknowledge his obligation to him 
for an abiding interest in the subject. For the critical exam- 
ination of portions of the manuscript bearing upon fields in 
which they are acknowledged authorities, grateful acknowledg- 
ment 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. Wood- 
ward 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. 



IIntroductoryI 9 

IThe Form of the EarthI 23 

IThe Rotation of the EarthI 44 





IThe Earth's Revolution] 105 




ISeasonsI 147 




ITidesI 176 

IMap Projections! 190 


IThe United States Government Land SurveyI . . . 227 


ITriangulation in Measurement and SurveyI .... 238 

IThe Earth in SpaceI 247 

IHistorical Sketch] 267 

IAppendix] 278 

IGlossary] 313 

IIndexI 322 



Observations and Experiments 

Observations of the Stars. On the first clear evening, 
observe the "Big Dipper"* and the polestar. In September and 
in December, early in the evening, they will be nearly in the 
positions represented in Figure [I] Where is the Big Dipper later 
in the evening? Find out by observations. 

Learn readily to pick 
out Cassiopeia's Chair 
and the Little Dipper. 
Observe their apparent 
motions also. Notice the 
positions of stars in dif- 
ferent portions of the sky 
and observe where they 
are later in the evening. 
Do the stars around the 
polestar remain in the Fig. 1 

same position in relation 

to each other, — the Big Dipper always like a dipper, Cassiopeia's 
Chair always like a chair, and both always on opposite sides of 

*In Ursa Major, commonly called the "Plow," "The Great Wagon," or 
"Charles's Wagon" in England, Norway, Germany, and other countries. 




< '•-••■"■■■., 4 / 


Jt» ta D '">P», * North Star. ' 

5 «/ 

* *. o. 

s \\ 







the polestar? In what sense may they be called "fixed" stars 

Make a sketch of the Big Dipper and the polestar, record- 
ing 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. |2J). Now lay the dividers on a sheet 
of paper and mark the angle thus formed, representing the posi- 
tions 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 first-hand observations. If the ques- 
tions 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 


Fig. 2 



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 the 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 

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 mea- 
surement each week. Mark this Exhibit 2. 

On a south-facing window 
sill, strike a north-south line 
(methods for doing this are dis- 
cussed on pp. 

59, 130) 


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 

Noon shadow 

Fig. 3 



1 o'clock, p.m. (see Fig. |3J). 

Carefully cut from cardboard a semicircle and mark the de- 
grees, beginning with the middle radius as zero. Fasten this 
upon the window sill with the zero meridian coinciding 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 Exhibit 3. Continue the observations every few 

The Sun's Meridian Al- 
titude. When the shadow is 
due north, carefully measure 
the angle formed by the shadow 
and a level line. The sim- 
plest way is to draw the win- 
dow 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 an- 
gle being the one sought (see 
Fig. 111). Another way is to drive 
a pin in the side of the win- 
dow 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.[TT2" 

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 

Altitude of Sun at noon* 

Fig. 4 



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 cur- 
rent meaning. It comes 
from the Latin centrum, 
center; and fugere, to flee. 
A centrifugal force is one 
directed away from a cen- 
ter. When a stone is 
whirled at the end of a 
string, the pull which the 
stone gives the string is 
called centrifugal force. 

Because of the inertia of _. 

Fig. 5 

the stone, the whirling motion given to it by the arm tends to 

Tends to fly off 

make it fly off in a straight line (Fig. pi), — 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 stone whirled, the scale will show the 
amount of the centrifugal 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 


*This is explained on p. 
^On the use of symbols, sue 

i as C for centrifugal force, <fi for latitude, 

etc., see Appendix, p. 306 


circle (r) in which it moves, in the following ratio: 

mv 2 
\j = . 


The instant that the speed becomes such that the available 


strength of the string is less than the value of , however 

slightly, the stone will cease to follow the curve and will imme- 
diately 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 cen- 
trifugal 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 

At the equator centrifugal force (C) amounts to about ^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 = — — (that is, centrifugal force is ^ the 


force of the earth's attraction); at 45°, C = ; at 60°, C = 



For any latitude the "lightening effect" of centrifugal force 

due to the earth's rotation equals times the square of the 

H 289 


cosine of the latitude (C = x cos 2 0). 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 .7660 2 = — . 

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 an- 
gles to the direction of motion of a body. To distinguish between 
centrifugal force and centripetal force, the student should real- 
ize 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 action (A on B) 
there corresponds an equal and opposite reaction (B on A). 

Centrifugal force is the reaction of the body against the cen- 
tripetal 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 

* These calculations are based upon a spherical earth and make no al- 
lowances for the oblateness. 



of that which is required, being, in fact, 289 times the amount 
needed. The centripetal force in this case is that portion of the 
attraction 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 equa- 
tor 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 is the 
all-pervasive force by 
virtue of which every 
particle of matter 
in the universe is 
constantly drawing 
toward itself every 
other particle of mat- 
ter, however distant. 
The amount of this 
attractive force ex- 
isting 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 attrac- 

More rays intercepted when near the /feme" 
Fig. 6 



tion, 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 general terms the law is that 
the nearer an object is, the greater is its attractive 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 attraction. The ratio of the increase 
of the power of gravitation as distance decreases, may be seen 
from Figures [7] and [8j 




Fig. 7 

** - * 

_^-— ^TJi^^ v 

— — -""isj 

j "4^ 









Fig. 8 

Two lines, AD and AiJ (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 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. [7J 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 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 diffi- 
cult to realize that what we call the weight of an object is simply 
the excess of attraction which the earth possesses for it as com- 
pared with other forces acting upon it, and that it is a purely 
relative affair, the same object having a different weight in dif- 
ferent 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 1 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 

'For a more accurate and detailed discussion of gravity, see p. 278 


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 distances 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 gj 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 

1. Earth's attraction : Moon's attraction : : 1 : ^-. 

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 : : 
We have then the two proportions: 

1. Att. Earth : Att. Moon : : 1 : i. 

2. Att. Earth : Att. Moon : : 

1 1 

40002 ' 10812 

4000 2 1081 2 
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, rep- 
resents 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 p. 263). 

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 

circles. Wherever used in this book, unless otherwise stated, 

the term circle refers to the circumference. 

Every circle is con- 
ceived to be divided into 
360 equal parts called de- 
grees. 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 constitute a 

Circles on a sphere di- 
viding it into two hemi- 
spheres are called great cir- 
cles. Circles on a sphere di- 
viding it into unequal parts 

Ac- A'. Foci. C D„ Minor Axis, 
XY, Major Axis. A toA'. Focal 
Distance. AM* AM' AN* AN 

Fig. 9 

are called small circles. 



All great circles on the same sphere bisect each other, re- 
gardless of the angle at which they cross one another. That 
this may be clearly seen, with a globe before you test these two 

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 some- 
where, 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 estimating 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 
formed by cutting a 
right cone by a plane 
passing obliquely through its opposite sides (see Ellipse in 

To Construct 
An Ellipse 

Fig. 10 



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 represented 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 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 Spheroid in Glossary). The diameter 

upon which it rotates is called the axis. The ends of the axis are 
its poles. Imaginary lines on the 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 opposite meridian. 



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 circular," 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 

all sides of the earth are turned toward the sun during different 





Almost Uniform Surface 
Gravity. An object has al- 
most exactly the same weight 
in different parts of the earth 
(that is, on the surface), show- 
ing a practically common dis- 
tance from different points on 
the earth's surface to the cen- 
ter of gravity This is ascer- 
tained, not by carrying an ob- 
ject all over the earth and weigh- 
ing it with a pair of spring scales 
(why not balances?); but by not- 
ing 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 wa- 
ter, the lower part is hidden by 
the intervening curved surface. 

Fig. 11. Ship's rigging 
distinct. Water hazy. 

Fig. 12. Water distinct. 
Rigging ill-defined. 

sunset at the higher elevation. 

(Figs. 11, 12 ) This has been ob- 
served in many different places, 
and the rate of curvature seems 
uniform everywhere and in every 
direction. Persons ascending in 
balloons or living on high eleva- 
tions note the appreciably earlier 
time of sunrise or later time of 


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 disappear- 
ance 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 po- 
sition 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 different 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 tele- 
phone 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 ex- 
tending 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 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 trav- 
elers 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 dif- 
ferent places and in nearly every direction shows a practically 
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 l|, 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-^height (feet). 



Ht. ft. 

Dist. miles 

Ht. ft. 

Dist. miles 

Ht. ft. 

Dist. miles 







































































































The Earth an Oblate Spheroid 

Richer's Discovery. In the year 1672 John Richer, the as- 
tronomer to the Royal Academy of Sciences of Paris, was sent by 
Louis XIV to the island of Cayenne to make certain astronom- 
ical 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. 

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 elsewhere, was the explana- 
tion. 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, Ap- 

pendix, p. 278; also Historical Sketch, pp. 272-274). 

Repeated experiments show it to be a general fact that pen- 
dulums swing faster on the surface of the earth as one approaches 
the poles. Careful measurements of arcs of meridians prove be- 
yond question that the earth is flattened toward the poles, some- 
what 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 gener- 
ally believed to have been at one time. The reason why rotation 
causes an equatorial bulging is not difficult to understand. Cen- 
trifugal force increases away from the poles toward the equator 
and gives a lifting or lightening influence to portions on the sur- 
face. 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. 279). The form given the earth 

by its rotation is called an oblate spheroid or an ellipsoid of 

Amount of Oblateness. To represent a meridian cir- 
cle accurately, we should represent the polar diameter about 
t^ part shorter than the equatorial diameter. That this differ- 
ence 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 op- 
posite direction. The oblateness of Saturn is easily perceptible, 


being thirty times as great as that of the earth, or one tenth (see 

p. 258). Thus an ellipsoid nine inches in polar diameter (minor 
axis) and ten inches in equatorial diameter (major axis) would 
represent the form of that planet. 

Although the oblateness of the earth seems slight when rep- 
resented on a small scale and for most purposes may be ignored, 
it is nevertheless of vast importance in many problems in sur- 
veying, astronomy, and other subjects. Under the discussion of 
llatitudel 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 equatorial 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 of 1866. 

A. Equatorial diameter 7, 926.614 miles 

B. Polar diameter 7, 899.742 miles 

O blateness ; — 

A 295 

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 addi- 
tional information, and gave the following dimensions, though it 


is doubtful whether these approximations are any more nearly 
correct than those of 1866. 

Clarke Spheroid of 1878. 

A. Equatorial diameter 7, 926.592 miles 

B. Polar diameter 7. 899.580 miles 

A ~ B l 

Oblateness ; — 

A 293.46 

Another standard spheroid of reference often referred to, 
and one used by the United States Governmental Surveys be- 
fore 1880, when the Clarke spheroid was adopted, was calculated 
by the distinguished Prussian astronomer, F. H. Bessel, and is 
called the 

Bessel Spheroid of 1841. 

A. Equatorial diameter 7, 925.446 miles 

B. Polar diameter 7, 898.954 miles 

A ~ B l 


A 299.16 

Many careful pendulum tests and a great amount of scien- 
tific triangulation surveys of long arcs of parallels and meridi- 
ans 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 from Cape May, New Jersey, to Point Arena, Cali- 
fornia, through 48° 46' of longitude, or a distance of about 2, 625 
miles. This is the most extensive piece of geodetic surveying ever 
undertaken by any nation and was so carefully done that the to- 
tal 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° 1' 
of latitude and 22° 47' of longitude, a distance of 1,623 miles. 


Another long arc will soon be completed along the 98th merid- 
ian 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 par- 
allel, with which he was closely identified, said:* 

"Abundant additional means for improving the existing de- 
ductions concerning the earth's figure are now at hand, and it is 
perhaps not too much to expect that the International 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 spheroid 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 

*In his Transcontinental Triangulation and the American Arc of the 


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 dis- 
tributed 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 

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 pass- 
ing through the earth from 15° E. to 165° W. and its short- 
est axis from 105° E. to 75° W. The amount of this oblate- 
ness of the equator is estimated at about j^ or a difference of 
two miles in the lengths of these two diameters of the equator. 
Thus the meridian circle passing through central Africa and cen- 
tral 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 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 to a re- 
markable 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 av- 
erage length of degrees of longitude from the Atlantic coast for 



160 190 

»i) 1M ICO 

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

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. 

Diff. in 


of 1° 



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


53.829 mi. 
53.822 mi. 

53.828 mi. 

53.821 mi. 

Earth not an Ellipsoid of Three Unequal Axes. This 
oblateness of the meridians and oblateness of the equator 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 meridi- 
ans are not true ellipses, for the amount of flattening northward 
is not quite the same as the amount southward, and the math- 
ematical center of the earth is not exactly in the plane of the 

Geoid Defined. The term geoid, which means "like the 
earth," is now applied to that figure which most nearly corre- 
sponds 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 conti- 
nents, is called the geoid. It does not coincide exactly with the 
earth's spheroid, but is a slightly wavy surface lying 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." 

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 il- 
lustrated 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 

* Encyclopaedia Americana. 

^Given at the International Geographic Congress, 1904. 


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 suppos- 
ing that the workman is given a piece of sheet metal cut to the 
outline of the continuous triangulation which is supplied with 
necessary astronomic observations, and accurately molded to fix 
the curvature of the geoid, as shown by the astronomic observa- 
tions, and that the workman is then requested to construct 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 con- 
siderable width which is now utilized fully. It is obvious that the 
workman would succeed much better in constructing 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 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 ob- 
servations will, even without 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 partic- 
ular area of the earth as having its own peculiar curvature and 

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 calcula- 
tions it is always so treated. For more exact calculations, 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. 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 to- 

ward the north pole (Fig. 14). Whatever your latitude, drop 

your arm half as many degrees from the northern horizon as 



■5 Hariioft 



you are degrees from the pole, and you will point directly toward 

that pole.* 

You may be so accus- 
tomed 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 hori- 
zon. This is because one is 
liable to forget that he is 
living on a ball. To point 

Horizon Line 

Fig. 14 

to the horizon is to point away from the earth 

A Pointing Exer- 
cise. It may not be easy 
or even essential to learn 
exactly to locate many 
places in relation to the 
home region, but the abil- 
ity to locate readily some 
salient points greatly clar- 
ifies one's sense of loca- 
tion and conception of the 
earth as a ball. 

The following exercise 
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 

Fig. 15 

adapted for the north temperate latitudes (Fig. 15). Persons 

*The angle included between a tangent and a chord is measured by one 
half the intercepted arc. 



living in the southern states may use Figure [16j based on the 
30th parallel. The student should make the necessary readjust- 
ment for his own latitude. 

Horizon Line Dro P the arm from the 

northern horizon quarter 

way down, or 22 1°, and 

you are pointing toward 

the north pole (Fig. 


Fig. 16 

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 equa- 
tor, on the same parallel but 
on the opposite side of the 
earth, in opposite longitude. 
Were you to travel half way 
around the earth in a due easterly or westerly direction, you 
would be at that point. Drop the arm 22|° more, or 67|° from 
the horizon, and you are pointing 45° farther south or to the 
equator on the opposite side of the earth. Drop the arm 
22|° more, or 90° from the horizon, toward your feet, and you 
are pointing toward our antipodes, 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 22~°, and you are 
pointing to a place having the same longitude but on the equa- 
tor. Drop the arm 22 1° 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 remem- 
bering 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 eas- 
ily remember that longitude (from the Latin longus, long) is 
measured east and west. The word latitude is from the Latin 
latitude/, which is from latus, wide, and was originally used to 
designate measurement of the "width of the earth," or north and 

Antipodal Areas. From a globe one can readily ascertain 
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 mid- Atlantic, the point antipo- 
dal 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 merid- 
ian. 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 French that of the Paris Observatory, the Greeks 



Fig. 17. Map of Antipodal Areas 

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 reck- 
oned both from Greenwich and Washington. The latter city be- 
ing a trifle more than 77° west of Greenwich, a meridian num- 
bered 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 Observatory, 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 na- 
tions 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 denned 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 

Comparative Lengths of Degrees of Latitude. If the 
earth were a perfect sphere, meridian circles would be true 
mathematical 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 

Fig. 18 


thus shorter near the 

equator. Conversely, 

the meridian being slightly flattened 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 degrees 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 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. 











































































































































































































































































































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. 

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

Fig. 19 

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 distance, a line 

*No allowance is here made for the refraction of rays of light or the 
slight curvature of the globe in the locality. 




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 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 ap- 
parently 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 indicated by degrees in distance from established circles or 

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 turn- 
ing 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. 

Fig. 20 



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 reference to the Big Dipper, the South star is 
located by reference to a group of stars known as the Southern 

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 prolonged, 
would pierce the celestial poles, almost pierce the North and 
South stars, and the equator of the earth if extended would 
coincide with the celestial equator. 

At the North Pole. 
An observer at the north 
pole will see the North 
star almost exactly over- 
head, and as the earth 
turns around under 
his feet it will remain 
constantly overhead 

(Fig. |21j). Half way, or 
90° from the North star, 
g ' is the celestial equator 

around the horizon. As the earth rotates, — though it seems to 
us perfectly still, — the stars around the sky seem to swing in 
circles in the opposite direction. The planes of the star paths 
are parallel to 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 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 over- 
head, the celestial equa- 
tor 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 

Fig. 22 



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 

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 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 
directly 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 observer 
is liable to feel that 
a star rising due east 
ought to pass the zenith 
about six hours later in- 
stead of swinging slant- 
ingly around as it actu- 
ally seems to 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 

Fig. 23 


the line of one's body when erect. The 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 Copernicus 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 unthinkable rates of speed, 
we see that it is far easier to believe that it is the earth and 
not the celestial sphere that rotates 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 

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 ^ 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 demonstrating it. With 
his clear vision he said that if the earth rotates and an object 
were dropped from a considerable height, instead of falling di- 
rectly 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 Michi- 
gan, one of which (Shaft No. 3 of the Tamarack mine) is the 
deepest in the world, having a vertical depth of over one mile 
(and still digging!), ft was stated that objects dropped into such 
a shaft never reached the bottom but always lodged among tim- 
bers on the east side. Some papers added a touch of the grew- 
some by implying that among the objects found clinging to the 

*The slight geocentric deviations of the plumb line are explained on 
p. 12801 


east side are "pieces of a dismembered human body" which were 
not permitted to fall to the bottom because of the rotation of the 
earth. Following is a portion of an account* by F. W. McNair, 
President of the Michigan College of Mines. 

McNair 's Experiment. "Objects dropping into the shaft un- 
der 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 com- 
mon 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 convenient 
object at hand, was suspended about one foot from the timbers 
near the western corner of the compartment. The compartment 
stands diagonally with reference to the cardinal points. Forty- 
two hundred feet below 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 indi- 
cated by a prearranged signal transmitted by telephone to the 

*In the Mining and Scientific Press, July 14, 1906. 



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 com- 
plicated than the foregoing dis- 
cussion implies. It has received 
mathematical treatment from the 
great Gauss. According to his re- 
sults, the deviation to the east for 
a fall of 5, 000 feet at the Tama- 
rack mine should be a little un- 
der three feet. Both spheres had 
that much to spare before strik- 
ing the timbers. It is almost cer- 
tain, therefore, that others of the 
causes mentioned in the begin- 
ning acted to prevent a vertical 
lg- 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 vertically. 

"If the timbering permits lodgment, as is the case in most 
shafts, it may truthfully be said that if a shaft is deep in pro- 
portion to its cross section few indeed will be the objects falling 



into it which will reach the bottom, and such objects are more 
likely to lodge on the easterly side than on any other." 

The Foucault 
Experiment. An- ^ KOTIOH 

other simple demon- 
stration of the earth's 

rotation is 
the French 
M. Leon 

by the 


In 1851, 



from the 

Interesting 1 Experiment in the Dome 
of the Pantheon. 

V*w Tnrh Sttn Special Ktrrlce 

Parts, Oct. 23.— An Interesting experi- 
ment under the auspice* of the- astro- 
nomical society of France took place yes- 
terday afternoon when ocular proof of the I 
revolution of the earth was given by I 

meana 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. Chaumle, minister 
of public Instruction, who presided, 
burned a string that tied the weight to a 
polar and the immense pendulum began 
Its Journey. ". Sand bad 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 awing remained unchanged. 
The experiment was completely success- 
ful, -no* 

Fig. 25 

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 
swinging. It was found that the path gradually moved around 
toward the right. Now either the pendulum 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 change the plane of the 
swinging pendulum. If the pendulum swings back and forth in 
a north and south direction, the ruler can be entirely turned 
around without changing 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 pendulum 
would seem to deviate toward the right or clockwise. 

Conditions for Success. The Foucault experiment has been 
made in many places at different times. To be successful 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 should be protection 
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 deviation may 
be calculated. 

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 rotation 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 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. 310). 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 

* Strictly speaking, in one sidereal day. 


of the latitude. 

d = deviation 
(j) = latitude 
d = sin0 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 pendulum 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 37| 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 = ts or .6000. 


From the |table of sines] we find that this sine, .6000, corresponds 
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 -r- 
9 = 40) for the pendulum to make the entire circuit. 

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. 





Circuit of 



Circuit of 



275 hrs. 



31 hrs. 

















































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 rota- 
tion. Cyclonic winds in the northern hemisphere 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 (p. 108) in 
connection with the proofs of the earth's revolution. 

Velocity of Rotation 

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







































































































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 vari- 
ation has been discovered. Before attacking the somewhat com- 
plex 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 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 the 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 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. 


True Pole 

True Poie 



C 6 True Pole 

. I 


Fig. 26 

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 observation is made 
when the Big Dipper and Cassiopeia are 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 l|° below the true 
pole, hence l|° 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 Cassiopeiae. Methods of deter- 


mining latitude will be further discussed on pp. |172j - |175} The 
instrument commonly used in observations for determining lati- 
tude is the meridian circle, or, on shipboard, the sextant. Read 
the description of these instruments in any text on astronomy. 


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? 



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 variation will 

be taken up in the |chapter| on the earth's revolution. 

Mean Solar Day. By a mean solar day is meant the av- 
erage 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 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 the 
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 Lon- 
don, 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. As- 

*Many thoughtlessly assume that the fact of day and night is a proof 
of the earth's rotation. 


suming that both the watch and the sun were "on time" we 
readily see that since London time was two and one half 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 according 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 reckoned. 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 difference in time, the difference in 
longitude must be l| 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 de- 
termine the time it is in those places when it is noon at home. 
Many other problems may also be suggested. 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 by the 


sundial. All places on the same meridian have the same local 
time. Places on different meridians must of necessity have dif- 
ferent 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 incon- 
venience was occasioned by the prevalence of so many time sys- 
tems. But as transportation and communication 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 

"Before the adoption of standard time there were sometimes 
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 

*The Scrap Book, May, 1906. 






suggestions and the subject was soon taken up in an official way 
by the railroads of the country under the leadership of William 
F. Allen, then secretary of the General Time Convention of Rail- 
road Officials. As a result of his untiring efforts the railway asso- 
ciations 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 Philadel- 
phia, so the time used throughout 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 conse- 
quence 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 
convenience 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 re- 
sult, are discussed under the topic "Four Kinds of Time around 

El Paso" (p.[74j). 

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 timepieces 
differ in time by whole hours. In addition to astronomical obser- 
vatory 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 coun- 
tries 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, Luxemburg, 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. 80-87 


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 estab- 
lished 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 re- 
ceive their time corrections from other observatories. Goodsell 
Observatory, Carleton College, Northfield, 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 Pennsyl- 
vania system and the Lick Observatory to the Southern Pacific 

How Time is Determined at the United States Naval Obser- 
vatory. 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 situated 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 de- 
gree to the east or west. Other instruments used are the chrono- 
graph 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 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 automatically 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 clock error when 
last corrected and the rate at which it varies. 

A while before noon each day the exact sidereal time is cal- 
culated; this is converted into local mean solar time and this 


into standard time. The Washington Naval Observatory con- 
verts this into the standard time of the 75th meridian or East- 
ern 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 tele- 
graph 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 cor- 
rect time accessible to sixty millions of our population living 
east of the Rockies. 

Not only are the time signals sent to the telegraph sta- 
tions and thence to railway offices, clock makers and repairers, 
schools, court houses, etc., but the same telegraphic signal that 
marks noon also actually sets many thousands of clocks, their 
hands whether fast or slow automatically 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 elec- 
tric noon signal. The time ball in Washington is conspicuously 
placed on the top of the State, War, and Navy building and may 
be seen at considerable distances from many parts of the city. 

A few minutes before noon each day, one wire at each tele- 
graphic office is cleared of all business and "thousands of tele- 
graph operators sit in silence, waiting for the click of the key 
which shall tell them that the 'master clock' in Washington has 




I - - 


20 30 


SO 60 

SS** Minute before noon 

. . i . . 

5 Sec omitted 

56 * Minute before noon 

. . 1 . . . . . 

5 Sec omitted 
.... 1... . ( 

57*" Minute before noon 

5 Sec omitted 

58'* Minute before noon 

5 Sec omitted 
....1 No0n l 

53 •* Minute before noon 
Fig. 28 

10 Sec omitted 

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, Appendix IV, United States Naval Obser- 
vatory Publications) 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 
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 


*From "What's the Time," Youth's Companion, May 17 and June 14, 


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 distant points as 
Madras, Mauritius, Cape Town, Pulkowa (near St. Petersburg), 
Rome, Lisbon, Madrid, Sitka, Buenos Ayres, Wellington, Syd- 
ney, and Guam. Upon these occasions "our standard clock may 
fairly be said to be heard in 'the remotest ends of the earth,' 
thus anticipating 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 sta- 
tions within a few seconds, being received at the Lick Observa- 
tory in 0.05 s , Manila in 0.11 s , Greenwich in 1.33 s , and Sydney, 
Australia, in 2.25 s . 

Confusion from Various Standards 

Where different time systems are used in the same commu- 
nity, confusion must of necessity result. The following editorial 
comment in the Official Railway Guide for November, 1900, very 
succinctly sets forth the condition 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 throughout the 
country about seventeen years ago, when standard 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 

* "The Present Status of the Use of Standard Time," by Lieut. Com- 
mander E. E. Hayden, U. S. Navy. 


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 newspapers that they probably will do so af- 
ter 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 en- 
countered, 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 
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, while the Southern rail- 
way passing through the city uses Eastern time and the Georgia 
railway uses Central time. 

Fig. 29 

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 employed. 
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. 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 

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, 

"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 
comparison 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, 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:* 

* Several of the following quotations are taken from the "Present Status 
of the Use of Standard Time," by E. E. Hayden. 


"We, Wilhelm, by the grace of God German Emperor, King of Prussia, 
decree in the name of the Empire, the Bundesrath and Reichstag concur- 
ring, 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 Ire- 
land, 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 Parliament, deeds, and other legal instruments. 

Whereas it is expedient to remove certain doubts as to whether ex- 
pressions of time occurring in acts of Parliament, deeds, and other legal 
instruments relate in England and Scotland to Greenwich time, and in Ire- 
land 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 temporal, 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 Parlia- 
ment, 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 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 

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 prescribed 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 of- 
fices 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 otherwise expressed." In Penn- 
sylvania 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, com- 
mercial, 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, dif- 
ficulties may arise, as the following clipping from the New York 
Sun, November 25, 1902, illustrates: 

WHAT'S NOON IN A Whether the word "noon," 

FIRE POLICY? which marks the beginning and ex- 

piration of all fire insurance policies, 

Solar Noon or Standard Time means noon b y standard time, or 
Noon — Courts Asked to Say. noon b Y solar time > is a question 

which is soon to be fought out in 

the courts of Kentucky, in thirteen 

suits which have attracted the at- 

Fire in Louisville at 11:45 a.m., tention of fire insurance people all 

Standard Time, Which Was over the world. The suits are be- 

12:02 1-2 p.m. Solar Time— ing brought by the Peaslee-Gaulbert 

Policies Expired at Noon and Company and the Louisville Lead 

13 Insurance Companies Wont and Color Company of Louisville, 

Pay. 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 reck- 
oned in. In Louisville the solar noon 
is 17^ mm utes earlier than the stan- 
dard noon, so that a fire occurring 
in the neighborhood of noon on the 
day of a policy's expiration, may eas- 
ily be open to attack. 

The records of the Louisville 
fire department show that the fire 
that destroyed the buildings of the 
two companies was discovered 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 factory 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 thirteen 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 de- 
clined to pay, and when asked for 

an explanation declared that noon 
in the policies meant noon by so- 
lar 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 Louisville, but it is also the 
time upon which all business engage- 
ments and all domestic and social 
engagements are reckoned. They 
state further that they are prepared 
to show that in 1890 the city of 
Louisville passed an ordinance mak- 
ing standard time the official time of 
the city, that all legislation is dated 
according to standard time, and that 
the governor of the state is inaugu- 
rated 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 institutions 
which got charters many years ago 
that compel them to use solar time 
to this day. Most banks, they say, 
operate on standard time, although 
they keep clocks going at solar time 
so as to do business on that ba- 
sis if requested. Judging by stan- 
dard 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. 


Iowa Case. An almost precisely similar case occurred at 
Creston, Iowa, September 19, 1897. In this instance the insur- 
ance 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 meridian and 
meridian, or 12 o'clock post meridiem. An arbitrary and artificial 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 sometimes varies as 
much as fifteen minutes faster or slower than the average, op- 
portunities for controversies are multiplied 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 case 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 meridian 
of Cordoba. At 11 o'clock, A.M., a daily signal is telegraphed from the 
Cordoba Observatory. 

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 to 24 hours, zero corresponding 
to midnight at Greenwich. The Royal Observatory at Brussels commu- 
nicates 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, 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. 

New South Wales, Queensland, Tasmania and Victoria, 10 h. fast. 
South Australia and Northern Territory, 9 h. 30 m. fast. 

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 tele- 
graphed daily all over the country; but for strictly local use it is gener- 
ally 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 m. 34.1 s. 
slow, of the observatory at the Naval School located there. 

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 Bo- 
gota 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 

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, 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 h. 


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 trans- 
mitted from the observatory by telegraph to the towns of the kingdom. 

Holland. The local time of Amsterdam, h. 19 m. 32.3 s. fast, is gen- 
erally used, but Greenwich time is used by the post and telegraph ad- 
ministration and the railways and other transportation companies. The 
observatory at Leyden communicates the time twice a week to Amster- 
dam, The Hague, Rotterdam and other cities, and the telegraph bureau 
at Amsterdam signals the time to all the other telegraph bureaus every 

Honduras. In Honduras the half hour nearest to the meridian of Teguci- 
galpa, 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 In- 
dustrial School, where in turn it is indicated to the public by a steam 
whistle. The central telegraph office communicates 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 throughout 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 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, the Yaeyama, and the Miyako groups, and shall be called 


western standard time. This ordinance shall take effect from the first of 
January, 1896." This is 8 h. fast. 

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 Japanese 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 stan- 
dard 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 Washington (see 
page 70 1 and by a similar signal from the observatory at St. Louis, Mis- 
souri. 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 throughout 
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 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, h. 36 m. 44.7 s. slow. Standard time is in use throughout 
Portugal and is based upon the meridian of Lisbon. It is established 
by the Royal Observatory in the Royal Park at Lisbon, and from there 
sent by telegraph to every railway station throughout 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 na- 
tional 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, telegraph 
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 governmental 
offices in Spain and the Balearic Islands, railroad and telegraph offices. 
The hours are numbered from 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. 

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 theoretical sunrise, from tables showing the times 
of sunset, but the tower clocks are set only two or three times a week. 
The government 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 

United States. Standard time based upon the meridian of Greenwich, vary- 
ing 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 Pa- 
cific coast from the observatory at Mare Island Navy Yard, California. 
For further discussions of standard time belts in the United States, see 

pp. |64f|67| 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 established 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 Montevideo. 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 telegraphed 
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. 

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 "O" standing for observatory, "C" for 
cathedral, etc. 


Longitude from 

Adelaide, S. Australia, 

Snapper 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 

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 





50" S 

40" N 
43" N 
30" N 
28" N 
56" S 
21" N 
20" N 

10" N 
40" N 
37" N 
17" N 
45" N 
07" N 
11" N 
30" S 
40" N 
38" N 










30' 39" E 

58' 58" E 

51' 40" E 

53' 04" E 

24' 44" E 

44' 56" W 

43' 55" E 

28' 42" E 

10' 55" E 

48' 25" E 

20' 15" E 

23' 44" E 

48' 58" E 

31' 23" W 

22' 18" E 

22' 14" W 

12' 20" W 

17' 14" E 



Longitude from 

Calcutta, Ft. Wm. 


Canton, China, Dutch Light . 
Cape Horn, South Summit . . 

Cape Town, S. Africa, O 

Cayenne, Fr. Guiana, 


Christiania, Norway, O 

Constantinople, Turkey, C. . . 
Copenhagen, Denmark, New 


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 

Jerusalem, Palestine, Ch 

Leipzig, Germany, O 

Lisbon, Portugal, O. (Royal) 

Liverpool, England, O 

Madras, India, O 

Marseilles, France, New O. . . 

Melbourne, Victoria, O 

Mexico, Mexico, O 

Montevideo, Uruguay, C 

Moscow, Russia, O 

Munich, Germany, O 

Naples, Italy, O 

Panama, Cent. Am., C 

Para, Brazil, Custom H 

Paris, France, O 

Peking, China 

22° 33' 25' 

23° 06' 35' 

55° 58' 41' 

33° 56' 03' 

4° 56' 20' 

59° 54' 44' 

41° 00' 16' 


41' 14' 

23' 13' 

57' 23' 

46' 04' 

07' 10' 

52' 43' 

04' 40' 

33' 07' 

09' 21' 

16' 52' 

46' 45' 

20' 06' 

42' 31' 

24' 04' 

04' 06' 

18' 22' 

49' 53' 

26' 01' 

54' 33' 

45' 20' 

08' 45' 

51' 46' 

57' 06' 

26' 59' 

50' 11' 

56' 00 

88° 20' 11' 

113° 16' 34 

67° 16' 15 

18° 28' 40'' 

52° 20' 25 

10° 43' 35'' 

28° 58' 59'' 















34' 47'' 

20' 30'' 

10' 54'' 

15' 22 

21' 17'' 

17' 39'' 

18' 30'' 

58' 25'' 

21' 30'' 

09' 31'' 

13' 25'' 

23' 30'' 

11' 10'' 

04' 16'' 

14' 51' 

23' 43' 

58' 32' 

06' 39' 

12' 15' 

32' 36'' 

36' 32'' 

14' 44 

32' 12'' 

30' 01'' 

20' 14'' 

28' 54 



Longitude from 

Pulkowa, Russia, O 

Rio de Janeiro, Brazil, O 

Rome, Italy, O 

Rotterdam, Holl., Time Ball. 
St. Petersburg, Russia, see 


Stockholm, Sweden, O 

Sydney, N. S. Wales, O 

Tokyo, Japan, O 

Valparaiso, Chile, Light 


59° 46' 19" N 

22° 54' 24" S 

41° 53' 54" N 

51° 54' 30" N 

59° 20' 35" N 

33° 51' 41" S 

35° 39' 17" N 

33° 01' 30" S 

30° 19' 40" E 

43° 10' 21" W 

12° 28' 40" E 

4° 28' 50" E 

18° 03' 30" E 

151° 12' 23" E 

139° 44' 30" E 

71° 39' 22" W 

United States 

Aberdeen, S. D., N. N. & I. S. 

Albany, N. Y., New O 

Ann Arbor, Mich., O 

Annapolis, Md., O 

Atlanta, Ga., Capitol 

Attu Island, Alaska, 

Chichagoff Harbor 

Augusta, Me., Baptist Ch. . . . 

Austin, Tex 

Baltimore, Md., Wash. Mt. . . 
Bangor, Me., Thomas Hill . . . 

Beloit, Wis., College 

Berkeley, Cal., O 

Bismarck, N. D 

Boise, Idaho, Ast. Pier 

Boston, Mass., State House.. 

Buffalo, N. Y 

Charleston, S. C, Lt. House. 
Cheyenne, Wyo., Ast. Sta. . . . 

Chicago, 111., O 

Cincinnati, Ohio 

Cleveland, Ohio, Lt. H 







45" W 







42" W 







48" W 







08" W 







29" W 







24" E 







37" W 







35" W 







59" W 







59" W 







46" W 







41" W 







08" W 







04" W 







50" W 







42" W 







58" W 







52" W 







36" W 







00" W 







10" W 




Longitude from 

Columbia, S. C 

Columbus, Ohio 

Concord, N. H 

Deadwood, S. D., P. O 

Denver, Col., O 

Des Moines, Iowa 

Detroit, Mich 

Duluth, Minn 

Erie, Pa., Waterworks 

Fargo, N. D., Agri. College . . 

Galveston, Tex., C 

Guthrie, Okla 

Hartford, 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., O 































































































































































Longitude from 

Newark, N. J., M. E. Ch 

New Haven, Conn., Yale 

New Orleans, La., Mint 

New York, N. Y., City Hall. . 

Northfield, Minn., O 

Ogden, Utah, O 

Olympia, Wash 

Omaha, Neb 

Pago Pago, Samoa 

Philadelphia, Pa., State 


Pierre, S. D., Capitol 

Pittsburg, Pa 

Point Barrow (highest 

latitude in the United 


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 




06" N 

28" N 
46" N 
44" N 
42" N 
08" N 
00" N 
50" N 
06" S 






10' 12" W 

55' 45" W 

03' 28" W 

00' 24" W 

08' 57" W 

59' 45" W 

57' 00" W 

57' 33" W 

42' 31" W 

39° 56' 53" N 
44° 22' 50" N 
40° 26' 34" N 

75° 09' 03" W 

100° 20' 26" W 

80° 02' 38" W 







00" W 







30" W 







24" W 







20" W 







00" W 







02" W 







27" W 







00" W 







16" W 







00" W 







32" W 







28" W 







45" W 







26" W 

47° 35' 54" N 

57° 02' 52" N 

30° 25' 00" N 

40° 13' 14" N 

122° 19' 59" W 

135° 19' 31" W 

84° 18' 00" W 

74° 46' 13" W 




Longitude from 

Virginia City, Nev 

Washington, D. C, O 

Wheeling, W. Va 

Wilmington, Del., Town Hall 
Winona, Minn 





06" W 

06" W 
30" W 
03" W 
00" W 



Magellan's Fleet. When the sole surviving ship of Mag- 
ellan's fleet returned to Spain in 1522 after having circumnavi- 
gated 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 mistake, 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 commandante 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 length- 
ened as compared with the day at any fixed place. When one 
has traveled 15° westward, at whatever 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 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 throughout 
the twenty-four belts of 15° each, losing one hour in each belt; 
by the time he arrives at his starting point again, he has set his 
hour hand back twenty-four hours and has lost a day. 

Fig. 30 

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 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 pro- 
cess is repeated. The traveler 
having 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, however, says Wednesday, 2 
o'clock, P.M. — two days and two 
hours since he left. The third day 
this occurs again, the traveler los- 
ing 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 Lon- 
don again he finds what to him 
was twenty- four days is, in real- 
ity, twenty-five days, for he has set 
his watch back an hour each day 
for twenty-four days, or an entire 

Fig. 31 


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 ar- 
rived at London again, he would have set his hour hand ahead 
twenty-four hours or one full day. To correct his calendar, some- 
where on his voyage he would have to repeat a day. 

The International Date Line. It is obvious from the 
foregoing explanation that somewhere and sometime in circum- 
navigation, a day must be omitted in traveling westward and a 
day repeated in traveling eastward. Where and when the change 
is made is a mere matter of convenience. The theoretical loca- 
tion of the date line commonly used is the 180th meridian. This 
line where a traveler's calendar 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 would make a serious inconvenience were the change 
made in some places. Imagine, for example, the 90th meridian, 
west of Greenwich, to be the line used. When it was Sunday in 
Chicago, New York, and other eastern points, it would be Mon- 
day 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 holidays 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 line one must set his calendar ahead a day; 
going eastward, back a day. 

As shown in Figures 31 and 32, this line 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 
line extends to the 180th meridian which it follows southward, 
sweeping somewhat eastward to give the Fiji and Chatham is- 
lands the same day as Australia and New Zealand. The following 
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 line. The older ge- 
ographies 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 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 annexation of the Samoan Islands 
by the United States and by Germany. The date in Samoa is, therefore, 
the same date as in the United States, 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, 

Mr. Willis E. Johnson, 

Vice President Northern Normal and Industrial School, 
Aberdeen, South Dakota. 

"It is fortunate that the 180th meridian falls where it does. 
From Siberia to the Antarctic continent this imaginary line tra- 
verses 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 fortu- 
nate was it, indeed, that opposite the spot where man was most 
destined to think there should have been placed so little to think 

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, suppose we wished to be the first on earth to hail the 

*From Choson, by Percival Lowell. 


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 

The Total Duration of a Day. While a day at any par- 
ticular 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 be- 
gins 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 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 vari- 
ation, the total duration of a day would be exactly forty-eight 
hours as just explained. But that line is quite irregular, as pre- 
viously 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 Desh- 



nef, 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 seventeen hours 
and forty- four minutes since it began at Cape Deshnef. When 
it reaches our most western point on this continent, Attu Is- 
land, 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. 

Fig. 32 


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, Tues- 
day 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 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 unen- 
viable position, as the line 180° passes through Loma-Loma. I 
went from Fiji to Tonga in Her Majesty's ship Nymph and ar- 
rived at our destination on Sunday, according to our reckoning 
from Fiji, but on Saturday, according to the proper computation 
west from Greenwich. We, however, found the natives all keep- 
ing Sunday. On my asking the missionaries about it they told 
me that the missionaries to that group and Samoa having come 
from 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 

*Mr. E. L. Layard, at the British Consulate, Noumea, New Caledonia, 
as quoted in a pamphlet on the International Date Line by Henry Collins. 


'This being Monday there will be no services in this cabin this 

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 passage, 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 hav- 
ing one time standard to a place having a different 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 
modification that as all events on a ship must be regulated by a 
common timepiece, changed according to longitude, so the com- 
munity 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 English 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 
indefinite and irregular. In 1845 by a decree of the Bishop of 
Manila, who was also Governor- General, Tuesday, December 31, 
was stricken from the calendar; the day after Monday, Decem- 
ber 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 

For some time after the acquisition of Alaska the people liv- 
ing 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 the United States, 
our system gradually was extended, but for a time both sys- 
tems 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 Rus- 
sian, 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 ordinance 
enacted by the governor of the colony of Fiji, with the advice and consent 
of the legislative council thereof, to provide for a universal 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 situated to the 
west of such meridian; and 

Whereas, in order to preclude uncertainty for the future it is expedient 
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 not- 
ing 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? 



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 contra- 
diction. The proofs rest upon three sets of astronomical obser- 
vations, 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 standing 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 carried 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, the amount varying with (a) the rate 
of its onward motion, and (b) 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 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 

Fig. 33 

Each of the millions of stars has its own apparent aberra- 
tional 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 revolves around the sun, 
the precise form of this aberrational orbit of any star can be 
calculated, and observation invariably confirms the calculation. 
Rational minds cannot conceive that the millions of stars, at 
varying distances, 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 prob- 

ably all of them, see p. 265) have. The discovery and explanation 

of these facts in 1727 by James Bradley (see p. 277), the En- 
glish 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 stationary, 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 of 

*This illustration is adapted from Todd's New Astronomy, p. 432. 


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 astronomical instrument which 
spreads out the line 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 farther 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, gradually 
stop, and six months later as rapidly recede, and stars away from 
this plane approach and recede at rates diminishing exactly in 
proportion to their distance from this plane, or the earth itself 
swiftly moves about the sun. 

*The rate of vibration per second for each of the colors in a ray of light 
is as follows: 

Violet 756.0 x 10 12 Yellow 508.8 x 10 12 

Indigo 698.8 x 10 12 Orange 457.1 x 10 12 

Blue 617.1 x 10 12 Red 393.6 x 10 12 

Green 569.2 x 10 12 

Thus the violet color has 756.0 millions of millions of vibrations each second; 
indigo, 698.8 millions of millions, etc. 


Proof of the Rotation of the Earth. The same set of 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 over- 
head, 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 (Cal- 
cutta 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 approaching 
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 cen- 
tury, demonstrates, with mathematical certainty, the rotation of 
the earth, and multiplies millionfold the certainty of the earth's 

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 observations at long intervals show 
other changes in the positions of stars. The latter motion of a 
star is called its proper motion to distinguish it from the appar- 


ent motion it has in common with other stars due to the motions 
of the earth. The spectroscope also assists in the demonstration 
that the sun with the earth and the rest of the planets and their 
attendant satellites is moving rapidly toward the constellation 

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 calculations of Pro- 
fessor Kiistner, Director of the Bonn Observatory. Extending 
from June 24, 1904, to January 15, 1905, he made careful ob- 
servations and photographs of the spectrographic lines shown 
by Arcturus. He then made calculations based upon a micro- 
scopic examination of the photographic plates, and was able to 
determine (a) the size of the earth's orbit, (b) 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 astronomers, rejected this the- 
ory 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 get- 
ting 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 parallaxes, thus 
multiplying the proofs of the motion of the earth around the 



Displacement of a Star Varies 
with its Distance. Figure 34 
shows that the amount of the dis- 
placement of a star in the back- 
ground 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 displacement; in ev- 
ery instance, however, this appar- 
ent shifting of a star is exceed- 
ingly minute, owing to the great 

distance (see pp. 44 247) of the 

very nearest of the stars. 

Since students often confuse 
the apparent orbit of a star de- 
Fig. 34 scribed under aberration of light 
with that due to the parallax, we 
may make the following comparisons: 

Aberrational Orbit 

1. The earth's rapid motion 
causes the rays of light to slant (ap- 
parently) 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 max- 
imum 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 background of the 
celestial sphere. 

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


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 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. 


Fig. 35 

If you have made a record of the observations suggested in 
Chapter III 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 have been gaining a little on the sun 
in the apparent daily swing of the celestial sphere around the 



Fig. 36 

The reasons for this may be understood from a careful study 
The outer circle, which should be indefinitely 

of Figure 36 

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 stars. When the earth is at point A 
the sun is seen among the stars at a; 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 mo- 
tions, — 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 di- 
rection 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 ap- 
parent 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 

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 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 ^ 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 noon by the sun, is called a solar 
day.* Its length varies somewhat, for reasons to be explained 

*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 corre- 
sponding to February 12, May 15, July 27, or November 3, is almost exactly 
twenty- four hours. The time intervening between sunrise and sunrise again 



later, but averages 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. 

£clip f J$ 

Aug 2/ 

/a 2 2 

Aug. 23 

Fig. 37 

Causes of Apparent Motions of the Sun. The apparent 
motions of the sun are due to the real motions of the 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 rotation 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 orbit is called 

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. 



Fig. 38. Celestial sphere, showing zodiac 

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 earth's orbit. The earth's equa- 
tor and parallels, if extended, 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 

*So called because eclipses can occur only when the moon crosses the 
plane of the ecliptic. 


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 ecliptic, 
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. 291). 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 correct 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 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 students 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 halfway 
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 per- 
pendicular to the plane of its orbit. 

Since the plane of the ecliptic 
forms an angle of 23 1° with the 
plane of the equator, the sun in 
its apparent annual course around 
in the ecliptic crosses the celes- 
tial equator twice each year, and 
at one season gets 23 1° 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 66 1°. On 

Fig. 39 

*The exact amount varies slightly from year to year. The following 
table is taken from the Nautical Almanac, Newcomb's Calculations: 

1903 23° 27' 6.86'' 

1904 23° 27' 6.39'' 

1905 23° 27' 5.92'' 

1906 23° 27' 5.45" 

1907 23° 27' 4.98" 

1908 23° 27' 4.51" 


March 21 and September 23 the 
sun is on the celestial equator. On 

June 21 and December 22 the sun is 23|° from the celestial equa- 

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 
materially from a circle, the difference is sufficient to make an 
appreciable difference in the rate of the earth's motion in differ- 

ent parts of its orbit. Figure 113 p. 284, 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 1 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 perihe- 
lion, 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 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 celestial sphere are par- 
allel and are at right an- 
gles to the axis. The sun in 
its annual path creeps diag- 
onally across their courses. 
When farthest from the ce- 
lestial equator, in June and in December, the sun's movement in 

Fig. 40 

the ecliptic is nearly parallel to the courses of the stars (Fig. 40); 
as it gets nearer the celestial equator, in March and in Septem- 
ber, the course is more oblique. Hence in the latter part of June 
and of December, the sun, creeping back in 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). 

I 9 

Some solar days being 
longer than others, and the 
sun being sometimes slow 
and sometimes fast, to- 
gether with standard time 
adoptions 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 is about De- 
cember 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 Jan- 
uary. 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 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. 


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 



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 

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 Septem- 
ber 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 fictitious 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 completing its annual course around the celestial sphere in 
the same time in which the sun apparently makes its circuit 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 difference 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 apparent 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 "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. Sup- 
pose 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^ 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 ac- 

curate data obtainable. Table II for each month gives in the 

column "Equation of Time" the number of minutes and seconds 
to be added to or subtracted from 12 o'clock noon at Green- 
wich 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 Nau- 
tical Almanac it may be of interest to notice the use of this 

* Prepared annually three years in advance, by the Professor of Mathe- 
matics, United States Navy, Washington, D. C. It is sold by the Bureau of 
Equipment at actual cost of publication, one dollar. 











of Time 
to be 



1 hour 









or Right 

Ascension of 





1 Hour 

1 Hour 


Mean Sun 

h m s 


o / // 


m s 


h m s 



18 42 9.88 


S. 23 5 47.3 


3 10.29 


18 38 59.60 



18 46 35.09 


23 1 6.3 


3 38.93 


18 42 56.16 



18 50 59.99 


22 55 57.7 


4 7.28 


18 46 52.71 



18 55 24.54 


22 50 21.8 


4 35.27 


18 50 49.27 



18 59 48.70 


22 44 18.6 


5 2.87 


18 54 45.83 



19 4 12.45 


22 37 48.2 


5 30.06 


18 58 42.39 



19 8 35.74 


22 30 51.0 


5 56.80 


19 2 38.94 



19 12 58.56 


22 23 27.1 


6 23.06 


19 6 35.50 



19 17 20.85 


22 15 36.8 


6 48.79 


19 10 32.06 



19 21 42.61 


22 7 20.2 


7 13.99 


19 14 28.62 



19 26 3.79 


21 58 37.7 


7 38.62 


19 18 25.17 



19 30 24.39 


21 49 29.5 


8 2.66 


19 22 21.73 

Part of a page from The American Ephemeris and Nautical Almanac, 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 m. 10.29 s., 
and is losing 1.200 s. each hour from that moment. We 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 be- 
fore 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 standard 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 equa- 
tor). Since our year has 365 1 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 approxi- 
mately 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 Central United States in 1909. Since for the 
ordinary purposes 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 number of 
minutes the apparent sun is slow or fast as compared with the 
mean sun. For example, the dot representing 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 merid- 
ian, before 12 o'clock four minutes for each degree. Since the 
analemma shows how fast or slow the sun is each day, it is obvi- 
ous that, knowing one's longitude, one can set his watch by the 
sun by reference to this diagram, or, having correct clock time, 
one can ascertain his longitude. 




*»■ M 


K. • 




SJEBA* •"■ • 

K, Id 


M, 12 

M< 14 




a, 1fr"7 








f s 








1 H 




it so 































ji 15 


Showing position of apparent sun ami 
Its declination every nay of the year. 

\ 1 1 
















1 ^v 






— i. 










Fig. 42 

the earth's revolution 129 

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. As- 
certain 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 11 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. 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 whe- 
ther 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. Multiplying 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 

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 explanation 
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 as- 
tronomical 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 

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. 

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. 



"In the early days of mankind, it is not probable that there 
was any concern at all about dates, or seasons, or years. Herodo- 
tus 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 Ge- 
ography 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 Equinoxes, p. 285), 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 

*R. W. Faiiand in Popular Astronomy for February, 1895. 



is what we mean when we say "a year."* Since it is the tropical 
year that we 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 adjustment 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 1 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 

*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, 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. 


were ignorant as to the astronomical data and by more ignorant 
kings often resulted in great confusion. The very 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 December, 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 alternately, 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 
collection 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, Sosi- 
genes, reformed the calendar. He decreed that beginning with 
January the months should have alternately 31 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 1 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 

Julian Augustan Durin § the rei § n of Augustus another 

day was taken from February 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 Septem- 
ber, from having 31 days each, such an ar- 
rangement being considered unlucky, Au- 
gustus 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 disarranged, 
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 when the sun 
crosses the celestial equator, occurred the 11th of March. In 




Feb. 29 


































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 Nicaea 
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 December 
9th as December 20th. It was adopted by Poland in 1786, and 
by Hungary in 1787. Protestant Germany, Denmark, and Hol- 
land 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 Gregorian sys- 
tem was effected in 1752 by having the date following Septem- 
ber 2d read September 14. The change was violently opposed by 
some who seemed to think that changing the number assigned 
to a particular day modified time itself, and the members of 
the Government are said to have been 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 usu- 
ally kept as they were 
then written, the let- 
ters O.S. sometimes be- 
ing 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 ex- 
ample, Columbus dis- 
covered land Oct. 12, 
1492, O.S. According 
to the Gregorian calen- 
dar 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 ending 
1500, only 9 days dif- 
ference had been made. 
So the discovery of 
America occurred Oc- 
tober 12, O.S. or October 

bhail Frjiii. wfi.cfi r.cflt. bui bruiil E/*» (nivtf, 
Umoueh'd £fow t<pt. unuftfd drop away/ 
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Lord ii a'tt SiOfri by H<a»'n foi Mm d«fg«"d.i 
And trample whn mild Sani bfingnly raife. ■ 
Wftitf Man mufl lofr iht Ulr. and Hra«'n <h( Pfljft,' 
Shall it thtft be f* (Indignant here (tit toft, 
tnd'gnani, ytt huminr, her Bofum alow*) 



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1 1 


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t 17 *> <*ft 9 *c 

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•M... ..ts M..k,„, 7.,.,, .n C M „„ „,f 01tfi y ^ „ c ; ^ 
»«o ««r, ift» .■ 10 (.,, Il,,,, D,,. i )Ir , ,h,„ , h( fcmt ^^^ 

Fig. 43. Page from Franklin's Almanac 

Showing Omission of Eleven Days, 


21, N.S. English 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 

July 27 

such dates as the following: Avaux, — , 1689. (Vol. III.) 

Aug. 6 

A few dates in American history 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 Colum- 
bus 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, Novem- 
ber, 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 com- 
monly 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 begin- 
ning of the year from March 25th. In 1563, by an edict of 
Charles IX, France changed the time of the beginning 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 Scot- 
land 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 
January 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 

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 div- 
idends 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. 

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 

*Under the date of September 10, 1906, the same authority says that 
the facts above quoted obtain in England at the present time. 


recorded by Protestants or Roman Catholics or written for gen- 
eral circulation are commonly recorded in both styles by placing 
the Gregorian date under the Julian date. For example, the 

date we celebrate as our national holiday would be written by 

June 21 

an American m Russia as . The day we commemorate 

July 4 J 

as the anniversary of the birth of Christ, Dec. — ; the day they 

Dec. 25,1906 T , ,, , , , , 

commemorate . It should be remembered that it 

Jan. 7, 1907 

the date is before 1900 the difference will be less than thirteen 
days. Steps are being taken in Russia looking to an early revi- 
sion 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 354 1^ 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 622 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, what we called May 1, 
1900, they called the first day of their first month, Muharram, 

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 conse- 
quently have 29 and 30 days alternately. To correct the error due 
to so short a year, seven out of every nineteen years have thir- 
teen months each. This still leaves the average year too short, 


so in every cycle of sixty years, twenty-two extra months are 

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 modifications 
became the basis of the Roman calendar which we have seen 
was handed down through successive generations 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 

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 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 








1. Sunday 


Dies Solis 




2. Monday 


Dies Lunas 




3. Tuesday 


Dies Martis 


Mythical God 

Tiew or Tuesco 



4. Wednesday 


Dies Mercurii 




5. Thursday 


Dies Jovis 


Thor (thunderer) 


6. Friday 


Dies Veneris 




7. Saturday 


Dies Saturni 



Samstag or 

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 Mohammedan religious year, a Greek 
or Eastern year, and a European 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 Turkey still further complicate the calendars in customary 
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 remember them. This page represents one day. 



Numbering the compartments in it from left to right, it reads as 


*-. - 









ip*a 2 cflipyi 1 

27 9 


5. Huguts- 

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 (Arme- 

nian) . 

9. April, Wednesday (French) 

10. March, Wednesday (Greek) 

11. Ecclesiastical Day (French 

R. C. Church). 

12. March, Wednesday (Rus- 


13. Month Day (Hebrew). 

14. Month Day (Old Style). 

15. Month Day (New Style). 

16. Ecclesiastical Day (Arme- 

nian) . 

17. Ecclesiastical Day (Greek) 

18. Midday, 5:35, 1902; Midday, 


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 according 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 
steamers on the Bosphorus run according to Turkish time, and 

■U,a w 5- ib **-. 1902 IMidt 5 2t Jr 30 


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 lengthened 
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 beginning of their year 5661. 

Many remedies have been suggested for readjusting our cal- 
endar so that the same date shall always recur on the same day 
of the week. While it is interesting for the student to specu- 
late 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 

If the student becomes impatient with the complexity of the 
problem, he may recall with profit these words of 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 

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 Decem- 
ber 25th of our calendar is not the greatest of the three. It is, 

*The Impressions of a Careless Traveler, by Lyman Abbott. — The Out- 
look, Feb. 28, 1903. 


at least, the first. Thirteen days after the Latin has burned his 
Christmas incense in the sacred shrine, the Greek Church patri- 
arch, 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. 



Vertical and Slanting Rays of the Sun. He would be 
unobservant, indeed, who did not know from first-hand experi- 
ence that the morning and evening rays of the sun do not feel so 
warm as those of midday, and, if living outside 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 

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 pro- 



duce a corresponding change in the apparent revolution of the 

Were the sun to pass among the stars at right angles to 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 ver- 
tical ray never gets farther north or south of the equator than 
about 23 |°, or nearer the poles than about 66 1°. The plane of 
the ecliptic or of the earth's orbit is, then, inclined at an angle 
of 66^° to the axis, or at an angle of 23|° to the plane of the 
equator. This obliquity of the ecliptic varies slightly from year 

to year, as is shown on pp. 119, 286 


Equinoxes. The sun crosses the celestial equator twice a 
year, March 20 or 21, and September 22 or 23,* varying from year 
to year, the exact date for any year being easily found by refer- 
ring to any almanac. These dates are called equinoxes (equinox; 
dsquus, 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 equa- 
tor March 21. Bear in mind the fact that the North star (strictly 
speaking, the north pole of the celestial sphere) is on the north- 
ern 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 horizon; the shadow it casts is due west and remains 

*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 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. 



due west until noon, getting shorter and shorter as the sun rises 

Fig. 45. Illumination of the earth in twelve positions, 
corresponding to months. The north pole is turned toward us. 

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 shadow is due east, lengthen- 
ing 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 
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, gradually, 
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 ecliptic far- 
thest from the celestial equator, about 23^° 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 mark- 
ing 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 23^° beyond the north 
pole, and all of the parallels north of 66^° from the equator 
are entirely within this circle of illumination and have daylight 
during the entire rotation of the earth. At this time the cir- 
cle 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 of it 23 1°, and consequently all parallels south of 66 \° 


are entirely in the dark portion of the earth, and it is contin- 
ual 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 illumination 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 

On December 22 the sun has reached the most distant point 
in the ecliptic from the celestial equator toward the South star, 
23^° from the celestial equator and 66^° from the South star, 
the vertical ray on the earth being at corresponding distances 
from the equator and the south pole. The sun is now near the 
constellation Capricorn, and everywhere 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 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 23 1°, 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 23 1° beyond 
each pole, and the frigid zones are thus 23 1° 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 ex- 
actly overhead and the celestial equator always on the horizon. 
On March 21 the sun is on the celestial equator 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."'" After the spring equinox, the sun gradually rises higher 

* 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. 161 

tThe 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 pole the compass 
points due south, and at points between the magnetic pole and the geo- 
graphical 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, 


and higher in a gently rising spiral until at the summer solstice, 
June 21, it is 23 1° above the horizon. After this date it gradually 
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 reflection 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. 165). 
On December 22 the sun is 23 1° 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 di- 
rection; 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. 

Parallelism of the Earth's Axis. Another condition of 
the earth in its revolution should be borne in mind in explain- 
ing change of seasons. The earth might rotate on an axis and 


"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 direction, 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 


revolve around the sun with the axis inclined 23 1° and still give 
us no change in seasons. This can easily be demonstrated by 
carrying a globe around a central object representing the sun, 
and by rotating the axis one can maintain the same inclination 
but keep the vertical 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 discussed in the Appendix under "Precession of 
the Equinoxes," "Nutation of the Poles," and "Wandering of the 

Poles" (p.|285j). 

Experiments with the Gyroscope. The gyroscope, 
probably familiar to most persons, admirably illustrates the 
causes of the parallelism of the earth's axis. A disk, supported 
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 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 |46f is one used by Professor 
R. S. Hoi way 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 advantages over those commonly 
used are its simplicity, the ball bearings, and its greater weight. 

Fig. 46 

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 the vertical ray is at the 
equator, and the circle of illumination 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 

'Taken, by permission, from the Journal of Geography for February, 



and twelve hours night. (No allowance is made for refraction or 

Day's Length after the Equinoxes. After the vernal 
equinox the vertical ray moves northward, and the circle of il- 
lumination 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 
hemisphere. The farther from the equator, too, the greater is 

the difference, as may be observed from Figure [50j page |163 
Parallels near the equator are always nearly bisected by the circle 
of illumination, and hence day nearly equals night there the year 

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, con- 
sequently 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 intersection. 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 
longest day, that is, from sunrise to sunset, in different latitudes 
is as follows: 










12 h. 


13 h. 34 m. 


16 h. 9 m. 


65 days 


12 h. 17 m. 


13 h. 56 m. 


17 h. 7 m. 


103 " 


12 h. 35 m. 


14 h. 22 m. 


18 h. 30 m. 


134 " 


12 h. 53 m. 


14 h. 51 m. 


21 h. 09 m. 


161 " 


13 h. 13 m. 


15 h. 26 m. 

66° 33' 

24 h. 00 m. 


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 everywhere. 

Refraction of Light 

Fig. 47 

The rays of light on entering the atmosphere are bent 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 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 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. 

Fig. 48 

Amount of Refraction 
Varies. The amount of re- 
fraction depends upon the 
difference in the density of 
the media and the oblique- 
ness with which the rays en- 
ter. Rays entering perpen- 
dicularly are not refracted at 
all. The atmosphere differs 
very greatly in density at dif- 
ferent 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 

Fig. 49 



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 ap- 
proaches the earth. 

Mean Refraction Table 
(For Temperature 50° Fahr., barometric pressure 30 in.) 




















1' 58.9" 








1 40.6 








1 9.4 









































Effect of Refraction on Celestial Altitudes. Thus, re- 
fraction increases the apparent altitudes of all celestial objects 

excepting those at the zenith (Fig. 49). The amount of refrac- 

tion at the horizon is ordinarily 36' 29"; that is to say, a star 
seen on the horizon is in reality over one half a degree below the 
horizon. The actual amount of refraction varies with the tem- 
perature, humidity, and pressure of the air, all of which affect 
its density and which must be taken into consideration in accu- 
rate 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 almanacs as to 
the exact length of the day at the equinoxes. Theoretically the 


day is twelve hours long then, but practically it is a few minutes 
longer. Occasionally there is an eclipse of the moon observed 
just before the sun has gone down. The earth is exactly be- 
tween 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. 


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 dis- 
appear 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 are broken up and reflected to the lower air. This, in brief, 
is the explanation of twilight. There being practically no atmo- 
sphere 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 p. |263 

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 atmo- 
sphere, the latitude, and the season of the year. There is thus 
a twilight zone immediately beyond the circle of illumination, 

and outside of this zone is the true night. Figure 50 represents 



these three portions: (1) the hemisphere receiving direct rays 
(slightly more than a hemisphere owing to refraction), (2) the 
belt 18° from the circle of illumination, and (3) the segment 
in darkness — total save for starlight or moonlight. The height 
of the atmosphere is, of course, greatly exaggerated. The atmo- 
sphere above the line AB receives direct rays of light and reflects 
and diffuses them to the lower layers of atmosphere. 

Twilight Period 
Varies with Sea- 
son. It will be seen 

Fig. 50 

from Figure [50] that 
the fraction of a par- 
allel in the twilight 
zone varies greatly 
with the latitude and 
the season. At the 
equator the sun drops 

down at right angles to the horizon, hence covers the 18° twilight 

zone in of a day or one hour and twelve minutes. This re- 

360 y 

mains practically 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 ap- 
parent if we remember that the sun, rising north of east, swing- 
ing 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 sol- 
stice that it does not sink 18° below the horizon at midnight, and 
stays within the twilight 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 midnight 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 midnight sun." 

Twilight Near the Equator. "Here comes science now 
taking from us another of our cherished beliefs — the wide su- 
perstition 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 obser- 
vation 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 disappeared. 
On August 27, 1899, the following observations 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 disap- 
pearance.' So the common superstition about no twilight in the 
tropics goes to join the William Tell myth." — Harper's Weekly, 
April 5, 1902. 

Twilight Near the Pole. "It may be interesting to relate 
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 estab- 
lished 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 midnight, 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 hun- 
dred 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 follow- 
ing 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 twilight; 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 sur- 
rounded, 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. 

Fig. 51 

Vertical Rays and Insolation. The more nearly verti- 
cal the rays of the sun are the greater is the amount of heat 
imparted to the earth at a given place, not because a vertical 
ray is any warmer, but because more rays fall over a given area. 

In Figure 51 we notice that more perpendicular rays extend 

over a given area than slanting ones. 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 traverse through more of the atmo- 

At the summer sol- 
stice the sun's rays 
are more nearly verti- 
cal over Europe and the 
United States than at 
other times. In ad- 
dition to the greater 
amount of heat re- 
ceived because of the 
less oblique rays, the Fig- 52 

days are longer than nights and consequently more heat is re- 
ceived during the day than is radiated off at night. This increas- 
ing 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 interesting 
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 pe- 
culiar heat-receiving and heat-transmitting properties 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 dispense 
with windows in cold weather. Stating this in more technical 
language, transparent glass is diathermanous 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 be- 
ing absorbed as they pass through. About three fourths of the 
heat the atmosphere 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 altitudes 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, to 
a very limited extent, of water, the heat is so distributed and 
accumulated that the hottest weather is in the month following 
the summer solstice (July in the northern hemisphere, and Jan- 
uary 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 summer 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 physical geography, seasons vary greatly 
in number and length with differing conditions of topography 
and position in relation to winds, mountains, and bodies of wa- 
ter. 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 De- 
cember, January, and February, winter months. In the southern 
states and in western Europe the seasons just named begin ear- 
lier. In California 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 hemi- 

Spring: Vernal equinox March 21 | 

\ 92 days^j 
Summer solstice June 21 J j g ummer na jf 

Summer: Summer solstice June 21 1 ( 186 da y s 

} 94 days J 
Autumnal equinox .... Sept. 23 I 

Autumn: Autumnal equinox .... Sept. 23 1 

\ 90 days^j 
Winter solstice Dec. 22 J I Winter half 

Winter: Winter solstice Dec. 22) j 179 days 

} 89 days J 
Vernal equinox March 21 I 

Hemispheres Unequally Heated. For the southern 
hemisphere, spring should be substituted for autumn, and sum- 
mer 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 northern summer when the earth is far- 
ther 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. 283 

Determination of Latitude from Sun's Meridian 


In Chapter [IT] we learned how latitude is determined by as- 
certaining 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 Celes- 
tial 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 conclu- 
sions are consequently hazy. 

Zen it ft 

Horizon Line at Latitude 4-Q'N 

Fig. 53 
1. The celestial equator is always 90° from the celestial pole. 


2. An arc of the celestial sphere from the northern horizon 
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°. 

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 equator 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 lati- 
tude must be 40° 29'. 

/. If the altitude of the celestial equator is 21° 24', the lati- 
tude 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 celestial equator, and its noon altitude 
subtracted from 90° equals the latitude of the place where the 
observation is made. 

*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. 



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 |128[ gives the 
approximate declination of the sun for every day in the year. The 

Nautical Almanac, |Table 1[ for any month gives the declination 
very exactly (to the tenth of a second) at apparent sun noon 
at the meridian of Greenwich, 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. 
Without good instruments, however, the proportion of error of 
observation is so great that the analemma will answer ordinary 

How to Determine the 
Latitude of Any Place. By 
ascertaining the noon altitude 
of the sun, and referring to the 
analemma or a declination ta- 
ble, one can easily compute the 
latitude of a place. 

1. First determine when the 
sun will be on your meridian 
and its shadow strike a north- 
south line. This is discussed on 
p. [129} 

2. By some device mea- 
sure the altitude of the sun 
at apparent noon; i.e., when 
the shadow is north. A card- 
board placed level under a win- 
dow shade, as illustrated in 

Fig. 54 

Figure 54 , will give surprisingly 

accurate results; a carefully mounted quadrant (see Fig. 55) 



however, will give more uniformly successful measurements. An- 

gle 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 ascer- 
tain the declination of 

the sun. Add this 

to the sun's altitude if 

south declination, and 

subtract it if north 

declination. If you 

are south of the equa- 
tor you must subtract 

declination south and 

add declination north. 

(If the addition makes 

the altitude of the 

sun more than 90°, 

subtract 90° from it, 

as under such circum- 
stances you are north 

of the equator 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 re- 
mainder is your latitude. 

Example. For example, 
say you are at San Francisco, 
October 23, and wish to ascer- 
tain your latitude. 

Fig. 55 



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.) 

2. The altitude of the sun 
when the shadow is north is found to be 41°. 

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 


lemma and the table of latitudes many interesting problems will 

suggest themselves, as the following examples illustrate. 

Problem. 1. How high above the 
horizon does the sun get at St. Peters- 
burg on December 22? 

Solution. The latitude of St. Pe- 
tersburg 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 ce- 
lestial equator at St. Petersburg, the al- 
titude 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 

Solution. The latitude of Chicago 
is 41° 50', and the altitude of the celes- 
tial equator, 48° 10'. The declination of 
the sun June 21 is 23° 27' N. North be- 
ing higher than the celestial equator at 


Fig. 57. Taking the 
altitude of the sun 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 altitude 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. 

Latitude from Moon or Stars. With a more extended 
knowledge of astronomy and mathematics and with suitable in- 
struments, we might ascertain the position of the celestial equa- 
tor in the morning or evening from the moon, planets, or stars 
as well as from the sun. At sea the latitude is commonly as- 
certained by making measurements of the altitudes of the sun 
at apparent noon with the sextant. 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 possi- 
ble, it is usually ascertained from star observations with a zenith 
telescope or a "meridian circle" telescope, and is verified in many 



Tides and the Moon. The regular rise and fall of the 

level of the sea and the accompanying inflows and outflows of 
streams, bays, and channels, are called tides. Since very ancient 
times this action of the water has been associated 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 some- 
what 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 as- 
sociations 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 conditions 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 pro- 


TIDES 177 

duces changes in level of adjacent estuaries. 

c. Variability in wind direction, especially strong and con- 
tinuous seasonal winds like monsoons, lowers the level on the 
leeward of coasts and piles it up on the windward side. 

d. Earthquakes sometimes cause huge waves. 

A few preliminary facts to bear in mind when considering 
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 conjunction 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. 

TIDES 178 

Moon's Orbit. The moon's orbit is an ellipse, its 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 (diameter 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. 


Laws Restated. This force was discussed in the first chap 

ter where the two laws of gravitation were explained and illus- 

trated. The term gravity is applied to the force of gravitation 

exerted by the earth (see Appendix, p. 278). Since the explana- 
tion 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 as the 
square of the distance of the object. 

TIDES 179 

Sun's Attraction Greater, but Moon's Tide-Produc- 
ing 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 attraction for the earth is approximately 
176 times that of the moon.* 

The reasoning which often leads to the erroneous conclusion 
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 attrac- 

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'^ 

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 unbalanced 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, gy; 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 



To Sun or Moon 
>»? > 

Fig. 58 

Portions of the earth toward the moon or sun are 8, 000 miles 
nearer than portions on the side of the earth opposite the at- 
tracting 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, 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 directly opposite to that of gravity. The 

TIDES 181 

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 somewhat 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 ex- 
actly 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 attrac- 
tion 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 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 

TIDES 182 

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 



.-. T: t :: : 1. 

390 3 

Combining the two proportions, we get, 

T : t :: 2 : 5. 

It has been shown that, owing to the very nearly equal at- 
traction 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 ; that is, an 

6 ' J J 20,000,000 

object weighing a ton varies in 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. 

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 

*The presence of oceans or an atmosphere is not essential to the theory, 
indeed, is not usually taken into account. It seems most certain that the 
earth is not perfectly rigid, and the theory assumes that the planets and 
the moon have sufficient viscosity to produce body tides. 



tidal waves swinging around in the opposite direction to its ro- 
tation caused a gradual retardation of its rotation until, as 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. 

Fig. 59 

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. 284). The sun pulls the earth toward 

itself about one ninth of an inch every second. If the earth were 
nearer, its revolutionary motion would be faster. In case of plan- 
ets having several satellites it is observed that the nearer ones 

revolve about the planet faster than the outer ones (see p. 255). 

TIDES 184 

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 stretching apart would cause 
them to separate entirely. The 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 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. 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 
gj 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 conjunc- 
tion, 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 oppo- 
site 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 opposition. 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 

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. 

Fig. 60. Co-tidal lines 

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 

TIDES 186 

the lifting progresses under or opposite the moon or sun from 
east to west. Thus a huge 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 miles per hour. The southern por- 
tion passes across the Indian Ocean, being retarded in the north 
so that the southern portion is south of Africa when the north- 
ern portion 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 north- 
ern portion of the wave assumes an almost northerly 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 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 move- 
ment 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 

TIDES 187 

crest, and by length the distance from crest to crest). Out in mi- 
docean 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 mo- 
tion 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. |188 

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, in contradis- 
tinction 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 ^ of a day or 51 min- 
utes. 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 1 h. 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 

*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. 


Fig. 61. Low tide 

day to day. The establishment of a port, as previously explained, 
is the average time 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 midocean, av- 
eraging only l| feet in the somewhat sheltered Gulf of Mexico, 
but averaging 37 feet in the Bay of Fundy. The shape and situ- 
ation 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 Dordogne, 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 century, gives 
a good idea of this phenomenon. "During three days before the 

TIDES 189 

new and full moons, the period of the highest tides, the sea, in- 
stead of occupying six hours to reach its flood, swells to its high- 
est 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 an- 
other, and another, and sometimes by a fourth. These watery 
mountains spread across the whole channel, and advance with 
a prodigious rapidity, rending and crushing everything in their 
way. Immense trees are instantly uprooted by it, and sometimes 
whole tracts of land are swept away." 



To represent the curved 
surface of the earth, or any 
portion of it, on the plane 
surface of a map, involves se- 
rious mathematical difficul- 
ties. Indeed, it is impos- 
sible to do so with perfect 
accuracy. 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 rep- 
resentation on a plane of 
parallels and meridians of 
the earth. 

Fig. 62 


map projections 
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 represents 
the relative positions of points on the earth's surface. 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 farther apart than those near the poles. 
This is not the way they are on the globe. The orthographic 
projection is the representation of the globe as a photograph 
would show it from a great distance. 

Parallels and 

Meridians Farther 
Apart in Center 
of Map. Viewing 
a globe from a dis- 
tance, we observe 
that parallels and 
meridians appear 

somewhat as repre- 
sented in Figure 63 
being farther apart 
toward the center and 
increasingly nearer 
toward the outer por- 
tion. Now it is obvious 
from Figure 64 that 


Equatorial orthographic 

Fig. 63. 

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 distortion. 



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. 

Fig. 64 

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 distance from D 
to E, to the distant eye A' and B' will appear 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 Projec- 
tion. 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 in the equatorial or- 
thographic projection is apparent if one remembers that if the 



Fig. 65 

eye is in the plane of the equator and is at an infinite distance, 
the parallels will lie in practically 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 meridians 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 cen- 
tral meridian as major axis 
as shown in Figure 66 
While somewhat more dif- 
ficult, the student should 
learn how thus to lay them 
off. To construct the el- 
lipse, one must first locate 
the foci. This is done by 
taking half the major axis 

Fig. 66. Western hemisphere, in 
equatorial orthographic projection 



(central meridian) as radius and with the point on the equa- 
tor 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 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 


How to Lay off a Po- 
lar Orthographic Pro- 
jection. This is laid off 
more easily than the for- 
mer projection. Here the 
eye is conceived to be di- 
rectly above a pole and the 
equator is the boundary of 
the hemisphere seen. It is 
apparent that from this po- 
sition the equator and par- 
allels will appear as circles 
and, since the planes of the 
meridians pass through the 
eye, each meridian must 
appear as a straight line. 

Fig. 67. Polar orthographic 

Lay off for the equator a circle the same size as the preceding 

one (Fig. 65), subdividing 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 pro- 

jection (Fig. 65). Through these points draw circles to repre- 


sent 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 latitudes other than 
0° or 90°. With the parallels and 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 
projection 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. 

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 hemisphere. 
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. 

Fig. 68 



How to Lay off 
an Equatorial Stereo- 
graphic Projection. In 

E represents 
at the equator, 

Figure [68 
the eye 
A and iV are the poles and 
A'N' is the correspond- 
ing meridian of the pro- 
jection with B' , C, etc., 
as the points where the 
parallels cross the merid- 
ian. Taking the line A'N' 

of Figure 68 as diameter, 

Fig. 69. Equatorial stereographic 

construct upon it a circle 

(see Fig. 69). 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 meridians as arcs of circles and the projection is 

The Polar Stereographic Projection is made on the 
same plan as the polar orthographic projection, excepting that 
the parallels have the distances from the pole that are repre- 

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

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 is so 
easily laid off. 



Fig. 70. Polar 
stereographic projection 

Fig. 71. Northern hemisphere in 
polar stereographic projection 

Fig. 72. Hemispheres in equatorial stereographic projection 


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 


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. 

With the eye at an infinite dis- 
tance (as in the orthographic pro- 
jection), parallels and meridians are 
nearer together toward the outside 
of the map; with the eye on the sur- 
face (as in the stereographic pro- 
jection), they are nearer together 
toward the center of the map. It 
would seem reasonable to expect 
that with the eye at some point in- 
termediate between an infinite dis- 
tance from the surface and the sur- 
face itself that the parallels and meridians would be equidistant 
at different 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 projection, make a circle of the same 
size as the one which is the basis of the map to be made, draw 
two radii at an angle of 45° (one eighth of the circle) and draw a 
line, AB, from the extremity of one radius perpendicular to the 
other radius. The length of this perpendicular is the distance 

Fig. 73 

sought (AB, 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. 

How to Lay off 
an Equatorial Glob- 
ular Projection. As 
in the orthographic 
or stereographic pro- 
jections, a circle is 
divided into equal 
parts, according to the 
number of parallels 
desired, the central 
meridian and equator 
being subdivided into 
half as many equal 
parts. Parallels and 
meridians may be 
drawn as arcs of cir- 
cles, being sufficiently 

Fig. 74 

accurate for ordinary purposes (see Fig. 75). 

Fig. 75. Hemispheres in equatorial 
globular projection 

The polar globular 
projection is laid off pre- 
cisely like the orthographic 
and the stereographic pro- 
jections having the pole 
as the center, excepting 
that the concentric circles 
representing parallels are 

equidistant (see Fig. 76). 
By means of starlike 

additions to the polar globular projection (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 projection 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, pro- 
portional areas are well represented, South America and Africa 
being shown with little distortion of area and outline. 

The globular projection is much 
used to represent hemispheres, or 
with the star map to represent the 
entire globe, because the parallels on 
a meridian or meridians on a par- 
allel are equidistant and show little 
exaggeration of areas. For this rea- 
son it is sometimes called an equidis- 
tant projection, although there are 
other equidistant projections. It is 
also called the De la Hire projection 
from its discoverer (1640-1718). 

Fig. 76. Polar globular 


In the globular projection: 

The eye is conceived to be at a cer- 
in distance from the globe (sine 45 

re divided equidistantly by 
llels, and parallels are divided 
quidistantly by meridians. 

^en a pole is the center of the map, 
| meridians are straight lines. 

*le is little distortion of areas. 

Fie. 77. World in nolar globular 



Gnomonic Projection 

When we look up at 
the sky we see what ap- 
pears to be a great dome in 
which the sun, moon, plan- 
ets, and stars are located. 
We seem to be at the cen- 
ter of this celestial sphere, 
and were we to imagine 
stars and other heavenly 
bodies to be projected be- 
yond the dome to an imag- 
inary plane we should have 
a gnomonic projection. Be- 
cause of its obvious conve- 
nience in thus showing the 
position of celestial bodies, 
this projection is a very old 
one, having often been used 
by the ancients for celestial 

Since the eye is at the 
center for mapping the ce- 
lestial sphere, it is con- 
ceived to be at the center of the earth in projecting parallels 

Fig. 78 

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 represented as straight 
lines. This will be apparent if one imagines himself at the center 



of a transparent globe having parallels 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 because of this fact that sailors 
frequently use maps made on this projection. 

To Lay off a Polar 
Gnomonic Projection. 
Owing to the fact that par- 
allels get so much farther 
apart away from the center 
of the map, the gnomonic 
projection is almost never 
made with any other point 
than the pole for center, 
and then only for lati- 
tudes about forty degrees 
from the pole. The po- 
lar gnomonic projection is 

made like the polar projec- 

, . . ii-ii Fig. 79. Polar gnomonic projection 

tions 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 concentric 

Great Circle Sailing. 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 parallel, that is, would 
sail due eastward. This, however, would not be its shortest 
course. The solution of the following little catch problem in 
mathematical 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 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.|80j). 

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 paral- 
lel on which both cities are 
located. Thus it is that 
the course of vessels plying 
between the United States 
and Europe curves, some- 
what 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 advantage in departing from 
the course of a parallel. Besides this, the trade winds and dol- 
drums control the choice of routes in that region and the Merca- 
tor projection is always used in sailing there. In higher latitudes 
the gnomonic projection is commonly used. 

Although the gnomonic projection is rarely used excepting by 
sailors, it is important that students understand the principles 
underlying its construction since the most important projections 

Fig. 80 


yet to be discussed are based upon it. 

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 center of 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. 

The Homolographic Projection 

The projections thus far discussed will not permit the repre- 
sentation of the entire globe on one map, with the exception of 
the starlike extension of the polar globular projection. The ho- 
molographic projection is a most ingenious device which is used 
quite extensively to represent 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 these points of subdivision and the 
poles, draw ellipses for meridians. 

— > 


Fig. 81. Homolographic projection 

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 page 22 This process must be repeated for each 
pair of meridians. 

The parallels are straight lines, as in the orthographic pro- 
jection, somewhat nearer together toward the poles. If nine par- 
allels are drawn on each side of the equator, they may be drawn 
in the following ratio of distances, beginning at the equator: 2, 
If, if, if, if, l|, l|, if, l|- This will give an approximately 
correct representation. 

One of the recent books to make frequent use of this projec- 
tion is the "Commercial Geography" by Gannett, Garrison, and 



Fig. 82. World in homolographic projection 

Houston (see Fig. 82) 

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. 






















































The homolographic projection is sometimes called the Moll- 
weide 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 parallels 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 

In the homolographic projection: 

1. The meridians are semi-ellipses, drawn as in the globular projection, 
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. 

The Van der Grinten Projection 

The homolographic projection was invented early in the nine- 
teenth century. At the close of the century Mr. Alphons Van der 
Grinten of Chicago invented another projection by which the 
entire surface of the earth may be represented. This ingenious 
system reduces greatly the angular distortion incident to the ho- 
molographic projection 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°. 



Fig. 83. World in Van der Grinten projection 

A modification of 
this projection is shown 

in Figure [83] In this 
the central meridian is 
only one half the length 
of the equator, and 
parallels are at uni- 
form distances along 
this meridian. 

Fig. 84. World in Van der Grinten 

map projections 
Cylindrical Projections 


Gnomonic Cylindrical Projection. In this projection 
the sheet on which the map is to be made is conceived to be 
wrapped as a cylinder around the globe, touching the equator. 
The eye is conceived to be at the center of the globe, projecting 
the parallels and meridians upon the tangent cylinder. Figure 85 
shows the cylinder partly unwrapped with meridians as parallel 
straight lines and parallels also as parallel straight lines. As in 
the gnomonic 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. 
Beyond 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 represented on the map as though 
they were just as far apart there as at the equator. Because 

Fig. 85 



60' N 

of the rapidly increasing distances of parallels, to represent 
higher latitude than 60° would require a very large sheet, so 
the projection is usually modified for a map of the earth as a 
whole, sometimes arbitrarily. 

G'H' is the distance from the equator to the first parallel, and 
since a degree of latitude is about equal to a degree of longitude 
there, this distance may be taken between meridians. 

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 with- 
out great distortion. 
Such a reduction is 
well shown in the 
stereographic projec- 
tion. In this the 
eye is conceived to be 
on the equator, pro- 
jecting each meridian 
from the view point 
of the meridian oppo- 
site to it. Figure 87 
shows the plan on 
which it is laid off, 
meridians being paral- 
lel straight lines and 
equidistant and paral- 
lels being parallel straight lines at increasing distances away from 
the equator. 

The Mercator (Cylindrical) Projection. In the ortho- 
graphic, stereographic, globular, gnomonic, homolographic, and 


— -^ ' 

* *k 














Vss ^ •» ,_ 







x \7 

— — "^ \ s 











Fig. 86 



Van der Grinten projections, 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 merid- 
ians. 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 













*«-•"■""' °\ 



/ ^> i * ^ -~ 













are pointing in almost 
exactly opposite di- 
rections and yet each 
is pointing due north. 
The arrows at the 
bottom point opposite 
each other, yet both 
point due south. The 
arrows pointing 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, paral- 
lels 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 projections 
studied thus far represent directions in a most unsatisfactory 
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 represented by a curved 

Fig. 87 



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 fifteenth 
century, sailors did not 
cover very great por- 
tions of the earth's 
surface in continuous 
journeys out of sight 
of land where they 
had to be guided al- 
most wholly by the 
stars. Mathematical 
accuracy in maps was 
not of very great im- 
portance in navigation 
until long journeys had 
to be made with no opportunity for verification of calculations. 
Various roughly accurate map projections were made. The map 
sent to Columbus about the year 1474 by the Italian astronomer 
Toscanelli, with which he suggested sailing directions across the 
"Sea of Darkness," is an interesting illustration of a common 
type of his day. 

The long journeys of the Portuguese along the coast of Africa 
and around to Asia and the many voyages across the Atlantic 


7S 1 



4 5* 


/ l\ /T^ 




/ / /K/wJ^Tx 




Zyfe^-"*^"** * 



F C H 













Fig. 89 




early in the sixteenth century, made accurate map projection 
necessary. About the middle of that century, 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 German, "retail merchant," and this trans- 
lated 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 projections, excepting 
as to the distances between parallels. The meridians are rep- 
resented 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 propor- 
tional to the meridional error. Thus at latitude 60° the merid- 
ians on the earth are almost exactly half as far apart as at the 
equator, but being equidistant on the map, they are represented 
as twice as far apart as they should be. The parallels in that 
portion of the Mercator map are accordingly made twice as far 
apart as they are near the equator. Since the distortion in lat- 
itude exactly equals the distortion 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 according to the 
bearings thus obtained. 

To Lay off a Mercator Projection. Figure [89] shows the sim- 
plest 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 AK . The meridians are equidistant 
and are the same distance apart as the first parallel is from the 

Fig. 90. World in mercator projection 

The table of meridional 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, is 

4 ' 5944.3 

the scale to be used in making the map; that is 1 inch on the 

map will be represented by 10 inches -j- 5944.3. Suppose we 

wish to 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 

F J 5944.3 

*Technically speaking, the distance is the tangent of the angle of lati- 

tude 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. 



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 multiplied by the scale 

F J 5944.3 

gives 2.03 inches from the equator. The 30th parallel has a 

meridional distance 1876.9, this multiplied by the scale gives 

3.15 inches. In like manner the other parallels are laid off. The 

meridians will be x 60 or 600 inches -f- 5944.3 for every 

5944.3 J 

degree, or for ten degrees 6000 inches -j- 5944.3, which equals 

1.01 inches. This makes the map 36.36 inches long (1.01 inches x 

36 = 36.36 inches). 

Table of Meridional Parts* 











































































































































































We see, then, that the same scale of miles cannot be used 
for different parts of the map, though within 30° of the equator 

'From Bowditch's Practical Navigator. 


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 distance apart as the meridians, 
the inaccuracy not being very great. 


In the cylindrical projection: 

1 . A cylinder is conceived to be wrapped around the globe, tangent to the 


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 merid- 

ional 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. 

4. There is great distortion of areas and outlines of continents in high 

latitudes; Greenland appears larger than South America. 

5. The entire globe may be represented in one continuous map. 

6. The same scale of miles cannot be used for high latitudes that is used 

near 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 conical 
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. Parallels 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, therefore, near the tangent parallel, 
there will be very little distortion in lati- 
tudes 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 mod- 
ification of it, is almost always employed 
Fig. 91 in representing small areas of the earth's 


To Lay off a Conic Pro- 
jection. If the forty-fifth paral- 
lel 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 par- 
allels as in the gnomonic pro- 
jection. With the apex as cen- 
ter, draw arcs of circles through 
these points for parallels. Merid- 
ians are straight lines meeting 
at the apex and are equidistant 

Fig. 92 



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 pro- 
jection 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 parallel. Because 
of this lengthening of parallels, meridians are sometimes curved 
inwardly to prevent too much distortion of areas. The need for 
this will be apparent if one draws parallels beyond the equator, 
for he will find they are longer than the equator itself unless 
meridians curve inwardly there. 

By taking the 
tangent 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 
distortion of parallels 
south of the equator, 

but since meridians Fig- 93 

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 projection is much 
better suited to represent the entire world than the Mercator 



T^T -' V>" 


C / 1/ 

71 ^\C^ 

\ /^" 

-*"*HJ ^ 1 



Fig. 94. North America in conic 

tween parallels. 

Having laid off the cen- 
tral meridian and marked 
off the arcs for parallels, 
the true distance of the 
meridian on each parallel 
is laid off and the merid- 
ian is drawn through these 
points. This gives a gen- 
tle inward curve for merid- 
ians toward the outside of 
the map of continents. In- 
stead of following Bonne's 

Bonne's (Conic) Pro- 
jection. This is a mod- 
ification of the conic pro- 
jection as previously de- 
scribed to prevent exaggera- 

■<( tion of areas away from the 

parallel which is conceived 
to be touching the globe. 
The central meridian is a 
straight line and parallels 
are concentric equidistant 
circles. The distance be- 
tween 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 be- 



/x/ / /p-^ii \/\ 

l TJjl5 


•i fr~~l f l TtTrr 


/** / */?' j~~f~X-AL. 

*^/ jt7fr%L lfr\ jf* 

fA \ L-- 




' /"-"*-£ 1 / 

+*\ VA^T^-k. ' 


f%^sl*m -T*°t~fir-T*c& 




Fig. 96. Europe in conic projection 



Fig. 95. The world in conic projection 

system with strict accuracy, the map maker sometimes makes 
the curve a little less in lower latitudes, allowing a slight exag- 
geration of areas to permit the putting in of more details where 
they are needed. 

Intersecting Conic Projec- 
tion. Where a considerable extent 
in latitude is to be represented, the 
cone is sometimes 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 paral- 
lels (A and B) there is little distor- 
tion, this plan is better adapted for 
a map showing greater width north 
and south than is the conic projec- 

The map of Europe well illus- 

Fig. 97 



trates this difference. Europe lies between 35° and 75° north lat- 
itude. 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 inter- 
secting 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 any- 

It should be noticed that the 
region between the intersections 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 de- 
fect, from the practical map maker's 
point of view, to minify it. 

Polyconic Projection. This 
differs from the conic projection 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 


jt\ \ \ fSf^ 1 

s ■■ 

— l — T 

Fig. 98. Africa and 

Europe in polyconic 



as in the conic projection or along two as in the intersecting 
conic projection. For representing small areas this is decidedly 
the most accurate projection known. 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 parallel 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 


In the conic projection: 

A cone is conceived to be fitted about a portion of the globe, tangent 
to some parallel. 

The tangent parallel shows no distortion and portions near it have but 
little. 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 
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. 

b. In the Bonne projection parallels are drawn at equidistant intervals 

from a common center and meridians are slightly curved to prevent 
distortion in longitudes. 

c. 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, . A scale of two inches to the mile 

yi 63,360 

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 pro- 
jections 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. 

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 dis- 
tances 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 merid- 
ians 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 

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 



Allowance for Curvature. One of the best proofs that 
the earth is a sphere is the fact that in all careful measurements 
over any considerable area, allowance must be made for the cur- 
vature 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 to- 
gether at the northern extremities than they are at the southern 

Origin of Geometry. One of the greatest of the practical 
problems of mathematics and astronomy has been the system- 
atic location of lines and points and the measurement of surfaces 
of the earth by something more definite, more easily described 
and relocated than metes and bounds. Indeed, geometry is be- 
lieved to have had its origin in the need of the ancient Egyptians 
for surveying and relocating 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 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 in- 
numerable 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 Do- 
mini one thousand seven hun- 
pj„_ 99 dred and thirty-six and in the 





































* 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 performed, the 
bridal party returning — they cared not to which state, for the festivities. 


tenth year of the reign of our 
sovereign Lord George the Second, King." In this, Emma Blow- 
ers deeds to William Stanley, "A certain parcel 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 Manchester 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 Southeast- 
erly 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, 
practically all of the terri- 
tory 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 expe- 
riences resulting from hastily 
marked boundaries. Thomas 
Hutchins was appointed Geog- 
rapher of the United States, 
and after the selection of thir- 
teen assistants, he was instructed to begin its survey. Starting in 
1786 from the southwest corner of Pennsylvania, he laid off a line 
due north to a point on the north bank of the Ohio River. From 





































Fig. 100 


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 spread- 
ing a huge "gridiron" over the land. The larger squares were 
called "townships," an adaptation of the New England "town." 
They are commonly called "Congressional townships" in most 
parts of the United States, to distinguish them from the politi- 
cal subdivision of the county called the "civil township" or the 
"municipal township." 

Jefferson is believed to 
have suggested this general 
plan which with some varia- 
tions has been continued over 
the major portion of the 
United States and the western 
portion of Canada. Hutchins 
and his crew laid off the "ge- 
ographer's line" only forty-two 
miles, making seven ranges of 
townships west of the Penn- 
sylvania state boundary, when 
they were frightened away by 
the Indians. The work was 
continued, however, on the same general plan one exception be- 
ing the method of numbering the sections. In these first "seven 





































Fig. 101 

ranges" the sections are numbered as in Figure 99 , elsewhere in 

the United States they are numbered as in Figure 100 and in 

western Canada as in Figure |101[ Each of the square miles is 
commonly called a "section." 

The law passed by Congress May 20, 1785, provided that, 


"The surveyors . . . shall proceed to divide the said territory 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 impossibility; "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 principal meridians have been sur- 
veyed at irregular intervals and of varying lengths. Some of 
these are known by numbers and some by names. The first 
principal meridian 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 Tallahassee princi- 
pal 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. 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 

*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 ac- 
tual cost, eighty cents, from the Commissioner of the General Land office, 
Washington, D. C. 


meridian and where two systems come together irregularities are 

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 geographic 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 township 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 

The fourth principal meridian in western Illinois and Wiscon- 
sin has two base lines, one at its southern extremity extending 
westward to the Mississippi River and the other constituting the 
interstate boundary line between Wisconsin and Illinois. The 
townships of western Illinois are numbered from the southern 
base line, and all of those in Wisconsin and northeastern Min- 
nesota are numbered from the northern base line. The fourth 
principal meridian is in three sections, being divided by an east- 
ern 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 extends north- 
ward from the mouth of the Arkansas River until it again in- 
tersects the Mississippi River in northeastern Missouri and then 
again it appears in the big eastern bend of the Mississippi River 
in eastern Iowa. Its base line passes a few miles south of Lit- 
tle Rock, Arkansas, from which fact it is sometimes called the 


r 7/ / / i r t 

g ft a s s s s; 


Little Rock base line. From this meridian and base line all of 
Arkansas, Missouri, Iowa, North Dakota, and the major por- 
tions 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 sections of tier 164 with odd lottings lies be- 
tween 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, mak- 
ing 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 sev- 
eral tiers, their conver- 
gence becomes consider- 
able. At latitude 40° 
the convergence is about 
6.7 feet per mile or some- 
what more than 40 feet to 
each township. To pre- 
vent this diminution in 

Fig. 103. 


size of townships to the north of the base line, standard par- 
allels are run, along which six-mile 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 upon was the seventh standard parallel 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 geographic parallel because it was the boundary line be- 
tween farms, sections, townships, and, to a considerable extent, 
counties. The boundary line between Minnesota and Iowa is 
what is called a secondary base line and corresponds to a stan- 
dard parallel between tiers 100 and 101 north of the base line of 
the fifth principal meridian. 

The standard parallels have been run at varying intervals, the 
present distance 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. 

" Lot 4, Section 7 

Xlot /.Section!. 

Fig. 104. 


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 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 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 sections 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 remaining 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 

N. '/£ 



80 A. 

160 Acres 







Legal Subdivisions 
of a Section. The legal 
subdivisions of a section 
are by halves, quarters, 
and half quarters. The 
designation of the portions 
of a section is marked in 

Figure |105| The abbrevi- 
ations look more unintelli- 
gible than they really are. 
Thus N. E. \ of S. E. \ of 
Sec. 24, T. 123 N. R. 64 W. 
5 P.M. means the north- 

_. ,„ east quarter of the south- 

Fig. 105. n 

east 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 



Fig. 106 

The ability to measure the dis- 
tance 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 impossibil- 
ity. Uninformed persons sometimes 
contend that astronomers only guess 
at the distances and dimensions of 

the sun, moon, or a planet. The principle of such measurement 
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 cross- 


ing it. First make a triangle 

having two equal sides and 




Fig. 107 

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 sighting you 
see that the long side is in line with B when the short side is in 
line with X. You will then form the triangle BAX or BCX. It 



is apparent (by similar triangles) that AB or CB equals BX. 
Measure off AB or BC and you will have BX, the distance 
sought. If you measure both to the right and to the left and 
take the average of the two you will get a more nearly correct 

„ To Measure the 

Height of an Ob- 
ject. In a similar 
manner one may mea- 
sure the height of a 
flagstaff or building. 
Let X represent the 
highest point in the 

\ \ X v 

\ v s s .\ 
\ x v x ^ *\ 

v v x V x \\ 

X v 

\ \ \ . 


N S S 

V \ 



Fig. 108 

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 point of sighting was less, by measuring that angle and 
looking up the value of its tangent in a trigonometrical table, 
one could as easily calculate the height or distance. The angle 
of the triangle from which sighting was done is 45°, its tangent 
is 1.0000, that is, XB equals 1.0000 times BC. If the angle 
used were 20°, instead 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 multi- 
plying this by its natural tangent get the height of the object. 


If it is a building that is thus measured, the distance should be 
measured from 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 Itablel 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'- L 

5 r 
The Itablel 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 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. 


Fig. 109 

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 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 hold the line even with his eye. You then 


have three elements of the following proportion: 

Dist. to disk : dist. to moon : : width of disk : width of moon. 

From these elements, multiplying extremes and means and di- 
viding, it is not difficult to get the unknown element, the diame- 
ter of the moon. If the student is careful in his measurement and 
does not forget to reduce all dimensions to the same denomina- 
tion, 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 accu- 
rate 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 mea- 
surement 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 apparently parallel. It is 
only by making observations six months apart, the diameter 
of the earth's orbit forming the 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 
center of the sun. 

Survey by Triangulation 

A method very extensively employed for exact measurement 
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 accurately computed. Im- 
mense areas in India, Russia, and North America have been thus 
surveyed. The triangulation surveys of the United States com- 
prise nearly a 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 topog- 
raphy and making geological maps, and by the United States 
Coast and Geodetic Survey. 









i 1 t 3 i b 10 

Bildwinw 1 


Morgan ' 



W '*---^^_^ 










Fig. 110 

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 be- 
ing used to make it as correct as possible, for this is the base line 
and all calculations rest for their accuracy upon this distance as 
it is the only line measured. The following extracts from the 
Bulletin of the United States Geological Survey on Triangula- 
tion, No. 122, illustrate the methods employed. "The Albany 
base line (in central Texas) is about nine miles in length and 
was measured 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 di- 
rect 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; temperature 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 Gunnison line (Utah) 
was measured under the direction of Prof. 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 an- 
gle CAB is determined and at B the angle CBA is determined. 
Then by trigonometrical tables the lengths of lines C A and BC 
are exactly 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 correct, and the angles are determined accurately, and 
proper allowances are made for elevations and the curvature 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, some- 
times 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 trian- 
gulation much of the observing has 
been done at night upon acety- 
lene lamps; directions to the distant 
light keepers have been sent by the 
telegraphic alphabet and flashes of 
light, and the necessary observing 
towers have been built by a party 
expert in that kind of work in ad- 
vance of the observing party"* 
Survey of Indian Territory. In March, 1895, Congress 
provided for the survey of the lands of Indian Territory and the 

Fig. Ill 

*John F. Hayford, Inspector of Geodetic Work, United States Coast 
and Geodetic Survey, in a paper relating to Primary Triangulation before 
the Eighth International Geographic Congress, 1904. 


work was placed in charge of the Director of the Geological Sur- 
vey instead of being let out on contract as had been previously 
done. The system of running principal 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 direction, a survey by triangulation was made 
at the same time. The generally level character of the coun- 
try made it possible to make triangles wherever desired, so the 
"checkerboard" system of townships has superimposed upon it 
triangles diagonally across the townships. In this way the ac- 
curate 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. 



The Solar System. The group of heavenly bodies to which 
the earth belongs is called, after its great central sun, the so- 
lar 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 oc- 
casional 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 rep- 
resent the orbit of the earth, an object over seventy miles away 
would represent 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 distance of a mile. 
Do not imagine that the brightest stars are nearest. 



From the foregoing one should not fail to appreciate the im- 
mensity of the earth's orbit. It is small only in a relative 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 alto- 
gether probable that each star has its own motion in the uni- 

verse. Now the sun is simply one of the stars (see p. 265), and 
it has been demonstrated that with its system of planets it is 
moving rapidly, perhaps 40, 000 miles per hour, toward the con- 
stellation 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 perform;" and 
to assign to natural causes and conditions what had been at- 
tributed to God's fiat was thought sacrilegious. It is hoped that 
day has forever passed. 

This great theory as to the successive stages and conditions 
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 foundation of all the current specula- 
tions on the subject." It gives the facts of the solar system a 
unity and significance scarcely otherwise obtainable. 

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 direction. 

3. Planetary rotation and revolution are in the same direc- 
tion, 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 

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 characteristics 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. According to the the- 
ory, at one time, countless ages ago, all 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 gen- 
erate by friction the flash of light, sometimes called falling or 


shooting stars, so the clashing of particles in this nebulous 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 condensed 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 
centrifugal 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 in- 
creased at a greater rate than did the power of gravitation due 
to its lessening diameter. Hence, a time came when the centrifu- 
gal 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 essen- 
tially the same plane. 

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 consolidated. 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 preponderating 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 displace it. Perhaps even the best of re- 
cent 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 mathematics and astrophysics. It assumes 
the systems to have been evolved from a spiral nebula, similar to 
the most common form of nebulae 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 tremendous 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 for- 
ward by the passing star and given a revolutionary course about 
the sun. The arms of spiral nebulae 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 ma- 


terial thus shot out was very hot at first, but soon cooled into 
discrete bodies or particles which moved independently about 
the sun like planets (hence the term planetesimal) . 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 neces- 
sary to form the planets and satellites. 

This hypothesis differs from the nebular hypothesis in a num- 
ber 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 measur- 
ably 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, according to 
the old theory, were condensed from the great masses of origi- 
nal aqueous vapors surrounding the earth; according to the new 
theory the water was derived from the same sources as the at- 
mosphere. 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 rea- 
sonably certain, the solar system is not an eternal one. When we 
endeavor to extend our thought and imagination backward to- 
ward "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 geography 
of the planets give their conditions in terms corresponding 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 inhabitancy on 
other celestial bodies than the earth, or suggests interplanetary 

locomotion (see p. 304). Such conditions exist only in the imag- 
ination. An attempt to exclude astronomical facts not bearing 
upon the topic in hand and not consistent with the purpose of 
the study, makes necessary the omission of some of the most 
interesting facts. For such information the student should con- 
sult an astronomy. 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. 


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 lati- 


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 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 

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 Mar- 
tian days,^ and is nearly twice as long as ours. The orbit is 
much more elliptical than that of the earth, perihelion 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 

*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. 

t Mars, by Percival Lowell. 


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 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. 


Form and Dimensions. Venus is very nearly spherical 
and has a diameter of 7, 700 miles, very nearly that of the earth, 
so its latitude and longitude are very similar to ours. Its sur- 
face gravity is about ^ 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, there- 
fore, 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 den- 
sity 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. 


Form and Dimensions. After Venus, this is the brightest 
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 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, occu- 
pying 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 verti- 
cal 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. 


Form and Dimensions. The oblateness of this planet 
is even greater than that of Jupiter, being the greatest of 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, varying, 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 elliptic- 
ity. 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° be- 
tween 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. 

* These terms are purely relative, meaning, simply, the zone on Jupiter 
corresponding in position to the temperate zone on the earth. The inap- 
propriateness 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. 


Saturn has surrounding its equator immense disks, of thin, 
gauzelike rings, extending out nearly 50, 000 miles from the sur- 
face. 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 appreciable attractive influence upon the planet. 

In addition to the rings, Saturn has ten moons. 


Form and Dimensions. This planet, which is barely vis- 
ible to the unaided eye, is also decidedly oblate, nearly 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) fig- 
ures, 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 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 backward motion suggests a similar 
explanation, the inclination of its axis is more than 90° from the 
plane of its ecliptic. 


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 

Form and Dimensions. Mercury has about three eighths 
the diameter of the earth, one seventh of the surface, 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 ellip- 
tical 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 23|° 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 terrestrial 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 1° and back again in the short summer of 
about 30 days, and westward a similar extent during the longer 
winter of about 58 days. Two and one half times as much heat is 
received 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 Mercury, there is no 
twilight there), each extending from pole to pole and 47° wide. 
The eastern of these zones has hotter summers and cooler win- 
ters 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 ^, and its 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 revolution 
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 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 perihelion, 
aphelion, longer and shorter days, etc., as that of the earth. 
The fact that the moon goes around the earth does not materi- 
ally 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 ma- 
terially 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 trop- 
ical year must contain as many of those days as that number is 
contained times in 365.25 days or about 12.4 days. The calen- 
dar for the moon does not have anything 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 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 crescent 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 corresponding 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 crim- 
son 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 

* 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. 


get entirely above the horizon on the equator, from the time 
the first glint of light appears, and the time of sunset is equally 
prolonged; as on the earth, the time occupied in rising or set- 
ting is longer toward the poles of the moon. The stars 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 volcanoes are very dark and also cold. In the tropical por- 
tion the temperature probably varies from two or three hundred 
degrees below zero at night to exceedingly high temperatures 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 at- 
mosphere 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 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 hundred 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 27| 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 meaning. 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- 
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 neigh- 
bor should equal one hundred miles. The insects would fly in 
every possible direction of their own volition. Suns move in 
every conceivable direction, not as they will, but in abject servi- 
tude to gravitation. They must obey the omnipresent force, and 
do so with mathematical accuracy." From "New Conceptions in 
Astronomy," by Edgar L. Larkin, in Scientific American, Febru- 
ary 3, 1906. 

Solar System Table 








As compared with the earth* 







cd >> 
u +^ 
"3 'S 
I* 03 

"3 , 

CD ^ 
^ TO 




to a 

S 3 





88 days 










225 days 




















24h 37m 22.7s 









88, 000 

9h 55m 









73, 000 

lOh 14m 



















32, 000 










25d 7h 48m 







27d 7h 43m 




'The dimensions of the earth and other data are given in the table of 

geographical constants p. 308 



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 prim- 
itive man was that, save for the irregularities of mountain, hill, 
and valley, the surface of the earth is flat. In all the earliest 
literature that condition seems to be assumed. The ancient nav- 
igators 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 spherical 
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 Ptolemy s, kings of Egypt, he relates the incident of 

*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. 



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, 
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 heavens as they 
traveled southward again. In traveling eastward or westward 
there was no displacement of the heavens 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. Anax- 
imander (about 611-547 B.C.), indeed, did teach that it is a 
cylinder* and thus prepared the way for the more nearly correct 

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 precession 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 doubt- 
less many of the astronomers 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 

* 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. 


(about 582-507 B.C.), a Greek* philosopher, seems to have been 
the first to advance the idea that the earth is a sphere. His the- 
ory being based largely upon philosophy, nothing but a perfect 
sphere would have answered for his conception. 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 inhabited. 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 as- 
tronomical measurements and calculations of very great value, 

* Sometimes called a Phoenician. 

^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. 


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 care- 
fully at Alexandria, he found the noon sun to be one fiftieth 
of the circumference to the south of overhead. He then multi- 
plied 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 ac- 
curately and his calculation probably contained 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 with- 
out 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, Posidonius, 
a contemporary of Strabo, made another measurement, 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 estimates as to the size of the earth, but we have no 
record of any further measurements with a view to exact calcula- 
tion until about 814 a.d. when the Arabian caliph Al-Mamoum 
sent astronomers and surveyors northward and southward, care- 
fully measuring the distance until each party found a star to have 
shifted to the south or north one degree. This distance of two 

*The most reliable data seem to indicate the length of the stadium was 
606 1 feet. 


degrees was then multiplied by 180 and the whole circumference 

The period of the dark ages was marked by a decline in 
learning and to some extent a reversion to primitive concep- 
tions concerning the size, form, or mathematical properties 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 dis- 
covery 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 be- 
lieved by practically all educated people for centuries before the 
days of Columbus. The Greek astronomer Cleomedes, writing 
over a thousand years before Columbus was born, said that all 
competent persons excepting the Epicureans accepted the doc- 
trine of the spherical form of the earth. 

In 1615 Willebrord Snell, professor of mathematics at the 
University of Leyden, made a careful triangular survey of the 
level surfaces about Leyden and calculated the length of a de- 
gree of latitude to be 66.73 miles. A recalculation of his data 
with corrections which he suggested gives the much more accu- 
rate 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 

It was about 1660 when Isaac Newton (1642-1727) discov- 
ered the laws of gravitation, but when he applied the laws to 
the motions of the moon his calculations did 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 lon- 
gitude and obtained a close approximation to its length. New- 
ton, 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 contributed 
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 contended 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. 27), 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 sig- 
nificance of Richer's discovery was apparent to all or generally 
accepted. An instance of a commonly accepted reason assigned 
for the shorter equatorial pendulum 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 sufficiently 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 correction for the expansion of the pendu- 
lum in consequence of the higher temperature." It is interesting 
to note that if this explanation were the true one, the average 
temperature at Cayenne would have to be 43° above the boiling 

Early in the eighteenth century Giovanni Cassini, the as- 
tronomer 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 measurements in polar and equato- 
rial regions. 

In 1735 two expeditions were sent out, one into Lapland and 
the other into Peru. Their measurements, while not without 
appreciable errors, showed the decided difference of over half a 
mile for one degree and demonstrated conclusively the oblate- 
ness 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 occupied 
the attention of many since the time of Newton. His calculation 
was 2§o' th & t is? the polar diameter was ^ shorter than the 
equatorial. Huygens estimated the flattening to be about ^. 
The most commonly accepted 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. 29). 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. Professor 
William Harkness, formerly astronomical director of the United 


States Naval Observatory, calculated it to be very nearly ^. 

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 ascertain the force of 
gravity. These measurements demonstrate 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 [n] 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 philoso- 
phy that the earth rotates on its axis and revolves about the sun. 
Basing their theory upon a priori reasoning, they had little bet- 
ter 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 sys- 
tematic manner and one based upon careful observations and 
calculations. From this time, however, until the time of Coper- 
nicus, 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 compiled so much of the observations of other 
astronomers who had preceded him and invented a most inge- 
nious system of "cycles," "epicycles," "deferents," "centrics," 
and "eccentrics" (now happily swept away by the Copernican 
system) by which practically all of the known facts of the celes- 
tial 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 geography 
were his employment of the latitude and longitude of places to 
represent their positions on the globe (a scheme probably in- 
vented 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 subdivisions of degrees that we get 
the terms "minutes" and "seconds" (for centuries the division 

had been followed, originating with the Chaldeans. See p. 142). 
The sixty subdivisions he called first small parts; in Latin, "min- 
utcE primtE," whence our term "minute." The sixty subdivisions 
of the minute he called second small parts; in Latin, u minut(E 
secundds" whence our term "second." 

The Copernican theory of the solar system, which has uni- 
versally displaced all others, gets its name from the Polish as- 
tronomer Nicolas Copernicus (1473-1543). He revived the the- 
ory 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 con- 
ception 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 dis- 
senters 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 apparently change their positions in relation to 
the earth (technically, 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. 110). 

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 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. 
Galilei was a thorough-going Copernican in private belief, but 
was not permitted to teach the doctrine, as it was considered 

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 concerning three comets which 
appeared in 1618. He based his argument as to their motions 

*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. 


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, ac- 
commodates 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 gen- 
eration was Kepler (1571-1630), the German astronomer, and 
friend and successor of Brahe. His laws of planetary motion (see 

p. 283) 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. 105), 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. 



Gravity is frequently denned as the earth's attractive in- 
fluence for an object. Since the attractive influence of the mass 
of the earth for an object on or near its surface 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. 16) 

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 
between 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 



(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 discussion of tides that, although 
the tidal lessening of the weight of an object is as yet an immea- 
surable quantity, it is a calculable one and produces tides (see 


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 influence on 
objects, increasing toward the equator. This lightening influence 
is sufficient to decrease the weight of an object at the equator 
by 2§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 ^ as compared with the weight at the poles. Thus a 
body at the equator has its weight lessened by ^9 because of 
rotation and by ^ because of greater distance from the center, 
or a total of j|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 centrifugal 
force. If the earth rotated seventeen times as fast, this force at 
the equator would exactly equal the earth's attraction,* objects 

* Other things equal, centrifugal force varies with the square of the ve- 

locity (see p. 13 I, and since centrifugal force at the equator equals 289 times 

gravity, if the velocity of rotation were increased 17 times, centrifugal force 



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 pendulum 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 to- 
ward the center of the mass 


of the earth at C (Fig. [112]). 
The plumb line would then 
be PC. But centrifugal 
force is exerted toward CF, 
and the resultant of the at- 
traction toward C and cen- 
trifugal force toward CF 
makes the line deviate to a 

■pip" 1 1 o 

point between those direc- s ' 

tions, as CG, the true center of gravity, and the plumb line 
becomes P'CG. The amount of the centrifugal 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 

would equal gravity (17 2 = 289). 


deviation 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". 


In Chapter |TT] 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 understand astronomical and 
geocentric latitude, and how geographic 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 

meridian near the pole (see p. 41 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 ascertains lati- 
tude from celestial measurements by reference 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 centrifugal 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 correspond- 
ingly the astronomical latitude of a place. Thus at latitude 45°, 
astronomical latitude 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 astronomical in- 
struments, 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 lati- 
tude, corrected for the deviation of the plumb line. Were it not 
for these deviations the latitude of a place would be determined 
within a few feet of perfect accuracy. As it is, errors of a few 

hundred feet sometimes may occur (see p. 287). 

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 lat- 
itude is distance in degrees of a meridian 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 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, measured 
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 

(a) Its declination from the celestial equator and position 
in relation to hour circles, as celestial meridians are commonly 

called (see Glossary). 

(b) Its celestial latitude from the ecliptic and celestial longi- 
tude from "ecliptic meridians." 


These three laws find their explanation in the laws of grav- 
itation, although Kepler discovered them before Newton 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 ar- 

eas. In the discussion of seasons we observed (p. 169) that when 
in perihelion, in January, the earth receives more heat each day 
than it does when in aphelion, in July. The northern hemi- 
sphere, 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 cor- 
responding days, not for the seasons as a whole. According to 
Kepler's second law the earth must receive exactly the same to- 
tal amount of heat from the vernal equinox (March 21) to the 
autumnal equinox (Sept. 23), when farther from the sun, as from 



the autumnal to the vernal equinox, when nearer the sun. Dur- 
ing the former period, the northern summer, the earth receives 
less heat day by day, but there are more days. 



Fig. 113 

Third Law. The squares of the lengths of the times (sidereal 
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 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 



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 23 1° 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 acceleration of the equator is to cause 
the axis to rotate slowly, keeping the same angle to the ecliptic, 

At the time of Hipparchus (see p. 274), who discovered this 
rotation of the axis, the present North star, Alpha Ursa Mi- 
noris, was about 12° from the true pole of the celestial sphere, 
toward which the axis points. The course which the pole is tak- 
ing 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 23|°. 

As the axis rotates about the pole of the ecliptic, the point 
where the plane of the equator intersects the plane 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. 133) is the time required for the earth 

to make a complete revolution in its orbit. A solar or tropical 


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 per- 
ihelion will be reached in July, and thus terrestrial climate is 
gradually changing. This perihelion point (and with it, aphe- 
lion) has a slight westward motion of its own of 11.25" each 
year, making, with the addition of the precession of the equi- 

noxes of 50.2", a total shifting of the perihelion point (see ' Ap 

sides' in the Glossary) of V 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 decreas- 
ing, 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 
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 satisfactory. 

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 mo- 
tion, 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. 119 , 148 ) 
gradually increases and decreases again. The rate of this nuta- 
tion 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. 278), it was shown that any change in the position of parti- 
cles 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, meteoro- 
logical conditions 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 "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 corresponding decrease is observed in the latitude of an ob- 
servatory 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."* 

*Todd's New Astronomy, p. 95. 


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 se- 
lected, two in the United States, one in Sicily, and the other in 

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 eclip- 

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. 248). 


The explanation of the cause of tides in the |chapter| on that 
subject may be relied upon in every particular, although math- 
ematical details are omitted. The mathematical treatment is 
difficult to make plain to those who have not studied higher 
mathematics and physics. Simplified 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. Sur- 



face gravity over the globe varies slightly in different places, be- 
ing less at the equator and greater toward the poles. As shown 
elsewhere, the force of gravity at the equator is less for two rea- 

a. Because of greater centrifugal force. 

b. Because of the oblateness of the earth. 

(a) Centrifugal force being greater at the equator than else- 
where, 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 corresponding shifting of the accu- 
mulated 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. 

(6) Exactly the same course of reasoning applies to the unbal- 
anced 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 necessarily shift as 
different portions of the earth are successively turned toward or 
from them. Our problem, then, is to find the cause and direction 
of the unbalanced forces produced by the moon or sun. 

Fig. 114 

In Figure 114, let CA be the acceleration toward the moon 


at C, due to the moon's attraction. Let BD be 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 parallelogram, 
we have BFDE. Now it is a simple demonstration 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 repre- 
sented in the figures is greatly diminished. The distance CA is taken arbi- 
trarily, 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 relations would remain unchanged. 
Because of the short distance given CM in the figures, the difference be- 
tween 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 1. 


Fig. 115 



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 exceedingly slight, and the 

tide on the opposite side is practically equal to the tide on the 
side toward the attracting body. 


Fig. 116 

Combining the arrows showing the directions of the unbal- 
anced forces in the two figures, we have the arrows shown in 
Figure 116 The distribution and direction of the unbalanced 

forces may be thus summarized: "The disturbing force produces 
a pull along AA' and a squeeze along BB' ."* 


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 planets known to them were always 

* Mathematical Astronomy, Barlow and Bryan, p. 377. 



seen within 8° of the pathway 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 
northern United States or central Europe. 

Fig. 117 

Signs. As the sun "creeps backward" in the center of the 
zodiac, one revolution each year, the ancients divided its path- 
way into twelve parts, one for each month. To each of these sec- 


tions of thirty degrees (360° -j- 12 = 30°) names were assigned, 
all but one after animals, each one being considered appropriate 

as a "sign" of an annual recurrence (see p. 118). Aries seems 
commonly to have been taken as the first in the series, the be- 
ginning 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 equi- 

noxes (p. 285), the point in the celestial equator where the cen- 
ter 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 westward travel of this point all the 
signs have moved back into a constellation of a different name. 
Another difference 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 constellations 

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 represented by a small 
picture of a ram ifriF or by a hieroglyphic ( V ) . 

Taurus, the second sign (pujf), 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 ( ff ) 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 (HBS), 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 (KSP) 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 constellations. 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 balancing 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 (HIE)- 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 Decern- 


ber 22. 

Capricorn, signifying goat, is often represented as having 
the tail of a fish (a2&). It probably has its origin as the mytho- 
logical 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 countries. 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 ( 5? ). 
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 completed the "labors" of its circuit, 
only to begin over again. 

The twelve signs of the ancient Chinese zodiac were dedi- 
cated 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 

Myths and Superstitions 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; counting 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 be- 
lief, at once their science and their religion, it is not strange that 
when a child was born the parents hastened to the astrologer 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 29 1 days or spend 
about 2 1 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 constellations of the zodiac. Beginning 
with the sign Aries, he dedicated that to the head, the neck he 
assigned to 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. Moreover, 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 immemorial 
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 be- 
ginning 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 cal- 
endar 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 Capricorn, 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 crossing 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 rheumatism 
or the "evil" eye. 

"The fault, dear Brutus, is not in our stars, 
But in ourselves, that we are underlings." 

— Shakespeare. 



Concrete work in this subject has been suggested directly, 
by implication, or by suggestive queries and problems through- 
out the book. No instruments of specific character 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 purchased 
and used. 

To Make a Sundial 

Fig. 118 

This is not espe- 
cially difficult and 
may be accomplished 
in several ways. A 
simple plan is shown 
in Figure 118 An- 
gle 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 equation 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. 
If you mark the hour lines at any other date; ascertain the 

equation of time (see p. 128) and make allowances accordingly. 
Suppose the date is October 27. The analemma 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, marking 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 merid- 
ian upon which the clock time is based. The difference 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 notic- 
ing 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 looking up the equation of time and making allowances ac- 
cordingly. 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. 63 ), set the XII hour line due north. 

Note accurately the clock time when the 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 p. 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 latitude 

were shown in the chapter on seasons (see p. 172). Dr. J. Paul 

Goode, of the University of Chicago, has designed a very con- 
venient little instrument which answers well for this and other 

Fig. 119 

A vertically placed quadrant enables one to ascertain the al- 
titude of the sun for determining latitude and calculating the 



heights of objects. By means of a graduated circle placed hor- 
izontally the azimuth of the sun (see Glossary) may be ascer- 

tained. A simple vernier gives the azimuth readings to quarter 
degrees. It also has a device for showing the area covered by a 
sunbeam of a given size, and hence its heating power. 

The Heliodon 

Fig. 120 

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 solstices at any latitude. The 


points of sunrise and sunset can also be shown and hence the 
length of the longest day or night can be calculated. 




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 diago- 
nal AA'. When it reaches A' the force AB still acts (not con- 
sidering the friction of the air), impelling it in the line A'B 1 . 
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) AA'A". It is obvious from this di- 
agram that if the impelling 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. 

Fig. 121 

The amount of such a projectile force near the surface of the 
earth at the equator as would thus keep an object at an unvary- 
ing 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. Di- 
viding the circumference 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 in- 
creased to 37, 000 feet 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, (b) the centrifugal 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 absolute 
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 discussion of Mars (see p. 255), 



37,000 f»nt a second. 

Re turn of th e pullet^ of 2$ t /OOft 

Fig. 122. Paths of Projectiles of Different Velocities (Scientific 

American Supplement, Sept. 22, 1906. Reproduced by 


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 prin- 
ciple 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. 



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 sym- 

bols is shown on page [13j where G represents universal gravita- 
tion and g represents gravity; C represents centrifugal force and 
c centrifugal force due to the rotation of the earth. 

<f> (Phi), latitude. 

£ (Epsilon), obliquity of the ecliptic, also eccentricity of an 

7r (Pi), the number which when multiplied by the diameter of 
a circle equals the circumference; it is 3.14159265, nearly 
3.1416, nearly 3±. tt 2 = 9.8696044. 

5 (Delta), declination, or distance in degrees from the celes- 
tial equator. 

oc, "varies as;" x oc 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. 


The Circle and Sphere 

r = radius. c = circumference. 

d = diameter. a = area. 

ird = c. 

C - = d. 



7rr 2 = area. 

4irr = surface of sphere. 


4 -jrr 3 = volume of sphere = 4.1888r 3 (nearly). 

The Ellipse 

a = \ major axis. o = oblateness. 

b = \ minor axis. e = eccentricity. 

trab = area of ellipse. 

a — b 
o = 

a 2 -b 2 
a 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 = ^§. 

' 3959 

P 2 = surface of body as compared with that of the earth. 
P 3 = volume of body as compared with that of the earth. 

— — = surface gravity as compared with that of the earth. 

Centrifugal Force 

c = centrifugal force. r = radius. 

v = velocity. m = mass. 

mv 2 


Lessening of surface gravity at any latitude by reason of the 

centrifugal force due to rotation. 
g = surface gravity. 


9 o 

c at any latitude = x cos 6. 


Deviation of the plumb line from true vertical by reason of 

centrifugal force due to rotation. 

d = deviation. 

d = 357" x sin 2(f). 

Rate of swing of pendulum varies inversely as the square root 

of the surface gravity, r = . 



Density of a body = . 


Hourly deviation of the plane of a pendulum due to the ro- 
tation of the earth = sin latitude x 15° (d = sin0 x 15°). 
Weight of bodies above the surface of the earth. 
w = weight, 
d = distance from the center of the earth. 

W0C Jr 

Weight of bodies below the surface of the earth, w oc d. 


Equatorial semi-axis: 

in feet 20, 926, 062. 

in meters 6, 378, 206.4 

in miles 3, 963.307 

* Dimensions of the earth are based upon the Clarke spheroid of 1866. 


Polar semi- axis: 

in feet 20,855,121. 

in meters 6, 356, 583.8 

in miles 3, 949.871 

Oblateness of earth 1 4- 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 119) 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. 


Statute mile 5, 280.00 feet 

Nautical mile,* 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 

*As defined by the United States Coast and Geodetic Survey. 





















































































































































































































































































































































































































































































































































































Curvature of earth's surface 



Day, length of longest day at different latitudes 

Declination of the sun, see analemma 

Deviation of freely swinging pendulum due to earth's rotation 

Distances, etc., of planets 

Equation of time, see analemma 

Earth's dimensions, etc 

Latitudes, lengths of degrees 

of principal cities of the world 

Longitudes, lengths of degrees 

of principal cities of the world 

Measures of length 

Meridional parts 

Sines, natural 

Solar system table 

Standard time adoptions 

Tangents, natural 

Time used in various countries 

Velocity of earth's rotation at different latitudes 




















Vertical ray of sun, position on earth, see analemma 128 




Aberration, the apparent displacement of sun, moon, planet, or star pro- 
duced as a resultant of (a) the orbital velocity of the earth, and (fr) 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 (b) the 
mean declination of the sun for each day of the year. 

Aphelion (a 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 

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 (az' 1 mirth) 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) 

Centrifugal force (sen trif u gal), a force tending away from a center. 
Centripetal force (sen trip' e tal), a force tending toward a center. 
Colures (ko lurz'), the four principal meridians of the celestial sphere, two 

passing through the equinoxes and two through the solstices. 
Conjunction, see |Syzygy| 

Copernican system (ko per' ni can), the theory of the solar system ad- 
vanced 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] 
Co-tidal lines, lines passing through places that have high tide at the 

same time. 

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 reck- 
oned 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. 
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 average 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 (ek sen tris' 1 ty), see Ellipse 

Ecliptic (e 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 (pronounced fo' 
si; singular, focus) that the sum of the distances is constant. 
Eccentricity (ek sen 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 

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 (el 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 (e kwa' ter) , when not otherwise qualified means terrestrial equa- 

Celestial equator, the great circle of the celestial sphere in the plane 
of the earth's equator. Declination is measured from the celestial equa- 
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 celestial 
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 reck- 
oned. 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 sen' 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 6d' e 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. 
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 (b) is so small that it is usually ignored and 

we commonly speak of gravity as the earth's attraction for an object. 

Gravity is still more accurately defined in the Appendix. 
Heliocentric (he li o sen' 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 Copernican 

system ) 

Parallax, see Parallax 

Horizon (ho rl' 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 I 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 geographical 

latitude are nearly the same (see discussion of Latitude in Appendix) . 
Local time, see [Time] 

Celestial longitude, the distance in degrees of a celestial body from 

lines passing through the poles of the ecliptic (see Ecliptic I , called 
ecliptic meridians; the zero meridian, from which celestial longitude 
is reckoned, is the one passing through the First point of Aries (see 

Equinox I 

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 |Timc] 


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 


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 Febru- 
ary 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 averages 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 several 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 princi- 
pally 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 side of the plane of 
the ecliptic there is a tilting tendency given the bulging equatorial region. 
The inclination of the earth's axis, or the 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 |Ellipsc| 
Oblate spheroid, see |Sphcroid| 

Obliquity (6b hk' wi ty), of the ecliptic, see Ecliptic 
Opposition, see |Syzygy| 

Orbit, the path described by a heavenly body in its revolution about an- 
other 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 (per' 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 l he' li on), the point in a planet's orbit which is nearest 

to the sun. 

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. 
Of 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' lk) , 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 I . 

Radius (plural, radii, ra' di l), half of a diameter. 

Radius Vector, a line from the focus of an ellipse to a point in the bound- 
ary line. 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 used in as- 
tronomy 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 IMonthl 

Sidereal time, see |Timc 

Signs of the zodiac, its division of 30° each, beginning with the vernal 
equinox or First point of Aries. 

Solar times, see |Timc} 

Solstices (sol' 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 (sfe' 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 I 

Prolate spheroid, a mathematical solid produced by rotating an el- 

lipse 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. 

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 I 

Astronomical time, the mean solar time reckoned by hours numbered 

up to twenty- four, beginning with mean solar noon (see Day, astro- 

nomical ) 

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 and from noon, 
and is based upon a meridian prescribed by law or accepted as legal 

(see Day, civil I 

Equation of 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 I 

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] 


Astronomical, the two small circles of the celestial sphere parallel to 
the celestial equator and 23° 27' from it, marking the northward 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| 

Anomalistic year (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 coun- 
tries 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 re- 
lation to a star; one exact revolution about the sun; length, 365.2564 d. 
Tropical year, the 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' mth), 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. 


Abbe, Cleveland, [6| [64] 
Abbott, Lyman, [75] [143HT45 
Aberdeen, S. D., [89} 
Aberration of light (see Glossary, 




Acapulco, Mexico, 102 

105 107 



Acceleration (see Glossary, p. 3131 

Adelaide, Australia, [87 
Aden (a' den), Arabia, 




186 201 

213 218 

Akron, O., [80 
Al-Mamoum, [27! i 
Alabama, 89 

Alaska, k>8 

Albany, N. Y. 
Albany, Tex., [244 

^91 ml ^91 H03 

Alberta, Canada 
Aleutian Is., 97 


Alexandria, Egypt, [STJ [82] [87] [270] 

Algeria, [82] 

Allegheny Observatory, Allegheny, 

Pa., [68] 
Allen, W. F. 


Allowance for curvature of earth's 

227 231 234 

Almanac, [119] [L24]^T26] [T72] 

Altitude, of noon sun, [12] [1701^175" 

of polestar or celestial pole, [57 

PI [T70l^T75l 
Amazon, bores of, |188| 

American Practical Navigator, 217 
Amsterdam, Holland, |83[ [87 
Analemma (see Glossary, p. 
description of, |127[ 

representation of, |127| 

uses of. 


129 131 171 


(an ax l man'der), 


Ann Arbor, Mich 
Annapolis, Md., [89] 
Antipodal (an tip'o dal) areas 

map showing, [39 
Antwerp, Belgium, 
Aphelion (see Glossary 

[1201 [253] [286 
Apia (a, pe'a, 




Apogee (see Glossary, p 
Apsides (see Glossary, p 
Aquarius (a kwa'ri us 
Arabia, [87] 
Arcturus (arktu'rus), 







54 110 

Area method of determining geoid, 


Area of earth's surface, 13091 


Arequipa (arake'pa), Peru, 164 



Argentina (ar jen te'na) 
Aries (a'riez), constellation 


first point of, [293] [315 

sign of zodiac 
Aristarchus (ar is tar'kus) 

Aristotle (ar'is tot 1) 
Arkansas (ar'kan sa 
Arkansas River 



, 269 





Armenian Church 
Asia, [32] [39 


186 yim 

247 253 



Astronomical day, 

Athens, Greece 

Atlanta, Ga., ^ 

Atlantic Ocean 

Atmosphere, [162 

absence of, on moon 
how heated, |167| 

T3i~] [314 


292 313 




263 264 

on Jupiter, 257| 

on Mars, [255 

on Mercury, |261| 

on Venus, 256| 

origin of, |252 
Attu Island, [89] [TOO] 
Auckland, New Zealand, 
Augusta, Me., [89] 
Augustan calendar, |136| 
Austin, Tex., [89] 
Australia, f39l f97l [T02] 

Austria-Hungary, [67] [80] 
Axis, changes in position of, |286| 
defined, [22] [313] 
inclination of, see |Obliquity of| 

parallelism of, 154 
Azimuth, [30ll [3131 


Babinet, [207 
Bailey, S 




Balearic Is., |85[ 

Balkan States, [140] [146] 

Ball's History of Mathematics, [268] 

Baltimore, Md., [89] 

Bangor, Me., [89] 

Bankok, Siam, |87[ 

Barcelona, Spain, |87[ 

Barlow and Bryan's Mathematical 

Astronomy, |291 

Base line, [232 

Batavia, Java, 
Behring Strait 
Belgium, [67 
Beloit, Wis. 

Barometer, 162 |176 



235] [243] [245] 


Benetnasch, |60[ 
Bergen, Norway, [87 
Berkeley, Calif., [88] 
Berlin, Germany, 
Bessel, F. W., [30 
Bethlehem, 11451 


31 33 110 

Big Dipper, [9] [47] [59] |60] [PH 

Bismarck Archipelago, |82[ 
Bismarck, N. P., [89] 
Black Hills Meridian, [231] 
Bogota (bogota/), Columbia, 82 
Boise (boi'za), Ida 

Bombay (bomba'), India, 87 

Bonn Observatory, |1 10 

Bonne's projection, |221~] [224 

Bordeaux (bordo'), France, 
Bores, tidal, [1861 [1881 

Bosphorus, |145 


Boston, Mass., [89} 



Bowditch, Nathaniel, 
Bradley, James, |107| 



276 277 

Brahe (bra), Tycho, 110 

Brazil, [88} 

British Columbia, |81| 

British Empire, |81| 

Brussels, Belgium, [ST] [87] 

Budapest (boo' da pest), Hungary, 

Buenos Aires (bo'mis a'riz), Ar- 
gentina, [72] [87] 

Buffalo, N. Y., |9} 

Bulgaria, [67] 

Bulletin, U. S. G. S., [244] 

Burmah, 18 1 1 

Cadiz (ka'diz), Spain, 

Caesar, Augustus, |136 


Julius, 135 139 

Cairo (kl'ro), Egypt, [82] 87 
Calais, Me., [30] 
Calcutta, India, [87] [109 
Calendar, [l33lpl6] 

ancient Mexican, 142 

Augustan, |136| 
Chaldean, [Till 

Chinese, 1141 

early Roman, 135| |139| 
Gregorian, 137| 
Jewish, [145] [189] 
Julian, fT36l fliol 

Mohammedan, [TIT] pl3]|T45] 

on moon, 


[9TJ [156] [T69 

30] [33] [35] [67] |8] [89] 

Callao (kalla'o), Peru, 85 
Canada, [47] [23UJ [234] 

Cancer, constellation of, 1151 

sign of zodiac, 294 

tropic of, [151] [268 

Canterbury Tales, quoted, 
Canton, China, |88| 
Cape Colony, Africa, |81[ 
Cape Deshnef, Siberia, [97 
Cape Horn, [39] 
Cape May, N. J., |30l [33 

Cape Town, Africa, 31 72] 

Capricorn, constellation of, |152| 
sign of zodiac, 
tropic of, |152| 

295 297 

Caracas (kara'kas), Venezuela, 87 
Carleton College, Northfield, 

Minn., [68] 
Carnegie Institution of Washington, 


Carolines, the, |82| 

Cassini (kasse'ne), G. D., and J., 

Cassiopeia (kas si o pe'ya), p\ |59| 


Cayenne (kien'), French Guiana, 

[27] [2H] [87] 
Celestial equator, [46] [151] [170HT75] 

[282] [2^3] [3T5] 

Celestial latitude, [282] [283] 
longitude, [282] [317] 

meridians, 282 283 

pole, [46] [57] [59] p7o]|T75] 

sphere, N^HS^ WA ^13 

tropics, [151] [152] [320 

^ ^51 

^6i m 

Central time, in Europe 

in the United States, 
[73}{75] [1291 
Centrifugal force (see Glossary, 
p. [314]), pip] [27] |50] [278 



[282] [2891 [3071 

Centripetal force (see Glossary, 


Ceres, 1135 

314), 15 

Ceylon (selon'), 181 
Chaldeans, [134] [142 
Chamberlin, T. C, [6] [25T " 
Charles IX., King of France, |139[ 
Charles V., Emperor of Spain, 215[ 
Charleston, S. C, [89] 
Chatham Islands, [Hi] [97] 
Chaucer, quoted, |118[ 
Cheyenne (shlen'), Wyo., |89[ 
Chicago, 111., [62] [66] [89] [96] [99] 

[T30] [T741 [2081 
Chile (che'la), MT 
China, [81] [M] [89] [ID I 1.64. 20N 
Chinese calendar, 11411 

zodiac, 12951 

Choson, Land of the Morning Calm 
(Korea), [98 

Christiania (kris te a'ne a), Nor- 
way, [57} 

Christmases, three in one year , |145[ 

Chronograph (kron'o graph) , 

Chronometer (kro nom'e ter), 63 
Cincinnati, O., [73] [89J 
Circle defined, [20} 
Circle of illumination, or day circle, 
[T5T1 [T52] [T55HT58] 

Circumference of earth, 29 |30[ |309[ 
Ciudad Juarez, see |Juarez[ 
Civil day, [1311 [314[ 

Clarke, A. R., [29}|33j [36] [42] [273] 

Cleomedes (kle om'e dez), 271 
Cleveland, O., [73] [89 
Clock, sidereal, 1681 

Co-tidal lines, [18§ [314] 

Collins, Henry, [TOT] 

Colombia, [82] 

Colon (kolon'), Panama, 85 

Color vibrations, |107[ 

Colorado, [6§ [89] 

Columbia, S. C, [90} 

Columbus, Christopher, [138] [139] 

[2T3| [27T1 
Columbus, O., [73] [89] 
Comets, [49] [247] [2761 

Compass, magnetic, or mariner's, 

[T52] [153] [228] 
Concord, N. H., [90} 
Congressional township, |230[ 
Conic projection, |218^225"] 
Conjunction, [178] [184] [320] 
Connecticut, |89[ 
Constantinople, Turkey, |88| 
Convergence of meridians, |231[ |234[ 
Copenhagen, Denmark, |88[ 
Copernican system (see Glossary, 


314), 275 277 

Copernicus (ko per'm kus) , 49 110 

Cordoba (kor'doba), Argentina, 

Corinto (ko ren'to), Nicaragua, 84 
Correction line, |235[ 
Cosines, natural, table of, |310| 
Costa Rica, |82| 

Cotangents, natural, table of, |311[ 

Crepusculum, the, |165[ 

Creston, Iowa, |79| 

Cuba, [68] [82] [8j 

Curvature of surface of earth, rate 
of, [26] [41] [42] [227] [234} 

Cygnus (sig'nus; plural and pos- 
sessive singular, cygni) 


Cylindrical projection, |2 10 218] 





Dakotas, division of, 235| 
Danish West Indies, 82] 
Date line, see [International date| 

Day (see Glossary, p. 314 1, astro- 


circle, see |Circle of illumination| 
civil, [131] 

length ol 

, 155 159 

lunar, 187 




solar, 61 



total duration of a, 99 

week, 143 

la Hire, Phillippe (fe lep' 


delaer'), [201 
Deadwood, S. D., [90] 
Declination (see Glossary, p. 314 1 

[125] [127] [17THT751 [283] 

Deimos (di'mus), | 255 
Denmark, [67] ^ [ 


Density of earth, |309| 
Density, formula for, |308| 

Denver, Col., |5| [26] [66] [90] [98] 
Des Moines (demoin'), Iowa, 
Deshnef, Cape, [97] [TtjO 
Detroit, Mich., [62] [72] [90 

Deviation, of pendulum, 53]|56| 

of plumb line, [280H282] 
Dewey, George, |104[ 
Diameter of earth, [2&H30] [44] [308] 
Dimensions of earth, |308[ 
Dip of horizon, (see Glossary, 


Distances, of planets, |266| |309 
of stars, [44] [247] 

District of Columbia, [76] [91] [125] 

Diurnal (dlur'nal), motion of 
earth, see 


Division of Dakotas, [235 
Dryer, Charles R., 
Dublin, Ireland, [63 
Duluth, Minn., [Ml 


Earth in Space, 247] 266 

Earth's dimensions, |308[ 
Eastern time, in Europe, |67| 

in the United States, |66[ |70[ [74 


Eastward deflection of falling ob- 
jects, |50"H54"1 
Eclipse, [23] [117] [162] [178 

Ecliptic (see Glossary, p. 315), 116 
[T20] [282] [285] 
obliquity of, [119] [148] [286] [309] 

Edinburgh (ed'in birr ro), Scotland, 

Egypt, [82] [87] [134] [268] 

El Castillo (el kas tel'yo), 

Nicaragua, |84[ 
El Ocotal (elokotal'), Nicaragua, 


El Paso, Tex., 67 74 

Ellipse (see Glossary, p. 315 1, 21 

[22] [192] [206] [3TJ71 
Ellipsoid of rotation (see Glossary, 

p. [319]), [35] 

Encyclopaedia Britannica, 102[ 
England, [9] [79] |l] |8] [99] [137] 
[1391 [272] 

Ephemeris (efem'eris), see Nauti- 

Ical almanacl 
Epicureans, |271[ 
Equador, |82[ 



Equation of time (see Glossary, 

315), 123 127 

Equator (see Glossary, p. 315) 


celestial, [46] [151] [170H175 


length of day at, |157| 
terrestrial, [25] [32] [48] [119] [149 

[1531 [271) HZH [308" 


Equinox (see Glossary, p. 315 I 
[H9] [T55] [T56] [1%] |169| [283 
precession of, [285^286] [30l] 

Eratosthenes (er a tos' the nez 
[269] [270 

Erie, Pa., 90 

Establishment, the, of a port, |180| 
Eudoxus, 12691 

Euripides (u rip'i dez), [122 

Europe, [102] [167] |168| |220| [223] 
[227] [2861 [292] 

Fargo, N. D., [90 
Farland, R. W 


Faroe (fa'ro), Islands, 


Fathom, length of, |309 
Fiji Islands, [97] 
Fiske, John, [145 



109 110 265 266 

Fixed stars, [10 

Florence, Italy, 

Florida, |9] |I] 

Form of the earth, [23f[42] 

Formosa, |83[ 

Formulas, I306H308] 

Foucault (fooko'), experiment, 

with gyroscope, 155 


with pendulum, 53 -[56] 
France, [M] [63] |f 
pT] [139] [T86] 

Franklin's almanac, 11371 


Fundy, Bay of, [188] 


Gainesville, Ga., |74( 

Galilei, Galileo (gal l le'o gal l la'e), 

Galveston, Tex., [74] [90| 
Gannett, Garrison and Houston's 

Commercial Geography, |206| 
Gauss, [52~1 


Genesis, [T42] 
Genoa, Italy, |75| 

Geocentric, latitude, see |Latitude] 


theory, (see Glossary, p. 316 1 
[275] [2771 

Geodesy (see Glossary, p 

Geodetic Association, Interna- 
tional, [2881 

Geographical constants, 308 

Geoid (je'oid), 32 36 

Geometry, origin of, |227[ 

George II., King of England, |229[ 

Georgia, [6Tl [74l [79l f88l f9ll 

German East Africa, 83 

Germany, [67] [75] |82] [83] [87] [88] 

[97] [1371 HEH 
Gibraltar, Spain, [8l] [88] 
Glasgow, Scotland, |88( 
Globular projection, [T99}[20T1 [2TT] 
Glossary, [311H32T1 
Gnomonic (no mon'ik) , cylindrical 

projection, [2T0H2T2] 
Gnomonic projection, |202^205"] 


Goode, J. Paul, [3001 



Goodsell Observatory, Northfield, 

Minn., [68] 
Gravimetric lines, map showing, 

Gravitation, [16]^T8] [178] [179] [271] 


Gravity, [l8j [24} [26j [28j [182HT861 
[278H28T1 [2891 [3031 ED [3071 


on Jupiter, [THJ [257] 
on Mars, |254| 
on Mercury, |2601 
on moon, |19| |261| 

on Neptune, 260 

on Saturn, |258 

18] [2651 

on Uranus, 259 

on Venus, |256| 
Great Britain, [63] [67] [7§ [79] [18T] 

Great circle sailing, [203] [204] [213] 

Greece, ^31 87 

Greenland, [2181 

Greenwich (Am. pron., gren' wich; 

Eng. pron., grm'ij; or gren'ij), 

England, [39l |40l 

H3 O HI1 H3 EI 

[961 013 P5HT27 



mi Hum 



Gregorian calendar, 136 139| 
Guam, [72] [86] 
Guaymas, Mexico, |84| 
Guiana, French, |7j J272] 
Gulf of Mexico 
Gunnison, Utah, |244 
Guthrie, Okla., |9] 
Gyroscope (jl'roskop), 154 155 

35 188 

Hague, The, Holland, |83[ 

Hamburg, Germany, 
Harkness, William, 




Harper's Weekly, [163] [164 
Harte, Bret, [93] 
Hartford, Conn., [90| 
Harvard Astronomical Station 


Havana, Cuba, [82] [86l [88| 

Hawaiian (Sandwich) Islands, 86 


Hayden, E. E., [6] [72] [75} 
Hayford, J. F., [34] [245] 
Hegira, |1411 
Helena, Mont., |90] 
Heliocentric theory (see Glossary, 

316), 275 276 

Heliodon (he'liodon), 301 

Hemispheres unequally heated, [169| 

Heraclitus (her a kh'tus) , 269 

Hercules (her'cu lez), constellation, 

Herodotus (he rod' o tus), 133 267 
Herschel, John, |32| 
Hidalgo, Mexico, [84} 
Hipparchus (hip ar'kus) 
Historical sketch 

Holland, [67J [831 [87 
Holway, R. S 
Homer, [2671 

267 277 

270 285 

Homolographic projection, 
[208] [2TT 

Honduras, 83 



Hongkong, [|T] |2] 

Honolulu, Hawaiian Islands, |89| 


Horizon (see Glossary, p. 316) 


[39] |^6] [T52] [T53] [T581 fTTO] [T75] 

Hungary, 137 



Hutchins, Thomas, [229] [230} 
Huygens (hl'gens), Christian, 272 

Iceland, |82| 
Idaho, [89] 
Illinois, |89l [2311 [2321 

Impressions of a Careless Traveler, 

quoted, 75 143 

India, [31] [81] [87[ 
Indian Ocean 


186 243 

Indian principal meridian, |231 

Indian Territory, survey of, |245| 
Indiana, [89] [23T] 
Indianapolis, Ind., |90| 
Insolation, [166HT70] 
International date line, 1961 11021 

International Geodetic Association, 

Intersecting conic projection, |222| 

Iowa, [79[j90[_ 

Ireland7]3lf|63l [76] |8l] [88] [234 

Isle of Man, 81 

Isogonal (l sog'on al) line, 228 

Italy, 67 83 



Jackson, Miss., [90 

Jacksonville, Fla. 

James II., King of England, |272| 

Japan, [83} [84] [89] [288] 

Java, [87] 

Jefferson, Thomas, |230| 

Jerusalem, |88| 

Journal of Geography, |156| 

Juarez (hoo a'reth), Mexico, [74 

Julian calendar, n35l \\M\ \U0 

Jupiter, [18] [251] [253] [256] [257] 
[2661 [272] [276] 

Kiistner, Professor, |110| 
Kamerun, Africa, |83| 
Kansas, |33| 

Kansas City, Mo., [90] [96} 
Keewatin, Canada, |81| 
Kentucky, [77] [89] 
Kepler, Johann, |277| 

laws of, [283H284} 
Key West, Fla., [90} 
Kiaochau (ke a, o chow'), China, 82 
Korea (ko re' a 


Kramer, Gerhard, 215 

La Condamine (la kon'da men) , 


Lake of the Woods, [234] 
Lake Superior, |232| 
Landmarks, use of, in surveys, |227| 

Lansing, Mich., 90 232 

Lapland, 273 

Larkin, E. L., 265 266 

Latitude (see Glossary, p. 317 1 

astronomical, 281 

celestial, 282 |283 
geocentric, 281] 

geographical, 41 

determined by altitude of cir- 
cumpolar star, [57]]60] 

determined by altitude of noon 
sun, [170HT75] 

determined by Foucault exper- 
iment, |54| 

lengths of degrees, |41]^43| 



of principal cities, |87||92| 

origin of term, |39| 
Law Notes, quoted, |80| 
Layard, E. L., [TOT] 
Leavenworth, Francis P., [6j 
Legal aspect of standard time, |75f 

Leipzig, Germany, 


Length of day, | 156f|159| 
Leo, [294] [296] 
Lewis, Ernest I.. 
Lexington, Ky.. 
Ley den, Holland 
Libra (li'bra) 


83] [27T[ 



68] [72] 


Lick Observatory. 

Lima (le'ma), Peru 

Lincoln, Neb., [90} 

Link of surveyor's chain, |309| 

Lisbon, Portugal, [39] [72] [85] [88] 
Little Dipper, [9] 
Little Rock, Ark., [90] [232] 
Liverpool, England, |88| 
London Times, |140| 
London, England, [39] [62] [63] [94 
[971 [TOOl [T3 

Longitude (see Glossary, p. 317) 
and time, J6Tf[92[ 
celestial, [2821 

how determined, |62||66| |129[ 
lengths of degrees, |42[ 

of principal cities, p7 [92 


Longitude, origin of term, 

Los Angeles, Calif., [89} 

Louis XIV., King of France, [27l 



Louisville, Ky., [77j [90] 
Lowell, Mass., [90] 
Lowell, Percival, [98} [254] 
Luxemburg, |67[ |84[ 

Luzon, MM 


Macaulay's History of England, 

[139] [HO] 
Madison, Wis., [66] [90] 

Madras (madras'), India, 72 81 

Madrid, Spain, [72] 

T52] [T53] [228] 

Magellan's fleet, 93 

Magnetic compass, 

Magnetic pole, |153 
Maine, [30] [35] 
Malta, |8T[ 
Managua (mana'gua), 


Manila, Philippine Is., 72 




change of date at, |102[ 
Manitoba, Canada, 


Map, [39] [40] [23TJ [237 

Map projections, [190H226] 

Mare Island Naval Observatory, |68| 


Mariane Islands, |82[ 
Markham, A. H., [TOJ 


Mars, [253H255] [266] [284] [305 

Marseilles, France, 88 
Maryland, [88[ 
Massachusetts, [88] [89] [228] 

Mauritius (ma rish' l us) Island, 72 
McNair, F. W., [6] [51] 
Mean solar day, see |Day[ 

Measures of length, |309 

Measuring diameter of moon, |241| 

Measuring distances of objects, |238| 

heights of objects, |239[ 



Mediterranean, |102| 
Melbourne, Australia, 
Memphis, Tenn., |90| 

Mercator projection, [204] [2TT}J220l 

Mercedonius, 1135 

Mercury, \\M\ 
Meridian, [22l 

253 260 261 

28] [31] [36] [96 

[T871 [T89][226l [282] [3T7] 



282 283 317 


circle, G2 

length of degrees of, |42| 

prime, [39] [40] 

principal, for surveys, 231^237] 

rate for convergence, 

standard time, |66f|67[ 
[80}]86] [3001 
Meridional parts, table of, |217| 
Meteors, |49l [2491 




Meter, length of, [309 
Metes and bounds 
Mexico, |84l f 


Gulf of, 188 

227] [229] 

671 m\ m\ 


Michigan, |50[ 

College of Mines, [51] 
Midnight sun, |163[ 
Mile, in various countries, |309[ 
Milwaukee, Wis., [90] [104] 
Mining and Scientific Press, |51f 54 

Minneapolis, Minn., [62] [90] [96 
Minnesota, [89] [91] [232] [234 
Mississippi, |89[ 

River, [232] 
Missouri, [89] [OTJ 

River, [228] 
Mitchell, Frank E., [6] 
Mitchell, S. D., m 

Miyako (me ya'ko) Islands, 83 
Mobile, Ala., [90] 

Mohammedan calendar, |141[ |143j - 


Mollendo, Peru, 85 

Mollweide projection, 207 
Montana, [H9] 
Montevideo, Uruguay 
Montgomery, Ala 


Month (see Glossary, p. 317 1, 134 







Moon or satellite, 

[175141861 [241] [2J2] [2471 
[258^264] [2661 [2761 [2871 

Moore, C. B. T., [97] 

Morse, J. F. 





Moscow, Russia, 

Motion in the line of sight, 1 1 07] 


Motions of the earth, [2881 

Motions of the earth's axis, |285| 

Moulton, F. R. 


Mount Diablo meridian, 
Mountain time belt, |66[ 
Munich (mu'mk), Germany, 
Myths and superstitions of the zo- 

67] [741 

diac, 295 297 


Nadir (see Glossary, p. 318), 36 

Naples, Italy, [88] 

Nash, George W., [5] 

Nashville, Tenn., [91] 

Natal, Africa, [81] 

Nautical almanac, [119] [125] [171} 

Neap tides, [185] [W] 

Nebraska, ^0 89^ 91 ^28 



Nebulae, [249] [251] 

Nebular hypothesis, [248f[252| 

Nehemiah, 11421 

Neptune, mm MM [266] 

Neuchatel, Switzerland, |86| 
Nevada, |<JlJ 

New Brunswick, Canada, |81[ 

New Caledonia, 
New Guinea, p2 
New Hampshire, |89| 
New Haven, Conn., |91| 
New Jersey, [30j [77] [89] [9T 
New Mexico, [67] [74 M [231] 
New Orleans, La., 58 


75 76 

New York 

[TOOl [20 
New York Sun, [77 

New South Wales, [8 1] [88] 
New Style 

91 109 


92 96 98 

New Zealand, \5T\ |97l 
Newark, N. J. 


Newchwang, China, |82| 

Newcomb, Simon, 1119 

Newfoundland, |81| 
Newton, Isaac, [15] [50j [139] [271] 
Nicaea, Council of, 137| 
Nicaragua, |84| 
Nile, [227] 

North America, [102] [184] [213] [219] 
[2l3] [286] 


North Carolina, [91 
North Dakota, 
North Sea, [186 
North, line 

89 235 

59 130 

on map, [212] [218] [225] 

pole, [22] [46] [152] [153] [287] 
star, g2] g6] g8] [57] [H9] [285] 

Northfield, Minn., [68} |9TJ 

Northwest Territory, survey of, 

Norway, [67] [84] [87] 

Norwood, Richard, |271[ 
Noumea, New Caledonia, [TOT 
Nova Scotia, Canada, [81 
Numa, [T~35l 

Nutation of poles, [286] [318] 

Oblateness of earth, [26}{32] [3§ [41] 

Obliquity of the ecliptic, |119[ |148 

[287] [309] [3T5] 
Observations of stars, [9 
Official Railway Guide, 72 74 

Ogden, Utah, 91 

Ohio, 89 





a, 89 

Old Farmer's Almanac, 

Old Style, 138 140 144 

Olympia, Wash., [91 
Omaha, Neb 



Ontario, Canada, 81 

Oporto, Portugal, |204 
Opposition, [177] [178 

Orange River Colony, [81 
Orbit, of earth 

[133] [T48] [T52] [247] 
of moon, [177] [l7| [262] [287] 
Oregon, [M] 

Origin of geometry, |227[ 
Orion (orl'on), 112 
Orkneys, The, M~ 

Oroya, Peru, |85[ 

Orthographic projection, |191f|195"] 
[199] [200] [211] 

Outlook, The, [75] [T43]|T45" 



Pacific Ocean, [67] [97] [98] [185] [243] 

Pacific time belt, 67 

Pago Pago (pron. pango, pango), 

Samoa, |9l] [97l 

Pallas, [292] 
Panama, |68[ 
Para, Brazil, 


Parallax, [TlO] [242] [276] [318] 

Parallelism of earth's axis, |154| 
Parallels, [22] [190H225] 
Paris, France, 27 


39 53 63 

Parliament, |76| 
Pegasus (peg' a siis 
Peking, China, [89] 
Pendulum clock 

Square of, 47 

Pennsylvania, 77 89] 229 ^30 

27 53 308 

Perigee, [178] [318 




[85] [273 


10 263 

Pescadores (peskador'ez) Is. 
Phases of the moon, 
Philadelphia, Pa., [25 


Philippine Is. 




Phobos (fo'bus), [255 




Photographing, |49| 
Picard (pekar'), Jean 
Pierre, S. P., |<JT} 
Pisces (pis'sez) 
Pittsburg, Pa., 

271 273 

293 295 


Planetesimal hypothesis, |251 

Planets, [19] [49] [247] [284] [304 



Pleiades (ple'yadez), 47 
Plumb line, [TT] [50] [280T [281 

Plutarch, ^69^ 

Point Arena, Calif., [30] 
Point Barrow, |91| 
Pointing exercise, |37]^39| 
Poland, [31] [137] 
Polar diameter of earth, |308| 
Polaris, see |Polestar 

Pole, celestial, [46] [l70] [m] [283] 
[285] [3T8] 

magnetic, 153 
nutation of, 


of the ecliptic, ^85 287 

terrestrial, 22 36 46 53H55 59 

[T52]jT55] [T571 [T92}[2T2] [2791 
[285] [2871 [289] 

Polestar, (see |North star| , [9] [To] 

Polyconic projection, 223f|224] 

Popular Astronomy, |133 

Port Said (saed'), Egypt, [82 
Portland, Ore., M\ 

Porto Rico, [86] [91 
Portugal, [39] [85] [88] 
Posidonius (pos'i do m us) , 270 
Practical Navigator, 217| 

Practical work, [298H302 


Precession of equinoxes, |285f|286" 
Prince Edward Island, Can., 
Princeton, N. J., [91] 
Principal meridian, 231f|23T] 
Projectiles, [303H305 

Projections, map, [T89H226[ 

Proofs, form of earth, [23}§8] [M]- 
[34] [2731 
revolution of earth, [105HTT21 [275] 

rotation of earth, [50}{56] [61] [109] 
Proper motion of stars, |110| 



Providence, R. I., 1911 

Psalms, [252] 

Ptolemaic system, |274[ [319 

Ptolemy Necho, of Egypt, 267| 

Ptolemy, Claudius, 270 

Pulkowa, Russia, [72] [85] [89] 
Pythagoras (pi thag'o ras), 

Quebec, Canada, |81| 
Queensland, Australia, |81| 
Quito (ke'to), Equador, [82| 



Radius vector, 
Raleigh, N. C. 


E831 m^ 


Ranges of townships, [23T}]234] [236] 

Rapid City, S. D., [244] 

Rate of curvature of earth's surface 

HU Eg ng 

Refraction of light 

44 159 319 

Revolution (see Glossary, p. 319 1 

[105HT32] [147] [154] [184] [247] 
[2491 [251] [2541 [257H263] [2761 

Rhode Island, [91] 

Rhodesia, Africa, |811 

Richer (re shay'), John 

Richmond, Va., |91| 

Rio de Janeiro, Brazil, 

Rochester, N. Y 

Roman calendar 

27 272 



135 139 

Rome, Italy, [72] [89 
Rotation of earth 
Rotation, proofs of, 
Rotterdam, Holland 
Roumania, 1671 



50 155 

Russia, |31| 

[85] [88} [97] [103] [140] 




Sacramento, Calif., |91| 
Sagittarius (sag it ta'ri iis 
Salvador (sal va, dor), I 
Samoa, [M] |7] [9l] [97] 
San Bernardino, Calif.. 
San Francisco, Calif., [91 [ |173[ 
San Jose (hosa'), Costa Rica 
San Juan del Sur, Nicaragua, 
San Juan, Porto Rico, |91| 
San Rafael (rafael'), Mexico, 
San Salvador, Salvador, |85| 
Santa Fe, N. M., [9T[ 
Santiago (san te a/ go 
Santo Domingo, |85| 
Saskatchewan, Canada, [81 
Satellite, see 
Saturn, [29] [249 
Savannah, Ga., 
Scale of miles, 
Schott, C. A., 




Chile, 81 


253 258 266 


195] [2171 [225] 


Scientific American, |266| |3041 

Scorpio, [294} 

Scotland, [8TJ [88} [139] [186] 

Scrap Book, |64| 

Seasons, |147| - 175 

Seattle, Wash. 




Seoul (saool'), Korea, 84 

Servia, [67] |5} 

Seven motions of earth, 288| 

Seven ranges of Ohio, |230 

Sextant, |60| 

Shakespeare, |297| 

Shanghai (shang'hl), China, 81 

Shetland Is., IHTl 



Siam, [87] 

Siberia, |98l flOOl [103 

Sicily, [288 

Sidereal, clock 
day, [54] [314] 


month, [177] [317] 
year, [133] [285] [309 

293 319 

Signals, time, 70] 72] 
Signs of zodiac, |116| 
Sines, natural, table of, |310 
Sirius (sir'i us) 



Sitka, Alaska, [721 [91 
Snell, Willebrord 
Solar day, see Day[ 

Solar system, [247 266 

table, [266 

Solstices, [149] [167] [301] |3l"9 

Sosigenes (so sig'e nez), 
South America, [32] [jM 
[2T3] [2T8] [272] 


186] [201] 


South Australia, [81 
South Carolina 
South Dakota, 

[235] [24^] 
South, on map, [201] [212] [225] 

pole, |153[ 

star, g§{|8] [57] [58] [l|9] [T52] 
Southern Cross, |46[ 
Spain, [67] [85] [87] [93] [129] [215] 


Spectrograph, 110 

Spectroscope, p6 
Sphere, defined 

Spheroid, M\ 27 

108] [110) 
32l [3191 

Spitzbergen, k}T 
Spring tides, [184] [187 

Square of Pegasus, 47 

St. John's, Newfoundland, |81[ 
St. Louis, Mo., [25] [26] [66] [84] [91] 
St. Paul, Minn., [9~T1 [Ml 



109 110 

234 235 

St. Petersburg, Russia, [72 
[1591 [1741 

Stadium (sta' di um 
Standard parallel 
Standard time 
Star, distance of a 
motions of. 






195 200 

sun a, [265H266] 
Stereographic projection, 


Stockholm, Swed en, [86] [89] 
Strabo (stra'bo) , |270[ 
Strauss, N. M., \$4\~ 
Sun, [10]4T2] [18] [162] [247][249l [25T] 

a star, [26&H266] 

apparent motions of, 

declination of, 

113 293 

127] pTTIpTB" 

fast or slow, [61] [T231J13T 
Sun Board, [300] [301] 
Sundial, |l] |i] [l3l] [298] [300 
Survey, [29] [3TJ [34] [227] [271} 
Surveyor's chain, [2281 [3091 
Sweden, [M] [67] [8( 
Switzerland, |67| [8^ 
Sydney, Australia, 
Syene, Egypt, 



Syzygy, [178] [185] [3201 

Tables, list of, [312] 

Tacubaya (ta, kdo ba'ya), Mexico, 

91 231 

Tallahassee, Fla. 

Tamarack mine, mO p2 

Tangents, natural, table of, |311| 
Tasmania, |81| 
Taurus, [293l [2961 



Tegucigalpa, Honduras, [83] 
Telegraphic time signals, [6i 
Tennessee, |89| 
Texas, |88l |89l [244 


288 291 

Thales (tha'lez), 
Thompson, A. H., |244J 
Thucydides (thu sid'i dez) 
Tidal wave, bore, etc 
Tides, [176}[T89| [279[ ' 
Tientsin (teen'tsen), China 
Tiers of townships 
Time (see Glossary, 
parent solar, |61| 
ball, [70] [81] 
confusion, [64] [72}{75 
how determined, 1681 

184 188 


232 235 236 

p. 320), ap 

in various countries, |80||87| 
local, [63] 


68 72 



Times, London 
Titicaca, Lake, 
Todd, David 
Toga Is., [83] 
Tokyo, Japan, [89 
Toledo, O 
Tonga Is., 



, Peru, [85] 
107] [287] 




Township, [230][237[ 
Transit instrument, 
Transvaal, |81| 
Trenton, N. J 
Tropics, |151[ [ 
Tunis, [82] 
Turkey, pi |88| |.l 43 1.45 

., 92 






Turkish calendar, 143H145 
Tutuila (too twe'la), 

86 97 

Twilight, [162}|T66] 


Unequal heating, 169 

United States, [29H32 
[501 [6 

34] [40] [47] 

68] [70] [76] [86] [89] [97] 
99] [T02]|T03l [T27] [T29] [TFT] 
T68] [155] [225] [227f[2371 [2881 

United States Coast and Geodetic 

Survey, [29] [31] [34] [243] [245] 

United States Geological Survey, 

[29] [2|3] [24^] 
United States Government Land 

Survey, [227] [237] 
United States Naval Observatory, 

[68}{72] [79] 
University of California, 156 

253 259 



University of Chicago, |251 |300 
Ur, ancient Chaldean city, 
Uranus (u'ra nus) 
Ursa Major, [9] 
Uruguay, [86] [88] 
Utah, [9ll |244l 


Valparaiso (val pa ri'so), Chile, 81 

Van der Grinten, Alphons, |208[ 
Velocity of rotation, |56[ 

Venus, [183 


E5m E56 

266] [276] 


Vernal equinox, see 
Vertical ray of sun, |147[ |148| [152] 
[155] [L56J [165] ~ 

Vibrations, color, 107[ 
Victoria, Australia, 18 1 1 

166] [312 



Vincocaya (vin ko ka'ya) , 


Virginia, |92| 

Virginia City, Nev., [92] 

Virgo, [294 


Voltaire, E73 

Volume of earth, 13091 

Wady-Halfa (wa'de hal'fa) , Egypt, 

Wallace, Kan., \33\ 

Wandering of the poles, 287 
Washington, [92] 

Washington, D. C, [34] gOj [66] [70j 
[T25] [23T] 


Washington, D. C, 92 

Washington, George, 139 


Watch, to set by sun, 129 
Weight, see |Gravity] 
Wellington, New Zealand, 72 
West Virginia, [91] 
Western European time, |67[ 
What keeps the members of the 
solar system in their orbits?, 


Wheeling, W. Va. 
Wilhelm II., Emperor, |76| 
Wilmington, Del., [92] 
Winona, Minn., [92l 

Winter constellations, |1 12 
Wisconsin, [6§ [Hi 
Woodward, R. S 

89UT04J [232 


World Almanac, |124| 
Wyoming, [89] 

Xico, Mexico, [84] 


Yaeyama (ye ya'ma) Is 
Yaqui River, Mexico, 84] 


Year, [133] [134] [285] [309] [321] 

Young, C. A. 


Youth's Companion, |71[ |153[ |166 
Ysleta (is la'ta), Tex 


Zikawei (zi ka'we) , China 



Zones, [153] [255>[259] [261] [263 






End of Project Gutenberg's Mathematical Geography, by Willis E. Johnson 


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