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Full text of "Elementary mathematical astronomy, with examples and examination papers"

tCbe TIim\>er0it tutorial Scried. 

ELEMENTARY 

. MATHEMATICAL ASTRONOMY, 

EXAMPLES AND EXAMINATION PAPERS. 



C. W. 0. ^ABLOW, M.A., B.Sc., 

GOLD MEDALLIST IN MATHEMATICS AT LONDON M.A., 

SIXTH WRANGLER, AND FIRST CLASS FIRST DIVISION PART II. MATHEMATICAL 
TRIPOS, CAMBRIDGE, 

AND 

GK H. BBYAN, D.So., M.A., F.E.S., 

SMITH'S PRIZEMAN, LATE FELLOW OK ST. PETER'S COLLEGE, CAMBRIDGE, 

JOINT AUTHOR OF " COORDINATE GEOMETRY, PART I.," " THE TUTORIAL ALGEBRA, 

ADVANCED COURSE," ETC. 

Third Impression (Second Edition). 




LONDON: W. B. OLIVE, 



(University Correspondence College Press], 

13 BOOKSELLEB.S Row, STKAND, W.C. 
1900. 



p 






PREFACE TO THE FIRST EDITION. 



FOR some time past it has been felt that a gap existed between 
the many excellent popular and non-mathematical works on As- 
tronomy, and the standard treatises on the subject, which involve 
high mathematics. The present volume has been compiled with 
the view of filling this gap, and of providing a suitable text-book 
for such examinations as those for the B.A. and the B.Sc. degrees of 
the University of London. 

It has not been assumed that the reader's knowledge of mathe- 
matics extends beyond the more rudimentary portions of Geometry, 
Algebra, and Trigonometry. A knowledge of elementary Dynamics 
will, however, be required in reading the last three chapters, but 
all dynamical investigations have been left till the end of the book, 
thus separating dynamical from descriptive Astronomy. 

The principal properties of the Sphere required in Astronomy 
have been collected in the Introductory Chapter ; and, as it is 
impossible to understand Kepler's Laws without a slight knowledge 
of the properties of the Ellipse, the more important of these have 
been collected in the Appendix for the benefit of students who have 
not read Conic Sections. 

All the more important theorems have been carefully illustrated 
by worked-out numerical examples, with the view of showing how 
the various principles can be put to practical application. The 
authors are of opinion that a far sounder knowledge of Astronomy 
can be acquired with the help of such examples than by learning 
the mere bookwork alone. 

Feb. 1st, 1892. 



PREFACE TO THE SECOND EDITION. 



THE first edition of Mathematical Astronomy having run out of 
print in less than eight months, we have hardly considered it 
advisable to make many radical changes in the present edition. 
We have, however, taken the opportunity of adding several notes at 
the end, besides answers to the examples, which latter will, we 
hope, prove of assistance, especially to private students ; our readers 
will also notice that the book has been brought up to date by the 
inclusion of the most recent discoveries. At the same time we 
hope we have corrected all the misprints that are inseparable from 
a first edition. Our best thanks are due to many of our readers for 
their kind assistance in sending us corrections and suggestions. 

Nov. 1st, 1892. 



CONTENTS. 



INTRODUCTORY CHAPTER. 

PAOB 

ON SPHERICAL GEOMETRY i 

Definitions ii 

Properties of Great and Small Circles iii 

On Spherical Triangles v 

CHAPTER I. 
THE CELESTIAL SPHERE. 

/Sect. I. Definitions Systems of Coordinates 1 

II. The Diurnal Rotation of the Stars 13 

III. The Sun's Annual Motion in the Ecliptic 

The Moon's Motion Practical Applications 20 

CHAPTER II. 
THE OBSERVATORY. 

Sect. I. Instruments adapted for Meridian Observations 35 
II. Instruments adapted for Observations off the 

Meridian 54 

CHAPTER III. 
THE EARTH. 

Sect. I. Phenomena depending on Change of Position 

on the Earth 63 

II. Dip of the Horizon 73 

III. Geodetic Measurements Figure of the Earth 77 

CHAPTER IV. 

THE SUN'S APPARENT MOTION IN THE ECLIPTIC. 

Sect. I. The Seasons 87 

II. The Ecliptic 99 

III. The Earth's Orbit round the Sun 105 

CHAPTER V. 
ON TIME. 

^/Sect. I. The Mean Sun and Equation of Time 115 

II. The Sun-dial 125 

III. Units of Time The Calendar 127 

IV. Comparison of Mean and Sidereal Times 129 



CONTENTS. 

CHAPTER VI. 

PACK 

ATMOSPHERICAL REFRACTION AND TWILIGHT 140 

CHAPTER VII. 
THE DETERMINATION OF POSITION ON THE EARTH. 

Sect. I. Instruments used in Navigation 153 

^X, II. Finding the Latitude by Observation 102 

^ HI. To find the Local Time by Observation 171 

IV. Determination of the Meridian Line 175 

CXJ, V. Longitude by Observation 177 

VI. Captain Sumner's Method 187 

CHAPTER VIII. 
THK MOON. 

Sect. I. Parallax The Moon's Distance and Dimensions 191 
II. Synodic and Sidereal Months Moon's Phases 

Mountains on the Moon 200 

III. The Moon's Orbit and Rotation 209 

CHAPTER IX. 
ECLIPSES. 

Sect. I. General Description of Eclipses 219 

,, II. Determination of the Frequency of Eclipses 224 
III. Occultations Places at which a Solar Eclipse 

is visible 232 

CHAPTER X. 
THE PLANETS. 

Sect. I. General Outline of the Solar System ... ... 238 

II. Synodic and Sidereal Periods Description of 
the Motion in Elongation of Planets, as 

seen from the Earth Phases 244 

III. Kepler's Laws of Planetary Motion 253 

IV. Motion relative to Stars Stationary Points ... 258 
V. Axial Rotations of Sun and Planets 264 

CHAPTER XL 

THE DISTANCES OF THE SUN AND STARS. 

Sect. I. Introduction Determination of the Sun's 
Parallax by Observations of a Superior 

Planet at Opposition 267 

II. Transits of Inferior Planets 271 

,, III. Annual Parallax, and Distances of the Fixed 

Stars 283 

IV. The Aberration of Light ... 293 



CONTENTS. 

DYNAMICAL ASTRONOMY. 

CHAPTER XII. PAOR 

THE ROTATION OF THE EARTH 315 

CHAPTER XIII. 
THE LAW OP UNIVERSAL GRAVITATION. 

Sect. I. The Earth's Orbital Motion Kepler's Laws 

and their Consequences 337 

II. Newton's Law of Gravitation Comparison of 

the Masses of the Sun and Planets 352 

III. The Earth's Mass and Density 362 

CHAPTER XIV. 
FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 

Sect. I. The Moon's Mass Concavity of Lunar Orbit... 371 

II. The Tides 375 

,, III. Precession and Nutation 392 

IV. Lunar and Planetary Perturbations 406 

NOTES. 

Diagram for Southern Hemisphere 421 

The Photochronograph 421 

Effects of Dip, &c., on Rising and Setting 422 

APPENDIX. 

Properties of the Ellipse 423 

Table of Constants 426 

ANSWERS TO EXAMPLES AND EXAMINATION QUESTIONS 428 

INDEX 434 




INTRODUCTORY CHAPTER, 



ON SPHERICAL GEOMETRY. 

Properties of the Sphere which will be referred to in the course of the 

Text. 

(1) A Sphere may be defined as a surface all points on which are 
at the same distance from a certain fixed point. This point is the 
Centre, and the constant distance is the Radius. 

(2) The surface formed by the revolution of a semicircle about 
its diameter is a sphere. For the centre of the semicircle is kept 
fixed, and its distance from any point on the surface generated will 
be equal to the radius of the semicircle. 




FIG. 1. 

(3) Let PqQP' be any position of the revolving semicircle whose 
diameter PP' is fixed. Let OQ be the radius perpendicular to PP', 
Cq any other line perpendicular to PP', meeting the semicircle in 
q. (We may suppose these lines to be marked on a semicircular disc of 
cardboard.) As the semicircle revolves, the lines OQ, Cgwill sweep 
out planes perpendicular to PP', and the points Q, q will trace out 
in these planes circles HQRK, hqrJc, of radii OQ, Cq respectively. 
From this it may readily be seen that Every plane section of a 
sphere is a circle, 

4-STKON, 5 



ii 



ASTRONOMY. 



Definitions. 

(4) A great Circle of a sphere is the circle in which it is cut by 
any plane passing through the centre (e.g., HQRK, PqQP' or PrRP ). 
A small circle is the circle in which the sphere is cut by any plane 
not passing through the centre (e.g., hqrk). 

(5) The axis of a great or small circle is the diameter of the 
sphere perpendicular to the plane of the circle. The poles of the 
circle are the extremities of this diameter. (Thus, the line PP is 
the axis, and P, P' are the poles of the circles HQK and hqJc). 

(6) Secondaries to a circle of the sphere are great circles passing 
through its poles. (Thus, PQP' and PRP" are secondaries of the 
circles HQK, hqk). 




FIG. 2. 

(7) The angular distance between two points on a sphere is 
measured by the arc of the great circle joining them, or by the angle 
which this arc subtends at the centre of the sphere. Thus, the dis- 
tance between Q and Bis measuredeither by the arc QE, or by the angle 
QOR. Since the circular measure of L QOR = (arc Qft) -f- (radius of 
sphere), it is usual to measure arcs of great circles by the angles 
which they subtend at the centre. This remark does not apply to 
small circles. 

(8) The angle between two great circles is the angle between 
their planes. Thus, the angle between the circles PQ, PR is the angle 
between the planes PQP', 7'EP'. It is called "the angle QPR." 

(9) A spherical triangle is a portion of the spherical surface 
bounded by three arcs of gr.eat_circles. Thus, in Fig. 2, PQR is a 
spherical triangle, but Pqr is not a spherical triangle, because qr is 
not an arc of a great circle. We may, however, draw a great circle 
passing through q and r, and thus form a spherical triangle Pqr. 



SPHERICAL GEOMETRY. ill 

Properties of Great and Small Circles. 

(10) All points on a small circle are at a constant (angular)) 
distance from the pole. 

For, as the generating semicircle revolves about PP 7 , carrying g 
along the small circle hk, to r, the arc Pq = arc Pr, and Z POq = L POr. 

The constant angular distance Pq is called the spherical, or 
angular radius of the small circle. The pole P is analogous to the 
centre of a circle in plane geometry. 

(11) The spherical radius of a great circle is a quadrant, or, 
All points on a great circle are distant 90 from its poles. 

For, as Q, by revolving about PP', traces out the great circle 
HQRK, we have L POQ = L POR = 90, and therefore, PQ, PE are 
quadrants. 

(12) Secondaries to any circle lie in planes perpendicular to 
the plane of the circle. 

For PP' is perpendicular to the planes of the circles HQK, liqk, 
therefore any plane through PP / , such as PQP' or PEP', is also per- 
pendicular to them. 

(13) Circles which have the same axis and poles lie in parallel, 
planes. For the planes HQK, hqk are parallel, both being perpen- 
dicular to the axis PP'. Such circles are often called parallels. 

(14) If any number of circles have a common diameter, their 
poles all lie on the great circle to which they are secondaries, and 
this great circle is a common secondary to the original circles. 

For if OA is the axis of the circle PQP', then OA is perpendicular- 
to POP'. Hence, if the circle PQP 7 revolves about PP', A traces out. 
the great circle HQRK, of which P, P 7 are poles. We likewise see that 

(15) If one great circle is a secondary to another, the latter is 
also a secondary to the former. 

This is otherwise evident, since their planes are perpendicular. 

(16) The angle between two great circles is equal to 

(i.) The angle between the tangents to them at their points 

of intersection ; 
(ii.) The arc which they intercept on a great circle to which 

they are both secondaries ; 
(iii.) The angular distance between their poles. 
Let Ft, Pu be the tangents at P to the circles PQ, PE, and let A, B 
bo the poles of the circles. If we suppose the semicircle PQP' to 
revolve about PP' into the position PEP', the tangent at P will 
revolve from Pt to Pu, the radius perpendicular to OP will revolve 
from OQ to 07?, and the axis will revolve from OA to OB. All these 
lines will revolve through an angle equal to the angle between 
the planes PQP', PRP / , and this is the angle QPE between the 
circles (Def. 8). BLenee, 

le between circles PQ, PR = L tPu = L QOR 



{y ASTEONO^TT. 

(17) The arc of a small circle subtending a given angle at the 
pole is proportional to the sine of the angular radius. 

Let qr be the arc of the small circle hqrJc, subtending L qPr at P, 
and let G be the centre of the circle. Evidently L qCr = L QOR 
(since Cq, Gr are parallel to OQ, OB). Hence, the arcs qr, QR are 
proportional to the radii Cq, OQ, 

. arc qr = G = Gq_ = ghl pQq = gin p^ 

arc QR OQ Oq 

But QR is the arc of a great circle subtending the same angle at the 
pole P hence the arc qr is proportional to sin Pq, as was to be shown. 
Since qQ = 90 - PQ, therefore sin Pq - cos gQ, so that the arc qr is 
proportional to the cosine of the angular distance of the small circle 
(jr from the parallel great circle QR. 




FIG. 3. 



FIG. 4. 



(18) Comparison of Plane and Spherical Geometry. 

It may be laid down as a general rule that great circles and small 
circles on a sphere are analogous in their respective properties to 
straight lines and circles in a plane. Thus, to join two points on a 
sphere means to draw the great circle passing through them. 

Secondaries to a great circle of the sphere are analogous to per- 
pendiculars on a straight line. The distance of a point from any 
great circle is the length of the arc of a secondary drawn from the 
point to the circle. Thus, rR is the distance of the point r from the 
great circle HQRK. 



SPHEEICAL GEOMETftf. V 

On Spherical Triangles* 

(19) Parts of a Spherical Triangle. A spherical triangle, like a 
plane triangle, has six parts, viz., its three sides and its three angles. 
The sides are generally measured by the angles they subtend at the 
centre of the sphere, so that the six parts are all expressed as angles. 

Any three given parts suffice to determine a spherical triangle, 
but there are certain " ambiguous cases " when the problem admits 
of more than one solution. The formulge required in solving 
spherical triangles form the subject of Spherical Trigonometry, 
and are in every case different from the analogous f ormulaj in Plane 
Trigonometry. There is this further difference, that a spherical 
triangle is completely determined if its three angles are given. 

Thus, two spherical triangles will, in general, be equal if they 
have the following parts equal : 



(i.) Three sides. 

(ii.) Two sides andincluded angle. 

(iii.) Two sides and one opposite 

angle. 



(iv.) Three angles, 
(v.) Twoanglesandadjacentside. 
(vi.) Two angles and one opposite 
side. 



Cases (iii.) and (vi.) may be ambiguous. 

(20) Right-angled Triangles. If one of the angles is a right 
angle, two of the remaining five parts will determine the triangle. 

(21) Triangle with two right angles. The properties of a 
spherical triangle, such as PQR, Fig. 3, in which one vertex P is 
the pole of the opposite side QR, are worthy of notice. Here two 
of the sides, PQ, PR, are quadrants, and two angles Q, R are right 
angles. The third side QR is equal to the opposite angle QPR, 

(22) Triangle with, three right angles. If, in addition, the angle 
QPR is a right angle (Fig. 4), QR will be a quadrant. The triangle 
PQR will, therefore, have all its angles right angles, and all its sides 
quadrants, and each vertex will be the pole of the opposite side. 

The planes of the great circles forming the sides, are three planes 
through the centre mutually at right angles, and they divide the 
surface of the sphere into eight of these triangles ; thus the area of 
each triangle is one-eighth of the surface of the sphere. 

(23) The three angles of a spherical triangle are together 
greater than two right angles. 

[For proof, see any text-book on Spherical Geometry.] 

(24) If the sides of a spherical triangle, when expressed as angles, 
are very small, so that its linear dimensions are very small com- 
pared with the radius of the sphere, the triangle is very approxi- 
mately a plane triangle. 

Thus, although the Earth's surface is spherical, a triangle whose 
sides are a few yards in length, if traced on the Earth, will not be 
distinguishable from a plane triangle. If the sides are several 
miles in length, the triangle will still be very nearly plane. 



vi AJSTKONOMY. 

(25) Any two sides 6f a spherical triangle are together 
greater than the third side. For if we consider the plane angles 
which the sides subtend at the centre of the sphere, any two of 
these are together greater than the third, by Euclid XL, 20. 

(26) The following application of (25) is of great use in astronomy, 
and is analogous to Euclid III., 7, 8. 

Let AHBK be any given great or small circle whose pole is P, 
Zany other given point on the sphere, and let the great circle ZP 
meet the given circle in the points A, B. Then A, B are the two 
points on the given circle whose distances from Z are greatest and 
least respectively. 

For let H be any other point on the circle. Join ZH, HP. 

Then, in spherical A ZPH, ZP + PH> ZH. But PH = PA ; 

/. ZP + PA > ZH, 
i.e., ZA>ZH. 

Also, if Z is on the opposite side of the circle to P, then 

ZII+PH>PZ', .:ZH + PB>PZ; .:ZH>PZ-PB, 
i.e., ZH>ZB. 




If Z' be a point on the same side of the circle as P, then PZ' + Z'H 
>PH. But PH - PB. .'. PZ'-t Z'H^PB. 

.-. Z'H>PB-PZ', 
i.e., Z'H>Z'B, as before. 

lie nee, A is further from Z, Z', and B is nearer to Z, Z', than any 
other point on the circle. 

(27) If H, K are the two points on the circle equidistant from Z, 
the spherical triangles ZPH, ZPK have ZP common, ZH = ZK (by 
hypothesis^), and PH = PK [by (10)], hence they are equal in all 
respects ; thus L ZPH = L ZPK, and L PZH = L PZK. 

Hence PH, PK are equally inclined to PB, as are also ZH, ZK. 

Similar properties hold in the case of the point Z'. These pro- 
perties are of frequent uw. 



ASTRONOMY. 



CHAPTEE I. 

THE CELESTIAL SPHERE. 

SECTION I. Definitions Systems of Co-ordinate*. 

1 . Astronomy is the science which deals with the celestial 
bodies. These comprise all the various bodies distributed 
throughout the universe, such as the Earth (considered as a 
whole), the Sun, the Planets, the Moon, the comets, the fixed 
stars, and the nebulae. It is convenient to divide Astronomy 
into three different branches. 

The first may be called Descriptive Astronomy. It is 
concerned with observing and recording the motions of the 
various celestial bodies, and with applying the results of 
such observations to predict their positions at any subsequent 
time. It includes the determination of the distances, and the 
measurement of the dimensions of the celestial bodies. 

The second, or Gravitational Astronomy, is an appli- 
cation of the principles of dynamics to account for the motions 
of the celestial bodies. It includes the determination of their 
masses. 

The third, called Physical Astronomy, is concerned 
with determining the nature, physical condition, temperature, 
and chemical constitution of the celestial bodies. 

The first branch has occupied the attention of astronomers 
in all ages. The second owes its origin to the discoveries of 
Sir Isaac Newton in the seventeenth century ; while the 
third branch has been almost entirely built up in the present 
century. 

In this book we shall treat exclusively of Descriptive and 
Gravitational Astronomy. 



ASTRONOMY. 



: -;2: :The ;C.elesti.al Sphere. On observing the stars it is 
' not^ 'difficult to imagine that they are bright points dotted 
about on the inside of a hollow spherical dome, whose centre 
is at the eye of the observer. It is impossible to form any 
direct conception of the distances of such remote bodies ; all 
we can see is their relative directions. Moreover, mof-t 
astronomical instruments are constructed to determine only 
the directions of the celestial bodies. Hence it is important 
to have a convenient mode of representing directions. 




FIG. 6. 

The way in which this is done is shown in Figure 6. Let 
be the position of any observer, A, , C, &c., any stars or 
other celestial bodies. About 0, as centre, describe a sphere 
with any convenient length as radius, and let the lines joining 
to the stars A, J3, C meet this sphere in a, ft, c respectively. 
Then the points a, I, c will represent, on the sphere, the 
directions of the stars A, H, C, for the lines joining these 
points to will pass through the stars themselves. In this 
manner we obtain, on the sphere, an exact representation of 
the appearance of the heavens as seen from 0. Such a 
sphere is called the Celestial Sphere. 

This sphere may be taken as the dome upon which the stars 
appear to lie. But it must be carefully borne in mind that 
the stars do not actually lie on a sphere at all, and that they 
are only so represented for the sake-of convenience. 



THE CELESTIAL SPHERE. 

3. Use of the Globes. The representation of directions 
of stars by points on a sphere is well exemplified in the old- 
fashioned star globes. Such a globe may be used as the 
observer's celestial sphere ; but it must be remembered that 
the directions of the stars are the lines joining the centre to 
the corresponding points on the sphere ; for in every case the 
observer is supposed to be at the centre of the celestial 
sphere. 

The properties given in the Introduction on Spherical Geo- 
metry are applicable to the geometry of the celestial sphere. 
A knowledge of thorn will be assumed in what follows. 

4. Angular Distances and Angular Magnitudes. 

Any plane through the observer will be represented on the 
celestial sphere by a great circle. The arc of the great circle 
a b (Fig. 6) represents the angle a 01 or A OB which the stars 
A, subtend at 0. This angle is generally measured in 
degrees, minutes, and seconds, and is called the angular 
distance between the stars. This angular distance must 
not be confused with their actual distance AB. In the same 
way, when we are dealing with a body pf perceptible dimen- 
sions, such as the Sun or Moon (DF, Fig. 6), we shall define 
its angular diametsr as the angle DOF, subtended by a 
diameter at the observer's eye. This angular diameter is 
measured by the arc df of the celestial sphere, that is, by the 
diameter of the projection of the body on the celestial sphere. 
From the figure it is evident that 



Od 01)' 

Since DF is the actual linear diameter of the body, mea- 
sured in units of length, the last relation shows us that the 
angular diameter (df) of a body varies directly as its linear 
diameter DF, and inversely as OD, the distance of the body 
from the observer's eye. 

As the eye can only judge of the dimensions of a body 
from its angular magnitude, this result is illustrated by the 
1'act that the nearer an object is to the eye the larger it looks, 
and vice versd. Thus, if the distance of the object be doubled, 
it will only look half as large, as may be easily verified. 




4 ASTRONOMY. 

5. The Directions of the Stars are very approxi- 
mately independent of the Observer's Position on 
the Earth. 

This is simply a consequence of the enormously great dis- 
tances of all the stars from the Earth. Thus, 
let x (Fig. 7) denote any star or other celestial 
body, S, JZtwo different positions o^ the observer. 
If the distance SJ be only a very small fraction 
of the distance Sx, the angle Ex 8 will be very 
small, and this angle measures the difference be- 
tween the directions of x as seen from ^and from 8. 

In illustration, if we observe a group of objects 
a mile or two off, and then walk a few feet in any 
direction, we shall observe no perceptible change FIG. 7. 
in the apparent directions or relative positions of the objects. 

If Ex be drawn parallel to Sx, the angle xEx will be 
equal to ExS, and will therefore be very small indeed. 
Hence, Ex will very nearly coincide in direction with Ex'. 
Thus, considering the vast distances of the stars, we see that 

The lines joining a Star to different points of the 
Earth may be considered as parallel.* 

The stars will, therefore, always be represented by the 
same points on a star globe, or celestial sphere, no matter 
what be the position of the observer. The great use of the 
celestial sphere in astronomy depends on this fact. 

6. Motion of Meteors. The projection of bodies on the 
celestial sphere is well illustrated by the apparent motion 
of a swarm of meteors. Where such a swarm is moving 
uniformly, all the meteors describe (approximately) parallel 
straight lines. II we draw planes through these lines and 
the observer, they will intersect in a common line, namely, 
the line through the observer parallel to the direction of the 
common motion of the meteors. The planes will, therefore, 
cut the celestial sphere in great circles, having this line as 
their common diameter. These great circles represent the 
apparent paths >i (he meteors on the celestial sphere. The 
paths appear, therefore, to radiate from a common point, 
namely, one of the extremities of this diameter. 

This point is called the Radiant, and by observing its 
position the direction of motion of the meteors is determined. 

* This is not true in the case of the Moon. 



tHE CELESTIAL StHE&E. 6 

7. Zenith and Nadir. Horizon. If, through the 
observer, a line be drawn in the direction in which gravity 
acts (i.e., the direction indicated by a plumb-line), it will meet 
the celestial sphere in two points. One of these is vertically 
above the observer, and is called the Zenith; the other is 
vertically below the observer, and is called the Nadir. (Fig. 
6, and Z, N, Fig. 8.) 

If the plane through the observer parallel to the surface 
of a liquid at rest be produced, it will cut the celestial 
sphere in a great circle. This great circle is called the 
Celestial Horizon. (Fig. 6, and sEnW, Fig. 8.) 

It is proved in Hydrostatics that the surface of a liquid at 
rest is a plane perpendicular to the direction of gravity. 
Hence, the celestial horizon is the great circle whose poles 
are the zenith and nadir. "We might have defined the 
horizon by this property. 

From the above definition, it is evident that, to an observer 
whose eye is close to the surface of the ocean, the celestial 
horizon forms the boundary of the visible portion of the 
celestial sphere. On land, however, the boundary, or visible 
horizon (as it is called), is always more or less irregular, 
owing to trees, mountains, and other objects. 

8. Diurnal Motion of the Stars. If we observe the 
sky at different intervals during 

the night, we shall find that the 

stars always maintain the same 

configurations relative to one 

another, but that their actual 

situations in the sky, relative to 

the horizon, are continually 

changing. Some stars will set 

in the west, others will rise in 

the east. One star which is 

situated in the constellation called 

the l< Little Bear," remains almost FlG - 8 - 

fixed. This star is called Polaris, or the Pole Star. All the 

other stars describe on the celestial sphere small circles 

(Fig. 8) having a common pole P very near the Pole Star, 

and the revolutions are performed in the same period of time, 

namely, about 23 hours 56 minutes of our ordinary time. 




6 



ASTEONOMt. 



9. Celestial Poles, Equator, and Meridian. The 

common motion of the stars may most easily be conceived by 
imagining them to be attached to the surface of a sphere 
which is made to revolve uniformly about the diameter PP'. 

The extremities of this diameter are called the Celestial 
Poles. That pole, P, which is above the horizon in northern 
latitudes is called the North Pole, the other, P\ is called 
the South Pole. 

The great circle, JEQR W, having these two points for its 
poles, is called the Celestial Equator. It is, therefore, the 
circle which would be traced out by the diurnal path of a 
star distant 90 from either pole. 




The Meridian is the great circle (PZP'N, Fig. 9) passing 
through the zenith and nadir and the celestial poles. It cuts 
both the horizon and equator at right angles [by Spher. 
Geom. (12), since it passes through their poles]. 



THE CELESTIAL SPHEKE. 7 

10. The Cardinal Points. The East and West 
Points (J, W, Eig. 9) are the points of intersection of the 
equator and horizon. The North and South Points 

(&, S) are the intersections of the meridian with the horizon. 

Verticals. rSecondaries to the horizon, i.e., great circles 
through the zenith and nadir., are called Vertical Circles, 
or, briefly, Verticals. Thus, the meridian is a vertical. 
The Prime Vertical is the vertical circle (ZENTF) passing 
through the east and west points. 

Since P is the pole of the circle QERW, and ^is the pole 
of nEsWy therefore E, W are the poles of the meridian 
PZP'N. Hence the horizon, equator, and prime vertical 
which pass through E, W, are all secondaries to the meridian ; 
they therefore all cut the meridian at right angles. 

11. Annual Motion of the Sun. The Ecliptic. 

The Sun, while participating in the general diurnal rotation 
of the heavens, possesses, in addition, an independent 
motion of its own relative to the stars. 

Imagine a star globe worked by clockwork so as to revolve 
about an axis pointing to the celestial pole in the same peri- 
odic time as the stars. On such a moving globe the directions 
of the stars will always be represented by the same points. 
During the daytime let the direction of the Sun be marked on 
the globe, and let this process be repeated every day for a year. 
We shall thus obtain on the globe a representation of the 
Sun's path relative to the stars, and it will be found that 

(i.) The Sun moves from west to east, and returns to the 
same position among the stars in the period called a year ; 

(ii.) The relative path on the celestial sphere is a great 
circle, inclined to the equator at an angle of about 23 27f. 

This great circle (CTL ===, Fig. 9) is called the Ecliptic. 
"We may, therefore, briefly define the ecliptic as the great 
circle which is the trace, on the celestial sphere, of the Sun's 
annual path relative to the stars. 

The intersections of the ecliptic and equator are called 
Equinoctial Points. One of them is called the First 
Point of Aries ; this is the point through which the Sun 
passes when crossing from south to north of the equator, and 
it is usually denoted by the symbol T The other is called 
the First Point of Libra, and is denoted by the symbol =0=, 



ASTKONOMY. 



12. Coordinates. In Analytical Geometry, the position 
of a point in a plane is denned by two coordinates. In like 
manner, the position of a point on a sphere may be denned by 
means of two coordinates. Thus, the position of a place on 
the Earth is denned by the two coordinates, latitude and 
longitude. For fixing the positions of celestial bodies, the 
following different systems of coordinates are used. 

13. Altitude or Zenith Distance and Azimuth. Let 
Fig. 10 represent the celestial sphere, seen from overhead, and 
lot x be any star. Draw the vertical circle ZxX. Then the 
position of x may be defined by either of the following pairs 
of coordinates, which are analogous to the Cartesian and 
polar coordinates of a point in a plane respectively : 

(a) The arc s X and the arc Xx ; 

(b) The arc Zx and the angle sZx. 

Practically, however, the two systems are equivalent ; for, 
since Z is the pole of sX, ZX = 90, therefore 

Zx = 90 xXj and angle sZx = arc sX, 




FIG. 10. 

The Altitude of a star (Xx} is its angular distance from 
the horizon, measured along a vertical. 

The Zenith Distance (abbreviation, Z.D.) is its angular 
distance from the zenith (Zx) , or the complement of the altitude. 

The Azimuth (sX or sZx) is the arc of the horizon inter- 
cepted between the south point and the vertical of the star, 
or the angle which the star's vertical makes with the meridian 



THE CELESTIAL SPHERE. 9 

*14. Points Of the Compass. In practical applications of Astro- 
nomy to navigation, it is usual to measure the azimuth in "points" 
and " quarter points " of the compass. The dial plate of a mariner's 
compass is divided into 32 points, by repeatedly bisecting the right 
angles formed by the directions of the four cardinal points. Thus 
each point represents an angle of Hi degrees. The points are again 
subdivided into " quarter points " of 2\ degrees. Starting from the 
north and going round towards the east, the various points are denoted 
as follows : 

N., N. byB., N.N.E., N.E. by N., N.E., N.E. by E., E.N.E., E. by N. 

E., E. byS., E.S.E., S.E. by E., S.E., S.E. by S., S.S.E., S. by E. 

S., S. by W. S.S.W., S.W. by S., S.W., S.W: by W., W.S.W , W. by S. 

W., W. by N., W.N.W. N.W. by W., N.W., N.W. by N., N.N.W., N. by W. 

The quarter points are denoted thus : E.N.B. E. means one 
quarter point to the eastward of E.N.E., that is, 6 points, or 
70 18' 45", from the north point, taken in an easterly direction. 

So, too, S.S.W. W. meafli 2J points, or 28 7' 30' , measured from 
the south point westwards. 

15. Polar Distance, or Declination, and Hour Angle. 

From the pole P, draw through x the great circle PxM-, this 
circle is a secondary to the equator EQ, W. 

Then we may take for the coordinates of x the arc Px and 
the angle sPx. Or we may take the arc x3f, which is the 
complement of Px, and the arc QM, which = angle QPx. 

The North Polar Distance of a star (abbreviation, 
N.P.D.) is its angular distance (Pa;) from the celestial pole. 

The Declination (abbreviation, Decl.) is the angular 
distance from the equator (xM), measured along a secondary, 
and is, therefore, the complement of the N.P.D. 

The great circle PxM through the pole and the star is 
called the star's Declination Circle. 

The Hour Angle of the star (ZPx] is the angle which 
the star's declination circle makes with the meridian. 

The declination may be considered positive or negative, 
according as the star is to the north or south of the equator, 
but it is more usual to specify this by the letter N. or S., as 
the case may be, and this is called the name of the declination. 

The hour angle is generally measured from the meridian 
towards the west, and is reckoned from to 360. 

Either the declination and hour angle or the N.P.D. and 
hour angle may be taken as the two coordinates of a star. 



10 



ASTBONOHY. 



16. Declination and Right Ascension. The position 
of a celestial body is, however, more frequently defined by 
its declination and right ascension. 

'The declination has been already defined, in 15, as the 
angular distance of the star from the equator, measured along 
a secondary. (xM, Fig. 11.) 

The Right Ascension (E.A.) is the arc of the equator 
intercepted between the foot of this secondary and the First 
Point of Aries. Thus, T^, Fig. 11, is the E.A. of the star a:. 

The E.A. of a star is always measured from T eastwards 
reckoning from to 360. Thus the star w Piscium, whose 
declination circle cuts the equator 1 34' 18" west of T, has 
the E.A. 360 1 34' 18", or 358 25' 42". 




FIG. 11. 

17. Celestial Latitude and Longitude. The position 
of a celestial body may also be referred to the ecliptic instead 
of the equator. 

The Celestial Latitude is the angular distance of the 
tody from the ecliptic, measured along a secondary to the 
ecliptic. (Hx, Pig. 11.) 

The Celestial Longitude is the arc of the ecliptic inter- 
cepted between this secondary and the first point of Aries, 
measured eastwards from T- (T#, Pig. 11.) 



tflE CELESTIAL SPHERE. ll 

18. Latitude of the Observer. The celestial latitude 
and longitude, defined in the last paragraph, must not be 
confounded with the latitude and longitude of a place on the 
Earth, as there is no connection whatever between them. 

The Latitude of a place is the angular distance of its 
zenith from the equator, measured along the meridian. 

Thus, in Pig. 1 1 , ZQ, is the latitude of the observer. 

Since PQ nZ 90 ; .-. ZQ = nP, or in other words, 
The latitude of a place is the altitude of the Celestial Pole. 

The complement of the latitude is called the Colatitude. 

Hence, in Pig. 11, PZ is the colatitude of the observer, 
and is the angular distance of the zenith from the pole. 

In this book the latitude of an observer will generally be 
denoted by the symbol /, and the colatitude by c. 

The longitude of a place will be defined in Chapter III. 

19. Obliquity of the Ecliptic. The inclination of the 
ecliptic to the equator is called the Obliquity. In Pig. 11, 
Q T C is the obliquity. As stated in 1 1 , this angle is about 
23 27-'. We shall generally denote the obliquity by i. 

20. Advantages of the Different Coordinate 
Systems. The altitude and azimuth of a celestial body 
indicate its position relative to objects on the Earth. Owing, 
however, to the diurnal motion, they are constantly changing. 

The N.P.D. and hour angle also serve to determine the 
star's position relative to the earth, and have this further 
advantage, that the N.P.D. is constant, while the hour angle 
increases at a uniform rate. 

Since the equator and first point of Aries partake of the 
common diurnal motion of the stars, the declination and right 
ascension of a star are constant. These coordinates are, there- 
fore, the most suitable for tabulating the relative positions of 
the various stars on the celestial sphere. 

The celestial latitude and longitude of a celestial body are 
also unaffected by the diurnal motion. They are most useful in 
defining the positions of the Sun, Moon, planets, and comets, 
for the first always moves in the ecliptic, while the paths 
described by the others are always very near the ecliptic. 

21. Recapitulation. Por the sake of convenient refer- 
ence, we give on the next page a list of all the definitions of 
this chapter, with references to Pigs. 11, 12. 

ASTRON. c 



12 



ASTRONOMY. 



GREAT CIRCLES. 
Horizon, nEsW. 
Equator, EQWR. 
Meridian, ZsZ'n. 
Prime Vertical, ZEZ'W. 



THEIR POLES. 
Zenith, Z-, Nadir, Z '. 
North Pole, P ; South Pole, P. 
East Point, E\ West Point, W. 
NorthPoint, n ; South Point, s. 

Ecliptic, T i:Z ; Equinoctial Points, T, =2=, viz. : Eirst 
Point of Aries, T , and Eirst Point of Libra, b ; Yertical of 
Star, ZxX-, Declination Circle of Star, Pxlf. 




FIG. 12. 
COORDINATES. 

Altitude, Xx ; '") 

or Zenith Distance, Zx. ) 
North Polar Distance, Px. 
Declination, MX. 
Celestial Latitude, Hx. 



Azimuth, sX = sZx. 
Hour Angle, QM = ZPx. 



Bight Ascension, T ^ 
Celestial Longitude, 

OTHER ANGLES. Obliquity of Ecliptic (t) CT Q- 
Observer's Latitude (1) = ZQ = nP. Colatitude (c) = PZ. 
Notice that the circles on the remote side of the celestial sphere 
are dotted. 



CELESTIAL SPHEKE. 13 

SECTION II. The Diurnal Rotation of the Stars. 

22. Sidereal Day and Sidereal Time. A Sidereal 

Day is the period of a complete revolution of tlie stars about 
the pole relative to the meridian and horizon. Like the 
common day it is divided into 24 hours (h.), and these are 
subdivided into 60 minutes (m.) of 60 seconds (s.) each. 
The sidereal day commences at "Sidereal Noon," i.e., the 
instant when the first point of Aries crosses the meridian. 

The Astronomical Clock, which is the clock used in 
observatories, indicates sidereal time. The hands should 
indicate Oh. Om. Os. when the first point of Aries crosses the 
meridian, and the hours are reckoned from Oh. up to 24h., 
when T again comes to the meridian and a new day begins. 

From the facts stated in 8, it appears that the sidereal 
day is about 4 minutes shorter than the ordinary day. The 
stars are observed to revolve about the pole at a perfectly 
uniform rate, so that the sidereal day is of invariable length, 
and the angles described by any star about the pole are pro- 
portional to the times of describing them. Thus, the hour 
angle of a star (measured towards the west) is proportional 
to the interval of sidereal time that has elapsed since the star 
was on the meridian. 

Now, in 24 sidereal hours the star comes round again to 
the meridian, after a complete revolution, the hour angle 
having increased from to 360. Hence the hour angle in- 
creases at the rate of 15 per hour. Hence, also, it increases 
15' per minute, or 15" per second. 

The hour angle of a star is, for this reason, generally 
measured by the number of hours, minutes, and seconds of 
sidereal time taken to describe it. It is then said to be 
expressed in time. Thus, 

The hour angle of a star, when expressed in time* 
is the interval of sidereal time that has elapsed 
since the star was on the meridian. 

In particular, since the instant when T is on the meridian 
is the commencement of the sidereal day, we see that 

The sidereal time is the hour angle of the first 
point of Aries when expressed in time. 



14 ASTHONOMY. 

23. To reduce to angular measure any angle ex- 
pressed in time. Multiply ~by 15. The hours, minutes, and 
seconds of time will thus be reduced to degrees, minutes, and 
seconds of angle. 

Conversely, to reduce to time from angular measure 
we must divide by 15, and for degrees, minutes, and seconds, 
write hours, minutes, and seconds. 

EXAMPLES. 1. To find, in angular measure, the hour angle of a 
star at 15h. 21m. 50s. of sidereal time after its transit. The process 
stands thus 

15 21 50 



230 27 30 

/. the angular measure of the hour angle is 230 27' ?0" 
2. To find the sidereal time required to describe 230 27' 30" 
(converse of Ex. 1). 

15 ) 230 27 30 

15 21 50 ; .-. required time = 15h. 21m. 50s. 




24. Transits. The passage of the star across the meri- 
dian is called its Transit. 

Let x be the position of any star in transit (Fig. 13). 

The star's E.A. = T Q or rPQ = hour angle of T 
= sidereal time expressed in angle. 

Hence, the right ascension of a star, when ex- 
pressed in time, is equal to the sidereal time of its 
transit. 

In practice the R.A. of a star is always expressed in time. 
Thus, the R.A. of a Lyrse is given in the tables aa 
18h. 33m. 14-8s., and not as 278 18' 42". 



THE CELESTIAL SPHEEE. 15 

Again, let 2 be the meridian zenith distance Zx, considered 
positive if the -star transits north of the" zenith, d the star's 
north declination Qx, and I the north latitude QZ. Wo 
have evidently - 

Qx = QZ+Zx; 

d = i+*c 

or (star's N. decl.) 

= (lat. of observer) + (star's meridian Z.D.) 

This formula will hold universally if declination, latitude, 
and zenith distance are considered negative when south. 

Hence the R. A. and decl. of a star maybe found by observing 
its sidereal time of transit and its meridian Z.D., the latitude of 
the observatory being known. 

Conversely, if the R.A. and decl. of a star are known, we 
can, by observing its time of transit and meridian Z.D., deter- 
mine the sidereal time and the latitude of the observatory. 

By finding the sidereal time we may set the astronomical 
clock. 

A star whose E.A. and decl. have been tabulated, is called 
a known star. 

In Chapter II. we shall describe an instrument called the 
Transit Circle, which is adapted for observing the times of 
transit and meridian zenith distances of celestial bodies. 

25. General Relation between R.A. and hour 
angle. Let x l (Fig. 13) be any star not on the meridian. 
Then 

z Qp Xl = L QPr- t rP^ = ^ QPr rM] 

hence, if angles are expressed in time, 

(star's hour angle) = (sidereal time) (star's H.A.). 

Hence, given the 11. A. and decl. of a star, we can find its hour 
angle and N.P.D. at any given sidereal time, and by this means 
determine the star's position on the 'observer's celestial sphere. 
Or we can construct the star's position thus On the equator, 
in the westward direction from Q, measure off Q T equal to 
the sidereal time (reckoning 15 to the hour). Prom T east- 
wards, measure f M equal to the star's It. A.; and from 3f, in 
the direction of the pole, measure off Mx l equal to the star's 
declinatiqn. We thus find the star x r 



1 6 ASTRONOMY. 

*26. Transformations. If the R.A. and decl. of a star are 
given, its celestial latitude and longitude may be found, and vice 
versti ; but the calculations require spherical trigonometry. The 
process is analogous to changing the direction of the axes through 
an angle i, in plane coordinate geometry. Again, the Z.D. and 
azimuth may be calculated from the N.F.D. and hour angle, by 
solving the triangle ZPx^ We know the colatitude PZ, Px^ and 
L ZPx t , and we have to determine Zxi and L QZx } (= ISO PZxJ. 

In the last article we showed how to find the hour angle in 
terms of the R.A., or vice versA, the sidereal time being known. 
Hence we see that, given the coordinates of a star referred to one 
system, its coordinates referred to any other of the systems can bo 
calculated at any given instant of sidereal time. 

27. Culmination and Southing of Stars. A celestial 
body is said to culminate when its altitude is greatest or 
least. 

Since the fixed stars describe circles about the pole, it 
readily follows, from Spherical Geometry (26), that a star 
attains its greatest or least zenith distance when on the meridian, 
and, therefore, that its culmination is the same as its transit. 

This is not strictly the case with the Sun, because, owing to 
its independent motion, its polar distance is not constant ; 
hence it does not describe strictly a small circle about the pole. 

When a star transits S. of the zenith it is said to south. 

28. Circumpolar Stars. A Circumpolar Star at any 

place is a star whose polar distance is less than the latitude 
of the place. Its declination must, therefore, be greater 
than the colatitude. 

On the meridian let Px and Px' be measured, each equal to 
the KP.D. of such a star (Fig. 14). Then x and x' will be 
the positions of the star at its transits. Since Px < Pn, both 
x' and x will be above n. Hence, during a sidereal day a cir- 
cumpolar star will transit twice, once above the pole (at x) 
and once below the pole (at x'), and both transits will be 
visible. The two transits are distinguished as the upper 
and lower culminations respectively, and they succeed one 
another at intervals of 12 sidereal hours ( since xPx' = 180). 
The altitude of the star is greatest at upper, and least at 
lower culmination, as may easily be seen from Sph. Geom. 
(26) by considering the zenith distances. Hence the altitude 
is never less than nx, and the star is always above the horizon. 



Since 



THE CELESTIAL SPHEBE. 

nx-nP=Px = Px = nPnaf, 



17 



that is, 

The observer's latitude is half the sum of the 
altitudes of a circumpolar star at upper and lower 
culminations. 

Also, Px \ (nx nx) ; 

that is, 

The Star's N.P.D. is half the difference of its 
two meridian altitudes. 




These results will require modification if the upper culmi- 
nation takes place south of the zenith as at 8. The meridian 
altitude will then be measured by sS, and not nS. Here, 
nS = 180 sS, and we shall, therefore, have to replace the 
altitude at upper culmination by its supplement. 

South Circumpolar Stars. If the south polar dis- 
tance of a star is less than the north latitude of the observer, 
the star will always remain below the horizon, and will, 
therefore, be invisible. Such a star is called a South Cir- 
cumpolar Star. 

EXAMPLE. The constellation of the Southern Cross ( Crux) 
is invisible in Europe, for its declination is 62 30' S ; there- 
fore its south polar distance is 27 30', and it will, therefore, 
pot be visible in north latitudes higher than 27 30'. 



18 



ASTBONOMY. 



29. Rising, Southing, and Setting of Stars. If the 

N. and S. polar distances of a star are both greater than the 
latitude, it will transit alternately above and below the 
horizon. This shows that the star will be invisible during a 
certain portion of its diurnal course. Astronomically, the 
star is said to rise and set when it crosses the celestial 
horizon. 

Let J, V be the positions of any star when rising and setting 
respectively. 




FIG. 15. 



In the spherical triangles Pnb, 

PI = Pb' (each being the star's KP.D.), 
right L Pnb = right L Pnb', 

and Pn is common. 
Hence the triangles are equal in all respects ; therefore 

Z nPb = Z nPb', 
and the supplements of these angles are also equal, that is, 

L sPb = L sPb'. 

But the angle sPb, when reduced to time, measures the 
interval of time taken by the star to get from b to the meri- 
dian, and sPV measures the time taken from the meridian to 
b'. Hence, 

The interval of time between rising and southing 
is equal to the interval between southing and setting. 



THE CELESTIAL SPHERE. 19 

Thus, if , f are the times of rising and setting, and T the 
time of transit, we have T t tfT. 



The time of transit is the arithmetic mean between 
the times of rising and setting. 

In order to facilitate the calculations, tables have been constructed 
giving the values of T t for different latitudes and declinations. 

If the observer's latitude Pn and the star's polar distance Pb are 
known, it is possible (by Spherical Trigonometry) to solve the right- 
angled triangle PZm, and to calculate the angle nPb, and therefore 
also the angle &Ps. This angle, when divided by 15, gives the time 
T t. Moreover, the sidereal time of transit T is known, being equal 
to the star's R.A. Hence the sidereal times of rising and setting can 
be found. 

If the star is on the equator, it will rise at E and set at W. 
Since JSQWis a semicircle, exactly half the diurnal path will 
be above the horizon, and the interval between rising and 
setting will be 12 sidereal hours. If the star is to the north 
of the equator, it will rise at some point b between E and , 
so that 

L IPs > Z JEPs, 

i.e., / bPs > 90, 

and the star will he above the horizon for more than 12 hours. 
Similarly, if the star is south of the equator, it will rise at a 
point c between E and *, and will be above the horizon for 
less than 12 hours. 

Prom the equality of the triangles bPn, b'Pn (Pig. 15), we 
also see that 

nb = nb', and sb = sb'. 

Hence the diameter (ns) of the celestial sphere, joining the 
north and south points, bisects the arc (W) between the 
directions of a star at rising and setting. 

This gives us an easy method of roughly determining, by 
observation, the directions of the cardinal points ; but, owing 
to the usual irregularities in the visible horizon, the methoij 
is not very exac. 



20 



ASTRONOMY. 



SECTION III. The Sun's Annual Motion in the Ecliptic 
The Moon's Motion Practical Applications. 

30. The Sun's Motion in Longitude, Bight Ascen- 
sion and Declination. In 11, we briefly described 
the Sun's apparent motion in the heavens relative to the 
fixed stars. "We defined a Year as the period of a complete 
revolution, starting from and returning to any fixed point 
on the celestial sphere. The Ecliptic was defined as the 
great circle traced out by the Sun's path, and its points of 
intersection with the Equator were termed the First Point 
of Aries and First Point of Libra, or together, the 
Equinoctial Points. 

We shall now trace, by the aid of Pig. 16, the variations 
in the Sun's coordinates during the course of a year, starting 
with March 21st, when the Sun is in the first point of Aries. 
We shall, as usual, denote the obliquity by i, so that 
i = 23 27' nearly. 




FIG. 16. 

On March 21st the Sun crosses the equator, passing 
through the first point of Aries (r). This is the Vernal 
Equinox, and it is evident from the figure that 

Sun's longitude = 0, B.A. = O, Decl. = 0. 
Prom March 21st to June 2 1st the Sun's declination is 
north, and is increasing. 



THE CELESTIAL SPHEEE. 21 

On June 21st the Sun has described an arc of 90 from r 
on the ecliptic, and is at C (Fig. 16). This is called the 
Summer Solstice. If we draw the declination circle 
PCQ, the spherical triangle T OQ is of the kind described in 
Sph. Geom. (21), and CP is a secondary to the ecliptic. 
Hence (Sph. Geom. 26) the Sun's polar distance CP is a 
minimum and therefore its decl. a maximum. 



Also r Q = 90 and CQ = tCrQ = i. Hence 

Sun's longitude = 90, B.A. = 90 - 6h., 
N. Decl. = /, (a maximum). 

From June 21 to September 23 the Sun's declination is 
still north, but is decreasing. 

On September 23rd the Sun has described 180, and is 
at the first point of Libra (=), the other extremity of the 
common diameter of the ecliptic and equator. This is the 
Autumnal Equinox, and we have 

Sun's long. = 180, R.A. = 180 = 12h., Decl. = 0. 

From Sept. 23 to Dec. 22 the Sun is south of the equator, 
and its south declination is increasing. 

On December 22ud the Sun has described 270 from T, 
and is at L (Fig. 16). This is called the Winter Solstice. 
We have t L = 90, and the triangle . RL has two right 
angles at R, L (Sph. Geom. 21). The Sun's polar dis- 
tance LP is a maximum (Sph. Geom. 26), and 

*R = L = 90, LR = / L^R = i. Hence 
Sun's longitude = 270, R.A. = 270 = 18h., 
S. Decl. = i, (a maximum). 

From December 22 to March 21 the Sun's declination is 
still south, but is decreasing. 

Finally, on March 21, when the Sun has performed a com- 
plete circuit of the ecliptic, we have . 

Sun's long. = 360, B.A. = 360 = 24h., Decl. = 0. 

The longitude and R.A. are again reckoned as zero, and 
they, together with the declination, undergo the same cycle 
of changes in the following year. 



22 



ASTEONOMT. 



31. Sun's Variable Motion in R.A. We observe that 
the Sun's right ascension is equal to its longitude four times 
in the year, viz., at the two equinoxes and the two solstices. 

At other times this is not the case. 

For example, between the vernal equinox and summer 
solstice we have T-3f< T$, .'. Sun's E.A. < longitude. 

Hence, even if the Sun's motion in longitude be supposed 
uniform, its R.A. will not increase quite uniformly. There 
is a further cause of the want of uniformity, namely, that 
the Sun's motion in longitude is not quite uniform ; but this 
need not be considered in the present chapter. 




32. Direct and Retrograde Motions. The direction 
of the Sun's annual revolution relative to the stars, i.e., motion 
from west through south to east, is called direct. The 
opposite direction, that of the diurnal apparent motions of the 
stars or revolution from east to west, is called retrograde. 

The revolutions of all bodies forming the solar system, 
with the exception of some comets and one or two small 
satellites, are direct. 

We shall see in Chapter III. that the apparent retrograde 
diurnal motion may be accounted for by the direct rotation 
pf the Earth about its polar axis, 



THE CELESTIAL SPHERE. 23 

33. Equinoctial and Solstitial Points Colures. 

From 30 it appears that the Summer and Winter Solstices 
may be defined as the times of the year when the Sun attains 
its greatest north and south declinations respectively. The 
corresponding positions of the Sun in the ecliptic ((7, Z, 
Fig. 17) are called the Solstitial Points. In the same way 
the Equinoctial Points (T, ) are the positions of the 
Sun at the Vernal and Autumnal Equinoxes when its 
declination is zero. 

The declination circle PTP'^j passing through the equi- 
noctial points, is called the Equinoctial Colure. The 
declination circle PCP'L, passing through the solstitial points, 
is called the Solstitial Colure. The latter passes through 
the poles of the ecliptic (7T, K'). 

34. To find the Sun's Right Ascension and Decli- 
nation. In the "Nautical Almanack,"* the Sun's R.A. 
and declination at noon are tabulated for every day of the 
year. Their hourly variations are also given in an adjoining 
column. To find their values at any time of the day, 
we only have to multiply the hourly variation by the 
number of hours that have elapsed since the preceding noon, 
and add to the value at that noon. 

EXAMPLE. Tfl find the Sun's R.A. and decl. on September 4, 1891 
at 5h. 18m. in^gjs^ afternoon. We find from the Almanack for 1891 
under Septembers : 

Sun's R.A. a*oon = lOli. 52m. 15s., hourly variation 9'04s. 
N. Decl. at noon = 7 12' 12" 55'4" 

(1) RA. at noon = lOh. 52m. 15s. 

Increase in 5h. = 9'04s. x 5 = 45*2 

18m. = 27 



.-. R.A. at 5h. 18m. - lOh. 53m. 3s. 

(2) From the Almanack, decl. is less on September 5, and is 
therefore decreasing. 

N. Decl at noon = 7 12' 12" 
Decrease in 6h. = 55'4" x 5 = 4' 37" \ To be 

18m. - 17") subtracted. 



N. Decl. at 6h. 18m. = 7 T 18 ' 



* Also in " Whitaker's Almanack," which may be consulted with 
advantage. 



24 ASTRONOMY. 

35. Rough Determination of the Sun's R.A. "We 

can, without the "Nautical Almanack," find to within a 
degree or two, the Sun's E. A. on any given date, as follow^ : 

A year contains 365 days. In this period the Sun's E.A. 
increases by 360. Hence its average rate of increase is very 
nearly 30 per month, or 1 per day. 

Knowing the Sun's E.A. at the nearest equinox or solstice, 
we add 1 for every day later, or subtract 1 for every day 
before that epoch. If the E.A. is required in time, we allow 
for the increase at the rate of 2h. per month, or 4m. per day. 

EXAMPLES. 1. To find the Sun's R.A. on January 1st. On 
December 22nd the R.A. = 18h. Hence on January 1st, which is 
ten days later, the Sun's R.A. = 18h. 40m. 

2. To find on what date the Sun's R.A. is lOh. 36m. On Sep- 
tember 23rd the R.A. is 12h. Also 12h.-10h. 36m. = 84m., and 
the R.A. increases Sim. in 21 days. Hence the required date is 21 
days before September 23, i.e., September 2nd, 

36. Solar Time. Apparent Noon is the time of the 
Sun's upper transit across the meridian, that is, in north 
latitudes, the time when the Sun souths. Apparent Mid- 
night is the time of the Sun's transit across the meridian 
below the pole (and usually below the horizon). 

An Apparent Solar Day is the interval between two 
consecutive apparent noons, or two consecutive midnights. 

Like the sidereal day, the solar day is divided into 24 hours, 
which are again divided into 60 minutes of 60 seconds each. 
For ordinary purposes the day is divided into two portions : 
the morning, lasting from midnight to noon ; the evening, 
from noon till midnight ; and in each portion times are 
reckoned from Oh. (usually called 12h.) up to 12h. For 
astronomical purposes we shall find it more convenient to 
measure the solar time by the number of solar hours that 
have elapsed since the preceding noon. Thus, 6.30 A.M. on 
January 2nd will be reckoned, astronomically, as 18h. 30m. 
on January 1st. On the other hand, 12.53 P.M. will be 
reckoned as Oh. 53m., being 53 minutes past noon. 

During a solar day the Sun's hour angle increases from 
to 360. It therefore increases at the rate of 15 per hour. 
Hence 

The apparent solar time = the Sun's hour angle 
expressed in time. 



THE CELESTIAL SPHERE. 25 

At noon the Sun is on the meridian. The sidereal time, 
being the hour angle of T, is the same as the Sun's H.A., i.e., 
Sidereal time of apparent noon Sun's R. A. at noon. 

At any other time, the difference between the sidereal and 
solar times, being the difference between the hour angles of 
T and the Sun, is equal to the Sun's E.A. Hence, as in 
25, we have 
(Sidereal time) (apparent solar time) = Sun's R.A. 

If a and a + x are the right ascensions of the Sun at two 
consecutive noons, then, since a whole day has elapsed between 
the transits, the total sidereal interval is 24h. +#, and exceeds a 
sidereal day by the amount x. But the interval is a solar day. 

Hence, the solar day is longer than the sidereal 
day, and the difference is equal to the sun's daily 
motion in R.A.* 

37. Morning and Evening Stars. Sunrise and 
Sunset. "When a star rises shortly before the Sun, and in 
the same part of the horizon, it is called a Morning Star. 
Such a star is then only visible for a short time before sunrise. 
When a star sets shortly after the Sun, and in the same part 
of the horizon, it is called an Evening Star. It is then 
only visible just after sunset. 

It will be readily seen from a figure, that a star will be a 
morning star if its decl. is nearly the same as the Sun's, while 
its E/.A. is rather less. Similarly, a star will be an evening 
star if its decl. is nearly the same as the Sun's, but its RA. 
somewhat greater. Thus, as the Sun's R.A. increases, the 
stars which are evening stars will become too near the Sun to 
to be visible, and will subsequently reappear as morning stars. 

The times of sunrise and sunset are calculated in the 
manner described in 29. The hour angles of the Sun, when 
crossing the eastern and western horizons, determine the 
intervals of solar time between sunrise, apparent noon, and 
sunset. The two intervals are equal, if the Sun's decl. be 
supposed constant from sunrise to sunset a result very 
approximately true, since the change of decl. is always very 
small. 

* Owing to the sun's variable motion in R. A., the apparent solar day is not quite 
of constant length. In the present chapter, however, it may be regarded as 
approximately constant. 




26 ASTRONOMY. 

38. The Gnomon. Determination of Obliquity of 
Ecliptic. The Greek astronomers observed the Sun's 
motion by means of the Gnomon, an instrument consisting 
essentially of a vertical rod standing in the centre of a hori- 
zontal floor. The direction of the shadow cast by the Sun 
determined the Sun's azimuth, while the length of the shadow, 
divided by the height of the rod, gave the tangent of the 
Sun's zenith distance. To find the meridian line, a circle was 
described about the rod as centre, and the directions of the 
shadow were noted when its extremity just touched the circle 
before and after noon. The sun's Z.D.'s at these two 
instants being equal, their azimuths were evidently (Sph. 
Geom. 27) equal and opposite, and the bisector of the angle 
between the two directions was therefore the meridian line. 

The Sun's meridian zenith distances were then observed 
both at the summer solstice, when the Sun's IS", decl. is i and 
meridian Z.D. least, and at the winter solstice, when the Sun's 
S. decl. is i and meridian Z.D. greatest. Let these Z.D.'s be z l 
and s 2 respectively, and let I be the latitude of the place of 
observation. From 24, we readily see that 
2 t = l-i, 2 2 = Z+t, 

/: *=*(.+*,), * = i(v-i);. 

thus determining both the latitude and the obliquity. 

39. The Zodiac. The position of the ecliptic was defined 
by the ancients by means of the constellations of the Zodiac, 
which are twelve groups of stars, distributed at about equal 
distances round a belt or zone, and extending about 8 on 
each side of the ecliptic. The Sun and planets were observed 
to remain always within this belt. The vernal and autumnal 
equinoctial points were formerly situated in the constellations 
of Aries and Libra, whence they were called the First Point 
of Aries and the First Point of Libra. Their positions are very 
slowly varying, but the old names are still retained. Thus, 
the " First Point of Aries" is now situated in the constel- 
lation Pisces. 

The early astronomers probably determined the Sun's 
annual path by observing the morning and evening stars. 
After a year the same morning and evening stars would be 
observed, and it would be concluded that the Sun performed 
a complete revolution in the year. 



THE CELESTIAL SPHEEE. 27 

40. Motion of the Moon. The Moon describes among 
the stars a great circle of the celestial sphere, inclined to 
the ecliptic at an angle of about 5. The motion is direct, 
and the period of a complete " sidereal " revolution is about 
27 days. 

In this time the Moon's celestial longitude increases by 360. 

"When the Moon has the same longitude as the Sun, it is 
said to be New Moon, and the period between consecutive 
new Moons is called a Lunation. AVhen the Moon has 
described 360 from new Moon, it will again be at the same 
point among the stars ; but the Sun will have moved forward, 
so that the Moon will have a little further to go before it 
catches up the Sun again. Hence the lunation will be rather 
longer than the period of a sidereal revolution, being about 
29 \ days. 

The Age of the Moon is the number of days which have 
elapsed since the preceding new Moon. Since the Moon 
separates 360 from the Sun in 29j days, it will separate at 
the rate of about 12, or more accurately 12-|- , per day, 
or 30' per hour. This enables us to calculate roughly the 
Moon's angular distance from the Sun, when the age of the 
Moon is given, and conversely, to determine the Moon's age 
when its angular distance is given. 

EXAMPLE. On September 23, 1891, the Moon is 20 days old. 
To find roughly its angular distance from the Sun and its longitude 
on that day. 

(1) In one day the Moon separates 12^- from the Sun; therefore, 
in 20 days it will have separated 20 x 121, or 244, and this is the 
required angular distance from the Sun. 

(2) On September 23 the Sun's longitude is 180 ; therefore the 
Moon's longitude is 180 + 244 = 424 = 360 + 64, or 64. 

This method only gives very rough results; for the Moon's 
motion is far from uniform, and the variations seem very 
irregular. 

Moreover, the plane of the Moon's orbit is not fixed, but 
its intersections with the ecliptic (called the Nodes) have a 
retrograde motion of 19 per year. Hence, for rough pur- 
poses, it is better to neglect the small inclination of the Moon's 
orbit, and to consider the Moon in the ecliptic. If greater 
accuracy be required, the Moon's decl. and R.A. may be 
found from the Nautical Almanack. 



28 ASTRONOMY. 

41. Astronomical Diagrams and Practical Applica- 
tions. We can now solve many problems connected with 
the motion of the celestial bodies, such as determining the direc- 
tion in which a given star will be seen from a given place, at 
a given time, on a given date, or finding the time of day at 
which a given star souths at a given time of year. 

"We have, on the celestial sphere, certain circles, such as 
the meridian, horizon, and prime vertical, also certain points, 
such as the zenith and cardinal points, whose positions relative 
to terrestrial objects always remain the same. Besides these, 
we have the poles and equator, which remain fixed, with 
reference loth to terrestrial objects and to the fixed stars. 
"We have also certain points, such as the equinoctial points, 
and certain circles, such as the ecliptic, which partake of 
the diurnal motion of the stars, performing a retrograde 
revolution about the pole once in a sidereal day. Lastly, 
we have the Sun, which moves in the ecliptic, performing 
one retrograde revolution relative to the meridian in a solar 
day, or one direct revolution relative to the stars in a year, 
and whose hour angle measures solar time. 

In drawing a diagram of the celestial sphere, the positions 
of the meridian, horizon, zenith, and cardinal points should 
first be represented, usually in the positions shown in Pig. 
18. Knowing the latitude nP of the place, we find the 
pole P. The points Q, ft, where the equator cuts the meri- 
dian, are found by making PQ = PR = 90 ; and the points 
Q, Ii, with E, W, enable us to draw the equator. 

We now have to find the equinoctial points. How to do 
this depends on the data of the problem. Thus we may 
have given 

(i.) The sidereal time ; 

(ii.) The hour angle of a star of known E.A. and decl ; 
(iii.) The time of (solar) day and time of year. 

In case (i.), the sidereal time multiplied by 15 gives, in 
degrees, the hour angle (Qf) of the first point of Aries. 
Measuring this angle from the meridian westwards, we find 
Aries, and take Libra opposite to it. Any star of known 
decl. and R.A. can be now found by taking on the equator 
= star's R.A., and taking on MP, MX = star's decl. 



THE CELESTIAL SPHERE. 



29 



The ecliptic may be drawn passing through Aries and 
Libra, and inclined to the equator at an angle of about 23 \ 
(just over right angle). As we go round from west to east, or 
in the direct sense, the ecliptic passes from south to north of 
the equator at Aries ; this shows on which side to represent 
the ecliptic. Knowing the time of year, we now find the 
Sun (roughly) by supposing it to travel to or from the 
nearest equinox or solstice about 1 per day from west to east. 
Finally, if the Moon's age be given, we find the Moon by 
measuring 12-i- per day, or 30' per hour eastwards from the 
Sun. 




P' 



FIG. 18. 



In case (ii.), we either know the hour angle, QMoi QPMof. 
a known star (#), or, what is the same thing, the sidereal 
interval since its transit ; or, in particular, it is given that the 
star is on the meridian. Each of these data determines J/~, 
the foot of the star's declination circle. From M we measure 
westwards equal to the star's R.A. This finds Aries. 



80 ASTRONOMY. 

fn case (iii-)> the solar time multiplied by 15 gives the- 
Hun's hour angle QPS in degrees. From the time of year 
we can find the Sun's R.A., TJPS. From these we find 
Q,PT and obtain the position of Aries just as in case (ii.) 

It will be convenient to remember that azimuth and hour 
angle are measured from the meridian westwards, while 
right ascension and celestial longitude are measured from the 
first point of Aries eastwards. Thus, since the Sun's diurnal 
motion is retrograde, and its annual motion direct, the Sun's 
azimuth, hour angle, R.A., and longitude are all increasing. 

Most problems of this class depend for their solution chiefly 
on the consideration of arcs measured along the equator, or 
(what amounts to the same) angles measured at the pole. 

In another class of problems depending on the relation be- 
tween the latitude, a star's decl. and meridian altitude ( 24), 
we have to deal with arcs measured along the meridian. 
These two classes include nearly all problems on the celestial 
sphere which do not require spherical trigonometry. 

EXAMPLES. 

1. To represent, in a diagram, the positions of the Sun and Moon, 
and the star Herculis as seen by an observer in London on Aug. 19, 
1891, at 8 p.m., the following data being given : Latitude of London- 
= 51, Moon's age at noon on Aug. 19 = 14 days 19 hours, Moon's 
latitude = 2 S., K.A. of (Herculia = 16h. 37m., decl. = 31 48' N. 

The construction must be performed in the following order : 

(i.) Draw the observer's celestial sphere, putting in the meridian, 
horizon, zenith Z, and four cardinal points n, E, s, W. 

(ii.) Indicate the position of the pole and equator. The observer' s- 
latitude is 51. Make, therefore, nP = 51. P will be the pole. Take 
PQ = PR = 90, and thus draw the equator, QERW. 

(Hi.) Find the declination circle passing through the Sun. The- 
time of day is 8 p.m. Therefore the Sun's hour angle is 8 x 15, or 
120. On the equator measure QK = 120 westwards from the- 
meridian. Then the Sun Q will lie on the declination circle PK. 
Since QW = 90, we may find K by taking WK = 30 = $ WR. 

(iv.) Find the first points of Aries and Libra. The date of obser- 
vation is August 19. Now, on September 23 the Sun is at =2=. Also- 
from August 19 to September 23 is 1 month 4 days. In this- 
interval the Sun travels about 34 from west to east. Hence the 
Sun is 34 west of rO=. And we must measure K* = 34 eastwards^ 
from 8, and thus find z. 

The first point of Aries ( T ) is the opposite point on the equator.. 



THE CELESTIAL SPHERE. 



31 



(v.) We may now draw the ecliptic Cri^= passing through the 
first points of Aries and Libra, and inclined to the equator at an 
angle of about 23 (i.e., slightly over of a right angle). The Sun 
is above the equator on August 19; hence the ecliptic cuts PK above 
K. This shows on which side of the equator the ecliptic is to be 
-drawn ; we might otherwise settle this point by remembering that 
the ecliptic rises above the equator to the east of T . 

The intersection of the ecliptic with PE determines Q, the position 
of the Sun. 




FIG. 19. 



ascenfion is 16h. 37m., in time, = 249 15' in angular measure. On 
the equator measure off T M = 249 15' in the direction west to east 
(i.e., the direction of direct motion) from T ; we must, therefore, 
take ^=M = 69 15'. On the declination circle HP, measure off 
MX = 31 48' towards P. Then x is the required position of 
Herculis. 

(vii.) Find the Moon. At 8 p.m. the Moon's age is 14d. 19h + 8h. 
= 15d. 3h. Hence, the Moon has separate/! from the Sun by 
about 185 in the direction west to east. Measure off }) = 185 
from west to east, and put in }) about 2 below the ecliptic. The 
Moon's position is thus found. 



32 



ASTRONOMY. 



a/- 



2. To find (roughly) at what time of year the Star o Cygni 
(R.A. = 20h. 38m., clecl. = 44 53' N.) souths at 7 p.m. 

Let o be the position of the star on the meridian (Fig 20). At 
7 p.m. the Sun's western hour angle (QS or QPS) = 7h. = 105. 

Also TEQ, the Star's R.A. = 20h. 
38m. Hence rRS, the Sun's R.A. 
= 20h. 38m. - 7h. = 13h. 38m. ; or, 
in angular measure, Sun's R.A. 
= 204 30'. Now, on September 23, 
Sun's R.A. = 180, and it increases at 
about 1 per day. Hence the Sun's 
R.A. will be 204 about 24 days later, 
i.e., about October 17th. 

3. At noon on the longest day (June 
21) a vertical rod casts on a horizontal 

plane a shadow whose length is equal p IG 20 

to the height of the rod. To find 

the latitude of the place and the Sun's altitude at midnight. 





FIG. 21. 



From the data, the Sun's Z.D. at noon, Z, evidently = 45. 
Also, if QR be the equator, 0Q = Sun's decl. = i = 23 27' (approx.); 

.-. latitude of place = ZQ = 45 + 23 27' = 68 27'. 
If ' be the Sun's position at midnight, 

P0' = PQ = 90-2.327' = G6 33'. 
But Pn = lat. = 68 27'. 

... Q' w = 68 27' -66 33' = 1 54'; 

and the Sun will be above the horizon at an alt. of 1 54' at 
midnight. 




THE CELESTIAL SPHERE. 



EXAMPLES. I. 

1. Why are the following definitions alone insufficient? Tlie zenith 
and nadir are the poles of the horizon. The horizon is the great 
circle of the celestial sphere whose plane is perpendicular to the 
line joining the zenith and nadir. 

2. The R.A. of an equatorial star is 270 ; determine approximately 
the times at which this star rises and sets on the 21st June. In 
what quarter of the heavens should we look for the star at mid- 
night ? 

3. Explain how to determine the position of the ecliptic relatively - 
to an observer at a given hour on a given day. Indicate the position . 
of the ecliptic relatively to an observer at Cambridge at 10 p.m. at 
the autumnal equinox. (Lat. of Cambridge = 52 12' 51'6".) 



VV! 

i 



4. Prove geometrically that the least of the angles subtended at 
an observer by a given star and different points of the horizon 
that which measures the star's altitude. 

5. Show that in latitude 52 13' N. no circumpolar star when 
southing can be within 75 34' of the horizon. 

C. Represent in a figure the position of the ecliptic at sunrise on 
March 21st as seen by an observer in latitude 45. Also in lati- 
tude 67. , 

7. If the ecliptic were visible in the first part of the preceding 
question, describe the variations which would take place during the 
day in the positions of its points of intersection with the horizon. 

8. Determine when the star whose declination is 30" N. and whose . 
E.A. is 356 will cross the meridian at midnight. 

9. The declination and R.A. of a given star are 22 N. and 
6h. 20m. respectively. At what period of the year will it be (i.) a 
morning, (ii.) an evening star ? In what part of the sky would you 
then look for it ? 

10. Find the Sun's R.A. (roughly) on January 25th, and thus de- 
termine about whatxtime Aldebaran (R.A. 4h. 29m.) will cross the 
meridian that night. 

11. Where and at what time of the year would you look for 
Fomalhaut ? (R.A. 22h. 51m., decl. 30. 16' S.) 

12. At the summer solstice the meridian altitude of the Sun is 
75. What is the latitude of the place ? What will be the meridian 
altitude of the Sun at the equinoxes and at the winter solstice ? 

~ 



34 ASTRONOMY. 



EXAMINATION PAPER. I. 

1. Explain how the directions of stars can be represented by 
means of points on a sphere. Explain why the configurations of 
the constellations do not depend on the position of the observer, 
and why the angular distance of two different bodies on the celestial 
sphere gives no idea of the actual distance between them. 

2. Define the terms horizon, meridian, zenith, nadir, equator, 
ecliptic, vertical, prime vertical, and represent their positions in a 
figure. 

3. Explain the use of coordinates in fixing the position of a body 
on the celestial sphere, and define the terms altitude, azimuth t 
polar distance, hour angle, right ascension, declination, longitude, 
latitude. Which of these coordinates alwa3 T s remain constant for 
the same star ? 

4. Define the obliquity of the ecliptic and the latitude of the 
observer. Give (roughly) the value of the obliquity, and of the latitude 
of London. Indicate in a diagram of the celestial sphere twelve 
different arcs and angles which are equal to the latitude of the 
observer. 

5. What is meant by a sidereal day and a sidereal hour ? How 
could you find the length of a sidereal day without using a tele- 
scope ? Why is sidereal time of such great use in connection with 
astronomical observations ? 

6. Show that the declination and right ascension of a celestial 
body can be determined by meridian observations alone. 

7. What is meant by a circumpolar star ? What is the limit of 
declination for stars which are circumpolar in latitude 60 N. ? 
Indicate in a diagram the belt of the celestial sphere containing all 
the stars which rise and set. 

8. Define the terms year, equinoxes, solstices, equinoctial and 
solstitial points, equinoctial and solstitial colures. What are the 
dates of the equinoxes and solstices, and what are the corresponding 
values of the Sun's declination, longitude, and right ascension? 
Find the Sun's greatest and least meridian altitudes at London. 

9. Why is it that the interval between two transits of the Sun or 
Moon is rather greater than a sidereal day ? Show how the Sun's 
R.A. may be found (roughly) on any given date, and find it on 
July 2nd, expressed in hours, minutes, and seconds. 

10. Indicate (roughly) in a diagram the positions of the following 
stars as seen in latitude 51 on July 2nd at 10 p.m, : Capella (R.A. 
5h. 8m. 38s., decl. 45 53' 10" N.), a Lyras (R.A. 18h. 33m. 14s., 
decl. 38 40' 57" N.), a Scorpii (R.A. 16h. 22m. 43s., decl. 26 11' 
22" S.), a Ursse Majoris (R.A. lOh. 57m. Os., dec!. 62 20' 22" N.) 



CHAPTER II. 



THE OBSERYATOHY. 

SECTION I. Instruments adapted for Meridian Observations. 

42. One of the most important problems of practical astro- 
nomy is to determine, by observation, the right ascension and 
declination of a celestial body. We have seen in Chapter I. 
that these coordinates not only suffice to fix the position of a 
star relative to neighbouring stars, but they also enable us to 
find the direction in which the star may be seen from a given 
place at a given time of day on a given date (41). More- 
over, it is evident that by determining every day the decli- 
nation and right ascension of the Sun, the Moon, or a planet, 
the paths of these bodies relative to the stars can be mapped 
out on the celestial sphere and their motions investigated. 

In Section II. of the preceding chapter we showed that 
the right ascension and declination of a star can be deter- 
mined by observations made when the star is on the meridian. 
We proved the following results : 

The star's R.A. measured in time is equal to the time of 
transit indicated by a sidereal clock ( 24). 

The star's north decl. d can be found from z its meridian 
zenith distance, and I the latitude of the observatory by the 

iormula d = l+z, 

where if the decl. is south d is negative, and if the star tran- 
sits south of the zenith z is negative (24). 

Lastly, I can be found by observing the altitudes of a 
circumpolar star at its two culminations, and is therefore 
known ( 28). 

Hence the most essential requisites of an observatory must 
include (i.) a clock to measure sidereal time, (ii.) a telescope 
so fitted as to be always pointed in the meridian, provided 
with graduated circles to measure its inclination to the ver- 
tical, and with certain marks to fix the position of a star in 
its field of view. 



36 



ASTRONOMY. 



43. The Astronomical Clock is a clock regulated to 
indicate sidereal time. It should be set to mark Oh. Om. Os. 
at the time when the first point of Aries crosses the meridian. 
It will therefore gain about 4 minutes per day 
on an ordinary clock, or a whole day in the 
course of a year ( 22, 36). 

The clock is provided with a seconds hand, and 
the pendulum beats once every second, produc- 
ing audible "ticks"; hence an observer can 
estimate times by counting the ticks, whilst he 
is watching a star through a telescope. 

The pendulum is a compensating pendu- 
lum, or one whose period of oscillation is un- 
affected by changes of temperature. The form 
most commonly used is Graham's Mercurial 
Pendulum, in which the bob carries two glass 
cylinders containing mercury (Fig. 22). If the 
temperature be raised, the effect of. the increase 
in length of the pendulum rod is compensated 
for by the mercury expanding and rising in the 
cylinders. The same result is also effected in 
Harrison's Gridiron Pendulum, described in 
Wallace Stewart's Text-Boole of Heat, page 37. 

The clock is sometimes regulated by placing 
small shot in a cup attached to the pendulum. 





FIG. 23. 



THE OBSERVATORY. 37 

44. The Astronomical Telescope (Fig. 23) consists 
essentially of two convex lenses, or systems of lenses, and 
0', fixed at opposite ends of a metal tube, and called the 
object-glass and eye-piece respectively. The former lens 
receives the rays of light from the stars or other distant objects, 
and forms an inverted " image " (al) of the objects. The 
centre of the round object-glass is. called its " optical 
centre," and the image is produced as follows: Let AAA 
be a pencil of rays from a distant star. By traversing the 
object-glass these rays are refracted or bent towards the 
middle ray A 0, which alone is unchanged in direction. The 
rays all converge to a common point or "focus'' at a point a 
in A produced, and, if received by the eye after passing #, 
they would appear to emanate from a luminous point or 
" image " of the star at a. 

Similarly, the rays BBB, coming from another distant star, 
will converge to a focus at a point b in BO produced, and 
will give the effect of an " image" of the star at b. All 
these images (a, b) lie in a certain plane FN, called the focal 
plane of the object-glass, and they form a kind of picture or 
image of such stars as are in the field of view. 

The eye-piece 0' acts as a kind of magnifying glass, and 
enlarges the image ab just as if it were a small object placed 
in the focal plane FN. The figure shows how a second image 
A'B' is formed by the direction of the pencils of light after 
refraction through (/. This is the final image seen on looking 
through the telescope. The eye must be placed in the plane 
EE, so as to receive the pencils from A', B'. 

If, now, a framework of fine wires or spider's threads 
(Fig. 25) be stretched across the tube in the focal plane 
FNj these wires, together with the image (#J), will be 
equally magnified by the eye-piece. They will thus be 
seen in focus simultaneously with the stars, and the field 
of view will appear crossed by a series of perfectly distinct 
lines, which will enable us to fix any star's position, and 
thus determine its exact direction in space. Suppose, for 
example, that we have two wires crossing one another at the 
point F', and the telescope is so adjusted that the image of a 
star coincides with F', then we know that the star lies in the 
line joining F' to the optical centre of the object-glass. 



00 ASTRONOMY. 

45. The Transit Circle (Figs. 24, 26) is the instrument 
used for determining both right ascension and declination. It 
consists of a telescope, ST, attached perpendicularly to a 
light, rigid axis, WPPE, hollow in the interior. The ex- 
tremities of this axis are made in the form of cylindrical pivots, 
E, W, which are capable of revolving freely in two fixed forks, 
called Y's, from their shape. These Y's rest on piers of solid 
stone, built on the firmest possible foundations, and they are 
carefully fixed, so as always to keep the axis exactly hori- 
zontal and pointing due east and west. 




FIG. 24. 



In order to dimini?0i the effect of friction in wearing away 
the pivots, the axis is also partially supported at P, P upon 
friction rollers (not represented in the figure) attached to a 



THE OBSERVATORY. 3<> 

system of levers ( Q, Q) and counterpoises (R, R) placed within 
the piers. These support about four-fifths of the weight of 
the telescope, leaving sufficient pressure on the Y's to ensure- 
their keeping the axis fixed. 

Within the telescope tube, in the focal plane of the object- 
glass ( 44), is fixed a framework of cross wires, presenting^ 
the appearance shown in Fig. 25. Five, or sometimes seven, 
wires appear vertical, and two appear horizontal. Of the 
latter, one bisects the field of view ; the other is movable up 
and down by means of a screw, whose head is divided by 
graduation marks which indicate the position of the wire. 

The line joining the optical centre of the object-glass to 
the point of intersection of the middle vertical wire with the- 
fixed horizontal wire is called the Line of 
Colliinatiou. The wires should be so 
adjusted that the line of colliination is per- 
pendicular to the axis about which the 
telescope turns. For this purpose the 
framework carrying the wires can be moved 
horizontally, by means of a screw, into the 
right position. If the Y's have been accu- 
rately fixed, then, as the telescope turns, 
the line of collimation will always lie in the plane of the 
meridian. Hence, when a star transits we shall, on looking 
through the telescope, see it pass across the middle vertical, 
wire. 

Attached to the axis of the telescope, and turning with it, 
are two wheels, or graduated circles, GH, having their 
circumferences divided into degrees, and further subdivided 
by fine lines at (usually) intervals of 5'. By means of these 
graduations the inclination of the line of collimation to the 
vertical is read off by aid of sevi ral fixed compound micro- 
scopes, A, /, JB, pointed towards the circle. One of these 
microscopes (7), called the Pointer or Index, is of 
low magnifying power, and shows by inspection the number 
of degrees and subdivisions in the mark of the circle, which 
is opposite a wire bisecting its field of view. The pointer 
should read zero when the line of collimation points to the 
zenith, and the graduations increase as the telescope is. 
turned northwards. 




40 




FIG. 26, 

46. Beading Microscopes. In addition to the pointer 
there are four (sometimes six) other microscopes, called 
Reading Microscopes, arranged symmetrically round each 
circle, as at ABCD (Fig. 26). These serve to determine the 
number of minutes and seconds in the inclination of the tele- 
scope, by means of the following arrangement. Inside the 
tube of each microscope in the focal plane of its object- 
glass* is fixed a graduated scale NL (Fig. 27) in the form of 
a strip of metal with fine teeth or notches. This scale, and 
the image of the telescope circle, formed by the object-glass of 
the microscope, are simultaneously viewed by the eye-glass, 
and present the appearance shown in Fig. 27. 




FIG. 27. 

A small hole O marks the middle notch, and 5 notches 
correspond to a division of the telescope circle, hence the 
number of notches from the hole to the next division of the 
circle gives the number of minutes to be added to the pointer 
reading. 

* A compound microscope, like a telescope, consists of an object- 
glass, which forms an image of an object, and an eye-piece which 
enlarges this image. A scale or wires fixed in the plane of the 
the image will, therefore, be seen in distinct focus, like the wires 
in the telescope. 



THE OBSERVATORY. 41 

To read off the number of seconds, a pair of parallel 
wires, Sit, are attached to a framework, and can be moved 
across the field of view by means of a screw. One whole 
turn takes the wires from one notch of the metal scale to 
the next, i.e., over a space representing 1' on the telescope 
circle ; and the head of the screw is divided into 60 parts, 
each, therefore, representing V. The wires are adjusted 
so that the graduation on the telescope circle appears midway 
between them, and the reading of the screw-head then gives 
the number of seconds. With practice, tenths of a second 
can be estimated. 

The four microscopes of one of the circles are all read, and 
the best result is obtained by taking the mean of the readings. 

47. Clamp and Tangent Screw. When it is required 
to rotate the telescope of the transit circle very slowly, this 
is done by means of the bar represented at LK in Fig. 24. 
The telescope axis may be firmly clamped to this bar by 
means of a clamp (not represented in the figure), which 
grips the rim of one of the circles as in a vice. When this 
has been done, the bar JTZ, and with it the telescope, may be 
slowly turned by means of a horizontal screw at Z, called 
the Tangent Screw, and provided with a long handle 
attached to it by a " universal joint." This handle is held 
by the observer, and he can thus turn the tangent screw 
without ceasing to watch the stars. 

48. Arrangements for Illumination. As most obser- 
vations are conducted at night, the wires in the telescope and 
the graduations of the circles must be illuminated. This is 
done by a lamp placed exactly in front of one of the pivots, 
the light from which is concentrated by means of a bull's-eye 
lens in front and a mirror behind. Part of the rays are 
reflected, by a complicated arrangement of mirrors and 
prisms, so as to illuminate the parts of the graduated circle 
viewed by the microscopes. The rest of the light passes 
through a plate of red glass down the hollow axis to a ring- 
shaped mirror, whence it is reflected up to the wires ; thus 
the wires appear as dark lines on a dull red ground. There 
is also another arrangement for illuminating the wires from in 
front, if desired, so that they appear bright on a dark ground 



42 ASTRONOMY. 

49. Taking a Transit. Eye and Ear Method. If 

a star is to be observed with the transit circle, its R.A. 
and decl. must have been roughly estimated beforehand ; 
hence, its meridian Z.D. [= (star's decl.) (observer's lat.)} 
is known roughly. Before the star is expected to- 
cross the meridian, the telescope is turned by hand 
until the pointer indicates this roughly determined Z.D. ; 
this adjustment is sufficiently accurate to ensure the- 
star traversing the field of view. The telescope is then 
clamped ( 47). The observer now " takes a second" from 
the astronomical clock, i.e., he observes and writes down the- 
hour and minute, observes the second, and begins counting 
seconds by the clock's ticks. Thus, if he sees the time to be- 
llh. 23ni. 29s., he writes down "llh. 23m.," and at the- 
subsequent ticks he counts " 303132 33 " and so on ; 
in this way he knows, during the rest of the observation, t he- 
exact time at every clock -beat without looking at the clock. 

The star soon approaches the first vertical wire, and passes 
it, usually between two successive ticks. With practice, the 
observer is able to estimate fractions of a second as follows : 
Suppose the star crosses the wire between the 34th and 35th 
tick. The positions of the star are noticed at tick 34 and at 
tick 35, and by judging the ratio of their distances from the 
wire on the two sides, the observer estimates the time of 
crossing the wire by a simple proportion, and writes down, 
this time, say 34'6. The estimate is difficult to make,, 
because the two positions of the star are not visible simulta- 
neously, and the star does not stop at them, but moves 
continuously; hence to estimate tenths of seconds (as is 
usually done) requires much training and practice. 

Moreover, the observer must not lose count of the ticks of' 
the clock, for when he has written down the instant of transit. 
over the first wire the star will be nearing the second wire.* 

The time of transit over the second vertical wire is now 
estimated in the same way, and the process repeated at 
each wire. The average of the times of crossing the five 
or seven wires is taken as the time of transit ; in this way, 

* In most instruments the wires are placed at such a distance- 
that a star in the equator takes about 13 seconds from one wire- 
to the next. 



THE OBSERVATORY. 4$ 

the effect of small errors of observation will be much smaller 
than if the transit over one wire only were observed. 

This method of taking the time of transit is called the 
" Eye and Ear Method." 

While observing the transit, the observer turns the tele- 
scope by means of the tangent screw, until the horizontal 
wire bisects the image of the star ; during the rest of the 
observation the star will appear to run along the horizontal 
wire. After the observation, one of the circles is read by the 
pointer and the four microscopes. If the circle reads 0' 0", 
when the line of collirnation points to the zenith, the reading 
for the star will determine its meridian Z.D., in other cases 
we must subtract the zenith reading. Prom the meridian 
Z.D. the declination can be found. 

50. The Chronograph. To obviate the difficulty of 
observing tr?nsits by the eye and ear method, an instrument 
called the Chronograph is now frequently used. A cylin- 
drical barrel, covered with prepared paper, is made to turn 
slowly and uniformly by clockwork about an axle, on which 
a screw is cut. In this way the barrel is made to move 
forward in the direction of its axis, about one-tenth of an 
inch in every revolution. The observer is furnished with a 
key or button, which is in electric communication with a pen 
or marker. At the instant when the star crosses one of the 
vertical wires, the observer depresses the key, and a mark is 
made upon the paper of the barrel. The astronomical clock, 
also, has electric communication with the marker, and marks 
the paper once every second, the beginning of a new minute 
being indicated, in some instruments, by the omission of the 
mark, in others, by a double mark. In this way, a record is 
made of the times of transit over the wires, the marks being 
arranged in a spiral, owing to the forward motion of the 
barrel. The distance of the beginning of any transit-mark 
from the previous second-mark can be measured at leisure 
with very great accuracy, and the time of transit may thus 
be readily calculated. Indeed, there is no difficulty in 
recording, by this method, the transits of two, or even more, 
near stars which are simultaneously in the field of view of 
the telescope, for the transit-marks of the different stars can 
be readily distinguished from one another afterwards. 

ASTRON. E 



44 ASTRONOMY. 

51. Corrections. After the transit of a star has been 
observed, certain corrections have to be allowed for in practice 
before its true B. A. and decl. are obtained. These corrections, 
which depend on errors of observation, may be conveniently 
classified as follows : 

(a) Corrections required for the Right Ascension : 

1 . Error and rate of the astronomical clock. 

2. Personal equation of the observer. 

3. Errors of adjustment .of the transit circle, including 

(a) Collimation error. 
(5) Level error. 

(c) Deviation error. 

(d) Irregularities in the form of the pivots. 

(e) Corrections for the " vertically" and " wire 

intervals." 
(5) Corrections required in finding the Declination : 

1. Beading for zenith point, or for the nadir, hori- 

zontal or polar point. 

2. Errors of imperfect centering of the circles. 

3. Errors of graduation. 

4. Errors of " runs " in the reading microscopes. 
Besides these corrections, which we now proceed to de- 
scribe, there are others of a physical nature, such as refraction, 
parallax, aberration, the description of which will be given 
later. A correction is always regarded as positive when it 
must be added to the ol served value of a quantity in order 
to get the true value, negative if it has to be subtracted. 

(a) CORRECTIONS REQUIRED FOR THE RIGHT ASCENSION. 
. 52. Clock Error and Hate. A good astronomical clock 
can generally be regulated so as not to gain or lose more than 
about 2s. in a sidereal day. But to estimate times with 
greater accuracy, it is necessary to apply a correction to the 
time indicated, owing to the clock being either fast or slow. 

The Error of a clock is the amount by which the clock is 
sloiv when it indicates Oh. Om. Os. Thus, the error must be 
added to the indicated time in order to obtain the correct 
time. If the clock is fast, its error is negative. 

The Rate of the clock is the increase of error during 24 
hours. It is, therefore, the amount which the clock loses in 
the 24 hours. If the clock gains, the rate is negative. 



THE OBSERVATORY. 45 

The rate of a clock is said to be uniform or constant 

when the clock loses equal amounts in equal intervals of 
time. In a good astronomical clock, the rate should remain 
uniform for several weeks. 

53. Correction for Error and Hate. If the error of a 
clock and its rate (supposed uniform) are known, the correct 
time can be readily found from the time shown by the clock. 

The method will be made clear by the following example : 

EXAMPLE. If the error of an astronomical clock be 2'52s., and its 
rate be O44s., to find to the nearest hundreth of a second the correct 
time of a transit, the observed time bythe clock being 19h.23m.25'44s. 
Here in 24h. the clock loses 0'44s. 

.-. in Ih. it loses -^ x 0'44s. = 0'0183s. 
Hence, loss in 19h. = 0'0183s. x 19 = 0'348s., 
and loss in 23m. = O'OOTs. 
At Oh. Om. Os. the clock error is = 2'52s. ; 

/. at 19h. 23m. 25'44s., clock is too slow by 2'52s. +0'355s. = 2'88s., 
/. the correct time = 19h. 23m. 25'44s. + 2'88s. 
= 19h. 23m. 28-32s. 

54. Determination of Error and Rate of Clock. 

The clock error is found by observing the transit of a known 
star, i.e., a star whose R.A. and decl. are known. 

If the clock were correct, the time of transit (when cor- 
rected for all other errors) would be equal to the star's R.A. 
(see 24). If this is not the case, we have evidently 

(Clock error) = (Star's R.A.) 

(observed time of transit). 

This determines the clock error at the time of transit. 

To find the rate, the transits of the same star are observed 
on two consecutive nights. 

Let t and t x be the observed times of transit ; then x is 
the amount the clock has lost in 24 hours, i.e., the rate of the 
clock. Therefore 
(Bate of Clock) = (observed time of Isb transit) 

(observed time of 2nd transit). 

Having found the rate of the clock and its error at the 
time of transit, the error at Oh. Om. Os. may be found by 
subtracting the loss between Oh. Om. Os. and the transit. 

Stars used in finding clock error arc known as "Clock 
Stars." 



46 ASTBONOMY. 

55. Personal Equation is the error made by any par- 
ticular observer in estimating the time of a transit. 

Of two observers, one may habitually estimate the transit 
too soon, another may estimate it too late, but experience 
shows that the error made by each observer in taking times 
of transit by the same method is approximately constant. 

If all observations are made, by the same individual there 
will be no need to take account of personal equation, because 
the error made in taking a transit will be compensated by the 
error made in observing the clock stars to set the clock. If 
the two operations are performed by different observers, we 
must allow for the difference of their personal equations. 

Personal equation may be measured by an apparatus for 
observing the transit of a fictitious star, .<?., a bright point 
moved by clockwork ; in this case the actual time of its transit 
is known, and can be compared with the observed time. 
Personal equation is positive if the observer is too quick, 
so that the correction must be added to the observed time to 
get the true time, as in 51. 

56. Errors of Adjustment of the Transit Circle. 

If the transit circle is in perfect adjustment, the line of colli- 
mation of the telescope must always lie in the plane of the 
meridian. If not, we must correct for the small errors of 
adjustment. The conditions required for perfect adjustment, 
together with the corresponding corrections when these con- 
ditions are not fulfilled, may be classified as follows : 

(a) The line of collimation should be perpendicular to the 
axis about which the telescope rotates. If not, the corre- 
sponding correction is called Collimation Error. 

(b) The axis of rotation must be horizontal. Level Error. 
(0) The axis must point due east and west. Deviation 

(or Azimuthal) Error. 

(d) The pivots resting on the Y's must be truly turned, 
and form parts of the same circular cylinder. Correction for 
shape of pivots. 

(e) The vertical wires in the transit must be truly vertical 
(i.e., parallel to the meridian) and equidistant. Verticality 
and Thread Intervals. 



THE OBSERVATOBT. 47 

*57. Collimation Error. We have seen ( 45) that the frame- 
work carrying the vertical wires in the transit telescope can be 
adjusted by a screw, so that collimation error can be corrected. 
Suppose, for simplicity, that no other error is present. Then the 
line of collimation will always make a constant small angle with the 
meridian, and this angle will measure the collimation error. 

To correct this error, two telescopes, called Collimators, are 
pointed towards each other, one due north, the other due south of 
the instrument (n, s, Fig. 26). Both contain adjustable " collimating 
marks," formed by cross wires in their focal planes. The transit 
telescope being first pointed vertically, and two apertures in the 
side of its tube being uncovered, the observer looks through the 
telescope s, and sees through the apertures into the telescope n. 
He then brings the wires in s into coincidence with the images of 
the wires in n ; he then knows (from the optical theory of the tele- 
scope) that the lines of collimation of n, s are parallel. Suppose 
(e.g.) that they make a small unknown angle x" W. of S., and E. of 
N., respectively. 

He now looks through the transit telescope into the collimator s. 
He adjusts the middle vertical wire of the transit to coincide with 
the image of the cross mark in s, reading the graduated screw by 
which the adjustment is made. The line of collimation of the 
transit is now x" we*t of the meridian. He points the telescope into 
n, and similarly adjusts the wires : the line of collimation is now x" 
east of the meridian. He now turns the adjusting screw to a reading 
midway between the two observed readings ; the line of collimation 
is then in the meridian, and collimation error has been removed. 

*58. Level Error is measured by the inclination to the horizon of 
the axis of rotation of the telecope. It causes the line of collima- 
tion to trace out, on the celestial sphere, a great circle inclined to 
the meridian at an angle equal to the level error. 

Level error is found by pointing the telescope (corrected for 
collimation error) downwards over a trough of mercury (N, Figs. 24, 
26, 28). 

An eye-piece is provided, called a " collimating eye-piece " (EF, 
Fig. 28, p. 49), containing a plate of glass M, which reflects the 
light from a lamp straight down the tube. The mercury will 
form a reflected image of the telescope, which may be treated just 
as it' it ",vere a real telescope or collimator ; the wires in the actual 
telescope will appear bright, and those in the image will appear 
dark. By the law of reflection, if the middle wire coincide with its 
image, the line of collimation will be vertical, and (since there is no 
collimation error) there will be no level error. If not, the wires 
are moved by the screw until the vertical wire coincides with its 
image. The observer reads the angle through which the screw has 
been turned, and thus measures the level error. The wires are then 
replaced (otherwise collimation error would be introduced) and 
level error is corrected by adjusting the Y's ( 59). 



48 ASTBOtfOMY. 

*59. Deviation Error is measured by the small angle which the 
axis of rotation of the telescope makes with the plane of the prime 
vertical. It causes the line of collimation of an otherwise correctly 
adjusted transit circle to describe a great circle through the zenith 
whose inclination to the meridian is equal to the deviation error. 

Deviation error can be discovered by observing the times of upper 
and lower transit of a circumpolar star, such as the pole star. 
Suppose (e.g.) that the telescope axis points slightly south of east; 
then it is readily seen by a diagram that when the telescope is 
pointed north of the zenith, the line of collimation will be slightly 
east of the meridian. Then, at upper transit, if the observed cir- 
cumpolar star is north of the zenith it will reach the middle wire 
before reaching the meridian. At lower transit it will not reach the 
wire till after passing the meridian. Hence, the time from upper to 
lower transit will be rather greater than 12h., and the time from 
lower to upper transit will be rather less than 12h. By observing 
the difference of the intervals the deviation error can be found. 

In many observatories, the Y's of the transit circle can be adjusted 
by screws, one moving vertically, to correct for level error, the 
other horizontally, to correct for deviation error. 

When these errors are corrected, the cross wires of the collimators 
are brought into coincidence with the middle wire of the telescope 
when pointed horizontally. 

*60. The correction for the shape of the pivots is rather compli- 
cated, but, in a good instrument, it should be very small. When 
the pivots are much worn by friction, they should be re-turned. 

The errors may be measured by making a small mark on the end 
of each pivot, and observing, by means of reading microscopes, the 
motions of the marks as the instrument is slowly turned round. If 
the pivots are true, the marks should remain fixed, or describe circles. 

*61. Verticality of the Wires maybe tested by observing one of 
the collimators, whose cross wires are adjusted as in 69. If the 
cross wires always appear to intersect on the middle wire of the 
transit when the instrument is turned through any small angle, we 
know that the middle wire is vertical. 

*62. Wire Intervals By "Equatorial Wire Intervals" are 
meant the intervals of time taken by a star on the equator in pass- 
ing from one vertical wire of the transit to the next. 

If the intervals between successive wires are unequal, the mean 
of the times of transit over the wires will not in general be the 
same as the time of transit over the middle wire. We may imagine 
a straight line so drawn across the field of view that the time of 
transit across it is exactly equal to the mean of the times of transit 
over the five or seven wires. This line is called the Mean of the Wires. 

By carefully determining the equatorial wire intervals, the very 
small interval between the transits over the mean of the wires and 
over the middle wire can be found. 

For a star not in the equator, the wire intervals are proportional 
to the secant of the declination. This follows from Sph. Gcom. (17). 



THE OBSERVATORY. 



(6) CORRECTIONS REQUIRED IN FINDING THE DECLINA- 
TION OF A STAR. 

63. Zenith Point. In 45 we stated that the pointer 
of the transit circle is usually adjusted to read 0' when 
the line of collimation is pointed to the zenith. Eut it would 
be very difficult to adjust the microscopes to give a mean 
reading of exactly 0' 0" for the zt nith. Hence it is neces- 
sary to determine the zenith point, or zenith reading, and 
in calculating the meridian Z.D. of any star, this must be 
subtracted from the reading for the star. 

Let ^and -ZVbe the readings when the telescope is pointed 
to the zenith and nadir, respectively, ZTand H' the readings 
for the north and south points of the horizon ; then evidently, 

Z=. H-90 = ^-180 = #"'-270. 

Also, if x is the reading for the meridian transit of any star, 
then star's meridian Z.D.= # Z, if north of the zenith, 
or, = 360 (xZ\ if south of the zenith. 

64. To find the Nadir Point, use is made of the Colli- 
mating Eye Piece, already mentioned in 56, and 
represented in Pig. 28. It consists of 

two lenses J2, F, between which is a 
plate of glass, l/~, inclined at an angle of 
45 to the axis. This plate illuminates 
the wires from above by partially re- 
flecting the light from a lamp on them, 
at the same time allowing them to be 
seen through the eye-glass, E, 

The telescope is pointed downwards 
over the trough of mercury, N\ and 
the rays of light from any one of the 
wires, Q, will produce by reflection a 
distinct image of the wire at q in the 
focal plane. Ey turning the telescope 
with the tangent screw, the fixed hori- 
zontal wire may be made to coincide 
with its image ; it will then be verti- 
cally over the " optical centre" of the 
object-glass ( 44). The line of colli- 
mation will, therefore, point to the 
nadir, and the nadir reading is given by 
the pointer and microscopes. Subtracting 
the zenith reading. 




FIG. 28. 
180, we have the 



50 



ASTRONOMt. 



65. Determination of Horizontal Point. Method 
of Double Observation. Both the horizontal reading and 
the meridian altitude of a star can be determined by observ- 
ing the star, both directly and by reflection, in a trough of 
mercury placed in a suitable position (M, Pigs. 26, 29). 




FIG. 29. 

Fig. 29 illustrates the method of double observation. Let 
PZ be the direction of the line of collimation corresponding 
to the zero reading, PR the horizontal direction, PS and 
HTP the directions of the star viewed directly and its image 
viewed by reflection. The reading of the circle for the direct 
observation is the angle ZPS, the reading for the reflection 
is the angle ZPM. 

Since the angles of reflection and incidence S'MZ', TMZ' 
at the mercury are equal, and MS', PS are parallel, we have 
evidently L SPH= S'MS' = TMJT = MPH-, 
.-. star's altitude, SPH= f 8PM- 
= \ (ZPN-ZPS) 

= half the difference of the two readings. 
Also : Horizontal reading, ZPH \ (ZPM+ZPS) ; 

=: half the sum of the two readings. 

Subtracting 90 from the north horizontal point, the zenith 
point is found. 

*66. In using this method with the transit circle of a fixed 
observatory, the star will remain sufficiently long in the field of 
view to allow of both observations being made at the same transit, 
and the fact of the star not being quite on the meridian will not 



THE OBSEBVATOBY. 51 

affect the results perceptibly. But there will not be time to read 
the circles by means of the four microscopes, between the two 
observations. This difficulty is obviated by proceeding thus : 
Before the first observation, point the telescope (by means of the 
pointer) in such a direction that the reflection of the star in the 
mercury will cross the field of view during fhe transit; for this 
purpose the star's meridian altitude must be known approximately. 
(Jlamp the telescope, and read the microscopes. When the star 
appears in the field of view, adjust the moveable horizontal wire (by 
means of its graduated screw) till it crosses the star, keeping the 
telescope fixed. Now un clamp the telescope, and point it to the star 
direct, turning it with the tangent screw until the moveable horizontal 
wire again crosses the star. After the observation, read the graduated 
screw of the horizontal wire, and also the pointer and microscopes. 
Since the star is bisected by the same wire at each observation, 
the difference in the readings gives the angle through which the 
telescope was rotated, and this angle is evidently double the star's 
altitude. Half the sum of the readings gives what would be the 
reading if the moveable wire were pointed horizontally. This must 
be corrected by adding the angular interval between the moveable 
and fixed wires as determined from the graduated screw, and we 
then have the reading for the horizon point when the fixed wire is used. 

67. Polar Point. In order to find the declination of a 
star by means of the transit circle, it is necessary to know 
the reading when the telescope is pointed to the pole. This 
may be found, just as in 28, by observing the upper and 
lower transits of a circumpolar star. The mean of the two 
readings gives the polar point. 

The N.P.D. of any star is found by taking the difference 
of the readings for the star and the polar point. The decli- 
nation is, of course, the complement of the N.P.D. 

We may also find declinations thus : Since angles are 
measured from the zenith northwards, it is evident (by draw- 
ing a figure or otherwise) that the reading for the point of 
the equator above the horizon is given by 

Equatorial point = (Polar point) +270. 
Since the decl. is the angular distance from the equator, we have 

(North Decl.) = (Reading for star) (Equatorial point). 
If the star transits north of the zenith, its reading must be 
increased by 360. 

The latitude of the observatory is given by 
Latitude = Altitude of pole 

= (North horizontal point) (Polar point). 



52 ASTEONOMT. 

*68. Errors of Graduation. The operation of testing the accuracy 
of the graduations on the circles of the transit circle is very long 
and laborious. One of the two graduated circles is so attached to 
its axis, so that it can be turned through any angle relative to the 
telescope. Then, by reading the microscopes belonging to both 
circles, every graduation on one circle is compared with every 
graduation on the other circle, and any errors of graduation are thus 
detected and measured. The effect of such errors is much reduced 
by using all the four microscopes, and taking the mean of their 
readings. 

*69. Errors due to Imperfect Centering of the Circles. By 
taking the mean of the microscope readings, all errors due to imper- 
fect centering are eliminated. In proof, let us suppose that only 
two microscopes (A, C, Fig. 26) are used, but that these are opposite 
to one another. If the circle is truly centred, with its centre on 
the line AC, the two readings will differ by 180. If, now, the gradu- 
ated circle is displaced, without being rotated, till its centre is at a 
distance h from AC, then the points of the scale, now under AC, 
will be at distances h from the points formerly under AC, both being 
displaced in the same direction. Hence, since both readings are 
measured the same way round the circle, one will be increased 
and the other will be decreased by the same angle. The arithmetic 
mean of the two readings will, therefore, be unaltered by the dis- 
placement of the centre, and will be independent of any small error 
due to imperfect centering. The same is, of course, true of the 
mean reading for the other pair of microscopes, B, D. 

The error in centering may be discovered by taking the difference 
of the readings of a pair of opposite microscopes. This difference 
should be 180 ' if the circle is properly centred ; if not, the amount 
by which it differs from 180 will determine how much the centre of 
the circle is to one side or the other of the line joining the centres 
of the pair of microscopes. 

*70. Error of Runs. In the reading microscopes, one turn of the 
micrometer screw should move the parallel wires over a space corre- 
sponding to exactly 1' on the graduated circle, so that the wires 
should be brought from one mark of the circle to the next by exactly 
five turns of the screw. In practice it will probably be found that 
rather more or rather less than five turns will be necessary. In this 
case the readings of the teeth and of the micrometer screw-head will 
differ slightly from true minutes and seconds of arc on the circle, 
and a correction will be required. This error is called Error of 
Runs. 

*71. Collimation, Level and Deviation Errors have no appre- 
ciable effect on observations for declination, provided that such 
errors are small compared with the star's N.P.D. Hence, they may 
be left out of account, except in observations of the Pole Star. 



1BE OBSEKVATOKI. 53 

72. General Remarks. We first described the Transit 
Circle, and the methods of " taking a transit" ; we afterwards 
described the corrections which must be applied to the results 
of the observations in finding the right ascension and decli- 
nation of a star. But in practical work the various errors 
must be determined before any observation can be made. 
Among these, collimation, level and deviation error, and the 
nadir point should be found daily, as they may be affected by 
heat or cold, or by shaking the instrument. 

Clock error and rate are also determined daily by observing 
certain " clock stars." The accuracy of the corrections may 
be tested by observing various "known stars" of different 
declinations. If the corrections have been accurately made, 
the observed right ascensions and declinations should agree 
with their values as given in astronomical tables. 

Before determining clock error and rate by nieuns of a 
11 clock star," the R.A. of one such star must be known. 
Since the R.A. is measured from the first point of Aries, that 
point must first be found. The method of finding it will be 
described in Chap. IY. 



73. Observations on the Sun, Moon, and Planets. 

The positions of the Sun, Moon, and Planets are defined by 
the coordinates of their centres. In finding these, the 
angular diameters must be taken into account. 

In observing the Moon or a planet, the fixed horizontal wire 
is adjusted to touch the illuminated edge of its disc, and the 
times at which its edge touches the vertical wires are ob- 
served. To find the coordinates of the centre, a correction 
is made for the angular semi-diameter of the body, which 
must be determined independently. It must not be forgotten 
that the image formed by the telescope is inverted. 

In observing the Sun, the semi-diameter may be found 
during the observation by adjusting the moveable horizontal 
wire to touch one edge of the disc, while the fixed wire 
touches the other edge. The reading of the micrometer 
screw gives the Sun's angular diameter. In finding the time 
of transit, the times of contact of the disc on arriving at and 
leaving each wire are separately observed ; their arithmetic 
mean for any wire is the time of transit of the centre. 



54 



ASTRONOMY. 



SECTION II. Instruments adapted for Okxertalions off the 
Meridian. 

74. The Transit Circle can only be used to observe celestial 
bodies during the short period before and after their transit 
that they remain in the field of view. It is, therefore, un- 
suited for continuous observation of a celestial body, such as 
is required more particularly in Physical Astronomy. Eor 
this purpose, a telescope must be mounted in such a way that 
it can be pointed in any required direction, or moved so as to 
keep the same body always in the field of view. There are 
two such forms of mounting, and the telescopes thus mounted 
are called the Altazimuth and the Equatorial. 




FIG. 30. 



75. The Altazimuth, In this instrument, a telescope, 
ST, is supported so that it can turn freely about a horizontal 
axis, CD, sometimes called the secondary axis. This 
secondaiy axis, with the attached telescope, is capable of 
turning about a fixed vertical axis, AB, sometimes called the 
primary axis, which is supported at its upper and lower 
ends as shown in the figure. 

Both axes are provided with graduated circles, GIT, 



TTTF. OBSERVATORY. 55 

attached to, and turning with them. Each circle is read 
by means of one or more "pointer" microscopes, M and TV. 
There are also clamps, furnished with tangent screws, hy 
means of which the circles may be fixed in any desired posi- 
tion, or rotated slowly if required. At C is a counterpoise, 
which balances the telescope and the circle 7F, and so 
prevents their weight from bending the axis AB. 

By rotating the whole instrument about the vertical axis 
AS, the telescope can be brought to any required azimuth. 
If now the circle GH\)Q clamped, the telescope can be turned 
about CD to any required altitude. The microscope N 
should indicate zero when the telescope is pointed in the 
plane of the meridian, and the microscope M should indicate 
zero when the telescope is horizontal. If now the telescope 
be pointed so that a star is in the middle of its field of view, 
the readings of the two microscopes TV, M will give the star's 
azimuth and altitude respectively. The time of observation 
being also known, the position of the star on the celestial 
sphere is completely determined, and its R.A. and decl. can 
be calculated if required. But for observations of this class, 
the altazimuth is not nearly so reliable as the transit circle. 

As the altazimuth possesses two independent motions, while 
the transit circle possesses only one, the former instrument 
is liable to a far greater number of errors of adjustment; 
moreover, its telescope is far less firmly and rigidly supported, 
and the instrument is therefore more liable to bend. 

A large altazimuth in Greenwich Observatory is used for 
observing the Moon's motion, when it is so near the Sun 
that it cannot be accurately investiga 4 o 1 by meridian observa- 
tions alone. 

A portable telescope, mounted on a tripod stand, such as is 
commonly used for observing the stars at night, is an altazi- 
muth unprovided with graduated circles. 

A Finder (F) is usually attached to a large altazimuth, 
whose field of view is of small angular breadth. This is a 
small telescope of lower magnify ing-power, with a larger field 
of view, the centre of which is marked by cross wires. To 
point the large telescope to any celestial body, the altazimuth 
is so adjusted that the body is seen in the centre of the finder. 
It will then be in the field of view of the large telescope. 



56 



ASTRONOMY. 



76. The Equatorial (Fig. 31). If we suppose an alta- 
zimuth inclined so that its primary axis, instead of being 
vertical, is pointed in the direction of the pole, we shall have 
an Equatorial. In this instrument the framework carrying 
the telescope turns as a whole about about the primary axis 
A JB, which is supported at A and J?, so as to point towards 
the pole. Attached perpendicularly to this axis, and turning 
with it, is a graduated circle, called the Hour Circle, which 
read by a " pointer " microscope N. 

The framework AB carries a secondaiy axis perpendi- 
cular to the primary axis, and the telescope ST\& attached 
perpendicularly to this secondary axis, about which it 
is free to turn. The axis of the telescope carries another 
graduated circle called the Declination Circle which is 
read by the " pointer" microscope M. 




FIG. 31. 



The declination circle should read zero when the telescope 
is pointed in the plane of the equator, and the hour circle 
should read zero when the telescope is in the plane of the 
meridian. If now the telescope is pointed towards any 
celestial body, the readings of the two microscopes will 
give, respectively, the declination and hour angle of the 
body. 

When it is required to observe the same body continuously 
with the equatorial, the declination circle is clamped, and 
the observer must slowly rotate the hour circle by hand, so 
as to keep the body observed in the field of view. 



THE OBSERVATORY. 57 

In large instruments the hour circle can be attached to a 
clamp which is worked by clockwork in such a manner that 
the whole framework turns uniformly round the primary axis 
AB once in a sidereal day. This motion will ensure that the 
star under observation shall always remain in the centre of 
the field of view. 

The pointer-microscope of the hour circle may be made to 
revolve with the clamp, and to mark zero when the telescope is 
pointed towards the first point of Aries ; its reading will then 
give the right ascension of any observed star. But the decli- 
nation and right ascension cannot be determined with any 
great degree of accuracy by reading the circles of the equa- 
torial. There are the same difficulties as in the altazimuth ; 
moreover, the primary axis, being inclined to the vertical, is 
more liable to bend under the weight of the telescope. 

The clockwork by which the equatorial is driven could not 
be regulated by an ordinary pendulum, as this would make 
the telescope move forward in a series of jerks, one at every 
beat. For this reason, a conical pendulum revolving uniformly 
must be used. The reader will find the principle of the 
conical pendulum explained in most text-books on elementary 
dynamics; a working example maybe seen in the "Watt's 
Governor" of a steam-engine. 

In most modern equatorials, the primary axis is not sup- 
ported as in Fig. 31, but on a pillar just underneath the 
secondary axis. The advantage is that the primary axis is 
less liable to bend than when supported at its two ends A, B. 

77. Uses of the Equatorial. Amongst these the fol- 
lowing may be mentioned : 

(i.) " Differential " observations, i.e., micrometric obser- 
vations of the relative distances and positions of two near 
stars simultaneously visible. 

(ii.) Observations of the appearance, structure, and magni- 
tude of the celestial bodies. 

(iii.) Stellar photography, 
(iv.) Spectroscopic analysis. 



58 



ASTRONOMY. 



78. Micrometers. Any instrument used for measuring 
the small angular distance between two bodies simultaneously 
visible in the field of view of a telescope is called a 
Micrometer. Thus the moveable horizontal wire in the 
transit circle, with its graduated screw, is a micrometer, for 
if the instrument be so adjusted that the fixed wire crosses 
one star, while the moveable wire crosses another neighbouring 
star, the distance between the wires, as read off on the screw 
head, gives the difference of declination of the stars. The 
moveable wire in the field of view of the reading microscope 
is identical in principle with a micrometer. 

79. The Screw and Position Micrometer (Fig. 32) 
serves to fmdboth the angular distance bet ween two neighbour- 
ing stars and the direction of the line joining them. It contains 
a framework of wires placed 

in the focal plane of the tele- 
scope. Two of these wires 
are parallel, and one of them 
can be separated from the 
other by turning a screw with 
a graduated head. A third 
wire, which we will call the 
" transverse wire," is fixed 
in the framework perpendi- 
cular to the two former. The 
whole apparatus, together 
with the eye piece of the 
telescope, can be rotated so 
that the wires may appear in any required direction across the 
field of view. A graduated circle, called the Position Circle, 
is attached to the eye-piece, and measures the angle through 
which it has thus been turned. Besides the wires, the frame- 
work contains a transverse strip of metal marked with notches, 
at distances apart corresponding to complete turns of the micro- 
meter screw, an arrangement similar to that employed in the 
reading microscope ( 45). 

In observing two stars, the equatorial and micrometer are 
so adjusted that one of the stars may appear at the inter- 
section of the two fixed wires, while the other appears at the 
intersection of the fixed and moveable wires. 




FIG. 32. 



THE OBSEBYATORY. 59 

Hie number of notches of the scale, together -with the 
reading of the screw-head, determine the distance "between 
the images of the stars in turns and parts of a turn of the 
screw-head. To find the angular distance "between the stars, 
we only require to multiply by the known angular distance 
corresponding to one turn of the screw. 

The reading of the position circle determines the direction 
of the small arc joining the stars. The position-circle should 
read zero if the stars have the same R.A. Then the reading 
in any other position will determine their position angle, 
i.e., the angle which the line joining the stars makes with a 
declination circle through one of the stars. 

*80. Dollond's Heliometer is another form of micrometer, de- 
pending on the principle that if the object-glass of an astronomical 
telescope be cut across in two, each half will form an image of the 
whole field of view, in the same way as if the lens were still com- 
plete.f In the Heliometer one half of the object-glass can be made 
to slide along the other by means of a graduated screw. 




Fm. 33. 

Suppose that we want to measure the angular diameter of the Sun 
(8, Fig. 33). When the halves of the object-glass are together, so 
that their optical centres coincide, one image of the Sun will be 
formed. When the two halves are separated, two separate images 
will be formed in the focal plane of the telescope, and will be seen 
simultaneously. The half -lenses are separated, till the two images 
touch, as db and be. Let 0, 0' be the optical centres of the two 
halves of the objective. The distance 00' is read off on the screw- 
head ; from this reading the Sun's angular diameter may be found. 

For at b, the point of contact of the images, the half-lens forms 
an image of the lower limb B, and the half -lens 0' forms an image of 
the upper limb A. Hence, BOb and AO'b are straight lines, and ObO' 
is the angular diameter Bb A. But the focal length 06 is known 
Hence, if 00' is also known, the angular diameter 060' can be found. 

t To show this, it is only necessary to cover up half the object- 
glass of an astronomical telescope. (N.B. Not an opera-glass.) 



60 ASTRONOMY. 

In measuring the angular distance between two stars, the helio- 
meter is adjusted so that the image of one star formed by one half- 
lens coincides with the image of the other star formed by the 
other half -lens 0'. The principle is the same as before. 

*81. To find the angular distance corresponding to a revolution 
of the micrometer screw, the simplest plan is to observe the Sun's 
diameter, and to compare the reading with its known value. The 
latter is given in the Nautical Almanack for every day at noon. 

To test the zero reading of the position circle, the equatorial 
is pointed to a star near the equator, and fixed, and the micrometer 
is turned till the diurnal rotation causes the star to run along the 
transverse wire. The circle should then read 90. 

82. Stellar Photography. For photographic purposes, 
the equatorial is driven by clockwork, carrying with it a 
sensitized plate, on which an image of the heavens is projected. 
In this way a photograph of part of the sky is obtained, and 
on such a photograph the distances and relative positions of 
the various stars, nebulaB, &c., can be accurately measured. 
Moreover, by continuing the exposure sufficiently long, even 
the faintest rays of light will produce an impression on the 
photographic plate ; and it is thus possible to detect stars and 
nebulaB which would be invisible to the eye. 

*83. Spectrum Analysis. A description of the spectrum is given 
in Wallace Stewart's Text-Book of Light, Chap. VIII., and the spec- 
troscope is described in 91 of the same treatise. 

A detailed account of the methods of spectrum analysis would be 
out of place in this book, as the subject belongs to the domain of 
Physical Astronomy. The general principle is this : We can, by 
means of the spectroscope, analyse the constituent waves of the 
light rays which reach us from the Sun and stars. We can compare 
these constituents with those emitted or absorbed by the various 
chemical elements in a state of vapour. Such comparisons enable 
us to infer what chemical elements are present in different celestial 
bodies. 

84. Other Instruments. The instruments described in 
this chapter are all such as are used in fixed observatori3S. 
Besides these, certain portable instruments are used in astro- 
nomical observations. Among the latter class the Zenith 
Sector will be described in the next chapter, in connection 
with the determination of the Earth's form and radius ; and 
the Sextant and Chronometer will be explained in treating of 
the methods of finding latitude and longitude at sea. 



THE OBSEEYATOBY. 61 



EXA.MPLES.II. 

1. Describe the Altazimuth. Why is it not so well suited for 
continuous observations as the equatorial, and, in particular, why is 
it quite unsuitable for stellar photography? 

2. Show that the altitude of a star is greatest when the star is on 
the meridian. 

3. From the result of Question 2, show how the meridian zenith 
distance of a star might be found by observing its altitude with an 
altazimuth. 

4. How may we most easily set the astronomical clock ? 

5. Show that the rate of a clock might be found by observations 
on successive nights with any telescope provided with cross wires, 
and pointed constantly in a fixed direction. 

6. Distinguish, with examples, direct and retrograde angular 
motion. Is R.A. measured direct or retrograde ? 

7. Show that in latitude 45 the interval between the time of 
any star's passing due east and its time of setting is constant. 

8. Show that, if a transit circle be not centred truly, the con- 
sequent error can be eliminated by taking the mean of the readings 
of the microscopes. 

9. In a double observation made with the transit circle, the 
readings of the pointer directly and by reflection are 59 35' and 
125 20' ; the means of the microscope readings are in the two cases 
3' 42" and 1' 13''. The moveable wire reads t 2", and the reflected 
star runs along the fixed horizontal wire. Find the zenith reading. 

10. Explain how it is that photography has revealed the existence 
of stars which are so faint as to be invisible. 

11. Find the decl. of a Ophiuchi from the following observations, 
made at Greenwich (lat. 51 28' 31" N.) -.Pointer reading 321 10', 
microscope readings, 1' 2", 0' 50", 0' 40", 0' 58", the zenith reading 
being 0' 16". 

12. Find also the R.A. of a Ophiuchi. Given : Time by sidereal 
clock = I7h. 29m., the numbers of seconds at the transits over the 
five wires being 37'4s., 50'2s., 1m. 2'9s., 1m. 15'2s., 1m. 27'4s. Clock 
error = 10'Gs. ; personal equation = + 0'4s. 



62 ASTBON01TY. 



EXAMINATION PAPER. II. 

1. Classify the various observations which are taken in astro- 
nomical investigations, and state the respective instruments which 
may be used for those observations. 

2. Define the right ascension and declination of a star, and describe 
shortly the principles of the methods of finding them. 

3. Describe how the time of transit of a star across each of the 
five or seven wires of a transit instrument is observed, and explain 
how the time of transit across the meridian is deduced. Define the 
equatorial interval of two wires. 

4. Describe the Reading Microscope, and show how the zenith 
distance of a star may be found by direct observation with the 
transit circle. 

5. Enumerate the errors of a transit instrument, and explain how 
level error may be measured and corrected. 

6. Explain what is meant by collimation error, and draw a diagram 
showing the circle traced out on the celestial sphere by the line of 
collimation in an instrument which has a small collimation error 
east of .the meridian. Is the correction, to be applied to the times 
of transit, positive or negative in such a case ? 

7. Describe the Equatorial, and explain the adjustments and 
principal uses of the instrument. 

8. Describe the Screw and Position Micrometer, and explain how 
the value of a turn of the screw may be found. 

9. What is meant by the error and rate of a clock, and the personal 
equation of an observer? How are they usually found ? 

10. On 1st March, 1872, the time of transit of j8 Librae, at Green- 
wich, was observed to be 15h. 9m. 615s., and on the 3rd March the 
observed time was 15h. 9m. 4'73s. The tabular R..A. of the star was 
15h. 10m. 7'25s. Find the error and rate of the clock on 3rd March. 



CHAPTER 111. 



THE EARTH. 

SECTION I. Phenomena depending on Change of Position on 
the Earth. 

85. Early Observations of the Earth's Form. One 

of the first facts ascertained by the early Greek astronomers 
was that the Earth's surface is globular in form. Even 
Homer (B.C. 850 circ.) speaks of the sea as convex, and 
Aristotle (B.C. 320) gives many reasons for believing the 
Earth to be a sphere. Among these may be mentioned the 
appearances presented when a ship disappears from view. If 
the surface of the ocean were a plane, any person situated 
above this plane would (if the air were sufficiently clear) see 
the whole expanse of ocean extending to the furthermost 
shores, with all the ships sailing on its surface. Instead of 
this, it is observed that as a ship begins to sail away its 
lowest part will, after a time, begin to sink below the appa- 
rent boundary of the surface of the sea ; this sinking will 
continue till only the masts are visible, and, finally, these 
will disappear below the convex surface of the water between 
the ship and the observer. 

Another reason is suggested, by observing the stars. If 
the Earth's surface were a plane, any star situated above the 
plane would be seen simultaneously from all points of the 
Earth, except where concealed by mountains or other 
obstacles, and any star below the plane would be everywhere 
simultaneously invisible. In reality, stars may be visible 
from one place which are invisible from another ; and all the 
appearances presented were found by the Greeks to agree 
with what might be expected on a spherical Earth. Eratos- 
thenes even made a calculation of the Earth's size from the 
distance between Alexandria and Assouan and their latitudes 
(91) deduced from the Sun's greatest meridian altitudes. 
He found the circumference to be 250,000 stadia, or furlongs. 

Lastly, the Earth's spherical form will account for the 
circular form of the Earth's shadow in a lunar eclipse. 



64 

86. General Effects of Change of Position. In 5, 

we showed that, owing to the great distance of the stars, they 
are seen in the same direction whatever be the position of 
the observer. In confirmation of this fact, it is found by 
observation that the angular distance between any two stars 
(after allowing for refraction) is observed to be independent 
of the place of observation. 

But the directions of the zenith and horizon vary with the 
position of the observer. If we suppose the Earth spherical, 
the vertical at any point on it will be the radius drawn from 
the Earth's centre, while the plane of the horizon will be 
a tangent plane to the Earth's surface ; both will depend 
on the place. This circumstance accounts for the difference 
in appearance of the heavens as seen simultaneously from 
different places. 

87. Earth's Rotation. The apparent rotation of the 
heavens is accounted for by supposing that the stars are at 
rest, and that the Earth rotates once in a sidereal day, from 
west to east, about an axis parallel to the direction of the 
celestial pole. The observer's zenith, horizon and meridian 
turn about the pole from west to east, relatively to the stars, 
and this causes the hour angles of the stars to increase by 360 
in a sidereal day, in accordance with observation. 

It is impossible to decide from observations of the stars 
alone whether it is the Earth or the stars which rotate, just 
as when two railway trains are side by side it is very difficult 
for a passenger in one train, when observing the other, to 
decide which train is in motion. That the Earth rotates has, 
however, been conclusively proved by means of experiments, 
which will be described when we come to treat of dynamical 
astronomy. 

88. Definitions. The Terrestrial Poles are the two 

points in which the Earth's axis of rotation meets its surface. 

The Terrestrial Equator is the great circle on the 
Earth whose plane is perpendicular to the Earth's axis. 

A Terrestrial Meridian is the section of the Earth's 
surface by a plane passing through its axis. If we suppose 
the Earth to be a sphere, a meridian will be a great circle 
passing through the terrestrial poles. 



1HE EAUTH. 65 

89. Phenomena depending on Change of Latitude. 

A ssuming the Earth to be spherical, let p Oqp'r be a meridian 
section, C being the Earth's centre, p, p the poles, q, r points 
on the equator. Then, if an observer is situated on the 
meridian at 0, the direction of his celestial pole P will be 
found by drawing .OP parallel to the Earth's axis^' Cp ( 87), 
while his zenith Z will lie in GO produced. 







Since OP is parallel to CpP lt therefore, 
angle ZOP = OCp, 

.'. altitude of pole at = WZOP = 90- OCp = qCO. 
But the latitude of has been shown to be the altitude of 
the pole ; therefore 

The latitude of a place on the Earth is the angle 
subtended at the Earth's centre by the arc of the 
meridian drawn from the place to the equator. 

Since the angle qCO is proportional to the arc qO, 

The latitude of a place is proportional to its 
distance from the equator. 

Suppose the observer to go northwards along the meridian 
from to 0', then, from what has just been shown, the 
altitude of the pole increases from qCO to Z.qCO\ hence 

The increase in the altitude of the pole (= /. OCO'} 
is proportional to the arc 00', i.e., to the distance 
travelled northwards. 



66 AStfRONOMt. 

90. Southern Latitudes. To an observer situated in tlio 
southern hemisphere of the Earth, as at 0", the North Pole of 
the heavens is below, and the South Pole, p" is above the 
horizon. The South Latitude of the place is measured by 
the altitude of the South Pole, p", and is equal to the 
angle qCO". 

At the terrestrial equator, the altitude of the pole is 
zero ; hence the pole is on the horizon. At the terrestrial 
North Pole p, the altitude of the celestial pole is 90, there- 
fore the celestial pole coincides with the zenith. Hence, 
also, an altazimuth, if taken to the North Pole, would there 
become an equatorial. 




PIG. 35. 



At the Earth's North Pole, those stars are only visible 
which are north of the equator, and they always remain 
above the horizon. 1 travelling southwards, other stars, 
whose declination is south, are seen in the south parts of the 
celestial sphere, and on reaching the Earth's equator all the 
stars will be above the horizon at some time or other, but the 
Pole Star will only just rise above the horizon, near the 
north point. After passing the equator, the Pole Star and 
other stars near the North Pole disappear. 



THE EABTfl. 6? 

91. Radius of the Earth. The Earth's radius 
may be found by measuring the distance between 
two places on the same meridian, and finding their 
difference of latitude. 

Let the places of observation be 0, 0' (Fig. 35). Let the 
latitudes qCO, qCO' be I and I' degrees respectively, and let 
the length 00' = s. We have, supposing the Earth spherical, 

ansle OCO' arc 00' 



360 circumference of Earth ' 

Of*(\ 

.'. Earth's circumference = s x = ; 
and Earth's radius = circumference = 180 . 

2?T 7T / I 

which determines the Earth's radius in terms of the data. 

By observations of this kind the Earth's radius is found to 
be very nearly 3,960 miles. For many purposes it will be 
sufficiently approximate to take the radius as 4000 miles. 
Its circumference is found by multiplying the radius by 27r, 
and is about 24,900 miles, or,' roughly, 25,000 miles. 

Conversely, knowing the Earth's radius, we can find the 
length of the arc of the meridian corresponding to any 
given difference of latitude. 

92. Metre, Nautical Mile, Geographical Mile, 
Fathom. The French Metre was originally defined as the 
ten-millionth part of the length of a quadrant of the Earth's 
meridian. 

A Nautical mile is defined as the length of a minute of 
arc of the meridian. Thus a quadrant of the meridian con- 
tains 90 x 60, or 5,400 nautical miles, and the Earth's 
circumference contains 21, GOO nautical miles. 

A Fathom is the thousandth part of a nautical mile. It 
contains almost exactly six feet. 

A Geographical Mile is defined as the length of a minute 
of arc measured, on the Earth's equator. Taking the Earth 
as a sphere, the nautiral mile and geographical mile are equal. 



68 ASTEOftOMT. 

93. The "Knot." Use of the Log Line in Naviga- 
tion. A nautical mile is sometimes called a knot. But the 
Knot is more correctly the unit of velocity used in navigation, 
being a velocity of one nautical mile per hour. Thus, a ship 
sailing 12 knots travels at 12 nautical miles an hour. 

The velocity of a ship is measured by means of the Log 
Line. This consists of a "log," or float, attached to a cord 
which can unwind freely from a small windlass. The log is 
"heaved " or dropped into the sea, and allowed to remain at 
rest, the cord being " paid out " as the ship moves away. By 
measuring the length paid out in a given interval of time 
(usually half a minute), the velocity of the ship may be 
found. To facilitate the measurement, the line has knots 
tied in it at such a distance apart that the number of knots 
paid out in the interval of time is equal to the number of 
nautical miles per hour at whioh the ship is sailing. It is 
from these that the unit of velocity derives the name of knot. 

Now one nautical mile per hour = nautical mile per 
half-minute. Hence, for this interval, the knots should be 
tied on the line at intervals of of a nautical mile apart. 

94. From the definitions of 92, 93, it is easy to reduce 
metres or nautical miles to ordinary foot and miles, and 
conversely. 

EXAMPLES. 

1. To find the number of miles in an arc of 1. 

An arc of 1 = circumference of Earth = 24900 = 69 , miles> 

360 , 360 

2. To find the number of feet in one fathom. 

By Ex. 1, 60 nautical miles = 69 ordinary miles j i.e., 60,000 
fathoms = 69jt x 5280 feet ; 

/. 1 fathom = 69 * x 528 feet = 6'086 feet. 

3. To express a metre in terms of a yard. 

By definition, 40,000,000 metres = Earth's circumference =24,900 
miles ; 

.-. 1 metre = ^^S^Ur y ards = 1 '0956 yards. 



Tttfe EARTH. 69 

95. Terrestrial Longitude. The Longitude of a 
place on the Earth is the angle between the terrestrial 
meridian through that place, and a certain meridian fixed on 
the Earth, and called the Prime Meridian. 

Thus, in Eig. 36, if PEP' represents the prime meridian, 
the longitude of any place q is measured by the angle RPq. 

The longitude of q is also measured by R Q, the arc of the 
equator intercepted between the meridian of the place and 
the prime meridian. 




FIG. 36. 

Since the latitude of q is measured by the arc Qq, we see 
that latitude and longitude are two coordinates denning the 
position of a place on the Earth just as decl. and 11. A., or 
celestial latitude and longitude define the position of a star.* 

The choice of a prime meridian is purely a matter of con- 
venience. The meridian of Greenwich Observatory is univer- 
sally adopted by English-speaking nations. The Erench use 
the meridian of Paris, and the University of Bolognahas recently 
proposed the meridian of Jerusalem as the universal prime me- 
ridian. Longitudes are measured both eastward and westward 
from the prime meridian, from to 180, not from to 360. 

*Note, however, that terrestrial latitude and longitude, being 
referred to the equator, correspond more nearly to declination and 
right ascension than to celestial latitude and longitude. 



?0 ASTEOIfOMiT. 

96. Phenomena depending on Change of Longitude. 

(i.) Let q, r (Fig. 37) be two stations in the same latitude, 
and let the longitude of q be L west of r, so that Z rPq = L. 
As the Earth revolves about its axis at the rate of 360 per 
sidereal day, or 15 per sidereal hour, the points q, r will 
be carried forward in the direction of the arrow. After an 
interval of -^ L sidereal hours, q will have revolved through 
Z and will arrive at the position originally occupied by r. 
Hence the appearance of the heavens to an observer at q will 
be same as it was, -^ L sidereal hours previously, to an 
observer at r. The stars will rise, south, and set -^ L hours 
earlier at r than at q. 

(ii.) If Aj B be two places in different latitudes, whose 
difference of longitude is Z, the transits of a star at A and 
B will take place when the meridian planes PAP' and 
PBP' (which are evidently also the planes of the celestial 
meridians of A, B respectively), pass through the direction of 
the star. Hence, in this case also, the transits will occur 
J-g- L hours earlier at B than at A. 

Now an observer at B will set his sidereal clock to indicate 
Oh. Om. Os. when T crosses the meridian of B. When T 
transits at A, the clock at B will mark -fa L h., but an 
observer at A will then set his clock at Oh. Om. Os. Hence, 
if the two clocks be brought together and com] ared, the 
clock from B will be -^ L h. faster than the clock from A. 
This fact may be expressed briefly by saying that the 
" local " sidereal time at B is T y h. faster than the local 
sidereal time at A. 

Since the Earth makes one revolution relative to the Sun 
in a solar day, in like manner the local solar time at B 
will be -jig-Z solar hours faster than the local solar time at A. 

Therefore, whether the local times be sidereal or solar, we 

have Longitude of A west of B = long, of B east of A 

= 15 {(local time at .B) (local time at A)}. 

In particular, Long, west of Greenwich 

= 15 {(Greenwich time) (local time)} 
= 15 (Greenwich time of local noon). 



THE EARTH. 



71 



97. To find the length of any arc of a given 
parallel of latitude, having given the difference of 
longitude of its extremities. 

[A small circle of the Earth parallel to the equator is 
called a Parallel of Latitude.] 

Let qr be the given arc of the parallel hqrk, I its latitude, 
and let qPr, the difference of longitudes of q and r, be = Z. 
Let a be the radius of the Earth. 




If the meridians of q, r meet the terrestrial equator in 
Q, R, we have, by Sph. Geom. (17), 

arc qr = arc QR X sin Pq = arc QR x cos I. 
But arc QR : circumference of Earth = Z : 360; 

.-. arc QR = 27T0Z/360 = 
180 



/. arc qr = 



iraL cos I 
180 



COROLLARY. Since V of arc of the equator measures a 
geographical mile, it follows that 

In latitude ?, the arc of a parallel corresponding to 
1' difference of longitude is cos I geographical miles. 



72 ASTRONOMY. 

98. Changes of Latitude and Longitude due to a 
Ship's Motion. Suppose a ship, in latitude I, to sail 
m nautical miles in a direction A degrees west of north. 
If m is small, we may easily see (by drawing a diagram) 
that the ship would arrive at the same place by sailing 
m cos .4 nautical miles due north, and then sailing msinA 
nautical miles due west. Hence, 

The ship's latitude will increase by m cos^4 minutes ( 92). 
Its W. long, will increase by m sin^ sec I minutes ( 97, cor.). 

NOTE. The shortest distance between two points on a sphere is 
along a great circle. Hence, the shortest distance between two 
places in the same latitude is less than the arc of the parallel joining 
them (except at the equator). But the difference is imperceptible 
when the arc is small. 

99. To explain the Gain or Loss of a Day in going 
round the World. If a traveller, starting from a place A, 
go round the world eastward, and if, during the voyage, the 
Earth revolves n times relative to the Sun, the traveller will 
have performed one more revolution relative to the Earth in 
the same direction, and therefore n + 1 revolutions relative to 
the Sun. Hence, to a person remaining at -df, the voyage 
will appear to have taken n days, while to the traveller, 
n + 1 days will appear to have elapsed in other words, the 
traveller will, apparently, have " gained a day." 

But, as he goes eastward, he will find the local time con- 
tinually getting faster, and he will have to move the hands 
of his watch forward Ih. for every 15, or 4m. for every 1 
of longitude. Thus, by the end of the voyage he will have 
put his watch forward through 24h., and the day apparently 
gained will be made up of the times apparently lost every 
time the watch is put forward to local time. 

Similarly, a traveller going round the world westward, 
and starting and arriving back simultaneously with the first 
traveller, will have made n 1 revolutions relative to the Sun, 
instead of n. Hence, the journey will appear to have taken 
n 1 days, and he will apparently have lost a day. 

But, during the journey, he will have been continually 
moving the hands of his watch backwards, so that the 24h. 
apparently lost will be made up of the times apparently 
gained each time the watch is put back to local time. 



THE EAETH. 73 

SECTION II. Dip of the Horizon 

100. Definitions. Let be an observer situated above 
the surface of the land or sea. Draw OT, OT tangents to the 
surface. Then it is evident, from the figure, that only those 
portions of the Earth's surface will be visible whose distance 
from the observer is less than the length of the tangents 
OT, OT. 




FIG. 38. 

The boundary of the portion of the Earth's surface visible 
from any point is called the Offing or Visible Horizon. 
Hence, if OA CB be the Earth's diameter through 0, and the 
Earth be supposed spherical, the offing at is the small circle 
TtT, formed by the revolution of T about OB, and having 
for its pole the point A vertically underneath 0. If, however, 
the Earth be not supposed spherical, the form of the offing 
will, in general, be more or less oval, instead of circular. 

Conversely, since it is observed that the " offing " at sea is 
very approximately circular, whatever be the position of 
the observer, it may be inferred that the Earth is approxi- 
mately spherical. 

The Dip of the Horizon at is the inclination to the 
horizontal plane of a tangent from to the Earth's surface. 

Hence, if HOH' be drawn horizontally (i.e., perpendicular 
to OC\ the dip of the horizon will be the angle HOT. 




74 



ASTEONOMY. 



101. To determine the Distance and Dip of the 
Visible Horizon at a given height above the Earth. 

Let h = A = given height of observer ; 
a = CA = Earth's radius; 
d OT = required distance of horizon ; 
D = L HOT = required dip expressed in circular 



D" the number of seconds in the dip D. 
(i.) By Euclid III. 36, OT 2 = OA . OB 



This determines d accurately. But in practical applications 
h is always very small compared with 2a ; therefore A 2 may be 
neglected in comparison with 2ah, and we have the approxi- 
mate formula, rf 2 = 2ah .*. d = */ (2a7i). 

(ii.) Since CTO is a right angle, 

.-. z OCT= complement of L COT '= L TOR= D. 

Therefore, D being expressed in circular measure, we hav<j 



7)- 

~ 



AT 



radius CT 




FIG. b9. 



Now, in practical cases, where the dip is small, the -arc AT 
will not differ perceptibly in length from the straight line OT. 
We may, therefore, take arc AT= d ; 



__ I2h 
~ \ a' 



THE EAETlt. 75 

To reduce to seconds, we must multiply by 180 x 60 x GO/vr, 
tbe number of seconds in a unit of circular measurement, and 
we bave 

, 180 X 60 X 60 



/2h 
V ft ' 



COROLLARY 1. Let #, h, d be measured in miles, and let 
h' be tbe number of feet in tbe beigbt h. 

Then h' = 52807& ; and taking tbe Eartb's radius a as 3960 
miles, we bave 

2x3960xA' 



a very useful formula. 

COROLLARY 2. Since tbe offing is a circle whose radius is 
very approximately equal to OT QT d, we have 

Area of Earth's surface visible from = nd 2 = lirah = f ?r7i' 
in square miles. 

*102. Accurate Determination of Dip. The use of approxi- 
mations can be avoided by the exact formula : 

toD- 

which is adapted to logarithmic computation. 

In this, as in the preceding formulae, no account has been taken 
of the effect of refraction due to the atmosphere. 

For this reason it is important to determine dip of the horizon 
by practical observations. An instrument called the Dip Sector is 
constructed for this purpose. 

Tables have also been constructed, giving the dip of the horizon 
as seen from different heights. They are of great use at sea, 
where the altitude of a star is usually found by observing its angular 
distances from the offing. 

103. Disappearance of a Ship at Sea. Wben a ship 
has passed the offing, the lower part will be the first to dis- 
appear. Let A' 0' (Fig. 38) be the position of the ship ; let its 
distance 0' be s, and let k = A 0' be tbe height above sea 
level of the lowest portion just visible from 0. By the 
approximate formula we have OT= ^/(2,a?i), 0'T= ,y/(2#) 

This formula determines the distance s at which an object of 
given height k disappears below the hori/on. 

A.STKON. G 



ASTRONOMY. 




104. Effect of Dip on the Times of Rising and 
Setting. To an observer on land, the offing is generally 
more or less broken by irregularities of the Earth's surface. 
At sea, however, the offing is well denned, and if the dip of 
the horizon in seconds be D", the visible horizon, which 
bounds the observer's view of the heavens, is represented on 
the celestial sphere by a small circle parallel to the celestial 
horizon, and at a distance D" below it (n'E's, Pig. 40). 

Hence the stars appear to rise and set when they are at an 
nngular distance D" below 
the celestial horizon. Thus 
they will rise sooner and set 
later than they would if 
there were no dip. 

Taking the observer's lati- 
tude to be I, let x', x be the 
positions of a star of decli- 
nation d, when rising across 
the visible horizon n'E's and 
the celestial horizon nEs 
respectively. Draw x' ZTperpendicular to nEs, then x'H= D". 

Then, if the star rise t seconds earlier at x' than at x, we have 
15 t = Z x'Px (in seconds of angle) 

= arc xx> = arc **'. (Sph. Geom., 17.) 

sin xP cos d 

But treating the small triangle x'xH&s plane (Sph. Geom., 24), 
and remembering that Z Pxx = 90, we have 

cos nxP ' 

.. t = If' sec d . sec nxP. 

lo 

Evidently the acceleration at rising = retardation at setting. 
COROLLARY 1. To an observer at the Equator, P 
coincides with w, .'. Z nxP = 0, 
.-. the time of rising is accelerated by -^D" sec d seconds. 

COROLLARY 2. If the star is on the equator, d = 0, 
x coincides with E, and z nEP = nP = I, 

.-. the acceleration = -&D" sec I seconds. 



THE EARTH. 77 

SECTION III. Geodetic Measurements Figure of the Earth. 

105. Geodesy is the science connected with the accurate 
measurement of arcs on the surface of the Earth. Such 
measurements may be performed with either of the two 
following objects : 

(i.) The construction of maps. 

(ii.) The determination of the Earth's form and magnitude. 
Only the second application falls within the scope of this book. 

10G. Alfred Russell Wallace's Method of Finding 
the Earth's Radius. An approximate measure of the 
Earth's radius can be readily found by means of the following 
simple experiment, due to Mr. A. 11. Wallace. 




FIG. 41. 

Let Z, M, JV(Fig. 41) be the tops of three posts of the same 
height set up in a line along the side of a straight canal. 
Owing to the Earth's curvature the straight line LM will, if 
produced, pass a little above N. Hence, in order to see Z, M 
in a straight line, an observer at the post JV^will have to place 
his eye at a point JST, a little above JV, and the height -ZTJV 
may be measured. Let JL, .Of be also measured. 

Since the posts are of equal height, Z, Jf, N will lie on a 
circle concentric with, and almost coinciding with, the 
Earth's surface. Let the vertical KN meet this circle again 
in n. By Euclid III. 36, 

KL . EM = EN. Kn; .-. Kn = KL . EMI EN, 
and Radius of Earth = \ Kn (very approximately) 
_ EL . EM 
1EN 

This method cannot be relied on where accuracy is required, 
for the small height EN is very dim cult to measure, and a 
very slight error in its measurement would affect the final 
result considerably. Moreover the observations are consider- 
ably affected by refraction. 



78 ASTROXOMT. 

107. Ordinary methods of Finding the Earth's 
Radius. "Where greater accuracy is required, the radius of 
the Earth is obtained by measuring the length of an arc of 
the meridian and determining the difference of latitude of its 
extremities; the radius may then be calculated as in 91. 

The instruments required for the observations include 
(i.) Measuring rods, such as the double bar ; 
(ii.) A theodolite, for measuring angles ; 

(iii.) A zenith sector. 

108. Measurement of a Base Line. The first step is 
to measure, with extreme accuracy, the length of the arc 
joining two selected points, several miles apart, on a level 
tract of country ; this line is called a Base Line. A series of 
short upright posts are placed at equal distances apart along 
the base line, and they are adjusted till their tops are seen 
exactly in the same vertical plane, and are on the same level 
as shown by a spirit level. Across these posts are laid 
measuring rods of metal, whose length is very accurately 
known, and these are also adjusted in a line, and made level 
by the spirit level. These rods are not allowed to touch, 
but the small distances between their ends are measured with 
reading microscopes. In this way, a base line several miles 
long can be measured correctly to within a small fraction 
of an inch f 

*109. The Double Bar. 
If the measuring rods be made 

of a single metal, their length i>. ^ iron I \j' 

will vary with the tempera- } 
ture. This disadvantage is, c 'l 
however, sometimes obviated 
by the use of the double bar 
(Fig 42). 

It consists of two bars, al, cd, one of iron, the other of brass. 
These are joined together in the middle, and to their ends are 
hinged perpendicular pointers eac, fbd of such length that 
ea : ec = /& : fd 

= coefficient of linear expansion of iron : that of brass, 
= about 11 : 18.f 

If the temperature be raised, the rods will expand, say to a'b', 
c'd'. But aa' : cc' = ea : ec, therefore e, and similarly /, will remain 
fixed. Hence the distance ef will be unaffected by the changes of 

temperature. _ 

f Wallace Stewart's Heat, Table 22. 



Brass 



K 



THE EARTH. 79 

110. Triangnlation. When once a base line has been 
measured, the distance between any two points on the Earth 
can be determined by the measurement of angles alone. For, 
calling the base line AB, let C be any object visible from 
both A and B. If the angles CAB, CBA 

be observed, we can solve the triangle H - G 

ABC and determine the lengths of the ,+''* 
sides CA, CB. Either of these sides, say ^S s 
CA, may now be taken as the base of a new 
triangle, whose vertex is another point, D. 
Thus, by observing the angles of the tri- 
angle A CD we can determine DA, DC in 
terms of the known length of AC. Pro- 
ceeding in this way, we may divide any 
country into a network of triangles connect- 
ing different places of observation A, B, (7, D, 
and the distance between any two of the 
places calculated, as well as the direction of ^C / 
the line joining them. Finally, two stations ^' 

(7, H are taken, which lie on the same meri- 
dian, and the distance CU is calculated ; in IG ' 
this way it is possible to measure an arc of the meridian. 

111. The Theodolite. The measurement of the angles 
is far easier in practice than the measurement of a base line. 
The instrument used for measuring angles is called a Theo- 
dolite, and is really a portable form of altazimuth. It is 
provided with spirit-levels, by means of which the instrument 
fan be adjusted so that the horizontal circle is truly horizon- 
tal, and the vertical axis, therefore, truly vertical; the 
direction of the north point is usually found by means of a 
compass needle. Most theodolites are only furnished with a 
small arc of the vertical circle, sufficient for measuring the 
altitude of one terrestrial object as seen from another. 

By reading the horizontal circle of the theodolite, the azimuths 
of B, C, as seen from A, are found. By using the difference of 
azimuth instead of the angle ABC, it becomes unnecessary to take 
account of the height of the various stations above the Earth. For 
if A, B, C are replaced by any other points, A', B', C', at the sea 
level, and vertically above or below A, B, G t the vertical planes 
joining them will be unaltered in position, and therefore the 
azimuths will also be unaffected. 



80 



ASTRONOMY. 




112. Having thus found, with great accuracy, the length 
of the arc joining two stations on the same meridian, it only 
remains now to observe their difference of latitude. 

The Zenith Sector is the most useful instrument for 
this purpose. It consists essentially of a long telescope ST 
(Eig. 44), mounted so as to turn about a horizontal axis, A, 
near its object-glass ; this axis is adjusted to 
point due east and west (as in the transit 
circle). Attached to the lower end near the 
eye piece is a graduated arc of a circle GH, 
whose centre is at A. The line of collimation 
of the telescope is indicated by cross-wires 
placed in the field of view. A fine plumb- 
line, AP, is attached to the axis A, and hangs 
freely in front of the graduated arc. The 
plumb-line should mark zero when the line of 
collimation points to the zenith. When the 
instrument is pointed to any star, the reading 
opposite the plumb-line will be the star's zenith distance 
This reading can be determined with great accuracy by 
means of a reading microscope. 

113. A star is selected which transits near the zenith* 
and its meridian zenith distances are observed at the two 
stations. Let these be s and z' degrees. Then if /, and /. 2 
are the latitudes of the stations, and d the declination, 
we have, by 24, 

l'-l= (d-z')-(d-z) = z-z'. 

Hence, if s is the measured length of the arc of the meri- 
dian joining the stations, and r the radius of the Earth, 91 
gives 

18Q * _ 13 _ 



FIG. 44. 



whence the Earth's radius is found. 



* This position is chosen because the effects of atmospheric 
refraction are least in the neighbourhood of the zenith, 



THE EARTH. 81 

114. Exact Figure of the Earth. If the Earth were 
an exact sphere, the same value would be found for the 
radius r in whatever latitude the observations were made. 
But in reality the length of a degree of latitude, and therefore 
also r, is found to be larger when the observation is made near 
the poles than when made near the equator, and hence it is 
inferred that the meridian curve is somewhat oval. 

Let PQP'R represent the meridian curve, 00' two near 
places of observation on it. Then, if 0J5Tand O'K be drawn 
normal (i.e., perpendicular) to the Earth's surface at 0, 0', 
they will be the directions of the plumb lines of the zenith 
sectors at 0, 0'. Hence the observed difference of latitudes 
or meridian altitudes at 0, 0' will give the angle OKO'. 

Eegarding the small arc 00' as an arc of a circle whose 
centre is JT, we shall have approximately, 

Circular measure of OKO' = arc 00' -f- OJT, 
arc 00' 180 s 



_ 

circ. measure of OKO' TT I' V 
and hence r, calculated as in 113, is the length OK. 

The length OK is called the 
radius of curvature of the arc, 
and K is called the centre of 
curvature ; they are respec- 
tively the radius and centre of 
the circle whose form most nearly 
coincides with the meridian along 
the arc 00'. 

This radius of curvature OK 
is not, in general, equal to C, 
the distance from the centre of 
the Earth, owing to the Earth FlG - 45 - 

not being quite spherical. 

As the result of numerous observations, the meridian curve 
is found to be an ellipse (see Appendix), whose greatest 
and least diameters, called the major and minor axes, are 
the Earth's equatorial and polar diameters respectively. The 
Earth's surface is the figure formed by making the ellipse 
revolve about its minor axis POP'. This figure is called an 
oblate spheroid. 




ASTRONOMY. 



115. To find the Equatorial and Polar Radii of Cur- 
vature of the meridian curve, supposing 1 it to be an 
ellipse. Let PQP'R be the ellipse. Let 2, 2i be the 
lengths of its equatorial and polar diameters QCR, PCP'. 
Let r v r z be the required radii of curvature at Q and P 
respectively. 

Take any point on the ellipse, 
and let the normal at meet the 
two axes in G and g respectively. 

It is proved in treatises on 
Conic Sections* that 
OG : Og = CP* : C& = i 2 : a\ 

First take very near to Q. 
Then OG will become equal to 
the radius of curvature r^ ; also 
Og will evidently become ulti- 
mately equal to CQ or a. 

Therefore, ^ : a = b* : a? ; 

Next take very near to P. 
to I and Og to r%. 

Therefore, I : r 2 = W : 2 ; 
Thus r x , r 2 are found in terms of a, 




r = 



Then G will become equal 



r = 



and I may be found ; 
I r*r. 



Conversely, if r, and r 2 are known, 
for, by solving, we find a = %/(rfr 

~We notice that since a > J, .*. r^ < r r 

That the equatorial radius of curvature is less than the 
polar is also evident from the shape of the curve. This, as the 
figure shows, is most rounded at Q, It, and flattest or least 
rounded at P, P'. Hence it will require a smaller circle to 
fit the shape of the curve at the equator than at the poles. 

116. Exact Dimensions of the Earth. The lengths of 
the Earth's equatorial and polar semi-diameters, , i, are 
a = 3963-296 miles, I = 3949'791 miles. 

Thus, the Earth's equatorial semi-diameter exceeds its 
polar semi-diameter by 13-505 miles. 



* Appendix, Ellipse (9). 



THE EAETH. 83 

The mean radius of an oblate spheroid is the radius of a 
sphere of equal volume, and is equal to ^/(a-1}. Thus, the 
Earth's mean radius is approximately 3958-8 miles. 

The ellipticity or compression (0) is the fraction 



For the Earth, c = - nearly. 
293 

The eccentricity (e) is given by the relation 



a~ 

Hence L l = s (I-* 2 ) = 8 (1 e)*; 

.-. !-* = (! --<?)" = 1 



Since c is small, 2 c 2, approx. ; .'. e* = 20, approx., 
which gives the Earth's eccentricity e '0826. 

117. Geographical and Geocentric Latitude. The 
Geographical Latitude of a place is the angle which the 
normal to the Earth's surface at that place makes with the 
plane of the equator. It is the latitude denned in 18, 
Thus, L QGO (Fig. 46) is the geographical latitude of 0. 

The Geocentric Latitude is the angle subtended at the 
Earth's centre by the arc of the terrestrial meridian between 
the place and the equator. Thus, / QCO is the geocentric 
latitude of 0. 

*118. Relations between the Geocentric and Geographical 

Latitudes. Let / QGO = I, Z QGO = I'. Draw ONperp. to CQ. 

Then GN : CN = OG : Og = 6 2 : a 2 ; .'. NO/CN = (NOJON) x (& 2 /o-) { 
/. tan I' = tan I x & 2 /a 2 = (1- e 2 ) tan I. 

We deduce also tan (l-l f ) = ^ S1 ^ 2 ^ 2 = ie"sin2l (approx.), 
since e 2 is small. 



84 ASTRONOMY. 



EXAMPLES. III. 

1. Show that the locus of points on the Earth's surface at which 
the Sun rises at the same instant is half a great circle ; and state 
the corresponding property possessed by the other half. 

2. Find the least height of a mountain in Corsica in order that it 
may be visible from the sea-level at Mentone, at a distance of 80 



3. At the equator, in longitude L, a given vertical plane declines 
a from the north towards the west ; find the latitude and longitude 
of the places to whose horizon the given plane is parallel. 

4. Prove that, at either equinox, in latitude I, a mountain whose 
height is 1/n of the Earth's radius will catch the Sun's rays in the 

morning , / hours bei'ore he rises on the plain at the base. 

7T COSt Y n 

5. Estimate to the nearest minute the value of this expression for 
a mountain three miles high in latitude 45. 

6. Find the distance of the horizon as seen from the top of a hill 
1056 feet high. 

7. Find, to the nearest mile, the radius of the Earth, supposing the 
visual line of a telescope from the top of one post to the top of 
another post two miles off, cuts a post, half way between, 8 inches 
below the top, the posts standing at equal heights above the water 
in a canal. 

8. In Question 7, what would be the length of a nautical mile, 
adopting the usual definition. 

9. Supposing the Earth spherical, and of radius r, and neglecting 
the refraction of the air, show that, if from the top of a mountain 
of height a above the level of the sea, the summit of another 
mountain is seen beyond the horizon of the sea, and at an elevation 
e above the horizon, and if its distance be known to be D, its height is 
approximately given by 

a .ran. D (2-J*i 



10. A railway train is moving north-east at 40 miles an hour in 
latitude 60; find approximately, in numbers, the rate at which it is 
phanging its longitude. 



THE EARTH. 85 



MISCELLANEOUS QUESTIONS. 

1. Explain the different systems of coordinates by which a star's 
position is fixed in thb hcnvenn. 

2. Show, by a figure, where a star will be found at 9 p.m. on the 
5th of June in latitude 50N., if the star's right ascension is 12 hours 
and its declination 5 south. 

3. Define dip, azimuth, culmination, circumpolar, zenith. Why 
would it be insufficient to define the declination of a star as its 
distance from the equator measured along a declination circle ? 

4. Three stars, A, B, C, are on the same meridian at noon, JB being 
on the equator, and A and C equidistant from B on either side. 
Prove that the intervals between the setting-times of A and B and 
J? and C are equal. 

5. Show how to find approximately the Sun's R.A. at a given 
date. Obtain its approximate value for March 1, August 10, 
October 23, and January 15. 

6. Describe the transit circle. 

7. Define a morning and evening star. Show that on the 1st of 
September a star, whose declination is 0, and R.A. llh. 28m., is an 
evening star, but that it is a morning star three weeks Inter. 

8. Assuming the Earth to be a sphere, show how its radius may 
be practically measured. 

9. Explain clearly the nature and uses of the zenith sector. 

10. A, B, C are the tops of the masts of three ships in a line, and 
are at equal heights above the sea- level, and is the centre of the 
Earth. If the distance BC be x miles, and r is the Earth's radius 
in miles, show that L BAC = \ L BOG ; and hence deduce that 

zIU C= 18Qx6Qx6Q JL seconds. 
TT 2r 

Find this angle, having given so = 2, r = 3960, IT = 3f. 



86 A.STEONOMT. 



EXAMINATION PAPER. III. 

1. Assuming the Earth to be a sphere, show that, as we travel from 
the equator due north, our astronomical latitude (i.e., the altitude of 
the Pole) will increase. Taking this increase as 1 for every 
69 miles, find the circumference and the radius of the Earth. 

2. Define the metre, the nautical mile, and the knot, and calculate 
their values in feet and feet per second respectively, taking the 
Earth's radius as 3960 miles. 

3. How is the speed of a ship estimated ? Find, in feet, the dis- 
tance apart of the knots on a log line, so constructed that the 
number run out in half a minute measures the ship's velocity in 
nautical miles per hour. 

4. What are the difficulties in measuring an arc of the meridian 
and how are they met ? 

5. Find the Earth's radius in fathoms, and in metres. Express 
the nautical mile in French units of length. 

6. Obtain formulae for the distance of the visible horizon from a 
place whose height is given. Deduce that, if the height h be 

measured in inches, the distance in miles will be*/ , taking the 

V 8 
Earth's radjus as 3960 miles. 

7. Define the dip of the horizon, and show how to find it. Prove 
that the number of seconds in the dip is nearly 52 times the 
distance in miles of the offing. 

8. If A, B, and G be the tops of three equal posts arranged in 
order two miles apart along a straight canal, show that the straight 
line AB passes 5 feet 4 inches above C, and that AC passes 2 feet 
8 inches below B. 

9. Find the length of a given parallel of latitude intercepted 
between two given circles of longitude. 

10. Is the Earth an exact sphere ? Show that a degree of latitude 
increases in length as we go northward. Distinguish a nautical 
from a geographical mile. 



CHAPTER IV. 



THE SUN'S APPARENT MOTION IN THE ECLIPTIC. 

SECTION I. The Seasons. 

119. In Section III. of Chapter L* we described the Sun's 
annual motion among the stars, and showed how, in con- 
sequence of this motion, the Sun's right ascension increases 
at an average rate of nearly 1 per day, while his declination 
fluctuates between the values 23 27J' north, and 23 27J' 
south of the equator. We shall now show how this annual 
motion, combined with the diurnal rotation about the poles, 
gives rise to the variations, both in the relative lengths of day 
and night, and in the Sun's meridian altitude, during the 
course of the year ; how these variations are modified by the 
observer's position on the Earth ; and how they produce the 
phenomena of summer and winter. 

Although both the diurnal and annual apparent motions of 
the Sun are known to be really due to the Earth's motion, it 
will be convenient in this section to imagine the Earth to be 
fixed, while the Sun and stars are moving ; thus the zenith, 
pole, horizon, meridian, and equator will be considered fixed, 
as they actually appear to be to an observer on the Earth. 

As the change in the Sun's declination during a single day 
is very small, the Sun's apparent path in the heavens from 
morning till night is very approximately a small circle parallel 
to the equator, and may be regarded as such for purposes of 
explanation. The effects of the variation in the declination 
will, however, become very apparent when we compare the 
Sun's diurnal paths at different seasons of the year. 

Throughout this section we shall denote the obliquity of 
the ecliptic by ", the Sun's declination at any time by ^, his 
zenith distance at noon by z, and the observer's latitude by I. 

* The student will do well to revise Chapter I., Section III., 
before proceeding further. 



88 ASTRONOMY. 

120. Zones of the Earth. Definitions. From 24 
it is evident that if the Sun passes through the zenith at 
noon, d must = I. 

But d lies between i (north) and t (south). 

Therefore I must lie between the limits i N. and i S. 

Thus, if the Sun be vertically overhead at some time in the 
year, the latitude must not be greater than 23 27|' N. or S. 

Again, from 28 we sec that the Sun, like a circumpolar 
star, will remain above the horizon during the whole of its 
revolution provided that 90^ < I. 

This requires that I > 90- i. 

Thus, if the Sun be visible all day long during a certain 
period of the year, the latitude must be greater than 66 32^' 
K. or S. 

These circumstances have led to the following definitions. 

The Tropics are the two parallels to the Earth's equator 
in north and south latitude , or 23 27|-'. The northern 
tropic is called the Tropic of Cancer, the southern the 
Tropic of Capricorn. 

The Arctic and Antarctic Circles are respectively the 
parallels of north and south latitude 90 *, or 66 32f. 

These four parallels divide the Earth's surface into five 
regions or zones. 

The portion between the tropics is called the Torrid Zone. 

The portion between the tropic of Cancer and the arctic 
circle is called the North Temperate Zone. The portion 
between the tropic of Capricorn and the antarctic circle is 
called the South Temperate Zone. 

The portions north of the arctic circle, and south of the 
antarctic circle are called the Frigid Zones, and are distin- 
guished as the Arctic and Antarctic Zones. 

121. Sun's Diurnal Path at Different Seasons and 
Places. "We shall now describe the various appearances 
presented by the Sun's diurnal motion at different times of 
the year, beginning in each case with the vernal equinox. 
We shall first suppose the observer at the Earth's equator, 
and shall then, describe how the phenomena are modified as 
he travels northward towards the pole. 



SUN'S APPARENT MOTION IN THE ECLIPTIC. 



89 



122. At the Earth's equator, I = 0, and the poles of 
of the celestial sphere are on the horizon (P, P', Fig. 47). 
Hence, between sunrise and sunset, the Sun has always to 
revolve about the poles through an angle 180, and the days 
and nights are always equal, each being 12 hours long. 

On March 21 the Sun is on the celestial equator, and it 
describes the circle EZW, rising at the east point, passing 
through the zenith at noon, and setting at the west point. 

Between March 21 and Sept. 23, the Sun is north of 
the celestial equator; it therefore rises north of E., transits 
north of the zenith Z, and sets north of W. Its IS", meridian 
zenith distance 2 is always equal to its !N". declination d 
(since by 24, 2 d I and I = 0) . 

Hence, from March 21 to June 21, z increases from to 
i N. On June 21, z has its greatest JN". value f, and the 
Sun describes the circle E'QW, where ZQ' = i. 




From June 21 to Sept. 23, z decreases from i to 0. 

On Sept. 23, the Sun again describes the great circle EQ W. 

Between Sept. 23 and March 21, the Sun is south of the 
equator, and therefore it transits south of the zenith. "We 
now have z = d, both being S. 

From Sept. 23 to Dec. 22, the Sun's south Z.D. at noon, 
2, increases from to i. 

On Dec. 22, 2 has its greatest value i (south) and the Sun 
describes the circle E 'Q," W" where ZQ, " = i. 

From Dec. 22 to March 21, 2 diminishes again from to 0. 
On March 21, the Sun again describes the circle EQW, and 
the same cycle of changes is repeated the following year. 



90 ASlRONOM*. 

123. In the Torrid Zone North of the Equator". 

On March 21, the Sun describes the equator KQW (Fig, 
48), rising at ^and setting at W. Here L ZPE L ZPW 
90, and the day and night are each 12h. long. The 
Sun transits S. of the zenith at Q, where ZQ = z =7. 

From March 21 to June 21, d increases from to t, and 
the Sun's diurnal path changes from EQVto E'QW. 

The hour angles at rising and setting increase from ZPE 
and ZPWiQ ZPE' and ZPW, respectively ; hence the days 
increase and the nights decrease in length. The day is 
longest on June 21, when the hour angle ZPE' is greatest. 
The increase in the day is proportional to the angle EPE', 
and is greater the greater the latitude I. 

At first the Sun transits S. of the zenith, and z = ld. 

"When d = , z = 0, and the Sun is directly overhead at noon. 

After this, the Sun transits N. of the zenith, and z = d L 

On June 21,2 attains its maximum N. value ZQ' = il. 

From June 21 to Sept. 23, the phenomena occur in the 
reverse order. The diurnal path changes gradually back to 
EQW. The day diminishes to 12h. The Sun, which at first 
continues to transit N". of the zenith, becomes once more ver- 
tical at noon when d again = I, and then transits S. of the 
zenith. 

From Sept. 23 to Dec. 22, the Sun's path changes from 
EQWto E"Q'W". 

The eastern hour angle at sunrise decreases to ZPE"; thus 
the days shorten and the nights lengthen. The day is 
shortest on Dec. 22. 

Also z increases from I to 1 -f i. 

On Dec. 22, s attains the maximum value ZQ" = -f-, and 
the Sun is then furthest from the zenith at noon. 

From Dec. 22 to March 21, the length of the day increases 
again to 12 hours, and the Sun's meridian zenith distance 
decreases to z = L 

124. On the Tropic of Cancer, I = i. The variations 
in the lengths of day and night partake of the same general 
character as in tbe Torrid Zone. But the Sun only just 
reaches the zenith at noon once a year, namely, on the longest 
day, June 21. At other times the Sun is south of the zenith 
at noon, and z attains the maximum value 2* on December 22. 



TIIE SUN'S APPABENT MOTION IN THE ECLIPTIC. 9l 

Z Q' 2 





P' 



FIG. 49. 



125. In the North Temperate Zone I > i but < 90 - i. 

Here the variations in the lengths of day and night are 
similar, hut more marked, owing to the greater latitude. 

On March 21, the Sun describes the equator EQWR (Fig. 
49), which is bisected by the horizon ; hence the day is 1 2h . long. 

The length of the day increases from March 21 to June 21. 
The day is longest on June 21, when the jSun describes 
E'Q'WR', and the hour angles ZPE', ZPW are greatest. 

The days diminish to 12h. on Sept. 23, when the Sun again 
describes EQ, WE. The day is shortest on Dec. 22, when the 
Sun describes E"Q!'W"R". 

From Dec. 22 to March 21, the days increase in length, and 
on March 21 the day is again 12 hours long. 

The difference between the longest and shortest days is the 
time taken by the Sun to describe the angles E'PE", W"PTP', 
and is therefore 



= iV ( ^ E'PE" + L W'PW} = A . / E'PE". 

It will be seen that L E'PE" is greater in Fig. 49 than in 
Fig. 48, thus the variations are more marked in the tem- 
perate zone than in the torrid zone. The variations increase 
as the latitude increases. 

The Sun never readies the zenith' in the temperate zone, 
but always transits south of the zenith. The Sun's zenith 
distance at noon is least on June 21, when z = ZQ ' = li, 
and is greatest on Dec. 22, when % = ZQ" = l+i. At the 
equinoxes (March 21 and Sept. 23), z = ZQ = /. 

ASTEON. H 



92 ASTRONOMY. 

126. On the Arctic Circle, I = 90 t. Hence on June 
21, when the Sun's KP.D. = 90-*', the Sun at midnight 
will only just graze the horizon at the north point without 
actually setting. On Dec. 22 at noon, the Sun's Z.D. = 90, 
and the Sun will just graze the horizon without actually 
rising. As in the preceding case, the days increase from Dec. 22 
to June 21, and decrease from June 21 to Dec. 22; on 
March 21 and Sept. 23, the day and night are each 12h. long. 

127. In the Arctic Zone we have l> 90- 1, and the 
variations are somewhat different (Fig. 50). 

On March 21, the Sun describes the circle EQW, and the 
day is 12h. long. 

As d increases, the days increase and the nights decrease, 
and this continues until d = 90 I. When this happens, 
the Sun at midnight only grazes the horizon at n. 

Subsequently, while ^>90 I, the Sun remains above 
the horizon during the whole of the day, circling about the 
pole like a circumpolar star. This period is called the Per- 
petual Day. 

During the perpetual day, the Sun's path continues to rise 

higher in the heavens every twenty -four hours until June 21, 

when the Sun traces out the circle R' Q'. The Sun's least and 

. greatest zenith distances will then be ZQ! = I i , and 

ZR' 180 tZ respectively. 

After June 21, the Sun's path will sink lower and lower. 

When d is again 90 I the perpetual day will end. 
Subsequently, the Sun will be below the horizon during 
part of each day. The days will then gradually shorten and 
the nights lengthen. 

On Sept. 23, the Sun will again describe the circle EQ, W, 
and the day and night will each be 12 hours long. 

The days will continue to diminish till the Sun's south 
declination d' 90 L When this happens the Sun at noon 
will only just graze the horizon at s. 

While d' >90 Z, the Sun remains continually below the 
horizon. This period is called the Perpetual Night. 

On Dec. 22 the Sun traces out the circle R"Q" below the 
horizon. 

When d' is again = 90 /, the perpetual night will end. 

Subsequently, the day will gradually lengthen until 
March 21, when it will again be 12 hours long. 



THE SUN'S APPARENT MOTION IN THE ECLIPTIC. 98 

Z P 




FIG. 50. 




Sun's altitude mil attain its greatest val ,' on June 21 
when the Sun will trace out the circle QK ' 

21 ther night. 






. 

of the equator. In fact, if we consider two antipodal 
or places at opposite ends of a diameter of the Earth 
at one place will coincide with the night at the other 



l , equat r and antarctic c Me, 

the longest day, and June 21 the shortest. 

Within the antarctic circle there will be perpetual day for 
j certam penod before and after Dec. 22, and perpetual 
for a certain period before and after June 21. 

_ in _ _ _ _ 



r V OF THK 

UNIVERSITY 



94 ASTRONOMY. 

The variations in the Sun's north zenith distance at noon 
will be the same as the variations in the south zenith distance 
in the corresponding north latitude six months earlier.* 

130. The Seasons. Having thus described the variations 
in the Sun's daily path at different times and places, we shall 
now show how these variations account for the alternations 
of heat and cold on the Earth. 

Astronomically, the four seasons are denned as the portions 
into which the year is divided by the equinoxes and the 
solstices. Thus, in northern latitudes, 

Spring commences at the Yernal Equinox (March 21), 
Summer ,, ,, Summer Solstice (June 21), 

Autumn ,, ,, Autumnal Equinox (Sept. 23), 

Winter ,, Winter Solstice (Dec. 22). 

It is obvious that the temperature at any place will depend 
in a great measure upon the length of the day. While the 
Sun is above the horizon, the Earth is receiving a considerable 
portion of the heat of his rays, the remaining portion being 
absorbed by the Earth's atmosphere through which the rays 
have to pass. When the Sun is below the horizon, the 
Earth's heat is radiating away into space, although the heated 
atmosphere retards this radiation to a considerable extent. 
Thus, on the whole, the Earth is most heated when the days 
are longest, and conversely. 

The variations in the Sun's meridian altitude have a still 
greater influence on the temperature. When the Sun's rays 
strike the surface of the Earth nearly perpendicularly, the 
same pencil of rays will be spread over a smaller portion of the 
surface than when the rays strike the surface at a considerable 
angle ; hence the quantity of heat received on a square foot 
of the surface will be greatest when the Sun is most nearly 
vertical. By this mode of reasoning it is shown in Wallace 
Stewart's Text-Boole of Light, 10, that the intensity oi 
illumination of a surface is proportional to the cosine of the 
angle of incidence, and the same argument holds good with 

* The student will find it instructive to trace out fully the varia- 
tions in S. latitudes corresponding to those described in 122-128. 
See diagram, p. 421. 



IN THE ECLIPTIC. 95 

regard to radiant heat as well as light. Hence the Sun's heat- 
ing power when ahove the horizon is always proportional to 
the cosine of the Sun's zenith distance or the sine of its altitnde. 
In this proof, however, the absorption of heat by the 
Earth's atmosphere has been neglected. But when the 
Sun's rays reach the Earth obliquely, they will have to pass 
through a greater extent of the Earth's atmosphere, and 
will, therefore, lose more heat than when they are nearly 
vertical. This cause will still further increase the effect of 
variations in the Sun's altitude in producing variations in the 
temperature. 

131. Between the Tropics the combination of the two 
causes above described tends to produce high temperatures, 
subject only to small variations during the year. The Sun's 
meridian altitude is always very great, and the variations in the 
lengths of day and night are small. If the latitude be north, the 
Sun's heating power is greatest while the Sun transits north 
of the zenith. During this period the Sun's meridian 
altitude is least when the days are longest. Thus the effects 
of the two causes in producing variations in the Sun's heat 
counteract one another, to a certain extent, and give rise to 
a period of nearly uniform but intense heat. 

In the North Temperate Zone, the Sun is highest at 
noon when the days are longest, and therefore both causes 
combine to make the spring and summer seasons warmer 
than autumn and winter. But the highest average tempera- 
tures occur some time after the summer solstice, and the 
lowest temperatures occur after the winter solstice ; for 
the Earth is gaining heat most rapidly about the summer 
solstice, and it continues to gain heat, but less rapidly, for 
some time afterwards. Similarly, the Earth is losing heat 
most rapidly at the winter solstice, and it continues to lose 
heat, but less rapidly, for some time afterwards. Por this 
reason, summer is warmer than spring, and winter is colder 
than autumn. ' 

As we go northwards, the Sun's altitude at noon becomes 
generally lower throughout the year, and the climate therefore 
becomes colder. At the same time, the variations in the length 
of the day become more marked, causing a greater fluctua- 
tion of temperature between summer and winter. 



96 ASTRONOMY. 

Within the Arctic Circle there is a warm period during 
the perpetual day, but the Sun's altitude is never sufficiently 
great to cause very intense heat. During the perpetual night 
the cold is extreme ; and the low altitude of the Sun, when 
above the horizon at intermediate times, gives rise to a very 
low average temperature during the year. 

In the Southern Hemisphere the seasons are reversed ; 
for, in south latitude I, when the Sun's south declination is d, 
the same amount of heat will be received from the Sun as in 
north latitude I, when his north declination is d. Hence, the 
seasons corresponding to our spring, summer, autumn and 
winter will begin respectively on September 23, December 22, 
March 21, and June 21, and will be separated from the corre- 
sponding seasons in north latitude by six months. 

132. Other Causes affecting the Seasons and 
Climate. It is found (as will be explained in the next 
section) that the Sun's distance from the Earth is not quite 
constant during the year. The Sun is nearest the Earth 
about December 3 1 , and furthest away on July 1 (these are 
the dates of perigee and apogee respectively) . As shown in 
Wallace Stewart's Text-Book oj Light, 9, the intensity of 
illumination, and therefore also of heating, due to the Sun's 
rays, varies inversely as the square of the Sun's distance. 
Hence the Earth receives, on the whole, more heat from the 
Sun after the winter solstice than after the summer solstice. 
This cause tends to make the winter milder and the summer 
cooler in the northern hemisphere, and to make the summer 
hotter, and the winter colder in the southern hemisphere. 

The variations in the Sun's distance are, however, small, 
and their effect on the seasons is more than counter- 
acted by purely terrestrial causes arising from the unequal 
distribution of land and water on the Earth. The sea has a 
much greater capacity for heat than the rocks forming the 
land ; it is not so readily heated or cooled. In the southern 
hemisphere the sea greatly preponderates, the largest land- 
surfaces being in the northern hemisphere. Hence, the 
climate of the southern hemisphere is generally more equable, 
and the seasons are not so marked as in the northern hemi- 
sphere, quite in contradiction to what we should expect from 
the astronomical causes. 



THE SUN'S APPAKENT MOTION IN THE ECLIPTIC. 97 

133. Times of Sunrise and Sunset. The times of 
sunrise and sunset at Greenwich are given for every day 
of the year in Whitaker > & and other almanacks. For any 
other latitude, the Sun's declination must be found from the 
almanack, the times of sunrise and sunset can then be found 
by means of tables of double entry constructed for the pur- 
pose (29). These are called ''Tables of Semidiurnal 
and Seminocturnal Arcs." . They give, for different latitudes 
and declinations, the interval between apparent noon 
and sunset, .#., the apparent time of sunset, or half the 
length of the day. Subtracting this from 12 hours, the 
apparent time of sunrise is found, and is half the length 
of the night. 

If, as in 129, we consider two antipodal places A 
and S, the planes of their horizons will be parallel, and the 
Sun will be above the horizon at A when he is below the 
horizon at J3, and vice versd. Hence, the apparent time of 
sunrise (measured from noon) in N. latitude I will be the 
apparent time of sunset (measured from midnight) in S. 
latitude I on the same date. 

For this reason the tables are usually constructed only for 
N. latitudes. For S. latitudes they give the time of sunrise 
instead of sunset. 

The times found in this manner will be the local solar times. 
To reduce to Greenwich solar time we must add or sub- 
tract 4m. for each degree of longitude, according as the place 
is W. or E. of Greenwich. 

134. To find the length of the perpetual day and 
night at places within the Arctic or Antarctic 
Circles. 

The perpetual day lasts while the Sun's declination at local 
midnight is greater than the colatitude (or complement of the 
latitude), during spring and summer. The perpetual night 
lasts while the Sun's S. decl. at local noon is greater than the 
colat. during autumn and winter. The Sun's decl. at Green- 
wich noon being given for every day of the year, in the 
Nautical Almanack, it is easy to find, to within a day, 
the durations of the perpetual day and night in any given 
latitude greater than 66 32|'. 



98 ASTRONOMY. 

135. To find the time the Sun takes to rise or 
set. Let D" be the Sun's angular diameter, measured in 
seconds. When the Sun begins to rise, his upper limb just 
touches the horizon, and his centre is at a depth \D" below 
the horizon. When the Sun has just finished rising, his 
lower limb touches the horizon, and his centre is at an altitude 
|_D" above the horizon. During the sunrise, the centre rises 
through a vertical height D". The problem is closely similar 
to that of 104, where the effect of dip is considered. Hence 
if t seconds be the time taken in rising, d the declination of 
the Sun's centre, and x the inclination to the vertical of the 
Sun's path at rising (Hx'x or nxP, Tig. 40) we have 

t = -jV D" sec d sec #, 

= 4 sec d sec x x (O's angular diameter in minutes). 
As in 104, this gives, for a place on the equator, 

t -^D n sec d, 
and at an equinox in latitude ?, 

t = T V D" sec I. 

EXAMPLE. At an equinox in latitude 60, the O's angular 
diameter being 32', 

the time taken to rise will be = 4 x 32 x sec 60 seconds 
= 256s. = 4m. 16s. 

136. Note. It may be mentioned that, owing to atmos- 
pheric refraction, the Sun really appears to rise earlier and 
set later than the times calculated by theory. As the pheno- 
mena of refraction will be discussed more fully in Chapter 
VI., it will be sufficient to mention here that the rays of light 
from the Sun are bent to such an extent by the Earth's 
atmosphere that the whole of the Sun's disc is visible when it 
would just be entirely below the horizon if there were no 
atmosphere. 

Moreover, there is daylight, or rather twilight, for some 
time after the Sun has vanished, so that what is commonly 
called night does not begin for some time after sunset. 

For the same reasons, the perpetual day at a place in the 
arctic circle is lengthened, and the perpetual night shortened, 
by several days. 

The time taken in rising and setting is, however, prac- 
tically UTI affected. 



99 



SECTION II. The Ecliptic. 



137. The First Point of Aries. In determining the 
right ascensions of stars, the first step must necessarily be 
to find accurately the position of the first point of Aries, since 
this point is taken as the origin from which R.A. is measured. 
In other words, we must first find the R.A. of one star. 
When this is known we can use that star as a " clock star," 
to determine the sidereal time and clock error ; and, these 
being known, we can then find the R.A. of any other star, as 
explained in Chapter II. But until the position of T has 
been found, the methods of Chapter II. will only enable us 
to find the difference of R.A. of two stars by observing the 
difference of their times of transit, as indicated by the astro- 
nomical clock, and will determine neither the sidereal time 
nor the clock error, nor the R.A.'s of the stars. 

138. First Method. The position of T may be found 
thus : At the vernal equinox the Sun's declination changes 
from south to north, or from negative to positive. Let the 
Sun's declination be observed by the Transit Circle at the pre- 
ceding and following noons, and let the observed values be 
^and -f^ 2 (.*., ^ S., and d t IT.). Let t v 2 be the corre- 
sponding times of transit of the Sun's centre, as observed by 
the astronomical clock, and let T^ the time of transit of any 
star, be also observed. Then, 

T tfj = difference of R. A. of star and Sun at first noon, 
Tt z = ,, ,, ,, at second noon. 

Let T rfj = ^ and T t z = 2 . "We have 
Increase in Sun's decl. in the day = d% ( d l ) = d z + d ly 

,, ,, R.A. ,, = t t t l a l a. 2 , 
and both coordinates increase at an approximately uniform 
rate during the day. 

Therefore the Q's decl. will have increased from d l to 
in a time d l /(d l + d^ of a day, and the corresponding increase 
in R.A. will be 

fa-oa) x dj(d l + d,\ 
The Sun is now at T, .' O's R.A. is now = 0. Hence, 

The star's R.A. = a, - ^"^ *** + A 



100 A.STEONOMT. 

*139. Flamsteed's Method for finding the First 
Point of Aries. The principle of the method now to be 
described is as follows : Let 8 lt $ be two positions of the 
Sun shortly after the vernal and before the autumnal equinox 
respectively, and such that the declinations S l J/j and SM are 
equal. Then the right-angled triangles r^/"A and ^MS 
will be equal in all respects, and we shall therefore have 




FIG. 52. 

At noon, some day shortly after March 21, the Sun is 
observed with the Transit Circle, say when at 8 V We thus 
determine its meridian zenith distance z 15 and also the dif- 
ference between the times of transit of the Sun and some fixed 
star x, whose R.A. is required. This difference, which is the 
difference of E.A. of the Sun and star, we shall call a r If 
d l be the Sun's declination, and I the observer's latitude, we 
shall have 

= a 



We now have to determine J/7V, the difference of R.A. of the 
Sun and star shortly before September 23, when the Sun'g 
declination SMis again equal to d r But the Sun can only 
be observed with the Transit Circle at noon, and it is highly 
improbable that the Sun's declination will again be exactly 
equal to d 1 at noon on any day. We shall, however, find two 
consecutive days in September on which the declinations at 
noon, S 2 M 2 and $ 3 Jf 3 , are respectively greater and less than d^ 



THE SUN'S APPARENT MOTION^ Iff T3l2 



Let 2 2 and 2 8 be the observed meridian zenith distances at 
3 and S & ; d% and <? 8 the corresponding declinations S. 2 lf. 2 , 
(S' 3 I/3 ; 2 an ^ ^3 the observed arcs M^N and J/g-ZV, being the 
differences of R. A. of the Sun and star on the two days. 

During the day which elapses between the observations at 
$a> $s> we ma y assume that the Sun's decl. and R.A. both 
vary at a uniform rate, so that the change in the decl. is 
always proportional to the corresponding change in R.A.* 

Therefore, 



and MN= M^N-M,M= a,- 
Now we have shown that 



-M 1 N= HN- ^ 



-12 hours; 
= 6h. + 



This determines T-ZV, the star's R. A., in terms of 15 a v 3 , the 
observed differences between the times of transit of the Sun 
and star, and d lt d^ d^ the Sun's declinations at the three 
observations. But we need not even find the declinations, for 

d l = l-z v d, = l-z v d s = l-% ; 
therefore, substituting, we have 
The star's R.A., r^= 6h.+f j ^-f^-^^ (a^-a,) } . 

2 3~ 2 2 

In applying either of the above methods to the numerical calcula- 
tion of the right ascension of any star, it is advisable to follow the 
various steps as we have described them, instead of merely sub- 
stituting the numerical values of the data in the final formulas. 

* In other words, we assume, as in Trigonometry, that tho 
" principle of proportional parts " holds for the small variations in 
decl. and E.A. during the day. 



10 V 2 ASTRONOMY. 



*140. The Advantages of Flamsteed's Method. Among these 
the following may be mentioned. 

1st. The method does not require a knowledge of the latitude, for 
we do not require to find the Sun's declination. Hence, errors 
arising from inaccurate determination of the latitude are avoided. 

2nd. One great source of error in determining Z.D.'s is the refrac- 
tion of the Earth's atmosphere. Since the Sun is observed each 
time in the same part of the sky, z lt z 2 , 3 will be nearly equally 
affected by refraction. Hence, the " principle of proportional 
parts" will hold, so that the small differences in the true Z.D.'s 
are proportional to the differences in the observed Z.D.'s. Hence 
we may use the observed Z.D.'s uncorrected for refraction. 

EXAMPLE. 

To find the Right Ascension of Sirius and the clock errors in 
March and Sept., 1891, from the following data, the rate of the clock 
being supposed correct. (Decl. of Sirius = 16 34' 2" S.) 





Mar. 25, 1891. 


Sept. 18. 


Sept. 19 


Decl. of Sun at noon... 
Time of transit of Sun 
Time of transit oiSirius 


1 48' 56" 
Oh. 15m. 36s. 
6h. 39m. 10s. 


1 53' 0" 

llh. 42m. 42s. 
6h. 40m. 25s. 


1 29' 43" 
llh. 46m. 17s. 
6h. 40m. 25s. 



OnMar.25,(R.A.ofSmws)-(Sun'sR.A.)=6h 39m. 10s. -Oh. 15m. 36s. 

=6h.23m.34s. 

Hence, in angular measure, the difference of R.A. is about 96. 
Draw the diagram as in Fig. 52, but make the angle SiPN = 96; iV 
will therefore lie between M l and J5f 2 , instead of where represented. 
Also, since Sirius is south of the equator, it should be represented 
at a point x on PN produced through N. In this figure we shall have 
8^ = 148'56"; MiN = 6h.39m.10s. -Oh.15m.36s. = 6h.23m.34a. 
S 2 3f 2 = 153' 0"; NM. 2 = llh.42m.42s.-6h.40m.25s. = 5h. 2m. 17s. 
= 129'43"; NM 3 = Ilh.46m.l7s.-6h.40m.25s. = 5h. 5m.52s. 
Also, SM is by construction equal to S^M } . 
Hence, applying the principle of proportional parts, we have 
SoMg-giJf! = 4' 4" = 244 
S. 2 M 2 -S 3 M 3 23' 17" 1397' 
and M%M 3 = 3m. 35s. = 215s. ; 
.-. M*M = 215 x 244/1397 = 37'5 seconds ; 
.-. NM = 5h. 2m. 17s. + 37s. = 5h. 2m. 54s. 
Now, NMt-NAI = NT -N~ = 2Nr -12h. 
bonce, TN = 6}i. + %(NMi-NM) = 6h. + i(6h.23m.34s.-5h.2m.54s.) 

= 6h. + (lh. 20m. 40s.) = 6h. 40m. 20s. 
Thus the right ascension of Sirius = 6h, 40m. 20s. 
Also, clock error in March = 6h.40m.20s.-6h.39m.10s. = + 1m. 10s. 
Sept. = 6h.40m.20s.-6h.40m.25s. = 5s. 



103 

141. Precession of the Equinoxes. Thus far we have 
treated the first point of Aries as being fixed, and this will 
evidently be the case if the equator and ecliptic are fixed in 
direction. But if the right ascensions of various stars are 
observed over an interval of several years, it will be found 
that the position of the first point of Aries is slowly changing, 
and that it moves along the ecliptic in the retrograde direc- 
tion at the rate of about 50-2" in a year. This motion is 
called Precession of the Equinoxes, or, briefly, Precession. 

Precession is found to be due almost entirely to gradual 
changes in the direction of the plane of the equator, the 
ecliptic remaining almost fixed among the stars. Its effect is 
to produce a yearly increase of 50-2" in the celestial longi- 
tudes of all stars, their latitudes being constant. 

In a large number of years the effect of precession will be 
considerable. Thus, T will perform a complete revolution 
in the period 

360x60x60 years, i.e., about 25,800 years. 
o(j' 2i 

At the present time the vernal equinoctial point has moved 
right out of the constellation Aries into the adjoining con- 
stellation Pisces. It still, however, retains the old name of 
lt First Point of Aries." Similarly, the autumnal equinoctial 
point is in the constellation Virgo, but it is still called the 
" First Point of Libra." 

The rate of precession can be found very accurately by 
observations of the first point of Aries separated by a con- 
siderable number of years. The larger the interval, the 
larger is the change to be observed, and the less is the result 
affected by instrumental errors. 

*142. Correction for Precession in using Flamsteed's Method. 
During the interval that elapses between the two observations in 
Flamsteed's method, the right ascension of the observed star will have 
increased slightly, owing to precession, and the E.A. given by the 
formula will be the arithmetic mean of the E.A.'s at the times of 
the two observations.f As the change in E.A. is very approximately 
uniform, this mean will be the star's E.A. at a time exactly half 
way between the two observations, i.e., at the summer solstice. 

t This may be most readily seen by imagining the equator and 
ecliptic to be at rest, and the change in E.A. to be due to motion of 
the star. 



104 



ASTROJDMT. 



143. Determination of Obliquity of Ecliptic. The 

method now used for finding the obliquity of the ecliptic is 
similar in principle to that of 38, hut the Sun's meridian 
zenith distance is observed by means of the transit circle 
instead of the gnomon. 

The obliquity is equal to the Sun's greatest declination at 
one of the solstices. Since observations with the Transit 
Circle can only be performed at noon, while the maximum 
declination will probably occur at some intermediate hour of 
the day, it will be necessary, in exact determinations, to 
make observations of the Sun's decl. for several days before 
and after the solstice. Prom these it is possible to determine 
the maximum decl. ; the method is, however, too complicated 
to be described here. For rough purposes the Sun's greatest 
noon decl. may be taken as the measure of the obliquity. 




144. When the position of T has been determined, the 
obliquity can also be found by a single observation of the 
Sun's E-.A. and decl. For we thus find the two sides T-3/i 
MS of the spherical triangle T^S, and these data are 
sufficient to determine both the obliquity flfTS, and the 
Sun's longitude T S. 



THE SUN'S APPARENT MOTION IN THE ECLIPTIC. 105 



SECTION III. The Earth's Orbit about the Sun. 

145. Observations of the Sun's Relative Orbit. By 

daily observations with the Transit Circle, the decl. and R. A. 
of the Sun's centre at noon are found for every day of the 
year. From these data the Sun's long, is calculated, as 
in 144, by solving the spherical triangle T SM (Fig. 53). 
If the obliquity of the ecliptic is also known, we have three 
data, any two of which suffice to determine the long., T$. 
Thus the accuracy of the observations can be tested, and the 
Sun's motion at various times of the year can be accurately 
determined. 

Although the determination of the Sun's actual distance 
from the Earth in miles is an operation of great difficulty, it 
is easy to compare the Sun's distance from the Earth at dif- 
ferent times of the year, for this distance is always inversely 
proportional to the Sun's angular diameter. This property is 
proved in 4, but numerous simple illustrations may also 
be used to show that the angular diameter of any object varies 
inversely with its distance (see 4). 

The Sun's angular diameter may be readily observed by 
means of the HeHometer ; or, if preferred, any other form of 
micrometer may be used. The Sun's distances at two different 
observations will be in the reciprocal ratio of the corresponding 
angular diameters. Thus, by daily observation, the changes 
in the Sun's distance during the year may be investigated. 

If the circular measure of the Sun's angular diameter is 
2r, then Trr 2 is called the Sun's apparent area. In fact, 
this is the area of a disc which would look the same size as 
the Sun if placed at unit distance from the eye. 

EXAMPLE. 

The Sun's angular diameter is 31' 32" at midsummer, and 32' 36" 
at midwinter. To find the ratio of its distances from the Earth at 
these times. 

The distances being inversely proportional to the angular dia- 
meters, we have 

Dist. at midsummer = 82' 36" = 1956 = 489 _ , 1 , 

Dist. at midwinter 31' 32" 1892 473 "** 

Hence the Sun is further at midsummer than at midwinter, in the 
proportion of very nearly 81 to 30. 



106 ASTRONOMY. 

146. Kepler's First and Second Laws. We may no\\ 

construct a diagram of the Sun's relative orbit. Let E repre- 
sent the position of the Earth, ET the direction of the first 
point of Aries. Then, by making the angle TES equal to the 
Sun's longitude at noon, and ES proportional to the Sun's 
distance, we obtain a series of points S, S'... , 8 r .. , representing 
the Sun's position in the plane of the ecliptic, as seen from 
the Earth at noon on different days of the year. Draw the 
curve passing through the points S, S'... , S r .. ; this curve will 
represent the Sun's orbit relative to the Earth, and it will be 
found that 

I. The Sun's annual path is an ellipse, of which the 
Earth is one focus. 

II. The rate of motion is everywhere such that the 
radius vector (i.e., the line joining the Earth to the 
Sun) sweeps out equal areas in equal intervals of time. 

These laws were discovered by Kepler for the motion of 
Mars about the Sun, and he subsequently generalized them by 
showing that the orbits of all the other planets, including the 
Earth, obeyed the same laws. In their general form they are 
known as Kepler's First and Second Laws. [See p. 253.] 




FIG. 54. 

147. Perigee and Apogee. When the Sun's distance 
from the Earth is least, the Sun is said to be in perigee. 
When the distance is greatest, the Sun is said to be in apogee 

The positions of perigee and apogee are called the two 
Apses of the orbit ; they are indicated at p, a in Fig. 54. 
The line pEa joining them is the major axis of the ellipse 
(Ellipse, 4), and is sometimes also called the apse line. 



107 

148. Verification of Kepler's First Law. The Sun's 
angular diameter is observed to be greatest on Dec. 31, 
and least on July 1 ; we therefore conclude that these are the 
days on which the Sun passes through perigee and apogee 
respectively. The positions of perigee and apogee being thus 
found, the angle TEp is known, which is the long, of perigee. 

From the winter solstice to perigee is about 10 days. 
Hence, during this interval the Sun will have moved through 
an angle of about 10 ; 

.-. longitude of perigee = 270 + 10 = 280 roughly. 

To verify that the orbit is an ellipse, it is now only neces- 
sary to show that the relation connecting ES and the angle 
pES is the same as that which holds in the case of the ellipse. 
If the orbit is an ellipse of eccentricity <?, we must have 
ES x (1 +0 cospJSS) = I (a constant). (Ellipse, 3.) 
Therefore the Sun's angular diameter must be always pro- 
portional to 1 + * wspES. 

As the result of numerous observations, it is found that 
this is actually the case, and the truth of Kepler's Eirst Law 
for the Sun's orbit relative to the Earth is confirmed. 

149. To find e, the eccentricity of the ellipse, the 

best plan is to compare the greatest and least angular dia- 
meters of the Sun, i.e., the diameters at perigee and apogee. 
Since at these positions pES becomes and 180 respectively, 
we have, from above, 
ang. diam. at p : ang. diam. at a = IjEp : IjEa 

= l+0cosO : 1+* cos 180 = \+e : l-e. 

from which proportion e can be found. 

Taking the angular diameters at perigee and apogee to be 
3,2' 36" and 31' 32" (as in the Ex. of 145), the Sun's distances 
at those times are in the ratio of 1956" : 1892", or 489 : 473 ; 

1+g = 489 _ 489-473 _ 16 _ 8 

' \-e 473 ~ 489+473 ~ 962 ~~ 481* 

Hence e is very nearly equal to 1/60. 

The Nautical Almanack contains a table giving the Sun's 
angular diameter daily throughout the year. The average 
angular diameter may be taken as 32' approximately. 

Owing to the smallness of 0, the orbit is very nearly circular, 
being, really, much more nearly so than is shown in Eig. 54 

ASTEON. i 



108 ASTEONOMY. 

150. Verification of Kepler's Second Law. It is 

found, as the result of observation, that the Sun's increase in 
longitude in a day, at different times of year, is always pro- 
portional to the square of the angular diameter, and is, there- 
fore, inversely proportional to the square of the Sun's distance. 
From this it may he deduced (as follows) that the area de- 
scribed by the radius vector in one day is always constant. 




PIG. 55. 

Let SS' represent the small arc described by the Sun 
in a day in any part of the orbit. Then the sector US 8' is 
the area swept out by the radius vector. This sector does 
not differ perceptibly from the triangle JESS' ; therefore, by 
trigonometry, 

area JS88' = %ES . US' . sin 8E8'. 

Since the change in the Sun's distance in one day is imper- 
ceptible, we may write JES for JES' in the above formula 
without materially affecting the result ; also, since the angle 
SES' is small, the sine of SES' is equal to the circular 
measure of the angle SES'. 

Therefore, area JESS' = \E& x L SES '. 

But, by hypothesis, the change of longitude SES' varies 
inversely as ES*, so that US' 2 x L SES' is constant ; 

area ESS' is constant, 

that is, the area described by the radius vector in a day is 
constant. Thus, the area described in any number of days 
is proportional to the number of days, and generally the areas 
described in equal intervals of time are equal. 



109 

151. Deductions from Kepler's Second Law. 

(i.) If the circular measure of the Sun's angular diameter 
is 2r, then Trr 2 is the Sun's apparent area ( 145). Hence 
the Sun's daily rate of change of longitude is proportional to 
the apparent area of its disc. 

(ii.) If T, K, :, L represent the Sun's positions at the 
equinoxes and solstices, we have 

Z rEK = z.KE = tEL LLEv = 90, 
and it is readily seen from the figure that 

area LET < area lEL < area TEE < area KE<, 
and the lengths of the seasons, being proportional to these 
areas, are unequal, their ascending order of magnitude being 

Winter, Autumn, Spring, Summer. 

Their lengths are, at the present time (1891), about 
39d. 0|h., 89d. IS^h., 92d. 20h., 93d. 14|h. 

(iii.) Since the intensity of the Sun's heat ( 131) and its 
rate of motion in longitude both vary as the inverse square of 
its distance, they are proportional to one another. Hence 
the Earth, as a whole, receives equal amounts of heat while the 
Sun describes equal angles. In particular, the total quantities 
of heat received in the four seasons are equal. 

(iv.) The Sun's longitude changes most rapidly on Decem- 
ber 31, and least rapidly on July 1. 

(v.) Since the apse line, or major axis, pSa, bisects the 
ellipse, the time from perigee to apogee is equal to the time from 
apogee to perigee. 

*152. To find the Position of the Apse Line. 
The Sun's distance remains very nearly constant for a short 
time before and after perigee and apogee, hence it is difficult 
to tell the exact instant when this distance is greatest or least. 
For this reason, the following method is generally used : 

The Sun's long, is observed at two points, S, S v before and 
after the apse, when its angular diameters, or its rates of 
motion in long., are found to be equal. Then ES = ES^ 
and the symmetry of the ellipse shows that JLpES = LpES^ 
and L aES = L aES r Hence the long, of the apse is the 
arithmetic mean of the Sun's longitudes at the two observations. 

153. Progressive Motion of Apse Line. Prom such 
observations, extending over a long period of years, it is found 
that the apse line is not fixed, but has a forward or direct 
motion in the ecliptic plane of 1 1 "25" in a year. 



1 10 ASTRONOMY. 

154. The Sun's apparent annual motion may be 
acco anted for by supposing the Earth to revolve 
roun$ the Sun. 

The annexed diagram will show how the Sun's annual 
motion in the ecliptic, as well as the changes in the seasons, 
may be accounted for on the theory that the Sun remains at 
rest while the Earth describes an ellipse round it in the 
course of the year in a plane inclined at an angle 23 27' to 
the plane of the Earth's equator. 



Mar 21 




FIG. 56. 

The distance of the nearest of the fixed stars is known to 
be over 200,000 times as great as the Earth's distance from 
the Sun. Hence, 5 shows that the directions of the fixed 
stars will not change to any considerable extent, as the 
Earth's position varies. "We shall, therefore, in the present 
description, consider the directions of the stars to be fixed. 
The directions of the various points and circles of the celestial 
sphere, such as the first point of Aries, will also be fixed. 

On March 21, the Earth is at JS lt and the Sun's direction 
determines the direction of T, the First Point of Aries. 



Ill 

The Sun is vertical at a point Q on the equator, and as the 
Earth revolves about its axis through P, all points on the 
equator will come vertically under the Sun. There is night 
all over the shaded portion of the Earth, day over the rest. 
The great circle bounding the illuminated part passes through 
the pole P, and, therefore, bisects the small circle traced out 
by the daily rotation of any point on the Earth ; thus, the 
day and night are everywhere equal. At the pole P the Sun 
is just on the horizon. 

On June 21, the Earth is at E^ and the Sun's longitude 
TE^S = 90. The Sun is vertical at a point on the tropic 
of Cancer. Since the arctic circle is entirely in the illumi- 
nated part there is perpetual day over the whole arctic zone. 

On September 23, the Earth is at E%, and the Sun's longi- 
tude TE S S is 180. The Sun is once more vertical at a 
point JR on the equator, and the day and night are everywhere 
12 hours long, as they are at E r 

On December 22, the Earth is at E, and the Sun's longi- 
tude vEfi (measured in the direction of the arrow) is 270. 
The Sun is now at its greatest angular distance south of the 
equator, and overhead at a point on the tropic of Capricorn ; 
this tropic is not represented, being on the under side of the 
sphere. Since the arctic circle is entirely within the shaded 
part there is perpetual night over the whole arctic zone. 

155. New Definitions and Pacts. According to the 
theory of the Earth's orbital motion, Kepler's First and 
Second Laws must be re-stated thus for the Earth. 

I. The Earth describes an ellipse, having the 
Sun in one focus. 

II. The radius vector joining the Earth and Sun 
traces out equal areas in equal times about the Sun. 

The ecliptic is now definedasthe great circle of the celestial 
sphere, whose plane is parallel to that of the Earth's orbit. 

The Earth is nearest the Sun on December 31, and is then 
said to be in perihelion. The Earth is furthest from the 
Sun on July 1, and is then said to be in aphelion. Thus, 
when the Sun is in perigee the Earth is in perihelion, when 
the Sun is in apogee the Earth is in aphelion. The positions 
of perihelion and aphelion are indicated by the letters p, a in 
Fig. 56. The line joining them is the apse line. 



112 ASTROWOttt. 

156. Geocentric and Heliocentric Latitude and 
Longitude. Hitherto we have been dealing only with the 
directions of the celestial bodies as seen from the Earth. 

In dealing with the motion of the planets, it is more con- 
venient, as a rule, to define their positions by the directions 
in which they would be seen by an observer situated at the 
centre of the Sun. 

In every case, the direction of a celestial body may be 
specified by the two coordinates, celestial latitude and longi- 
tude, which measure respectively the arc of a secondary from 
the body to the ecliptic and the arc of the ecliptic between 
this secondary and the first point of Aries ( 17). 

These coordinates are called the Geocentric Latitude 
and Longitude when employed to define the body's geocen- 
tric position, or position relative to the centre of the Earth. 
The names Heliocentric Latitude and Longitude are 
given to the corresponding coordinates when employed to 
define the body's heliocentric position, or position relative 
to the Sun's centre. 

"When the distance of a fixed star is immeasurably great 
compared with the radius of the Earth's orbit, its geocentric 
and heliocentric directions coincide, and there is no difference 
between the two sets of coordinates. There is a slight differ- 
ence between the geocentric and heliocentric positions of a 
few of the nearest fixed stars. But, in the case of the 
planets, and of comets, the heliocentric latitude and 
longitude differ entirely from the geocentric, and laborious 
calculations are required to transform from one system of 
coordinates to the other. 

One fact may, however, be noted. The direction of the 
Earth as seen from the Sun is always opposite to the direction 
of the Sun as seen from the Earth. Hence, 

The Earth's heliocentric longitude differs from the 
Sun's geocentric longitude by 180. 

This may be illustrated by referring to Pig. 56. We see 

thatr&E^oyr/SLE^ 90, r&Ei = 180, TSJS, = 270; 

thus, the Earth's longitude is on September 23, 90 on 
December 22, 180 on March 21, and 270 on June 21. 



THE SUN'S APPARENT MOTION IN THE ECLIPTIC. 113 



EXAMPLES. IV. 

1. Describe the phenomena of day and night at a pole of the 
Earth. 

2. Show how to find how long the midwinter Moon when full is 
above the horizon at a place within the arctic circle of given 
latitude. 

3. Show that the ecliptic can never be perpendicular to the 
horizon except at places between the tropics. 

4. Show that for a place on the arctic circle the Sun always rises 
at 18h. sidereal time from December 21 to June 20, and sets at the 
same sidereal time from June 20 to December 21. 

5. Find the angle between the ecliptic and the equator in order 
that there should be no temperate zone, the torrid zone and the 
frigid zone being contiguous. 

6. Show how, by observations on the Sun, taken at an interval of 
nearly six months, the astronomical clock may be set to indicate 
Oh. Om. Os. when T is on the meridian. 

7. On March 24, 1878, at noon, the Sun's declination was 
1 29' 5*1", and the difference of right ascension of the Sun and a 
star 6h. 1m. 34'45s. On September 18, 1878, at noon, the Sun's 
declination was 1 49' 30'2", and it was distant from the star 
5h. 27m. 32'97s. in right ascension. On September 19, 1878, at 
noon, the Sun's declination was 1 26' 12'8", and it was distant from 
the star 5h. 31m. 8'3s. in right ascension. Find the right ascension 
of the star and that of the Sun at the first observation. 

8. Describe the appearance presented to an observer in the Sun 
of the parallels of latitude and the meridians of the Earth, any day 
(i.) between the vernal equinox and the summer solstice, 
(ii.) between the autumnal equinox and the winter solstice. 

9. If a sunspot be situated near the edge of the Sun's disc, 
describe how its position, relative to the horizon, will change between 
sunrise and sunset. 

10. Describe how the Sun's apparent velocity in the ecliptic 
varies throughout the year; and give the dates of apogee and 
perigee. Compare the daily motion in longitude at these dates, 
having given that the eccentricity of the Earth's orbit is ^5. 



114 



ASTRONOMY, 



EXAMINATION PAPEK. IV. 

1. What is the astronomical reason for the Earth being divided 
into torrid, temperate, and frigid zones ? 

2. Assuming your latitude to be 52, show by a figure the daily 
path of the Sun as seen by you on June 21, December 22, and 
March 21 respectively. 

3. Explain the causes of variation in the length of the day on the 
Earth. Give the dates at which each season begins, and calculate 
their lengths in days. 

4. Discuss the variations in the length of the day at points within 
the arctic circle ; and show how to find, by the Nautical Almanack, 
the length of the perpetual day. 

5. Prove that, in the course of the year, the Sun is as long above 
the horizon at any place as below it. 

6. Explain how it is that winter is colder than summer, although 
the Sun is nearer. 

7. Investigate Flamsteed's method of determining the first point 
of Aries. 

8. From the following observations calculate the Sun's R.A. on 
March 30, 1872 : 





Sun's 
declination. 


Sun crossed 
meridian. 


a Serpentis 
crossed meridian. 


March 30, 1872... 
Sept. 11, 1872 ... 
Sept. 12,1872 ... 


4 (/ 8-1" 
4 20' 58-8" 
3 58' 3-0" 


Oh. 1m. 4'47s. 
Oh. 1m. 4-09a. 
Oh. 1m. 4-07s. 


15h. 1m. 5476s. 
4h. 19m. ll-38s. 
4h. 15m. 49-33s. 



9. State Kepler's First Law for the orbit of the Earth relative to 
the Sun, and explain how the eccentricity of the orbit can be found 
by observations of the Sun's angular diameter. 



10. State Kepler's Second Law, and find the relation between the 
Sun's angular velocity and its apparent area. 



CHAPTER V. 

ON TIME. 
SECTION I. The Mean Sun and Equation of Time. 

157. Disadvantages of Sidereal and Apparent Solar 
Time. In Chapter L, Sections II., III., we explained two 
different ways of reckoning time. One of these, called 
Sidereal Time, was denned by the diurnal motion of the first 
point of Aries ; the other, called Apparent Solar Time, was 
defined by the Sun's diurnal motion. We shall now show 
that neither of these measures of time is suitable for every- 
day use. 

If we were to adopt sidereal time, the time of apparent 
noon on any day of the year would be measured by the Sun's 
K.A. on that day, and therefore would get later and later by 
24h. during the course of the year. 

Thus (0.^.), the time of noon would be Oh. on March 21, 
6h. on June 21, 12h. on September 23, and 18h. on Decem- 
ber 22, and the phenomena of day and night would bear no 
constant relation to the time. 

Apparent solar time is free from these disadvantages, but 
it cannot be measured by a clock whose rate is uniform, 
because the length of the solar day is not quite invariable. 
In 36 we showed that the difference between a solar and 
a sidereal day is equal to the Sun's daily increase in R.A., 
and in 31 we showed that this increase takes place at a rate 
which is not quite the same at different times of the year. 
Hence, the difference between a solar and a sidereal day is 
not quite constant. But the length of a sidereal day is con- 
stant ( 22). Hence the solar day is not quite constant, 
and a clock cannot be regulated so as to always mark exactly 
Oh. Om. Os. when the Sun crosses the meridian. 



116 ASTRONOMY. 

158. The Mean Sun. Definitions. To obviate these 
disadvantages, another kind of time, called Mean Time, has 
been introduced, and this is the time indicated by clocks, and 
used for all ordinary purposes. Mean Time is defined by 
means of what is CAiled the Mean S&o. This is not really 
a Sun at all, but simply a point, which is imagined to move 
round the equator on the celestial sphere.* The hour angle 
of this moving point measures mean time, just as the hour 
angle of T measures sidereal time ; and the mean Sun has to 
satisfy the following requirements : 

1st. It must never be very far from the Sun. 

2nd. Its R.A. must increase uniformly during the year. 

Now the inequalities in the motion in R.A., which render 
the true Sun unsuitable as a timekeeper, are due to two 
causes. 

1st. The Sun does not move uniformly in the ecliptic, its 
longitude increasing less rapidly in summer than in winter 
( 151). t 

2nd. Since the Sun moves in the ecliptic, and not in the 
equator, its celestial longitude is in general different from its 
R.A. ( 31 ). Hence, even if the Sun were to revolve uni- 
formly, its R.A. would not increase uniformly. 

In defining the mean Sun, or moving point which measures 
mean time, these two causes of irregularity are obviated 
separately as follows : 

The Dynamical Mean Sun is defined to be a point which 
coincides with the true Sun at perigee, and which moves 
round the ecliptic in the same period (a year) as the true 
Sun, but at a uniform rate. 

Thus, in the dynamical mean Sun, irregularities due to 
the Sun's unequal motion in longitude are removed, but those 
due to the obliquity of the ecliptic still remain. 

The Astronomical Mean Sun is defined to be a point 
which moves round the equator in such a way that its R.A. 
is always equal to the longitude of the dynamical mean Sun. 

* The conception of the mean Sun as a moving point is important. 
It would be physically impossible for a body to move in this manner. 



Ott TIME. 117 

Since the longitude of the dynamical mean Sun increases 
uniformly, the R.A. of the astronomical mean Sun increases 
uniformly. Hence the motion of the latter point does give 
us a uniform measure of time. 

The astronomical mean Sun is, therefore, the moving point 
chosen in denning mean time. It is usually called simply 
the Mean Sun. 

159. Mean Noon and Mean Solar Time. Equation 
of Time. 

Mean Noon is defined as the time of transit of the mean 
Sun. 

A Mean Solar Day is the interval between two successive 
mean noons. Like the apparent and sidereal days, it is 
divided into 24 mean solar hours. During this interval, the 
hour angle of the mean Sun increases from to 360. 
Hence the mean solar time at any instant is measured by 
the mean Sun's hour angle, converted into time at the rate 
of Ih. per. 15, or 4m. per 1. 

The Sun itself is frequently spoken of as the True Sun, 
or Apparent Sun, to distinguish it from the mean Sun. 
As explained in 36 the hour angle of the true Sun measures 
the apparent solar time, and its time of transit is called 
apparent noon. 

The Equation of Time* is the name given to the amount 
which must be added to the apparent time to obtain the mean 
time. 

Thus, the time indicated by a sun-dial ( 167) is determined 
by the position of the shadow thrown by the true Sun, and is 
the apparent solar time ; while a clock, which should go at a 
uniform rate, is regulated to keep mean time. The equation 
of time will then be defined by the relation, 
(Time by clock ) = (Time by dial) + (Equation of time). 

At apparent noon the sun-dial will indicate 12h., or, as it 
is more conveniently reckoned, Oh. Hence, 

Equation of time = Mean time of apparent noon. 

* Thus, " equation of time " is not an equation at all in the 
generally accepted sense of the word, but an interval of time (posi- 
tive or negative). 



118 ASTRONOMY. 

The equation of time is positive if the Sun is " after the 
clock," or the true Sun transits after the mean Sun. If the 
Sun is "before the clock," or the true Sun transits first, the 
equation of time is negative. The value of the equation of 
time for every day in the year is given in most almanacks, 
under the heading " Sun before clock," or " after clock." 

160. The equation of time is divided into two parts. The 
first, which is called the equation of time due to the eccen- 
tricity, or to the unequal motion, is measured by the 
difference between the hour angles of the true and dynamical 
mean Suns. The second, or the equation due to the 
obliquity, is measured by the difference of hour angle 
between the dynamical and astronomical mean Suns. 

161. Equation of Time due to Unequal Motion. 

"We shall now trace the variations during the year of that 
portion of the equation of time which is due to the Sun's 
unequal motion in the ecliptic. We shall denote this portion 
by^. 

Let the true Sun be denoted by 8, and the dynamical mean 
Sun (which moves in the ecliptic) by 8 r If angles are 
measured in time, then 

E l (hour angle of SJ (hour angle of S) = L SPS l ' t 
.-. E l = (RA. of S) -(R.A. of 8J ; 

since R.A. and hour angle are measured in opposite directions. 

When the Sun is in perigee (p) (on December 31), /S t coin- 
cides with S by definition ; .*. E l = O. 

From perigee (p} to apogee (a), the Sun, has described 180, 
and the time taken is ( 151, v.) half that of a complete 
revolution. Hence, S l will also have described 180 ; 

.*. at apogee (July 1), E l is again O. 
Now ( .151, iv.) S is moving most rapidly at perigee, and 
most slowly at apogee. Hence, after perigee, S will have got 
ahead of S v and after apogee, S will have got behind 8 V 
Thus : From perigee to apogee, E l is positive, 

From apogee to perigee, E l is negative. 
and E v vanishes twic a year, viz., at perigee and apogee. 



ON TIME. 



119 



162. Equation of Time due to Obliquity. Let the 

portion of the equation of time due to the obliquity be 
denoted by E v 

Take S 3 on the equator so that r 8 t = T 8 r Then 2 
will be the astronomical mean Sun. Draw PS^, the 
secondary to the equator through 8 r Then 

EI = hour angle of $ 2 hour angle of ^ 

= L SJP&i (taken positive if 8 9 is west of SJ 

= L rPS^ z rPS* = rM- rS, = r Jf- r ^, 

all angles being supposed converted into time at the rate 
of 15 to the hour. 




At the vernal equinox,* when 8 l is at T , 2 will also be 

of 1 */* /'' ~~ o 

ai T , -^z u - 

Between the vernal equinox and summer solstice, the angle 
will be < 90, and, therefore, < T-WSiJ hence, 



is negative. 



* The vernal and autumnal equinoxes are, strictly, the times when 
fif, and not 8 l} coincides with the equinoctial points, but, as Sj is 
always near 8, the distinction need not be considered here. The 
same remarks apply to the solstices. 



120 



ASTRONOMY. 



At the summer solstice, 8 l is at C, and S 9 at Q, where 
r Q = r C = 90. Hence (Sph. Geom., 21), r QC = 90; 
and Mis also at Q ; .-. _# 2 o. 

Between the summer solstice and autumnal equinox we 
shall have M= < 8^. But rM^ = rS^ = 180; 
.:TM>r8 l i .:rK>r8 t ; .-.Dispositive. 

At the autumnal equinox, since fC = TQ==z 180, 
S lt &J will both coincide with b; .. JEJ 8 = 0. 




In a similar manner we may show that : 

From the autumnal equinox to the winter solstice, U 3 is 
negative. 

At the winter solstice, 1 2 as O. 

From the winter solstice to the vernal equinox, E% is 
positive. 

Collecting these results, we see that 
(i.) From equinox to solstice /^ is negative. 

(ii.) From solstice to equinox J 3 is positive. 

(iii.) Es vanishes four times a year, viz., at the 
equinoxes and solstices. 



ON TIME. 121 

163. Graphic Representation of Equation of Time. 

The values of the equation of time at different seasons may 
now be represented graphically by means of a curved line, in 
which the abscissa of any point represents the time of year, 
and the ordinate represents the corresponding value of the 
equation of time. 

In the accompanying figure (Fig. 59) the horizontal line 
or axis from JE^ to E^ represents a year, the twelve divisions 
representing the different months as indicated. The thin curve 
represents the values of E^ the portion of the equation of 
time due to the unequal motion ; this curve is obtained by 
drawing ordinates perpendicular to the horizontal axis and 
proportional to E r Where the curve is below the horizontal 
line E i is negative. 




FIG. 59. 

The thick curved line is drawn in a similar manner, and 
represents, on the same scale, the values of E v the equation 
of time due to the obliquity. 

In drawing the diagrams to scale, it is necessary to know 
the maximum values of JEJ, E v These can be calculated, 
but the calculations do not depend on elementary methods 
alone. We shall therefore have to assume the following 
facts : 

The greatest value of E l is about 7 minutes. 



Hence the greatest distances of the thin and thick curves 
from the horizontal axis should be taken to be about 7 and 1 
units of length respectively. 



122 



ASTRONOMY. 



We may now draw the diagram representing JE, the total 
equation of time. We have 



Hence, at every point of the horizontal line we must erect 
an prdinate whose length is equal to the algebraic sum of the 
ordinates (taken with their proper sign) of the two curves 
which represent E^ and E The extremities of these ordi- 
nates will determine a new curve which represents E. 





FlG. 60. 

This curve is drawn separately in the annexed diagram 
(Fig. 60). It cuts the horizontal axis in four points. At 
these points the ordinate vanishes, and E is zero. Hence, 

The Equation of Time vanishes four times a year. 

164. Alternative Proof. But without representing the 
values of the equation of time graphically, it can he readily 
proved that E vanishes four times a year. The proof 
depends on the fact stated in the last paragraph, that 

The greatest equation of time due to the obliquity is greater 
than the greatest equation due to the eccentricity. 



Off TIME. 123 

From 162 it is evident that JS t must attain its greatest 
positive value some time between a solstice and the following 
equinox, and its greatest negative value between an equinox 
and the following solstice. These maxima occur, in fact, in 
the months : 

February, May, August, November. 

Their values, with the proper signs, are respectively about 
-flOm., 10m., 4- 10m., 10m. 

Now, E l is never greater than the maximum value of 7m. ; 
hence, whether E^ is positive or negative, the total equation, 
E^ -f EX corresponding to either of these maxima, must have 
the same sign as E Hence, in the year beginning and ending 
with the date of the maximum value of E^ in February, E 
will have the following signs alternately : 

+ + + 

Thus, 73 changes sign, and thereJx^e vanishes, four 
times in the year. 

165. Miscellaneous Remarks. "From Pig. 59 it will 
be seen that the largest fluctuations in the equation of time 
occur in the autumn and winter months ; during spring and 
summer they are much smaller. 

The days on which the equation of time vanishes are about 
April 16, June 15, September 1, and December 25. 

Between these days E increases numerically, and then 
decreases, attaining a positive or negative value at some inter- 
mediate time. These maxima are : 
+ 14m. 28s. on February 11 ; 3m. 49s. on May 14 ; 
-f-6m. 17s. on July 26 ; 16m. 21s. on November 3. 

166. Inequality in the Lengths of Morning and 

Afternoon. If we neglect the small changes in the Sun's 
declination during the day, the interval from sunrise to 
apparent noon is equal to the interval from apparent noon to 
sunset ( 37). But by morning and afternoon are meant the 
intervals between sunrise and mean noon, and between mean 
noon and sunset respectively. Hence, unless mean and appa- 
rent noon coincide, i.e., unless the equation of time vanishes, 
the morning and afternoon will not be equal in length. 

4-STKOF, K 






124 ASTBONOMT. 

Let r, * be the mean times of sunrise and sunset, E the 
equation of time. Then 

12h. r = interval from sunrise to mean noon. 
But apparent noon occurs later than mean noon by E\ 

.'. 12h. r-\-E interval from sunrise to apparent noon. 
Similarly, sJE= interval from apparent noon to sunset; 

.-. 12h.-r+.# =*-.#, 
or r + s= 12h. + 2^, 

so that the sum of the times of sunrise and sunset 
exceeds 12 hours by twice the equation of time. 

The length of the morning is 12h. r, and that of the 
afternoon is . Now the last relation gives 

2J0 = -(12-r); 
.-. 2 (equation of time) 

= (length of afternoon) (length of morning). 

About the shortest day (December 22) the curve represent- 
ing the equation of time is going upwards, hence E is 
increasing. But the length of day is changing very slowly 
(because it is a minimum), hence, for a few days, the half 
length, jE", may be regarded as constant. Hence, must 
increase, and, therefore, the mean time of sunset is later 
each day. Similarly, it may be shown that sunrise is also 
later. The afternoons, therefore, begin to lengthen, while 
the mornings continue to shorten. 

Similarly, about June 21, the afternoons continue to 
lengthen after the longest day, although the mornings are 
already shortening. 

EXAMPLE. On Nov. 1, the sun-dial is 16m. 20s. before the clock. 
Given that the Sun rose at 6h. 54m., find the time of sunset. 
Time from sunrise to mean noon = 12h. 6h. 54m. = 5h. 6m. 
apparent noon to mean noon = Oh. 16m. 20s. 

sunrise to apparent noon = 4h. 49m. 40s. 

apparent noon to sunset = 4h. 49m. 40s. 

mean noon to sunset 

= 4h. 49m. 40s. - 16m. 20s. = 4h. 33m. 20s. 

Hence, the time of sunset was 4h. 33m., correct to the nearest 
minute. 



ON TIME. J 25 

SECTION II. The Sun-dial 

essentially of a rod or flat 



the direction of the celestial pole The shadow 




Pia. 61. 



The plane through OA, the edge of the style, and throueh 
the edge of the shadow, evidently passes through the Sun 
also it passes through the celestial pole, therefore it will meet 



' "Declination ce 



shln t plane, which is the plane of the 

known t,? p P arent f 00 ?' and whose position is supposed 

123' oli e A 1 Y- rder , to graduate the p late for *~ 

1, ^, do clock, it is only necessary to determine the posi- 




esof no,-' " - c "' w 

l ' 1 ' ' , 45 . ' &0 '' Wlth the meridian plane. Since 



be thn - S P erour ' te e P^e* wffl 

the planes bounding the shadow at 1, 2 3 o'clock 
rcspecfavely. If we join the points 0,, On. dm *'"&* , hese 



, 

lines of shadow in the plane of the 
m * * he cirram ^ of the dial-plate 



126 ASTRONOMY. 

168. Geometrical Method of Graduating the Dial- 
plate. To find the planes OA i., OAn., &c., suppose a plane 
AKR drawn through A perpendicular to OA, meeting the 
plane of the dial-plate in KR and the meridian plane in A~s.ii. 
If, in this plane, we take the angles xn.-4i., i.^n., n.^tm., 
&c., each = 15, the points i., n., m...., &c., will evidently 
determine the directions of the shadow at 1, 2, 3,... o'clock 
respectively. 





FIG. 62 

But in practice it is much more convenient to perform the 
construction in the plane of the dial itself. Imagine the 
plane AKR of Fig. 62 turned about the line KR till it is 
brought into the plane of the dial, the point A of the plane 
being brought to U (Fig. 62). Then, by making the angles 
xii. 7i., i. ^n., n..Z7"m., &c., each = 15, we shall obtain the 
same series of points i., n., m. as before. 

If the dial -plate is horizontal, and I is the latitude of the 
place (xn. OA), we have evidently therefore the following 
construction : 

On the meridian line, measure xn. = OA sec I, and 
xn. U = xn. A = Oxn. sin I. Draw -ZTxn. R perpendicular 
to OU. Make the angles xn.ZTi., i.Z7n., n.JTni., &c., 
each = 15, taking i., n., m., &c., on KR. Join 0i., On., 
0m., &c., and let the joining lines meet the circumference 
of the dial in 1, 2, 3, &c. These will be the required 
points of graduation for 1, 2, 3,... o'clock respectively. 



127 

SECTION III. Units of Time The Calendar. 

169. Tropical, Sidereal, and Anomalistic Years. 

Hitherto we have defined a year as the period of a complete 
revolution of the Sun in the ecliptic. In order to give a 
more accurate definition, however, it is necessary to specify 
the starting point from which the revolution is measured. 
"We are thus led to three different kinds of years. 

A Tropical Year is the period between two successive 
vernal equinoxes, or the time taken by the Sun to perform a 
complete revolution relative to the first point of Aries. 

The length of the tropical year in mean solar time is very 
approximately 365d. 5h. 48m. 45 -5 Is. at the present time. 
For many purposes it may be taken as 365 days. 

A Sidereal Year is the period of a complete revolution 
of the Sun, starting from and returning to the secondary to 
the ecliptic through some fixed star. Thus, after a sidereal 
year the Sun will have returned to exactly the same position 
among the constellations. 

If T were a fixed point among the stars, the sidereal and 
tropical year would be exactly of the same length. But T 
has an annual retrograde motion of 50-22" among the stars 
( 141). Consequently, the tropical year is rather shorter 
than the sidereal. 

An Anomalistic Year is the period of the Sun's revo- 
lution relative to the apse line in other words, the interval 
between successive passages through perigee. 

Owing to the progressive motion of the apse line, the positions 
of perigee and apogee move forward in the ecliptic at the rate 
of 11-25" per annum ( 153). Hence the anomalistic year is 
rather longer than the sidereal. 

It is easy to compare the lengths of the sidereal, tropical, 
and anomalistic years. For, relative to the stars, 

In the sidereal year the Sun describes 360, 

In the tropical year it describes 360 - 50-22", 

In the anomalistic year it describes 360 -f 11-25" ; 
.'. (Sidereal year) : (tropical year) : (anomalistic year) 
= 36O : 360 50-22": 360+11'25". 

From this proportion it will be found that the sidereal year 
is about 20 m. longer than the tropical, and 4 Jin. shorter than 
the anomalistic. 



128 AST&ONOMf. 

170. The Civil Tear. For ordinary purposes, it is 
important that the year shall possess the following qualifications : 

1 st. It must contain an exact (not a fractional) number of days. 

2nd. It must mark the recurrence of the seasons. 

Now the tropical year marks the recurrence of the seasons, 
but its length is not an exact number of days, being, as we 
have seen, about 365d. 5h. 48m. 45 -5 Is. To obviate this 
disadvantage, the civil year has been introduced. Its length 
is sometime? 365, and sometimes 366 days, but its average 
length is almost exactly equal to that of the tropical year. 

Taking an ordinary civil year as 365d., four such years 
will be less than four tropical years by 23h. 15m. 2'04s. ? or 
nearly a day. To compensate foi this dilierence, every fourth 
civil year is made to contain 366 days, instead of 365, and is 
called a leap year. For convenience, the leap years are chosen 
to be those years the number of which is divisible by 4, such as 
1892, 1896. 

The introduction of a leap year once in every four years 
is due to Julius Caesar, and the calendar constructed on this 
principle is called the Julian Calendar. 

Now three ordinary years and one leap year exceed four 
tropical years by 24h. 23h. 15m. 2'04s., t.g.!44m. 57j96a^ 
Thus, 400 years of the Julian Calendar will exceed 400 
tropical years by (44m. 57'96s.) x 100, i.e., by 3d._2h^56m.36s. 

To compensate for this difference, Pope ~Gregory~XlII. 
arranged that three days should be omitted in every 400 years. 
This correction is called the Gregorian correction and is 
made as follows : Every year whose number is a multiple 0/100 is 
taken to be an ordinary year of 365 days, instead of being a leap 
year of 366, unless the number of the century is divisible by 4; 
in that case the year is a leap year. 

EXAMPLES. (i.) 1892 is divisible by 4, .*. the year 1892 is a 
leap year, (ii.) 1900 is a multiple of 100, and 19 is not divisible 
by 4, .'. 190O is not a leap year. (Hi.) 2000 : the number of tho 
century is 20, and is divisible by 4, .'. 2OOO is a leap year. 

The Gregorian correction still leaves a small difference 
between the tropical year and the average length of the civil 
year, amounting to only Id. 5h. 26m. in 4,000 years. 

171. A Synodic Year is a period of 12 lunar months, 
being nearly 355 days. The name is, however, rarely used. 



OK TIME. 129 

SECTION IV. Comparison of Mean and Sidereal Times. 

172. Relation between Units. One of the most 
important problems in practical astronomy is to find the 
sidereal time at any given instant of mean solar time, and 
conversely, to find the mean time at any given instant of 
sidereal time. Before doing this it is necessary to compare 
the lengths of the mean and sidereal days. 

We have seen ( 169) that a tropical year contains abont 
365| mean solar days. In this period both the true and 
mean Sun describe one complete revolution, or 360 from 
west to east relative to T ; or, what is the same thing, T 
describes one revolution from east to west relative to the 
mean Sun. But the mean Sun performs 365 revolutions 
from east to west relative to the meridian at any place. 
Therefore T performs one more revolution, i.e., 366 revo- 
lutions, relative to the meridian. 

Now, a sidereal day and a mean solar day have been defined 
( 22, 159) as the periods of revolution of the mean Sun 
and of T relative to the meridian ; 

.-. SOS} mean solar days = 366| sidereal days. 

Prom this relation we have, 
One mean solar day = ( 1 + ~ ) sidereal days 

\ ODO^-/ 

= (1 + '002738) sidereal days 
= 24h. 3m. 56'5s. sidereal time 
= 1 sidereal day + 4m. 4s. nearly; 

.. one mean solar hour = Ih. + 10s. s. sidereal time, 
and 6m. of mean solar time = Gin. + Is. sidereal time nearly. 
In like manner we have 

One sidereal day = ( 1 ) mean solar days 

\ o66f/ 

-- (1 -002730) mean days 
= 23h. 56m. 4' Is. mean time 
= 1 mean day 4m. -f 4s. nearly ; 

.*. one sidereal hour = Ih. 10s. +-J-S. of mean time, 

and 6m. sidereal time = 6m. Is.meansolartimenearly. 



L 



130 tf __ |4 

173. From the results of the last paragraph we have the 
following approximate rules : 

(i.) To reduce a given interval of mean time to 
sidereal time, add 10s. for every hour, and Is. for every 
6m. in the given interval. For every minute so added, sub- 
tract Is. 

(ii.) To reduce a given interval of sidereal time 
to mean time, subtract 10s. for every hour, and Is. for 
every 6m. in the given interval. Then add Is. for everr 
minute so subtracted. 

EXAMPLE 1. Express in sidereal time an interval of 13h. 23m. 25s. 
mean time. 

The calculation stands as follows : H. M. s.^ 

Mean solar interval =13 23 25 

Add 10s. per hour on 13h.... ' 2 10 

Is. per 6m. on 23m. ... ... ... 4 

13 25 39 
Subtract Is. per 1m. on 2m. 13'8s. ... 2 

.-. Required sidereal interval = 13 25 37 

EXAMPLE 2. Find the mean solar interval corresponding to 
14h. 45m. 53s. of sidereal time. 

The calculation stands as follows : H. M. s. 

Given sidereal interval ... ... ... ... = 14 45 53 

Subtract 10s. per hour on 14h. = 2m. 20s. \ o 28 
Is. per 6m. on 46m. (nearly )= 8s. / 

14 43 25 
Add Is. per 1m. on 2m. 28s. .... ... 3 

.. Required interval of mean time =14 43 28 

If accuracy to within a few seconds is not required, the 
second correction of Is. per 1m. may be omitted. On the 
other hand, if the interval consists of a considerable number 
of days, or if accuracy to the decimal of a second is needed, 
the results found by the rules will no longer be correct. 
"We must, instead, add 1/365^- of the given mean solar interval 
to get the sidereal interval, or subtract 1/366 J of the given 
sidereal to get the mean solar interval. 

In order to still further simplify the calculations, tables 
have been constructed ; in most cases, these give the quantity 
to be added or subtracted according as we are changing from 
mean to sidereal, or from sidereal to mean time. 



Off TIME. 131 



174. To find the sidereal time at a given instant 
of mean solar time on a given date at Greenwich. 

The Nautical Almanack* gives the sidereal time of mean 
noon at Greenwich on every day of the year. 

Now the given mean time represents the number of hours, 
minutes, and seconds which have elapsed since mean noon, 
expressed in mean time. Convert this interval into sidereal 
time ; we then have the sidereal interval which has elapsed 
since mean noon. Add this to the sidereal time of mean 
noon ; the result is the sidereal time required. 

Thus, let m he the mean time at the given instant, mea- 
sured from the preceding mean noon, 

* the sidereal time of mean noon from the Nautical Almanack, 
and let k = l/365 ; so that l+ is the ratio of a mean solar 
unit to the corresponding sidereal unit. 

Then, from mean noon to given instant, 

Interval in mean time = m 

.*. interval in sidereal time = m+lan 

But, at mean noon, sidereal time = s 

.*. at given instant, 

required sidereal time, s-=s Q +m+km. 

If the result he greater than 24h., we must subtract 24h., for 
times are always measured from Oh. up to 24h. 

EXAMPLE. Find the sidereal time corresponding to 8h. 15m. 40s. 
P.M. on Dec. 20, given that the sidereal time of mean noon was 
I7h. 55m. 8s. 

From mean noon to the given instant, the interval in mean time 
is 8h. 15m. 40s. 

Converting this interval to sidereal time, by the method of 173, 
we have Mean solar interval = 8h. 15m. 40s. 

Add 10s. per hour on 8h. 1m. 20s. 

Add Is. per 6m. on 15m. 40s. 3s. 

8h. 17m. 3s. 

Subtract Is. per 1m. on 1m. 23s. ls._ 

.*. Sidereal interval since mean noon = 8h. 17m. 2s. 

But sidereal time of mean noon = I7h. 55m. 8s. 

.*. Sidereal time at instant required = 26h. 12m. 10s. 

Or, deducting 24h., sidereal time is = 2h. 12m. 10s. 

* Or Whitaker's Almanack, which may be used if the Nautical is 
not at hand. 



132 ASTEONOMf. 

175. To find the mean solar time corresponding to 
a given instant of sidereal time at Greenwich. 

Subtract the sidereal time of mean noon from the given 
sidereal time ; this gives the interval which has elapsed since 
mean noon, expressed in sidereal time. Convert this interval 
into mean time ; the result is the mean time required. 

Let It = 1/366 J ; so that 1 k' is the ratio of a sidereal to a 
mean solar unit. 

Let the given sidereal time = , 

and let the sidereal time of the preceding mean noon = ; 
Then, from mean noon to given instant, 

Interval in sidereal time = s S Q 

.'. interval in mean time = (s s ) '(s s ). 

.'. required mean time m = (s * ) M> ). 

If s be less than , we must add 24h. to s in order that the 
times s, s may be reckoned from the same transit of T 

EXAMPLE. Find the solar time corresponding to 16h. 3m. 42s. 
sidereal time on May 6, 1891, sidereal time at mean noon being 
2h. 52m. 17s. 

Sidereal interval since mean noon 

= 16h. 3m. 42s. -2h. 52m. 17s. = 13h. llm. 25s. 

/. Mean solar interval ( 173) 

= 13h. llm. 25s. -2m. 10s. 2s. + 2s. = 13h. 9m. 15s. 

Hence, 13h. 9m. 15s. is the mean time; which, in our usual 
reckoning, would be called Ih. 9m. 15s., on the morning of May 6 
( 36). The sidereal timo was also 16h. 3m. 42s. a sidereal day 
or 23h. 56m. 4s. previously, i.e., Ih. 13m. 11s. a.m. on the morning 
of May 5. % 

176. To find the mean time corresponding to a 
given instant of sidereal time at Greenwich (alterna- 
tive method). The Nautical Almanack also contains the 
mean time of tl Sidereal Noon," i.e., the mean time when T 
is on the meridian, and when the sidereal clock marks 
Oh. Om. Os. Let this be m , and let s be the given sidereal 
time, k' the factor l/366 as before. Then 

From sidereal noon to given instant, sidereal interval = s ; 
.-. ,, ,, mean solar,, = s-k's. 

But, at sidereal noon, mean time = m ; 

/. at given instant, 

The required mean time = m Q +s Jc's. 



Oft TIME. 133 

177. To find the sidereal time from the mean solar, 
or the mean time from the sidereal, in any given 
longitude. If the longitude is not that of Greenwich, the 
ahove methods will require a slight modification, because the 
sidereal time of mean noon and mean time of sidereal noon 
are tabulated for Greenwich. 

In such cases, the safest plan is as follows : Find the 
Greenwich time corresponding to the given local time ( 96). 
Convert this Greenwich time from mean to sidereal, or sidereal 
to mean, as the case may be, and then find the corresponding 
local time again. 

Let the longitude be L west of Greenwich (Z being nega- 
tive if the longitude is east), 
lot m l be the mean and s l the sidereal local time, 

m, s the corresponding times at Greenwich, 
and let /c, &', w> , have the same meanings as in 172-4. 

By 96 we have, whether the times be local or sidereal, 
(Greenwich time) (local time in long. Z W.) = T 1 - F Zh. 
= 4Z m. Therefore, s s l ^L = m m r 

(i.) If m l is given and s l is required, we have (in hours), 



By 174, 8 = 8H + 

s l = srfa = 
(ii.) If s 1 is given and m l is required, we have 



By 175, 176, m = (-* ) - #(-) or = i +-#, 
i.e. m *-*~^ s -* > + L 



EXAMPLE. Find the solar time when the local sidereal time is 
5h. 17m. 32s. on March 21, the place of observation being Moscow 
(long. 37 34' 15" B.) ; given that sidereal time of mean noon was 
23h. 54m. 52s. at Greenwich. 

Eeduced to time ( 23), 37 34' 15" is 2h. 30m. 17s. 

/. Greenwich sidereal time at instant required 

= 5h. 17m. 32s. -2h. 30m. 17s. = 2h. 47m. 15a. 

Sidereal interval since Greenwich noon 

= 2h. 47m. 15s. + 24h -23h. 54m. 52s. = 2h. 52m. 23s. 
.'. Greenwich mean time = 2h. 52m. 23s. -20s. -9s. = 2h. 51m. 54s. 
.'. Moscow mean time = 2h. 51m. 54s. + 2h. 30m. 17s. - 5h. 22m. 11s 



134 AStKONOiTT. 

178. Equinoctial Time. For the purpose of comparing 
the times of observations made at different places on the 
Earth, another kind of time has been introduced. 

The Equinoctial Time at any instant is the interval of 
time that has elapsed since the preceding vernal equinox, 
measured in mean solar units. 

The advantage of equinoctial time is that it is independent 
of the observer's position on the Earth, since the instant when 
the Sun passes through T is a perfectly definite instant of 
time, and is independent of the place of observation. On the 
other hand, mean time and sidereal time, being measured 
from the transits of the mean Sun and of T across the 
meridian, depend on the position of the meridian that is, on 
the longitude of the observer. 

The chief disadvantage of equinoctial time is that since the 
tropical year contains 365d. 5h. 48m. 46s., and not exactly 
365 days, the vernal equinox will occur 5h. 48m. 46s. later 
in the day every year, so that at the end of each tropical year 
the equinoctial clock will have to be put back 5h. 48m. 46s. 
Hence also the same equinoctial time will represent a different 
time of day on the same date in different years. 

The disadvantages of using local time are obviated in Great 
Britain by the universal use of " Greenwich Mean Time." 

179. Practical Applications. In 41 we showed how 
to determine roughly the time of night at which a given star 
would transit on a given day of the year. "With the intro- 
duction of mean time, in the present chapter, we are in a 
position to obtain a more accurate solution of the problem. 

Por the R.A. of any star (expressed in time) is its sidereal 
time of transit. If this be given, we only have to find the 
corresponding mean time ; this will be the required time 
of transit, as indicated by an ordinary clock. 

In the calculations required in converting the time from 
one measure to the other, it is advisable not to quote the 
formula? of 174-177, but to go through the various steps 
one by one. 

If neither the sidereal time of mean noon nor the mean 
time of sidereal noon is given, we must fall back on the 
rough method of 35. 



ON TIME. 135 



EXAMPLES. 

1. Find the solar time at 5h. 29m. 28s. sidereal time on July 1, 1891 ; 
mean time of sidereal noon being 17h. 20m. 8s. 

Sidereal interval from sidereal noon to the given instant = 5h.29m.28s. 

.-. Mean solar interval = 5h. 29m. 28s. 50s. 5s. + Is. = 5h.28m.34s. 
i.e., Mean solar time = 5h. 28m. 34s. + I7h. 20m. 83. =22h. 48m. 42s. ; 

or, lOh. 48m. 42s. A.M., July 2. 

It was also 5h. 29m. 28s., a sidereal day or 23h. 56m. 4s. pre- 
viously, i.e., lOh. 52m. 38s. a.m. July 1. ^ 

2. To find the mean time of transit of Aldelaran at Greenwich on 
December 12, 1891. Given H M s 

R.A. of Aldelaran = 4 29 40 ; 

Sidereal time of noon, December 12, 1891 = 17 23 56. 

Since the star's R.A. is less than the sidereal time of noon, we 
must increase the former by 24h., in order that both may be mea- 
sured from the same " sidereal noon." H. M. s. 
Sidereal time of transit + 24h. = 28 29 40 

Subtract noon = 17 23 56 



/. Sidereal interval from noon to transit =11 5 44 

To convert into mean solar units, subtract 1 49 

.'. Mean Solar interval from noon to transit =11 3 55 
.'. Aldelaran transits at llh. 3m. 55s. mean time. 7~&~*ii- 

3. To find the (local) sidereal time at New York at 9h. 25m. 31g. 
(local mean time) on the morning of September 1, 1891. 
Longitude of New York = 74 W. 

Sidereal time of mean noon at Greenwich, Sept. 1 = lOh. 42m. 24s. 

The given local mean time is measured from midnight, therefore 

we must take the time measured from noon as H. M. s. 

August 31, 1891. = 21 25 31 

Add for 74 west longitude reduced to time = 4 56 

.*. Greenwich mean time is, August 31, 26 21 31 

or, September 1, 2 21 31 

To convert this interval to sidereal units, add 24 

/. Sidereal time elapsed since Greenwich noon = 2 21 55 

But at Greenwich noon, sidereal time (by data) = 10 42 24 

.-. Sidereal time at Greenwich is 13 4 19 

Subtract for 74 west longitude, 4 56 

.-. Sidereal Time at New York = 8 18 9 



136 ASTRONOMY. 

4. To find the Paris mean time of transit of Regulus at Nice on 

December 26, 1891. H. M. s. 

Longitude of Paris = 2 21' E.E.A. of Regulus =10 2 34 

Nice = 7 18' E. 

Sidereal time at Greenwich noon = 18 18 48 

Here local sidereal time of transit at Nice 10 2 34 

Subtract east longitude of Nice, T 18', in time 29 12 

V. Greenwich sid. time of transit at Nice -I- 24h. C3 33 22 

Subtract Greenwich sidereal time at noon, 18 18 48 

/. Sidereal interval since Greenwich noon 15 14 34 

To convert to mean solar units, subtract 2 30 



.'. Greenwich mean time = 15 12 4 

Add east longitude of Paris, expressed in time = 9 24 

.-. Paris mean time of transit = 15 21 28 

That is, 3h. 21m. 28s. in the morning on December 27. 

5. Find the E.A. of the Sun at true noon on October 8, 1891, given 
that the equation of time for that day is 12m. 24s., and that the 
sidereal time of mean noon on March 21 was 23h. 54m. 52s. 

Mean solar interval from mean noon March 21 to mean noon Oct. 8 

= 201 days. 

Mean solar interval from mean noon to apparent noon on Oct. 8 

= -12m. 24s. 

/. interval from mean noon on March 21 to apparent noon on Oct. 8 

= 201d.-12m. 24s. 

Now, in 365 days the mean Sun's E.A. increases 24h., and the 
increase takes place quite uniformly. 

.'. increase in mean Sun's E.A. in 201 days H. M. s. 

= 24h.x 201 -=-365| = 13 12 27 
Add mean Sun's E.A. on March 21 

( = sidereal time of mean noon) = 23 54 52 



.'. mean Sun's E.A. at mean noon Oct. 8 = 37 7 19 

or, subtracting 24h., =13 7 19 

Subtract change of E.A. in 12m. 24s. = 2 

.'. mean Sun's E.A. at apparent noon Oct. 8 =13 7 17 
But true Sun's E.A. mean Sun's E.A 

= equation of time = 12 24 

/. True Sun's E.A. at apparent noon Oct. 8 = J2h. 54m. 53s, 



ON TIME. 137 



EXAMPLES. -V. 

1; To what angles do Sidereal Time, Solar Time, and Mean Time 
correspond on the celestial sphere ? Are these angles measured 
direct or retrograde ? 

2. Draw a diagram of the Equation of Time, on the supposition 
that perihelion coincides with the vernal equinox. 

3. On May 14 the morning is 7'8 minutes longer than the after- 
noon : find the equation of time on that day. 

4. On a sun-dial placed on a vertical wall facing south, the 
position of the end of the shadow of a gnomon at mean noon is 
marked on every day of the year. Show that the curve passing 
through these points is something like an inverted figure of eight. 

5. Why are not the graduations of a level dial uniform ? Show 
that they will be so if the dial be fixed perpendicular to the index. 

6. Show that if every 5th year were to contain 366 days, every 
25th year 367 days, and every 450th year 368 days, the average 
length of the civil year would be almost exactly equal to that of the 
tropical year. How many centuries would have to elapse before the 
difference would amount to a day ? 

7. Give explicit directions for pointing an equatorial telescope to 
a star of R.A. 22h., declination 37 N., in latitude 50 N., longitude 
25 E., at lOh. Greenwich mean time, when the true Sun's E.A. is 
14h. 47m. 17s., and the equation of time is 16m. 14s. 

8. If the mean time of transit of the first point of Aries be 
9h. 41m. 24*4s., find the time of the year, and the sidereal time of 
an observation on the same day at Ih. 22m. 13'5s. 

9. At Greenwich, the equation of time at apparent noon to-day is 
- 3m. 39'42s., and at apparent noon to-morrow it will be 3m. 35'39s. 
Prove that the mean solar time at New York corresponding to ap- 
parent time 9 A.M. there this morning is 8h. 56m. 2O9s., having given 
that the longitude of New York is 74 I' W. 

10. Find the sidereal time at apparent noon on Sept. 30, 1878, at 
Louisville ( long. 85 30' W.) having given the following from the 
Nautical Almanack : 

At mean noon. 



Sun's apparent right 



ascension. 



Sept. 30. 12h. 26m. 23'16s. 
Oct. 1. 12h. 30m. 0'51s. 



Equation of time 

to be added to mean time. 

10m. 0'77s. 

10m. 19-98s, 



138 ASTBONOMT. 



MISCELLANEOUS QUESTIONS. 

1. Explain how to determine the position of the ecliptic relatively 
to an observer in S. latitude at a given time on a given day. 

2. Indicate the position of the ecliptic relatively to an observer 
at Cape Town (lat. 33 56' 3'5" S.) at noon on August 3. 

3. Explain why a day seems to be gained or lost by sailing round 
the world. State which way round a day seems to be lost, and give 
the reason why. 

4. If the inclination of the ecliptic to the equator were 60, instead 
of 23 27^', describe what would be the variations in the seasons to 
an observer in latitude 45, illustrating your description with a 
diagram. 

5. Describe the changes of position in the point of the Sun's 
rising at different times of the year, and at different points on the 
Earth's surface. 

6. If the equator and ecliptic were coincident, what kind of curve 
would be described in space by a point on the Earth's surface, say 
at the equator, daring the course of the year ? 

7. Examine when that part of the equation of time due to the 
eccentricity of the Earth's orbit is positive. 

8. On September 22, 1861, the times of transit of a Lyrse and of 
the Sun's centre over the meridian of Greenwich were observed to 
be 18h. 32m. 51'3s. and 12h. Om. 23'3s. by a sidereal clock whose 
rate was correct. Given that the R. A. of a Lyrae was 18h. 31m. 43'9s., 
find the Sun's B..A. and the error of the clock. 

9. Define mean time and sidereal time, and compare the lengths 
of the mean second and the sidereal second. 

10. If a, a' are the hour angles in degrees of the Sun at Greenwich, 
at t and t' hours mean time, show that the equations of time at the 
preceding and following mean noons, expressed in fractions of an 
hour, are respectively 

a't-at' 21 .X(24-Q-a(24 f) 



oir TIME. 



EXAMINATION PAPER. v. 

1. Define tiie dynamical mean Sun and the mean Sun, stating at 
what points they have the same R.A., and when the former coin- 
cides with the true Sun. Show that the mean Sun has a uniform 
diurnal motion, and state how it measures mean time. 

2. Define the equation of time. Of what two parts is it generally 
taken to consist? State when each of these parts vanishes, is 
positive, or negative. Give roughly their maximum values, and 
sketch curves showing their variations graphically. 

3. Show that the equation of time vanishes four times a year. 

4. If, on a certain day, the sun-dial be 10 minutes before the clock, 
what is the value of the equation of time on that day ? Will the 
forenoon of that day or the afternoon be longer, and by how much ? 

5. Define the terms solar day, mean solar day, sidereal day. 
What is the approximate difference and the exact ratio of the 
second and third ? 

6. Define the terms civil year, anomalistic year, equinoctial 
time. Why was this last introduced ? 

7. Show how to express mean solar time in terms of sidereal 
time, and vice versd. 

8. If the mean Sun's R.A. at mean noon at Greenwich on June 1 
be 4h. 36m. 54s., find the sidereal time corresponding to 2h. 35m. 45s. 
mean time (1) at Greenwich, (2) at a place in longitude 25 E. 

9. On what day of the year will a sidereal clock indicate lOh. 20m. 
at 4 P.M. ? 

.10. In what years during the present century have there been 
five Sundays in February ? When will it next happen ? 
ASTBON. L 



CHAPTER VI. 



ATMOSPHERICAL KEFBACTION AND TWILIGHT. 

180. Laws of Refraction. It is a fundamental prin- 
ciple of Optics that a ray of light travels in a straight line, 
so long as its course lies in the same homogeneous medium ; 
but when a ray passes from one medium into another, or 
from one stratum of a medium 
into another stratum of dif- 
ferent density, it, in general, 
undergoes a change of direction 
at their surface of separation. 
This change of direction is 
called Refraction.* 

Letarayof light S0(Fig. 64) 
pass at from one medium into 
another, the two media being 
separated by the plane surface 
AB, and let OT be the direc- 
tion of the ray after refraction 
in the second medium. Draw 
ZOZ' the normal or perpendicular to the plane AB at 0. 
Then the three laws of refraction may be stated as follows : 

I. Thfrancident and refracted rays SO, OTand the normal 
ZOZ 1 all lie in one Diane. 




. . 

is a constant quantity, being tlie same for all directions of the 
rays, so long as the two media are the same.] 

This constant ratio is called the relative index of 
refraction of the two media, and is usually denoted by the 
Greek letter fi. 

* For a fuller description, see Stewart's Light, Chap. YI. 
f The value of the ratio varies slightly for rays of different colours, 
but with this we are not concerned in the present chapter. 



ATMOSPHERICAL REFRACTION AND TWILIGHT. 141 

Thus, if TO be produced backwards to S', 

sin ZOS = /i sin Z' OT = p sin ZOS r , 

The angles ZOS and Z' OT are usually called the angle of 
incidence and the angle of refraction respectively. 

III. When light passes from a rarer fo a denser medium, the 
angle of incidence is greater than the angle of refraction. 

Since Z ZOS> L Z'OT, sin ZOS > sinZ'OT and /. /i > 1. 

181. General Description of Atmospherical Refrac- 
tion. If the Earth had no , 
atmosphere, the rays of light 
proceeding from a celestial 
body would travel in straight 
Lines right up to the obser- 
ver's eye or telescope, and we 
should see the body in its 
actual direction. 

But when a ray Sa (Fig. 65) 
meets the uppermost layer 
AA' of the Earth's atmo- 
sphere, it is refracted or bent 
out of its course, and its direc- FlG - 

tion changed to aft. On passing into a denser stratum of aiv 
at BB', it is further bent into the direction be, and so on ; 
thus, on reaching the observer, the ray is travelling in 
a direction OT, different from its original direction, but 
(by Law I.) in the same vertical plane. 

The body is, therefore, seen in the direction OS', 
although its real direction is aS or OS. Also, since the 
successive horizontal layers of air A A', BB', CC', ... 
are of increasing density, the effect of refraction is to 
bend the ray towards the perpendicular to the surfaces of 
separation, that is, towards the vertical. 

Hence : The apparent altitudes of the stars are 
increased by refraction. 

In reality, the density of the atmosphere increases gradually 
as we approach the Earth, instead of changing abruptly at 
the planes A A', BB', .... Consequently, the ray, instead of 
describing the polygonal path Sabc 0, describes a curved path, 
but the general effect is the same. 




142 



ASTRONOMY. 



182. Law of Successive Refractions. Let there be 
any number of different media, separated by parallel planes 
JLA', ', CO', HH' (Fig. 66), and let Sale OT represent 
the path of a ray as refracted at the various surfaces. Then 
it is a result of experiment that the final direction S'T 
of the ray is parallel to what it would have been if the ray 
had been refracted directly from the first into the last medium 
without traversing the intervening media. 

Thus, if a ray SO, drawn parallel to Sa, were to pass 
directly from the first medium to the last by a single refrac- 
tion at 0, its refracted direction would be the same as that 
actually taken by the ray Sa, and would coincide with OT. 



S' 




FIG. 66. 



FIQ. 67. 



183. The Formula for Astronomical Refraction. 
"We shall now apply the above laws to determine the change 
in the apparent direction of a star produced by refraction. 

Since the height of the atmosphere is only a small fraction 
of the Earth's radius, it is sufficient for most purposes 
of approximation to regard the Earth as flat, and the surfaces 
of equal density in the atmosphere as parallel planes. "With 
this assumption, the effect of refraction is exactly the same 
( 182) as if the rays were refracted directly into the lowest 
stratum of the atmosphere, without traversing the intervening 
strata. 



ATMOSPHERICAL EEFEACTION AND TWILIGHT. 143 

Let OS (Fig. 67) be the true direction of a star or other 
celestial body. Then, before reaching the atmosphere, the 
rays from the star travel in the direction SO. Let their 
direction after refraction be S'OT, then OS' is the 
apparent direction in which the star will be seen, and the 
angle SOS' is the apparent change in direction due to 
refraction. The normal OZ points towards the zenith. 
Hence ZOS is the star's true zenith distance, and ZOS 1 
or Z'OT is its apparent zenith distance, and the first and 
third laws of refraction show that the star's apparent direction 
is displaced towards the zenith. 

Let L ZOS' = 8, tS'OS = u, and .-. L ZOS = z + u ; 
and let /j be the index of refraction. 

By the second law of refraction, 

sin (a -f u) = p sin s. 
sin 2 cos w-j-cos s sin u = yu sin z. 

Now the refraction u is in general very small. Hence,' if 
u be measured in circular measure, we know by Trigonometry 
that sin u = , and cos u = 1 very approximately. Therefore 

we have 

sin 2 -f- u cos 8 = (j. sin 8 ; 



Let U be the amount of refraction in circular measure 
when the zenith distance is 45. Putting s 45, we have 



.-. u = 

Thus the amount of refraction is proportional to 
the tangent of the apparent zenith distance. 

The last result does not depend on the fact that the refrac- 
tion is measured in circular measure. Hence, if w", U" be 
the numbers of seconds in u, U, we have 

u" = U" tan a. 

The quantity U" is called the coefficient of refraction. 

Since U is the circular measure of Z7", we have 

V" = 180X60X6 . V= 206265 (,,-1), 

7T 

whence, if U" is known, p cm be found, and conversely, 



144 ASTEONOMT. 

184. Observations on the preceding Formula. In 

the last formula u" represents the correction which must be 
added to the apparent or observed zenith distance in order to 
obtain the true zenith distance. By the first law, the azimuth 
of a celestial body is unaltered by refraction. 

Thus the time of transit of a star across the meridian, or 
across any other vertical circle, is unaltered by refraction. 
In using the transit circle, there will, therefore, be no cor- 
rection for observations of right ascension, but in finding the 
declination the observed meridian Z.D. will require to be 
increased by U" tan z. 

A star in the zenith is unaffected by refraction, and the 
correction increases as the zenith distance increases. When 
a star is near the horizon, the formula u" = U" tan z fails, 
since it makes u" = co, when 2 = 90. In this case u is no 
longer a small angle, so that we are not justified in putting 
sin u = u and cos u = 1. But there is a more important reason 
why the formula fails at low altitudes, namely, that the rays 
of light have to traverse such a length of the Earth's atmo- 
sphere that we can no longer regard the strata of equal density 
as bounded by parallel planes. In this case, it is necessary to 
take into account the roundness of the Earth in order to obtain 
any approach to accurate results. 

For zenith distances less than 75, the formula is found to 
give fairly satisfactory results ; for greater zenith distances it 
makes the correction too large-. 

The coefficient of refraction U" is found to be about 57", 
when the height of the barometer is 29 -6 inches and the 
temperature is 50. But the index of refraction depends on 
the density of the air, and this again depends on the pressure 
and temperature. Hence, where accurate corrections for 
refraction are required, the height of the barometer and 
thermometer must be read. Any want of uniformity in the 
strata of equal density, or any uncertainty in determining 
the temperature, will introduce a source of error ; hence it is 
desirable that the corrections shall be as small as possible. 
For this reason observations made near the zenith are always 
the most reliable, 



ATMOSPHERICAL EEFEACTION AND TWILIGHT. 



145 



*185. Cassini's Formula. The law of refraction was also investi- 
gated by Dominique Cassini on the hypothesis that the atmosphere 
is spherical but homogeneous throughout ; in this way he obtained 
the approximate formula 

u = (ju1) tans (1 n sec 2 s), 

where n is the ratio of the height of the homogeneous atmosphere 
to the radius of the Earth. 

Cassini's formula may be proved as follows : Let SO'O be the 
path of a ray of light from a star 8. 
By hypothesis this ray undergoes a 
single refraction on entering the homo- 
geneous atmosphere at 0'. Let be 
the position of the observer, G the 
centre of the Earth. Produce 00' 
to 8', CO to Z, and GO' to Z'. Let 
u = L SOS' (in circular measure), 



Then, by 183, if u is small, we have 

u = (jti 1) tans'; 

but here z' is not the apparent zenith 
distance, so that we must express tan z' 
in terms of tan z. 

Draw CT perpendicular to O'O pro- 
duced, and O'N perpendicular to COZ. 
Then O'T tans' = TG = OTtansj 
tan_s_ = OT = 1 + ^0 
tans'" Or OT 

OIV sec z __ , O# gec 2 
00 




= 1 + ' 



FIG. 68. 



00 cos z 

But ON is very approximately the height of the homogeneous 
atmosphere OH, and is therefore = n . OG ; 

tan 2 . , tans 



tans' 



= 1 + n sec 2 z ; 



tans' 



1 + Ti sec 2 z 



whence, by substituting in the formula, we have 

, .. , tan z 

1 4 n sec 2 z 
= (fj. 1) tans {l wsec 2 s + 'n- 2 sec 4 s n 3 sec 6 s, &c.} 

Now n is very small ; we may therefore neglect its square and higher 
powers; hence we obtain approximately 

u= (/* !) tan z (1Ti sec- s), 

which is Cassini's formula. 

If the value of n be properly chosen, Cassini's formula is found 
to give very good results for all zenith distances up to 80. 



146 . A8TBONOMY. 

186. To determine the Coefficient of Refraction 
from Meridian Observations. Assuming the " tangent 
law," u= 7"tanz, the coefficient of refraction U may be 
found from observations of circumpolar stars as follows. 

Let Z D z 2 , the apparent zenith distances of a circumpolar 
itar, be observed at upper and lower culminations respectively. 
Then the true zenith distances will be 

%i -f 7~tan z l and z 2 + 7"tan z 2 . 

Now, the observer's latitude is half the sum of the meri- 
dian altitudes at the two culminations ( 28), hence if I be 
the latitude, we have 



or 90-? = i(2 1 + 2 2 ) + J^(tanz 1 +tanE 2 ) ...... (i.). 

Now let a second circumpolar star be observed. Let its 
apparent zenith distances at upper and lower culminations be 
z' and % '. Then we obtain in like manner 

90-Z = i (a'+a") + i T(tan z' + tan 2") ...... (ii.). 

Eliminating I from (i.) and (ii.) by subtraction, we have 



(tan Zj + tan z 2 ) (tan z' + tan z")* 

If the two stars have the same declination, we shall have 
Zj = z and z 2 = z", and the above formula will fail. Hence 
it is important that the two observed stars should differ con- 
siderably in declination ; the best results are obtained by 
selecting one star veiy near the pole (e.g-, the Pole Star) and 
the other about 30 from the pole. 

187. Alternative Method (Bradley's). Instead of 
using a second circumpolar star, Bradley observed the Sun's 
apparent Z.D.'sat noon at the two solstices. Let these be Z v Z y 

By 38, since the true Z.D.'s are 

Z^ + 7tan Z t and Z^ + U tan Z v 

Z^ + Z7tan Z l = l- i, Z^ + Vtan Z z = l + i; (i = obliquity.) 
.-. 2Z = ' 1 + 4+7(tan;+tan 2 ) ......... (iii.). 

Eliminating ?from (i.), (iii.), we have 



^hence 7" is found. 



UNIVERSITY 



TWILIGHT,. 147 

188. Other Methods of finding the Refraction. 

Suppose that at a station on the Earth's equator, either a 
star on the celestial equator, or the Sun at an equinox, is 
observed during the day. Its diurnal path from east to west 
passes through the zenith, and during the course of the 
day its true zenith distance will change uniformly at the 
rate of 15 per hour. Thus the true Z.D. at anytime is 
known. Let the apparent Z.D. be observed with an altazi- 
muth. The difference between the observed and the calcu- 
lated Z.D. is the displacement of the body due to refraction. 

By this method we find the corrections for refraction at 
different zenith distances without making any assumptions 
regarding the law of refraction. 

Except at stations on the Earth's equator, it is not possible 
to observe the refraction at different zenith distances in such 
a simple manner. Nevertheless, methods more or less similar 
can be employed. For this purpose the zenith distances of a 
known star are observed at different times. The true zenith 
distance at the time of each observation can be calculated 
from the known R.A. and declination ( 26). Hence 
the refraction for different zenith distances of the star 
can be determined. This method is very useful for verifying 
the law of refraction after the star's declination and the 
observer's latitude have been found with tolerable accuracy. 
Moreover, it can be employed to find the corrections for 
refraction at low altitudes when the " tangent law " ceases 
to give approximate results. 

189. Tables of Mean Refraction. From the results 
of such observations tables of mean refraction have been con- 
structed by Bessel,* and are now used universally. These 
are calculated for temperature 50 and height of barometer 
29*6 inches ; they give the refraction for every 5' of altitude 
up to 10, for larger intervals at altitudes between 10 
and 54, and for every 1 at altitudes varying from 54 to 
90. Other tables give the "Correction for Mean Refraction," 
which must be added to or subtracted from the mean refrac- 
tion given in the first table in allowing for differences in the 
temperature and barometric pressure. The corrections for 
temperature and pressure are applied separately. 

* See any book of Mathematical Tables, such as Chambers'^. 



148 ASTBOffOMY. 

190. Effects of Refraction on Rising and Setting. 

At the horizon the mean refraction is about 33' ; con- 
sequently a celestial body appears to rise or set when it 
it is 33' below the horizon. Thus, the effect of refraction is 
to accelerate the time of rising, and to retard, by an equal 
amount, the time of setting of a celestial body. In particular, 
the Sun, whose angular diameter is 32', appears to be 
just above the horizon when it is really just below. 

The acceleration in the time of rising due to refraction can 
be investigated in exactly the same way as the acceleration 
due to dip ( 104). If u" denotes the refraction at the hori- 
zon in seconds, d the declination, x the inclination to the 
vertical of the direction in which the body rises, the accelera- 
tion in the time of rising in seconds 

= u" sec x sec d. 
lo 

Taking the horizontal refraction as 33', or 1980", and 
putting x = 0, d = 0, we see that at the Earth's equator at 
an equinox, the time of sunrise is accelerated by about 
2m. 12s. owing to refraction, 

"When the Sun or Moon is near the horizon, it appears 
distorted into a somewhat oval shape. This effect is due to 
refraction. The whole disc is raised by refraction, but the 
refraction increases as the altitude diminishes ; so that the 
lower limb is raised more than the upper limb, and the 
vertical diameter appears contracted. The horizontal dia- 
meter is unaffected by refraction, since its two extremities 
are simply raised. Hence, the disc appears somewhat flat- 
tened or elliptical, instead of truly circular. 

According to the tables of mean refraction, the refraction 
on the horizon is 33', while at an altitude 30', the refraction 
is only 28' 23", and at 35' it is 27' 41". Hence, taking the 
Sun's or Moon's diameter as 32', the lower limb when on the 
horizon is raised about 5' more than the upper. The con- 
traction of the vertical diameter, therefore, amounts to 5', 
i.e., about one-sixth of the diameter itself, so that the appa- 
rent vertical and horizontal angular diameters are approxi- 
mately in the ratio of 5 to 6. 



ATMOSPHERICAL REFRACTION AND TWILIGHT. 149 

191. Illusory Variations in Size of Sun and Moon. 

The Sun and Moon generally seem to look larger when 
low down than when high up in the sky. This is, however, 
merely a false impression formed by the observer, and is not 
in accordance with measurements of the angular diameter 
made with a micrometer. When near the horizon, tho 
eye is apt to estimate the size and distance of the Sun and 
Moon by comparing them with the neighbouring terres- 
trial objects (trees, hills, &c.). When the bodies are at 
a considerable altitude no such comparison is possible, and a 
different estimate of their size is instinctively formed. 

192. Effect of Refraction on Dip, and Distance of 
the Horizon. Since refraction increases as we approach 
the Earth, its effect is always to bend the path of a ray of 
light into a curve which is concave downwards (Fig. 69). 




FIG. 69. 

Let be any point above the Earth's surface, and let T' 
be the curved path of the ray of light which touches the Earth 
at T' and passes through 0. Then OT' is the distance of 
the visible horizon. Draw the straight tangent OT, then 
OT would be the distance of the visible horizon if there 
V^ere no refraction; hence, it is evident from the figure that 

The Distance of the horizon is increased by 
refraction. 

Draw OT", the tangent at to the curved path OT', then 
OT" is the apparent direction of the horizon. Hence, from 
the figure we see that 

The Dip of the horizon is diminished by refraction. 

Both dip and distance are still approximately proportional 
to the square root of the height of the observer. 



150 ASTBONOMY. 

193. Effect of Refraction on Lunar Eclipses and 
on Lunar Occupations. In a total eclipse the Moon's 
disc is never perfectly dark, but appears of a dull red colour. 
This effect is due to refraction. The Earth coming between 
the Sun and Moon prevents the Sun's direct rays from reach- 
ing the Moon, but those rays which nearly graze the Earth's 
surface are bent round by the refraction of the Earth's 
atmosphere, and thus reach the Moon's disc. 

From observing the " occultations " of stars when the 
unilluminated portion of the Moon passes in front of them, 
we are enabled to infer that the Moon does not possess an 
atmosphere similar to that of our Earth. For the directions 
of stars would be displaced by the refraction of such an 
atmosphere just before disappearing behind the disc, and just 
after the occultation ; and no such effect has been observed. 

194. Twilight. The phenomenon of twilight is also due 
to the Earth's atmosphere, and is explained as follows : 
After the Sun has set, its rays still continue to fall on the 
atmosphere above the Earth, and of the light thus received 
a considerable portion is reflected or scattered in various 
directions. This scattered light is what we call twilight, 
and it illuminates the Earth for a considerable time after 
sunset. Moreover, some of the scattered light is transmitted 
to other particles of the atmosphere further away from the 
Sun, and these reflect the rays a second time ; the result of 
these second reflections is to further increase the duration of 
twilight. Twilight is said to end when this scattered light has 
entirely disappeared, or has, at least, become imperceptible. 
From numerous observations, twilight is found to end when 
the Sun is at a depth of about 18 below the horizon. 

If the Sun does not descend more than 18 below the 
horizon, there will be twilight all night. 

Let I = latitude, d = Sun's declination, then it is easily 
seen by a figure that the Sun's depth below the horizon 
at midnight = 90 dl. 

This depth is less than 18, if I > 72 <?. 

But the greatest value of d is , or nearly 23J (mid- 
summer). Hence, there is twilight all the night about 
midsummer, at any place whose latitude I is not less than 
72-23j, or48i. 



ATMOSPHERICAL KfcFKACTiON AND TWILIGHT, 151 



EXAMPLES. YI. 

1. What would IDG the effect of refraction on terrestrial objects as 
seen by a fish under water ? 

2. For stars near the zenith show that the refraction is approxi- 
mately proportional to the zenith distance, and that the number of 
seconds in the refraction is equal to the number of degrees In the 
zenith distance. (Take coefficient of refraction = 57".) 

3. From the summit of a mountain 2400 feet above the level of 
the sea, it is just possible to see the summit of another, of height 
3450 feet, at a distance of 143 miles. Find approximately the radius 
of the Earth, assuming that the effect of refraction is to alter the 
distance of the visible horizon in the ratio 12 : 13. 

4. Trace the changes in the apparent declination of a star due to 
refraction in the course of a day, at a place in latitude 45 N"., the 
actual declination being 50 N. 

5. At Greenwich (latitude 51 28' 31" N.) the star o Oygni was 
observed to transit 6 34' 57" south of the zenith. Find the 
star's declination, employing the results of Question 2. 

6. Prove that if the declination of a star observed off the meridian 
is unaffected by refraction, the star culminates between the pole 
and the zenith, and that the azimuth of the star from the north 
is a maximum at the instant considered. 

7. Show how the duration of twilight gives a measure of the 
height of the atmosphere. 

8. What is the lowest latitude in the arctic circle at which there 
is no twilight at midwinter, and what is the corresponding distance 
from the North Pole in miles ? 



152 ASTRONOMt. 



EXAMINATION PAPEB.-VL 

1. What effect Las refraction on the apparent position of a star ? 
Show that the greater the altitude of the star the less it is displaced 
by refraction, and that a star in the zenith is not displaced at all. 

2. Prove (stating what optical laws are assumed) that, if the 
Earth and the layers of the atmosphere be supposed flat, the 
amount of refraction depends solely on the temperature and pressure 
at the Earth's surface. 

3. Prove the formula for refraction, r = (/j. 1) tan z. Is this 
formula universally applicable ? Give the reason for your answer. 

4. Given that the optical coefficient of refraction of air (u) 
= r0003, find the astronomical coefficient of refraction (U) in 
seconds. 

5. What is the refraction error? How may we approximately 
determine the correction for refraction from observations made 
on the transits of circumpolar stars ? 

6. Show how the constant of refraction (on the usual assumption 
that the refraction is proportional to the tangent of the zenith 
distance) might be determined by observing the two meridian alti- 
tudes of a circumpolar star whose declination is known. 

7. Assuming the tangent formulas for refraction, find the latitude 
of a place at which the upper and lower meridian altitudes of a cir- 
cumpolar star were 30 and 60 ( ^ 3 = T732), the coefficient of 
refraction being 57". 

8. Why is the Moon seen throughout a total eclipse ? 

9. In the Scientific American, June 18, 1887, it was stated by the 
editor that " The atmosphere by its refraction acts as a lens, pro- 
ducing an apparent increase in the diameter (of the Sun and Moon) 
near the horizon. When we consider that the atmosphere, as seen 
from the surface of the globe, is a section of a vast lens whose radius 
is the semi-diameter of the Earth, it is reasonable to assume a small 
increase in the size of the objects seen through it, and a still greater 
increase when seen in the obliquity of the horizon." Why is the 
above statement altogether incorrect? 

10. Find the duration of twilight at the equator at an equinox. 



CHAPTER VII. 



THE DETERMINATION OF POSITION ON THE 
EARTH. 

SECTION I. Instruments used in Navigation. 

195. Among the different uses to which Astronomy has 
been put, perhaps the most important of all is its application 
to finding the geographical latitude and longitude of any 
place on the Earth from observations of celestial bodies. Such 
observations may be made for either of the following purposes : 

1 . The determination of the exact latitude and longitude 
of an observatory. These must be known accurately before 
the coordinates of a star can be found or observations taken 
at different observatories can be compared. 

2. The construction of maps. The geographical latitude 
and longitude of a place form a system of coordinates which 
enable us to represent its exact position on a map. 

3. The determination of the exact position of a ship in 
mid-ocean. This is the most useful application of all ; on a 
long sea voyage it is necessary to calculate daily the ship's 
latitude and longitude correct to within a mile or so. 

Now, owing to the motion and rocking of a ship, all the 
astronomical instruments hitherto described are useless at 
sea. The mariner is therefore obliged to have recourse to 
others which are unaffected by the unsteadiness of the vessel. 
The two instruments best fulfilling this condition are the 
Sextant and the Chronometer, which we shall now describe. 



154 

196. The Sextant. The use of the Sextant is to measure 
the angular distance between two objects by observing them 
both simultaneously. It consists of a brass framework form- 
ing a sector CDE graduated along the circular arc or limb 
DE\ the angle DCE is usually about 60 or rather more. 
To the centre C of the arc is fixed an arm BI, capable of 
turning about C, and which carries the small mirror B, called 
the index glass. Another small mirror A, called the 
horizon-glass, is fixed to the arm CD, making an angle of 
about 60 with BD. Of this mirror half the back is usually 
silvered, the other half being transparent. Finally, at T is 
fixed a telescope, pointed towards A in such a manner as to 
receive the rays of light from the mirror B after reflec- 
tion at A (Pigs. 70, 71). 




FIG. 70. 



On looking through the telescope T we shall see two sets 
of images, for objects at -ZTwill be seen directly through the 
unsilvered part of the mirror A, while objects at S will be 
seen after two reflections at the mirrors B and A. The 
mirror is so near the object glass of the telescope as to ba 
quite out of focus ; hence these two sets of images will not 
appear separate, but will overlap one another. 



THE DETERMINATION OF POSITION ON THE EARTH. 155 

The arm BI carries at / an index mark or pointer by which. 
its position can be read off on the graduated scale DE. The 
pointer should read zero when the mirrors A, B are parallel 
(as in the position B'E, Fig. 70). When this is the case, the 
two images of any very distant object H will coincide. For 
when a ray of light is reflected in succession at two parallel 
mirrors, its final direction is parallel to its initial direction.* 
Hence if H' CA T represents the path of a ray of light from 
the object H, as reflected in succession at B' and A, the por- 
tion AT is parallel to H'C, and therefore coincides with the 
ray HAT, by which the object is seen directly. 

Now let it be required to find the angular distance between 
the two objects .ZTand 8. To do this, the mirror B is rotated 
by means of the arm BI until the image of S (formed by 
the two reflections) is seen to coincide with H. The angle 
EC1, through which the mirror B has been turned from its 
original position, is then half the required angular distance 
between H, S. 

For draw CN', CN perpendicular to the two positions 
JB'j B of the mirror respectively. Since in reflection at a 
plane mirror the angles of incidence and reflection are equal, 

and .-. ACH'= 2 



also L NCS = A CN and .-. L A CS 2 z A ON. 
Hence /.ACS- ACH'= 2(^ACJV- LACN'\ 
i.e., LH'CS = 2. LN'CN 

= 2 . L ECI-, 

or the angular distance between the objects is double the 
angle ECL 

On the scale ED, every half -degree is marked as 1. The 
reading of the pointer / will therefore give double the angle 
ECI, and this is the angular distance required. 

The coincidence of the two images in the field of view of 
the sextant will not be affected by any small displacement of 
the instrument in its own plane. This peculiarity renders 
the sextant particularly useful on board ship, where it is 
impossible to hold the instrument perfectly steady. 

* See Stewart's Text-Book of Light, Chap. IV. 
ASTRON. M 



156 ASTRONOMY. 

197. Shades, Clamp and Tangent Screw, Beading 
Glass, Vernier. 

For viewing the Sun, the sextant is provided with 
shades. These consist simply of plates of glass blackened 
for the purpose of reducing the great intensity of the Sun's 
rays. There are two sets of shades, G, 6r, hinged to the 
frame CE in such positions that one set can be inserted 
between A and (7, to deaden the rays from S, while the other 
set can be turned behind A to deaden the rays from H. 
They are called respectively the " index shades" and 
"horizon shades." 




FIG. 71. 

The arm or index bar B C is furnished with a clamp, by 
means of which it can be clamped at any desired part of the 
graduated limb DJE. When this has been done the arm can 
be moved slowly by means of a tangent screw -ZT, and in 
this way can be adjusted with great precision. 

The arc D12is usually graduated to divisions of 10',* and 
is used by means of the lens Jf, called the " reading glass." 
But the index bar also carries a scale V called a Vernier 
( 198) which, sliding beside the scale on the limb, enables 
us to read off observations to within 10". 

*0f course these divisions are only 5' apart, but in what follows 
we shall speak of half-minutes as minutes. 



THE DETEKMINATION OF POSITION ON THE EABTH. 157 

198. The Vernier is a scale the distance between whose gradua- 
tions is 10' 10", i.e., 9' 50", or 10" less than the distance between 
the graduations on the limb. These graduations are marked 0", 
10", 20", &c., being measured in the same direction as on the limb. 
For example, let us suppose the zero point on the vernier is between 
the marks 26 20' and 26 30' on the limb. We take the reading by 
the limb as 26 20 7 . We then look along the vernier scale until we 
find that one of the marks on it exactly coincides ivith one of the marks 
on the limb. Suppose that this is the 25th graduation from tne 
zero point of the vernier, i.e., the point marked 4' 10". We add 
this 4' 10" to the 26 20' read on the limb, and the sum gives the 
correct reading, namely, 26 24' 10". 

The principle is as follows. Let us denote by P the mark which 
coincides on the two scales. 

Then from zero of vernier scale to P is 25 divisions of vernier, 
i.e., an arc of 25 x (10' -10"). 

Also from 26 20' of scale on limb to P is 25 divisions of limb, i.e.> 
an arc of 25 x 10'. 

.*. from 26 20' on limb to of vernier, represents an arc of 
25 x 10' -25 x (10' -10") ; i.e., 25 x 10", or 4' 10". 

Hence the zero mark of the vernier scale is at a distance 26 20' 
+ 4' 10" from the zero on the limb, and the reading is 26 24' 10". f 

199. The Errors of the Sextant need not be described in detail. 
If the sextant does not read zero when the two mirrors are parallel, 
it is said to have an Index Error, and a constant correction for 
index error must be added to all readings made with the instrument. 
There are also errors due to eccentricity or want of coincidence 
between the centre about which the index bar turns, and the 
centre of the limb, errors of graduation, &c. 

200. To determine the Index Error of the Sextant, In all goou 
sextants the graduated limb is continued backwards for about 5 
behind the zero point. This portion of the limb is called the "arc 
of excess," and is used for finding the index error, as follows. The 
Sun or full Moon is observed ; the two images of its disc are 
brought into contact. Let e be the index-error, r the sextant reading, 
D the angular diameter of the disc, then we have evidently D = r + e. 
Now let the index bar be moved along the arc of excess until the 
images again touch, the image which was before uppermost being 
undermost. If the reading on the arc of excess be r', we have 
now D = r' + e, or D = / e. 

Hence, 2e = r'r. 

f The simpler forms of mercurial barometer are provided with a 
vernier by means of which the height of the mercury is read off to 
the nearest hundredth of an inch. The student will find it of great 
assistance to carefully examine the vernier in such an instrument 



158 ASTEONOMY. 

201. To take altitudes at Sea by the Sextant. 

The principal use of the sextant is for finding altitudes. 
Now the altitude of a star is its distance from the nearest 
point of the celestial horizon. To find this, the sextant is so 
adjusted that the reflected image of the star appears to lie on 
the ofiing or visible horizon ; when the plane of the sextant 
is slightly turned, the image of the star should just graze the 
horizon without going below it. The sextant reading then 
gives the star's angular distance from the nearest point of the 
"offing." Subtract the dip of the horizon and the correc- 
tion for refraction, both of which are given in books of 
mathematical tables. The star's true altitude is thus 
obtained. 

202. To take the Altitude of the Sun or Moon. 

In observing the Sun's altitude, the " index " shades must be 
turned into position between the two mirrors, and the instru- 
ment adjusted so that the Sun's lower limb appears just to 
graze the horizon. The reading of the sextant, when 
corrected for dip and refraction, gives the altitude of the 
Sun's lower limb. Add the Sun's angular semi-diameter, 
which is given in the Nautical Almanack ; the altitude of the 
Sun's centre is then obtained. 

Both the Sun's altitude and its angular diameter may be 
obtained by observing the altitudes of the upper and lower 
limbs. The difference of the two corrected readings gives the 
Sun's angular diameter, and half the sum of the readings 
gives the altitude of the Sun's centre. 

If this method is used, allowance must be made for the 
change in the Sun's altitude between the observations. For 
this purpose, three observations must be made. First take 
the altitude of the Sun's lower limb, then of the upper limb, 
and lastly, again of the lower limb. Also note the time 
of each observation. The difference between the first and 
third readings determines the Sun's motion in altitude ; from 
this, by a simple proportion, the change in altitude between 
the first and second observations is found, and thus the alti- 
tude of the lower limb at the second observation is known. 
We can now find the Sun's angular diameter, and the altitude 
of its centre at the second observation. 



THE DETERMINATION OP POSITION ON THE EAETH. 159 

Let , = time of 1st observation, when a = alt. of lower limb ; 

3 = time of 2nd observation, when I = alt. of upper limb ; 

# 3 = time of 3rd observation, when a' = alt. of lower limb ; 
Then in time t s t v the alt. of lower limb increases a' a. 

.'. in time 2 ^ it increases (a a) x f ~. 

h~h 
Hence if 2 denote the alt. of lower limb at second observation, 



tn - t j 69 - TI 

This finds # 2 , and we then have 

Sun's angular diameter = l a. r 

Alt. of Sun's centre at second observation = 

In taking the altitude of the Moon, the altitude of the 
illuminated limb must be observed, an'l the angular semi- 
diameter, as given in the " Nautical Almanac," must be 
added or subtracted, according as the lower or upper limb is 
illuminated. 

203. Artificial Horizon for Land Observations. 

Owing to the absence of a well-defined offing on land, an 
artificial horizon must be used. This is simply a shallow 
dish of mercury, protected in some manner from the disturbing 
effect of the wind. The sextant is used to observe the 
angular distance between a star and its image as reflected in 
the mercury. Half this angular distance is the star's apparent 
altitude ; correcting this for refraction, the true altitude is 
obtained (<?/. 65). 

As the limb of the sextant is generally an arc of not more 
than 70, the instrument will not measure angular distances 
of more than 140, and it can, therefore, only be used with an 
artificial horizon for altitudes of under 70. For greater 
altitudes the zenith sector must be used. 

At sea, where altitudes are measured from the offing, this 
objection does not apply. On account of the motion of the 
vessel an artificial horizon is useless; hence, no observations 
can be taken when the offing is ill- defined, which fre- 
quently happens, especially at night. The mariner is, 
for this reason, chiefly dependent upon observations of the 
Sun and Moon, and such stars of the first magnitude, or 
planets, as are visible about dusk. 



160 ASTRONOMY. 

204. The Chronometer is the form of timepiece used on 
board ship, and in all observations in which clocks are un- 
available, owing to their want of portability. In principle, 
the chronometer is simply a large and very accurately con- 
structed watch ; its rate of motion being controlled, not by a 
pendulum, but by a balance-wheel, which oscillates to and 
fro under the influence of a steel hair-spring. In order that 
the chronometer may go at a uniform rate, the balance-wheel 
is constructed in such a manner that its time of oscillation is 
unaffected by changes of temperature. If the wheel were 
made of one continuous piece of metal, any increase of tem- 
perature would cause the whole to expand, and the couple 
exerted by the spring would not reverse its motion so readily, 
so that the time of oscillation would be increased. To 




FIG. 72. 

obviate this, the rim of the wheel is made in several (generally 
three) disconnected arcs, each being formed of steel within 
and of brass without. When the temperature rises, the sup- 
porting arms or spokes expand, pushing the arcs outward; 
but in each arc the outer half of brass expands more than the 
inner half of steel, and this causes it to curl inwards, 
bringing the extremity actually nearer the centre than it was 
before. The arcs carry small screw weights, and by adjusting 
these nearer to or further from the supports, the compensa- 
tion can be arranged with great accuracy.* 

* The student who has read a little Rigid Dynamics will notice 
that the compensation must be so arranged that the " moment of 
inertia " of the balance-wheel is unaffected by the temperature. 



ttE DETERMINATION OF POSITION ON THE EALITII. 161 

Another peculiarity of the chronometer consists in the 
"detached escapement." The action of the main spring, 
while keeping up the oscillations, must not affect their 
periodic time, and to secure this condition the escapement is 
so arranged that the balance wheel is only acted on during a 
very small portion of each oscillation. 

The chronometer is usually suspended in a framework, in 
such a manner that when the vessel rolls the instrument 
always swings into a horizontal position ; the framework 
also serves to protect it from violent shaking. 

205. Error and Bate of the Chronometer. A chrono- 
meter is constructed to keep Greenwich mean solar time. As 
in the case of the astronomical clock, the amount that a chrono- 
meter is slow when it indicates noon is called its error, and 
the amount which it loses in 24 hours is called its rate. If 
the chronometer is fast, the error is negative ; if it gains, the 
rate is negative. 

The essential qualification of a good chronometer is that 
its rate must be quite uniform. It is not necessary that the 
rate shall be zero, provided that its amount is known, since 
a correction can easily be applied to obtain the correct 
time from the chronometer reading. During sea voyages 
extending over a large number of days, the correction for rate 
may become considerable, and there is no very satisfactory 
method of finding the chronometer error at sea ; for this 
reason the instrument is rated, i.e., has its rate determined 
by comparisons with a standard clock, whenever the ship is in 
port. Moreover, many ships carry several chronometers, which 
serve to check each other ; if the rate of one should vary slightly, 
this change would be detected by comparison with the others. 

Many of the best chronometers used in the !N~avy arid 
elsewhere are tested at the Greenwich Observatory. They 
are there kept in a special room, in which they can be 
subjected to artificial variations of temperature, with a view 
of ascertaining whether the compensation for temperature is 
perfect or not. The chronometers are compared daily with 
the standard clock. The process of rating is performed by 
two assistants who have acquired the power of counting the 
beats of the clock while reading off the errors of one chrono- 
meter after another. In this manner, about a hundred 
chronometers can be rated in half an hour. 



1 62 ASTBONOMY. 

SECTION II. Finding the Latitude by Observation. 

206. The methods of finding latitude may be conveniently 
classified as follows : 

A. Meridian Observations. 

(1) By a single meridian altitude of the Sun or a known 

star. 

(2) By meridian altitudes of two stars, one north and one 

south of the zenith, taken with the sextant. 

(3) By two observations of a circumpolar star. 

B. Observations not made on the Meridian. 
(" Ex-meridian Observations") 

(4) By a single observed altitude, the local time being known. 

(4A) By " circum-meridian altitudes." 

(4s) By observing the altitude of the Pole Star. 

(5) By observations of two altitudes. 

(6) By the Prime Vertical instrument. 

We now proceed to examine the various methods in detail, 
but it must be premised that the " ex-meridian " methods 
cannot be thoroughly explained without spherical trigo- 
nometry. 




207. Latitude by a Single Meridian Altitude. Let 

S (Pig. 73) represent the position of the Sun or a star of 
known declination when southing. 

Let the meridian altitude sS be observed, and let it be = a\ 
also let % be the meridian Z.D. ZS, so that z = 90- a. Let 
tfbe the known K". decl. QS, and / the required N. latitude QZ. 



THE DETEKMINATION OF POSITION ON THE EARTH. 166 



EXAMPLE. 

On April 11, 1891, in longitude 80 12' E. (roughly) with an 
artificial horizon, the meridian reading of the sextant for the Sun's 
lower limb was observed to be 107 59' 48". Barometer 307 inches, 
thermometer 72. Find the latitude, having given the following 
data : 

Q i II 

's (Sun's) decl. at Greenwich noon, Ap. 11 =8 19 4 "I From 

Hourly variation of decl = 55'1 \ Nautical 

O's semi-diameter = 15 59 J Almanack. 

Mean refraction at altitude 54 = 41 \ from 

Correction for barometer = +1 I m ^ 

for thermometer = 2 J 

The calculation is best arranged as follows : 

O I II 

(i.) Double observed alt. of lower limb = 107 59 48 

.'. observed alt = 535954 

Corrected refraction at this alt. 

(which is nearly 54) = 40 (-) 

.'. true alt. of lower limb = 53 59 14 

Aug. semi- diam = 1559( + ) 

Merid. alt. O's centre =, 54 15 13 

Subtract from 90 



Mcrid. ZJD. of Q's centre = 35 44 47 S (i.) 

M. 8. 

(ii.) Long. 8 12' E. in time ... = 32 48 

.'. tune of observation = 32 48 before Greenwich noon. 

O / II 

's decl. at Greenwich noon April 11 = 8 19 4 N. (increasing). 
Variation in 30m. before noon ... = 27 ( ) 

2m. 48s. (about) ... = 3(-) 

.'. 's decl. at time of observation ... = 8 18 34 N. 
Add O's merid. Z.D. from (i.) = 35 44 47 S. 



Required north latitude = 44 3' 21". 



166 ASTKONOMT. 

211. To find the latitude by sextant observations 
of the meridian altitudes of two stars which culmi- 
nate on opposite sides of the zenith. This is really only 
a modification of the first method. Two stars of known 
declination are selected which culminate, one south and the 
other north of the zenith, at very nearly the same altitude. 
The latitude is calculated independently from ohservations of 
the meridian altitudes of either star, and the mean of the two 
results is taken as the correct latitude. 

This method possesses the following advantages : 

1st. There is no need to correct the observed altitudes for 
dip of the horizon ; 

2nd. The result is unaffected by any constant instrumental 
errors (index error, &c.) which affect both altitudes equally; 

3rd. The correction for refraction is reduced to a minimum, 
or even entirely eliminated, if the altitudes are almost equal. 

.For let rfj, d. 2 be the north declinations of the two stars ; 

Zj (south) and z 2 (north) their true meridian Z.D.'s ; 

#! and # 2 their observed meridian altitudes ; 

t and u. 2 the corrections for refraction; 

D the dip of the horizon ; 

e the correction for constant instrumental errors. 

For true meridian altitudes of the two stars we have 
90 z l a^e D-u v 90 s 2 = a ^eDu y 

The two observations give, therefore, for the latitude (by 204) 
I ^+ 2l = ^ + 90- 0! -<? + ./) + !, 
1= d 2 -% = d 3 9Q+a z + eD-u 2 . 
Therefore, taking the mean of the two results, 



a result involving no corrections beyond the difference of 
refractions, w 2 u r 

Moreover, if the altitudes a^ and # 2 are greater than 45, 
and their difference (0 a 0,) is less than a degree, then 
\ ( 2 - MJ) is < 1", and therefore the refraction correction 
may be entirely neglected. 



THE DETERMINATION OP POSITION ON THE EARTH. 167 

212. Latitude by Circumpolars. This method has 
already been mentioned in 28, but we will here repeat 
the investigation for convenience. 

Let x, x 1 (Fig. 74) represent the positions of a circumpolai 
star at its upper and lower transits. Let its meridian 
altitudes nx and nx 1 be observed, and let their corrected 
values be a^ and tf a respectively. Since 

Px = star's N.P.D. = P^ 



or 




In this formula no knowledge of the star's declination is 
required, but the observed altitudes require to be corrected 
for refraction, dip, &c. 

The circumpolar method is most useful in determining the 
latitude of a fixed observatory, because this must be done 
before the declination of any star can be determined. The 
transit circle is used to determine the meridian altitudes at 
the two culminations. 

By observing two or more circumpolars the correction for 
refraction may be found, as in 186, and the observed alti- 
tudes may then be corrected for refraction. 



168 ASTRONOMY. 

As the declinations of a large number of stars are given 
in astronomical tables, the circumpolar method is never 
used at sea. It would possess no advantage, and would have 
the disadvantage of requiring a correction for the change in 
the ship's place between the two culminations. 

EXAMPLES. 

1. The observed meridian altitude of /3 Ceti (decl. 18 36' 44'5" S.) 
is 36 43' 12", and that of a Ursse Minoris (decl. 88 41' 53'1" N.) at its 
upper culmination is 30 9' 57", both altitudes being measured from 
the " offing," and the dip being unknown. Find the latitude, given 
Refraction at alt. 36 = 1' 20" ; at alt. 37 = 1' 17". 

This is an example of the method of 211. The calculation stands 
thus : 



Ceti (south). 
36 43' 12" 
- 1 18 


a 
Observed altitudes 
Refraction corrections 

Corrected Altitudes 

Zenith Distances 
Declinations 


Ursse Minoris (north). 
36 9' 57" 
- 1 19-5 


36 41 54 
90 


36 8 37-5 
90 


+ 53 18 6 S. 
-18 36 44-5 S. 


-53 51 22-5 N. 
+ 88 41 53-1 N. 



34 41 21-5 N. Calculated Latitudes 34 50 30 6 N. 

Thus, lat. by star north of zenith = 34 50 7 30'6" N. 
south = 34 41 21-5 N. 

2)69 31 52-1 
Mean latitude = 34 45' 56" N? 

Here, owing to dip, one of the calculated latitudes is 4' 34'6" too 
great, and the other is 4' 34'5" too small, but the mean of the two 
results is the correct latitude. 

2. The observed altitudes of /3 Ursa? Minoris at lower and upper 
culmination are 29 58' 16" and 60 45' 3". Find approximately the 
latitude, assuming the coefficient of refraction to be 57". 

By the " tangent formula," refraction at altitude 30 (approx.) 

= 57" tan 60 = 57" x A/3 = 57" x 1-732 = 1' 39". 
Refraction at alt. 60 = 57" tan 30 = 57" x A/3/3 = 1' 39"-=-3 - 33". 
Hence truealt. at lower culmination = 29 58' 15" 1' 39" = 29 56' 36" 
upper =60 45 3 - 33" =60 44 30 

2) 90 41 6 
.-. Required North Latitude= 45 20' 33" 



THE DETERMINATION OF POSITION ON THE EARTH. 1 GO 

LATITUDE BY EX-MERIDIAN OBSERVATIONS. 

213. To find the latitude "by a single altitude, the 
local time being known. If the local time be known, a 
single altitude of the Sun or a known star is sufficient to 
determine the latitude. 

For let S be the observed body, ^the zenith, P the pole.f 
Then in the spherical triangle PZS, the known local time 
enables us to find the hour angle ZPS. For, if the Sun be 
observed, its hour angle ZPS 

= 15 x (apparent local time) 
= 15 x (mean local time equation of time) ; 
and if a star be observed, its hour angle ZPS 

= 15 x (local sidereal time star's R. A.). 
Also ZS = observed body's Z.D. = 90 (observed altitude) ; 

PS= ,, N.P.D. = 90 -(known dccl.). 

Hence, ZS, PS, and the angle ZPS are known. These 
data completely fix the spherical triangle ZPS, and from 
them ZP can be found by Spherical Trigonometry. 
Hence the latitude is found, being = 90 ZP. 

*214. By Circum-meridian Altitudes. This is a par- 
ticular case of the method last described. In attempting to 
find the latitude by meridian observations, it may happen 
that passing clouds prevent the body from being observed at 
the instant of transit. In this case the latitude can be found 
from the observed altitude when very near the meridian. 
The hour angle ZPS is then small, and the difference 
between the observed and meridian altitudes is also 
small. This difference is called the " Itoduction," and is 
found by approximate methods. 

The best results are obtained by taking a number of alti- 
tudes of the body before and after passing the meridian. 

*215. By a Single Altitude of the Pole Star. The 
N.P.D. of Polaris is only about 1 16'. Hence, if its alti- 
tude is observed at any time, the latitude may be found by 
adding to, or subtracting from, this altitude, a small correc- 
tion, never greater than about 1 16J'. 

iThe student will have no difficulty in illustrating 213-216 
with diagrams. For 213, Fig 75 may be copied. 




1 70 ASTRONOMY. 

This correction consists of three parts, which are given by 
three tables in the Nautical Almanack. The first two cor- 
rections depend on the sidereal time, and on the observed 
altitude ; the third is due to variations in the R.A. and 
N.P.D. of Polaris, due to precession ( 141), etc. 

*216. Latitude by observation of Two Altitudes. By observing 
the altitudes of two known stars, both the latitude and the local 
sidereal time can be found. 

The same method can be employed to determine the latitude by 
two observations of the Sun's altitude, separated by a known interval 
of time. 

The necessary calculations are very complicated, involving 
Spherical Trigonometry, and they cannot be materially simplified 
even by the use of tables. 

A very useful geometrical construction, enabling us, from the two 
observed altitudes, to indicate the exact position of a ship on a 
globe without calculation, will be detailed in Section VI. of this 
chapter. 

217. Latitude by the Prime Vertical Instrument. 

The latitude of a fixed observatory may be found by means 
of an instrument similar to the Transit Circle, but whose 
telescope turns in the plane of the prime vertical instead of 
the meridian. A star will cross the middle wire of such an 
instrument when its direction is either due east or west ; 
the times of the two transits are observed. Let S, S' be the 
positions of a known star at its eastern and western transits, 
Z the zenith. P the pole. The sidereal interval between the 
two transits determines the angle SPS', and this is evidently 
twice the angle ZPS. Hence z ZPS is known. Also PS, the 
star's N.P.D., is known, and PZS is a right angle. Therefore, 
the spherical triangle ZPS is completely determined, and the 
colatitude ZP can be found. 

The times of the transits are unaffected by refraction, and 
this fact constitutes the principal advantage of the method. 

The observations may be performed by an altazimuth, whose 
horizontal circle is clamped so that the telescope moves in 
the prime vertical. The instrument must be so adjusted that 
the interval of time between the first transit and culmination 
is equal to the interval between culmination and the second 
transit. The culmination must be observed with a Transit 
Circle. 



THE DETERMINATION OF POSITION ON THE EARTH. 171 

SECTION III. To find the Local Time by Observation. 

218. In determining the longitude of a place on the Earth, 
the first step is to find the local time by observations of the 
hour angle of a known celestial body. If the time indicated 
by a chronometer or clock at the instant of observation be 
also noted, we shall find the difference between the true local 
time and the indicated time. This difference is the error of 
the clock on local time. 

In 167 we described one instrument for observing local 
time the Sun-dial. This cannot, however, be used except 
for very rough observations, as the boundary of the shadow 
cast by the style is not sufficiently well defined to admit of accu- 
rate measurements. Moreover the Sun-dial is not portable. 

For this reason the local time is usually found by one or 
other of the following methods : 

1st. By meridian observations. 

2nd. By equal altitudes. 

3rd. By a single altitude, the latitude being known. 

4th. By observation of two altitudes. 

219. Local Time by Meridian Observations. In a 

fixed observatory, the local sidereal time is found by means of 
the Transit Circle, as explained in 24, 54. The transit of 
a known star is observed ; the local sidereal time of transit is 
equal to the star's E.A., and is therefore known. 

Or by observing the transit of the Sun's centre, the time 
of apparent local noon may be found. The equation of time 
is the mean time of apparent noon, and is given in the 
" Nautical Almanack " ; ht-ncc the local mean time is found. 

These methods are not available at sea, as the Transit 
Circle cannot be used. It might be thought that we could 
use a soxtant to ascertain the instant when the body's altitude 
is greatest, bat, for a short interval before and after the transit, 
the altitude remains very nearly constant ; it is therefore 
impossible to tell with any degree of accuracy when it is 
a maximum. 

On the other hand, a slight error in the time of observation 
does not affect the altitude perceptibly, so that the meridian 
altitude may be observed with great accuracy, as in 208. 

ASTRON. N 



172 



ASTRONOMY. 



220. Method of Equal Altitudes. "When it is required 
to find the local time from observations taken with a sextant, 
the simplest method is as follows : Observe the altitude of 
any celestial body some time before it culminates. After the 
body has passed the meridian, observe the instant of time 
when its altitude is again the same as it was at the first 
observation. Half the sum of the times of the two observa- 
tions gives the time of transit. 




FIG. 75. 

For let S, S' be the two observed positions of the body, Z 
the zenith, and P the pole. 

The altitudes of SX, S'X' being equal, the zenith distances 
are equal ; 

.-. ZS = ZS'. 

Also PS = PS', 

and the spherical triangles ZPS, ZPS' have ZP in common. 

.-. tSPZ= Z ZPS'. 

Now let ^ and 2 be the times of the two observations, 
t the time of transit. 

Then tt^ is the time taken to describe the angle SPZ-, 
a t ,, ,, ,, ,, ZPS'. 

Since the two angles are equal, 

.-. t-t^ t^-t-, 

.-. * = }(, +3). 

From the time of transit the local time can be found, as in 
the last article. 



THE DETERMINATION OF POSITION ON THE EARTH. 173 

221. In observing the Equal Altitudes with a 
Sextant, the following method is used : At the first ob- 
servation clamp the index bar at an altitude slightly greater 
than that of the body. Continue to observe the body as it 
rises, till its image is in contact with the horizon, and note 
the instant of time (^) at which this happens. Keep the 
index bar clamped until the second observation ; commence 
observing the body again just before it has reached the same 
altitude again, and note the instant of time ( 2 ) when its 
image is again in contact with the horizon. The two observed 
times (t v tf s ) are the times of equal altitude. 

If an artificial horizon le used, we must observe the two 
instants of time (t v 2 ) when the two images are in contact. 

222. Equation of Equal Altitudes. If the Sun be the 

observed body, its declination will, in general, change 
slightly between the two observations ; hence PS will not be 
exactly equal to PS', and the angles SPZ, ZPS' will not be 
quite equal. For this reason a small correction must be 
applied, in order to allow for the effect of the change of 
declination. This correction is called the Equation of 
Equal Altitudes, and may be found from tables which 
have been calculated for the purpose. 

At Sea allowance must also be made for the change of 
position of the ship between the two observations, and thk 
correction is also effected by means of tables. 

223. The method of Equal Altitudes possesses the 
following advantages : 

1st. The results are unaffected by errors of graduation of 
the sextant, for the actual readings are not required. 

2nd. The semi-diameter of the observed body need not be 
known. 

3rd. The observed altitudes, being equal, are equally affected 
by refraction, and no refraction correction need therefore be 
made. 

4th. The dip of the horizon need not bo known, provided 
that it is the same at both observations. 



1 74 ASTBONOMY. 

224. With, a Gnomon, the time of apparent noon can be 
roughly found in a very simple manner. A rod is fixed 
vertically in a horizontal plane, and on the latter are 
drawn several circles, concentric with the base of the rod. 
Let the times be observed, before and after noon, when 
the extremity of the shadow cast by the rod just touches 
one of these circles. At these two instants the Sun's alti- 
tudes are, of course, equal, and therefore the time of apparent 
noon is the arithmetical mean between the observed times. 

EXAMPLE. The shadow of a vertical stick at Land's End (long. 
5 40' W.) is observed to have the same length at 9h. 27m. A.M. and 
3h. 1m. 40s. P.M., Greenwich time. Find the equation of time on 
the day of observation. 

Greenwich mean time of local apparent noon is 

i { 9h. 27m. Os. + 3h. 1m. 40s. 12h. } = 14m. 20s. 

But, by 96, Greenwich mean time of local mean noon = 22m. 40s. 

.'. Eqn. of time = local mean time of apparent noon = 8m. 20s. 

*225. The Latitude may also be found by the method of 
equal altitudes, though the calculations require Spherical 
Trigonometry. For this purpose, the altitude at either 
observation must be read off' on the sextant, and corrected for 
refraction, dip, &c. The zenith distance SZ is therefore 
known. The angle SPZis also known, being half the angle 
described in the interval t^ t lt and PS, being the comple- 
ment of the declination, is also known. The spherical triangle 
ZP8 is therefore completely determined, and ZP, which is 
the complement of the latitude, can be found. 

226. Local Time by a Single Altitude, the Latitude 
being known. This is the converse of the method for 
finding the latitude described in 213. If the altitude of a 
known body, S, be observed in known latitude, we know 
ZS, SP, PZ, which are the complements of the observed 
altitude, the declination, and the latitude respectively ; hence 
the hour angle SPZ, and therefore also the local time, may 
be found. 

*227. Local Time by Two Altitudes. The method of 216 
determines, not only the latitude, but also the hour angles of the 
bodies at the two observations, and these determine the local time, 
The method of equal altitudes is in reality only a particular case 



THE DETERMINATION OF POSITION ON THE EARTH. 



175 



SECTION IY. Determination of tTie Meridian Line. 

228. Before setting up a transit circle or equatorial in a 
fixed observatory, it is necessary to know with considerable 
accuracy the direction of the meridian line, i.e., the line 
joining the north and south points of the horizon. At sea, 
the directions of the cardinal points are determined by a 
mariner's compass ; but here, too, it is of great use, on long 
voyages, to determine the variation of the compass, or 
the deviation of the magnetic needle from the meridian line. 
This deviation is different at different parts of the Earth. 

There are three ways of finding the meridian line : first, 
by two observations of a celestial body at equal alti- 
tudes ; second, by a single observation of the azimuth ; third, 
by one or more observations of the Pole Star. 

229. By Equal Altitudes. When a body has equal 
altitudes before and after culmination, the corresponding 
azimuths are equal and oppo- 
site. 

For if 8, S' denote the two 
positions of tho body, the tri- 
angles ZPS, ZPS' are equal in 
all respects ; 

.-. Z PZS = Z PZS' and 
/. Z sZS = z sZS'. 

230. At Sea, the Sun's azi- 
muth, or compass bearing, may 
be observed when rising and 

when setting; the meridian FlG - 

line bisects the angle between the two directions ( 29). 

231. On Land, we may observe tho directions of the 
shadow cast by a vertical rod on a horizontal plane when it 
has equal lengths ; for this purpose we mark the points at 
which the end of the shadow just touches a circle concentric 
with the base of the rod (of. 224). Bisecting the angle 
between the two directions, the north and south points are found. 

If greater accuracy is required, an altazimuth may be used. 
The readings of the horizontal circle arc taken when tho 
altitudes of a star are equal ; the meridian reading is the 




176 VSTBONOMT. 

arithmetical mean of the two readings. While observing th 
equal altitudes, the vertical circle must be kept clamped. 

*232. By a Single -Observation. If the direction o 
the vertical plane through a single celestial body S b 
observed at any instant, the direction of the meridian lin 
may be found by means of Spherical Trigonometry. 

For if any three parts of the triangle ZPS are known, th 
triangle is completely determined, and the angle PZS can b 
found. 

The azimuth sZS = 180 PZS, and is then known 
hence the meridian line ZS is found. 

Now the sides PS, ZS, ZP are the complements of th 
declination, the altitude, and the latitude ; and the hour angl 
ZPS is known, if the local time be known. Any three o 
these data are sufficient to determine the angle PZS. 

Thus, for example, the Sun's direction, either at sunrise fl- 
at sunset, determines the meridian line, if either the loca 
time or the latitude is known. 

233. By Observations of the Pole Star. The direc 
tion of the meridian may be very accurately determined b 
observations of the star Polaris. If the azimuthal readings o 
this star be observed at the two instants when it i 
furthest from the meridian, east and west, respectively 
the reading for the meridian is half their sum. Th 
observations maybe made with an altazimuth. The azimut 
at either observation is a maximum, and it remains ver 
nearly constant for a short interval before and after attainin 
its maximum. Hence, a slight error in the time of observe 
tion will not perceptibly affect the azimuth. The sam 
method is applicable to any star which culminates betwee 
the pole and the zenith. 

The most accurate method is, however, that employed i 
finding the deviation error of the Transit Circle ( 59 
If the telescope always moves in the plane of the meridiai 
the interval from upper to lower culmination, and tb 
interval from lower to upper culmination, will both I 
exactly twelve sidereal hours. If not, the small amount b 
which the vertical plane swept out by the telescope is east c 
west of the meridian, can be found by observing the amouni 
by which the two intervals are greater and less than 12h. 



THE DETERMINATION OF POSITION ON THE EARTH. 177 

SECTION Y. Longitude ly Observation. 

234. In Section III. of the present chapter we showed 
how the local time can he found hy observing \he celestial 
bodies. "When this has been done, the longitude of the place 
of observation may be fonnd by comparing the observed local 
time with the corresponding Greenwich time. 

For in 96 we showed that if the longitude of a place west 
of Greenwich be Z, then 

(Greenwich time) (local time) = -^L h. = 4Z m. ; 
whence, knowing the difference of the two times, L may be 
found. 

The methods of finding Greenwich mean time, and henco 
longitude, may be classified as follows : 

A. Methods available at Sea. 

(1) By the chronometer. 

(2) By the method of lunar distances. 

(3) By celestial signals. 

B. Methods suitable for Land Observations. 

(4) By repeated transmission of chronometers. 

(5) By the chronograph. 

(6) By terrestrial signals. 

(7) By Moon culminating stars or by the Moon's meridian 

altitude. 

235. Longitude by the Chronometer. By reading 
the chronometer used on board ship, and making the necessary 
corrections for error and rate, the Greenwich mean time at 
any instant may be found. If, then, the local mean time is 
determined by observing the Sun, or one of the other 
celestial bodies, and the observations are timed by the chrono- 
meter, the difference between the local and Greenwich mean 
times will be found, and this determines the ship's longitude 
measured from Greenwich. 

EXAMPLE 1. At apparent noon a chronometer indicates 
19h. 33m. 25.s., Greenwich mean time, and the equation of time is 
- 2m. Is. To find the longitude. 

Here the local mean time is 2m. Is. 

.'. Greenwich mean time local mean time ... = 191i. 35m. 26s. 
Mult, by 15, we have long. W. of Greenwich ... = 293 51' 30" 
or sub. from 360, long. E. of Greenwich =66 8' 30" 



178 ASTROtfOMT. 

EXAMPLE 2. Find the longitude, from the following data : 
Sun's computed hour angle = 75E. Time by chronometer = 23h.7m.31s. 
Equation of time = + 3m. 55s. Correction for error and rate, Im.lSs. 

(i.) Here 0's hour angle in time = 5h. before noon 

/. apparent local time = 19h.0m. Os. 

Equation of time = 3 55 

/. mean local time =19h.3m.55s. 

(ii.) Observed time = 23h. 7m. 31s. 

Correction = 1 18 



Greenwich time = 23 6 13 

19 3 55 



W. Long, in time =42 18 

15 

.-. required long ... = 60 34' 30" W. 

EXAMPLE 3. On June 29, from a ship in the North Atlantic 
Ocean, the Sun was observed to have equal altitudes when the 
chronometer indicated llh.27m. 26s. and 6h. 48m. 32s. At noon on 
June 25, the chronometer was 3s. too fast, and it gains 8s. a day. 
Tho equation of time on June 29 at 3 p.m. was -f 2m. 58s. To 
find the ship's longitude. 

The process stands as follows : H. M. s. 

Chronometer time of first observation =11 27 26 

second observation + 12h. ., =18 48 32 



2)30 15 58 



15 7 59 
Hence the chronometer time of local apparent noon =3 7 59 

Correction forchronometer error June 25 = 3s. \ 

rate in 4 days^= 32s. j- = 36 

,,3 hours =- Is.J 



/. Greenwich time of local apparent noon =3 7 23 

Subtract equation of time (since mean noon occurs 

first) = -2 58 



/. Greenwich time of local mean noon =3 4 25 

15 



/. longitude west of Greenwich ... ... ... ... = 46 6' 



THE DETERMINATION OF POSITION ON THE EARTH. 179 

236. Method of Lunar Distances. If from any cause 
the ship's chronometer should stop, or its indications should 
become unreliable, the Greenwich time may be found by 
observations of lunar distances. In this method the Moon, 
by its rapid motion among the stars, takes the place of a 
chronometer, its position relative to the neighbouring stars 
determining the Greenwich time. The Moon moves through 
360 in 27 days ; hence it travels at the relative rate of about 
33' per hour, or rather over 1" in every 2s., and this motion 
is sufficiently rapid to render it available as a timekeeper. 

For this purpose, tables of lunar distances are given in the 
Nautical Almanack. These tables give the angular distances 
of the Moon's centre from the Sun or from such bright stars or 
planets as are in its neighbourhood, calculated for every third 
hour of Greenwich mean time, and for every day of the year. 

The angular distance of the Moon's bright limb from one 
of the given stars may be observed by means of a sextant. 
By adding or subtracting the Moon's semi-diameter, as given 
in the Nautical Almanack, and correcting as explained below, 
the angular distance of its centre may be found. During the 
interval of three hours between the times given in the 
Nautical Almanack, the angular distance changes at an 
approximately uniform rate, and therefore the Greenwich 
time of the observation may be computed by proportional parts. 

237. Clearing the Distance. One of the great draw- 
backs of the lunar method consists in the laborious calculations 
necessary for what is called "clearing the distance." The 
angular distance between the Moon and the star will be 
affected by refraction, and this alone requires a correction to 
be applied to the observed lunar distance ; but there is another 
correction, for what is called parallax, which is equally 
important. This latter correction depends on the fact 
that the Moon's distance from the Earth is only about 60 
times the Earth's radius, and at this comparatively small 
distance the direction of the Moon cannot be considered as 
independent of the observer's position on the Earth, as has 
been done with the fixed stars* (5). 

* Indeed, if a star happens to be behind the Moon's disc, it may 
sometimes appear on opposite sides of the Moon to two observers at 
nearly opposite points on the Earth. 



180 ASTRONOMY. 

For this reason, the lunar distances of a star, as tabulated 
in the Nautical Almanack, are the angles which the Moon 
and star subtend at the centre of the Earth. They are, 
therefore, sometimes called the geocentric lunar distances. 
Hence it is necessary to calculate the Moon's geocentric 
position from that observed, before the Greenwich time of the 
observation can be determined. 

The correction for parallax, will be dealt with more fully 
in the next chapter. Suffice it to mention here that the 
parallax, like the refraction correction, depends only on 
the Moon's zenith distance, and therefore, the only data 
needed for clearing the distance are the altitudes of the two 
bodies at the time of observation. The calculations are then 
greatly simplified by the use of tables. 

238. Advantages and Disadvantages of the Lunar 
Method. The method of lunar distances was introduced at 
a time when chronometers were very imperfectly constructed, 
and could not be relied on during a moderate voyage. At the 
present time, owing to the high degree of accuracy attained 
in the construction of chronometers, combined with the 
reduction in the length of sea voyages since the introduction 
of steam, the lunar method has been almost entirely super- 
seded by the use of chronometers. It is still used, however, 
for the occasional corre-ction of a chronometer if the voyage 
be extremely long ; and explorers rely upon it mainly. 

The principal disadvantages of using lunar distances are : 

1st. The calculations necessary for clearing the distance 
are very tedious, and not such as could be performed readily 
by a seaman possessing little or no knowledge of mathematics. 
Moreover, the corrections are often considerable. 

2nd. A slight error in the observed lunar distance would 
introduce a considerable error in the estimated longitude. 
The best sextants are only divided to every 10", and an error 
of 10" in the observed lunar distance would introduce an error 
of 20s. in the computed Greenwich time. This would give, 
in the longitude, an error of 5', or of 5 geographical miles at 
the equator. Even this degree of accuracy would be difficult 
to attain in practice, while the rate of a well-constructed 
chronometer can be depended upon to within Is. per day. 



THE DETERMINATION OP POSITION ON THE EARTH. 181 

EXAMPLE. On 'Nov. 14, the cleared angular distance of the 
Moon's centre from Aldebaran was found to be 32 44' 52". Find 
the Greenwich time, having given the following data : 

ANGULAR DISTANCE OF THE MOON FROM Aldebaran. 



Date. 


Position of Star. 


6 P.M. 


9 P.M. 


Midnight. 


Nov. 14. 


East. 


33 32' 57' 


31 44' 14" 


29 55' 32" 



The calculation stands as follows : 



Ang. dist. at 6 P.M. =33 32' 57" 
at observation = 32 44 52 

Decrease since 6 P.M. = 48 5 



Ang. dist. at 6 P.M. = 33 32' 57" 
at 9 P.M. = 31 44 14 

Decrease in 3 h. = 1 48 43 



/. In 3h. the Moon's angular distance from Aldebaran decreases 
148'43", or 6523"; 

.*. the time in which it decreases 48' 5", or 2885", ia 
- 3h. x _ lh. 19m. 37s. 



. Greenwich time of observation = 6h. + lh. 19m. 37s. 

= 7h. 19m. 37s. 

239. Longitude by Celestial Signals. The eclipses of 
Jupiter's satellites begin and terminate at times which can 
be calculated beforehand ; it would, therefore, appear 
possible to ascertain the Greenwich time by observing the 
instants at which a satellite disappears into, or emerges 
from, the shadow cast by the planet. .But, as the dis- 
appearance and emergence take place gradually, it is im- 
possible to employ this method with accuracy to the 
determination of longitude. The same objection applies still 
more forcibly in the case of eclipses of the Moon. 

By observing the occultations of stars behind the disc of 
the Moon, we have another way of determining the Greenwich 
time and finding the longitude. This is merely a particular 
case of the method of lunar distances, since at the instant of 
disappearance, the star's apparent (unconnected) distance from 
the Moon's centre is equal to the Moon's semi-diameter. 



182 ASTRONOMY. 



METHODS OF FINDING LONGITUDE ON LAND. 

240. Longitude by repeated transmission of Chro- 
nometers. The chronometer method of comparing longitudes 
can be employed with far greater accuracy on land, on account 
of the possibility of taking repeated journeys to and fro 
in order to effect the comparison of the local times. The 
rate of the chronometer is determined by observing its 
error at the first station, both before and after taking it 
to the second. 

Suppose, for example, that it is required to find the differ- 
ence of longitude between two stations, A and B. A chrono- 
meter is compared with the standard clock at A, and its 
error is noted. It is then carried to JB, and its indications are 
compared with those of a clock regulated to keep local time. 
It is then again brought back to A, and compared a 
second time with the standard clock. The increase in the 
chronometer error during the whole interval serves to 
determine the rate of the chronometer. Wo can now 
correct for error and rate the time indicated by the 
chronometer at A, and thus determine the difference 
between the local times at A and . By converting 
this difference into angular measure at the rate of 15 
to the hour, the required difference of longitude of the two 
stations is determined. 

It is probable that the rate of the chronometer may not be 
the same while it is being shaken about on its journey 
as while it is at rest. This difference of rate may be 
allowed for by comparing the chronometer with the local 
clock soon after arrival, and again before departing. The 
total loss while at rest is thus found, and by subtract- 
ing we have the total loss during the two journeys. The 
only assumption which it is necessary to make is that 
the rate is the same on the outward journey as on the 
return journey. 

In order to obtain a result as free from error as possible, 
a number of journeys to and fro are performed, and several 
chronometers are used on each journey. The most accurate 
result is found by taking the mean of the calculated values 
for the difference of longitude. 



THE DETEllMINATIOff OF POSITION OK THE EAUTH. 183 



EXAMPLES. 

At I7h. by a chronometer, the Greenwich mean time was found 
fco be 16h. 59m. 57'2s. It was taken to a place A, and indicated 4h., 
when the local mean time was 3h. 47m. 46'9s. ; and when it indicated 
llh., the Greenwich time was llh. Om. 9'7s. To find the longitude 
of A in time and in angle. 

Here, at I7h., the chronometer error by Greenwich time was 2'8s. 
24 + llh. +9'7s 

.'. in 18h. the chronometer lost 12'5s. ; 

/. the loss in llh. = x 12'5s. = 7'64s. nearly ; 
18 

.'. the Greenwich time, when the chronometer indicated 4h., was 

= 4h.-2'8s. + 7'64s. = 4h. Om. 4'84s., 

and the local time at the same instant was = 3h. 47m. 46'9s. 
/. required longitude = 12m. 17'9s. W. = 3 4' 28" W. 

2. As a ship starts from Liverpool, its chronometer indicates Oh., 
and is correct by Greenwich mean time. After 16 days, as it reaches 
Quebec, the chronometer indicates 7h. Om. 23s., and Quebec time is 
2h. 5m. 42s. Nearly seven days afterwards, the ship departs at 
Quebec noon, the chronometer then reading 4h. 54m. 39s. ; and when 
it reaches Liverpool, after a voyage of just over fourteen days, it is 
found to be 17s. slow by Greenwich mean time. Find the longitude 
of Quebec. 

By Quebec time, the ship stayed in port 7d. 2h. 5m. 42s. 

= 6d. 21h. 54m. 18s. 
By chronometer, the ship stayed in port 7d.4h.54in.39s. 7h.0m.23s. 

= 6d. 21h. 54m. 16s. 

.-. in 7 days in port, chronometer lost 2s. 

But in 37 days altogether, 17s. 

/. in 30 days at sea, 15s. 

/. in 16 days, from Liverpool to Quebec, it lost 8s. 

But chronometer time on arrival was 7h. Om. 23s. 

.'. Greenwich time was 7h. Om. 31s. 

And local time was 2h. 5m. 42s. 

The difference = longitude of Quebec (in time) = 4h. 54m. 49s. 

/. Longitude of Quebec (in angle) = 73 42' 15" W. 



184 ASTRONOMY. 

241. Longitude by the Chronograph. When two 
observatories are in telegraphic communication, the local 
time maybe readily signalled from one to the other by means 
of the electric current, and the difference between the longi- 
tudes thus determined. 

This method is employed in connection with the chrono- 
graphic method of recording transits, the chronographs being 
connected by the telegraph line, so that a transit is recorded 
nearly simultaneously at both stations. 

Let us call the two stations A and B. When the star 
crosses the meridian at A, the observer presses the button of 
his chronograph. Let t% be the times of transit at A as 
thus recorded at A and B respectively. When the same star 
crosses the meridian at B, the times of transit are again 
recorded at A and B. Let these recorded times be T^ and 
T 2 respectively. 

The transmission of the signal from one station to the 
other is not quite instantaneous, because a small interval of 
time must always elapse before the current has attained 
sufficient strength to make the signal at the distant station. 
Let this interval be x. Then the transit at A will be recorded 
too late at B by the amount x, and the transit at B will be 
recorded too late at A by the same amount x. 

When this correction is applied, the true times of the two 
transits, as determined by the chronograph record at A, will 
be j and T^x. Hence, if L denote the difference of longi- 
tude in time measured westwards from A to , the chrono- 
graph record at A gives 

L T^-x-t r 

Again, the true times of the two transits, as determined by 
the chronograph record at B, will be 2 x and T y Hence 
the chronograph record at B gives 

z = r t -(v-*)==zi 

By addition, we have 

2L=T l -t l + T,-t s ; .-. L = \(T^t^T,-t,\ 
a result which does not involve x. 

Thus we see that, by using both chronograph records, and 
taking the mean of the separately calculated differences of 
longitude, the corrections due to the time occupied by the 
passage of the signals are entirely eliminated. 



THE DETEKMINATION OF POSITION ON THE EAETH. 185 

*242. Elimination of Personal Equation. In the 

above investigation we have taken no account of the personal 
equations of the two observers. But if e is the correction for 
personal equation of the observer at A, and E is that of the 
observer at B, the observed times t lt 2 must both be increased 
by e, and 2^, jF 2 must both be increased by E. Introducing 
these corrections, the formula gives 



To eliminate the corrections, let the two observers change 
places, and repeat the operations, and let the new recorded 
times of transit be denoted by accented letters. The cor- 
rection E must now be applied to the times /, 2 ', and the 
correction e must be applied to T^ and T 2 '. Therefore 



By again taking the mean of the two results we get 



a result in which personal equation is eliminated. 

243. Longitude "by Terrestrial Signals. Before the 
introduction of the electric telegraph and the chronometer, 
other signals had to be used. Among such signals may 
be mentioned flashes of light and rockets visible simul- 
taneously from two stations at a considerable distance apart. 
The heliograph, in which signals are transmitted by flashes 
of reflected sunlight, forms another means of determining 
differences of longitude between two stations visible one from 
the other ; and this method is still often found very useful 
in surveying a country. A flash of lightning and the bursting 
of a meteor have also occasionally been used, but they are 
far too uncertain in their occurrence to be of much value. 
The local time of the signal is noted at each place, and the 
difference of these times gives the difference of longitudes. 

The signals must in every case be seen, not heard, as an ex- 
plosion, even if audible at two distant stations, would not be 
heard simultaneously at both, owing to the comparatively small 
velocity of sound. Where the distance between the two stations 
is great, a chain of intermediate stations must be established, 
and the local time of each station compared with that of the 
next ; this method was used in most of the earliest determina- 
tions of longitude. Now such methods are entirely superseded 
by the use of the chronometer and the electric telegraph. 



186 



ASTRONOMY. 



244. Longitude by Moon culminating Stars. Here, 

as in the method of lunar distances, the Moon's position 
determines the Greenwich time, but instead of observing the 
Moon's angular distance from a neighbouring star, we 
observe the difference of right ascension between the 
Moon and the star by taking their times of transit with a 
tran.-it circle. 

The method is not available at sea, because transits cannot 
be taken with a sextant. It can be used to determine, by 
means of a portable transit circle, the longitude of a tem- 
porary observatory set up in a country where there is no 
means of telegraphic communication with the outer world. 
Its great advantage over the method of lunar distances is that 
it does not involve the laborious, process of " clearing the 
distance," because the times of passage across the meridian 
are unaffected by parallax and refraction. 

The necessary data for the calculations are given in the 
Nautical Almanack. The time of transit of the star deter- 
mines the local sidereal time at the place, and when the 
observatory clock is thus corrected, the time of the Moon's 
transit is its R.A. The tables in the Nautical Almanack give 
the Moon's R.A. at the time of its transit at Greenwich. 
The increase of R.A. is proportional to the time which elapsed 
between the transits at Greenwich and at the place of 
observation, and hence the Greenwich time of the local 
transit is known. Hence, the longitude may be found. 

*245. Longitude by Meridian Altitude of the Moon. Another 
method of finding the longitude is sometimes used, namely to find 
the Greenwich time by observations of the Moon's declination. For 
this purpose, the Moon's meridian altitude is observed with a 
transit circle and its declination deduced ( 24). The Nautical 
Almanack contains the Moon's declination for every 3h. of Green- 
wich time ; from this the Greenwich time of observation may be 
found by proportional parts. But the method is difficult to 
employ, because the observations are affected by the same 
sources of error, arising from parallax and refraction, as in the 
method of lunar distances, and there is also a correction for dip in 
observations made at sea. Moreover, the Moon's daily motion in 
declination is so small (the greatest variation being about 5 per 
day), that a slight error in the computed declination would very 
considerably affect the calculated value of the longitude. 



THE DETERMINATION OF POSITION ON THE EARTH. 187 

SECTION VI. Captain Sumner's Method. 

246. "We shall now show that, by taking two altitudes of 
the Sun with a sextant, and noting the Greenwich times of 
observation with a chronometer, we can construct a ship's 
position on a terrestrial globe geometrically. 

The Sub-Solar Point. We can at once find the position 
on the terrestrial globe of a place at which the Sun is in the 
zenith on a given day, at a given instant of Greenwich time. 
For, evidently, the latitude of the place is equal to the Sun's 
declination, and is, therefore, known ; while the longitude 
west of Greenwich is equal to the Greenwich apparent time, 
which may be found by subtracting the equation of time from 
the mean time. The place is called the Sub-Solar Point. 

The Circle of Position. Assuming the Earth to be 
spherical, the Sun's Z.D. at any place is equal to the angular 
distance of the place from the sub-solar point. (For it is 
evidently the angle between the directions of the zeniths at 
the given place and at the sub-solar point.) Hence, the 
places at which the Sun has a given Z.I), all lie on a small 
circle of the terrestrial globe, whose pole is at the sub -solar 
point, and whose angular radius is equal to the Sun's Z.D. 
This circle is the circle of position. 

Geometrical Construction for the Position of the 
Ship. If, then, two altitudes of the Sun be observed, 
and the Greenwich times noted with a chronometer, we can 
find the sub-solar points, and thus construct the circles of 
position, and we know that the ship lies on each circle. The 
ship must, therefore, be at one of the two points in which 
the two circles cut. To decide which is the actual position, 
the Sun's azimuth must be very roughly estimated at the 
two observations. On the globe it will be easy to see at which 
of the two places the Sun had the observed azimuths. Thus 
the ship' s exact position on the globe is found . It is easy to allow 
for the ship's motion between the observations. 

If two stars are observed, the two substellar points (or 
places at which the stars are in the zenith) can be con- 
structed. For the latitude of either is equal to the corres- 
ponding star's decl., and its longitude is equal to the star's 
hour angle at Greenwich = sidereal time star's R.A. 

The ship's place can now be found by drawing the circles 
of position as before. 

ASTRON. o 



188 ASTRONOMY. 

EXAMPLES. VII. 

1. At noon on the longest day a circumpolar star is passing over 
the observer's meridian, and its zenith distance is the same as that 
of the Sun's centre ; at midnight it just grazes the horizon. Find 
the latitude. 

2. On January 2, 1881, on a ship in the North Atlantic in longi- 
tude 48 W., it was observed that the Sun's meridian altitude was 
15 21' 45". The Sun's declination at noon at Greenwich on the 
same day was 22 54' 33", and the hourly variation 13'78". Find 
the ship's latitude. 

3. Show how to find the latitude by observing the difference of 
the meridian zenith distances of two known stars which cross the 
meridian on opposite sides of the zenith at nearly equal distances 
from it. Explain whether the stars chosen should be near to or 
remote from the zenith. Give also the advantages and disadvan- 
tages of this method of finding the latitude, as compared with the 
method of circumpolars. 

4. On a certain day the observed meridian altitude of a Cassiopeia 
(decimation 55 49' ll'l" N.) was 85 10' 18". The eye of the 
observer was 18 feet above the horizon, and the error for refraction 
for the altitude of the star is 5" ; determine the latitude. 

5. The deck of a ship (stationary) is 25 feet from the sea, and the 
dip of the horizon at 1 foot is 1' ; if the two meridian altitudes of a 
circumpolar star from the sea horizon be 60 2' and 29 58', find the 
latitude. 

6. At the winter solstice the meridian altitude of the Sun is 15. 
What is the latitude of the place ? What will be the meridian 
height of the Sun at the equinoxes and at the summer solstice ? 

7. Describe the altazimuth, and show how it can be used to find 
the time of apparent noon and the azimuth of the meridian by the 
method of equal altitudes. 

8. A vertical rod is fixed exactly in the centre of a circular foun- 
tain basin, and it is observed that on the 25th of July the extremity 
of the shadow exactly reaches the margin of the water at lOh. 7m. 
A.M., and at 2h. 25m. P.M. The equation of time on that day is 
+ 6m. What is the error, compared with local time, of the watch 
by which these observations were taken ? 

9. In the railway station at Ventimiglia is a clock one face of 
which indicates Paris time, the other Eoman time. It is observed 
that, when the former indicates 12h. 39m. 4s., the latter indicates 
Ih. 19m. 40s. The longitude of Paris being 2 21' E., find the i 
longitude of Borne, 



THE DETERMINATION OP POSITION ON THE EARTH. 189 

10. In Question 9, what is the corresponding local time at Venti- 
miglia, the longitude being 7 35' E. ? 

11. A chronometer is set by the standard clock at Greenwich at 
6 A.M. It is then taken to Shepton Mallet, and indicates noon when 
the local time is llh. 49m. 50s. The chronometer is then brought 
back to Greenwich, and indicates 9 P.M., when the correct time is 
8h. 59m. 55s. Find the longitude of Shepton, supposing the chrono- 
meter rate uniform. 

12. In applying the lunar method, find the error in the calculated 
longitude of the observer due to an error of 1' in the tables of the 
Moon's longitude. 

13. Amerigo Vespucci is said to have found his longitude in lati- 
tude 10 N. in the following manner. At 7.30 P.M. the Moon was 
1 E. of Mars, at midnight the Moon was 5^ E. of Mars. The 
Nuremberg time of conjunction of the Moon and Mars was midnight. 
Hence he calculated that his longitude was 82 W. of Nuremberg. 
Discuss the accuracy of the method, and point out the necessary 
corrections. 

14. A chronometer whose rate is uniform is found at Greenwich 
to have an error of Sj hours when the time which it indicates is t\. It 
is then taken to a place A t and when it indicates t 2 it is found that 
the excess of the observed local time of the place A over 2 is 5 2 hours. 
It is now again brought back to Greenwich, and the chronometer 
time and error are observed to be t z and S 3 hours respectively. 
Prove that the longitude of A east of Greenwich is 

15 (So^ + Mi + S^-Ms-^i-W/^-*,) degrees. 

15. The sidereal times of transit of a certain star across the 
meridian of an observatory A, as recorded at A, and by a telegraphic 
signal at B, are t\, t. 2 respectively. The sidereal times of transit of 
the same star across the meridian of -B, recorded by telegraphic 
signal at A, and at B, are T 1? T 2 respectively. If the signals take 
the same time to travel in either direction, show that the difference 
of the longitudes of B and A in angular measure 



16. The altitudes of two known stars are observed at a given 
instant of time. Show how to find on a terrestrial globe the places 
at which the stars are vertically overhead, and give a geometrical 
construction for the place of observation. 

17. In Question 16, find the condition that there should be two, 
one, or no possible positions of a ship at which the altitudes of the 
known stars have certain given values. 

18. If longitude is found by lunar distances, and latitude by 
meridian altitudes, find the latitude in which an error of 1' in the 
sextant reading will introduce the same error in both observations 
if estimated not in angle, but in miles on the Earth's surface. 



190 ASTBONOMY. 



EXAMINATION PAPER, VII. 

1. Give a description of the Sextant, and explain how to use it 
for taking altitudes (1) at sea, (2) on land. 

2. How does a Chronometer differ from an ordinary watch ? 
What are its error and rate ? 

3. Prove that a single meridian altitude of a star, whose declina- 
tion is known, will determine the latitude. Why is a zenith sector 
sometimes preferred to a transit circle for this purpose ? 

4. Show how the latitude is determinable by two meridian obser- 
vations of a circumpolar star. Why is this method not generally 
applicable on board ship ? 

5. Show how to find the latitude of a place (1) by observing the 
Sun's altitude at a given time ; (2) by the Prime Vertical Instru- 
ment. 

6. Describe the method of equal altitudes for finding the time of 
transit of a celestial body. If the times be observed by the ship's 
chronometer, show how to find the longitude. 

7. What methods are available for the determination of Greenwich 
time at sea ? Describe the method of taking lunar distances. 

8. How is the difference of longitude determined by electric 
telegraph ? Explain how the personal equation and the time of 
transmission of the signal are eliminated. 

9. Contrast the method of Moon-culminating Stars with that of 
Lunar Distances in respect of the instruments employed, and of the 
intricacy of the calculations involved. What other celestial signals 
have been proposed, and what is their disadvantage ? 

10. Knowing the Greenwich time, show how to construct graphi- j 
cally on a globe the position of the ship without any calculation 
whatever. 



CHAPTER VIII. 

THE MOON. 
SECTION I. Parallax The Moon's Distance and Dimensions. 

247. Definitions. By tlie Parallax of a celestial body 
is meant the angle between the straight lines joining it to 
two different places of observation. 

In 5 we stated that the fixed stars are seen in the same 
direction from all parts on the Earth ; hence such stars have 
no appreciable parallax. The Moon, Sun, and planets, on 
the other hand, are at a (comparatively) much smaller dis- 
tance from the Earth, and their parallax is a measurable 
quantity. The distance of the Moon from the Earth's centre 
is about 60 times the radius of the Earth. The effects of 
parallax in connection with the method of Lunar Distances 
have already been mentioned (237). 

To avoid the necessity of specifying the place of observa- 
tion, the direction of the Moon or any other celestial body is 
always referred to the centre of the Earth. The direction 
of a line joining the body to the Earth's centre is called the 
body's geocentric direction. The angle between the geo- 
centric direction and the direction of the body relative to 
any given observatoiy is called the body's Geocentric 
Parallax, or more shortly, its Parallax. Thus the 
geocentric parallax is the angle subtended at the body by 
the radius of the Earth through the point of observation. 

The Horizontal Parallax is the geocentric parallax of 
a body when on the horizon of the place of observation.* 



192 

248. General Effects of Geocentric Parallax. 

Assuming the Earth, to be spherical, let C (Fig. 77) 
be the Earth's centre, the place of observation, and M the 
centre of the Moon or other observed body. Then the angle 
MC is the geocentric parallax of M. 

Produce CO to Z\ then OZ is the direction of the zenith 
at 0, and ZOM is therefore the zenith distance of M as seen 
from (corrected of course for refraction). Now 

L ZOM = L ZCM+ L OMC ; 

therefore the apparent zenith distance of M is increased by 
the amount of the geocentric parallax. Conversely to find 
L ZCM we must subtract the parallax OMC from the 
observed zenith distance ZOM. 

The azimuth is unaltered by parallax, because OM } CM 
lie in the same plane through OZ. 




FIG. 77. 

249. To find the Correction for Geocentric Parallax. 

In Fig. 77, let 

a = CO J^vth's ladiuA, 

d = CM = Moon's (or other body's) geocentric distance, 

s = ZOM = observed zenith distance of M, 

p = OMC = parallax of M. 

By Trigonometry, since the sides of A OMC are propor- 
tional to the sines of the opposite angles, 

sin CMO _ CO 
' ' sin COM CM' 



THE MOON. 193 

sin a 



ji, 

that is , . 

sins a 

Therefore sinp = sin . 

Cv 

Let P be the horizontal parallax of Jf. Then, when 
2 = 90, p = P, and therefore the last formula gives 

sin P= sin 90 = . 
rt d 

Hence, by substitution, 

sinp = sin P . sin s. 

This formula is exact. But the angles p and P are in 
every case very small, and therefore their sines are very 
approximately equal to their circular measures. Hence we 
have the approximate formula 

p = JP . sin z, 

or, The parallax of a celestial body varies as the 
sine of its apparent zenith distance. 

The last formula holds good no matter what be the unit of 
angular measurement. Thus if p", P" denote the numbers 
of seconds in p, P respectively, we have, by reducing to 
seconds, p" = P" sin 2. 

EXAMPLES. 

1. Supposing the Sun's horizontal parallax to be 8'S", to find the 
correction for parallax when the Sun's altitude is 60. 

Here z = 90 -60 = 30, P" = 8'8", and therefore 
p" = P" sin 30 - 8-8" x i - 4'4". 

2. To find the corrections for the Moon's parallax for altitudes of 
30 and 45, the Moon's horizontal parallax being 57'. 

In the two cases wo have respectively z = 60 and z = 45, and 
the corresponding corrections are 



p" = 57' sin 60 = 57' x x /3 = 28' 30" x ^/3 

= 1710" x 1-7320 = 2961-7"= 49' 21-7", 
and p" = 57' sin 45 = 57 X x ^2 = 28'' 30" x v/2 

= 1710" x 1-4142 - 2418-3"= 40' 18'3". 



*94 ASTRONOMY. 

250. Relation between the Horizontal Parallax 
and Distance of a Celestial Body. In the last paragraph 

we showed that sin P = . 

a 

This formula may be proved independently by drawing 
MA to touch the Earth at A. M is on the horizon at A ; 
the Z CMA is therefore the horizontal parallax P, and we 
have immediately 

sin P = sin CMA = CA/CM= /d. 
Since P is small, we have approximately 

Circular measure of jP = a/d. 
and therefore in seconds 

p ,, = 180X60X60 ^ = 206265 
if d d 

which shows that, The horizontal parallax of a body 
varies inversely as its distance from the Earth. 




FIG. 78. 

If we know the Earth's radius a and the distarce d, the 
last formula enables us to calculate the horizontal j ara lax 
P". Conversely, if we know the horizontal parallax of a body 
we can calculate its distance. 

EXAMPLE 1. Given that the Moon's distance is 60 times the 
Earth's radius, to find the Moon's horizontal parallax. 

We have - = ; 

d 60 

circular measure of P = ^r approximately, 
bu 

"NTow the unit of circular measure = 57'2957 ; 

P (in angular measure) = x 57 - 2957 = 57'2957' 

= 57' 17-7", 
and this is tho required horizontal parallax. 



THE MOON. 195 

EXAMPLE 2. Given that the Sun's parallax* is 8'8", to find 
the Sun's distance, the Earth's radius being 3,960 miles. 

The circular measure of 8*8" is = ** , 

and, by the formula, we have, for the Sun's distance in miles, 
d = a = 3960x180x60x60 

circ. meas. of P 8'8 x TT 

Taking TT =3f , and calculating the result correct to the first three 
significant figures, we find the Sun's distance d 

= 92,8OO,OOO miles approximately. 

It would be useless to carry the calculations beyond the third 
figure, for, of course, the values of the Earth's radius and Sun's 
parallax are only approximate; moreover, we should have to use 
the more accurate value of TT, viz., 3'141592 

251. Comparison between Parallax and Refraction. 
It will be noticed that while parallax and refraction both 
produce displacements of the apparent position of a body along 
a vertical circle, the displacement due to parallax is directed 
away from the zenith, and is always proportional to the sine 
of the zenith distance, while that due to refraction is directed 
towards the zenith, and is proportional to the tangent of the 
zenith distance, provided the altitude is not small. Also the 
correction for parallax is inversely proportional to the distance 
of the body, and is imperceptible, except in the case of mem- 
bers of our solar system ; while the correction for refraction 
is independent of the body's distance, and depends only on 
the condition of the atmosphere. 

The Moon's horizontal parallax is about 57', while the 
horizontal refraction is only 33'. Hence, by the combined 
effects of parallax and refraction, the Moon's apparent 
altitude is diminished, or its Z.D. increased. The time of 
rising is, therefore, on the whole retarded, and the time of 
setting accelerated. The effect of parallax on the times of 
rising and setting may be investigated by the methods of 
104, 190. 

For all other bodies, including the nearest planets, the 
correction for refraction far outweighs that due to parallax. 

* When astronomers speak of the parallax of the Sun, Moon, or 
a planet, without further specifying the observation, the horinontal 
parallax is always to be understood. 



6 ASTRONOMY. 

252. To find the Moon's Parallax by Meridian 
Observations. The Moon's parallax may be conveniently 
determined as follows. Let A and B be two observatories 
situated on the same meridian, one north, the other sonth of 
the equator. Let M denote the Moon's centre, and let x be 
a star having no appreciable parallax, whose R.A. is approxi- 
mately equal to that of the Moon, their declinations being 
also nearly equal. 

Let the Moon's meridian zenith distances ZAM and Z'BM 
be observed with the transit circles at A and B, and let xA M 
and xBM, the differences of the meridian Z.D.'s of the Moon 
and star at the two stations, be also observed. 

Let 2, = / ZAM, 2 2 = L Z'BM. 

a, = L xAM, 3 = L xBM. 

P = Moon's required horizontal parallax. 
By 249, we have, approximately, 

/ A M C = P sin 54, Z BMC = P sin s 2 . 




FIG. 79. 

.-. Z AMB = P (sin 2, + sin 2 3 ) (i.). 

Moreover, if MX be drawn parallel to Ax or Bx, 
Z XMA = Z MAx = a, ; 
Z XMB Z J/#r = fl a ; 

.-. z ^J/7? = !-, '. (ii.), 

From'(i.) and (ii.), 

P (sin , + sin 2 2 ) = ^j 2 ; 

P = ^i- 3 . 

siu 2, -{-sins., 
whence the Moon's parallax, P, may be found. 



tSE Moott. 197 

253. If the two observatories are not on the same meridian, 
allowance must be made for the change in the Moon's 
declination between the two observations. Let the stations 
be denoted by A, JJ, and let S' be the place on the meridian 
of A, which has the same latitude as B. Then, if the Moon's 
meridian Z.D. be observed at B, we can, by adding or sub- 
tracting the change of declination during the interval, find 
what would be the meridian Z.D. if observed from B'. 
Moreover, the star's meridian Z.D. is the same both at B and 
at B'. Hence it is easy to calculate what would be the angles 
at B' corresponding to the observed angles at B. From the 
former, and the observed angles at A, we find the parallax 
P, as before. 

To ensure the greatest accuracy, it is advisable that the 
difference of longitude of the two stations should be so small 
that the correction for the Moon's motion in declination 
is trifling. It is necessary, however, that 0, # 2 should 
be large ; for this reason the stations should be chosen 
one as far north and the other as far south of the equator as 
possible. The observatories at Greenwich and the Cape of 
Good Hope have been found most suitable. 

The principal advantage of the above method is that the 
probable errors arising from any uncertainty in the corrections 
for refraction are diminished as far as possible. 

For, since the Moon and observed star have nearly the 
same declination, the corrections for refraction to be applied 
to #!, # 2 , their small differences of Z.D., are very small indeed. 
The errors are not of so much moment in the denominator 
sin z l -f- sin z 2 , as the latter is not itself a small quantity. 

From such observations, the mean horizontal parallax of 
the Moon has been found to be 57' 2 '70 7". 

This value corresponds to a mean distance of 60-27 times 
the equatorial radius of the Earth, or 238,840 miles. The 
distance and parallax of the Moon are not, however, quite 
constant ; their greatest and least values are in the ratio of 
(roughly) 19 : 17. For rough calculations, the Moon's 
distance may be taken as 60 times the Earth's radius. 

Neither this method nor the next ( 254) gives accurate 
resulls for the Sun, for the brilliancy of the rays renders all 
stars in its neighbourhood invisible 



198 ASTKONOMY. 

254. To find the Parallax of a rlanet from Observa- 
tions made at a Single Observatory. The parallax of 
Mars, when nearest the Earth, has also been determined by 
the following method, depending on the Earth's rotation. 

Since the apparent altitude of a body is always diminished 
by parallax, it can easily be seen by a figure, that, shortly after 
a planet has risen, its R.A. and longitude appear greater 
than their geocentric values (the planet being displaced east- 
wards), while shortly before setting they appear less 
than their geocentric values (the displacement being west- 
wards). The planet's position, relative to certain fixed stars, 
is observed soon after rising and before setting by means of 
an equatorial furnished with a micrometer or heliometer. 

The observed change of position is due partly to parallax 
and partly to the planet's motion relative to the Earth's 
centre during the interval between the observations, which 
produces displacements far greater than those due to 
parallax. But by repeating the observations on successive 
days, the planet's rate of motion can be accurately 
determined, and the displacements due to parallax can thus 
be separated from those due to relative motion. Refraction 
need not be allowed for ; because it affects those stars with 
which the planet is compared, as well as the planet itself. 

This method can be used for the Moon, but the Moon's 
motion is so rapid that the calculations are more complicated. 

*255. Effect of the Earth's Ellipticity. The effect of parallax 
is made rather more complicated by the spheroidal form of the 
Earth. For, by 249, the magnitude of the horizontal parallax at 
any place depends on its distance from the Earth's centre, and since 
this distance is not the same for all places on the Earth, the horizontal 
parallax is not everywhere the same. Again, the direction in which 
the body is displaced is away from the line (produced) joining the 
centre of the Earth with the observer ( 248). But this line does not 
pass exactly through the zenith ( 117). Hence the displacement 
is not in general along a vertical, so that the azimuth as well as 
altitude is very slightly altered by parallax. 

250. The Equatorial Horizontal Parallax is the geo- 
centric parallax of a body seen on the horizon of a place at the 
Earth's equator. It is generally adopted as the measure of 
the parallax of a celestial body. Its sine is equal to 

(Earth's equatorial radius)/(body's geocentric distance). 



THE M001T. 199 

257. Relation between Parallax and Angular 
Diameter. In Fig. 80 it will be seen that the angle CM A, 
which measures the parallax of M, also measures the Earth's 
angular semi-diameter as it would appear from M. Thus, 
the Moon's parallax is the angular semi-diameter of the Earth 
as it would appear if observed from the Moon. 




FIG. 80. 



258. To Find the Moon's Diameter. Let 0, c be the 

radii of the Earth and Moon respectively, measured in miles, 
d the distance between their centres, Pthe Moon's horizontal 
parallax, m the Moon's angular semi-diameter as it would 
appear if seen from the Earth's centre. Then, from Fig. 80, 



.. c : a = sin m : sin P = m : P approximately ; 
i.e. (rad. of Moon) : (rad. of Earth) 

= ( C 's ang. semi-diam.) : ( <T 's hor. parx.). 
Hence, knowing the Moon's horizontal parallax and its 
angular diameter, the Moon's radius can be found. 

The Moon's mean angular diameter 2m is observed to be 
about 31' 5". From this the Moon's actual diameter is readily 
found to be about 2160 miles, or T 3 T of the Earth's diameter. 

The surfaces of spheres are proportional to the squares, and the 
volumes to the cubes of their radii. Hence the Moon's superficial 
area is about -jf T , or $, and its volume about T f| T , or -^ of that 
of the Earth. 

EXAMPLE. To find the Moon's diameter in miles, given 

< 's angular diameter = 31' 7", 
([ 's equatorial horizontal parallax = 57' 2", 

Earth's equatorial radius = 3963 miles. 



.-. <t 's diameter 2c = a x ^ = 3963 x ^J^. = 3963 x -- = 2162. 
Thus the Moon's diameter is 2162 miles. 



200 ASTEONOMT. 

SECTION II. Synodic and Sidereal Months Moon's Phases 
Mountains on the Moon. 

259. Definitions. In 40 we defined the lunation as 
the period between consecutive new Moons, and showed that 
it was rather longer than the period of the Moon's revolution 
relative to the stars. We shall now require the following 
additional definitions, most of which apply also to the planets. 

The elongation of the Moon or planet is the difference 
between its celestial longitude and that of the Sun. If the 
body were to move in the ecliptic its elongation would be its 
angular distance from the Sun. 

The Moon or planet is said to be in conjunction when it 
has the same longitude as the Sun, so that its elongation is 
zero. The Moon is in conjunction at new Moon ( 40). The 
body is in opposition when its elongation is 180. In both 
positions it is said to be in syzygy. The body is said to be 
in quadrature when its elongation is either 90 or 270. 

The period between consecutive conjunctions is called the 
synodic period of the Moon or planet. The Moon's 
synodic period is, therefore, the same as a lunation; it is 
also called a Synodic Month. In this period the Moon's 
elongation increases by 360, the motion being direct. 

The period of revolution relative to the stars is called the 
sidereal period ; that of the Moon, the Sidereal Mouth. 

The average length of the Calendar Month in common 
use is slightly in excess of the synodic month (cf. 171). 

260. Relation between the Sidereal and Synodic 
Months. 

Let the number of days in a year be F, in a sidereal month 
J/j and in a synodic month S. 

In M days the Moon's longitude increases 360 ; 

.'. in 1 day the Moon's longitude increases 360/J/. 
Similarly in 1 day the Sun's longitude increases 360/F, 
and the Moon's elongation increases 360/& 

Now, from the definition, 

(Moon's elongation) = (Moon's long.) (Sun's long.), 
and their daily rates of increase must be connected by the 
same relation : 



THE MOON. 201 

360^360 360. 
8 " M Y ' 



=- or = 

8 " M ' M~~ 



1 



sider. month synod, month, year 

EXAMPLE. Find (roughly) the length of the sidereal month, given 
that the synodic month (8) = 29|d., and the year (Y) = 365id. 

Here we have i = i+ JL. 

To simplify the calculations, we put the relation into the form 




= 29 5 - 29-5 x = 29-5 - 2'20 = 27'3. 

1579 

Hence the sidereal month is very nearly 27 J days. 

261. To determine the Moon's Synodic Period. 

An eclipse of the Sun can only happen at conjunction, and 
an eclipse of the Moon at opposition, and the middle of the 
eclipse determines the exact instant of conjunction or oppo- 
sition, as the case may be. Hence, by observing the exact 
interval of time between the middle of two eclipses, and 
counting the number of lunations between them, the length 
of a single lunation, or synodic period, can be found with 
great accuracy expressed in mean solar units of time. 

The records of ancient eclipses enable us to find a still closer 
approximation to the mean length of the lunation. From 
modern observations, the length of a lunation has been found 
with sufficient accuracy to enable us to tell the exact number 
of lunations between these ancient eclipses and a recent lunar 
eclipse (this number being, of course, a whole number], ]3y 
dividing the known interval in days by this number, the 
mean length of the synodic period during the interval can be 
accurately found. At the present time the length of a 
lunation is 29-5305887 days, or 29d. 12h. 44m. 2'7s. nearly. 

Prom this the length of the Moon's sidereal period is cal- 
culated, as in 260, and found to be 27d. 7h. 43m- 11 -5s. 
nearly 



202 ASTBOUOMY. 

262. Phases of the Moon. The acccompanying dia- 
grams will show how the phases of the Moon are accounted 
for on the hypothesis that the Moon is an opaque body 
illuminated by the Sun. In the upper figure the central 
globe represents the Earth, the others represent the Moon in 
different parts of its orbit, while the Sun is supposed to be at 
a great distance away to the right of the figure.* The half 
of the Moon that is turned towards the Sun is illumi- 
nated, the other half 'being dark. The Moon's appearance 
depends on the relative proportions of the illuminated and 
darkened portions that are turned towards the Earth. 




FIG. 81. 

The lower figures, 0, b, c, d, e, /, g, h, represent the appear- 
ances of the Moon relative to the ecliptic, as seen from the 
Earth when in the positions represented by the corresponding 
letters in the upper figure. 

* The Sun's distance is about 390 times the Moon's. If the 
former be represented by an inch, the latter will be represented by 
about 11 yards. 



TKb MOON. 203 

At A, a the Moon is in conjunction, and only the dark 
part is towards the Earth. This is called M"ew Moon. 

At B, b a portion of the bright part is visible as a crescent 
at the western side of the disc. The Moon's appearance is 
known as horned. The points or extremities of the horns 
are called the cusps. 

At 7, c the Moon's elongation is 90, and the western half 
of the disc, or visible portion, is illuminated, the eastern half 
being dark. The Moon is then said to be dichotomized. 
This is called the First Quarter. The Moon's age is about 
7| days. 

At D, d more than half the disc is illuminated. The 
Moon's appearance is then described as gibbous. 

At E) e the Moon is in opposition. The whole of the disc 
is illuminated. This is called Pull Moon. The Moon's age 
is about 15 days. 

At F, f a portion of the disc at the western side is dark. 
The Moon is again gibbous, but the bright part is turned in 
the opposite direction to that which it has at D, d. 

At 6r, g the Moon's elongation is 270. The eastern half 
of the disc is illuminated, and the western half is dark. The 
Moon is again dichotomized. This is called the Last 
Quarter. The Moon's age is about 22 days. 

At -ZZ", h only a small crescent in the eastern portion is still 
illuminated. The Moon is now again horned, but the horns 
are in the opposite direction to those in , I. 

Finally, the Moon comes round to conjunction again at A, 
and the whole of the part towards the Earth is dark. 

From new to full Moon, the visible illuminated portion 
increases , and the Moon is said to be waxing 1 . From full to 
new, the illuminated portion decreases, and the Moon is said 
to be waning. 

It will be noticed from a comparison of the figures that 
the illuminated portion of the visible disc is always that 
nearest the Sun. Moreover, its area is greater the greater 
the Moon's elongation.* 

* The phases of the Moon may be readily illustrated experi- 
mentally, by taking an opaque ball, or an orange, and holding it in 
different directions relative to the light from the Sun or a gas- 
burner. 

ASTEON. P 



201 ASTRONOMY. 

263. Relation between Phase and Elongation. Let 

M (Fig. 82) "be the centre of the Moon, MS the direction of 
the Sun, E'ME that of the Earth. Draw the great circles 
AMB perpendicular to ME, and CMD perpendicular to MS ; 
the former is the boundary of the part of the Moon turned 
towards the Earth, and 'the latter is the boundary of the 
illuminated portion. Hence the visible bright portion is the 
lune A MC. The angle of the lune, L AMC, is equal to 
Z E'MS (Sph. Greom. 16). The area of a spherical lune is 
proportional to its angle. Hence, 

area of visible illuminated part _ Z AMC _ E'MS 
area of hemisphere 180 ' 180 

180 ~ 



180 





FIG. 83. 

But this does not give the " apparent area " of the bright 
part. For, as in 145, the apparent area of a body is the 
area of the disc formed by projecting the body on the celestial 
sphere. If IT denote the projection of the point C on the 
plane AMB (so that CN is perpendicular to BA\ the arc 
A C will be seen in perspective as a line of length AN, and 
the bright part will be seen as a plane lune (Fig. 83), whose 
boundary POP' optically forms the half of an ellipse whose 
major axis is PP' t and minor axis 



THE MOON. 205 

It may be shown that 

area of half-ellipse POP' : area of semicircle PAP 1 

and .-. area APCP' : area APBP' = AN~ AB 

= 1 - cos AMC : 2 = 1 cos E'MS \ 2. 
Hence the apparent area of the bright part is proportional to 




FIG. 84. 

m le 83E ' diffcrs from the Moo *'s elongation 
r \r C Smal ]. anglG ESK (Fig. 84); i.e., the Bangle 
which the Moon's distance subtends at the Sun. This angle 
is very small, being always less than 10'. Hence the 
area 01 the phase is very approximately proportional to 
1 cos(Moon's elongation). 

264. Determination of the Sun's Distance by Aris- 
tarchus. From observing the Moon's elongation when 
dichotomized Aristarchus (B.C. 270 circ.) made a computation 
ol the bun s distance in the following manner. When the 
^^/dichotomized, L SME = 90, the Moon's elongation 
t SEM= 90- z ESM, and cos SZM= JEM/US. Hence 
by observing the angle SEN, the ratio of the Sun's distance 
to the Moon's was computed. 

But this method is incapable of giving reliable results, owing 
to the impossibility of finding the exact instant when the Moon 
is dichotomized. The Moon's surface is rough, and covered 
with mountains, and the tops of these catch the light before 
the lower parts, while throwing a shadow on the portions 
behind them. Hence the boundaiy of the bright part is 
always jagged, and is never a straight line, as it would be at 
the quarters, if the surface of the Moon were perfectly smooth. 
In tact, Aristarchus estimated the Sun's distance as only 
about 19 times that of the Moon, whereas they are really in 
the proportion of nearly 400 to 1. 



206 ASTRONOMY. 

265. Earth-Shine on the Moon. Phases of the 
Earth. When the Moon is nearly new, the tmilluminated 
portion of its surface is distinctly visible as a disc of a dull- 
red colour. This appearance is due to the light reflected 
from the Earth as " Earth-shine," which illuminates the 
Moon in just the same way that the moonshine illuminates 
the Earth at full Moon. From 258, the Earth's superficial 
area is greater than the Moon's in the proportion of about 
40 : 3. Consequently the Earth-shine on the Moon is more 
than 13 times as bright as the moonshine on the Earth. 

The Earth, as seen from the Moon, would appear to pass 
through phases similar to those of the Moon, as seen from 
the Earth. The Earth's and Moon's phases are evidently 
supplementary. Thus, when the Moon is new the Earth 
would appear full, and vice-versd ; when the Moon is in the 
first quarter, the Earth would appear in the last quarter. 

Owing, however, to twilight, the boundary of the Earth's 
illuminated portion would not be so well denned as in the 
case of the Moon ; there would be a gradual shading off from 
light to darkness, extending over a belt of breadth 18 on 
beyond the bright part. The entire absence of twilight on 
the Moon is one of the strongest evidences against the exist- 
ence of a lunar atmosphere similar to that of our Earth. 

266. Appearance of Moon relative to the Horizon. 

"We are now in a position to represent, in a diagram, the 
Moon's position and appearance relative to the horizon at a 
given time of day and year when the Moon's age is given. 

The ecliptic having been found, as explained in 41, 
the age of the Moon determines the Moon's elongation, 
as in 40. Measuring this angle along the ecliptic, we find 
the Moon's position roughly j for the Moon is never very far 
from the ecliptic (cf. 40). The elongation also determines 
the phase, and enables us to indicate the appearance of 
the disc. The bright side or limb is always turned towards 
the Sun. The cusps, therefore, point in the reverse direction, 
and the line joining them is perpendicular to the ecliptic. 

We can also trace the changes in the direction of the 
Moon's horns relative to the horizon, between its time of 
rising and setting. 



THE MOON. 207 

Take, for example, the case when the Moon is a few (say 
three) days old. The Moon is then a little east of the Sun ; 
therefore the bright limb is at the western side of the disc, 
and the horns point eastward. Hence, at rising, the horns 
are pointed downwards, and at setting they are pointed 
upwards (Fig. 85). 

n 

FIG. 85. FIG. 86. 

When the Moon is waning, the reverse will be the case 
(Fig. 86). 

267 Heights of Lunar Llouutains. We stated m 
264 that the Moon's surface is covered with mountains, and 
that in consequence the bounding line between the illumi- 
nated and dark portions of the disc is always jagged and 
irregular , while the mountains themselves throw their 
shadows on the portions of the surface behind them. These 
circumstances have led to the two following different ways 
of measuring the height of the lunar mountains 

First Method. If a tower is standing in the middle 
of a perfectly level plain, it is evident from trigono~ 
nietry that the length of the shadow, multiplied by the 
tangent of the Sun's altitude, gives the height of the tower. 
The same will be true in the case of the shadow cast by a 
mountain, provided we measure the length of the shadow 
from a point vertically underneath the summit. Now, in 
the case of the Moon it is possible, from knowing the Moon's 
age, to calculate exactly what would be the altitude of the 
Sun as it would be seen from any point of the lunar surface. 
The apparent length of the shadows of the mountains can be 
measured, in angular measure, by means of a micrometer ; 
from this their actual length can be calculated, allowance 
being, of course, made for the fact that we are not looking 
vertically down on the shadows, and hence they appear fore- 
shortened. In this way. the height of the mountains can be 
found. 



208 ASTKONOMY. 

The principal disadvantage of this method is, that if the 
surface of the Moon surrounding the mountain should be less 
flat than it has been estimated, there will be a corresponding 
error in the height of the mountain. In particular, it would 
be impossible to apply the method to find the heights of 
mountains closely crowded together. 

268. Second Method. In treating of the Earth in 
104, we showed that one effect of the dip of the horizon is 
to accelerate the times of rising, and to retard the times of 
setting of the Sun and stars. We also showed how to calcu- 
late the amount of the acceleration if the dip be known. 
Conversely, if the acceleration in the time of rising be known, 
the dip of the horizon can be calculated, and from this the 
height of the observer above the general level of the Earth 
may be found. 

Tfow precisely the same method may be applied to measure 
the heights of lunar mountains. When the Moon is waxing 
the Sun is gradually rising over those parts of the Moon's 
surface which are turned towards the Earth. The tops of 
the mountains catch the rays before the lower parts, and, 
therefore, stand out bright against the dark background of 
the unilluminated parts below. Similarly, when the Moon 
is waning, the summits of the mountains remain as bright 
specks after the lower portions are plunged in shadow. By 
noticing the exact instant at which the Sun's rays begin or 
oeasc to illuminate the summit, this acceleration or retarda- 
tion, due to dip, may be calculated, and the height of the 
mountain determined. 

If the Moon's surface around the mountain is fairly level, 
the distance of the mountain from the illuminated portion at 
the instant of disappearance determines the distance of the 
visible horizon as seen from the mountain. This distance can 
be calculated from measurements made with a micrometer 
(proper allowance being made for foreshortening if the moun- 
tain is not in the centre of the disc). 

Hence the height (h) of the mountain may be calculated 
by the formula of 101 (i.), viz., h = d^/Za, where d is the 
estimated distance of the horizon, and a the Moon's radius, 



THE MOON. 209 

SECTION III. The Moon's Orbit and Rotation. 

269. The Moon's Orbit about the Earth can be inves- 
tigated by a method precisely similar to that employed in the 
case of the Sun (see 145). The Moon's E.A. and decl. 
may be observed daily by the Transit Circle. The observed 
decl. must be corrected for refraction and parallax (neither of 
which affect the R.A., since the observations are made on the 
meridian}. "We thus find the positions of the Moon on the 
celestial sphere relative to the Earth's centre for eveiy day 
at the instant of its transit across the meridian of the obser- 
vatory, 

Instead of observing the Moon's parallax daily, the Moon's 
distances from the Earth's centre on different days, may be 
compared by measuring the Moon's angular diameters, with 
the heliometer. Here, however, another correction for 
parallax is required. For the observed angular diameters 
are inversely proportional to the corresponding distances of 
the Moon from the observer, and not from the centre of the 
Earth. 

This correction is by no means inconsiderable. Thus, for 
example, if the Moon be vertically overhead, its distances from 
the observer and from the Earth's centre will differ by the 
Earth's radius, i.e., by about -^ of the latter distance, and 
its angular diameter will, therefore, be increased in the pro- 
portion of about 60 to 59. 

Having thus determined the direction and distance of the 
Moon's centre, relative to the Earth's centre, for every clay in 
the month, the Moon's orbit may be traced out in just the same 
way as the Sun's orbit was traced out in 146. It is thus 
found that the motion obeys approximately the following 
laws : 

(i.) The Moon's orbit lies in a plane through the Earth's 
centre, inclined to the plane of the ecliptic at an angle of about 
5 8'. 

(ii.) The orbit is an ellipse, having the- Earth's centre in 

one focus, the eccentricity of the ellipse being about . 

18 

(iii.) The radius vector joining the Earth's and Moon\ 
centres traces out equal areas in equal intervals of time. 



210 ASTRONOMY. 

The period of revolution is, of course, the sidereal lunar 
month, as denned in Section II., namely, about 27^ days. 

The laws which govern the Moon's motion are thus iden- 
tical with Kepler's laws for the Earth's orbital motion round 
the Sun ( 155). 

270. The Eccentricity of the Moon's Orbit is found 
by comparing the Moon's greatest and least distances, which 
are inversely proportional to its least and greatest (geocentric) 
angular diameters respectively. The latter are in the ratio 
of about 17 to 19, and it is inferred that the eccentricity is 

'about (19-17)/(19 + 17) 5 or ^ (ef. 149). 

The terms perigee, apogee, apse line are used in 
the same sense as in 147. Perigee and apogee are the 
points in the orbit at which the Moon is nearest to and 
furthest from the Earth respectively. Both are called the 
apses or apsides, the line joining them being called the apse 
line, apsidal line or line of apsides, according to choice. It is 
the major axis of the orbit. 

As in 151, it follows that the Moon's angular motion in 
its orbit is swiftest at perigee, and slowest at apogee. 

271. Nodes. The points in which the Moon's orbit, or its 
projection on the celestial sphere, cuts the ecliptic are called 
the Moon's Nodes (ef. 40). The line joining them is 
called the Nodal Line. It is the line of intersection of the 
planes of the Moon's orbit and ecliptic. That node through 
which the Moon passes in crossing from south to north of the 
ecliptic is distinguished as the ascending node, the other is 
distinguished as the descending node. 

272. Perturbations. As the result of observations extend- 
ing over a large number of lunar months, it is found that the 
Moon docs not describe exactly the same ellipse over and 
over again, and that, therefore, the laws stated in 269 are 
only approximate. The actual motion can, however, be 
represented by supposing the Moon to revolve in an ellipse, 
the positions and dimensions of which are very slowly vary- 
ing. This mode of representing the motion may be illustrated 
by imagining a bead to revolve on a smooth elliptic wire 
which is very slowly moved about and deformed. 



THE MOON. 211 

The complete investigation of these small changes or 
perturbations, as they are called, belongs to the domain of 
Gravitational Astronomy. It will be necessary here to 
enumerate the chief perturbations, on account of the important 
part they play in determining the circumstances of eclipses. 

273. Retrograde Motion of the Moon's Nodes. The 

Moon's nodes are not fixed, but have a retrograde motion 
along the ecliptic of about 19 in a year. This phenomenon 
closely resembles the retrograde motion of T (Precession, 
141), but is far more rapid. Its effect is to carry the line 
of nodes, with the plane of the Moon's orbit, slowly round 
the ecliptic, performing a complete revolution in 6793-391 
days, or rather over 18-6 years. 

One result of this nodal motion is that the angle of inclination 
of the Moon's orbit to the equator is subject to periodic 
variations. When the Moon's ascending node coincides with the 
first point of Aries, the angle between the Moon's orbit and the 
equator will be the difference of the angles they make with the 
ecliptic, i.e. about 23 28' - 5 8' or 18 20'. When, on the contrary, 
the ascending node coincides with the first point of Libra, the angle 
between the orbit and the equator will be the sum of the angles they 
make with the ecliptic, i.e., 23 28' + 5 8' or 28 36'. The period of 
fluctuation is the time of revolution of the Moon's nodes relative to 
the first point of Aries, and is a few days (nearly five) greater than 
their sidereal period of revolution, on account of precession. 

274. Progressive Motion of Apse Line. The line oi 
apsides is not fixed, but has a direct motion in the plane 
of the Moon's orbit, performing a complete revolution in 
3232-575 days, or about nine years. A similar progressive 
motion of the apse line of the Earth's orbit about the Sun 
was mentioned in 153. The latter motion is, however, 
much less rapid, its period being about 108,000 years. 

275. Other Perturbations. The inclination of the 
Moon's orbit to the ecliptic is not quite constant. It is 
subject to small periodic variations, its greatest and least 
values being 5 13' and 5 3'. 

In addition there are variations in the eccentricity of the 
orbit, in the rates of motion of the nodes, and in the length 
of the sidereal period. All of these render the accurate 
investigation of the Moon's orbit one of the most complicated 
problems of Astronomy. 



212 A8TEONOMT. 

276. The Moon's Rotation. It is a remarkable fact 
that the Moon always turns the same side of its surface to th 
Earth. Whether we examine the markings on its surface 
with the naked eye, or resolve them into mountains and 
streaks with a telescope, they always appear very nearly the 
same, although their illumination, of course, varies with the 
phase. 

From this it is evident that the Moon rotates upon its axis 
in the same "sidereal" period as it takes to describe its 
orbit about the Earth, i.e., once in a sidereal month. It 
might, at a first glance, appear as if the Moon had no rota- 
tion, but such is not the case. To explain this, let us consider 
the phenomena which would be presented to an observer if 
situated on the Moon in the centre of the portion turned 
towards the Earth. 

The Earth would always appear directly overhead, i.e., in 
the observer's zenith. But as the Moon describes its orbit 
about the Earth, the direction of tlie line joining the Earth 
and Moon revolves through 360, relative to the fixed stars, 
in a sidereal month. Hence the direction of the observer's 
zenith on the Moon must also revolve through 360 in a 
sidereal month, and therefore the Moon must rotate on its 
axis in this period. 

The Moon would be said to describe its orbit without 
rotation, if the same points on its surface were to remain 
always directed towards the same fixed stars. "Were this the 
case, different parts of the surface would become turned to- 
wards the Earth as the Earth's direction changed, and this is 
not what actually occurs. 

It thus appears that, to an observer on the Moon, the 
directions of the stars relative to the horizon would appear 
to revolve through 360 once in a sidereal lunar month. 
Thus, the sidereal month is the period corresponding to the 
sidereal day of an observer on the Earth. In a similar way, 
the Sun's direction would appear to revolve through 360 in 
a synodic month. This, therefore, is the period corresponding 
to the solar day on the Earth, as is otherwise evident from 
the fact that the Moon's phases determine the alternations of 
light and darkness on the Moon's surface, and that they 
repeat themselves once in every synodic month. 



THE MOON. 213 

277. Libratipns of the Moon. Libration in Lati- 
tude. If the axis about which the Moon rotates were per- 
pendicular to the plane of the Moon's orbit, we should not be 
able to see any of the surface beyond the two poles ('.*., ex- 
tremities of the axis of rotation). In reality, however, the 
Moon's axis, instead of being exactly perpendicular to its 
orbit, is inclined at an angle of about 6| to the perpendicular, 
just as the Earth's axis of rotation makes an angle of about 
23 28' with a perpendicular to the ecliptic. The conse- 
quence is that during the Moon's revolution the Moon's north 
and south poles are alternately turned a little towards and a 
little away from the Earth ; thus, in one part of the orbit we 
see the Moon's surface to an angular distance of 6 44' beyond 
its north pole, in the opposite part we see 6 44' beyond the 
southpolc. This phenomenon is called the Moon's libration in 
latitude. It makes the Moon's poles appear to nocl, oscillat- 
ing to and fro once in every revolution relative to the nodes. 

Libration in latitude may be conveniently illustrated by 
the corresponding phenomenon in the case of the Earth's 
motion round the Sun, as represented in Fig. 56 ( 154). At 
the summer solstice the whole of the Arctic circle is illumi- 
nated by the Sun's rays, and therefore an observer on the 
Sun (if such could exist) would see the Earth's surface for a 
distance of 23 28' beyond the north pole. Similarly, at the 
winter solstice an observer on the Sun would see the whole 
of the Antarctic circle, and a portion of the Earth's surface 
extending 23 28' beyond the south pole. 

278. Libration in Longitude. Owing to the elliptical 
form of the orbit, the Moon's angular velocity about the 
Earth is not quite uniform, being least at apogee and greatest 
at perigee. But the Moon rotates about its polar axis with 
perfectly uniform angular velocity equal to the average 
angular velocity of the orbital motion (so that the periods 
of rotation and of orbital motion are equal). 

Thus, at apogee the angular velocity of rotation is slightly 
greater than that of the orbital motion, and is, therefore, 
greater than that required to keep the same part of the 
Moon's surface always turned towards the Earth. In con- 
sequence, the Moon will appear to gradually turn round, so 
as to show a little more of the eastern side of its surface. 



214 ASTRONOMY. 

At perigee, the angular velocity of rotation is less than 
that of the orbital motion, and is, therefore, not quite suffi- 
cient to keep the same part of the Moon's surface always 
turned towards the Earth. In consequence we shall begin to 
see a little further round the western side of the Moon's disc. 

This phenomenon is called libration in longitude. Its 
maximum amount is 7 45' ; thus, during each revolution of 
the Moon relative to the apse line, we alternately see 7 45' 
of arc further round the eastern and western sides of the disc 
than we should otherwise. 

279. Diurnal Libration. The phenomenon known as 
diurnal libration is really only an effect of parallax. If 
the Moon were vertically overhead, and if we were to travel 
eastwards, we should, of course, begin to see a little further 
round the eastern side of the Moon's surface. If we were to 
travel westwards we should begin to see a little further round 
the western side. Now, the rotation of the Earth carries the 
observer round from west to east. Hence, when the Moon is 
rising wo see a little further round its western side, and 
when setting we see a little further round its eastern side, 
than we should from a point vertically underneath the Moon. 

Similarly an observer in the northern hemisphere would 
always see rather more of the Moon's northern portion, and 
an observer in the southern hemisphere would see rather more 
of the southern portion than an observer at the equator. 

The greatest amount of the diurnal libration is equal to the 
Moon's horizontal parallax, and is therefore about 57'. We 
see 57' round the Moon's western corner when rising, and 57' 
round the eastern corner when setting. 

An observer at any given instant sees not quite half 
(49-998 per cent.) the Moon's surface. The visible portion is 
bounded by a cone through the observer's eye enveloping the 
Moon, and is less than a hemisphere by a belt of breadth equal 
to the Moon's angular semi-diameter, i.e., about 16'. 

280. General Effects of Libration. In consequence 
of the three librations, about 59 per cent, of the Moon's sur- 
face is visible from the Earth at some time or other, instead of 
rather under 50 (49-998) per cent., as would be the case if 
there were no libration. At the same time only about 41 per 
cent, of the surface is always visible from the Earth. The 
remainder is sometimes visible, sometimes invisible. 



THE MOON. 215 

To an observer on the surface of the Moon the result of libra- 
tion in latitude and longitude would be that the Earth, instead 
of remaining stationary in the sky, would appear to perform small 
oscillations about its mean position. It would really appear to de- 
scribe a series of ellipses. The motion of the different parts of the 
Earth across its disc in the course of the Earth's diurnal revolution 
would be the only phenomenon resulting from the cause which pro- 
duces diurnal libration. 

281. Metonic Cycle. A problem of great historic interest in the 
study of the lunar motions is the finding of a method of ready pre- 
diction of the Moon's phases. From the earliest times there have 
been religious festivals regulated (as Easter still is) by the Moon's 
phases; but the direct calculation, from first principles, of the phase 
for a given day would be long and tedious. 

This difficulty was overcome by the discovery of the EO-called 
Metonic Cycle by Meton and Euctemon, B.C. 433. They found that 
after a cycle of nineteen years the new and full Moons recurred on the 
same days of the year. To show this it is necessary to prove that 
nineteen years is nearly an exact multiple of the synodic month. 
Now, 1 tropical year = 365'2422 days ; .'.19 years = 6939'60 days, 
and 1 synodic month = 29'5306 days ; /. 285months = 6939'69days; 
.'. 19 years differs from 235 lunations by '09 days, i.e., 2h. 10m. nearly. 

If we define the Golden Number of a year as the remainder when 
(1 + the number of the year A.D.) is divided by 19, and the Epact as 
the Moon's age on the 1st of January, we see that two years which 
have the same Golden Number have corresponding lunar phases on 
the same days, and in particular have the same epact. 

Hence, the Golden Number of the year 1 B.C. (which might be 
more consistently called A.D.) is evidently 1 ; and it happens that 
that year had new Moon on January 1, and, therefore, its epact is 
zero. But twelve lunar months contain 354'37 days, and fall short 
of the average year (365'25 days) by 10'88 days, which is nearly 
5^ lunations. Hence, the epact is greater by -^ of a lunation each 
year ; and since whole months are not counted in estimating the 
Moon's age, it is (in months) the fractional part of 

|i (Golden Number -1}; 
or, in days, the remainder when 11 {Golden No. 1 j is divided by 30. 

Thus the Golden Number of 1892 is the remainder when 1893 is 
divided by 19, i.e., 12. Hence, the epact is the remainder when 
11 {12 1} is divided by 30, i.e., 1 ; hence, the Moon is one day old 
on January 1, 1892, and new on December 31, 1891. 

In the epact, fractions of a day are never reckoned. Owing to 
the extra day in leap year, the rule is sometimes a day wrong; .but 
it is near enough for fixing the ecclesiastical calendar. 



ASTROXOMT. 

282. Harvest Moon. The full Moon which occurs 
nearest the autumnal equinox is called the Harvest Moon. 
Owing to the Moon's direct motion in its orbit the time of 
moonrise always occurs later and later every day, but in the 
case of the harvest Moon the daily retardation is less than 
in the case of any other full Moon, as we shall now show. 
To simplify our rough explanations we suppose the Moon to 
be moving in the ecliptic. 

The Moon's E.A. determines the time at which the Moon 
crosses the meridian (cf. 24). In consequence of the 
orbital motion the R. A. increases continuously, just as in the 
case of the Sun ( 30), only the increase is more rapid (360 
per month instead of per year). Therefore the Moon transits 
later and later every night. 

When the Moon is in the first point of Aries it is passing 
from south to north of the equator, and its declination is 
increasing most rapidly. Now, the arguments of 123-125 
are applicable to the Moon as well as the Sun, and they show 
that, as the declination increases, there is, in north latitudes, 
a corresponding increase in the length of time that the 
Moon is above the horizon. The effect of this increase is to 
lengthen the interval from the Moon's rising to its transit ; 
this lengthening tends to counterbalance, more or less, the 
retardation in the time of transit, thus reducing the retarda- 
tion in the time of moonrise to a minimum. 

Similarly it may be shown that whenever the Moon passes 
the first point of Libra, the daily retardation of moonrise will 
be a maximum, while that of the time of setting will be a 
minimum. These phenomena, therefore, recur once each 
lunar month. 

Now, at harvest time the Sun is near ; hence, when the 
Moon is near T it is full ; and the minimum retardation of the 
Moon's rising, therefore, takes place at full Moon. And since 
the Moon is then opposite the Sun, it rises at sunset. Both 
these causes make the phenomenon more conspicuous in itself 
than at other times, and as the continuance of light is useful 
to the farmers when gathering in their harvest, the name 
Harvest Moon has been applied. 

At the following full Moon the phenomena are similar but 
less marked. But as it is now the hunting season, the Moon 
is called the " Hunter's Moon." 



TOE MOON. 217 



EXAMPLES. VIII. 

1. If a, a 1 be the true and apparent altitudes of a body affected 
by parallax, prove the equation a = a' + P cos a'. 

2. If the Sun's parallax be 8'80", find the Sun's distance. 

3. If in our latitude, on March 21, the Moon is in its first quarter, 
about what time may it be looked for on the meridian, and how 
long does it remain above the horizon ? 

4. Show that from a study of the Moon's phases we can infer the 
Sun to be much more distant than the Moon. Prove that if the 
synodic period wore 30 days, and the Sun only twice as distant as 
the Moon, the Moon would be dichotomized after only 5 days 
instead of 7. 

5. Taking the usual values of the Sun's and the Moon's distances, 
calculate, roughly, the mean value of the angle ESM when the Moon 
is dichotomized. 

6. Under what conditions is the line of cusps perpendicular to 
the horizon ? Consider specially the appearance to an observer on 
the Arctic circle. 

7. There was an eclipse of the Moon on Jan. 28, 1888, central at 
11.10 in the evening. What is the Moon's age on May 21 of that 
year? 

8. Find approximately the position and appearance of the Moon, 
relatively to the horizon, in latitude 50 N., in the middle of Novem- 
ber at 10 P.M., when it is ten days old 

9. At a place in the temperate zone can the Sun or the Moon be 
longer above the horizon ? 

10. What would be the effect on the Harvest Moon (i.) if the 
polar axis of the Earth were perpendicular to the ecliptic, or (ii.) if 
the Moon were to move in the ecliptic ? 



218 ASTRONOMY. 



EXAMINATION PAPER. VIII. 

1. What is parallax, and under what conditions is the parallax of 
a heavenly body greatest ? Show by some simple illustrations that 
as the distance of an object increases, its parallax lessens. 

2. Prove the formula sin p = sin P sin z, where P is the Moon's 
horizontal parallax, and p its parallax when its zenith distance 
is z. 

3. How is the distance of the Moon determined by observations 
made in the plane of the meridian? Why cannot the Sun's parallax 
be accurately determined in this way ? 

4. Show that we can calculate the Moon's sidereal period given 
its synodic period and the length of the year. Find it, given that 
these are 29^ and 365J: days respectively. 

5. Describe the phases of the Moon, and find an expression for 
the phase when the Moon is at a given elongation. Show how ao 
observation of the Moon, when at its first quarter, would help ua to 
find the ratio of the distances of the Moon and the Sun. 

6. Describe some methods for determining the heights of lunar 
mountains. 

7. Describe the phenomena of the Moon's motion. Given that 
the Moon moves in a plane inclined at 5 to the ecliptic, find the 
lowest north latitude of a place where the full Moon can never rise 
at the summer solstice. 

8. Explain (and illustrate by figures) how it is that we see more 
than half the Moon's surface, and define the terms node, phase, 
libration. 

9. Describe the general appearance presented by the solar system 
to an observer situated at the centre of the Moon's hemisphere 
turned towards the Earth. When would the Earth be partially 
eclipsed to such an observer ? 

10. Explain the phenomenon called the Harvest Moon, and show 
that from a similar cause the daily retaliation in the sidereal time 
of sunrise is least at the vernal equinox. 



CHAPTER IX, 



ECLIPSES. 



SECTION I. General Description of Eclipses. 

283. Eclipses are of two kinds, lunar and solar. If at 
full Moon the centres of the Sun, Earth, and Moon arc very 
nearly in a straight line, the Earth, acting as a screen, will 
stop the Sun's rays from reaching the Moon, and the Moon 
will, therefore, be either wholly or partially darkened. This 
phenomenon is called a Lunar Eclipse. 

On the other hand, if the three centres are nearly in a 
straight line when the Moon is new, the Moon, by coming 
between the Earth and the Sun, will cut off the whole or a 
portion of the Sun's rays from certain parts of the Earth's 
surface. In such parts the Earth will be darkened, and the 
Sun will appear either wholly or partially hidden. This 
phenomenon is a Solar Eclipse. 

If the Moon were to move exactly in the ecliptic we 
should have an eclipse of the Moon at every opposition, and 
an eclipse of the Sun at every conjunction, for at either 
epoch the centres of the Earth, Sun, and Moon would be in 
an exact straight line. In consequence, however, of the 
Moon's orbit being inclined to the ecliptic at an angle of 
about 5i, the Moon at " syzygy " (conjunction or opposition) 
is generally so far on the north or south side of the ecliptic 
that no eclipse takes place. An eclipse only occurs when 
the Moon at syzygy is very near the ecliptic, and, 
therefore, not far from the line of nodes ( 271). 

ASTBOtf. Q 



220 * ASTRONOMY. 

284. Different Kinds of Lunar Eclipse. Eclipses of 
the Moon are of two kinds, total and partial. Let S, E be 
the centres of the Sun and Earth respectively. Draw the 
common tangents ^LSFand A'B'Vio the two glohes, meet- 
ing on SE produced in F, and draw also the other pair of 
tangents AB'K', A'BK cutting at Z7, between S and J If 
the figure be supposed to revolve about &#, the tangents 
will generate cones, enveloping the Sun and Earth, and 
having their vertices at U and F. The space BVB , inside 
the inner cone, is called the umbra ; the space between the 
inner and outer cone is called the penumbra.* The 
character of the lunar eclipse will vary according to the 
following conditions : 




FIG. 87. 

(i.) If at opposition, the Moon falls entirely within the 
umbra or inner cone B FZ?', as at J/J, no portion of the Moon's 
surface then receives any direct rays from the Sun, and the 
Moon is therefore plunged in darkness (except for the light 
which reaches it after refraction by the Earth's atmosphere, 
as explained in 193). The eclipse is then said to be total. 

(ii.) If the Moon falls partly within and partly without 
the umbra B VB\ as at J/~ 2 , the portion within the umbra 
receives no light from the Sun, and is, therefore, obscured, 
while the remaining portion receives light from part of the 
Sun's surface about A, and is, therefore, partially illuminated. 
The eclipse is then said to be partial. 

*For further description of the formation of the umbra and 
penumbra, see Wallace Stewart's Text-Book of Lie/lit, 5, 



ECLIPSES. 221 

(iii.) If the Moon falls entirely within the "penumbra," 
or outer cone, as at M z , it receives the Sun's rays from A, 
but not from A'. There is no true eclipse, but only a 
diminution of brightness (sometimes called a " penumbral 
eclipse"). 

A lunar eclipse is visible simultaneously from all places on 
that hemisphere of the Earth over which the Moon is above 
the horizon at the time of its occurrence. 

!N"ear the boundary of the hemisphere there are two strips 
in the form of lunes, comprising those places respectively at 
which the Moon sets and rises during the eclipse j at such 
places only its beginning or end is seen. 

285. Phenomena of a Total Eclipse of the Moon. 
As the Moon gradually moves towards opposition, the first 
appearance noticeable is the slight darkening of the Moon's 
surface as it enters the penumbra. This darkening increases 
very gradually as the Moon approaches the umbra, or true 
shadow. At "First Contact" a portion of the Moon 
enters the umbra, and the eclipse is then seen as a partial 
eclipse, the dark portion being bounded by the circular arc 
formed by the boundary of the umbra. As the Moon 
advances, the dark portion increases till the whole of the 
Moon is within the umbra, and the eclipse is total. "When 
the Moon begins to emerge at the other side of the umbra, 
the eclipse again becomes partial, and continues so until 
"Last Contact," when the Moon has entirely emerged 
from the umbra, after which the Moon gradually gets brighter 
and brighter till it finally leaves the penumbra. 

In the case of a partial eclipse, the umbra merely appears 
to pass over a portion of the Moon's disc, which portion is 
greatest at the middle of the eclipse* 

286. Effects of Refraction on Lunar Eclipses. In 

193 it was stated that, owing to atmospheric refraction, the 
Moon's disc appears of a dull-red colour during the totality 
of the eclipse. A still more curious phenomenon is noticed 
when an eclipse occurs at sunset or sunrise. The refraction 
at the horizon increases the apparent altitudes of the Sun and 
Moon in the heavens, so that both appear above the horizon 
when they are just below. Hence a total eclipse of the 
Moon is sometimes seen when the Sun is shining. 



222 



ASTRONOMY. 



287. Different Kinds of Solar Eclipse. An eclipse of 
the Sun may be either total, annular, or partial. To 
explain the difference between the first two kinds of eclipse, 
let us suppose that the observer is situated exactly in the 
line of centres of the Sun and new Moon, so that both bodies 
appear in the same direction. Then, if the Moon's angular 
diameter is greater than the Sun's, the whole of the Sun will 
be concealed by the Moon ; the eclipse is then said to be 
total. If, on the other hand, the Sun has the greater 
angular diameter, the Moon will conceal only the central 
portion of the Sun's disc, leaving a bright ring visible all 
round; under such circumstances, the eclipse is said to be 
annular. Lastly, if the observer is not exactly in the line 
of centres, the Moon .may cover up a segment at one side of 
the Sun's disc ; the eclipse is then partial. 

Now, the Moon's angular diameter varies, according to the 
distance of the Moon, from 28' 48" at apogee to 33' 22" at 
perigee, the corresponding limits for the Sun's diameter being 
31' 32" at apogee, and 32' 36" at perigee. Hence, both total 
and annular eclipses of the Sun are possible. Thus, when 
the Sun is in apogee and the Moon in perigee an eclipse must 
be either total or partial ; when the Sun is in perigee and 
the Moon in apogee, an eclipse must be annular or partial. 




FIG. 88. 

288. Circumstances of a Solar Eclipse. Fig. 88 

shows the different circumstances under which a solar eclipse 
is seen from different parts of the Earth. Draw the common 
tangents CDQ, C'D'Q, CRU, C'RD to the Sun and Moon, 
forming the enveloping cones DQD' and fRg\ these consti- 
tute respectively the boundaries of the umbra and penumbra 
of the Moon's shadow. First let the umbra DQD* meet the 
Earth's surface (E^ before coming to a point at Q, the curve 



223 

of intersection being de. Also let the penumbra fRg meet 
the Earth's surface in the curve fg. Then from anyplace on 
the Earth within the space de the Sun appears totally eclipsed. 
At a place elsewhere within the penumbra fg, the Sun appears 
partially eclipsed, a portion only being obscured by the Moon. 

Next let the umbra DQD' come to a point Q before 
reaching the Earth E y Then, if the cone of the umbra be 
produced to meet the Earth in d'e', an observer anywhere 
within the space d'e' sees the eclipse as an annular eclipse. 
At any place elsewhere within the penumbra /y, the eclipse 
appears partial, as before. At parts of the Earth which fall 
without the penumbra there is no eclipse. Hence a solar 
eclipse is only visible over a part of the Earth's surface, 
and its circumstances are different at different places. 

As the Sun and Moon move forward in their relative orbits, 
and the Earth revolves on its axis, the two cones of the 
Moon's shadow travel over the Earth, and the eclipse becomes 
visible from different places in succession The inner cone 
traces out on the Earth a very narrow belt, over which 
the eclipse is seen as a total or annular eclipse, according 
to circumstances. The outer cone, or penumbra, sweeps out 
a far broader belt, including that part of the Earth's surface 
where the eclipse is visible as a partial eclipse. 

A total or annular eclipse of the Sun, like a total eclipse 
of the Moon, always begins and ends as a partial eclipse, the 
totality or annular condition only lasting for a short period 
about the middle of the eclipse. The maximum duration 
of totality at the Equator is just under eight minutes. 

In the case of an annular eclipse, there are two internal, 
as well as two external, contacts, and the eclipse remains 
annular during the interval between the internal contacts. 
This may sometimes be rather more than twelve minutes. 

Owing to the limited area of the belt over which a solar 
eclipse is visible, the chance that any eclipse may be visible 
at any given place is far smaller than in the case of a lunar 
eclipse. The chance of an eclipse being total at any place is 
very small indeed. The last eclipse visible as a total eclipse 
in England occurred in 1724 ; the next will take place on 
June 29th, 1927. One or more partial eclipses are visible at 
Greenwich in nearly every year. 



224 



ASTROBOMf. 



SECTION IT. Determination of the Frequency of Eclipses. 
289. To Find the Limits of the Moon's geocentric 
position consistent with a Solar or Lunar Eclipse. 

In Fig. 89, let the plane of the paper represent any piano 
through the Sun's and Moon's centres; and let ABV and 
A 'B' ^"represent the common tangents bounding the cone of 
the Earth's true shadow. Let A UB' be the other common 
tangent, which goes (nearly) through B 1 ; and let the line SE, 
joining the centres of the Sun and Earth, meet the common 
tangents in Fand U. Let T, t, t' be those points on AB Fand 
AB' whose distance from E is equal to that of the Moon. 




FIG. 89. 



Then, if J/i, M. 2 denote the positions of the Moon's centre, 
when touching the cone B V externally and internally at T, 
it is evident that a lunar eclipse occurs whenever the full 
Moon is nearer the line of centres than N r Hence, if m 
denote the Moon's angular semi-diameter TEM^ the Moon's 
angular distance from JSVmnst be less than VEM or VET+ m. 

Similarly, the lunar eclipse is total when the Moon is not 
further from the line of centres than Jf 2 ; for this the Moon's 
(geocentric) angular distance from the line of centres must 
be not greater than VEN^ qr VET-m. 

Let m v m 2 be the centres of the Moon at internal and 
external contact with ^47? near t. There is evidently a solar 
eclipse visible at some point of the Earth's surface (such as 
Z?) as a partial eclipse, if the Moon's angular distance from 
the Sun is less than SEm^ or SEt+m. 

Supposing the Moon's distance to be such that its angular 
radius is less than that of the Sun, there is an annular 
eclipse whenever the Moon lies wholly within the cone A VA', 
as at m v This requires the Moon's geocentric angular dis- 
tance from the Sun to be less than SEm v or SEtm. 



ECLIPSES. 225 

If, however, the Moon is so near that its angular radius 
is greater than that of the Sun, the angle it subtends is 
greater than ABA', and therefore there is a total 
eclipse at B whenever the edge of the Moon reaches the 
internal tangent A'B. Taking m s to represent the corre- 
sponding position of the Moon when touching the other 
tangent AB' at t' (for the sake of clearness in the figure), 
we see that, in order that there may be a total eclipse 
fomewhere on the Earth's surface, the geocentric angular 
distance between the Moon's and Sun's centres must be less 
than SEm B or SEt' + m. 

Now, as the cone A VA' tapers to a point at V, the breadth 
of its cross section is greater near m v m v m s than near M ly M 9 , 
and when the Moon is in syzygy, its angular distance from EV 
or ES = its latitude. Hence the limits of latitude aro greater 
for a solar than for a lunar eclipse, and therefore the proba- 
bility of the occurrence of a solar eclipse is greater than the 
probability of a lunar eclipse. This explains why, on the 
whole, solar eclipses are more frequent than lunar. 

*290. We shall now calculate the angles VEM^ VEM. 2 , 
SEm^ , SEm^ SEm y Let p, P denote the horizontal parallaxes . 
of the Moon and Sun respectively; m, s their respeetive 
angular semi-diameters (Fig. 89). We have s= Z SEA, 

p-A BTE = z BtE= Z B't'E, P = z BAE=* Z B'AE, 
and m = Z TJEM l = Z TEM Z = z tEm l = Z tEm^ = Z t'Em y 
For the lunar eclipses we have, from the triangle TEA, 
Z ETJB+ Z EAB = 180- z TEA = Z VET+ Z SEA 
.-. VET = LETB+ LEAB- LSEA=p+P-s-, 
.-. z VEMi = ^ VET+ z TEM^ = p+ps+m ; 
and z rJSMt = Z VET- z TEM 9 =p+P-s-m ; 
For the solar eclipses we have, from the triangle tEA, 
Z EtB z EAB = z tEA - Z SEt z SEA 



.-. z SEin^ = p P + s + in , 
and z SEni^ =p JP + s m, 

Lastly, from the triangle t'EA we have 

Z Et'B'- /.EAB' z AEt' = Z AES+ Z SEf. 

:. L SEt 1 = Z B't'E- z B'AE- Z ^4^>S = p-P-s. 

. . z >S^i, = p -P s -i- w- 



226 

[As an examplo, the student may show that the greatest 
latitudes the Moon can have, in ordei that it may bo partially 
or wholly within the penumbra at opposition arc p + s + P-f m 
and p + s + P m respectively.] 

*291. Greatesb Latitudes of the Moon at Syzygy. 

Since S and V are in the ecliptic, it follows that when 
the Moon is in conjunction or opposition, the plane of the 
paper in !Fig. 89 is perpendicular to the ecliptic. Therefore 
the angles VEM^ VEM. 2 measure the Moon's latitude at con- 
junction, and SJEm v SEm. 2 , SEm 3 measure its latitude at 
opposition in the positions represented. The above expres- 
sions are, therefore, the greatest possible latitudes at syzygy 
consistent with eclipses of the kinds named. 

Now, taking the mean values we have, roughly, 

s=l6'- m=l5'; p = 57' ; P = 0' 8". 
Substituting these values, and collecting the results, we have, 
roughly, the following limits for the Moon's geocentric lati- 
tude, or angular distance from the line of centres : 

(1) For a lunar eclipse, VEM^ =p+ P-s+m = 56'; 

(2) Fora total lunar eclipse, VEM^p + Psm = 26'; 

(3) For a solar eclipse, SEm^ =zp P+s + m 88'; 

(4) For an annular eclipse, SEm z = pP+s m = 58'. 
Lastly, taking the Sun at apogee, and the Moon at perigee, 

we have, m = 17' and s = 16' nearly, whence we have, in the 
most favourable case, 

(40) For a total solar eclipse, SEm^ -=p P 8+ m = 58'. 

292. Ecliptic Limits. From the last results it appears 
that a lunar eclipse cannot occur unless at the time of oppo- 
sition the Moon's latitude is less than about 56', and that a 
solar eclipse cannot occur unless at conjunction the Moon's 
latitude is less than about 88'. Now the Moon's latitude 
depends on its position in its orbit relatively to the line of 
nodes ; hence there will be corresponding limits to the Moon's 
distance from the node consistent with the occurrence of 
eclipses. These limits are called the Ecliptic Limits. 

*The ecliptic limits may be computed as follows : Let the 
geocentric direction of the Moon's centre be represented on 
the celestial sphere by Jf. Let JV represent the node, 



ECLIPSES. 227 

secondary to the ecliptic. [The ecliptic limit, strictly speak- 
ing, means the limit of NH measured along the ecliptic, and 
not that of NMJ\ 

Now the limit of latitude MH lias been calculated in the 
last paragraph for the different cases. Let this be denoted by 
I. Also let I be the inclination of the Moon's orbit to the 
ecliptic. Then in the spherical triangle NHM, right-angled 
at H t we have HM = I, and z JINM= /; both of these are 
known, hence NIL can be calculated. 




FIG. 90. 



For rough purposes it will be sufficient either to treat the 
small triangle ITNMas a plane triangle (Sph. Geom. 24), or 
to regard Mil as approximately the arc of a small circle, 
whose pole is JV. The first method gives 



.-. NH= I cot /. 

Or, adopting the second method, we have (Sph. Geom. 17) 
I = MH= z MNITx sin NH = /sin Nil-, 

.-. sin NH =1/1, 
whence the ecliptic limit Nil is found. 

EXAMPLES. 

1. To find the Lunar Ecliptic Limit. For a lunar eclipse we have, 
by 291, I = 56'. Also, 1-5 roughly. 

Ilence 



= sin 11 (from table of natural sines) 
and the lunar ecliptic limit is about 11. 

2. To find the Solar Ecliptic Limit. For a solar eclipse we have 
I = 88'. Hence, taking I = 5 as before, we have 



= sin 17, roughly, 
and the solar ecliptic limit is about 17. 



228 ASTRONOMY. 

293. Major and Minor Ecliptic Limits. Owing to 

the variations in the distances of the Sun and Moon their 
parallaxes and angular semi-diameters are not quite constant. 
Hence the exact limits of the Moon's latitude /, as calculated 
by the method of 291, are subject to small variations. 

This alone would render the ecliptic limits variable. But 
there is another cause of variation in the ecliptic limits, 
arising from the fact that Z the inclination of the Moon's 
orbit, is also variable, its greatest and least values being about 
5 19' and 4 57'. 

The greatest and least values of the limits for each kind of 
eclipse are called the Major and Minor Ecliptic Limits. 

For an eclipse of the Moon the major and minor ecliptic 
limits have been calculated to be about 12 5' and 9 30' re- 
spectively at the present time. For an eclipse of the Sun the 
limits are 18 31' and 15 21' respectively. 

Thus a lunar eclipse may take place if the Moon, when 
full, is within 12 5' of a node; and a lunar eclipse must 
take place if the full Moon is within 9 30' of a node. 

Similarly, a solar eclipse may take place if the Moon, when 
new, is within 18 31', and a solar eclipse must take place if 
the new Moon is within 15 21' of a node. 

The mean values of the lunar and solar ecliptic limits are 
now 10 47' and 16 56'. But the eccentricity of the Earth's 
orbit is very slowly decreasing ; consequently the major limits 
are smaller and the minor limits larger than they were, say, 
a thousand years ago. 

294. Synodic Revolution of the Moon's Nodes. An 
eclipse is thus only possible at a time when the Sun is within 
a certain angular distance of the Moon's nodes. Hence the 
period of revolution of the Moon's nodes, relative to the Sun, 
marks the recurrence of the intervals of time during which 
eclipses are possible. This period is called the period of a 
synodic revolution of the nodes. 

In 273 it was stated that the Moon's nodes have a retro- 
grade motion of about 19 per annum, more exactly 19 21'. 
In one year (365d.) the Sun, therefore, separates from a node 
by 360+ 19 21' or 379*35, hence it separates 360 in 
(360 x 365)/379-35 days, or about 346'62d. This, then, 
is the period of a synodic revolution of the node. 



EClttSES. 229 

In a synodic lunar month (29| days), the Sun separates 
from the line of nodes by an angle 

379jx29j-h365, or 30 36', 
a result which will be required in the next paragraph. 

295. To find the Greatest and Least number of 
Eclipses possible in a Year. Let the circle in Fig. 9* 
represent the ecliptic, and let JVJ n be the Moon's nodes. 
Take the arcs NL, NL', til, nl' each equal to the lunar eclip- 
tic limit, and NS, JVb", ns, ns' each equal to the solar ecliptic 
limit. Then the least value of S3' or ss' is twice the minor 
solar ecliptic limit, and is 30 42', and this is greater than 
30 36', the distance traversed by the Sun relative to the 
nodes between two new Moons. Hence, at least one new 
Moon must occur while the Sun is travel- 
ling over the arc SS', and two may occur. 

Therefore there must be one, and there may 
le two eclipses of the Sun, while the Sun 
is in the neighbourhood of a node. 

Again, the greatest value of LL', IV is 
double the major lunar ecliptic limit, and 
is, therefore, 24 10'. This is consider- 
ably less than the space passed over by 
the Sun relative to the nodes between Fio.TJi. 

two full Moons. Hence, there cannot 
be more than one full Moon while the Sun is in the arc LL\ 
and there may be none. Therefore there cannot le more than 
one eclipse of the Moon while the Sun is in the neighbourhood of 
a node, and there may be none at all. 

296. The case most favourable to the occurrence of 
eclipses is that in which the Moon is new just after the Sun 
has come within the solar ecliptic limits, i.e., near S. There 
will then be an eclipse of the Sun. 

When the Moon is full (about 14| clays later) the Sun will 
be near N, at a point within the lunar ecliptic limits ; there 
will therefore be an eclipse of the Moon. 

At the following new Moon the Sun will not have reached 
S'-, and there will be a second eclipse of the Sun. 

In six lunations from the first eclipse the Sun will have 
travelled through just over 180, and will be within the space 
ss', near s ; there will therefore be a third eclipse of the Sun. 





2,30 ASTROttOMt. 

At the next full Moon the Sun will be near , and there 
will be a second eclipse of the Moon. 

The Sun may just fall within the space 88 near *' at the 
next new Moon ; there will then be & fourth eclipse of the Sun. 

In twelve lunations from the first eclipse, the Sun will 
have described about 368, and will, therefore, be about 8 
beyond its first position, and well within the limits ss' ; there 
will, therefore, be & fifth eclipse of the Sun. 

About 14f days later, at full Moon, the Sun will be well 
within the lunar ecliptic limits LL, and there will be a third 
eclipse of the Moon. 

All these eclipses occur in 12J lunations, i.e., 369 days, or 
a year and four days. "We cannot, therefore, have all the 
eight eclipses in one year, but 

There may be as many as seven eclipses in a year, namely, 
either five solar and tivo lunar, or four solar and three lunar. 

297. The most unfavourable case is that in which the 
Moon is full just before the Sun reachesthe ecliptic limits at L. 

At new Moon the Sun will be near N, and there will be 
one solar eclipse. 

At the next full Moon the Sun will have passed L', so that 
there will be no lunar eclipse. After 
six .lunations the Sun will not have 
arrived at I. 

At the next new Moon the Sun will 
be within the ecliptic limits, and there 
will be a second solar i-flijM-e. 

At the next full Moon the Sun will be 
again just beyond I', and at 12 lunations 
from first full Moon, the Sun may again 
not have quite reached L. 

At 12 lunations there will be a third solar eclipse. 

The interval between the first and third eclipses will be 12 
lunations, or about 354 days. If, therefore, the first eclipse 
occurs after the llth day of the year, i.e., January 11, the 
third will not occur till the following year. Therefore, 

The least possible number of eclipses in a year is two. These 
must both le solar eclipses. 




ECLIPSES. 231 

298. The Saros of the Chaldeans. The period of a 
synodic revolution of the nodes is ( 294) approximately 
346-62 days. Hence, 

19 synodic revolutions of the node take 6585-78 days. 
Also 223 lunar months = 6585*32 days. 

It follows that after 6585 days, or 18 years 11 days, the 
Moon's nodes will have performed 19 revolutions relative to 
the Sun, and the Moon will have performed 223 revolutions 
almost exactly. Hence the Sun and Moon will occupy almost 
exactly the same position relative to the nodes at the end of 
this period as at the beginning, and eclipses will therefore 
recur after this interval. 

The period was discovered by observation by the Chaldean 
astronomers, who called it the Saros. By a knowledge of it 
they were usually able to predict eclipses. Indeed, in the 
records of eclipses handed down to us in the form of cuneiform 
inscriptions, they invariably stated whether the circumstances 
accorded with prediction by the Saros or not. 

A "synodic revolution of the Moon's apsides," or the 
period in which the Sun performs a complete revolution 
relative to the Moon's apse line, occupies 411-74 days. 
Hence sixteen such revolutions occupy 6587*87 days, or 
about two days longer than the Saros. Therefore the Moon's 
line of apsides also returns to very nearly the same 
position relative to the Sun and Moon. Hence, the 
solar eclipses, as they recur, will be nearly of the same 
kind (total or annular) in each Saros. The whole number 
of eclipses in a Saros is about 70. The average of all 
eclipses from B.C. 1207 to A.D. 2162 shows that there are 
20 solar eclipses to 13 lunar. 

The present values of the mean solar and lunar ecliptic limits, 
16 56', and 10 47', are in the ratio of 31 : 18 very nearly. 
This ratio gives, on the whole, a higher average proportion 
of solar eclipses to lunar than that given above. It must, 
however, be remembered that all the angles used in calcu- 
lating the limits are subject to gradual changes. Con- 
sequently the numbers of eclipses in that period aro subject to 
very gradual variation ; after a large number of Saroses have 
recurred, the order of eclipses in each will have changed, 



232 ASTRONOMY. 

*SECTION III. Occultations Places at which a Solar Eclipse 
is visible. 

299. Occultations. When the Moon's disc passes in front 
of a star or planet, the Moon is said to occult it. 

An occultation evidently takes place whenever the ap- 
parent angular distance of the Moon's centre from the star 
becomes less than the Moon's angular semi- diameter. As the 
apparent position of the Moon is affected by parallax, the cir- 
cumstances of an occultation are different at different 
places on the Earth's surface. 




FIG. 



Let m denote Moon's angular semi-diameter, p its horizontal 
parallax. In the figure, let E and M be the centres of the 
Earth and Moon, and let s C, sC' represent the parallel rays 
coming from a star, and grazing the Moon's disc. These 
rays cut the Earth's surface along a curve 00*, and it 
is evident that only to observers at points within this curve 
is the star hidden by the Moon's disc. Let EC, Es, EM, 
EC' cut the Earth's surface in c, x, m, c' ; the rays EC, EC' 
cut the Earth's surface in a small circle cc, whose angular 
radius mEc = MEC = m. Let d be the geocentric angular 
distance SEM between the Moon's centre and the star. 
Then the angle ECO = angle subtended by the Earth's 

radius EO at C ; 

= parallax of C when viewed from ; 

= ^sin COZ(\ 249); 

= p sin OEx (by parallels). 
But ECO = CEs ; 

= angle subtended by ex ; 

.% sin OEx = aD " Ic **'. 
P 



ECLIPSES. 233 

Hence we have the following construction for the curve 
separating those points on the Earth's surface at which the 
occultation is visible at a given instant from those at which 
the star is not occulted. Taking the sublunar point m as 
pole, describe a circle cc on the terrestrial globe, with the 
Moon's angular semi-diameter (m) as radius. Through the sub- 
stellar point x draw any great circle, cutting this small circle 
in any point c. Measure along it an arc c such that sin c 

is always the same multiple f ) of me. The locus of the 

points 0, thus determined, is the curve required. 

Half of the circle cc' consists of points under the advancing 
limb of the Moon; hence, over the portion of the curve 00' 
corresponding to this half -circle, the occultation is just 
beginning. At points on the other half of cc the Moon's 
limb is receding ; hence over the other portion of 0' the 
star is reappearing from behind the Moon's disc. 

Since the greatest and least values of ex in any position 
are d+m and d m, it is evident that the greatest value of d 
for which an occultation can take is when 

d m=p; d=m+p. 

300. Occnltation of a Planet. If s be a planet, the 
lines Es, Os can no longer be regarded as rigorously parallel; 
but the angle between them, Es 0, 

= angle subtended at s by the Earth's radius EO 
= parallactic correction at ( 248) 
= P sin ZOs ( 249) = P sin OEx very nearly. 
As before, EGO - p sin OEx. But ECO = EsO + CEs; 

.-. p sin OEx = P sin OEx + ex ; sin OEx = -^. 

pP 

With this exception, the construction is the same as for a 
star. 

If the planet be so large that we must take account of its 
angular diameter, the method of the next paragraph must be 
used. 



234 ASTRONOMY. 

301. Eclipse of the Sun. There is a total eclipse of the 
Sun, provided the Moon's disc completely covers the Sun's; 
this occurs if the Moon's angular semi-diameter (m) is larger 
than the Sun's (s), and the apparent angular distance 
between the Sun's and Moon's centres (as seen from any 
point at which the eclipse is visible) is less than m s. 
ilcnce, if the Moon's angular semi-diameter were reduced 
to m s, the Sun's centre would then be occulted. Hence 
the points 0, whose locus encloses the places from which 
the eclipse is visible, can be found as follows : 

With centre m the sublunar point, and angular radius 
m s, describe a circle. Through the subsolar point x draw 
any arc of a great circle xc, cutting the circle in 0, and 
take 0, on xc produced, such that 



xo 



p-p 

For an annular eclipse m < *, and the apparent angular 
distance between the centres is s m\ hence the same con- 
struction is followed, save that s m is the angular radius of 
the small circle first described. For a partial solar eclipse, 
the angular radius is s + m. 

When a planet has a sensible disc, the beginning of its 
occupation may be compared to a partial eclipse of the Sun ; 
and the planet is entirely occulted when the conditions are 
satisfied corresponding to those for a total eclipse. 

EXAMPLE. Supposing the centres of the Earth, Moon, and Sun to 
be in a straight line and the Moon's and Sun's semi-diameters to be 
exactly 17' and 16', to find the angular radii of the circles on the 
Earth over which the eclipse is total and partial respectively, taking 
the relative horizontal parallax as 57'. 

At those points at which the eclipse is total, the apparent angular 
distance between the centres, as displaced by parallax, must be not 
greater than 17' - 16', or 1'. Hence, since the centres are in a line, 
with the Earth's centre, the parallactic displacement must be not 
greater than 1'. Hence, if z be the Sun's zenith distance at the 
boundary, then 57' sin z = 1' ; .*. sin z = ?, or approximately cir- 
cular measure of z = -^j-. But a radian contains about 57 ; .*. -gV of 
a radian = 1 approx. Hence the eclipse is total over a circle of 
angular radius 1 about the sub-solar point. 

Similarly, the eclipse is partial if 57' sin z < 16' + 17', or 33', or 
sin < ff, or '58. From a table of natural sines, we find that 
sin- 1 '58 = 35^ roughly ; therefore the angular radius is 35 3 . 



ECLIPSES. 235 

EXAMPLES ON ECLIPSES GENERALLY. 

1. To find (roughly) the maximum duration of an eclipse of tho 
Moon, and the maximum duration of totality. 

From 291 we see that a lunar eclipse will continue as long as the 
Moon's angular distance from the line of centres of the Earth and 
Sun is less than 58', and the eclipse will continue total while the 
angular distance is less than 26'. Hence, the maximum duration of 
the eclipse is the time taken by the Moon to describe 2 x 58', or 116', 
and the maximum duration of totality is the time taken to describe 
2 x 26', or 52'. 

Now the Moon describes 360 (relative to the direction of the Sun) 
in the synodic month, 29 -r days. Therefore, the times taken to 
describe 116' and 52'' respectively are 

29^x116 , 29jj<52 d 
360x60 360x60 ay8> ,. 

i.e. 3h. 48m. and Ih. 42m., 

and these are the maximum durations of the eclipse and of totality. 
The eclipse of Nov. 15, 1891, lasted 3h. 28m., and was total for Ih. 23m. 

2. To calculate roughly the velocity with which the Moon's 
shadow travels over the Earth. (Sun's distance = 93,000,000 miles.) 

The radius of the Moon's orbit being about 240,000 miles, its cir- 
cumference is about 1,508,000 miles. Relative to tho line of centres, 
the Moon describes the circumference in a synodic month, i.e., about 
29 days. Hence its relative velocity is about 1,508,000 -f- 29, or 
51,000 miles per day, i.e., 2,100 miles per hour. If q 'denote the 
point where the middle of the shadow reaches the Earth (Fig. 88), 
and if the Earth's surface at q is perpendicular to Sq, wo have 

velocity of q : vel. of M = Sq : 8M 
= 93,000,000 : 93,000,000-240,000 - 1-0026 nearly. 
Hence the velocity of the shadow at q = vel. of M very nearly 

= 2,100 miles an hour. 

To find the velocity of the shadow relative to places on the Earth, 
we must subtract the velocity of the Earth's diurnal motion. This, 
at the Earth's equator, is about 1,040 miles an hour. Hence, if the 
Earth's surface and the shadow are moving in the same direction, 
the relative velocity is about 1,060 miles an hour. 

3. To find the maximum duration of totality of the eclipse of 
the example on page 234, neglecting the obliquity of the ecliptic. 

The angular radius of the shadow being 1, or about 69 miles, its 
diameter is 139 miles. The obliquity of the ecliptic being neglected, 
the eclipse is central at a point on the equator, and the shadow and 
the Earth are therefore moving in the same direction with relative 
velocity 1,060 miles an hour (by Question 2). The greatest duration 
of totality is the time taken by the shadow to travel over a distance 
equal to its diameter, i.e., 139 miles, and is therefore 139 x 60/1000 
minutes, i.e., 7'9 minutes (roughly). 

ASTKON. E 



236 ASTEONOMT. 

EXAMPLES. IX. 

1. If a total lunar eclipse occur at the summer solstice, and at 
the middle of the eclipse the Moon is seen in the zenith, find the 
latitude of the place of observation. 

2. If there is a total eclipse of the Moon on March. 21, will the 
year b.e favourable for observing the phenomenon of the Harvest 
Moon? 

3. Having given the dimensions and distances of the Sun and Moon, 
show how to find the diameter of the umbra where it meets the 
Earth's surface. 

4. Calculate (roughly) the totality of a solar eclipse, viewed from 
the Equator at the Equinox, supposing 

Moon's diameter 2,1GO miles, Sun's diameter 400 times Moon's ; 
Distance of Moon from Earth 222,000 miles ; 
Distance of Sun from Earth 92,000,000 miles. 

6. If 8 is the semi-diameter of the Sun, and p, P the horizontal 
parallaxes of the Sun and the Moon at the time of a lunar eclipse, show 
that to an observer on the Earth the angular radius of the Earth's 
shadow at the distance of the Moon is P + p S, and that of the 
penumbra P + p + S. Determine, also, the length of the shadow. 

6. If the distance of the Moon from the centre of the Earth is 
taken to be 60 times the Earth's radius, the angular diameter of the 
Sun to be half a degree, and the synodic period of the Sun and 
Moon to be 30 days, show that the greatest time which can be 
occupied by the centre of the Moon in passing through the umbra 
of the Earth's shadow is about three hours, and explain how this 
method might be employed to find th<? Sun's parallax. 

7. If the distance of the Moon were diminished to 30 times the 
Earth's radius, what would be the time occupied in passing through 
the shadow ? 

8. Determine what length of the axis of the Earth's shadow is 
absolutely dark, having given that the horizontal refraction is about 
35' j and account for the copper colour often seen on the Moon 
when eclipsed. 

9. What kind -of eclipse is most suitable for the determination of 
longitude, and why? 

10. What would be the greatest possible inclination of the plane 
of the Moon's orbit to the ecliptic, that there might be a partial 
eclipse at each conjunction? 

(The greatest distance of the Moon - 60 x Earth's radius.) 



ECLIPSES. 237 



EXAMINATION PAPER. IX. 

1. What is the cause of eclipses of the Sun, and of the Moon ? 
Why is a solar eclipse visible over so small a portion, and a lunar 
eclipse over so large a portion of the Earth ? 

2. Account for the phenomenon called a Lunar Eclipse. Show 
that it begins and ends at the same instant at all places from which 
it is visible. 

3. Explain briefly the manner in which a solar eclipse passes over 
the Earth. 

4. Explain clearly how an annular eclipse of the Sun is produced. 
Why are there no annular eclipses of the Moon ? Explain why 
solar eclipses are sometimes total and sometimes annular. 

5. Explain why, though there are, on the whole, more eclipses of 
the Sun than of the Moon, many more of the latter than of the 
former are visible at Greenwich. 

6. Define umbra and pemnnbra. Calculate the lengths of the 
cones of shadow (umbra) cast by the Earth and Moon, and find the 
breadth of the Earth's umbra at the distance of the Moon. 

7. Define and roughly calculate the solar and lunar ecliptic limits. 
What is the greatest number of lunar eclipses which can occur in a 
year ? What is the least number of solar eclipses which can occur 
in the same interval ? 

8. What is the Saros ? State its length, and why it has to be 
an exact multiple of the synodic period of the Moon and nearly 
a multiple of that of the node. 

9. Do occultations of a star by the Moon occur at the same 
instant at all observatories ? 

10. Show how to find at what point (if any) of the Earth's 
surface a solar eclipse will be central. 



CHAPTER X. 



THE PLANETS. 
SECTION I. General Outline of the Solar System. 

302. The name planet, or " wanderer," was applied by 
the Greeks to designate all those celestial bodies, except 
comets and meteors, which changed their position relative to 
the stars, independently of the diurnal motion ; these included 
the Sun and Moon. At present, however, only those bodies 
are called planets which move in orbits about the Sun. The 
Sun itself is considered to be a star, while the Earth is 
classed among the planets, and the Moon, which follows the 
Earth in its annual path, and has an orbital motion about 
the Earth, is described, along with similar bodies which 
revolve about other planets, as a satellite or secondary. 

303. The Sun, 0, is distinguished by its immense size and 
mass. It forms the centre of the solar system, for, in spite 
of the great distances of some of the furthest planets, the 
centre of mass of the whole system always lies very near the 
Sun. The Sun resembles the other fixed stars in being self- 
luminous. 

Its diameter is 110 times that of the Earth, or nearly 
twice as great as the diameter of the Moon's orbit about the 
Earth. 

Erom observing the apparent motion of the spots or cavities 
which are usually seen on the Sun's disc, it is inferred that 
the Sun rotates on its axis in the sidereal period of about 
25 days. 



THE PLANETS. 239 

304. Bode's Law. The distances of the planets from the 
Sun have been observed to be approximately connected by a 
remarkable law known as Bode's Law. This law is purely 
empirical, that is, it is merely a result of observation, and it 
has not as yet been proved to be a consequence of any known 
physical principle. Moreover, it is only roughly true, giving, 
as it does, a result far too great for the furthest planet 
Neptune. 

The law is given by the following rule : Write down the 
series of numbers 

0, 3, 6, 12, 24, 48, 96, 192, 384, 
each number (after the second) being double the previous one. 
Now add 4 to every term ; thus we obtain 

4, 7, 10, 16, 28, 52, 100, 196, 388. 
These numbers represent fairly closely the relative distances 
of the various planets from the Sun, the distance of the Earth 
(the third in the series) being taken as 10. 

The planets all revolve round the Sun in the same direction 
as the Earth. Their motion is, therefore, direct. 

305. Mercury, > ^ s the planet nearest the Sun, its dis- 
tance on the above scale being represented by 4. It is 
characterized by its small size, the great eccentricity of its 
elliptical orbit, amounting to about 3-, and the great inclina- 
tion of the orbit to the ecliptic, namely, about 7. The 
sidereal period of revolution round the Sun is about 88 of our 
days. 

Thus, Mercury's greatest and least distances from the Sun 
ore in the ratio of 1+i : 1 i (cf. 149), 

or 3:2. 

Professor Schiaparelli, of Milan, has found that Mercury 
rotates on its axis once in a sidereal period of revolution ; 
consequently it always turns nearly the same face to the 
Sun, like the Moon does to the Earth ( 276). 

Owing, however, to the great 'eccentricity of the orbit, the 
" libration in longitude " is much greater than that of the 
Moon, amounting to 47. Consequently, rather over one 
quarter of the whole surface is turned alternately towards 
and away from the Sun, three-eighths is always illuminated, 
and three-eighths is always dork. 



240 ASTKONOMY. 

306. Venus, ? , is the next planet, its mean distance 
from the Sun being represented by about 7 (really 7-2). Its 
orbit is very nearly circular, arid is inclined to the ecliptic at 
an angle of about 3 23'. 

Yenus revolves about the Sun in a period of 224 days. 

307. The Earth, , comesnext, its mean distance being re- 
presented by 10, audits orbit very nearly circular (eccentricity 
^i_). Its period of revolution in the ecliptic is 365|- days, 
and its period of rotation is a sidereal day, or 23h. 56m. mean 
time. It is the nearest planet to the Sun having a satellite 
(the Moon, ([ ), which revolves about it in 27|- days. 

308. Mars, <?, is at a mean distance represented roughly 
by 16, or more accurately by 15 '2. Its orbit is inclined at 
less than 2 to the ecliptic, and is an ellipse of eccentricity 
about T a T . It revolves about the Sun in a sidereal period of 
about 686 days, and rotates on its axis in about 24h. 37m. 

Mars has two very small satellites, which revolve about it 
in the periods 7J and 30^ hours, roughly. The appearance 
which would be presented by the inner satellite, if observed 
from Mars, is rather interesting. As it revolves much faster 
than Mars, it would be seen to rise in the west and set in the 
cast twice during the night. The outer satellite would appear 
to revolve slowly in the opposite direction from east to west. 
The inner satellite is eclipsed often at opposition, and would 
appear to transit the Sun's disc often at conjunction. 

309. The Asteroids. The next conspicuous planet, 
Jupiter, is at a distance represented by 52 ; but, according 
to Bode's law, there should be a planet at the distance 28. It 
was for a long time thought that no planet existed at this 
distance, but the gap was filled, at the beginning of the cen- 
tury, by the discovery of a number of small planets, to which 
the name of Asteroids, or Minor Planets, was given. 
Since that time a few new asteroids have been discovered 
almost every year, the total number found up to October 15, 
1891, being 321. It is probable that this number will be 
very largely increased by stellar photography. 

The largest asteroid, Vesta, is just visible to the naked 
eye when in opposition ; and the length of its diameter is 



THE PLANETS. 24! 

between -1 and -2 of that of the Moon. Among the others 
Juno, Ceres, Pallas, and Astreea are the most con- 
spicuous telescopic objects. Many of the smaller asteroids 
are less than ten miles in diameter, and are probably simply 
masses of rock flying round and round the Sun. 

The periodic times of revolution of the asteroids vary con- 
siderably, but their average is about 1,600 days. The orbits 
are in many cases very oval, the eccentricity of one (Poly- 
hymnia) being over i, and they are often inclined at consider- 
able angles to the ecliptic, the inclination in the case of Pallas 
amounting to nearly 35, while that of Juno is 13. 

The planets outside the asteroid belt are distinguished from 
those hitherto described by their far greater dimensions and 
masses, and by their smaller densities. In this respect they 
resemble the Sun. They are also supposed to be at high 
temperatures, though not hot enough to emit light. 

310. Jupiter, I/, is at a mean distance almost exactly re- 
presented by 52. It revolves round the Sun in a period of 
twelve years, in an orbit nearly circular and inclined at only 
1J to the ecliptic. 

The diameter of Jupiter is about eleven times that of the 
Earth, and through a telescope the disc is seen to be encircled 
with a series of belts or streaks parallel to its equator. On 
account of their variability, these are supposed to be due to 
lelts of clouds in the atmosphere of the planet. 

Jupiter is now known to have five satellites. The four 
outer ones arc interesting as being the first celestial bodies 
discovered with the telescope by its inventor Galileo (A.D. 
1610). A fairly powerful opera glass will just show them. 
The outermost of all revolves in an ellipse of considerable 
eccentricity inclined to the ecliptic plane at about 8, its 
period being about lOd. 17h. The three next revolve in 
orbits nearly circular, and in the ecliptic, in periods of 7d. 4h., 
3d. 13h., and Id. 18jh. The fifth or innermost satellite 
has only just been discovered (1892) by Mr. Barnard with 
the great Lick telescope ; it revolves in a period of nearly 
12h., at a mean distance of 70,000 miles from the surface, or 
113,000 miles from the centre of Jupiter. Jupiter's satellites 
are frequently eclipsed by passing into the shadow cast by Jupi- 
ter, or occulted whenJupitercom.es between them and the Earth, 



242 ASTRONOMY. 

311.. Saturn, J? , is at a mean distance from the Sun of 95 J, 
taking the Earth's distance as 10. This is rather less than 
the distance given by Bode's Law. The periodic time of revo- 
lution is 29-^ years. The orbit is nearly circular, and inclined 
to the ecliptic at an angle of 2J. 

Saturn's rings are among the most wonderful objects 
revealed by the telescope. They appear to be three flat 
annular discs of extreme thinness, lying in a plane .inclined, 
to the ecliptic at an angle of about 28, and extcndingjto a 
distance rather greater than the radius of the planet ; the 
middle ring is by far the brightest, whilejthe inner ring is 
very_f aint. When the Earth is in the plane of the rings they 
are seen edgewise, and, owing to their very small thickness, 
they then become invisible except in the best telescopes. 

It is probable that the rings consist of a large number _oi 
small satellites or meteors. It is certain that they do not 
consist of a continuous ma?s of solid or liquid matter. The 
surface of the planet itself is encircled with belts similar to 
those on Jupiter. 

In addition to the rings, Saturn has at least eight satellites, 
all situated outside the rings. The seven nearest move in 
planes nearly coinciding with that of the rings, while the 
orbit of the eighth is inclined to it at an angle of 10. The 
sixth satellite is by far the largest, having a probable dia- 
meter not far short of that of the planet Mars. The seventh 
has been observed, like our moon, always to turn the same* 
side towards the planet. The distances of the satellites from 
Saturn range from 3 to 60 times the planet's semi-diameter, 
and the corresponding periods range from 22^ h. to 79 d. 

312. Uranus, IjJ, at mean distance 192, revolves in an 
approximately circular orbit, nearly coinciding with the 
ecliptic, in a period of 84 years. It was discovered in 1781 
by Sir William Herschel, who named it the Georgium Sidus 
in honour of the king. 

Uranus is attended by four satellites at least, and these 
possess the remarkable peculiarity of revolving, in a plant 1 
nearly perpendicular to the ecliptic and in a retrograde 
direction. In fact, the plane of their orbits makes an angle 
of 82 with the ecliptic. Their periods are 2|d., 4d., 8|d.. 
and 13|d. roughly. 



THE PLANETS. 



243 



313. Neptune, ^. The position of this planet was pre- 
dicted in 1846 almost simultaneously by Adams and Leverrier, 
from., the observed effects of its attraction on the orbital 
motion of Uranus. It was first actually seen by Galle, of 
Berlin, in September, 1846, very close to the position which 
had been computed beforehand. It has a mean distance 
300 (being considerably less than that which it would have 
according to Eode's Law), and it revolves in its orbit in about 
164 years. 

^Neptune has one satellite moving in a retrograde direction 
in a plane inclined to the ecliptic at about 35. 

The discovery of Neptune will be treated more fully in the 
chapter on Perturbations. 

314. Tabular View of the Solar System. For con- 
venient reference, the mean distances of the planets, measured 
in terms of the Earth's mean distance as the unit, and their 
periodic times, are given below, together with the inclina- 
tions and eccentricities of the orbits, and the numbers of 
their satellites. 



Name of 
Planet. 


Mean Dist. 
of Planet. 

of Earth.' 


Periodic Time. 


Inclination 
of Orbit. 


Eccen- 
tricity 
of 
Orbil. 


No. of 
Satel- 
lite* 


Mercury, $ 


0-38 


days = 
88 


= years 
0-24 


o / 

7 


206 




Venus, ? 


0-72 


224 


0-62 


3 23 


007 





Earth, 


1-00 


335 


100 





017 


1 


Mars, 3 


1-52 


687 


1-88 


1 51 


093 


2 


Ceres, (?) 


2-77 


1,681 


4-60 


10 37 


076 





Jupiter, "4 
Saturn, \i 
Uranus, fl 


5-20 
9-54 
19-18 


4,332 
10,759 
30,687 


11-86 
29-46 
84-02 


1 19 
2 30 
46 


048 
056 
046 


I 8 & 3 
I rings 
4 


Neptune, ttf 


30-05 


60,181 


164-78 


1 47 


009 


1 



244 ASTHONOMY. 

SECTION 11. Synodic and Sidereal Periods Description of 
Motion in Elongation of Planets as seen from the Earth 
Phases. 

315. Inferior and Superior Planets. Definitions. 

In describing the motions of the planets relative to the 
Earth, it is convenient to divide the planets into two classes, 
inferior and superior planets. 

An inferior planet is one which is nearer to the Sun than 
the Earth ; Mercury and Venus are the two inferior planets. 

A superior planet is one which is further from the Sun 
than the Earth : all the planets except Mercury and Yenus 
are superior. 

The angle of elongation is the difference between the 
geocentric (156) longitude of the planet and that of the 
Sun. It has the same meaning as in the case of the Moon 
(259). 

We shall now describe the changes in elongation of the 
inferior and superior planets, as seen from the Earth. It 
appears from the preceding section that 

(i.) The planets all revolve round the Sun in the same 
direction ; 

(ii.) The planets which are nearer the Sun travel at a 
greater speed than those which arc more remote. 

The second fact can be easily verified from comparing the 
distances and periods of the planets given in the previous 
section. Even if we take into account the fact that the more 
distant ones ha\c further to travel, we shall still find that 
they take longer to travel over the same distance. 

In order to further simplify the descriptions we shall 
assume that the planets all revolve uniformly in circles, about 
the Sun as centre, in the plane of the ecliptic. These 
assumptions are only roughly true, on account of the small 
eccentricities of the orbits and their small inclinations to 
the ecliptic ; hence our results will only agree roughly with 
observation. 

316. Changes in Elongation of an Inferior Planet. 

Let E be the Earth, Pan inferior planet moving in the orbit 
AUBU' about S the Sun. Since $F revolves more rapidly 
about S than SE, the motion of V relative to .#, as it would 
appear from S, is direct, 



THE PLANETS. 



245 



S V separates from SJS at a rate which is the difference 
of the rates at which J, V revolve in their orbits. The 
changes in the positions of the planet relative to the Sun 
are therefore the same as if E were at rest and V re- 
volved with an angular velocity equal to the excess of the 
angular velocity of the planet oVer that of the Earth. 




Let the line E& meet the orbit of V in A and B. "When 
V is at A or B it has the same longitude as S, and if the 
planet actually moved in the ecliptic it would be in front of 
the Sun at A, behind the Sun at B. In reality, owing 
to the inclination of the orbits, this but rarely happens. 

At A, the planet is said to be in inferior conjunction 
with the Sun ; it has the same longitude and is nearer the 
Earth. At B the planet is said to be in superior conjunc- 
tion with the Sun ; it has the same longitude but is further 
away. If we consider the appearances which would be pre- 
sented on the Sun, the planet is in " heliocentric conjunction " 
with the Earth at A and in " heliocentric opposition " at B. 

After inferior conjunction at A, the pianet is seen on the 
westward side of the Sun, as at V r The elongation SEV 
gradually increases till the planet reaches a point 7 such 
that EU\$ a tangent to the orbit. The planet is then at its 
greatest elongation, the angle SEU being a maximum. 

Subsequently, as at V tt the elongation diminishes, and the 
planet approaches the Sun, until superior conjunction occurs, 
as at B. The planet then separates from the Sun, reappear- 
ing on the opposite (eastern) side, as at F~ 3 , attains its maxi- 
mum elongation at Z7 7 , and finally comes round again to 
inferior conjunction at A. 



246 AsriiONOMi. 

The time between two consecutive conjunctions of the 
same kind (superior or inferior) is called the synodic period 
of the planet (cf. 259), and is the period in which 
SV separates from SB through 360. 

317. To find (roughly) the Ratio of the Distance 
from the Sun of an Inferior Planet to that of the 
Earth, it is only necessary to observe the planet's greatest 
elongation. F.or if 7", E (Fig- 95) represent the planet and 
Eartli at the instant of greatest elongation, the angle EUS is 
a right angle, and therefore 




that is, 

Distance of planet . . 

Distance of Earth = S1UC f greatest elo ** at "> n - 

This method is, however, much modified by the fact that 
the real orbits are not circles, but ellipses. 

EXAMPLE 1. Given that the greatest elongation of Yenus is 45, 
find its distance from the Sun, that of the Earth being 93,000,000 
miles. 

Here distance of Venus = 93,000,000 sin 45 = 93,000,000 x V^ 
= 93,000,000 x -70711 = 65,760,000 miles. 

EXAMPLE 2. Taking the Earth's distance as unity, to find the 
distance of Mercury, having given that Mercury's greatest elonga- 
tion is 22i. 



The distance of Mercury = Ixsin 22 = ^{1(1 -cos 45)} 
= ^(2-^/2) = -38268. 




THE PLAN 



318. Changes in Elongation of a Superior Planet. 

Let us now compare the apparent motion of the superior planet 
7with that of Sun. Since it revolves about the Sun in the 
same direction as the Earth does, but more slowly, the line 
SJwill move, relative to SE, in the opposite or retrograde 
direction. Hence, in considering the changes in the position 
of the planet relative to the Sun, we may regard SE as a 
fixed line, and J must then revolve about 8 in the circle 
ARBTwifh a retrograde motion, i.e., in the same direction 
as the hands of a watch.* 

At A the planet is in opposition with the Sun, and its 
elongation is 180. At B it is in conjunction, and its 
elongation is 0. If, however, we were to refer the directions 
of the Earth and planet to the Sun, the planet would be in 
heliocentric conjunction with the Earth at A, and in helio- 
centric opposition at B. 

The planet is nearest the Earth at A, and since its orbital 
Telocity is constant, its relative angular velocity is then 
greatest, and the elongation SEJ is decreasing at its most 
rapid rate. As the planet moves round from opposition A to 




FIG. 96. 

conjunction B, the elongation SEJ decreases continuously 
from 180 to 0. 

At R the elongation is 90, and the planet is said to be in 
quadrature. 

* As a simple illustration, both the hour and minute hands of a 
watoh revolve in the same directions, but the minute hand goe* 
faster and leaves the hour hand behind. Hence the hour hand 
separates from the minute hand in the opposite direction to that in 
which both tare moving. 



248 



ASTRONOMY". 



At conjunction, J?, the elongation is 0; and we may also 
consider it to be 360. As the planet revolves from B to A, 
the elongation (measured round in the direction BRA} de- 
creases from 360 to 180. 

At T the elongation is 270, and the planet is again seid 
to be in quadrature. 

At A the elongation is again 180, the planet being once 
more in opposition. After this the elongation decreases from 
180 to as before, as the planet's relative position changes 
from A through R to B. 

The cycle of changes recurs in the synodic period, i.e., 
the period between two successive conjunctions or oppositions. 
"We see that the elongation decreases continually from 360 
to as the planet revolves from conjunction round to con- 
junction, and there is no greatest elongation. 




FIG. 97. 

319. To compare (roughly) the Distance of a 
Superior Planet with that of the Earth. Here there 
is no greatest elongation, and therefore we must resort to 
another method. 

Let the planet's elongation SEJ (Fig. 97) be observed at 
any instant, the interval of time which has elapsed since the 
planet was in opposition being also observed. Let this 
interval be , and let 8 denote the length of the planet's 
synodic period. Then, in time S the angle JSE increases 
from to 360 j therefore, if we assume the change to take 
place uniformly, the angle JSE at time t after conjunction 
is = 860 x tlS 



THE PLANETS. 249 

Hence, JSE is known. Also JJES has been observed, and 
SJE (= 18QJJSSJSE) is therefore also known. 

Therefore we have, by plane trigonometry, 

Distance of Planet _ SJ _ sin SEJ 
Distance of Earth " 8E sin SJE 

which determines the ratio of the distances required. 

This method is also applicable to the inferior planets. It 
is, however, not exact, owing to the fact that the planetary 
motions are not really uniform (see 327). 

*320. It U not necessary to observe the instant of conjunction or 
opposition. If 8 is known, t\ro observations of the elongation and 
the elapsed time are sufficient to determine the ratio of the distances. 
The requisite formulea are more complicated, but they only involve 
plane trigonometry. We, therefore, leave their investigation as an 
exercise to the more advanced student. 

EXAMPLE. To calculate the distance of Saturn in terms of that 
of the Earth, having given that 94 days after opposition the elonga- 
tion of Saturn was 84 17', and that the synodic period is 376 days. 
Given also tan 5 43' = !. 

Let the Sun, Earth, and Saturn be denoted by fif, E, J. In 376 
days / J8E increases from to 360. 

.*. in 94 days after opposition L JSE = 90 j 

also, by hypothesis, L JES = 84 17'. 
Distance of Saturn = SJ = ^ flj?/ = ^ 840 1? ,, 
Distance of Earth SE 

-r cot 6 43' = ~ = 10. 

Therefore the distance of Saturn, as calculated from the given 
data is 10 times that of the Earth. 



321. The synodic period of an inferior planet may 

be found very readily by determining the time between two 
transits of the planet across the Sun's disc and counting the 
number of revolutions in the interval. 

For a superior planet this is not possible, and we 
must, instead, find the interval between two epochs at which 
the planet has the same elongation. 



250 ASTRONOMY. 

322. Relations between the Synodic and Sidereal 
Periods. The relation between the synodic and sidereal 
periods is almost exactly the same as in the case of the Moon, 
the only difference being that the planets revolve about the 
Sun and not about the Earth. 

The sidereal period of a planet is the time of the planet's 
revolution in its orbit about the Sun relative to the stars. 
The synodic period is the interval between two conjunc- 
tions with the Earth relative to the Sun. It is the time in 
which the planet makes one whole revolution as compared 
with the line joining the Earth to the Sun. 

Let S be the planet's synodic period, 

P its sidereal period, 

Yihc length of a year, that is, the Earth's sidereal period, 
all the periods being supposed measured in days. 

Then, in one clay, 

the angle described by the planet about the Sun 360/P, 
the angle described by the Earth = 360/F, 

and the angle through which their heliocentric 

directions have separated = 860//S. 

If the planet be inferior, it revolves more rapidly 
than the Earth, and 360/ represents the angle gained by 
the planet in one day. 

360 360 360 



If the planet be superior, it revolves more slowly 
than the Earth, and 360/ is the angle gained by iheJZarth 
i-i one day. 

360 _ 360 360 
~^~ ~Y~ ~ ; 

or _i=i-JL. 
3 1 P 

From these relations, the sidereal period can be found if 
the synodic period is known, and vice vtrsd. 



THE PLANETS. 



323. Phases of the Planets. As the planets derive 
their light from the Sun, they must, like the Moon, pass 
through different phases depending on the proportion of their 
illuminated surface which is turned towards the Earth. 

Phases of an Inferior Planet. An inferior planet V 
will evidently be new at inferior conjunction A, dichotomized 
like the Moon at its third "quarter at greatest elongation 7", 
full at superior conjunction B, dichotomised like the Moon at 
first quarter when it again comes to greatest elongation at 
IT. Thus, like the Moon, it will undergo all the possible 
different phases in the course of a synodic revolution. 

There is, however, one important difference. As the 
planet revolves from A to B its distance from the Earth 
increases, and its angular diameter therefore decreases. Thus 
the planet appears largest when new and smallest when full, 
and the variations in the planet's brightness due to the differ- 
ences of phase arc, to a great extent, counterbalanced by the 
changes in the planet's distance. For this reason, Venus 
alters very little in its brightness (as seen by the naked eye) 
during the course of its synodical revolution. 





FIG. 98. 

The phase is determined by the angle 8 VE, and this is 
the angle of elongation of the Earth as it would appear from 
the planet. The illuminated portion of the visible surface 
of the planet at V is proportional to 180-F.#, and the 
proportion of the apparent area of the disc which is illumi- 
nated varies as 1 + cos S VE or 2 cos 2 \ S VE. ( Cf. 263). 

The phases of Venus are easily seen through a telescope. 



ASTROX. 



252 ASTRONOMY. 



324. Phases of a Superior Planet. For a superior 
planet J the angle SJE never exceeds a certain value. It is 
greatest when SJEJ = 90, being then the greatest elongation 
of the Earth as it would appear from the planet. Hence tho 
planet is always nearly full, being only slightly gibbous, and 
the phase is most marked at quadrature. 




FIG. 90. 

The gibbosity of Mars, though small, is readily visible at 
quadrature, about one-eighth of the planet's disc being 
obscured. The other superior planets are, however, at a 
distance from the Sun so much greater than that of the Earth 
that they always appear very approximately full. 

325. The "Phases" of Saturn's Kings are due to an entirely 
different cause. The plane of the rings, like the plane of the Earth's 
equator, is fixed indirection, and inclined to the ecliptic at an angle 
of about 28. Hence, during the course of the planet's sidereal 
revolution, the Sun passes alternately to the north and south side? 
of the rings (just as in the phenomena of the seasons on our Earth, 
the Sun is alternately N. and S. of the equator). The Earth, which, 
relatively to Saturn, is a small distance from the Sun, also passes 
alterrately to the north and south sides of the rings, and we see the 
rings first on one side and then on the other. At the instant of 
transition the rings are seen edgewise, and are almost invisible. 

Unless Saturn is in opposition at this instant, the Sun and Earth, 
do not cross the plane of the rings simultaneously, and between 
their passages there is a B'.iort interval during which the Sun and 
Earth are on opposite sides of the plane; and the unilluminaled 
side of the rings is turned towards the Earth. The last " dis- 
appearances" of the rings occurred in Sept., 1891 May, 1892, but 
they occur twice in each sidereal period, or once about every 15 years. 
Other interesting appearances are presented by the shadows 
thrown by the planet on the rings and by the rings o^ *he planet. 



THE PLANETS. 253 

SECTION III. Kepler's Laws of Planetary Motion. 

326. Kepler's Three Laws. We have already seen 
that the orbits of most of the planets are nearly circular, their 
-distances from the Sun being nearly constant and their 
motions being nearly uniform. A far closer approximation 
to the truth is the hypothesis held for a long time by Tycho 
Brahe and other astronomers, namely, that each planet re- 
solved in a circle whose centre was at a small distance from 
the Sun, and described equal angles in equal intervals of 
time about a point found by drawing a straight line from the 
Sun's centre to the centre of the circle and producing it for 
nn equal distance beyond the latter point. 

The true laws which govern the motion of the planets were 
discovered by the Danish astronomer Kepler, in connection 
with his great work on the planet Mars (De Motibus Stellae 
Jfartis). After nine years' incessant labour the first and second 
of the following laws were discovered, and shortly afterwards 
the third. 

I. Every planet moves in an ellipse, with the Sun 
in one of the foci. 

II. The straight line drawn from the centre of 
the Sun to the centre of the planet (the planet's 
"radius vector") sweeps out equal areas in equal 
times. 

III. The squares of the periodic times of the 
several planets are proportional to the cubes of their 
mean distances from the Sun. 

These laws are known as Kepler's Three Laws. We 
Tiave already proved that the first two laws hold in the 
case of the Earth. The third law is also found to hold good 
for the Earth as well as the other planets, and this fact alone 
.affords strong evidence that the Earth is a planet 



254 ASTRONOMT. 

By the mean distance of a planet is meant the arith- 
metic mean between the planet's greatest and least dis- 
tances from the Sun. If p, a (Fig. 100) be the planet' im- 
positions at perihelion and aphelion (i.e., when nearest and 
furthest from the Sun respectively), the planet's mean 
distance = (Sp + Sa) = \pa = \ (major axis of ellipse 
described) (147). 

The periodic times are, of course, the sidereal periods. 
Hence the third law is a relation between the sidereal periods- 
and the major axes of the orbits. 




FIG. 100. 

327. Verification of Kepler's First and Second 
Laws. We will now roughly sketch the principle of the- 
methods by which Kepler determined the orbit of Mars, and 
thus proved his Eirst and Second Laws. A verification of 
the laws in the case of the Earth. has already been given, and 
we have shown ( 145) how to determine exactly the position 
of the Earth at any given time ; we may regard this, there- 
fore, as known. We may also suppose the length of the 
sidereal period of Mars to be known, for the average length of 
the synodic period may be found, as in 261, and the sidereal 
period may be deduced by the formula of 322. 

Let the direction of the planet be observed when it is at 
any point J/ in its orbit, the Earth's position being E. When 
the planet has returned again to Jf after a sidereal revolution, 
the Earth will not have returned to the same place in it* 



'J HE PLAXETS. 



255 



orbit but will be in a different position, say F Let no 
the planet's new direction FM be observed * 

the "rie k ^T g F hG ^\ m0ti< 2' We knmv ^> ^ nd 
tne angle -^A Prom th 



the ieF 0< 2' We nmv ^> ^ nd 

tne angle -^A Prom the observations of the two directions 
of J/ we know the angles SEM ^\ SFM. These "* G 
sufficient to enable us to solve the quadrilateral 




Via. 101. 



. . 

We can thus determine SM and the angle .2SJ/; whence 
the dis ance and d lre ction of M from the Suu am found? 
Similarly, any other position of Mars in its orbit can be found 
by two observations o the planet's sidereal period separated 
by the interval of the planet's sidereal revolution. In 
s way, by a senes of observations of Mars, extending ovei 

dS 



Clirccti011 
d daity ' 



* For simplicity we suppose Mars to move in thelcliptic plane 
The methods require some modification when the inclination of the 
orbits 1S taken into account, but the general principle is the same. 

r A 






256 ASTEOXOMY. 

b28. Verification of Kepler's Third Law. Kepler '& 
Third Law can le verified much more easily, especially if we 
make the approximate assumption that the planets revolve 
uniformly in circles about the Sun as centre. The sidereal 
periods of the different planets can be found by observing the 
average length of the synodic period (the actual length of 
any synodic period is not quite constant, owing to the planet 
not revolving with exactly uniform velocity) and applying 
the equations of 322. The distance of the planet may be 
compared with that of the Earth, either by observing the 
greatest elongation (317) in the case of an inferior planet, 
or by the method of 319. It is then easy to verify the 
relation between the mean distances and periodic times of 
the several planets. 

In the table of 314, the student will have little 
difficulty in verifying (especially if a table of logarithms- 
be employed) that the square of the ratio of the periodic 
time of the planet to the year (or periodic time of the 
Earth) is in every case equal to the cube of the ratio of 
the planet's mean distance to that of the Earth.* The data 
being only approximate, however, the law can only be veri- 
fied as approximately true, although it is in reality accurate. 

Owing to the importance of Kepler's Third Law, we append 
the following examples as illustrations. 

EXAMPLES. 

1. Given that the mean distance of Mars is 1'52 times that of the 
Earth, to find the sidereal period of Mars. 

Let T be the sidereal period of Mars in days. Then, by Kepler'a 
Third Law, 



/. T - 305^ x A/(3-511S) - 305 L x T874 = 684'5. 

Hence, from the given data, the period of Mars is T874 of a year, 
or 684-5 days. 

Had we taken the more accurate value of the relative distance, 
viz., l - 5237, we should have found for the period the correct value, 
namely, 687 days. 

* In other words, 2 log (period in years) = 3 log (distance in terms- 
of Earth's distance). 



THE PLANETS. 257 

2. The synodic period of Jupiter being 399 days, to find its distance 
from the Sun, having given that the Earth's mean distance is 92 
million miles. 

Let T be the sidereal period of Jupiter. Then, by 322, 

JL = _1 1 33f 

T 365^ 399 36o x 399' 



. 

= 11-82, or nearly 12 years. 

Let a be the distance of Jupiter in millions of miles. Then by 
Kepler's Third Law, 



I SL V = ( Y - 144 

\ 92 ; IT I 



/. a =92 x3/(l44) = 92x5-24 = 482; 
that is, Jupiter's distance is 482 millions of miles. 

By taking T - 11'82 and the Earth's distance as 92'04, we should 
have found the more accurate value 477'6 for Jupiter's distance in 
millions of miles. 

329. Satellites. The motions of the satellites about any 
planet are found to obey the same laws as those which Kepler 
investigated for the orbits of the planets. For example, the 
Moon's orbit about the Earth is an ellipse, and (except so far 
as affected by perturbations) satisfies both of Kepler's First 
and Second Laws. When a number of satellites are revolv- 
ing round a common primary (i.e., planet) as is the case with 
Jupiter, the squares of their periodic times are found, in 
every case, to be proportional to the cubes of their mean 
distances from the planet.* 

EXAMPLE. To compare (roughly) the mean distances of its two 
satellites from Mars. The periodic times are 30^ h. and7|h. respec- 
tively, and these are in the ratio (nearly) of 4 to 1. 

Hence the mean distances are as 4^ : 1, or %/W : 1. 

Now, 2-yi6 = s/128 = 5 very nearly (since 5 :J - 125). Hence, 
the mean distances are very nearly in the ratio of 5 to 2. 



* Of course the relation docs not hold between the periodic times 
and mean distances of satellites revolving round different planets, 
nor between those of a satellite and those of a planet. 



258 ASTRONOMY. 

SECTION IV. Motions Relative to Stars Stationary Points. 

330. Direct and Retrograde Motion. We have 

described ( 316-318) the motion of a planet relative to the 
Sun. In considering- its motion relative to the stars we must 
take account of the Earth's motion. 





FIG. 103. 

An inferior planet moves more swiftly than the Earth. 
Hence at inferior conjunction the line ^^(Fig. 102) joining 
them is moving in the direction of the hands of a watch. The 
planet therefore appears to move retrograde. At greatest elonga- 
tion ( 7, U') the planet's own motion is in the line joining it to 
the Earth, and hence produces no change in its direction ; 
but the Earth's direct motion causes the line EU or EU' ta 
turn about U QT U' with a rotation contrary to that of the 
hands of a watch; and therefore the apparent motion is 
direct. Over the whole portion UBU' of the relative orbit 
both the Earth's motion and the planet's combine to make the 
planet's apparent motion direct. There must, therefore, be 
two positions, M between A and U and N between U' and 
A, at which the motion is checked and reversed. At these 
two positions the planet is said to be stationary. 

A superior planet moves slower than the Earth ; hence at 
opposition the line EA (Fig. 103) joining them is turning in 
the direction of the hands of a watch. The planet therefore 
appears to move retrograde. At quadrature (2t, T) the Earth is 
moving along RET] hence its motion produces no change in 
the planet's direction. Hence the planet's direct motion about 



THE PLANETS. 



259 



the Sun makes its apparent motion also direct. In all parts 
of the arc RBT the orbital velocities of Earth and planet 
conspire to produce direct motion. Hence the planet is 
stationary at If, between A and H, and at JV between 



In both cases the longitude increases from J/ to JV and 
decreases from Nto J/; hence it is a maximum at JV and a 
minimum at M. After a complete synodic revolution the 
planet's elongation is the same as at the beginning, and the 
Sun's longitude has been increased ; therefore the planet's 
longitude has also increased. Hence the direct preponderates 
over the retrograde motion. 





FIG. 105. 



FIG. 104. 



331. Alternative explanation. We may also proceed 
;as follows. Let E, J represent two planets at heliocentric 
conjunction. Let E^ E^ E z , ..., J l} J^ 7 8 , ..., be their 
successive positions after a series of equal intervals. To find 
the apparent motion of /among the stars, as seen from J2, 
take any point E, and let E\, E'l, JS3, ... (Fig. 105) be 
parallel respectively to E^'E^ E./^ .... Then the 
points 1, 2, 3, ... represent 7's direction as seen from ^at a 
'series of equal intervals, starting from opposition. 



260 



ASTKONOMY. 



Again, if Jl, <72, <73 be taken parallel to 
(Pig. 108), the points 1, 2 now represent j's direction as 
seen from J. 

We observe from Figs. 107, 108 that the relative motion is 
retrograde from 1 to 2, and becomes direct near 3. At the 
instant at which this takes place, either planet must be 
stationary, relative to the other. Since J 4 E 4 is nearly a tan- 
gent to JS's orbit, E is near its greatest elongation, and J 
is near quadrature at the positions 4 ; hence, E appears 
stationary from /between inferior conjunction and greatest 
elongation ; and J appears stationary between opposition and 
quadrature. 




FIG. 107. 



We notice that <71, J2, . . . are parallel to E 1, E<2, but 

measured in opposite directions, showing that the motion of 
E relative to J is the same (direct, stationary, or retrograde) 
as that of / relative to E. 



THE PLANETS. 261 

332. Effects of Motion in Latitude. Hitherto we 
have supposed the planet to move in the ecliptic. When, 
however, the small inclination of the orbit to the ecliptic is 
taken into account, it is evident that the planet's latitude is 
subject to periodic fluctuations. 

The points of intersection of the planet's orbit with the 
ecliptic are (as in the case of the Moon) called the Nodes. 
Whenever the planet is at a node its latitude is zero; and 
this happens twice in every sidereal period of revolution. 

A planet is stationary when its longitude is a maximum or 
mininium, but unless its latitude should happen to be a 
maximum at the same time, the planet does not remain 
actually at rest. When the change from direct to retrograde 
motion, and vice versa, is combined with the variations in lati- 
tude, the effect is to make the planet describe a zigzag curve, 
sometimes containing one or two loops, called " loops of 
retrogression." This is readily verified by observation. 



t Ecliptic 




FIG. 109. 

Fig. 109 is an example of the path of Venus in the neigh- 
bourhood of its stationary points, the numbers representing 
its positions at a series of intervals of ten days. Here,, 
the planet is stationary close to the node JV, between 4 
and 5, and it describes a loop in the neighbourhood of the- 
stationary point near 9, where its motion changes from re- 
trograde to direct. 

The student will find it an instructive exercise to trace out 
the path of any planet in the neighbourhood of its retrograde 
motion, using the values of its decl. and R.A., at intervals of 
a few days, as tabulated in the Nautical or 
Almanack. 



262 



ASTRONOMY. 



333. To find the condition that two planets may be 
stationary as seen from one another, assuming the 
orbits circular and in one plane. Let P, Q be the 

positions of the planets at any instant ; P', Q' their position? 
after a very short interval of time. 

Then, if PQ and P'Q' are parallel, the direction of either 
planet, as seen from the other, is the same at the beginning 
and end of the interval ; that is, P is stationary as seen from 
Q, and Q is stationary as seen from P. 

Let u, v represent the orbital velocities of the planets P, Q ; 
a, b the radii $P, SQ respectively. 




FIG. 110. 

Draw P'J/, Q'N perpendicular to PQ. Then, in the 
stationary position, we must have P'M = Q'N. 

But PP', QQ', being the arcs described by the two planets 
in the same interval, are proportional to the velocities u, v. 
Therefore P'M, Q'N are proportional to the component 
velocities of the planets perpendicular to PQ. These com- 
ponent velocities must, therefore, be equal, and we have 

u sin P'PM= v sin Q'QN. 
"Whence, since P'P is perpendicular to SP and Q'Qto SQ, 

u cos SPQ = v cos SQN = vcos SQP (i.), 

and this is the condition that the planets may be stationary 
relative to one another. 



THE PLANETS. 263 

*334. To find the angle between the radii vectores in the station- 
ary position, and the period during which a planet's motion is 
retrograde. By projecting SQ, QP on SP, we have 

a = 6 cos PSQ + PQ cos SPQ. 
Similarly b = a cos PSQ + PQ cos SQP. 

.-. cos SPQ : cos SQP = a - b cos PSQ : b - a cos PSQ. 
Whence, by (i.), u (a-b cos PSQ) +v (b-acosPSQ) = 0; 



= "" '" . (ii.). 

ac -\-lni 

By means of Kepler's Third Law, we can express the ratio of u to 
v in terms of a and b. For if T l} To denote the periodic times, then. 
evidently uT v = 2-, vT 2 = 2.T& ; 

.-. u : v = aT 2 : Z-7',. 
But T, I T, = ^ ; bl ; 



Substituting in (ii.), we have 
cos PSQ = ^ 



[From this result it may be easily deduced that 

tan t PSQ = ( 1=? /SQ ) * 

\l + cosPSQ/ 

In the above investigation PSQ is .the angle through which SQ 1 
separates from SP between heliocentric conjunction and the station- 
ary point. Hence, since L PSQ increases from to 360 in the 
synodic period S, the time taken from conjunction to the stationary 

- 4. /P-SO 

point =Sx ^T' 

If L PSQi = L PSQ, there is another stationary point before con- 
junction, when the planets are in the relative positions P, Q. Hence, 
the interval between the two stationary positions is twice the time 
taken by the planets to separate through /PSQ, and is therefore 



This represents the interval during which the motion of either 
planet, as seen from the other, is retrograde. During the remainder 
of the synodic period the motion is direct, and the time of direct 
motion is therefore 



264 ASTROXOMY. 

SECTION Y. Axial Rotations of Sun and Planets. 

335. The Period of Eotation of the Sun can be found by 
observing the passage of sunspots across the disc. These spots, by 
the way, are very easily exhibited with any small telescope by 
focussing an image of the Sun on to a piece of white paper placed 
.a few inches in front of the eye-glass for to look straight at the 
Sun would cause blindness. As the Sun's axis of rotation is nearly 
perpendicular to the ecliptic, the rotation of the spots is seen in 
perspective, and makes them appear to move nearly in straight 
lines across the disc. From this observed apparent motion (as 
projected on the celestial sphere in a manner similar to that 
explained in 263) their actual motion in circles about the Sun's 
axis is readily determined. For example, if a spot move^ from the 
-centre of the disc to the middle point of its radius, we may readily 
see that the angle turned through = sin" 1 -| = 3C. 

The spots are observed to return to the same position in about 
.27| days, and this is their synodic period of rotation relative to the 
Earth. Call it 8, and let T be the time of a sidereal rotation, T the 
length of the year. Then, as in the case of an inferior planet 
( 322), we may show that 

J_ = 1 j_ m 1 = 1 t 1 

8 ~* T - Y ' T ~ 27i "*" 365^ ; 

whence the true period of rotation T = 25| days (roughly). 

It has been observed that spots near the Sun's equator rotate 
rather faster than those near the poles. This proves the Sun's surface 
-to be in a fluid condition, for no rigid body could rotate in this way. 

336. Periods of Rotation of Planets. The rotation period of a 
.superior planet is easily found by observing the motions of the 
markings across its di-c near opposition, allowance being made for 
the motions of the Earth and planet. The surface of Mars has well- 
-defined markings, which give the period 24h. 37m. The principal 
mark on Jupiter is a great red spot amid his southern belts, which 
rotates in the period of 9h. 56m. Saturn rotates in lOh. 14m. 

For an inferior planet, the period is more difficult to observe. 
There is still some uncertainty as to whether Venus rotates in about 
;23h. 21m., or whether, like Mercury, it always turns the same face 
to the Sun. There are no well-defined markings, and, as the 
greatest elongation is only 45, Venus can only be seen for part of 
the night as an evening or morning star, and in the most favourable 
positions only a portion of the disc is illuminated. Moreover, 
refraction, modified by air-currents, prevents the planet from being 
seen distinctly when near the horizon. If the same markings are 
:Been on the disc of a planet on consecutive nights, they may either 
hare remained turned towards the Earth, or they may have rotated 
through 360 during the day ; hence the difficulty of deciding between 
the two alternative hypotheses. Before the researches of Schiapa- 
j-elli ( 305), it was believed that Mercury also rotated in about 24h. 



THE PLANETS. 265 



EXAMPLES. X. 

1. The Earth revolves round the Sun in 365'25 days, and Venus 
in 224'7 days. Find the time between two successive conjunctions 
of Venus. 

2. If Venus and the Sun rise in succession at the same point of 
the horizon on the 1st of June, determine roughly Venus' elongation. 

3. Find tbe ratio of the apparent areas of the illuminated portions 
of the disc of Venus when dichotomized and when full, taking 
Venus' distance from the Sun to be T 8 T of that of the Earth. 

4. Mars rotates on his axis once in 24 hours, and the periods of 
the sidereal revolutions of his two satellites are 1\ hours and 3O 
hours respectively. Find the time between consecutive transits 
over the meridian of any place on Mars of the two satellites 
respectively. 

5. A small satellite is eclipsed at every opposition. Find an 
. expression for the greatest inclination which its orbit can have to 

the plane of the ecliptic. 

6. If the periodic time of Saturn be 30 years, and the mean dis- 
tance of Neptune 2,760 millions of miles, find (roughly) the mean 
distance of Saturn and the periodic time of Neptune. (Earth's mean 
distance is 92 millions of miles.) 

7. If the synodic period of revolution of an inferior planet were a. 
year, what would be its sidereal period, and what would be its mean 
distance from the Sun according to Kepler's Third Law ? 

8. Jupiter's solar distance is 5'2 times the Earth's solar distance ' t 
find the length of time between two conjunctions of the Earth and. 
Jupiter. 

9. Saturn's mean distance from the Sun is nine times the Earth's 
mean distance. Find how long the motion is retrograde, having 
given cos" 1 \ = 65. 

10. Show that if the planets further from the Sun were to move 
-with greater velocity in their orbits than the nearer ones, there 
would be no stationary points, the relative motion among the stars 
"being always direct. AVhat would be the corresponding phenomenon, 
if the velocities of two planets were equal ? 



266 ASTRONOMY. 



EXAMINATION PAPER. X. 

1. Explain the apparent motion of a superior planet. Illustrate* 
by figures. 

2. Describe the apparent course among the stars of an inferior 
planet as seen from the Earth, and the changes in appearance which 
the planet undergoes. 

3. Define the sidereal and synodic period of a superior or inferior 
planet, and find the relation between them. Calculate the synodic 
period of a superior planet whose period of revolution is thirty 
years. 

4. How is it that Venus alters so little in apparent magnitude (as- 
seen by the naked eye) in her journey round the Sun ? Why does- 
not Jupiter exhibit any perceptible phases ? 

5. State Bode's Law connecting the mean distances of the various; 
planets from the Sun. 

6. Prove that the time of most rapid approach of an inferior 
planet to the Earth is when its elongation is greatest, and that the-' 
velocity of approach is then that under which it would describe its- 
orbit in the synodic period of the Earth and the planet. Give the- 
corresponding results for a superior planet. (The orbits are to be- 
taken circular and in the same plane.) 

7. AVhat is meant by stationary points in the apparent motion of a 
planet ? Prove that, if a planet Q is stationary as seen from P> 
then P will be stationary as seen from Q. 

8. State Kepler's Three Laws, and, assuming the orbits of the- 
Earth and Venus to be circular, show how the Third Law might be 
verified by observations of the greatest elongation and synodic 
period of Venus. 

9. Find the periods during which Venus is an evening star and a 
morning star respectively, being given that the mean distance of 
Venus from the Sun is '72 of that of the Earth. 

10. Having given that there will be a full Moon on the 5th of June, 
that Mercury and Venus are both evening stars near their greatest 
elongations, that Mars changed from an evening to a morning star- 
about the vernal equinox, and that Jupiter was in opposition to the 
Sun on April 21st, draw a figure of the configuration of these 
heavenly bodies on May 1st. (All these bodies may be supposed to 
move in one plane.) 



CHAPTER XI. 



THE DISTANCES OF THE SUN AND STARS. 

SECTION I. Introduction Determination of the Surfs Parallax 
by Observations of a Superior Planet at Opposition* 

337. In Chapter VIII. , Section I., we explained the nature 
of the correction known as parallax, and showed how to find 
the distance of a celestial body from the Earth in terms of its 
parallax. We also described two methods of finding the 
parallax of the Moon or of a planet in opposition the first 
by meridian observations at two stations, one in the northern 
and the other in the southern hemisphere ( 252) ; the second 
by micrometric observations made at a single observatory 
shortly after the time of rising and shortly before the time 
of setting of the planet or observed body ( 254). 

In both methods the position of the body is compared with 
that of neighbouring stars. This is impossible in the case of 
the Sun, for the intensity of the Sun's rays necessitates the 
use of darkened glasses in observations of the Sun, and these 
render all near stars invisible. 

Of course the star could theoretically be dispensed with in the 
method of 252, but only (as there explained) at a great sacrifice 
of accuracy ; and if a star is used which crosses the meridian at 
night, the temperature of the air has changed considerably, and the 
corrections for refraction are therefore quite different, besides 
which other errors are introduced by the change of temperature 
of the instrument. 

* The student will find it of great advantag3 to revise Section I. 
of Chapter VIII. before commencing the present Section. 
A.STKON". T 



268 ASTRONOMY. 

In 264 we described a method, due to Aristarchus, in 
which the ratio of the Sun's to the Moon's distance was 
determined by observing the Moon's elongation when dicho- 
tomized, but this method was rejected, owing to the irregular 
boundary of the illuminated part of the disc, and the con- 
sequent impossibility of observing the instant of dichotomy. 

338. Classification of Methods. The principal prac- 
ticable methods of finding the Sun's distance may be con- 
veniently classified as follows : 

A. Geometrical Methods. 

(1) By observations of the parallax of a superior planet at 

opposition (Section I.). 

(2) By observations of a transit of the inferior planet 

Venus (Section II.). 

B. Optical Methods (Section IV.). 

(3) By the eclipses of Jupiter's satellites (Roemer's Method). 

(4) By the aberration of light. 

C. Gravitational Methods (Chapter XIV., Section IV.). 

(5) By perturbations of Venus or Mars. 

(6) By lunar and solar inequalities. 

339. To find the Sun's Parallax by Observation of 
the Parallax of Mars. By observing the parallax of 
Mars when in opposition, the Sun's parallax can readily be 
found. For the observed parallax determines the distance of 
Mars from the Earth, and this is the difference of the dis- 
tances of the Sun from the Earth and Mars respectively. 
The ratio of their mean distances may be found, if we assume 
Kepler's Third Law ( 326), by comparing the sidereal period 
of Mars with the sidereal year, and is therefore known. 
Hence the distance of either planet from the Sun may readily 
be found, and the Sun's parallax thus determined. 

The parallax of Mars in opposition may be observed by 
cither of the methods described in Chapter VIII., Section I. 
The method of 252 (by meridian observations at two 
stations) was employed by E. J. Stone in 1865. The observa- 
tions were made at Greenwich and at the Cape, and the Sun's 
parallax was computed as 8 -943". The method of 254 (by 
observations at a single observatory) was employed by Gill 
at Ascension Island in 1879, and the result was 8-783", 



THE DISTANCES OF THE SUN AND STAES. 269 

EXAMPLE. 

If the parallax of Mars when in opposition be 14", to find the 
Sun's parallax, assuming the distances of the Sun from the Earth 
and Mars to be in the ratio of 10 : 16. 

The distance of the Earth from Mars in opposition is the difference 
of the Sun's distances from the two planets. Hence 

Distance of Earth from Mars I Distance of Earth from Sun 

= 16 - 10 : 10 = 3 : 5. 

But the parallax of a body is inversely proportional to its dis- 
tance ( 250). 

.'. Parallax of Sun : Parallax of Mars = 3:5; 

.'. Sun's parallax = 8 * 14// = 8'4". 
5 

*340. Effect of Eccentricities of Orbits. Owing to the eccen- 
tricities of the orbits of the Earth and Mars, their distances from the 
Sun when in opposition will not in general be equal to their mean 
distances, and therefore their ratio will differ from that given by 
Kepler's Third Law. But, by the method of 145, the Earth's dis- 
tance at any time may be compared with its mean distance, and 
similarly, since the eccentricity of the orbit of Mars and the position 
of its apse line are known, it is easy to determine the ratio of Mars' 
distance at opposition to its mean distance, and thus to compare its 
distance with that of the Earth. 

341. Sun's Parallax by Observations on the Aste- 
roids and on Venus. The Sun's parallax may also be 
found by observing the parallax of one of the asteroids when 
in opposition, the method being identical with that employed 
in the case of Mars. In this way Galle, by meridian obser- 
vations of the parallax of Flora at opposition in 1873, com- 
puted the Sun's parallax at 8'873", and Lindsay and Gill, by 
observing the parallax of Juno in 1877, found the value 
8-765". 

The next planet, Jupiter, is too distant to be utilized in 
this way. Its parallax at opposition is less than a quarter of 
the Sun's parallax, and is too small to be observed with 
sufficient accuracy. 

The Sun's parallax might also be found by an observation of 
Venus near its greatest elongation. The ratio of its distance 
to the Sun's might be calculated and its parallax found by the 
method of 252, and that of the Sun deduced. The method 
of 254 could not be employed, because one of the observa- 
tions would have to be made in full sunshine. 



270 ASTLONOMT. 



EXAMPLES. 

1. Having given that the greatest possible parallax of Mars when in 
opposition is 21'OS", to find the Sun's mean parallax, the eccentri- 
cities of the orbits of the Earth and Mars being ^ and T y respec- 
tively, and the periodic time of Mars being 1'88 of a year. 

The parallax of Mars is greatest when Mars is nearest the Earth ; 
hence the greatest possible value occurs when, at opposition, Mars is 
in perihelion and the Earth is at aphelion. 

Let r, r' denote the mean distances of the Earth and Mars from 
the Sun respectively. By Kepler's Third Law we have 
r' 3 _ (1-88) 2 . . r' n .J , ~ 9q 

;. 3 - -p > - = 

(The calculation is most easily performed with a table of logarithms.) 

But since the Earth is in aphelion, its distance from the Sun at 
the time of observation is greater than its mean distance by Jj, 
and is therefore 

= r (1 + e^) = 1-017 r. 
Also the distance of Mars from the Sun at perihelion 

= r'(l-TV) = (l-iV)x 1-523 r 

= (1-523- -090) r- l'433r. 

Hence the least distance of Mars from the Earth at opposition 
= -416 r. 

Therefore, since r is the Sun's mean distance from the Earth, we 
have 

Observed parallax of Mars : mean parallax of Sun = 1 : '416; 

/. Sun's mean parallax = 21-08" x -416 = 8'77". 

2. To find the Earth's moan distance from the Sun, and its dis- 
tances at perihelion and aphelion, taking the Sun's parallax as 8"79". 

If a denote the Earth's equatorial radius, we have, approximately, 

r = __ a __ = <* - ^ a x 2Q6 ' 265 

sin 8*79" circ. meas. of 8'79" 879 * 

Taking a = 3963'3, this gives 

r (Earth's mean solar distance) = 93,002,000 miles, 
correct to the nearest thousand miles. 

Also, perihelion distance from Sun = 93,002,000 x (1-^) 

= 93,002,000-1,550,000 - 91,452,000 miles, 
and aphelion distance = 93,002.000 x (1 + -L) 

= 93,002,000 + 3,550,000 = 94,552,000 miles. 



DISTANCES OF THE SUN AND STARS. 271 

SECTION II. Transits of Inferior Planets. 

342. When Yenus is very near the ecliptic at inferior con- 
junction, it passes in front of the Sun's disc, appearing like 
a black dot on the Sun. Now the circumstances of such a 
transit are different at different places, for although both 
the Sun and planet are displaced by parallax, their displace- 
ments arc different, and their relative directions are therefore 
not the same. Now the ratio of the parallaxes of the Sun 
and planet at conjunction can be calculated from comparing 
their periodic times, or from the ratio of their distances, as 
determined by observations of the planet's greatest elonga- 
tion or otherwise. Hence, by comparing the circumstances 
of the transit at different places, it becomes possible to deter- 
mine the parallaxes of both the Sun and planet. 

The various methods of finding the Sun's parallax from 
observing transits of Venus may be classified as follows : 

(i.) By simultaneous observations of the relative position 
of the planet at different stations, either by micrometric mea- 
surements, or from photographs. 

(ii.) Delislis method, by comparing the times of the begin- 
ning or end of the transit at stations in different longitudes. 

(iii.) Halley's method, by comparing the durations of the 
transit at stations in different latitudes. 

Of these methods Halley's is the earliest, Delisle's the next. 

343. First Method. Let P and p be the horizontal 
parallaxes of the Sun and of Yenus respectively at the time of 
transit. Then, at a place where the planet's zenith distance 
is z, its direction is depressed by parallax through an angle 
p sin z ( 249) ; also the Sun is depressed through P sin z* 
Hence the planet appears to be brought nearer to the Sun's 
lower limb by an angle (pP) sin %. 

If, now, the positions of the planet relative to the Sun's disc 
be simultaneously observed at any two or more different 
places, and the Sun's zenith distances be also determined, 
the difference of parallaxes p P can be readily found. 
Thus, if one of the stations be chosen where the Sun is 

* Strictly speaking, this should be P sin z,, where z\ is the Z.D. of 
the Sun's centre, but z l is very nearly equal to 2, and no sensible 
error is introduced by taking z instead of K\. 



272 ASTBONOMt. 

vertical, and another where the Sun is on the horizon, the 
relative displacement will be zero at the former station, and 
p -P at the latter. Hence, the two directions of the planet 
relative to the Sun will he inclined at an angle p P. If 
two stations are at opposite ends of a diameter of the Earth, 
the angular distance between the relative positions will be 
'2 (pP). Hence, in either case, 2? P can be readily found. 
Let now / and r denote the distances of Yenus and the 
Earth from the Sun respectively. Then, if The the ratio of 
the sidereal period of Yenus to a year, we have, by Kepler's 
Third Law (assuming the orbits circular), 

r'/r = T\ 

whence the ratio of r' to r is found. Also, since Yenus is in 
conjunction, its distance from the Earth is = rr'. There- 
fore p : P= r : r r' t 

and ^_=n' = JL-l. 
p-P r V 

Whence, since the ratio of r to r is known, and P p has 

been observed, the Sun's horizontal parallax Pmay be found. 

We have roughly (by Bode's Law) r' =T*O r i an ^ therefore 



Hence the displacement of Yenus on the Sun's disc at a place 
where its zenith distance is 2, is about | P sin 2. 

The apparent position of Yenus on the Sun's disc may be 
observed either by measuring the planet's distance from the 
edge of the disc with a micrometer or heliometer, or by taking 
a photograph of the Sun. But the photographic method, 
though easier, does not give such accurate results. 

For, to obtain P correct to O'Ol", it would be necessary to find 
2(p-P) correct to ^-xO'Ol", or about 0'05". Since the Sun's dia- 
meter is 32', the greatest possible difference of positions would be 

20 x 32 x 60 ' r 37400' 

of the Sun's diameter. It is difficult to obtain a good photograph 
of the Sun more than 4J inches in diameter, and it would, therefore, 
be necessary to measure the planet's position correct to - & zoo f an 
inch, a degree of accuracy unattainable in practice. The slightest 
distortion or imperfection in the photographic plate would render 
the observations worthless. 



THE DISTANCES OF THE SUN AND STARS. 273 

344. Delisle's Method. In this method, the Sun's 
parallax is determined by observing the difference between 
the times at which the transit begins or ends at different 
places. Let A, B be two stations near the Earth's equator in 
widely different longitudes, say at the ends of the diameter 
of the Earth, and in the plane containing UV, the path of 
Yenus' relative motion. Draw AUL and BVL, touching 
the Sun in L and cutting the path of Yenus in &, V. Then, 
when Yenus reaches U the transit begins at A, the planet 
appearing to enter the Sun's disc at L t and when Yenus is at 
V the transit begins at B. In the interval between the 
times of commencement of the transit as seen from A and B, 
the planet moves through the angle ULVor ALB about the 
Sun relative to the Earth, and this angle, being the angle sub- 
tended at the Sun by the Earth's diameter AB, is twice the 
Sun 1 9 parallax. 




FIG. 111. 



But the rate of relative angular motion of Yenus is known, 
being 360 in a synodic period. Hence the angle TJLV, 
described in the observed interval, is known, and the Sun's 
parallax is thus found. 

In a similar way, the Sun's parallax may be determined by < 
observing the interval between the times at which the transit 
ends at two stations A, B. We should have to draw two 
tangents from A, B to the opposite side of the Sun (M ). As 
before, the angle described by Yenus in the observed interval 
is twice the Sun's parallax. 



274 ASTRONOMY. 

In employing Delisle's method, the observed times ol 
ingress or egress must be the Greenwich times, or must be 
reckoned from an epoch common to both observers. For this 
reason the difference of longitudes of the two stations must 
be accurately known. In the following example the ob- 
served interval 690s. corresponds to 8-86" of parallax, and it 
follows that an error of Is. in the estimated interval would 
give rise to an error of just over 0-01" in the computed 
parallax. Hence if the interval of time be estimated correct 
to the nearest second, the parallax will be correct to two 
decimals of a second. 

In practice it would be dim cult to make observations from 
the extremity of a diameter of the Earth, but the method is 
readily modified so as to be applicable when the stations are 
not so favourably situated. 

EXAMPLE. 

Given that the synodic period is 584 days, and that the difference 
between the times of ending of a transit, as seen from opposite ends 
of a diameter of the Earth, is llm. 30s., to find the Sun's parallax. 

In 584 days Venus revolves through 360 about the Sun relative 
to the Earth ; therefore its angular motion per minute 

360 x 60 x 60 . , 

= - seconds = T541". 

584 x 24 x 60 

Therefore in 11 Jm. Venus describes an angle T541" x 11^ = 1772". 
This angle is twice the Sun's parallax ; 

.'. Sun's parallax = 8'86". 

345. Halley's Method. The method now to be de- 
scribed was invented by Dr. Halley in 1716, and was first 
put into use at the transits in 1761 and 1769. In Halley's 
method the times of duration of the transits are observed 
from two stations A, B, one in north and the other in south 
latitude, in a plane as nearly as possible perpendicular to the 
ecliptic, or, more strictly, to the relative path of Venus. Take 
this plane as the plane of the paper in Fig. 112, and suppose 
also (for the purpose of simplifying the explanation) that 
-4, B are at the ends of a diameter of the Earth. Let LM 
be the diameter of the Sun's disc perpendicular to the line of 
centres, and let the directions of Yenus A V, BV, when pro- 
duced, meet the disc in a, I. Then a, b are the relative posi- 
tions of Yenus as seen at conjunction from A and 7>. 



THE DISTANCES OP TRK SUN A\0 STARS. 



275 



In Fig. 113 the Sun's disc is represented as seen from the 
Earth ; a, I are the positions of Venus as seen on the disc 
from A, B, projected on L1I, in Fig. 112, and PQR, PQR' 
are the apparent paths of Venus as it appears to cross the 
disc at B and A respectively. 

As in 343, the angular measure of the arc db or QQ! 
measures the sum of the displacement of Venus due to relative 
parallax at A and B, and this, in the circumstances here 
considered, is twice the difference of the parallaxes of the 
Sun and Venus. 




Now the observed times of duration of the transit ut A 
and B are the times taken to describe the chords P' Q-R' and 
PQR respectively. Knowing the synodic period of Venus 
and the ratio of its distances from the Sun and Earth, the 
rate at which Venus travels across the Sun's face can be 
found. Hence, the angular lengths of the chords PQR, 
P'Q'R' can be found. Also the Sun's angular diameter ZJ/ 
is known. Hence the angular distances OQ, OQ', QQ f can 
be calculated, for we have (very approximately) 



whence the 



and QQ' = OQ'-OQ.. 

Hence QQ' is known, and therefore the difference of 
parallaxes of Venus and the Sun is found ; 
Sun's parallax may bd found as in 343. 



276 ASTRONOMY. 

*34C. Or if AB be known in miles, the length cf ab in miles can 
be found from the proportion ab : AB = Va : VA, and then, the angle 
aAb being known (being the angular measure of QQ')> we can find 
the Sun's distance in miles, for we have 

circular measure of L aAb = t ; whence 
aA 

Sun's distance Aa (in miles) = lep S th ab (in mi]es > . 

circular measure of L aAb 

The working of Halley's method will be made much 
clearer by a careful study of the following numerical 
examples. The student should copy Pigs. 112 and 113. 

EXAMPLES. 

1. To find the angular rate at which Venus moves across the 
Sun's disc. 

Let 8, E, V denote the Sun, Earth, and Venus respectively 
(Fig. 112). 

From the example of 344, 8V separates from 8E with relative 
angular velocity, about 8, of T54" per minute, or 1' 32'4" per hour. 

But Venus is nearer the Earth than the Sun in the ratio 28 : 72 
(roughly). And we have 

angular velocity of EV I ang. vel. of 8V 



Therefore EV separates from E8 with angular velocity 
= ^ x 1' 32-4" per hour = 3' 57'6" per hour 
= V per minute very nearly. 

2. Neglecting the motion of the observatory due to the Earth's 
rotation, find the position on the Sun's disc of the chord PR, tra- 
versed by the planet, in order that the trsviisit may take four hours. 

Draw the figures as in 345. 

In four hours Venus moves 4x3' 58' , or very nearly 16' relative 
to the Sun (by Ex. 1) ; /. the chord PR must measure 10'. Hence 
PR is equal to the Sun's angular semi-diameter OP. 

Therefore, PR is a side of a regular inscribed hexagon in the Sun, 
and L MOP = 30. 

3. If, at A, B. at opposite ends of a diameter of the Earth perpen- 
dicular to the piano of the ecliptic, the durations of transit are 
3h. 21in. and 4h. respectively, to find the Sun's parallax. 



tHB DISTANCES OP THE SUN AND STARS. 2^7 

Here tlie arc PR takes 39m. longer to describe than P*R'. Hence 
it is longer by 39 x 4", or 156". Draw R'K perpendicular to PR. 
Then, KR = ^PR-Pit) = x 156" * 78". 

Now, by Example 2, 

Z Jf OR = 60? 

And JBE', being very small, is approximately a straight line perpen- 
dicular to OR ; .'. R'RK = 30 approximately. Hence 

Q'Q = R'K = RKtan 30 = RK^/% = ^V3" = 45" nearly. 
But 
angular measure of Q'Q : twice Sun's parallax 

= 8V:EV= 18:7; 

.*. twice Sun's parallax = 45'' x T ^ = 17'50"; 
.'. Sun's parallax = 8*75". 

4. A transit of Venus was observed from two stations selected as 
favourably as possible, one in N. the other in S. latitude, the zenith 
distances of the planet being 53 8' (sin 53 8' = '8) and 30 
respectively. Given that the times occupied by the planet in pass- 
ing across the disc were 4h. 52m. and 4h. 30m., to find the Sun's 
parallax, assuming the distances of Venus and the Earth from the 
Sun to be in the ratio of 18 : 25 and neglecting the rotation of the 
Earth. 

Venus moves nearly 4" per minute relative to the Sun; hence in 
4h. 30m. it moves through 18'. 

In 4h. 52m. it moves through 19 7 28" j 

/. in Fig. 113, P'Q' = 18' x = 9', 

PQ = 19' 28" x * = 9-73', 

and the Sun's semi-diameter SP ~ 16' nearly; 

.'. SQ =Vsp-'-pQ2 = v/256- 94-67 - 12W; 

SQ' = v/SP' 2 -P'Q 2 = v/256 -81 = 13-23 7 ; 

/. QQ' = -53 7 = 31-8". 

Now, if A and B be well chosen, QQ' is the sum of the relative 
displacements of Venus at the two stations. Let P be the Sun's 
parallax, p that of Venus ; then we have 

QQ' = (2>-P)( s in 2 + sin z'} = (p - P) x (sin 30 + sin 53 8') 



1*3 



Again, P:p= : 



/. P = 24-5" x T 7 ? = 9'5". 
Hence, with the given data, the Sun's parallax is 9'5". 



278 \STRONOMY. 

347. Difficulties of Observing the Duration of a 
Transit. In Examples 3, 4, above, the observed differences 
of duration were 39m. and 20m. respectively. An error of 
one second in the estimated durations of transit would give 
rise to an error of less than O'l per cent., and if we could be 
sure of observing the durations to within a second, the Sun's 
parallax could be found correct to two decimal places. But 
in practice it is extremely difficult to estimate the times of 
beginning and ending of a transit, even to the nearest second. 

For in the first place, Venus, when seen through the telescope, 
is not a mere point, but a disc of finite dimensions, its angular 
diameter at conjunction being about 67", or one-thirtieth of 
the diameter of the Sun. Hence its passage across the edge 
of the disc from external to internal contact occupies an 
interval which is never less than about 17s. (See Example 
on page 279.) 




est 



FIG. 114. 

Now, it is impossible to observe the first external contact ( U} 
of Venus with the Sun, because the planet is invisible until 
it has cut off a perceptible portion from the edge of the Sun's 
disc, and by that time it has advanced considerably beyond 
the point of contact. The last external contact ( F') at the end of 
the transit is also difficult (though rather less so) to observe, 
for a similar reason. 

For this reason, the internal contacts U', V. are alone 
observed, and a correction is applied for the angular semi- 
diameter of Venus. 

But in observing the first internal contact U\ when the 
planet's disc separates from the edge of the Sun, another 
difficulty, in the form of an optical illusion, makes itself 
manifest. 



THE DISTANCES OF THE SUN AND STARS. 279 

Instead of remaining truly circular, the planet's disc appears 
to become elongated towards the edge of the Sun, and remains 
for some time connected with the edge by a narrow neck 
called the " black drop." This breaks suddenly at last, but 
not until the planet has separated some distance from the 
Sun's edge.* Even if the "black drop" be remedied, the 
atmosphere surrounding the planet Venus renders the con- 
tacts uncertain and ill-defined. 

It is worthy of notire that in Dclisle's method the times of 
ingress and egress at both stations are equally affected by the 
"black drop" appearance, and therefore it has no effect on 
the computation, provided that both observers take the same 
stage of the phenomenon for the observed time of ingress. 
EXAMPLE. 

Having given that the angular diameter of Venus at conjunction 
is 67", to find the interval between external and internal contact (i.) 
when Venus passes across the centre of the Sun's disc, (ii.) in the 
circumstances of Example 2, 346. 

(i.) Between external and internal contacts the planet moves 
through a distance equal to its angular diameter; therefore, since its 
rate of motion is 4" per second, the time occupied = 67 -f 4s. = 17s. 
very nearly. 

(ii.) Here the planet is 67" nearer the centre at internal than at 
external contact. Now the planet's direction of motion UV is 
inclined at angle 60 to the radius through the centre of the disc (Fig. 
114). Hence the planet's .component relative velocity along the 
radius is 4" cos 60 per second, and therefore the interval required, 
in seconds, 

67 = 67 
4 cos 60 2 
= 33-5s. 

348. Recent Determinations of the Parallax of 
the Sun. Professor Arthur Auwers, the well-known Berlin 
astronomer, has recently (December 11, 1891) completed the 
calculations based on the observations in Germany of the 
transit of Venus in 1882. He finds that the parallax of the Sun 
is 8 800 seconds, with an error of 0*03 of a second at most. 
From the old observations of the transits of 1761 and 1709, 
Prof. Newcomb has lately computed the parallax at 8'79". 

* The " black drop " may be illustrated by holding two globes in 
the sunshine, at different distances from a white screen, and moving 
them until their shadows nearly touch. 



280 ASTRONOMY. 

349. Advantages and Disadvantages of H alley's 
and Delisle's Methods. In Halley's method the observed 
data are the intervals of time occupied by Yenus in crossing 
the Sun's disc at the two stations. It is not necessary to 
know the actual times of the transit ; hence neither the 
Greenwich time nor the longitude of the observatories need 
be known. In Delisle's method it is essential that the 
Greenwich times of the observations should be known with 
great accuracy, but it is not necessary to observe both the 
beginning and end of the transit at the two stations. Still, 
if these be both observed, we have two independent data 
for calculating the parallax, which afford some test of the 
accuracy of the computations. 

On the other hand, Delisle's method possesses the advan- 
tage that the places of observation mut be near the Earth's 
equator, and it may therefore be possible to select the stations 
nearly at opposite ends of a diameter of the Earth, and thus 
to get the greatest effect of parallax, while in Halley's 
method it is necessary that the stations shall be in as high 
latitudes as possible, and, owing to the practical difficulties 
of taking observations near the poles, the greatest effect 
of parallax cannot be utilized. 

Delisle's method is most easily employed if the transit is 
nearly central, i.e., if Venus passes nearly across the centre 
of the Sun's disc. This condition is fatal to the success of 
Halley's method ; here the best results are obtained when 
Yenus transits near the edje of the disc. 

For in Fig. 113 (page 27ij) we have 

OQ' 2 -OQ 2 = QP 2 -Q'P' 2 , 

PR-P'R' QP+Q^ 

2 OQ + OQ' 

Hence the effect on QQ' of a small error in the computed length of 
PR or PR' will be least when QP + Q'P' is smallest and OQ+O'Q' is 
largest, a condition satisfied when the transit takes place near the 
edge M of the disc. 

On the other hand, for a nearly central transit, OQ, O'Q' would be 
email, and very slight errors in the estimated lengths of PR, P'R' 
would produce such large errors in the computed displacement 
QQ' as to render the method practically worthless. 

The transits of 1874 and 1882 were both favourable to the 
use of ll'illey's method. 



THE DISTANCES OP THE SUN AND Sf AKS. 2S1 

*350. To determine the frequency of Transits of 
Venus. Since the Sun's angular semi-diameter is about 16', 
a transit of Yenus only occurs when the angular distance 
between the centres of the Sun and Yenus, as seen from some 
place on the Earth, is 1 6'. Hence, neglecting the effects of 
the relative parallax (P-p =. 23" by Ex. 3. 346, and this 
is small compared with 16'), Yenus must be at an angular dis- 
tance (SEV] < 16' from the ecliptic at the time of conjunc- 
tion. Hence the planet's heliocentric latitude JSSVmust be 
less than 16' xEVjSV, that is l6'x T 7 ? , or about 6'. Now 
the orbit of Yenus is inclined to the ecliptic at about 3 23', 
or 203'. Hence, by a method similar to that of 292, we see 
that the planet must be at a distance from the node of not 
more than about sin'^f-g = sin' 1 -^ (roughly) = 142', in 
order that a transit may take place. The smallncss of 
this limit alone shows that transits of Yenus are of rare 
occurrence. 

Now, a synodic period of Yenus contains about 584 days, 
that is, 1-599, or, more accurately, 1-598662 of a year. 
Hence five synodic revolutions occupy almost exactly eight 
years, the difference only amounting to T -^ of a year. This 
difference corresponds to an arc of f~f , or 2 24' on the 
ecliptic. This arc is much less than the doulle arc 3 24' 
within which transits take place. Hence it frequently 
happens that, eight years after one transit has taken place, 
the Sun and Yenus arc again at conjunction within the 
necessary limits, and another transit occurs near the same 
node. But after sixteen years, conjunction will occur at 4 48' 
from its first position ; this is greater than 3 24' ; hence 
there cannot be more than t\vo transits near the same node at 
intervals of eight years. And if a transit should be central, 
occurring almost exactly at the node, the conjunctions occurring 
eight years before and after would fall outside the required 
limits, and no second transit would then take place in eight 
years. 

Again, it maybe shown that 1-598662x147 = 235-003. 
Hence 147 synodic periods of Yenus occupy almost exactly 
235 years, the difference being only '003 of a year. Thus a 
transit of Yenus may recur at the same node at an interval 
of 235 years. And it is possible to prove that thore is no 



282 ASTRONOMY. 

intermediate interval between. 8 and 235 years at "which 
transits recur at the same node. 

If the orbits of the Earth and Venus were circular, a transit 
at one node would be followed by one at the opposite node in 
11 3^ or 121* years. For 
1-598662x71 = 113+'005; 1-598662x76 = 121*--002. 

But this result is modified by the eccentricities of the orbits 
(which now cause a difference of nearly a day in the times 
taken by the Earth to describe the two halves into which its 
orbit is divided by the line of nodes). 

In reality it is found that the intervals between transits of 
Venus occur at present in the following order : 
8, 105*; 8, 121*; 8, 105*; 8, 121*. 

Transits have occurred, and are about to occur, in 1761, 1769, 
1874, 1882, 2004, 2012 (the thick and thin type being 
used to distinguish the two different nodes). 



. Transits of Mercury occur much more frequently 
than transits of Venus. For although the orbit of Mercury 
is inclined to the ecliptic at about twice as great an angle as 
that of Venus, this cause is more than compensated for by 
the greater proximity of the planet to the Sun ; and since 
the synodic period of Mercury is only about % of that of 
Venus, conjunctions occur five times as often, so that we 
should ceteris paribus expect five times as many transits. By 
a method similar to that employed for Venus it is found that 
transits occur at the same node at intervals of 7, 13, 33, or 
-46 years. The next transit will occur in 1894. 

Although transits of Mercury thus occur far more often 
than transits of Venus, they cannot be used to determine the 
Sun's parallax with such accuracy, for Mercury is so near the 
Sun that the parallaxes of the two bodies are more nearly 
equal, and the planet's relative displacement is therefore 
much smaller than that of Venus. Moreover, Mercury moves 
much more rapidly across the Sun's disc, giving less time for 
accurate observations ; besides which, owing to the great 
eccentricity of the orbit, the ratio of Mercury's to the Earth's 
distance from the Sun cannot be so exactly computed. 



THE DISTANCES OF THE SUN AND STAES. 283 

SECTION III. Annual Parallax, and Distances of the. 
Fixed Stars. 

352. Annual Parallax, Definition. By Annual 
Parallax is meant the angle between the directions of a 
star as seen from different positions of the Earth in its annual 
orbit round the Sun. 

We haye several times ( 5, 247) mentioned that the fixed 
stars have no appreciable geocentric parallax. Their distances 
from the Earth are so great that the angle subtended at one 
of them by a diameter of the Earth is far too small to be 
observable even with the most accurately constructed instru- 
ments. But the diameter of the Earth's annual orbit is 
about 23,400 times as great as the Earth' R diameter, or about 
186 million miles (twice the Sun's distance), and this 
diameter subtends, at certain of the nearest fixed stars, an 
angle sufficiently great to be measurable, sometimes amounting 
to between \" and 2". 

Now, the Earth, by its annual motion, passes in six months 
from one end to the other of a diameter of its 
orbit ; hence, by observing the same star at an 
interval of six months, its displacement due to 
annual parallax can be measured. 

Since the Sun is fixed, the position of a star on 
the celestial sphere is correctedfor annual parallax 
by referring its direction to the centre of the Sun; 
this is called the star's heliocentric direction, as 
in 156. 

The correction for annual parallax is 
the angle between the geocentric and heliocen- 
tric directions of a star. Let S be the Sun, E the Earth, x 
the star (Fig. 115). Then Ex is the apparent or geocentric 
direction of the star, Sx its heliocentric direction, and z ExS 
is the correction for annual parallax. This angle is also equal 
to xEx ! where Ex 1 is parallel to Sx. 

We notice that the correction for annual parallax (ExS) 
is the angular distance of the Earth from the Sun as they would 
appear if seen ly an observer on the star. 

ASTRON. u 




284 ASTBONOMY. 

353. To find the Correction for Annual Parallax. 

Let r = JES = radius of Earth's orbit. 

= Sx = distance of star. 

E = L SEx = angular distance of star from Sun. 
p = z ExS = annual parallax of star. 

By trigonometry we have in the triangle SEx 



smSJEx Sx ' 
whence* sinp= sin.E ..................... (i.). 




Hence the parallactic correction p is greatest when 
E = 90. This happens twice a year, and the corresponding 
positions of the Earth in its orbit are evidently the inter- 
sections of the ecliptic with a plane drawn through S per- 
pendicular to Sx. Let this greatest value of p be denoted 
by P, then P is called the star's annual parallax, or 
simply the star's parallax. j 

Putting E= 90 in (i.), we have 



and therefore sin^? = sin P . sin E. 

* Notice the close similarity between the present investigation 
and that of 249. 

f There is no risk of confusion in the use of the term parallax 
alone, because a star has no geocentric parallax. The " parallax " 
of a body means its equatorial horizontal parallax if the body belongs 
to the solar system. If not, its " parallax " is its annual parallax. 



THE DISTANCES OF THE SUN AND STAES. 285 

But the angles P, p are always very small ; therefore their 
sines are very approximately equal to their circular measures. 
Thus we have approximately 

_P (in circular measure) = ~ , 

d 
# = JPsin E; 

and, if P", p" denote the numbers of seconds in P, p, 

P" = 180x60x60 -L = 206,265 r (approximately)) 
v d> d 

and p" = P" sin E. 

354. Relation between the Parallax and Distance 
of a Star. If a star's parallax be known, its distance from 
the Sun is given by the formula 

fll = 180X60X60 JL =ao6266 r 

TT d d 



whence d = S r = 206,265 , 

where r is the Sun's distance from the Earth. 

For most purposes r may be taken as 93 million miles. 



EXAMPLES. 

1. The parallax of Castor is 0'2" ; to find its distance. We have 

d = 206265 *- = 206,265x93,000,000 

P" 0-2 

= 5 x 206,265 x 93,000,000 
= 95,900,000,000,000, or 959 x 10 11 miles 

approximately. It would be useless to attempt to calculate more 
figures of the result with the given data, which are only approximate. 
It is most convenient (besides being shorter) to write the result in 
the second form. 

2. To find the distance of a Centauri (i.) in terms of the Sun's 
distance, (ii.) in miles, taking its parallax to be 0750". 



Here d = . r = 275,000r 

"75 



275 x 10 3 x 93 x 10 fi = 25,575 x 10 9 
256 x 10 11 miles approximately. 



286 ASTRONOMY. 

355. General Effects of Parallax. Since Ex is 

parallel to Sz, it is in the same plane as US and JSx. 
Hence the lines ES, Ex, Ex' cut the celestial sphere of E at 
points 8, x, a? , lying in one great circle, and we have the 
two following laws : 

(i.) Parallax displaces the apparent position of a star from 
its heliocentric position in the direction of the Sun. 

(ii.) The parallactic displacement of any star at different times 
varies as the sine of its angular distance from the Sun. 





FIG. 117. FIG. 118. 

Let Fig. 118 represent the observer's celestial sphere, 
S the Sun. Let x be the apparent or geocentric position 
of the star, whose parallax is P. Draw the great circle 
Sx and produce it to # , making 

XX Q = P sin Sx. 

Then # represents the star's heliocentric position, and this 
is its position as corrected for annual parallax. 

Conversely, if the star's heliocentric position # is given, 
we may obtain its geocentric or apparent position x by join- 
ing tf $, and on it taking 

xp = P sin 8x = P sin Sx Q very approximately 
(for the difference between P sin Sx and P sin Sx is 
exceedingly small, and may be neglected). 

The terms Parallax in Latitude and Parallax in 
Longitude are used to designate the corrections for parallax 
which must be applied to the celestial latitude and longitude 
of a star respectively. Similarly, the parallax in decl. and 
parallax in R. A. denote the corresponding corrections for 
the decl. and E-.A. 



THE DISTANCES OP THE SUN AND STARS. 28? 

356. To show that any star, owing to parallax, 
appears to describe an ellipse. 

In Fig. 117, Ex' is parallel to the star's heliocentric direc- 
tion; therefore, x is fixed, relative to the Earth. Moreover, 
x'x = ES. Hence, as the Sun 8 appears to revolve about 
the Earth in a year, the star x will appear as though it 
revolved in an equal orbit about its heliocentric position x ', in 
a plane parallel to the ecliptic. 




FIG. 119. 

Let the circle MN(Fig. 119) represent this path, which the 
star x appearsto describe in consequence of parallax. This circle 
is viewed obliquely, owing to its plane not being in general 
perpendicular to Ex'\ hence, if mn denote its projection on the 
celestial sphere, the laws of perspective show that mn is an 
ellipse. (Appendix, 12.) This small ellipse is the curve 
described by the star on the celestial sphere during the year. 

Particular Cases. A star in the ecliptic moves as if it 
revolved about its mean position in a circle in the ecliptic 
plane, hence its projection on the celestial sphere oscillates 
to and fro in a straight line (more accurately a small arc of a 
great circle) of length 2P. 

For a star in the poie of the ecliptic the circle MN is per- 
pendicular to Ed) hence Ex describes a right cone, and the 
projection x describes on the celestial sphere a circle, of 
angular radius P, about the pole K. 

If the eccentricity of the Earth's orbit be taken into account, the 
curve MN will be an ellipse instead of a circle, but its projection 
mn will still be an ellipse. 



288 



ASTRONOMY. 



357. Major and Minor Axes of the Ellipse. We 

shall now prove the following properties of the small ellipse 
described during the course of the year by a star whose 
parallax is P, and celestial latitude I. 

(i.) (A) The length of the semi-axis major is P. 
(B) The major axis is parallel to the ecliptic. 

(c) When the star is displaced along the major axis it 
has no parallax in latitude. 

(D) At these times the Sun's longitude differs from the 
star's by 90. 

(ii.) (A) The length of the semi-axis minor is Psinl. 

(B) The minor axis is perpendicular to the ecliptic. 

(c) When the star is displaced along the minor axis it 
has no parallax in longitude. 

(D) At these times the Sun's longitude is either equal to 
the star's, or differs from it ly 180. 




On the celestial sphere let a? denote the heliocentric 



position of the star, ABAB' the ecliptic, JTits pole, 
the secondary to the ecliptic through the star. 

Then, if S is the Sun, the star X Q is displaced to #, where 

x = P sin xS. 



THE DISTANCES OF THE SUN AND STAES. 289 

(i.) The displacement is greatest when sin x Q S is greatest, 
and this happens when 

Bin# /S=: 1, # =90. 
If, therefore, we take A, A' on the ecliptic so that 



A) A' are the corresponding positions of the Sun. 

JTow A, A are the poles of BKB* (Sph. Geom., 11, 14, 
15), and therefore the great circle Ax^A is a secondary to 
BKB'. Hence, if a, a' denote the displaced positions of the 
star, aa' is perpendicular to JK7?, and is therefore, parallel to 
the ecliptic. 

Also, x Q a = x$ = P sin 90 = P ; 

therefore the semi-axis major of the ellipse is P. 

Since AB = AB = 90, the star's longitude ( r B) differs 
from the Sun's longitude at A or A by 90. 

And since the star is displaced parallel to the ecliptic, its 
latitude, or angular distance from the ecliptic, is unaltered, 
and therefore the parallax in latitude is zero. 

(ii.) The parallactic displacement is least when sin xJ3 is 
least, and this happens when S is at B. For (Sph. Geom., 
26) B is the point on the ecliptic nearest to # . Also, since 

sin xj = sin (180 # #) = sin x^B, 
it follows that the parallactic displacement is also least when 
S is at B'. 

If, therefore, b, V be the extremities of the minor axis, the 
arc lit is along JO?, and is therefore perpendicular to the 
ecliptic. 

Also, xjb = x$ = P sin x B = P sin / ; 

therefore the semi-axis minor is P sin I. 

When the Sun is at B, it has the same longitude as the 
star ; when at ^, the longitudes differ by 180. 

And since the star is displaced in a direction perpendicular 
to the ecliptic, its longitude TB is unaltered; therefore the 
parallax in longitude is zero, 



290 ASTRONOMY. 

The parallax in latitude is evidently equal to the apparent 
angular displacement of the star resolved parallel to x Q K, and 
its maximum value is xjb, or xj)'. The parallax in longitude 
is not equal to the star's angular displacement perpendicular 
to Jx Q , but to the change of longitude thence resulting, and 
this is measured by the angle xKx^. Hence, in Tig. 120, 

(i.) The maximum parallax in latitude = x Q b = I* sin . 

(ii.) The maximum parallax in longitude = L x^Ka 
= x^Kd = x Q a/sin Kx Q (Sph. Geom. l7)=P/eos x Q B 

= P sec L 

358. To determine the Annual Parallax of any 
Star, the following methods have been employed : 

(i.) The absolute method, by the Transit Circle ; 

(ii.) Bessel's, or the differential method, by the micrometer 
or heliometer j 

(iii.) The photographic method. 

The absolute method consists simply in observing with 
the Transit Circle the apparent decl. and R.A. of a star at 
different times in the year. From the small variations in 
these coordinates it is possible to find the star's parallax. 

Although this method has been successfully employed, it 
possesses many disadvantages. For the observations are con- 
siderably affected by errors of adjustment of the Transit 
Circle and by refraction. Moreover, several other causes 
give rise to variations in the star's apparent decl. and R.A. 
during the year. These include aberration (vide Section I V.) , 
precession ( 141), and nutation, all of which produce dis- 
placements much larger than those due to parallax. 

In 372 we shall see that when either the latitude or longi- 
tude is most affected by parallax it is unaffected by aberra- 
tion. Hence the best plan is to find the changes in these 
coordinates when they are respectively most affected by 
parallax. These changes are P sin I and Psec I ( 357) and 
from them P may be found. 

359. Bessel's Method consists in observing with a micro- 
meter (^79) or heliometer ( 80) the variations in the 
angular distance and relative position of two optically near 
stars during the course of a year. 



THE DISTANCES OF THE STJN AND STARS. 291 

The stars, being nearly in the same direction, are very 
nearly equally affected by refraction, and we may also men- 
tion that the same is true of aberration, precession and 
nutation. These corrections do not therefore sensibly affect 
the relative angular distance and positions of the stars. On 
the other hand, the two stars may be at very different dis- 
tances from the Earth ; if so, they are differently displaced 
by parallax, and their angular distance and position undergo 
variations depending on their relative parallax or difference of 
parallax. Hence, by observing these variations during the 
year the difference of parallax can be found. 

This method does not determine the actual parallax of 
either star. But if one of the observed stars is very bright 
and the other is very faint, it is reasonable to assume that 
the former is comparatively near the Earth, while the latter 
is at such a great distance away that its parallax is insensible. 
Under such circumstances the observed relative parallax is 
the parallax of the bright star alone. By making compari- 
sons between the bright star and several different faint stars 
in its neighbourhood, this point may be settled. 

If a considerable discrepancy .is found in the observed 
relative parallaxes, one or more of the comparison stars must 
themselves have appreciable parallaxes, but since the vast 
majority of stars in any neighbourhood are too distant to have 
a parallax, we shall be able to find the parallax not only of 
the star originally observed, but of that with which we had 
first compared it. 

The parallax of a star can never be negative ; if the relative 
parallax should be found to be negative, we should infer that 
the comparison star has the greater parallax, and is therefore 
nearer the Earth. 

360. The Photographic Method is identical in prin- 
ciple with the last, but instead of observing the relative 
distances of different stars with a micrometer, portions of the 
heavens are photographed at different seasons, and the dis- 
placements due to parallax are measured at leisure by 
comparing the positions of any star on the different plates. 
This method has been used by Dr. Pritchard, of Oxford, and 
possesses the advantages of great accuracy, combined with 
convenience. 



292 ASTRONOMY. 

361. Parallaxes of certain Fixed Stars. The nearest 
stars are a Centauri, with a parallax of 0-75", and 61 Cygni, 
with parallax 0-54". Among others, the following may he 
mentioned: a Lyra, 0-18", Sirim, 0-2", Arcturm, 0-1 3", 
Polaris, 0-07", a Aquilce, 0-19". Of these, 61 Cygni is hy 
no means bright ; and a companion star to Sirius is invisible 
in all but two or three of the best telescopes. So it is not an 
invariable rule that faint stars are most distant, and have no 
appreciable parallax ; it is, however, true in the great majority 
of cases.* 

362. Proper Motions. Binary Stars. Many stars, 
instead of being fixed in space, are gradually changing their 
positions. They are then said to have a proper motion. 
This motion may partly belong to the star, but is also partly 
an apparent motion, due to the fact that the solar system is 
itself moving through space in the direction of a point in the 
constellation Hercules. The displacement due to this cause 
can be allowed for approximately. 

Many of these motions, like that of our own Sun, are apparently 
progressive ; i.e., the star moves with constant velocity and 
in the same direction. Others are orbital, i.e., the star 
revolves about some other star, or (more accurately) two 
stars revolve about their common centre of mass. Such a 
system of stars is called a Binary Star. It is usually seen by 
the naked eye as a single heavenly body, its components 
being too near to be distinguished. Frequently a system of 
stars has itself a progressive motion; and sometimes an 
apparently progressive motion may really be an orbital one, 
with a period so long that the path has not sensibly diverged 
from a straight line during the short period for which stellar 
motions have been watched. 

A progressive or orbital motion cannot be confounded with 
the displacement due to annual parallax, for the former is 
always in the same direction, and the latter has a period dif- 
fering from a year, while parallax always produces an annual 
variation. 

* These figures can only be regarded as very rough approxima- 
tions, for considerable discrepancies exist between the values foun4 
by different methods. 



THE DISTANCES OP THE SUN AND STAES. 293 

SECTION IV. The Aberration of Light. 

363. Velocity of Light. "We now come to certain 
methods of finding the Sun's distance which depend on the 
fact that light is propagated through space with a large but 
measurable velocity. 

The velocity of light has been measured by laboratory 
experiments in two different ways, invented by two French 
physicists, Fizeau and Foucault. For the description of these 
the reader is referred to "Wallace Stewart's Text Book of Light, 
Chapter IX.* The experiments give the velocity of light in 
air ; the velocity in vacuo can be obtained by multiplying 
this by the index of refraction of air.f The latter quantity 
may be found either by direct experiment or from the coeffi- 
cient of astronomical refraction (see 183). 

In 1876, Cornu, by employing Fizeau's method, found the 
velocity of light in vacuo to be 300,400,000 metres per second. 
Still more recently, Michelson, by a modification of Foucault's 
method, has found the velocity to be 299,860,000 metres, 
or 186,330 miles per second ; this may be taken as the 
most probable value. 

364. Roemer's Method. The Equation of Light. 

In the last chapter we stated that Jupiter has four satellites, 
which revolve very nearly in the plane of the planet's orbit. 
Consequently a satellite passes through the shadow cast by 
Jupiter once in nearly every revolution, and is then eclipsed, 
as is our Moon in a lunar eclipse. 

Since the orbits and periods of the satellites have been 
accurately observed, it is possible to predict the recurrence 
of the eclipses, so that when one eclipse has been observed 
the times at which subsequent eclipses will begin and end 
can be computed. 

]S"ow, the Danish astronomer Roemer in 1675 observed a 
remarkable discrepancy between the predicted and the 
observed times of eclipses. If of two eclipses one happens 
when Jupiter is near opposition, and the other happens near 
the planet's superior conjunction, the observed interval 

* The student will find it useful to read this chapter before com- 
mencing the present section, 
t Stewart's Light, 41. 



294 



ASTROXOMT. 



between the former and the latter is always greater than the 
computed interval ; similarly the observed interval between 
an eclipse near superior conjunction and the next eclipse 
near opposition is always less than the computed interval. 
The eclipses at conjunction arc thus always retarded, relatively 
to those at opposition, by an interval of time which is observed 
to be about 16m. 40s. As explained by Roemer, this apparent 
retardation is due to the fact that light travels from Jupiter 
to the Earth with finite velocity, and therefore takes 1 6m. 
40s. longer to reach the Earth when the planet is furthest 
away at superior conjunction (B} than when the planet is 
nearest the Earth at opposition (A). 




The relative retardation is the difference between the times 
taken by the light to travel over the distances AE and BE. 
But BE- AE = 2S. Therefore the retardation is twice the 
time taken by the light to travel from the Sun to the Earth. 

Taking the retardation as 16m. 40s., we see that light takes 
8m. 20s. to travel from the Sun to the Earth. 

This interval is sometimes called the " equation of light?' 

If we know the equation of light and the velocity of light, 
\ve may calculate the Sun's distance. Conversely, if the 
Sun's distance and the equation of light are known, the 
velocity of light can be determined. 

Knowing the Sun's distance, the Sun's parallax can be 
computed, as in Chapter VIII., Section I. The present 
method differs from those described in Sections I., II., in 
that it gives the distance instead of the parallax of the Sun. 



THE DISTANCES OF TIFE SUN AND STARS. 295 

EXAMPLE 1. To find the Sun's distance, having given that the 
velocity of light is 186,330 miles per second, and that eclipses of 
Jupiter's satellites which occur when the planet is furthest from 
the Earth, are retarded 16m. 40s. relatively to those which occur 
when the planet is nearest. 

Here the time taken by light to pass over a diameter of the 
Earth's orbit is 16m. 40s. ; therefore light travels from the Sun to 
the Earth in 8m. 20s., or 500 seconds. 

/. the Sun's distance = 186.330 x 500 miles 
= 93,165,000 miles. 

EXAMPLE 2. Taking the value of the Sun's distance calculated in 
the preceding example, the Sun's parallax will be found to be 
about 8-78". 

365. The Aberration of Light is a displacement of the 
apparent directions of stars, due to the effect of the Earth's 
motion on the direction of the relative velocity with which 
their light approaches the earth. 

The rays of light emanating from a star travel in straight 
lines through space* with a velocity of about 186,330 miles 
per second. We see the star when the rays reach our eye, 
and the appearance presented to us depends solely on how 
the rays are travelling at that instant. If the Earth were 
at rest, and there were no refraction, we should see the star 
in its true direction, "because the light would be travelling 
towards our eyes in a straight line from the star. But in 
every case the direction in which a star is seen is the direction 
of approach of the light-rays from the star at the instant of 
their reaching the eye. 

Now the velocity of approach is the relative velocity of the 
light with respect to the observer. If the observer is in 
motion, this relative velocity is partly due to the motion 
of the light and partly due to the motion of the ob- 
server. If the observer happens to be travelling towards 
or away from the source of light, the only effect of 
his motion will be to increase or decrease the velocity of 
approach of the light, without altering its direction, but if he 
be moving in any other direction, his own motion will alter 
the direction of the relative velocity of approach, and will 
therefore alter the direction in which the star is seen. 

* Of course the rays are refracted when they reach the Earth's 
atmosphere, but the effects of refraction can be allowed for separately. 



296 A8TEONOMY. 

Suppose the light to be travelling from a distant star x 
in the direction xO. Let T"be the velocity of light, and let 
it be represented by the length M 0. Suppose also that an 
observer is travelling along the direction NO with velocity w, 
represented by the straight line NO. Then, if we regard 
as a fixed point, the light is approaching with velocity re- 
presented by MO. Also since the observer is approaching 
with velocity represented by NO, the point is approaching 
the observer jVwith an equal and opposite velocity repre- 
sented therefore by ON. Hence the whole relative velocity 
with which the light is travelling towards the observer is 
the resultant of the velocities represented by M and ON. 




By the Triangle of Velocities this resultant velocity is repre- 
sented in magnitude and direction by MN. Hence MN 
represents the direction of approach of the light towards the 
observer's eye. Therefore when the observer has reached 
the star is seen in the direction Ox' drawn parallel to NM, 
although its real direction is Ox, 

In consequence, the star appears to be displaced from its 
true position x to the position x'. This displacement is 
called the aberration of the star, and its amount is, of 
course, measured by the angle xOx. This angle is sometimes 
called the angle of aberration or the aberration error. 

366. Illustrations of Eelative Velocity and Aberration. The 
following simple illustrations may possibly assist the reader in 
understanding more thoroughly how aberration is produced. 

(1) Suppose a shower of rain-drops to be falling perfectly 
vertically, with a velocity, say, of 40 feet per second. Then, 
if a man walk through the shower, say with a velocity of 4 feet 



THE DISTANCES OF THE SUN AND STABS. 297 

per second, the drops will appear to be coining towards him, and 
therefore to be falling in a direction inclined to the vertical. Here 
the man is moving towards the drops with a horizontal velocity of 
4 feet per second, and therefore the drops appear to be coming 
towards the man with an equal and opposite horizontal velocity of 
4 feet per second. 

Their whole relative velocity is the resultant of this horizontal 
velocity and the vertical velocity of 40 feet per second with which 
the drops are approaching the ground. By the rule for the compo- 
sition of velocities, this lelative velocity makes an angle tan~ l -$ or 
tan* 1 '1 with the vertical. Hence the man's own motion causes an") 
apparent displacement of the direction of the rain from the vertical Y" 
through an angle tan' 1 g l. This angle corresponds to the angle ofJ 
aberration in the case of light. 

(2) Suppose a ship is sailing due south, and that the wind is blow- 
ing from due west with an equal velocity. Then to a person on the 
ship the wind will appear to be blowing from the south-west, its 
southerly component being due to the motion of the ship, which is 
approaching the south. In this case the ship's velocity causes the 
wind to apparently change from west to south-west, i.e., to turn 
through 45. We might, therefore, consistently say that the 
" angle of aberration " of the wind was 45. 

367. Annual and Diurnal Aberration. A point on 
the Earth's surface is moving through space with a velocity 
compounded of 

(i.) The orbital velocity of the Earth in the ecliptic about 
the Sun ; 

(ii.) The velocity due to Earth's rotation about the poles. 

These give rise to two different kinds of aberration, known 
respectively as annual and diurnal aberration. Now the 
Earth's orbital velocity is about 2?r x 93,000,000 miles per 
annum, or rather over 18 miles per second, while the 
velocity due to the Earth's rotation at the equator is roughly 
2^x4000 miles per day, or 0*3 miles per second. The 
former velocity is about T oi^o ^ * ne velocity of light, and 
therefore the annual aberration is a small though measurable 
angle. The latter velocity is only -fa as great ; hence the 
diurnal aberration is much smaller and less important. For 
this reason the term " aberration" always signifies annual 
aberration, unless the word "diurnal" is also used. We shall 
now consider the effects of annual aberration, leaving diurnal 
aberration till the end of this section. 



298 * ASTRONOMY. 

368. To determine the correction for aberration 
on the position of a Star. Let Ox be the actual direction 
of a star x seen from the Earth at ; U the direction of the 
Earth's orbital motion at the time of observation. On Ox 
take OM representing on any scale the velocity of light, and 
draw MY parallel to 017, and representing on the same scale 
the velocity of the Earth. Then YO represents the relative 
velocity of the light in magnitude and direction, so that OYx 
is the direction in which the star x is seen (Mg. 123). 

[For if ON be drawn parallel and equal to YM, the parallelogram 
of velocities MNOT shows that 21/0, the actual velocity of the light- 
rays in space is the resultant of the two velocities TO and NO, or 
YO and MY, and therefore YO is the required relative velocity.] 





FIG. 124. 

Since Ox, Ox, and U all lie in one plane, it follows, by 
representing their directions on the celestial sphere, that a 
star is displaced by aberration along the great circle joining its 
true place to the point on the celestial sphere towards which the 
Earth is moving. 

The displacement xOx is called the star's aberration 
error. Let it be denoted by y, and let 

u = NO = velocity of Earth, 
V MO = velocity of light. 
Then the triangle OM Y gives 

sinMOY_ MY _ u t 
sin M YO ~~ MO ~~ V* 

or sin y = -^ sin HYO = ~ sin 



THE DISTANCES OF THE SUN AND STARS. 299 

The aberration error y is, therefore, greatest when UOx' 
90. Let its value, then, be k. Putting UOx = 90, we 
have sin H =s 



and .*. sin y = sin k sin UOoc. 

The angle UOx is called the Earth's Way of the star, 
and k is called the Coefficient of Aberration. Since a and 
k are both small, we have, approximately 

y = k sin (Earth's way), 

k (in circular measure) = u/ V\ 

and, therefore, if y", k" denote the number of seconds in y, k 
respectively 

?/' = k" sin (Earth's way), 

, 180x60x60 u 

~iT ' v 






= 206,265 x. 
velocity of light 

369. General effect of Aberration on the Celestial 
Sphere. Neglecting the eccentricity of the Earth's orbit, 
the direction of motion of the Earth, in the ecliptic 
plane, is always perpendicular to the radius vector drawn to 
the Sun. Hence, on the celestial sphere, the point 7", towards 
which the Earth is moving, is on the ecliptic, at an angular 
distance 90 behind the Sun. This point is sometimes called 
the apex of the Earth's Way. 

Let x' denote the observed position of the star. Draw the 
great circle x' 7, and produce it to a point x , such that 
xx' = k sin x' U. 

Then x represents the star's true position, corrected fV 
aberration. 

Conversely, if we are given the true position x, we can find 
the apparent position x' by joining #7 and taking 

xx' = k sinxV, 

for it is quite sufficiently approximate to use k sin xU instead 
of k sin x'U. 

A3TROX. X 



300 ASTRONOMT. 

"We thus have the following laws : 



(i.) Aberration produces displacement in the apparent 
position of a star towards a point U on the ecliptic, distant 90 
behind the Sun. 

(ii.) The amount of the displacement varies as the sine of the 
Earth's Way of the star, i.e., the star's angular distance 
from the point U. 




FIG. 125. 



FIG. 123. 



370. Comparison between Aberration and Annual 
Parallax. The student will not fail to notice the close 
analogy between the corrections for aberration and annual 
parallax. 

The point 7 for the former corresponds to the point S for 
the latter, in determining the direction and magnitude of the 
displacement. In fact, the aberration error of a star is exact/// 
the same as its parallactic correction would be three months earlier 
(when the Sun was at U) if the star's annual parallax ice re k. 

There is, however, this important difference that the annual 
parallax depends on a star's distance, whilst the constant of 
aberration k is the same for all stars. 

For k depends only on the ratio of the Earth's velocity to 
the velocity of light, and not on the star's distance. The 
value of k in seconds is about 20'492" ; for rough purposes it 
mnv he taken as 20'5". 



THE DISTANCES OF THE SUN AND STARS. 301 

371. To show that the aberration curve of a star is 
an ellipse. This result, which follows immediately from the 
analogy between aberration and parallax, may be proved inde- 
pendently as follows: On Ox (Fig. 125), the true direction 
of a star #, take Ox to represent the velocity of light, and 
xM to represent the Earth's velocity. Then M meets the 
celestial sphere in m, the star's apparent position. 

As the Earth's direction of motion in the ecliptic varies, 
while its velocity remains constant, Jfdescribes a circle, about 
x as centre in a plane parallel to the ecliptic plane. The 
projection of this circle on the celestial sphere is an ellipse 
(cf. 356), and this is the curve traced out by a star during 
the year in consequence of aberration. 

Particular Cases. A star in the ecliptic oscillates to 
and fro in a straight line, or more accurately an arc of a great 
circle of length 2/j. A star at the pole of the ecliptic revolves 
in a small circle of radius k (cf. 356). 

372. Major and Minor Axes of the Aberration 
Ullipse. By writing Z7"for S and k for P in the investiga- 
tion of 357, we obtain the analogous results relating to the 
ellipse described by a star in consequence of aberration, 
namely : 

(i.) (A) The length of the semi-axis major is k. 

(B) The major axis of the ellipse is parallel to the ecliptic. 

(c) When the star is displaced along the major axis it has no 
aberration in latitude. 

(D) At these times the Surfs longitude is either equal to the 
.star's, or differs from it by 180.* 

(ii.) (A) The length, of the semi-axis minor is k sin I. 

(B) The minor axis is perpendicular to the ecliptic. 

(c) When the star is displaced along the minor axis, it has no 
aberration in longitude. 

(D) At these times the Sun's longitude differs from the star's 
.by 90. 

COROLLARY. The maximum aberration in longitude = k sec? 
(cf. 357, ii.). 

* Note that (i., r) and (ii., D) are the reverse of the corresponding 
-properties in 357. 



302 ASTRONOMY. 

*373. Effect of Eccentricity of Earth's Orbit. Owing to the 
elliptic form of the Earth's orbit the Earth's velocity is not quite 
uniform, and therefore the coefficient of aberration is subject to 
small variations during the year. The earth's velocity is greatest 
at perihelion and least at aphelion. The angular velocities at those 
times are inversely proportional to the squares of the corresponding 
distances from the Sun, but the actual (linear) velocities are in- 
versely proportional to the distances themselves, and these are in 
the ratio of l-e : 1 + e, or 1 --fa : 1 + ^ ( 149). Since the coeffi- 
cient of aberration is proportional to the Earth's velocity, its 
greatest and least values are therefore in the ratio of 61 : 59, and 
are respectively and of its mean value. 

Moreover, the direction of the Earth's motion is not always 
exactly perpendicular to the line joining it to the Sun, hence the 
" apex of the Earth's way," towards which a star is displaced, may 
be distant a little more or a little less than 90 from the Suu at 
different seasons. 

The aberration curve is still an ellipse. The student who 
has read the more advanced parts of particle dynamics may know 
that the curve MN, tracecfout by M, is in this case the "hodograph " 
of the Earth's orbital motion. It is also known, in the case of 
elliptic motion, such as the Earth's, that this hodograph is a circle, 
whose centre does not, however, quite coincide with x. Hence the 
aberration-curve hTc is an ellipse. 

374. Discovery of Aberration. Aberration was dis- 
covered by Bradley, in 1725, in the course of a series of 
observations made with a zenith sector on the star y Draconis 
for the purpose of discovering its annual parallax. The star's 
latitude was observed to undergo small periodic variations 
during the course of the year, and these differed from the 
variations due to annual parallax in the fact that the dis- 
placement in latitude was greatest when the Sun's longitude 
differed from that of the stars ly 90 ; that is, at the time when 
the parallax in latitude should be zero ( 357, i., c.). The fact 
that the phenomenon recurred annually led Bradley to suppose 
that it was intimately connected with the Earth's motion 
about the Sun, and he was thus led to adopt the explanation 
which we have given above, It will be seen that the pecu- 
liarity which led Bradley to discard annual parallax as an 
explanation is quite in harmony with the results of 372. 

375. To Determine the Constant of Aberration by 
Observation. The constant k can best be found by observ- 
ing different stars with a zenith sector or transit circle, as in., 
the direct method of finding a star's parallax ( 358). 



THE DISTANCES OF THE SUN AND STAUS. 



303 



The differential method of 359 cannot be used, because 
the coefficient of aberration is the same for all stars. But 
aberration is much larger than parallax (the coefficient of 
aberration being 20-49", while the greatest stellar parallax 
is < I"), and can therefore be found directly with greater 
accuracy. Of course it is necessary to make corrections for 
refraction and precession. The former correction is the most 
liable to uncertainty, as it varies slightly according to atmo- 
spheric conditions. But, as all stars have the same constant 
of aberration, a star may be selected which transits near the 
zenith, and is therefore but little affected by refraction. 

This condition was secured by Bradley when he observed 
the star y Draconis. The star is very favourable in another 
respect, for its longitude is very nearly 270. It therefore 
lies very nearly in the " solstitial colure," its declination 
circle passing nearly through the pole^f the ecliptic. 



J, 




At the vernal equinox, the star's longitude is less than the 
Sim's by 90, and it is therefore displaced away from the 
poles of the ecliptic and equator through a distance k" sin ?, 
its decimation being therefore decreased by k" sin I. At the 
autumnal equinox its declination is increased by k" sin L 

Hence the difference of the apparent declinations = 2k' 1 sin ?, 
and this is also the difference of the star's apparent meridian 
zenith distances. By observing these, k" may be found, 
7 being of course known. 

The value of k" is very approximately 20 - 493". 



304 A.STBONOMY. 

376. Relation between the Coefficient of Aberration 
and the Equation of Light. "We have seen (368) that 

JU 180x60x60 u r x 

T~ 'T '. ................. W ' 

where V is the coefficient of aberration in seconds, u the 
velocity of the Earth, V that of light, hoth of which we will 
suppose measured in miles per second. 

Now let r represent the radius of the Earth's orbit (sup- 
posed circular) in miles. Then in one sidereal year, or 365 J 
days, the Earth travels round its orbit through a distance 
2irr miles. Hence the Earth's velocity in miles per second is 



365ix 24x60x60 

Substituting in (i.), we have 

jfc" - 15 JL 
365 V 

But r/ Fis the time taken by the light to travel from the 
Sun to the Earth, measured in seconds, or the " equation of 
light." Hence, 

The coefficient of aberration in seconds 

= --- x number of seconds taken by Sun's light to 
365 4 reach Earth. 

Thus, by observing the retardation of the eclipses of 
Jupiter's satellites at superior conjunction, the coefficient of 
aberration can be found independently of the methods of 
375, the number of days (365^) in the sidereal year being 
of course known. 

The close agreement between the values found thus and 
by direct observation affords the strongest evidence in support 
of Bradley 's explanation of aberration. 

EXAMPLE. To find the coefficient of aberration in seconds, having 
given that light takes 8m. 20s. to travel from the Sun to the Earth. 

Here the required coefficient of aberration 
,// 15 x 500 7oOO 



THE DISTANCES OF THE SUN AKD STARS. 305 

377. To find the time taken by the light from a 
star to reach the Earth. It is sometimes convenient to 
estimate the distance of a star by the number of years which 
the light from it takes to reach the Earth. This may he 
determined from a knowledge of the star's parallax, and of 
the coefficient of aberration, without knowing either the Snn's 
distance or the velocity of light. 

Let the parallax of a star he = P" in seconds = P radians, 
and let the coefficient of aberration = k" seconds = k radians. 

Then, if r, d be the Earth's and star's distances from the 
Sun, we have 

p _ r 7 _ velocity of Earth 
d ' " velocity of light " 

Now, in one year, the Earth travels over a distance 2?rr ; 

27Tf 

.-. in one year light travels a distance ; 

/J 

.-. the number of years taken by light to travel from the 
star (distance d) to the Earth 



' \ k ~ 27TT ~~ 27TP 27TP"' 

The distance travelled by light in a year is sometimes 
called a " light-year." Hence, 

The product of a star's parallax and its distance in light- 
years is equal to the coefficient of aberration divided by 2?r. 

EXAMPLES. 1. To find how long the light would take to reach U8 
from a star having a parallax Ol". 
The required time, in years, 

1 fc" 10x20-49x7 
== = --- approximately 

STT 0-1 2 x 22 

= 32-6. 

2. To find the time taken by the light from the nearest star, 
a Centauri, taking its parallax as 075". 

The parallax is 7'5 times that of the star in the last question, 
therefore its distance is 10/75 as great, and the time taken by the 

light = ^ = 4-35 years. 

7'5 



306 ASTRONOMY. 

378. Relation between the Coefficient of Aber- 
ration, the Sun's Parallax, and the Velocity of Light. 

-It follows from 376 that if the coefficient of aberration 
k" be determined by observation, the fraction rjV is also 
known, independently of observations of the eclipses of 
Jupiter's satellites. And if F, the velocity of light, be deter- 
mined experimentally by the method of Foucault or Fizeau, 
the Sun's distance r can be found. Thus the Sun's parallax 
can be calculated from the coefficient of aberration and the 
velocity of light. And generally, if, of the four quantities, 
Sun's parallax, coefficient of aberration, velocity of light, and 
length of sidereal year in days, any three are observed, the 
value of the fourth may be deduced from them. 

In this manner Foucault, by his determination of the 
velocity of light, in 1862, found the Sun's parallax to be 8'86". 
Cornu, by experiments in 1874 and 1877, combined with the 
values for k" determined by Struve, obtained the values 
8-83" and 8*80" respectively. Hichelson's experiments make 
the parallax 8- 793". 

EXAMPLE. If the velocity of light = 186,000 miles per second 
and the Earth's radius (a) = 3,960 miles, to prove that the product 
of the Sun's parallax and the coefficient of aberration, both measured 
in seconds, is 180'35. 

The Sun's parallax P" = 18 * G0 * 60 






n- r 

15 r GO 



P " fc" = 18 x G0 x ( > x 60 a 200205 x GO 3000 



14G1* V 14G1 186000 

= 180-35. 

379. Planetary Aberration. The direction of any 
planet is affected by aberration, which is due partly to the 
motion of the Earth, and partly to that of the planet itself. 

For, during the time occupied by the light in travelling 
from a planet to the Earth, the planet itself will have moved 
from the position which it occupied when the light left it. 

We shall, however, show that the direction in which a 
planet is seen at any instant was the actual direction of the 
planet relative to the Earth at the instant previously when the 
lujlt left the planet. 



THE DISTANCES OF THE SUN AND STARS. 307 

Let t be the time required by the light to travel from 
the planet to the Earth. Let P, Q be the positions of the 
planet and Earth at any instant ; P', Q' their positions after 
an interval t. 

The light which leaves the planet when at P reaches the 
Earth when it has arrived at Q' ; the direction of the actual 
motion of the light is, therefore, along PQ. Eut PQ' and 
Q Q' are the spaces passed over by the light and the Earth* 




FIG. 128. 

respectively in the time t (and QQ' is so small an arc that it 
may be regarded as a straight line). Therefore 

QQ' : PQ' = velocity of Earth : velocity of light. 

Hence it follows from 368 that the line PQ represents the 
direction of relative velocity of the light with respect to the 
Earth. Therefore, when the Earth is at Q' the planet is seen 
in a direction parallel to PQ, and its apparent direction 
is exactly what its real direction was at a time t previously. 

The same is true in the case of the Sun or a comet, or 
any other body, provided that the time taken by the light 
from the body to reach the Earth is so small that the Earth's 
motion doe. not change sensibly in direction in the interval. 

The aberration of the planet at any instant is the angle 
between the apparent direction PQ and the actual direction 
P'Q. 



308 ASTRONOMY. 

EXAMPLE. To find the effect of aberration on the positions of (i.) 
the Sun, (ii.) Saturn in opposition, taking its distance from the 
Sun to be 9^ times the Earth's. 

(i.) The light takes 8m. 20s. to travel from the Sun to the Earth 

therefore the Sun's apparent coordinates at any instant are its actual 

coordinates 8m. 20s. previously. Thus, its apparent decl. and R.A. 

at noon are its true decl. and R.A. at 23h. 51m. 40s., or llh. 51m. 

,40s. A.M. 

Now the Sun describes 360 in longitude in 365 days. Hence, in, 
500 seconds it describes 20'492", and the Sun's aberration in longi- 
tude is 20'492". This is otherwise evident from the fact that the- 
Earth's way of the Sun is 90 ; and it is at rest, consequently its 
aberration = fc. 

(ii.) The distance of Saturn from the Earth at opposition is. 
= 9| 1, or 8 times the Sun's distance. Light travels over this 
distance in 8m. 20s. x 8 = 500x8|s. = Ih. 10m. 50s. Therefore, 
the apparent coordinates are the actual coordinates Ih. 10m. 50s. 
previously. 

Thus the observed decl. and R.A. at midnight (12h. 0*m. Os.) are the- 
true decl. and R.A. at lOh. 49m. 10s. 



380. Diurnal Aberration is due to the effect of the 
Earth's diurnal rotation about the poles on the relative velo- 
city of light. 

As the Earth revolves from west to east, the portion of the- 
motion of an ohserver due to this diurnal rotation is in the- 
direction of the east point E of the horizon. 

The effect of diurnal aberration can thus be investigated 
by methods precisely similar to those of 368, Staking the- 
place of U.* 

Hence, every star x is displaced by diurnal aberration, 
towards the east point E. And if x' be its displaced position,, 
then 

the displacement xx' = A sin x E, 
where 

,-, i n . volocitv of observer 

Circular measure of A = ^-- . 

velocity ot light 



* The student will find it useful to go through the various steps- 
of 368-371, considering the diurnal motion. 



THE DISTANCES OF THE STTN AND STARS. 309" 

Taking a for the Earth's radius, V for the velocity of light, 
let the observer's latitude be I. 

In a sidereal day (86164-1 mean seconds) the Earth's 
rotation carries the observer round a small circle, whose dis- 
tance from the Earth's axis is a cos , and whose circumference 
is, therefore, 'lira cos I. Hence, the observer's velocity 

= miles per second : 

86164-1 ( F 

1-irtt COS / 



.*. circular measure of A = 



86164-1 x V 1 
.'. A" (number of seconds in A] 

__ 180x60x60 x 2Tra cos/ 
TT 86164-1 7' 

Ida cos I , t 

= approximately. 

Thus, the coefficient of diurnal aberration varies as the 
cosine of the latitude. If K" denote the coefficient of 
diurnal aberration at the equator in seconds, we therefore,, 
have 



K" = - l5 * 39fi3 = n-32" 

V 186,000 

A" = K" cos I = O'32' cos L 

* Effect of Diurnal Aberration on Meridian Observations. 

The correction for diurnal aberration is greatest when the star 
is 90 from the east point, i.e., is on the meridian. In this case, 
the displacement is perpendicular to the meridian, and is equal 
to A". 

The star's meridian altitude is thus unaffected, but its time of 
transit is somewhat retarded at upper culmination, and (for a cir- 
cumpolar star) accelerated at lower culmination, since the star 
appears on the meridian, when it is really A" west of the meridian- 
The effect of diurnal aberration on the time of transit is thus equi- 
valent to that of a small collimation error A" in the Transit Circle. 

For a star on the equator, seen from the Earth's equator, the- 
retardation of the time of transit would be -^ K " seconds, = -g^ 
of a second nearly, and it would be difficult to observe such a small 
interval. 



310 ASTRONOMY. 

381. To determine the Coefficient of Diurnal Aber- 
ration "by Observations of the Azimuths of Stars 
when on the Horizon. 

When a star is rising or setting it is evidently displaced by 
diurnal aberration along the horizon towards the east point. 
Consider two stars, one of which rises S. of E., and the other 
"N. of E. It is evident that their rising points are drawn 
towards one another. But the stars set S. of W. and "N. 
of W., and their displacements are still towards the E. 
point ; hence, their setting points are separated away from 
one another. And, if the stars, at rising and setting, be 
carefully observed with an altazimuth, the difference between 
their azimuths at setting will exceed that between their 
azimuths at rising by an amount proportional to the diurnal 
aberration. From this, the coefficient of diurnal aberration 
may be found. 

The azimuths are unaltered by refraction ( 184), but the times 
of rising and setting are slightly altered by refraction. If the co- 
efficient of refraction be the same at both observations, however, 
the acceleration in rising will be equal to the retardation at setting, 
.and the refraction will increase the azimuths at rising and setting 
by the same amount ; thus the data will be unaffected. If the tem- 
perature of the air has changed considerably between rising and 
setting, it is only necessary to make the observations at equal 
intervals before and after the stars transit. 

.382. Relation between the Coefficients of Aberra- 
tion and the Sun's Parallax. We have evidently 
7T" _ velocity of diurnal motion at equator 
k" velocity of Earth's orbital motion 
But the velocities in miles, per sidereal day, are 2-n-a and 



This gives the coefficient of diurnal aberration at the equa- 
tor in terms of the coefficient of annual aberration and the 
Sun's parallax. Conversely, if it were possible to observe 
the coefficient of diurnal aberration accurately, we should 
thus have another way of finding the Sun's parallax. 

But the smallness of the diurnal aberration renders it im- 
possible to obtain good results by this method. 



THE DISTANCES OF THE SUN AND STARS. 311 



EXAMPLES. XI. 

1. Prove that cosec 876" = 23546 approximately, and thence that 
the distance of the Sun is nearly 81 million geographical miles, the 
angle 8' 76" being the Sun's parallax, and a geographical mile sub- 
tending 1' at the Earth's centre. 

2. Find the Sun's diameter in miles, taking the Sun's parallax as 
8'8", its angular diameter as 32', and the Earth's radius as 3,960 
miles. 

3. A spot at the centre of the Sun's disc is observed to subtend 
an angle of 5". What is its absolute diameter? 

4. Show, by means of a diagram, that the general effect of the 
Earth's diurnal rotation is to shorten the duration of a transit of 
Venus, and that this circumstance might be used to find the Sun's 
parallax. 

5. Supposing the equator, ecliptic, and orbit of Venus all to lie in 
one plane, and that a transit of Venus would last eight hours, at a 
point on the Earth's equator, if the Earth were without rotation ; 
show that, if the Sun is vertically overhead at the middle of the 
transit, the duration is diminished by about 9m. 55?s. owing to the 
Earth's rotation, taking the Sun's parallax to be 8'8", and the syn- 
odic period of Venus to be 586 clays. 

6. If the annual parallax be 2", determine the distance of the star, 
taking the Sun's distance to be 90,000,000 miles. Hence, deduce 
the distance of a star whose pamllax is 0'2". 

7. Find, roughly, the distance of a star whose parallax is 0'5", 
given that the Sun's parallax is 9", and the Earth's radius is 4000 
miles. 

8. The parallax of 61 Cyyni is O'o", and its proper motion, per- 
pendicular to the line of sight, is 5" a year; compare its velocity in 
that direction with that of the Earth in its orbit round the Sun. 

9. Account for the following phenomena : (i.) all stars in the 
ecliptic oscillate in a straight line about their mean places in the 
course of the year ; (ii.) two very near stars in the ecliptic appear to 
Approach and recede from one another in the course of the year. 

10. Suppose the velocity of light to be the same as the velocity of 
the Earth round the Sun. Discuss the effect on the Pole Star as- 
seen by an observer at the North Pole throughout the year. 



312 ASTRONOMY. 

11. Sound travels with a velocity 1,100 feet per second. Deter- 
mine the aberration produced in the apparent direction of sound to 
a person in a railway train travelling at sixty miles an hour, if the 
source of sound be exactly in front of one of the windows of the 
carriage. 

12. Show that, in consequence of aberration, the fixed stars 
whose latitude is I appear to describe ellipses whose eccentricity 
is cos I. 

13. How must a star be situated so as to have no displacement 
-due to (i.) aberration, (ii.) parallax? "Where must a star be so that 
the effect may be the greatest ? 

14. On what stars is the effect of aberration or parallax to make 
them appear to describe (i.) circles, (ii.) straight lines? 

15. Show that the effect of annual parallax on the position of a 
fitar may be represented by imagining the star to move in an orbit 
equal and parallel to the Earth's orbit, and that the effect of aber- 
ration may be represented by imagining it to revolve in a circle 
whose radius is equal to the distance traversed by the Earth while 
the light is travelling from the star. 

16. Supposing the star 17 Virginia to be situated (as it nearly is) 
at the first point of Libra, find the direction and magnitude of its 
displacement due to aberration about the 21st day of every month 
of the year, taking the coefficient of aberration to be 20*5". When 
is its aberration greatest ? 

17. At the solstices show that a star on the equator has no aber- 
ration in declination. If its R.A. be 22h., show that its time of 
transit is retarded at the summer and accelerated at the winter 
solstice by "68 of a second. 

18. If the coefficient of aberration be 20", and an error of 2,000 
miles a second be made in determining the-velocity of light, find, in 
miles, the consequent error in the value of the Sun's mean distance 
as computed from these data. 

19. Show that when a planet is stationary its position is unaffected 
by aberration. 

20. Taking the Earth's radius as 4,000, velocity of light 186,000 
miles per second, show that the coefficient of diurnal aberration at 
the equator is about one-third of a second. 



THE DISTANCES OF THE SILX AND STARS. 313 



MISCELLANEOUS QUESTIONS. 

1. Explain the following terms: asteroid, libration, lunation 
parallax, perihelion, planet's elongation, right ascension, synodical 
period, gyxygies. ztn : .th. 

2. Given that the R.A. of Orion's belt is 80, show by a figure its 
position at different hours of the night about March 21 and 
September 23. 

3. Prove that the number of minutes in the dip is equal to the 
number of nautical miles in the distance of the visible horizon. 

4. Show how to determine the latitude of a place by meridional 
observations on a circumpolar star, taking into account the refraction 



5. Show how to find longitude from lunar distances. The cleared 
lunar distance of a star at 8h. 30m. local mean time is 150'45", and 
the tabular distances are 150'0" at 6h. and 151'30' / at 9h. of Green- 
wich mean time. Find the longitude. 

6. At what time of the year can the waning moon best be seen ? 

7. On July 21 at 2 A.M. the Moon is on the meridian. What is 
the age of the Moon ? Indicate the position on the celestial sphere 
of a star whose declination is and whose R.A. is 30. 

8. Taking the distance of Venus from the Sun to be f of that of 
the Earth, find the ratio of the planet's angular diameters at superior 
and inferior conjunction and greatest elongation, and draw a series 
of diagrams showing the changes in the planet's appearance during 
a synodic period, as seen through a telescope under the same 
magnifying power. 

9. Defining a lunar day as the interval between two consecutive 
transits of the Moon across the meridian, find its mean length in 
(i.) mean solar, and (ii.) sidereal units. 

10. At what season is the aberration of a star least whose R.A. is 
$0 and whose declination is 60 ? 

11. Show that the constant of aberration can be determined by 
observation of Jupiter's satellites, without a knowledge of the radius 
of the Earth's orbit. 

12. How is it possible to calculate separately the aberration the 
constant of aberration being supposed unknown annual parallax, 
and proper motion of a star, from a long series of observations of the 
apparent place of a star ? 



3H ASTRONOMY. 



EXAMINATION PAPEK. XL 



1. Why is the method for finding the Moon's parallax not available- 
in the case of the Sun? Show how the determination of the 
parallax of Mars leads to the determination of the Sun's parallax. 

2. Show how the Sun's parallax can be found by comparing the 
times of commencement or of termination of a transit of Venus at two- 
stations not far from the Earth's equator. 

3. Show how the Sun's parallax can be found by comparing the- 
durations of a transit of Venus at two stations in high N. and S. 
latitudes. Why is this method not available when the transit is- 
central ? 

4. Distinguish between solar and stellar parallax. Towards what 
point does a star seem to be displaced by heliocentric parallax ? 
Find an expression for the displacement. 

5. Describe Bessel's method of determining the annual parallax 
of a fixed star. 

6. How might the Sun's parallax be determined by observations 
of the eclipses of Jupiter's satellites? 

7. Explain the aberration of light, and investigate the direction- 
and magnitude of the displacement which it produces on the- 
apparent position of a star. 

8. Show that owing to aberration a star in the pole of the ecliptic 
appears to describe a circle, and that a star in the ecliptic appears-, 
to oscillate to and fro in a straight line during the course of the year^ 

9. Show how the velocity of light may be determined from the 
aberration of.a star when the Sun's mean distance is known. 

10. Investigate the general effects of diurnal aberration due to- 
the Earth's rotation about its axis. In what direction nre stars- 
displaced by diurnal aberration ? Show that the coefficient of 
diurnal aberration at a place in latitude I is K cos I, where K is the- 
coefficient at the equator. 



DYNAMICAL ASTRONOMY, 



CHAPTER XII. 



THE ROTATION OF THE EARTH. 

383. Introductory. In the preceding chapters we have 
shown how the motions of the celestial bodies can be determined 
by actual ob crvation, and we have also explained certain 
resulting phenomena. But no use has yet been made of the 
principles of dynamics ; consequently we have been unable 
to investigate the causes of the various motions. In par- 
ticular, while we have assumed that the diurnal rotation of the 
stars is an appearance due to the Earth's rotation, we have 
not as yet given any definite proof that this is the only pos- 
sible explanation. 

The ancient Greeks accounted for the motions of the solar 
system by means of the Theory of Epicycles, according to 
which each planet moved as if it were at the end of a system 
of jointed rods rotating with uniform but different angular 
velocities. Suppose AB, BO, CD to be three rods jointed 
together at II, C. Let A be fixed ; let AB revolve uniformly 
about A ; let BC revolve with a different angular velocity 
about B ; and let CD revolve with another different angular 
velocity about C. Then, by properly choosing the lengths and 
angular velocities of the rods, the motion of J), relative to A, 
may be made nearly to represent the motion, relative to the 
Earth, of a planet. 

Copernicus (A.D. 1500 eirc.) was the first astronomer who 
explained the motions of the solar system on tho theory that 
the diurnal motion is due to the Earth's rotation, and that 
the Earth is one of the planets which revolve round the Sun. 
This theory was adopted by Kepler (A.D. 1609 circ.) whose 
laws of planetary motion have already been mentioned ( S26). 

A.STRON. Y 



316 ASTRONOMY. 

These laws were, however, unexplained until their true cause 
was found by Newton (A.B. 1687) by his discovery of the 
law of gravitation. 

384. Arguments in Favour of the Earth's Rota- 
tion. Without appealing to dynamical principles, the pro- 
bability of the Earth's rotation about its axis (87) may be 
inferred from the following considerations : 

(i.) If the Earth were at rest, we should have to imagine 
the Sun and stars to be revolving about it with inconceivably 
great velocities. If the Earth rotates, the velocity of a point 
on its equator is somewhere about 1,050 miles an hour. But 
since the Sun's distance is about 24,000 times the Earth's 
radius, the alternative hypothesis would require the Sun a 
body whose diameter is nearly 110 times as great as that of 
the Earth to be moving with a velocity 24,000 times as 
great, or about 25,000,000 miles an hour; while most of the 
fixed stars are at such distances from the Earth that they 
would have to move with velocities vastly greater than the 
velocity of light. It is inconceivable that such should 
be the case. 

(ii.) The diurnal rotations all take place about the pole, 
and are all performed in the same period a sidereal day. 
This uniformity is a natural consequence of the Earth's rota- 
tion, but it' the Earth were at rest, it could only be explained 
by supposing the stars to be rigidly connected in some manner 
or other. Were such a connection to exist it would be difficult 
to explain the proper motions of certain fixed stars, and the 
independent motions of the Sun, Moon, and planets. 

(iii.) By observing the motion of the spots on the Sun at 
different intervals, it is found that the Sun rotates on its axis. 
Moreover, similar rotations may be observed in the planets ; 
thus, Mars is known to rotate in a period of nearly 24 
hours. There is, therefore, nothing unreasonable in suppos- 
ing that the Earth also rotates once in a sidereal day. 

(iv.) The phenomenon of diurnal aberration affords a proof 
of the Earth's rotation. Were it not for the difficulty of its 
observation, this proof alone would be conclusive. 

We may mention that diurnal parallax ^ould be equally well 
accounted for if the celestial bodies revolved round the Earth; not 
so, however, diurnal aberration, 



THE B.OTATIOTT OF THE EARTH. 317 

385. Dynamical Proofs of the Earth's Rotation. 

The following is a list of the methods by which the Earth's 
rotation is proved from dynamical considerations : 

(1) The eastward deviation of falling bodies. 

(2) Eoucault's pendulum experiment. 

(3) Foucault's experiments with a gyroscope. 

(4) Experiments on the deviation of projectiles. 

(5) Observations of ocean currents and trade winds. 

(6) Experiments on the differences in the acceleration of 
gravity in different latitudes, due to the Earth's centrifugal 
force, as observed by counting the oscillations of a pendulum ; 
combined with 

(7) Observations of the figure of the Earth. 

386. The Eastward Deviation of Falling Bodies. 

If the Earth is rotating p.Jbout its polar axis, those points 
which are furthest from the Earth's axis move with greater 
velocity than those which are nearer the axis. Hence the 
top of a high tower moves with slightly greater velocity than 
the base. If, then, a stone be dropped from the top of the 
tower, its eastward horizontal velocity, due to the Earth's 
rotation, is greater than that of the Earth below, and it falls 
to the east of the vertical through its point of projection. 
The same is true when a body is dropped down a mine. This 
eastward deviation, though small, has been observed, and 
affords a proof of the Earth's rotation. 

Consider, for example, a tower of height h at the equator. If a be 
the Earth's equatorial radius, the base travels over a distance 2ira in 
a sidereal clay, owing to the Earth's rotation, while the top of the 
tower describes 2ir(a + h) per sidereal day. Thus, the velocity at 
the top exceeds that at the bottom by 2irh per sidereal day. 

If h be measured in feet, the difference of velocities is irh/'SQOO indies 
per sidereal second, and is sufficiently great to cause a small but 
perceptible deviation when a body is let fall from a high tower. 

The earliest experiments were too rough to show this deviation, 
and were, therefore, used as evidence against, instead of for, the 
Earth's rotation. The deviation can only be observed in experi- 
ments conducted with very great care, and it is very difficult to 
measure. Its amount is largely modified by the resistance of the 
air and other causes, and therefore differs considerably from that 
by theory. 



318 



ASTRONOMY. 



387. Foucault's Pendulum Experiment. In 1 85 1 , M . 

Foucault invented an experiment by which the Earth's rota- 
tion is very clearly shown. A pendulum is formed of a large 
metal ball suspended by a fine wire from the roof of a high 
building, and is set in motion by being drawn on one side and 
suddenly released ; it then oscillates to and fro in a vertical 
plane. If now the pendulum be sufficiently long and heavy 
to continue vibrating for a considerable length of time, the 
plane of oscillation is observed to very gradually change its 
direction relative to the surrounding objects, by turning 
slowly round from left to right at a place in the northern 
hemisphere, or in the reverse direction in the southern. If 
the experiment is performed in latitude ?, the plane of 
oscillation appears to rotate through 15 x sin I in a sidereal 
hour, 360 sin lin a sidereal day, or 360 in cosec I sidereal 
days. This apparent rotation is accounted for by the Earth's 
rotation, as follows. 

(i.) Let us first imagine the experiment to be performed at 
the north pole of the Earth. Let the pendulum AB be 
vibrating about A in the arc BB' in 
the plane of the paper. The only forces 
acting on the bob are the tension of 
the string BA and the weight of the 
bob acting vertically downwards ; both 
are in the plane of the paper. The 
Earth's rotation about its axis CA pro- 
duces no forces on the bob. Hence 
there is nothing whatever to alter the 
direction of the plane of oscillation ; 
this plane therefore remains fixed in 
space. But the Earth is not fixed in 
space ; it turns from west to east, making 
a complete direct revolution in a sidereal 
day. Hence the plane of the pendulum's oscillation appears, 
to an observer not conscious of his own motion, as though it 
rotated once in a sidereal day, in the reverse or retrograde 
direction (east to west). If, however, he were to compare 
the plane of oscillation not with the Earth but with the 
stars, whose directions are actually fixed in space, he would 




FIG. 129. 



THE ROTATION OF THE EARTH. 



319 



see that it always retained the same position relatively to 
them. 

Since, then, the pendulum at the pole of the Earth appears to 
follow the stars, it evidently appears to rotate in the same 
direction as the hands of a watch at the north pole, and in 
the direction opposite to the hands of a watch at the south 
pole. 



7 



(ii.) Next suppose the experiment performed at the Earth's 
equator. If the bob be set swinging in 
the plane of the equator, take this as the 
plane of the paper (Fig. 130). The 
direction of the vertical AQC is now 
rotating about an axis through C per- 
pendicular to the plane of the paper ; 
hence it always remains in that plane. 
Hence there is nothing whatever to 
turn the plane of oscillation of the pen- 
dulum out of the plane of the Earth's 
equator. It therefore continues always 
to pass through the east and west points, 
and there is no apparent rotation of the 
plane of oscillation. 



FIG. 130. 



If the pendulum do not swing in the plane of the equator, 
the explanation is much more complicated. As the Earth 
rotates, the direction of gravity performs a direct revolution 
in a sidereal day. Hence, relative to the point of support, 
gravity is gradually and continuously turning the bob west- 
wards, in such a way as to keep its mean position always 
pointed towards the centre of the Earth. When the bob is 
south of its position of equilibrium, this westward bias tends 
to turn the plane of oscillation in the clockwise direction, 
but when the bob is north of the mean position, the west- 
ward bias has an equal tendency to turn the plane in the 
reverse direction. Consequently the two effects counter- 
act one another, and therefore produce no apparent 
rotation of the plane of oscillation relative to surrounding 
objects. 



320 



ASTRONOMY. 



Qu 



(in.} Lastly, consider the case of an observer in latitude 
I (Fig. 131). Let w denote the 
angular velocity with which the 
Earth is rotating about its polar axis 
CP. It is a well-known theorem 
in Rigid Dynamics that an angular 
velocity of rotation about any line 
maybe resolved into components about 
any two other lines, by the parallelo- 
gram law, in just the same way as a 
linear velocity or a force along that 
line; this theorem is called the 
Parallelogram of Angular Velocities. 
Applying it to the angular velocity 
n about CP, we may resolve it into 
two components 




FIG 131. 



and 



n cosPCO or n sin I about CO, 



n sin PGO or n cos I about a line CO' perpendicular to CO, 

and we may consider the effects of the two angular velocities 
separately. 

As in case (i.), the component nsin I causes the Earth to 
turn about CO, without altering the direction in space of the 
plane of oscillation ; this plane, therefore, appears to rotate 
relatively in the reverse or retrograde direction, with 
angular velocity n sin I. As in case (ii.)> the angular velocity 
n cos I about CO' produces no apparent rotation of the plane 
of oscillation relative to the Earth. Hence the plane of oscilla- 
tion appears to revolve, relative to the Earth, with retrograde 
angular velocity n sin I. 

But the angular velocity n = 15 per sidereal hour 
= 360 per sidereal day. 

Therefore the plane of oscillation turns through 

15 sin I per sidereal hour = 360 sin I per sidereal day, 

360 

fcnd its period of rotation = , - 

n sin I 

= cosec I sidereal days. 



THE IIOTATI02? OF THE EARTH. 321 

388. The Gyroscope or Gyrostat is another apparatus 
used by Foucault to prove the Earth's rotation. It is simply 
a large spinning-top, or, more correctly, a heavy revolving 
wheel IT (Fig. 132), whose axis of rotation AB is supported 
by a framework, so that it can turn about its centre of gravity 
in any manner. Thus, by turning the wheel and the inner 
frame A CBD about the bearings CD, and then turning the 
outer frame DECF about the bearings EF, the axis AB (like 
the telescope in an altazimuth or equatorial) can be pointed 
in any desired direction. The three axes A B, CD, EF all 
pass through the centre of gravity of the top ; hence its weight 
is entirely supported, and does not tend to turn it in any 
way; and the bearings A, B, C, D, E, JPare very light, and 
so constructed that their friction may be as small as possible. 
The top may be spun by a string in the usual way, and it 
continues to spin for a long time. 




FIG. 132. 

When a symmetrical body, such as the wheel H, is revolv- 
ing rapidly about its axis of figure, and is not acted on by 
any force or couple, it is evident that no change of motion 
can take place, and therefore the axis of rotation AB must 
remain fixed in direction. This is the case with the gyro- 
scope, for, from the mode in which the weight of the wheel 
is supported, there is no force tending to turn it round. 

When the experiment is performed it is observed that the 
axis AB follows the stars in their diurnal motion ; if pointed 
to any star, it always continues to point to that star, its posi- 
tion relative to the Earth changing with that of the star. 
Hence it is inferred that the directions of the stars arc fixed 
in space, and that the diurnal motion is not due to them, but 
to the rotation of the Earth. 



322 ASTllOXOMY. 

389. If while the gyroscope is spinning rapidly any attempt be made 
to alter the direction of the axis of rotation AB by pushing it in any 
direction, a very great resistance will be experienced, and the axis 
will only move with great difficulty. This shows that the small 
friction at the pivots CD, EF can have but little effect in turning 
the axis of the top, and therefore the gyroscope spins as if it were 
practically free, as long as its angular velocity remains considerable. 

The following additional experiments with the gyroscope can be 
also used to prove the Earth's rotation. 

Experiment 1. Let the hoop CEDF be steadily rotated about the 
line EF. The line AB is no longer free to take up any position, for 
the pivots and D obviously force it always to be in a plane through 
EF and perpendicular to plane CEDF. Hence the axis of rotation 
is no longer able to maintain always the same position, unless that 
position coincides with EF. The result 
is that the axis gradually turns about 
CD till it does coincide with EF, the di- 
rection of rotation of the wheel being 
the same as that in which frame is forced 
to revolve. It will then have no further 
tendency to change its place. Of course 
we suppose the hoop turned so quickly 
that the effect of the slow motion of the 
Earth is imperceptible. 

Experiment 2. We may now repeat 
Experiment 1, using the Earth's rota- 
tion. Let the framework CEDFbe fixed 
in a horizontal position, the line CD 

being held pointed due east and west. The axis AB is then 
free to turn in the plane of the meridian. Now, owing to the 
Earth's rotation, the framework carrying CD is turning about the 
Earth's polar axis, and this causes the top to turn till its axis points 
to il\e celestial poles. The result of experiment agrees with 
theory, thus affording a further proof of the Earth's rotation about 
the poles. 

Experiment 3. Let the framework CEDF be clamped in a vertical 
plane. The axis AB can then turn in a horizontal plane, but it cannot 
point to the pole. It will, however, try to point in a direction 
differing as little as possible from the direction of the Earth's axis, 
and will therefore turn till it points due north and south. This has 
also been verified by actual observation. 

Experiments 3 and 2, if performed with a sufficiently perfect 
gyroscope, would enable us to find the north point, and then to find 
the celestial pole, and thus determine the latitude without observing 
any stars. By means of Foucault's pendulum experiment we could 
also (theoretically) determine the latitude,. 







THE ROTATION OF THE EARTH. 323 

390. The Deviation of Projectiles. If we suppose a 
cannon ball to be fired in any direction, say from the Earth's 
North Pole, the ball will travel with uniform horizontal velocity 
in a vertical plane. But, as the Earth rotates from right to left, 
the object at which the ball was aimed will be carried round 
to the left of the plane of projection, and therefore the ball 
will appear to deviate to the right of its mark. At the South 
Pole the reverse would be the case, because in consequence 
of the direction of the vertical being reversed, the Earth would 
revolve from left to right ; hence the ball would deviate to 
the left of its mark. At the equator no such effect would 
occur. 

The deviation, like that in Foucault's pendulum, depends on 
the Earth's component angular velocity about a vertical axis 
at the place of observation, and this component, in latitude I, 
is n sin ( 387, iii.). Now the Earth rotates about the poles 
through 15" per sidereal second. Hence, if t be the time of 
flight measured in sidereal seconds, the deviation is 

= nt sin I = 15". t sin , 

and it is necessary to aim at an angle 15". t sin I to the left of 
the target in N. lat. /, or 15". t sin/ to the right in S. lat. I. 
The formula is sufficiently approximate even if t be measured 
in solar seconds. It is necessary to allow for this deviation 
in gunnery thus affording another proof of the Earth's 
rotation. 

391. The Trade Winds are due to a similar cause. The 
currents of air travelling towards the hotter parts of the 
Earth at the equator, like the projectiles, undergo a deviation 
towards the right in the northern hemisphere, and towards the 
left in the southern. This deviation changes their directions 
from north and south to north-east and south-east respectively. 
In a similar manner the Earth's rotation causes a deviation 
in the ocean currents, making them revolve in a direction 
opposite to that of the Earth's rotation, which is "counter 
clockwise " in the N. and " clockwise " in the S. hemisphere. 
The rotatory motion of the wind in cyclones is also due to 
the Earth's rotation. 



324 ASTEONOMY. 

392. Centrifugal Force. If a body of mass m is revolving 
in a circle of radius r with uniform velocity v under the action 
of any forces, it is known that the body has an acceleration v*/r 
towards the centre of the circle.* Hence the forces must 
have a resultant mv*/r acting towards the centre, and they 
would be balanced by a force mv 2 /r acting in the reverse 
direction, i.e., outwards from the centre. This force is called 
the centrifugal force. 

Thus, in consequence of its acceleration, the body appears to 
exert a centrifugal force outwards. If it be attached to the 
centre of the circle by a string, the pull in the string is mv*/r. 
If m be measured in pounds, r in feet, and v in feet per 
second, then mv^/r represents the centrifugal force in poundals. 
Similarly, in the centimetre-gramme-second system of units, 
mv*/r is the centrifugal force in dynes. 

If n represent the body's angular velocity in radians per 
second, v = nr, and the centrifugal force is therefore mn*r. 

393. General Effects of the Earth's Centrifugal 
Force. If the Earth were at rest the weight of a body 
would be entirely due to the Earth's attraction. But in con- 
sequence of the diurnal rotation the apparent weight is the 
resultant of the Earth's attraction and the centrifugal force. 

Let QOR represent a meridian section of the Earth 
(Fig. 134). Consider a body of mass m supported at any 
point on the Earth's surface. Since the Earth is nearly, 
but not quite, spherical, the force ^ of the Earth's attraction 
on a unit mass is not directed exactly to the Earth's centre, 
but along a line OK. But, owing to the body's central 
acceleration along OM, the force which it exerts on the 
support is not quite equal to the Earth's attraction mg^ 
but is compounded of mg Q acting along OJT, and the centri- 
fugal force m . ri* . MO acting along M 0. 

On KQ, take a point G such that 
KG 



* See any book on Dynamics. 



THE ROTATION OF THE EARTH. 



325 



then, by the triangle of force?, OG is the direction of the re- 
sultant force exerted by the body on its support, and this 
force is the apparent weight of the body. Hence, also OG 
represents the apparent direction of gravity, or the verti- 
cal as indicated by a plumb-line. Producing GO, KO to Z, 
Z", we see that the effect of centrifugal force is to displace the 
vertical from Z" towards the nearest pole (P). 

The angle ZGQ measures the (geographical) latitude of 
the place, and is greater than Z ' KQ, which would measure 
the latitude if the Earth were at rest. Hence the apparent 
latitude of any place is increased ly centrifugal force. 




I 



FIG. 134. 

Again, if the apparent weight be denoted by mg, we have, 
by the triangle of forces, 

g :y = GO: KQ-, 

now from the figure it is evident that G < IL(), and there- 
fore g < # . Hence the apparent iveight of a body is diminished 
by centrifugal force. 

394. Effect on the Acceleration of a Falling Body. 

If a body is falling freely towards the Earth near 0, the 
whole acceleration of its motion in space is due to the Earth's 
attraction, and is # , along OK. But the Earth at has 
itself an acceleration ri*OM to wards 31. Hence the accelera- 
tion of the body relative to the Earth is the resultant of </ 
along (7, and w 2 . M along J/0, and is therefore g along 
G. Hence the body approaches the Earth with acceleration 
g along OG. Therefore its relative acceleration is the accele- 
ration dve to its apparent weight, that is, to the resultant of 
the Earths attraction and centrifugal force. 



326 ASTEONOMT: 

395. To find the loss of weight of a body at the 
equator, due to centrifugal force. At the equator 
centrifugal force is directly opposed to gravity ; hence, if a 
denote the Earth's radius CQ, 

ff = &*-***> 

Now we have roughly 

ff = 32-18 feet per second per second, 
a = 3963 miles = 3903 x 5280 feet, 
and n = 2?r radians per sidereal day 

= radians per mean solar second. 

Hence A = 3963x5280x4.' = . 
86164 x 86164 

and therefore - = = nearly. 

ff 32-18 28J 

Hence </ = f/ -JL </ , 

or the effect of the Earth's rotation is to decrease the weight of a 

body by about of the whole. 

289 

For rough calculations it would be sufficient to take g = 32'2, 
a = 3960 miles, and to neglect the difference between a solar and a 
sidereal day. This would give -fa, as bei'ore. 

396. To find approximately the loss of weight of a 
body and the deviation of the vertical due to centri- 
fugal force in any given latitude. 

Let 1= QGO = astronomical latitude of 0; D GOK 
= ZOZ" = deviation of vertical from direction of Earth's 
attraction, or increase of latitude due to centrifugal force. 
We have OM = CO cos COM 

= a COB I approximately ; 

where a is the Earth's radius, since the Earth is very nearly 
spherical, and Z COM is therefore very nearly equal to the 
latitude 1. Therefore centrifugal force per unit mass at 
= n a . ON = n* . a cos I = ^ ^ cos I (from 395). 



THE ROTATION OF TFE EARTH. 



327 



Resolving along 06r, we have, if y be the Earth's attraction 
per unit mass at 0*, 

g = 0Q cos J)n* . OlTcos I 
= f7 cos 2 1 approximately 

(since D is small, and .*. cos D = 1 nearly). 

Hence, in latitude I, the Earth's rotation dim : nishes theweiyld 

of a body ly approximately cos 2 1 of itself. 

289 

Resolving perpendicular to G, 
we have 

g~ sin D w 2 M sin I = ; 

tfa cos I sin I 
.'. sin J> = - 



1 sin 2? 
~ 289 2 

Since d is small, this gives 
approximately 

1 sin 21 
circular measure ot d = -^- - 




FIG. 135. 



" (number of seconds in 
180x60x60 



289X27T 
206265 



sin 21 



578 



sin 21 = 357" sin 



Hence the deviation D = 5' 57". sin 21, and this is the in- 
crease of latitude due to centrifugal force. 

COROLLAEY. The deviation of the vertical due to centri- 
fugal force is greatest in latitude 45 (v sin 2? = 1), and is 
there 5' 57". 



* Since the Earth is not quite spherical, g is not the same at as 
at the equator. The difference may be neglected, however, when 
multiplied by the small constant jy. 



328 ASTRONOMY. 

397. Figure of the Earth. In 114 we stated that the 
form of the Earth has been observed to be an oblate spheroid. 
Now it has been proved mathematically that a mass of 
gravitating liquid- when rotating takes the form of an oblate 
spheroid whose least diameter is along its axis of rotation. 
Thus the Earth's form may be accounted for on the theory that 
the Earth's surface was formerly in a fluid or molten state, 
and that it then assumed its present form, owing to its diurnal 
rotation. We thus have another argument in favour of the 
Earth's rotation ; but it is only fair to say that this theory 
of the Earth's origin has not been satisfactorily demonstrated. 

It accounts satisfactorily, however, for the form of the 
surface of the ocean. 

This theory may be illustrated by the following general considera- 
tions. When a mass of liquid is acted on by no forces beyond the 
attractions of its particles, it is easy to realize that the whole is in 
equilibrium in a spherical form, being then perfectly symmetrical. 

If, however, the fluid be rotating about the axis PGP', the centri- 
fugal force tends to pull the liquid away from this axis and towards 
the equatorial plane. The liquid would, therefore, fly right off, but 
its attraction is always trying to pull it back to the spherical form. 
Hence, the only effect of centrifugal force (which, for the Earth, is 
small compared with gravity) is to distort the liquid from its spheri- 
cal form by pulling it out towards the equator ; and it is therefore 
reasonable to suppose that the fluid will assume a more or less oblate 
figure, whose equatorial is greater than its polar diameter. 

It may also be remarked that the form assumed by the liquid 
would be such that the effective force of gravity (i.e., the resultant 
of the attraction and centrifugal force) on the surface would every- 
where be perpendicular (i.e., normal) to the surface. 

*398. Gravitational Observations. If the Earth were a sphere, 
its attraction g would everywhere tend to its centre, and would be 
of the same intensity at all points on its surface, while the variations 
in g, the apparent intensity of gravity, would be entirely due to the 
Earth's centrifugal force, its value in latitude I being proportional 
to 1 -^-g cos- 1 ( 396). By comparing the values of g at different 
places, we should then be able to demonstrate the Earth's centri- 
fugal force, and hence prove its rotation. But, owing to the Earth's 
ellipticity, its attraction gr does not pass through the centre, except 
at the poles and equator, and its intensity is not everywhere con- 
stant. It is, therefore, important to determine experimentally the 
values of g at different stations. By allowing for centrifugal force, the 
corresponding values of the Earth's attraction g can be found, and 
the variations in its intensity at different places afford a measure of 



THE ROTATION OF THE EARTH. 329 

tlie amount by which the Earth differs from a sphere. We thus 
have a gravitational method of finding the Earth's ellipticity. 

But the Earth's ellipticity can also be determined by direct obser- 
vation, as explained in Chapter III., Section III. The agreement 
between the results thus independently obtained furnishes another 
proof of the Earth's rotation. 

In consequence of the EarthVellipticity it is found (by observa- 
tion) that the difference in the intensity of gravity between the polo 
and equator is increased from ^-g- to -3^-5 of the whole. 

399. To compare the Intensity of Gravity at different places. 
The intensity of gravity may be measured by the force with which 
a body of unit mass is drawn towards the Earth. This cannot be 
measured by weighing a body with a common balance, because the 
weights of the body and of the counterpoise, by means of which it is 
weighed, are equally affected by variations in the intensity of gravity, 
and two bodies of equal mass will, therefore, balance one another 
when placed in the scale pans, no matter what be the intensity of 
gravity. In fact, by weighing a body with weights in the ordinary 
way, we determine only its mass, and not the absolute force with 
which it is drawn to the Earth. 

We might determine the intensity of gravity by means of a 
" spring balance," for the elasticity of the spring does not depend on 
the intensity of gravity, and therefore the extension of the spring 
gives an absolute measure of the force with which the body is drawn 
towards the Earth. If the apparatus were to support a mass of one 
pound, first at the equator and then at the pole, the force on it 
would be greater at the latter place by about - l ^, and this spring 
would thcro be extended about -j--^ more. It would be very difficult 
to construct a spring balance sufficiently sensitive to show such a 
small relative difference of weight, but it has been done. 

Aticood's machine might be used to find g, but this method is not 
capable of giving very accurate results. 

The most accurate method of finding g is by timing the oscillations 
of a pendulum of known length. 

[* A theoretical simple pendulum, consisting of a mere heavy par- 
ticle of no dimensions, suspended by a thread without weight, is of 
course impossible to realize in practice, but the difficulty is over- 
come by the use of a pendulum called Captain Rater's Reversible 
Pendulum. This pendulum is a bar which can be made to swing 
ab ut either of two knife-blades fixed ?.t opposite sides of, but un- 
equal distances from, its centre of gravity, and it is so loaded that 
the periods of oscillation, when suspended from either knife-edge, 
are equal. It is then known that the pendulum will swing about 
either knife-edge in just the same manner as if it were a simple 
pendulum whose whole mass was concentrated at the other knife- 
edge. The distance between the knife-edges is, therefore, to be 
regarded as the length of the pendulum.'] 



330 

400. Oscillation? of a Simple Pendulum. In a simple 
pendulum, formed of a small heavy particle suspended by a 
fine light thread of length I, the period of a complete oscillation 
to and fro is 



the time of a single swing or " leat " being of course half of 
this. 

Hence by observing the time of oscillation t and measuring 
the length I, the intensity of gravity g can be found. 

By the " seconds pendulum " is meant a pendulum in 
which one beat occupies one second, hence a complete 
oscillation occupies two seconds. 

EXAMPLK. Having given that the length of the seconds pendulum 
is 99'39 centimetres, to find g in centimetres per second per second. 

t = 2nVZ/7 = 2 seconds, and I = 99'39 cm., 
.-. g =^i = 99-39 x (3-1416) 2 = 981. 

It is often necessary to compare the lengths of two 
pendulums whose periods of oscillation are very nearly equal, 
to find the effect of small changes in the length of a pendulum 
due to variations in temperature, or, in comparing the intensity 
of gravity at different places, to find the effect of a small 
alteration in the value of g on the period of oscillation and on 
the number of oscillations in a given interval. If the differ- 
ences are small, the calculations may be much simplified by 
means of the following methods of approximation.* 

401. To find the change in the time of oscillation of 
a pendulum, and in the number of oscillations in a 
given interval, due to a small variation in its length 
or in the intensity of gravity. 

If t be the time of a complete oscillation of a pendulum of 
length J, we have, by 400, 

? = 47T 2 - (i). 

* The same results can of course be obtained by means of the 
differential calculus. 



THE HOTATIOX OF THE EARTH. 331 

(i.) Suppose the length increased to 1' 9 and let t' be the new 
period of oscillation. We have 

t* = 47T 3 -. 

g 
Therefore, by division, 

*"_r 

*T*T 

and therefore also 

I'*-?-,, ..t + t I'-l 

~1T ~ t} ~T 

These formulae are exact. But if I' is very nearly equal to , 
t' is very nearly equal to t, and therefore, putting t + t'= 2t, 
we have approximately 

Q t'-t_l'-l 

T T' 

whence, if t, I be known, the change V t, consequent on the 
increase of length I' I, may be readily found approximately 
without the labour of extracting any square roots. 

(ii.) Suppose the intensity of gravity increased to g', the 
length I being unaltered, and let t' be the new period. Since 



we have, by division, 
and therefore also 



But, if , g are very nearly equal to t', /, this gives 
approximately 

2 """ = - . 
* f/ 

ASTEON. Z 



332 ASTEONOMf. 

(iii.) If I and g both vary, becoming V and g', we have, in 

like manner 



Therefore also 




I 9 

or approximately, if the variations are small, 

2 *-* = r ~ ? - ff '- ff 

t I <J ' 

showing that the effects of the two variations may be con- 
sidered separately. 

(iv.) If n, ri be the number of complete oscillations of the 
pendulum in a given interval T, and if, in consequence of the 
change, this number be altered to w', we have 
nt = nt'=T, 
n _ t 

IT-T', 

n '- n t t' 

whence - = -7- 

n t 

If t' is very nearly equal to t, this gives approximately 



which determines the number of beats gained by the pendulum 
in the time T, in consequence of the variations, the original 
number n being supposed known. 

EXAMPLE. To find the number of oscillations gained or lost in an 
hour by the pendulum of the Example of 400, supposing (i.) its 
length increased to 1 metre; (ii.) the acceleration of gravity in- 
creased to 982 ; (iii.) both changes made simultaneously. 

(i.) The pendulum beats seconds ; therefore it performs 3600 half 
oscillations or 1800 whole oscillations in an hour. Also V = lOO'OO 
l'-l = 0-61, g'-g = Q, 



THE ROTATION OF TUB EARTH. 333 

Hence, if n' be the new number of oscillations in an hour, 



'-1800 = _0^1 = _ 9^1, ) = _ OjGl 

1800 21 21' 200* 

.-. n' -1800 = -9 x -61 = -5'49. 

Hence the pendulum loses nearly 5 oscillations in an hour, and the 
number of oscillations is therefore 1794$. 

... . TT tt'-1800 g'-g 982-981 1 

-~ * = - = ~ ; 



'- 1800 = =-9 = 1 nearly. 



Hence the pendulum gains 1 oscillation in an hour, the total 
number of oscillations being 1801. 

(iii.) Since from the first cause the pendulum loses 5J oscillations 
and from the second it gains 1 oscillation, therefore on the whole it 
loses 5^ 1 or 4^ oscillations per hour. It therefore performs 1795^ 
oscillations or 3591 beats per hour. 

402. To compare the times of oscillations of two 
pendulums whose periods are very nearly equal. 

If two pendulums of nearly equal periods are simultaneously 
started swinging in the same direction, the one whose period 
is a little the shortest will soon begin to swing before the 
other. After some time it will gain a half oscillation, and 
the pendulums will then be swinging in opposite directions. 
After another equal interval, the quicker pendulum will have 
gained one whole oscillation on the slower, and both will 
be again swinging together in the same direction. Similarly, 
every time the quicker pendulum has gained an exact number 
of complete oscillations on the slower, both will be swinging 
together in the same direction. Thus, the number of coinci- 
dences, or the number of times that the two pendulums are 
together, in any interval, is equal to the number of complete 
oscillations (to and fro) gained by the quicker pendulum over 
the slower, i.e., the difference between the numbers of com- 
plete oscillations performed by the two pendulums. 

Thus, if n, n 1 be the number of oscillations of the slower 
and faster pendulums in any given interval, then n' n is the 
the number of oscillations gained by the latter, and is, there- 
fore, the number of " coincidences." If either of the num- 
bers n, n' is known, we can, by counting the coincidences, 
find the other number. 



334 ASTBONOMY. 

403. To find g, the acceleration of gravity, the 

simplest plan is to use a Captain Kater's pendulum, the beat 
of which is very nearly one second. By counting the 
"coincidences" of the pendulum with the pendulum of a 
clock regulated to beat seconds during, say, an hour (as 
shown by the clock) the exact time of oscillation can be 
found. Moreover, from the number of beats gained or lost, 
and the observed length of the pendulum, we may calculate 
the amount by which the length must be increased or decreased 
in order to make the pendulum beat seconds. The length of 
the seconds pendulum is thus known, and the value of g can 
be found. 

The reason for using two pendulums is that it would be extremely 
difficult to measure the length of the pendulum of the clock, and it 
would be equally difficult to find the period of oscillation of a pen- 
dulum without comparing it with that of a clock, whose rate can be 
regulated daily by astronomical observations. 

404. To compare the value of g at two different 
stations, the simplest plan is to determine the number of 
seconds gained or lost in a day by a clock after it has been 
taken from one station to the other, the length of the pen- 
dulum remaining the same. If n, ri be the number of 
seconds marked by the clock in a day at the two places, we 

have exactly =--, 

*' 9 



or approximately, 

n g 

whence the ratio of g' to g may be found. 

Here there is no necessity to use a Captain Kater's pendulum, 
because the length of the pendulum is not required ; hence the 
ordinary compensating pendulum of the clock answers the purpose. 
If a non-compensating pendulum were used, it would be necessary I 
to make allowance for any change in the length of the pendulum \ 
due to variations ID temperature. 



THE ROTATION OF THE EARTH. 335 

EXAMPLES. XII. 

iat af tPr n CaUl -rf Pen ^ lu bein S sefc vibrating in latitude 30, show 
at atter one sidereal day it is ao-ain vihr.r,nr ,v __ '_i7_. 



that af r n-rf n S se vrang n 

K?5 ^^ 



place m the northern hemisphere the pendulum which 
the same direction as the hands of a watch will have 
apparent angular velocity, and will gain two complete 

e 



sthrhsro p enuum 

the North p g nt A *l C n8ld f' '? the first P lace > the Phenomena at 
so^^mf 80 deSCnbeth P ondin g phenomena in the 



. 

3. If a railway is laid along a meridian, and a train is travelling 

anTast^To " ^^^ P ole > "yeBtigate whether it wHIexert 
an eastward or a westward thrust on the rails, and why 

4. A bullet is fired in N. latitude 45, with a velocity of 1 600 frpf 

5 Ihn ft ' f f?i I how many feet it will deviate to the right. 

a bodv 7ti * 6 Ear S W6re t0 r tate ^venteen times as fist 
a body at the equator would have no weight. 

6. If the Earth were a homogeneous sphere rotating so fast thaf 



^ f * he last q uest! . show that the Earth's 






336 ASTRONOMY. 



EXAMINATION PAPEB. XII. 

1. Give reasons for supposing that the diurnal rotation of th( 
heavens is only an appearance caused by a real rotation of the 
Earth. Name methods by which it has been claimed that this ii 
proved. 

2. Describe the gyroscope experiment, and the gyroscope. 

3. Give any theoretical methods of determining latitude withou 
observing a heavenly body. 

4. Describe Foucault's experiment for exhibiting the Earth' 
rotation ; and find the time of the complete rotation of the plane c 
vibration of a simple pendulum fieely suspended in latitude 60. 

5. Having given that the Earth's circumference is 40,000 kilc 
metres, find the acceleration of a body at the equator due to th 
Earth's rotation in centime bres per second per second, and takinj 
<7 , the acceleration of gravity, to be 981 of these units, deduce t 
ratio of centrifugal force to gravity at the equator. 

6. What is meant by the vertical at any point of the Earth 
surface ? Supposing the Earth to be a uniform sphere revolvir 
round a diameter, calculate the deflection of the vertical from t 
normal to the surface. 

7. State what argument is drawn from the Earth's form to suppo 
the hypothesis of its rotation. 

8. Why is it that the intensity of gravity is less at the equal 
than in higher latitudes ? Show that the alteration in the appare 
weight of a body due to centrifugal force varies nearly as cos 
where I is the latitude, and state the ratio of centrifugal force 
gravity at the equator. 

9. If a body is weighed by a spring balance in London and 
Quito, a difference of weight is observed. Why is this not observed" 
an ordinary pair of scales be used ? 

10. Show that an increase in the intensity of gravity will cat 
a pendulum to swing more rapidly, and vice vers&. If the accele: 
tion of gravity be increased by the small fraction l/r of its vali 
show that a pendulum will gain one complete oscillation in every 



CHAPTER XIII* 



THE LAW OF UNIVERSAL GRAVITATION. 

SECTION I.^T/ie Earttis Orlital Motion Kepler's Laws and 
their Consequences. 

405. Evidence in favour of the Earth's Annual 
Motion round the Sun. The theory that the Earth is a 
planet, and revolves round the Sun, was propounded by 
Copernicus (circ. 1530) and received its most convincing 
proof, over 150 years later from Newton (A.D. 1687), who 
accounted for the motions of the Earth and planets as a 
consequence of the law of universal gravitation. This proof 
is based on dynamical principles ; but the following arguments, 
based on other considerations, afford independent evidence in 
favour of the theory that the Earth revolves round the Sun 
rather than the Sun round the Earth. 

(i.) The Sun's diameter is 110 times that of the Earth's, 
and it is much easier to believe that the smaller body revolves 
round the larger, than that the larger body revolves round 
the smaller. 

If the dynamical laws of motion be assumed, it is impossible to 
gee how the larger body could revolve round the smaller, unless 
either its mass and. therefore its density were very small indeed, 
or the smaller one were rigidly fixed iu some way. 

(ii.) The stationary points, and alternately direct and retro- 
grade motions of the planets, are easily accounted for on the 
theory that the Earth and planets revolve round the Sun 
(Chap. X.) in orbits very nearly circular, and it would be 
impossible to give such a simple explanation of these motions 
on any other theory. It is true that we might suppose, with 
Tycho Erahe (circ. 1600), that the planets revolve round the 
Sun as a centre, while that body has an orbital motion round 
the Earth, but this explanation would be more complicated 
than that which assumes the Sun to be at rest. And it would 
be hard to explain how such huge bodies as Jupiter and 
Saturn could be brought to describe such complex paths. 



33$ ASTRONOMY. 

(iii.) As seen through a telescope, Venus and Mars are 
found to be very similar to the Earth in their physical charac- 
teristics, and their phases show that, like the Earth and 
Moon, they are not self-luminous. It is, therefore, only 
natural to suppose that their property of revolving round the 
Sun is shared by the Earth. Moreover, the Earth's relative 
distance from the Sun agrees fairly closely with that given 
by Bode's law ; hence there is a strong analogy between the 
Earth and the planets. 

(iv.) The orbital motion of the Earth is in strict accordance 
with Kepler's Laws of Planetary Motion. In particular, the 
relation between the mean distances and periodic times given 
by Kepler's Third Law ( 326) is satisfied in the case of the 
Earth's orbit. 

Moreover, a similar relation is observed to hold between the 
periodic times of Jupiter's satellites and their mean distances 
from Jupiter. Hence it is probable that the Earth and 
planets form, like Jupiter's satellites, one system revolving 
about a common centre. But it is improbable that the Sun 
and Moon should both revolve about the Earth, for their 
distances from it and their periods are not connected by this 
relation. 

(v.) The changes in the relative positions of two stars 
during the year in consequence of annual parallax can only 
be accounted for on the hypothesis either of the Earth's 
orbital motion, or of a highly improbable rigid connection 
between all the nearer stars and the Sun, compelling them 
all to execute an annual orbit of the same size and position. 

(vi.) The aberration of light affords the most convincing 
proof of all. In particular, the relation between the coefficient 
of aberration and the retardation of the eclipses of Jupiter's 
satellites has been fully verified by actual observations, and 
affords incontestible evidence that the phenomenon is actually 
due to the finite velocity of light, as explained in Chapter XI. 
And the alternative hypothesis which would account for 
annual parallax would not give rise to aberration, but would 
produce an entirely different phenomenon. Hence the evi- 
dence derived from the aberration of light, unlike the previous 
evidence, furnishes a conclusive proof, and not merely an 
argument, in favour of the Earth's orbital motion. 



THE LAW OF UNIVERSAL GRAVITATION. 339 

406. NEWTON'S THEORETICAL DEDUCTIONS 
FROM KEPLER'S LAWS. 

Kepler's Three Laws of planetary motion naturally suggest 

the following questions : 

(1) What makes the planets move in ellipses ? 

(2) Why does the radius vector from the Sun to any planet 
trace out equal areas in equal times ? 

(3) Why are the squares of the periodic times proportional 
to the cubes of the mean distances from the Sun ? 

These questions were first answered by Newton about 1687, 
or nearly sixty years after the death of Kepler. The theore- 
tical interpretation of the Second Law necessarily precedes 
that of the first; accordingly we now repeat the laws in their 
new order, together with Newton's interpretations of them. 

Kepler's Second Law. The radius vector joining 
each planet to the Sun moves in a plane describing 
equal areas in equal times. 

NEWTON'S DEDUCTION. The force under which a 
planet describes its orbit always acts along the 
radius vector in the direction of the Sun's centre. 

Kepler's First Law. The planets move in ellipses, 
having the Sun in one focus. 

NEWTON'S DEDUCTION. The force on any planet 
varies inversely as the square of its distance 
from the Sun. 

Kepler's Third Law. The squares of the periodic 
times of the several planets are proportional to the 
cubes of their mean distances from the San. 

NEWTON'S DEDUCTION. The forces on different 
planets vary directly as their masses, and inversely 
as the squares of their distances from the Sun, 
or, in other words, the accelerations of different 
planets, due to the Sun's attraction, vary inversely 
as the squares of their distances from the Sun. 



340 ASTRONOMY. 

.If, as we have every reason for believing, the planets are 
material bodies, Newton's laws of motion show that they 
cannot move as they do unless they are acted on by some 
force, otherwise they would either be at rest or move uni- 
formly in a straight line. Kepler's Second Law then enables 
us to determine the direction of this force, his First Law 
enables us to compare the force at different parts of the same 
orbit, and his Third Law enables us to compare the forces on 
different planets. 

407. We have seen that the orbits of most of the planets are 
nearly circular, the eccentricities being small, except in the 
case of Mercury. Before proceeding to the general discussion 
of the dynamical interpretation of Kepler's Laws, it will be 
convenient therefore to consider the case where the orbits are 
supposed circular, having the Sun for centre. Kepler's 
Second Law shows that under such circumstances the planets 
will describe their orbits uniformly, and it hence follows that 
the acceleration of a planet has no component in the direction 
of motion, but is directed exactly towards the centre of the 
Sun. The law of force can now be deduced very simply, as 
follows : 

KEPLER'S THIRD LAW FOE, CIRCULAR 
ORBITS. 

408. To compare the Sun's attractions on different 
Planets, assuming that the orbits are circular and 
that the squares of the periodic times are propor- 
tional to the cubes of the radii. 

Suppose a planet of mass J/is moving with velocity v in a 
circle of radius r. Let T be the periodic time, P the force to 
the centre. . .', . 

Since the normal acceleration in a circular orbit is 2 /r, 

therefore * 



In the period T 7 the planet describes the circumference lira ; 

.-. vT= 2vr. 
Substituting for v, we have 

P iii^: _ ^L ***** 

jf ~- r 2 -yT 



TfLE LAW OF UNIVERSAL GRAVITATION. $4l 

Let M 1 be the mass of another planet revolving in a cir- 
cular orbit of radius r', T its periodic time, P' the force of 
the Sun's attraction ; then we have in like manner 
p ,_ JT 4*V 
- ^ x r * ' 

By Kepler's Third Law, 



r- r 

Therefore the forces on different planets vary 
directly as their masses and inversely as the squares 
of their distances from the Sun. 

COEOLLAEY 1. Let P = CM/r* ; then C is called the abso- 
lute intensity of the Sun's attraction, and we see that 

The absolute intensity of the Sun's attraction is the 
same for all planets. 

For c 



The constant C evidently represents the force with which 
the Sun would attract a unit mass at unit distance, or the 
acceleration which the Sun would produce at unit distance. 

COROLLAS r 2. If another body be revolving in an orbit of 
radius / in a period T\ under a different central force, which 
produces an acceleration C"// 2 at distance r', we have 

tT=l=! and (7= 



.-. C'T : CT 2 = i : r 8 , 

a formula which enables us to compare the absolute intensities 
of two different centres of force, which attract inversely as 
the squares of the distances, when the periodic times and 
distances of two bodies revolving about them are known. 



342 ASTRONOMY. 

409. To compare the velocities and angular velo- 
cities of two planets moving in circular orbits. If 

v, v are the velocities, n, ri the angular velocities (in radians 
per unit time), we have 



Also v = rw, v = rn' ; 

.*. v:v'= r~*: r'~*. 

EXAMPLES. 

1. If the Earth's period were doubled, to find what would be its 
new distance from the Sun. 

If r, r' be the old and new distances, Kepler's Third Law gives 

r' 3 : r 3 = 2 2 : 12; 

/. S = r x */4 = 92,000,000 x T587 
= 146,000,000 miles. 

2. If the Earth's velocity were doubled, its orbit remaining cir- 
cular, to find its new distance. 

Here r' : r = v 2 : t/ 2 = 1 : 4 ; 

... r'=ir = 23,000,000 miles. 

3. If the Earth's angular velocity were doubled, to find its new 
distance. 

The new angular velocity being double the old, the new period 
would be half the old, and therefore 

r' 3 :*- 3 =()': I'; 

/. r' = r x */i = r/ V4 = 92,000,000 -f- T587 
= 92,000,000 x -63 = 58,000,000 miles. 

4. To find what would be the coefficient of aberration to an 
observer situated on Venus. 

The coefficient of aberration (in circular measure) is the ratio of 
the observer's velocity to the velocity of light ; hence, if fc, k' are the 
coefficients on the Earth and Venus, 

_ = t/ r^ = \r_ /100 
k v r-* V r' V 72 ; 

.. k' = 20-493" x A/(l-38*) = 20'493" x M785 
= 24-151". 



THE LAW OF UNIVERSAL GRAVITATION. 343 

We shall now prove Newton's deductions from Kepler's 
Laws, for the general case of elliptic orbits, employing, how- 
ever, different and simpler proofs to those used by Newton. 

410. Areal Velocity. definition. If a point P is 
moving in any path MPK about a centre S, the rate of 
increase of the area of the sector MSP, bounded by the fixed 
line SM and the radius vector SP, is called the areal 
velocity of P about the point S. 

If the radius vector SP describes equal areas in equal 
times, in accordance with Kepler's Second Law, the areal 
velocity of P about S is of course constant, and is then 
measured by the area of the sector described in a unit of time. 

If the rate of description of areas is not constant, we 
must, in measuring the areal velocity at any point, pursue a 
similar course to that adopted in measuring variable velocity 
at any instant, as follows : 




FIG. 136. 

If the radius vector describes the sector PSP' in the inter- 
val of time t, then the average areal velocity over the 
arc PP' is measured by the ratio 

area PSP' 

time t 

(Thus the average areal velocity is the areal velocity with 
which a radius vector, sweeping out equal areas in equal 
times, would describe the sector PSP' in the same time t.) 

The areal velocity at a point P is the limiting value of 
the average areal velocity over the arc PP when this arc 
is infinitisimally small. 



344 ASTRONOMY. 

411. Relation between the Areal Velocity and the 
Actual (linear) Velocity. Let PP' be the small arc 
described by a body in any small interval of time t. Let 
be the actual or linear velocity of the body, h its areal velocity. 
Since the arc PP is supposed small, we have 

PP'=vt, 
area PSP'=M. 

Draw S Y perpendicular on the chord PP' produced. Then 
&PSP'= | (base) x (altitude) 



or 




FIG. 137. 

But when the arc PP' is infinitesimally small, PFis the 
tangent at P, and SYis therefore the perpendicular from S 
on the tangent at P. If this perpendicular be denoted 
we have therefore 



or (areal vel. about S) 

= J (velocity) x (perp. from S on tangent). 

COROLLARY. Por planets moving in circular orbits 
of radii r, r, h = |IT, and h'= \v'r r . 
But v I v = r"* : r x ~ J ; 

A:A'=r:r'; 

hence the areal velocity of a planet moving in a circular orbit 
is proportional to the square root of the radius, . . 



THE LAW OP UNIVERSAL GRAVITATION. 345 

412. PROPOSITION I. If a particle moves in such a 
manner that its areal velocity about a fixed point is 
constant, to prove that the resultant force on the 
particle is always directed towards the fixed point. 
[Newton's Deduction from Kepler's Second Law.] 

Let a body be moving in the curve PQ in such a way that 
its areal velocity about S remains constant. Let v, v' be 
the velocities at P, Q, and let PF, QY, the corresponding 
directions of motion, intersect in R. Drop SY, S Y perpen- 
dicular on PF, QY. 

Since the areal velocities at P and Q are equal, 

.-. v.SY=v'. SY. 
But SY= Rsin&KF, 

SY = SItsmSltY. 
.-. v$wSRY=v sin 




FIG. 138. 

i.e.) Component velocity at P perpendicular to BR 

= component vel. at Q pcrp. to SB. 

Therefore, as the particle moves from P to Q, its velocity 
perpendicular to JRS is unaltered, and therefore the total 
change of velocity is parallel to ItS. 

This is true whether the arc PQ be large or small. But if 
the arc PQ be taken infinitesimally small, the average rate 
of change of velocity over PQ, measures the acceleration at 
P, and P coincides with P. 

Therefore the direction of the acceleration of the particle 
nt any point of its path always passes through S, and there- 
fore the force acting on the particle also always passes 
through S. 



346 ASTEONOMT. 

413. Conversely, if the force on the particle always passes through 8, 
the areal velocity about 8 remains constant. For in passing from P 
to Q, the direction of motion is changed from PR to EQ, and the 
same change of velocity could therefore be produced by a suitable 
single blow or instantaneous impulse acting at R. And since the 
force at every point of PQ always passes through 8, this equivalent 
impulse must evidently also pass through 8 ; it must therefore act 
along RS. Hence the velocity perpendicular to R8 is unaltered by 
the whole impulse, and is the same at P as at Q j therefore 




FIG. 139. 

v sin 8RT = v' sin 8RT j 
therefore v .SY = v' .SY 1 ; 

therefore areal vel. at P = areal vel. at Q. 

414. PROPOSITION II. A particle describes an ellipse 
under a force directed to wards the focus ; to show that 
the force varies inversely as the square of the dis- 
tance. 

[Newton's Deduction from Kepler's First Law.] 
If h is the constant areal velocity, we have, by (i.), 



We will now express the kinetic energy of the particle in 
terms of r, its distance from the focus. Let its mass be M. 

In the Appendix (Ellipse 11) it is proved that for the 
ellipse whose major and minor axes are 20, 2J, 



m, j. 2 4# 4tfa / 2 1 \ 

Therefore t? 8 = - = ^- ( --- J. 

jt? 2 i 2 \ r a r 

and kinetic energy at distance r 



THE LAW OP T7NIVEESAI, GBAVITATION. 347 

If v is the velocity at distance r', we have, similarly, 



and therefore, for the increase of kinetic energy, 



(in.). 




FIG. 140. 

Now the increase of kinetic energy is equal to the work 
done by the impressed force in bringing the particle from 
distance r to distance r. The resolved part of the displace- 
ment in the direction of the force is rr'. Hence if P' 
denote the average value of the force between the distances 
r and r', we have 

Work done = P' 



(r-r'} = JJf^-^W = ~^(~ - -M 



b* 



rr 
rr 



Put / = r ; then the average force P' becomes the actual 
force P at distance r. Therefore 



A. j- j. \ 

orce at distance r) = 2 

This is proportional to 1/r 2 . 

Therefore the force varies inversely as the square of the dis- 
tance. 

\STRON. 2 A 



348 ASTRONOMY. 

415. PROPOSITION III. Having given that the squares 
of the periodic times of the planets are proportional 
to the cubes of the semi-axes major of their orbits, to 
compare the forces acting on different planets. 
[Newton's Deduction from Kepler's Third Law.] 

Let T be the periodic time of any planet; then, by 
hypothesis, the ratio 



is the same for all planets. 

In the last proposition (vi.) we showed that the force at 
distance r is given by 

-p _ 

Let this be put = Jf(7/r 2 , where C is some constant ; then 
4h~a , .. N 

= ........................ ( yu -)- 

Now in the period T the radius vector sweeps out the area 
of the ellipse, and this area is nab (Appendix, Ellipse 13). 
Hence, since the areal velocity is h, we have 

hT= irab. 

Substituting the value of h from this equation in (vii.), we 
have 



But a*/T 2 is the same for all the planets ; therefore C is con- 
stant for all the planets, and since the force 



it follows that 

The forces on different planets are proportional to tlieir 
masses divided by the squares of their distances from the Sun. 

Or, as in 408, Cor. 1, 

27)e absolute intensity of the Sun's attraction ( C) is the same 
for all the planets. 

CoROLLAjtY. Let accented letters refer to the orbit of 
another particle revolving round a different centre of force of 
intensity C'. Then, by (viii.), 

FC: T'*C r = a s : a'\ 



THE LAW OP UNIVERSAL GRAVITATION. 349 

416. Other Consequences of Kepler's Laws. 

(i.) In 150 we showed that, in consequence of Kepler's 
Second Law being satisfied by the Earth in its annual orbit, 
the Sun's apparent motion in longitude is inversely propor- 
tional to the square of the Earth's distance from it. Since 
the areal velocity of any planet about the Sun always remains 
constant^ it may be shown in like manner that its angular 
velocity is inversely proportional to the square of its distance 
from the Sun. 




FIG. 141. 

For, if the planet's radius vector revolves from SP to SP 
in the time t, and if the arc PP' is very small, we have 

area SPP' = SP* x Z PSP' ( 150), 
the angle being measured in radians ; 

area SPP' _ i o p2 v tPSP 
___ = f * __, 

i.e., (areal velocity of P) = %SP* x (angular velocity of P), 
provided that the angular velocity is measured in radians per 
unit of time. 

If n denote the angular velocity, h the areal velocity, and r 
the distance SP, we have therefore 



And since h is constant, n is inversely proportional to r. 

* (ii.) If the mass of the planet is M, its momentum is Mv along 
PY, and the moment of this momentum about 8 is 

= Mv x 8T = Mvp = 2hM. ( 411.) 

This is the planet's angular momentum, and is constant, since 7i in- 
constant. 



350 



ASTRONOMY. 



*417. Having given, in magnitude and direction, the velocity of a 
planet at any point of its orbit, to construct the ellipse described 
under the Sun's attraction. 

Let the attraction at distance r be 0/r 2 per unit mass, where C is 
given. Suppose that at the point 
P of the orbit the planet is moving 
with velocity v in the direction 
PT. We have 
v x ST = 2h, which determines h. 

Also, from (vii.), 

G = 47i 2 a/6 2 . 

Substituting in (ii.), 



*-c(- 1)...(,). 

Hence, by considering the planet 
at P, we have 




SP 

Now v, G, and SP are known ; hence the last equation determines 
the semi-axis major a. If r = SP, we have 



2a 



2C-ru 2 ' 



Let H be the other focus of the ellipse. Then it is known 
(Ellipse 8) that HP, SP make equal angles with PT. Also SP + HP 
= 2a. Hence, we can construct the position of If by making 
/ TPI = / TPS, and producing IP to a point H such that 

PH = 2a-SP. 
The ellipse can now be constructed as in Appendix (Ellipse 2). 

COROLLARY 1. Since SP + HP = 2a, equation (x.) gives 



SP.a 

COROLLARY 2. Substituting for h in terms of G, we see from 
equation (iv.) that the work done when the body moves from dis- 
tance r to distance / is 

*jfafJL-4. 



* This result is also proved independently in many treatises on 
dynamics, but a fuller investigation would be out of place here. 



THE LAW OF UNIVERSAL GBAVITATION. 351 

Hence the work done by a mass M in falling from distance 2a to 
distance r is 



= MQ (--- = iMu 2 byfxi. 

\ r 2a/ 

kinetic energy of the planet when at distance r. 

Therefore, if a circle be described about the centre of force 8, with 
radius equal to the major axis 2a, the velocity at any point of the 
orbit is that which the planet would acquire by falling freely from 
the circle to that point under the action of the attracting force. 

COROLLARY 3. If the planet be revolving in a circle, r a, and 
therefore v 2 = C/r = C/a, as in 408. 

COROLLARY 4. If v 3 = 2C/r, (x.) gives I/a = 0; /. a = oo. 

Hence the velocity is that acquired by falling from an infinite 
distance. In this case, the orbit is not an ellipse, but a parabola, a 
conic section satisfying the " focus and directrix " definition of 
Appendix (1), but having its eccentricity equal to unity. 

If v z > 2C/r, the velocity is greater than that due to falling from 
infinity, a comes out negative, and the orbit is a hyperbola, a conic 
section satisfying the focus and directrix definition, but having its 
eccentricity e greater than unity. 

A few cornel- have been observed to describe parabolas and hyper- 
bolas .-ibout the Sun. In such a case the motion is not periodic; the 
comet gradually moves away TO an infinite distance, and is lost for 
ever, unless the attraction of some other heavenly body should 
happen to divert its course, and send it back into the solar system. 

EXAMPLE. To find how long the Earth would take to fall into 
the Sun if its velocity were suddenly destroyed. 

If the Earth's velocity were very nearly, but not quite destroyed, 
it would describe a very narrow ellipse, very nearly coinciding with 
the straight line joining the point of projection to the Sun. The 
major axis of this ellipse would be very nearly equal to the Earth's 
initial distance from the Sun, and therefore the Earth would have 
very nearly gone half round the narrow ellipse when it would 
collide with the surface of the Sun. 

Hence, if r denote the Earth's distance from the Sun, the semi- 
rnajor axis of the narrow ellipse is \r, and the periodic time in this 
ellipse would be ()* years. The Earth would therefore collide 
with the Sun in 

2 x (1)^ years = years = - years 



= x 1-4142 days = 64 days nearly. 
8 



352 ASTRONOMY. 

SECTION II. Newton's Law of Gravitation Comparison of 
the Masses of the Sun and Planets. 

418. In the last section we showed that the Sun attracts any 
planet of mass M at distance r with a force CM/r 1 , where C 
is a constant. If we assume the truth of Newton's Third 
Law of Motion (i.e., that action and reaction are equal and 
opposite), the planet must also attract the Sun with an equal 
and opposite force CM/r 2 . Since in the former case the 
force is proportional to the mass of the attracted body, and in 
the latter to the mass of the attracting body, it is reasonable 
to suppose that the attraction between two bodies is propor- 
tional to the mass of each. 

Moreover, the motions of the various satellites, such as the 
Moon, confirm the theory that they revolve in their orbits 
under the attraction of their respective primary planets. 
From evidence of this character Newton, after many years of 
careful investigation, enunciated his Law of Universal 
Gravitation, which he stated thus : 

Every particle in the universe attracts every other 
particle with a force proportional to the quantities 
of matter in each, and inversely proportional to the 
square of the distance between them. 

By quantity of matter is, of course, meant mass, and the 
word attracts implies that the force between two particles 
acts in the straight line joining them and tends to bring them 
together. 

If M, M' be the masses of two particles, and r the distance 
between them, the law asserts that either particle is acted on 
by a force, directed towards the other, of magnitude 



where k has the same value for all bodies in the universe. 
The constant is called the constant of gravitation. 

*419. Astronomical Unit of Mass. Taking any fundamental 
units of length and time, it is possible to choose the unit of mass 
such that fc = 1. This unit is called the astronomical unit of mass. 
Hence, if M, M ' are expressed in astronomical units, the force 
between the particles is equal to MH'jr". It is, however, usually 
more convenient to keep the unit of mass arbitrary, and to retain 
the constant fc. 



THE LAW OF UNIVERSAL GRAVITATION. 353 

420. Remarks on the Law of Gravitation. Newton's 
Law states that not only do the Sun, 'the planets and their 
satellites, and the stars, mutually attract one another, but 
every pound of matter on one celestial body attracts every 
other pound of matter, on either the same or another body. 
But it is well-known that two spheres attract one another 
in just the Fame way as if the whole of the mass of either 
were concentrated at its centre, provided that the spheres 
are either homogeneous or made up of concentric spherical 
layers, each of uniform density. Since the Sun and 
planets are very nearly spherical, and their dimensions are 
very small compared with their distances, we see that their 
attractions may be very approximately found by regarding 
them as mere particles, instead of taking separate account of 
the individual particles forming them. 

Moreover, every planet is attracted by every other planet, 
as well as by the Sun. But the mass of the Sun, and con- 
sequently its attraction, is so much greater than that of any 
other member of the solar system, that the planetary motions 
are only very slightly influenced by the mutual attractions. 
Kepler's Laws, therefore, still hold approximately, but the 
orbits are subject to small and slow changes or perturbations. 

The Moon , on the other hand, is far nearer to the Earth 
than to the Sun ; hence the Moon's orbital motion is mainly 
due to the Earth's attraction. The chief effect of the Sun's 
attraction on the Earth and Moon is to cause them together 
to describe the annual orbit ; but it also produces pertur- 
bations or disturbances in the Moon's relative orbit ( 272) 
with which we are not here concerned. 

The fixed stars also attract one another and attract the 
solar system, which in its turn attracts the stars. The 
proper motions of stars are probably due to this cause ; 
but when we consider the vast distances of the stars, and 
remember that the attraction varies inversely as the square 
of the distance, it is evident that the relative accelerations 
are mostly too feeble to have produced any sensible changes of 
motion within historic times, and that countless ages must 
elapse before such changes can be discerned. 



354 ASffcONOMf* 

42 1 . Correction of Kepler's Third Law. Prom the fact 
that a planet attracts the Sun with a force equal to that with 
the Sun attracts the planets, it may he shown that Kepler's 
Third Law cannot he strictly true, as a consequence of the 
law of gravitation. Not only will the planet move under 
the Sun's attraction, but the Sun will also move under the 
planet's attraction. Eut since the forces on the two hodics 
are equal, while the mass of the Sun is very great compared 
with the mass of any planet, it follows that the acceleration 
of the Sun is very small compared with that of the planet, 
and hence the Sun remains very nearly at rest. 

We may, however, obtain a modification of Kepler's Third 
Law, in which the planet's reciprocal attraction is allowed 
for as follows : ,, 

Let S, M be the masses of the Sim and planet; then the 
attraction betweeen them is 



This attraction, acting on the mass JJ/of the planet, produces 
an acceleration of the planet towards the Sun equal to 



The corresponding attraction on the mass 8 of the Sun pro- 
duces an acceleration, in the reverse direction, of 



Hence the whole acceleration of the planet relative to the 
Sun is iM, 

yd 

instead of kS/r z , as it would be if the Sun were at rest. 
Hence the absolute intensity of the planet's acceleration 
towards the Sun is k (S + M), and this depends on the values 
of both M and 8. Let now T be the periodic time, r the 
planet's mean distance from the Sun, or the semi-axis major 
of the relative orbit ; then, by 408 (for a circular orbit), or 
415 (for an elliptical orbit), 



1HE LAW OF UNIVERSAL GRAVITATION. 355 

If M 1 be the mass of another planet, we have in like manner 
for its orbit Jc (8 +11') T l = 4^ r'\ 

Therefore T 2 (8+3T) : T l (8 + M 1 ) = r 3 : r*, 
the correct relation between the periods and mean distances. 

It is known that different planets have different masses. 
Hence, the fact that Kepler's Third Law is approximately 
true shows that the masses of the planets are small compared 
with that of the Sun. 

422. Motion relative to Centre of Mass. The 

mutual attractions of the Sun and planet have no influence 
on the position of the centre of mass (commonly called the 
" centre of gravity ") of the solar system ; hence, in consider- 
ing the relative motions, that point may be treated as fixed. 
It is known from general dynamical principles that when a 
system of bodies are under the influence of their mutual 
reactions or attractions alone, the centre of mass of the whole 
system is not accelerated. But it may be interesting to prove 
independently that when two bodies, such as the Sun and a 
planet, attract one another, they both revolve about their 
centre of mass. 

Let us suppose (to take a simple case) the relative 
orbit circular and of radius (P=) r, the angular velocity 
being n. Then, if G be the point about which the planet 
(P) and Sun (S) revolve, individually, we have 

n*xGP = acccl. of planet = kS/r* ; 

w 2 x GS = acccl. of Sun = 



Hence MxGP= Sx GS ; 

which relation shows that G is the common centre of mass, 

as was to be proved. 

In the case of three or more bodies, such as the Sun and 
pLinets, the centre of mass is still the common centre about 
which they revolve, but the corresponding investigation is 
more difficult, owing to the effect of the mutual attractions 
of the planets in producing perturbations. 

It may be mentioned that the mass of the Sun is so large, 
compared with those of the planets, that, although the further 
planets arc so very distant, the centre of mass of the whole 
solar system always lies very near the Sun. 



356 ASTRONOMY. 

423. Verification of the Theory of Gravitation for 
the Earth and Moon. Before considering the motions of 
the planets about the Sun, Newton investigated the orbital 
motion of the Moon about the Earth, with the view of dis- 
covering whether the Earth's attractive force, which retains 
the Moon in its orbit, is the same force as that which pro- 
duces the phenomenon of gravity at the Earth's surface. 

If we assume that the force varies inversely at the square 
of the distance, and that the Moon's distance is 60 times the 
Earth's radius, the acceleration of gravity at the Moon should 
be (-frV) 2 g, where g is the acceleration of gravity on the 
Earth's surface. 

But the acceleration g = 32-2 feet per sec. per sec. ; 
.. accel. at Moon's distance = 32-2/3600 feet per sec. per sec. 
= 32-2 feet per min. per min. 

From the length of the lunar month and the Moon's dis- 
tance in miles, Newton calculated what must be the normal 
acceleration -of the Moon in its orbit. At the time of his first 
investigation (1666) the Earth's radius and the Moon's dis- 
tance were but imperfectly known, and the Moon's normal 
acceleration, as thus computed, came out only about 27 feet 
per minute per minute. Some fifteen years later, the Earth's 
radius, and consequently the Moon's distance, had been 
measured with much greater accuracy, and, working with the 
new values, Newton found that the Moon's normal accelera- 
tion to the Earth agreed with that given by his theory. 

Taking the lunar sidereal month as 27 -3 days, the Earth's 
radius as 3960 miles, and the radius of the Moon's orbit as 
60 times the Earth's radius, the angular velocity (n) of the 
Moon, in radians, per minute is 

27T 

27-3x24x60' 

The Moon's distance in feet (d) = 3960 X 60 x 5280. 
Hence the Moon's normal acceleration (tfd) in feet per 
minute per minute 

= 3 150 XJL XJ280 x 47T 2 _ 2xll0 2 X7r 2 
(27'3) 2 x 24 2 x 60 2 = (27-3) 2 x 10 
= 32 approximately, 
thus agreeing with that given by the law of gravitation. 



THE LAW OF UNIVERSAL GRAVITATION. 357 

EXAMPLE. Having given that a body at the Earth's equator loses 
1/289 of its weight in consequence of centrifugal force, 

(i.) To calculate the period in which a projectile could revolve in 
a circular orbit, close to, but without touching the Earth, and 

(ii.) To deduce the Moon's distance. 

(i.) The centrifugal force on the body would have to be equal to 
its weight, and would therefore have to be 289 times as great as that 
at the Earth's equator. 

Hence the projectile would have to move -v/289, or 17 times as 
fast as a point on the Earth's equator, and would therefore have to 
perform 17 revolutions per day.* 

Therefore the period of revolution = j\- of a day. 

(ii.) Assuming the law of gravitation, the periodic times and dis- 
tances of the projectile and Moon must be connected by Kepler's 
Third Law. Hence, taking the Moon's sidereal period as 27| days, 
we have, if a = Earth's rad., d = Moon's dist., 



.-. d z = a? x (17 x 27^) 2 = a 3 {^^} 2 = a 3 . 215915'i ; 

.-. d = a x 3/215915-1 = 59'99; 
.'. distance of Moon = 60 x Earth's radius almost exactly. 

424. Effect of Moon's Attraction. Moon's Mass. 

If we take account of the Moon's attraction on the Earth we 
must introduce a correction analogous to that made in Kepler's 
Third Law (421). If J/, m are the masses of the Earth 
and Moon, the whole relative acceleration is k(lf+m)l$, 
instead of kM/d*. But, if g n is the acceleration of gravity on 
the Earth's surface, ^ = 



and, if I 7 is the length of the sidereal month, then, by 421, 

jJf-fW/nj 

w-jr 
* 1H "^ :=r ^^' 

This formula might be used (and has been used by Airy) 
to find m/M, the ratio of the Moon's to the Earth's mass, in 
terms of the observed values of a, d, g^ T. It is not, how- 
ever, a very accurate method, owing to the smallness of M/Jf. 

* Relative to the Earth it would perform 16 or 18 revolutions per 
day, according to whether it was revolving in the same or the 
opposite direction to the Earth. 



358 ASTRONOMY. 

425. To find the ratio of the Sun's Mass to that 
of the Earth. 

Let /S, M, m be the masses of the Sun, Earth, and Moon, 
d, r the distances of the Moon and Sun from the Earth, T, Y 
the lengths of the sidereal lunar month and year respectively. 
Then, if k be the gravitation constant, the Earth's attraction 
on the Moon is = klfm/d', and its intensity is kM. 

The Sun's attraction on the Earth is = kSM/r 1 , and its 
intensity is kS. 

Therefore, by 415, Corollary, 

kM . T 2 = 47rd 3 , kS . F 2 = 47rV ; 



whence the ratio of the Sun's to the Earth's mass may be 
found. 

If we take account of the attraction of the smaller body 
on the larger, the whole acceleration of the Earth, relative to 
the Sun, is k (S + M+m)/^ (since the Sun is attracted by 
the Moon as well as the Earth), and that of the Moon, relative 
to the Earth, is k (M+m)/eP. Hence the corrected or more 
exact formula is 



Since the Moon's mass is about -fa of that of the Earth, 
the first or approximate formula can only be used if the cal- 
culations arc not carried beyond two significant figures. 

In this manner it is found that the Sun's mass is about 
331,100 times that of the Earth. 

EXAMPLES. 

1. To compare, roughly, the masses of the Earth and Sun, taking 
the Sun's distance to be 390 times the Moon's, and the number of 
sidereal months in the year to be 13. 

We have 8:M = ~:l^ t 

. mass of Sun 390? _ 2 _ 

mass of Earth 18* " l51 ' 000 - 

To the degree of accuracy possible by this method, the Sun's 
mass is therefore 350,000 times that of the Earth. 



THE LAW OP UNIVERSAL GRAVITATION. 359 

2. To find the ratio of the masses, taking the Moon's mass as Jj 
of the Earth's, and the number of sidereal months in the year as 13^. 

390 3 _ 390 3 x 3 2 _ 5338710 --,.._ 

oooooy ; 



.'. 8 = 333668 (M + m) = 333668 (1 + ff T ) 31 = 337,787 M. 

426. To determine the mass of a planet which has 
one or more satellites. 

The method of the last paragraph is obviously applicable 
to the case of any planet which has a satellite. We require 
to know the mean distance and the periodic time of the 
satellite. The former may be easily found by observing the 
maximum angular distance of the satellite from its primary, 
the distance of the planet from the Earth at the time of 
observation having been previously computed. The periodic 
time of the satellite may also be easily observed. 

Let M' , ml be the masses of the planet and satellite, d' 
their distance apart, r' their distance from the Sun, T' the 
period of revolution of the satellite, Y' the planet's period of 
revolution round the Sun. Using unaccented letters to re- 
present the corresponding quantities for the Earth and Moon 
we have, roughly, 

tE __ M'T" 2 __ SY'' 2 = SY 2 _ 31T* 
k " d'* ' r* r & d* ' 
or, more accurately, 

(S+M+m'^Y* 



k ' ~~d'~ 

_ (S+M+m) Y' 



whence the mass of the planet, or, more correctly, the sum of 
the masses of the planet and satellite, may be determined in 
terms of the mass of the Sun, or the sum of the masses of the 
Earth and Moon. We do not require to know the periodic 
time and mean distance of the planet from the Sun, since the 
above expressions enable us to express the required mass, 
M' + m' 9 in terms of the year and mean distance of the 
Earth, or in terms of the lunar month and the mean distance 
of the Moon. 



360 

EXAMPLE. To find the mass of Uranus in terms of that of the 
Sun, having given that its satellite Titania revolves in a period of 
8 days 17 hours at a distance from the planet = '003 times the 
distance of tho Earth from the Sun. 

Let M be the mass of Uranus, then we have 

d 3 . r 3 

and, by Kepler's Third Law, r*/Y' 2 is the same for Uranus as for the 
Earth. Hence 

M'8= C' 003 ) 3 . I 3 

(8d. 17h.) a ' (365d.6h.) 2? 

*L = I 3 Y /365d.6h. 
8 UOOO/ \8d.l7h. 

= ?7_ x /8766\ 2 
~ 10 9 \ 209 / 



Thus, the mass of Uranus is to that of the Sun in the ratio of 
1 to 21,Q53. 

*427. The Masses of Mercury and Venus (which have no satellites) 
could theoretically be found by determining their mean distances 
from the Sun by direct observation, and comparing them with 
those calculated from their periodic times by Kepler's Third Law. 
For, if 3T is the mass of such a planet, we have 

(S+_J)_r 2 = (8 + M + m) T- 

~ 



This enables ns to find the stun of the masses of the Sun and 
planet, and, the Sun's mass being known, the planet's mass could 
be found. 

This method is, however, worthless, because the masses of Mercury 
nd Venus are only about TOO^WO ^ *injW of that of the Sun > 
and in order to calculate one significant figure of the fraction M'/S 
it would be necessary to know all the data correct to about seven 
significant figures, a degree of ^Qewracy unattainable in practice. 



For this reason it is necesgar,y ?to Calculate the masses of these 
planets by means of the pefltwrbatiqns t they produce on one another 
and on the Earthy these p*L r All r ) ia ^9 I n j' vW^ ^ e discussed in the next 
chapter. 



T1TR LAW OP TTNTVERSAL GRAVITATION". 36 1 

428. Centre of Mass of the Solar System. When 
the masses of the various planets have been found in terms 
of the Sun's mass, the position of the centre of mass of the 
system can be found for any given configuration, and can 
thus be shown to always lie very near the Sun. 

EXAMPLES. 

1. To find the distance of the centre of mass of the Earth and 
Sun from the centre of the Sun. 

Here the mass of the Sun is 331,100 times the Earth's mass, and 
the distance between their centres is about 92,000,000 miles. Hence, 
the centre of mass of the two is at a distance from the Sun's centre 
of about 92,000,000 =278mil 

331,100 + 1 

2. To find the centre of mass of Uranus and the Sun, and to show 
that it lies within the Sun. 

The distance of Uranus from the Sun is 19'2 times the Earth's 
distance, and its mass is 1/21053 of the Sun's. Hence the C,M- ia 
at a distance from the Sun's centre of 
92,000,000 x 19-2 

21053 + 1 

The Sun's semi-diameter is 433,200 miles ; hence the centre of mass 
of the Sun and Uranus is at a distance from the Sun's centre of 
rather less than the radius. 

3. In the case of Jupiter, the mean distance is 5'2 times that of 
the Earth, and the mass is 1/1050 of that of the Sun j hence the 
C.M. is at a distance 

5-2 x 92,000,000 

1050 + 1 

This is just greater than the Sun's radius (433,200), showing that 
the centre of mass lies just without the Sun's surface. 



362 ASTBONOMY. 

SECTION III. The Earth's Mass and Density. 

429. The so-called " Weight of the Earth " really 
means the Earth's mass, and the operation called " weighing 
the Earth," in some of the older text-books, means finding the 
mass of the Earth. In the last section we explained how to 
compare the masses of the Sun and certain planets with that 
of the Earth, and in the next chapter we shall give methods 
applicable to a planet having no satellites. But before the 
masses can be expressed in pounds or tons it is necessary to 
determine the Earth's mass in these units. The methods of 
doing this all depend on comparing the Earth's attraction 
with that of a body of known mass and distance ; and the only 
difficulty lies in determining the \atter attraction, since the 
force between two bodies of ordinary dimensions is always 
extremely small. The following methods have been used. 
The first two are by far the best. 

(1) By the "Cavendish Experiment," or the balance. 

(2) By observations of the influence of tides in estuaries. 

(3) By the "Mountain" method. 

(4) By pendulum experiments in mines. 

430. The " Cavendish Experiment " owes its name to 
its having been first used to determine the Earth's mass by 
Cavendish, about the year 1798. The essential principle of 
the method consists in comparing the attractions of two heavy 
balls of known size and weight with the Earth's attraction. 
Since the attraction of a sphere at any point is proportional 
directly to the mass of the sphere and inversely to the square 
of the distance from its centre, it is evident that by comparing 
the attractions of different spheres such as the Earth and the 
experimental ball of metal we can find the ratio of their 
masses. 

The comparison is effected by means of a torsion "balance. 
Two equal small balls A, B are fixed to the ends of a light 
beam suspended from its middle point by means of a slender 
vertical thread or "torsion fibre" (in his recent experiments, 
Professor C. V. Boys has used a fine fibre of spun quartz), so 
as to be capable of twisting about in a horizontal plane 
(the plane of the paper in Fig. 143). Two heavy metal balls 
C, D, are brought near the small balls A, (as shown in the 



THE LAW OP UNIVERSAL GRAVITATION. 363 

figure), and their attraction causes the beam to turn about 0, 
say from its original position of rest XX' to the position AB. 
As the beam turns the fibre twists ; this twisting is resisted by 
the elasticity of the fibre, which produces a couple, propor- 
tional to the angle of twist XOA, tending to untwist it again. 
Let us call this couple /x L XOA, where / is a constant 
depending on the fibre, called its " torsional rigidity" 

The beam AB assumes a position of equilibrium when the 
moments about of the attractions of the large spheres (7, D 
on the balls A, B, just balance the "untwisting couple" 
/x Z XOA. The angle XOA being measured, and the 
dimensions of the apparatus being supposed known, the 
attractions of the spheres can now be determined in terms of 
the torsional rigidity. 





FIG. 143. 

The value of /is found in terms of absolute units of couple 
by observing the time of a small oscillation of the beam when 
the balls A, B have been removed. [The beam will then 
swing backwards and forwards like the balance wheel of a 
chronometer (204). The greater the torsional rigidity, the 
more frequently will it reverse the motion of the beam, and 
the more frequent will be the oscillations.*] 

Hence finally the attractions between the known masses 
C, D and A, B are found in terms of known units of force, 
and by comparing these attractions with that of gravity the 
Earth's mass is found. 



* The student who has read Kigid Dynamics should work out the 
formula. 

ASTRON. 2 B 



364 ASTRONOMY. 

In practice, instead of measuring the angle XOA, the masses C, I) 
are subsequently placed on the reverse side of the beam, say with 
their centres at c, fl, and they now deflect the beam in the reverse 
direction, say to ab. The angle measured is the whole angle aOA, 
and this angle is ticice the angle XOA, if the positions CD and cd 
are symmetrically arranged with respect to the line XOX'. 

In the earlier experiments the beam AB was six feet long, and the 
masses C, D were balls of lead a foot in diameter. Quite recently, 
however, Professor C V. Boys, by the use of a quartz fibre for the 
suspending thread, has performed the experiment on a much smaller 
scale, the whole apparatus being only a few inches in size and being 
highly sensitive. He uses cylinders instead of spheres for the 
attracting bodies, and this introduces extra complications in the 
calculations. 

Although the above description shows the general principle of the 
method, many further precautions are required to ensure accuracy. 
A full description of these would be out of place here. 

431. The common balance has also been used to deter- 
mine the Earth's mass. In this case the differences of weight 
of a body are observed when a large attracting mass is placed 
successively above and below the scale-pan containing it. 

EXAMPLE. To find the Earth's mass in tons, having given that the 
attraction of a leaden ball, weighing 3 cwt., on a body placed at a 
distance of 6 inches from its centre is '0000000432 of the weight of 
the body. 

Let M be the mass of the Earth in tons. 

The mass of the ball in tons is = /. 

The Earth's radius in feet = 3960 x 5280 = 20,900,000 roughly ; 
and the distance of the body from the ball in feet = ^. 

Hence, since the attractions of the Earth and ball are proportional 
directly to the masses and inversely to the squares of the distances 
from their centres, 



-0000000432 ; 1 = _ 
' 



' 2 (20,900,000) 
(20,900,000^ x J_ = 3 x 209- x 10*0 
x -0000000432 5 x 432 x 10 - 10 



5 432 2160 

= 6067 x 10 1S . 

Hence the mass of the Earth is (roughly) 6067 million billion 
tons. 



THE LAW OF UNIVERSAL GRAVITATION. 365 

432. To determine the Earth's Mass by observa- 
tions of the Attraction of Tides in Estuaries. A 

method which admits of very great accuracy is that in 
which the mass of the Earth is found hy comparing it with 
that of the water brought by the tide into an estuary. Con- 
sider an observatory situated (like Edinburgh Observatory) 
due south of an arm of the sea, whose general direction is 
east and west. The direction of its zenith, as shown either 
by a plummet or by the normal to the surface of a "bowl of 
mercury, is not the same at high tide as at low, because the 
additional mass of water at high tide produces an attraction 
which deflects the plummet and the nadir point northward, 
and hence displaces the zenith towards the south. Hence 
the latitude of the observatory is less at high tide than at 
low ; and the difference is a measurable quantity. The great 
advantage of this method is that the mass which deflects the 
plumb-line can be measured with great certainty ; for the 
density of the sea-water is exactly known (and, unlike that of 
the rocks in the next methods, is uniform throughout) and 
the shape and height of the layer of water brought in are 
known from the ordnance maps, and the tide measurements 
at the port. 

*433. In the Pendulum Method the values of g r the 
acceleration of gravity, are compared by comparing the oscil- 
lations of two pendulums at the top and bottom of a deep 
mine. The difference of the two values is due to the attrac- 
tion of that portion of the Earth which is above the bottom 
of the mine ; this exerts a downward pull on the upper pen- 
dulum, and an upward pull on the lower one. If the Earth 
were homogeneous throughout, the values of g at the top and 
bottom would be directly proportional to the corresponding 
distances from the Earth's centre. If this is not observed to 
be the case, the discrepancy enables us to find the ratio of the 
density of the Earth to that of the rocks in the neighbourhood 
of the mine. If the latter density is known, the Earth's 
density can be found, and knowing its volume, its mass can 
be computed. But this method is very liable to considerable 
errors, arising from imperfect knowledge of the density of the 
rocks overlying the mine. 



366 ASTEONOMT. 

*434. In the Mountain Method the Earth's attraction is com* 
pared with that of a mountain projecting above its surface. Suppose 
a mountain range, such as Schiehallien in Scotland, running due E. 
and W. ; then at a place at its foot on the S. side the attraction of 
the mountain will pull the plummet of a plumb line towards the N., 
and at a place on the N. side the mountain will pull the plummet to 
the S. Hence the Z.D. of a star, as observed by means of zenith 
sectors, will be different at the two sides, and from this difference 
the ratio of the Earth's to the mountain's attraction may be found. 

In order to deduce the Earth's density it is then necessary to 
determine accurately the dimensions and density of the mountain. 
This renders the method very inexact, for it is impossible to find 
with certainty the density of the rocks throughout every part of the 
mountain. 

435. Determination of Densities. Gravity on the 
Surface of the Sun and Planets. When the mass and 
volume of a celestial body have been computed, its average 
density can, of course, be readily found. By dividing the 
mass in pounds by the volume in cubic feet, we find the 
average mass per cubic foot, and since we know that the 
mass of a cubic foot of water is about 62 J Ibs., it is easy to 
compare the average density with that of water. The deter- 
mination of densities is particularly interesting, on account of 
the evidence it furnishes regarding the physical condition of 
the members of the solar system. The Earth's density is 
about 5*58. 

Prom knowing the ratios of the mass and diameter of the 
Sun or a planet to that of the Earth, we can compare the 
intensity of its attraction at a point on its surface with the 
intensity of gravity on the Earth. 

It may be noticed that attraction of a sphere at its surface is pro- 
portional to the product of the density and the radius. 

For the attraction is proportional to mass -*- (radius) 2 , and the 
mass is proportional to the density x (radius) 3 ; .*. the attraction 
at the surface is proportional to the density x radius. 

EXAMPLES. 

1. To find the Earth's average density and mass, having given 
that the attraction of a ball of lead a foot in diameter, on a particle 
placed close to its surface, is less than the Earth's attraction in the 
proportion of 1 : 20,500,000, and that the density of lead is 11'4 times 
that of water. 



THE LAW OF UNIVERSAL GRAVITATION. 367 

Let D be the average density of the Earth. Then, since the radii 
of the Earth and the leaden ball are | and 20,900,000 feet respectively, 
and the attractions at their surfaces are proportional to thei* 
densities multiplied by their radii, 

/. 1 : 20,500,000 = ll'4xi : -Dx 20,900,000; 
/. average density of Earth. D = 5'7x|^ = 5-6. 
Hence the average mass of a cubic foot of the material forming 
the Earth is 5 - 6 x 62'5 pounds. But the Earth is a sphere of volume 

|TT (20,900,000) 3 cubic feet. 
Hence the mass of the Earth, with these data, 

= -I* x 209 3 x 10 15 x 5-6 x 62-5 pounds 
= 1H38 x 10 22 pounds = 597 x 10 19 tons. 

2. To calculate the mean density of the Sun from the following 
data: 

Mass of O = 330,000 . (mass of $) ; 
Density of = 5 '58 ; 

Q's parallax = 8'8"j Q's angular semi-diameter = 16'. 
The radii of the Sun and Earth being in the ratio of the Sun's 
angular semi-diameter to its parallax ( 258), we have 

Q's radius 16' 960 inQ-1 
- = - = - = iuy j. j 

(p's radius 8'S 8'8 

/. volume of Sun = (109'1) 3 . (vol. of Earth) 

= 1,298,000 . (vol. of Earth) roughly. 
But mass of Sun = 330,000. (mass of Earth) ; 

. density of Sun ^ 330 = _1_ nearl 

density of Earth 1298 3'9 

/. density of Sun = 1'4. 

3. To find the number of poundals in the weight of a pound at the 
surface of Jupiter, taking the planet's radius as 43,200 miles and 
density 1^ times that of water. 

Taking the Earth's radius as 3960 miles and density as 5'58, we 
have 

(gravity at surface of Jupiter) : (gravity on Earth) 

= 1-33 x 43,200 : 5-58 x 3960. 
But at the Earth's surface the weight of a pound 

= 32-2 poundals ; 
therefore on the surface of Jupiter the weight of a pound 



= 83-7 poundals. 



368 ASTRONOMY. 



EXAMPLES. XIII. 

1. Taking Neptune's period as SO years, and the Earth's velocity 
as 91 miles per second, find the orbital velocity of Neptune. 

2. If we suppose the Moon to be 61 times as far from the Earth's 
centre as we are, find how far the Earth's attraction can pull the 
Moon from rest in a minute. 

3. If the Earth possessed a satellite revolving at a distance of only 
6,000 miles from the Earth's surface, what would be approximately 
its periodic time, assuming the Earth to be a sphere of 4,000 miles 
radius ? 

4. Assuming the distance between the Earth's centre and the 
Moon's to be 240,000 miles, and the period of the Moon's revolution 
28 days, find how long the month would be if the distance were 
only 80,000 miles. 

5. Calculate the mass of the Sun in terms of that of Mars, given 
that the Earth's mean distance and period are 92 x 10 6 miles and 
365i days, and the mean distance and period of the outer satellite 
of Mars are 14,650 miles and Id. 6h. 18m. 

6. Show that the periodic time of an asteroid is 3| years, having 
given that its mean distance is 2'305 times that of the Earth. 

7. Show that we could find the Sun's mass in terms of the Earth's, 
from exact observation of the periods and mean distances of the 
Earth and an asteroid, by the error produced in Kepler's Third Law 
in consequence of the Earth's mass. 

8. Show that an increase of 10 per cent, in the Earth's distance 
from the Sun would increase the length of the year by 56' 14 days. 

9. The masses of the Earth and Jupiter are approximately 
"iuroVoT) an d ToW respectively of the Sun's mass, and their distances 
from the Sun are as 1 : 5. Show that Kepler's Laws would give the 
periodic time of Jupiter too great by more than 2 days. 

10. Prove that the mass of the Sun is 2 x 10 27 tons, given that 
the mean acceleration of gravity on the Earth's surface is 9'81 
metres per second per second, the mean density of the Earth is 
5*67, the Sun's mean distance T5 x 10 s kilometres, a quadrant of the 
Earth's circumference 10,000 kilometres, and taking a metre cube of 
water to be a ton. 



THE LAW OF UNIVERSAL GRAVITATION. 369 

11. Having given that the constant of aberration for the Earth 
is 20'49", and that the distance of Jupiter from the Sun is 5'2 times 
the distance of the Earth from the Sun, calculate the constant of 
aberration for Jupiter. 

12. If the mass of Jupiter is T oVo f t ne m ass of the Sun, show 
that the change in the constant of aberration caused by taking into 
account the mass of Jupiter is 004" nearly (see Question 11). 

13. Find the centre of mass of Jupiter and the Sun. Hence find 
the centre of mass of Jupiter, the Sun, and Earth, (1) when Jupiter 
is in conjunction, (2) when in opposition. (Sun's mass = 1,048 
times Jupiter's = 332,000 times Earth's. Jupiter's mean distance 
= 480,000,000 miles ; Earth's = 93,000,000 miles.) 

14. If the intensity of gravity at the Earth's surface be 32 - 185 
feet per second per second, what will be its value when we ascend 
in a balloon to a height of 10,000 feet ? (Take Earth's radius = 4,000 
miles and neglect centrifugal force.) Would the intensity be the 
same on the top of a mountain 10,000 feet high ? If not, why not ? 

15. Show how by comparing the number of oscillations of a 
pendulum at the top and bottom of a mountain of known density, 
the Earth's mass could be found. 

16. How would the tides in the Thames affect the determination 
of meridian altitudes at Greenwich observatory theoretically ? 

. 17. If the mean diameter of Jupiter be 86,000 miles, and his mass 
315 times that of the Earth, find the average density of Jupiter. 

18. If the Sun's diameter be 109 times that of the Earth, his mass 
330,000 times greater, and if an article weighing one pound on the 
Earth were removed to the Sun's surface, find in poundals what its 
weight would be there. 

19. Taking the Moon's mass as ^ that of the Earth, show that 
the attraction which the Moon exerts upon bodies at its surface is 
only about l-5th that of gravity at the Earth's surface. 

20. If the Earth were suddenly arrested in its course at an 
eclipse of the Sun, what kind of orbit would the Moon begin ^0 
describe ? 



370 ASTRONOMY. 



EXAMINATION PAPER. XIII. 



1. State reasons for supposing that the Earth moves round the 
Sun, and not the Sun round the Earth. 

2. State Kepler's Laws, and give Newton's deductions therefrom. 

.3. If the Sun attracts the Earth, why does not the Earth fall into 
the Sun ? 

4. Show that the angular velocitiesof two planets are as the cubes 
of their linear velocities. 

5. State Newton's Law of Gravitation, and prove Kepler's Third 
Law from it for the case of circular orbits, taking the planets small. 

6. Explain clearly (and illustrate by figures or otherwise) what 
is meant by a force varying inversely as the square of the distance. 

7. Are Kepler's Laws perfectly correct ? Give the reason for your 
answer. What is the correct form of the Third Law if the masses 
of the planets are supposed appreciable as compared with the mass 
of the Sun ? 

8. How can the mass of Jupiter be found ? 

9. Show that if a body describes equal areas in equal times about 
a point, it must be acted on by a force to that point. 

10. Find the law of force to the focus under which a body will 
describe an ellipse ; and if C be the acceleration produced by the 
force at unit distance, T the periodic time, and 2a the major axis 
of the ellipse, find the relation between 0, a, T. 



CHAPTEE XIV, 



FURTHER APPLICATIONS OF THE LAW OF 
GRAVITATION. 

SECTION 1. The Hoori 's Mass Concavity of Lunar Orbit. 

436. The Earth's Displacement due to the Moon. 

In Section II. of the last chapter we saw that when two 
bodies are under their mutual attraction they revolve about 
their common centre of mass. Thus, instead of the Moon 
revolving about the Earth in a period of 27-^ days, both 
bodies revolve about their centre of mass in this period, 
although from the Moon's smaller size its motion is more 
marked. 

In this case both the Earth and Moon are under the 
attraction of a third body the Sun which causes them 
together to describe the annual orbit. But the Sun's dis- 
tance is so great compared with the distance apart of the 
Earth and Moon, that its attraction is very nearly the same, 
both in intensity and direction, on both bodies. To a first 
approximation, therefore, the resultant attraction of the Sun 
is the same as if the masses of both the Earth and Moon were 
collected at their common centre of mass. Hence it is strictly 
the centre of mass of the Earth and Moon, and not the centre 
of the Earth, which revolves in an ellipse about the Sun with 
uniform areal velocity, in accordance with the laws stated in 
155. And, owing to the revolution of the Moon, the Earth's 
centre revolves round this point once in a sidereal month, 
threading its way alternately in and out of the ellipse 
described, and being alternately before and behind its mean 
position. 



372 



ASTRONOMY. 



This displacement of the Earth has been used for finding 
the Moon's mass in terms of tho Earth's, by determining the 
common centre of mass of the Earth and Moon, as follows. 




FIG. H4. 

Let EV J/,, 6 r 1 (Fig. 144) be the positions of the centres of the 
Earth and Moon, and their centre of mass, at the Moon's last 
quarter, JE* 2 , M v 2 and E# M# G 3 their positions at new Moon 
and at first quarter respectively, S the Sun's centre. 

Then, at last quarter, E^ is behind # and the Sun's longi- 
tude, as seen from E^ is less than it would be as seen from G l 
by the angle E^SG^. At first quarter, JE 9 is in front of G 3 , 
and therefore the Sun's longitude is greater at E% than at G s 
by the angle G^SE y If, then, the observed coordinates of 
the Sun be compared with those calculated on the supposition 
that the Earth moves uniformly (i.e., with uniform areal 
velocity), its longitude will be found to be decreased at last 
quarter and increased at first quarter. 

From observing these displacements the Moon's mass may 
be found. For, knowing the angle of displacement E l SG l 
and the Sun's distance, the length E l G l may be found. Also 
the Moon's distance E^ is known. And, since G l is the 



FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 373 

centre of mass of the Earth and Moon, 

mass of Moon : mass of Earth = E l G l : G^J/j; 
whence the mass of the Moon can be found. 

The Sun's displacement at the quarters could he found by 
meridian observations of the Sun's R.A. with a transit circle. 
The displacement of the Earth will also give rise to an 
apparent displacement, having a period of about one month, 
in the position of any near planet ; this could be detected 
by observations on Mars, when in opposition, similar to those 
used in finding solar parallax ( 339). 

From this and other methods it is found that the mass of 
the Moon is about 1/81 of that of the Earth. The Moon's 
density, as thus deduced, is about 3 '44, or of that of the 
Earth. 

EXAMPLE. To compare the masses of the Moon and Earth, having 
given that the Sun's displacement in longitude at the Moon's 
quadratures is equal in f of the Sun's parallax. 

Since L E i SG i = f the angle subtended by Earth's radius at S, 
therefore E& = | (Earth's radius). 

But E^Mi = 60 (Earth's radius) ; 

.'. EjJf, = 80. #!(?,; 
.'. GjMi = 79..E? 1 G 1 , 

and mass of Moon : mass of Earth =E^G^ : O i M l = 1 : 79; 
/. the Moon's mass = 1/79 of the Earth's mass. 

437. Application to Determination of Solar Paral- 
lax. If the Moon's mass be found by any other method, the 
above phenomena give us a means of finding the Sim's 
parallax and distance. For we then know E^ G l : G^^ and 
therefore E& and the angle E.SG, is found by observation. 
But the exact ratio of IS 1 SG 1 to the parallax is known, for it 
is equal to that of ^ G l to tbe Earth's radius ; hence the Sun's 
parallax and distance can be found. Since the Moon's mass 
can be found with extreme accuracy by many different 
methods, this method is quite as accurate as many that have 
been used for finding the solar parallax. 

*438 Concavity of the Moon's Path about the Sun. - The Moon, 
by its monthly orbital motion about the Earth, threads its way alter- 
nately inside and outside of the ellipse which the centre of mass of 
the Earth and Moon describes in its annual orbit about the Sun, 



,374 ASTRONOMY. 

Hence the path described by the Moon in the course of the year is 
a wavy curve, forming a series of about thirteen undulations about 
the ellipse. It might be thought that these undulations turned 
alternately their concave and convex side towards the Sun, but the 
Moon's path is really always concave ; that is, it always bends 
towards the Sun, as shown in Fig. 145, which shows how the path 
passes to the inside of the ellipse without becoming convex. 

To show this it is necessary to prove that the Moon is always 
being accelerated towards the Sun. Let n, n' be the angular velo- 
cities of the Moon about the Earth and the Earth about, the Sun 
respectively. Then, when the Moon is new, as at M 2 (Fig. 145), its 
acceleration towards G 2 , relative to G 2 , is n 2 . MG 2 . But (? 2 has a 
normal acceleration n'' 2 G^S 'towards 8. Hetice the resultant accelera- 
tion of the Moon Jf 2 towards 8 is n'-G. 2 S- 



FIG. 145. 

Now, there are about 13 sidereal months in the year ; therefore 
TO = 13X- Also E^S is nearly 400 times E 2 M 2 , and therefore G 2 S is 
slightly over 400 times GM- 2 . Therefore roughly 

n'"G z S : n*M 2 Gz = 400 : 182 ; .'. ri-G^S > n 2 (? 2 3f 2 . 
Thus, the resultant acceleration of M is directed towards, not away 
from 8, even at Jf 2 , where the acceleration, relative to (? 2 . is directly 
opposed to that of G 2 . Therefore the Moon's path is constantly 
being bent (or deflected from the tangent at M 2 ) in the direction of 
the Sun, and is concave towards the Sun. 

*439. Alternate Concavity and Convexity of the Path, of a 
Point on the Earth.. In consequence of the Earth's diurnal rota- 
tion, combined with its annual motion, a point on the Earth's equator 
describes a wavy curve forming 365 undulations about the path 
described by the Earth's centre. In this case, however, it may be 
easily shown in the same way that the acceleration of the point 
towards the Earth's centre is greater than the acceleration of the 
Earth's centre towards the Sun. The path is, therefore, not 
always concave to the Sun, being bent away from the Sun in 
the neighbourhood of the points where the two component accelera- 
tions act in opposite directions. 



FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 375 

SECTION II. The Tides. 

In the last section we investigated the displacements due to 
the -Moon's attraction on the Earth as a whole. We shall 
now consider the effects arising from the fact that the Moon's 
attractive force is not quite the same either in magnitude or 
direction at different parts of the Earth, and shall show how 
the small differences in the attraction give rise to the tides. 

440. The Moon's or Sun's Disturbing Force. Let C, 

Jf be the centres of the Earth and Moon ; AC A' the Earth's 
diameter through M ; B, B' points on the Earth such that 
M C = MB = MB'. Let 3/, m denote the masses of the 
Earth and Moon, a the Earth's radius, d the Moon's distance. 
The resultant attraction of the Moon on the Earth as a 
whole is &Mm/ CM*, and the Earth is therefore moving with 
acceleration km/ CM* towards the common centre of mass of 
the Earth and Moon, as shown in 422, 424. 




FIG. 146. 

(i.) N"ow at the sublunar point A the Moon's attraction on 
unit mass is km/AM 2 and is greater than that at C (since 
AM < CM). Hence the Moon tends to accelerate A more 
than C and thus to draw a body at A away from the Earth, 
with relative acceleration F, where 

l CA 




d*(d-aj 2 d* (1-a/dy 

Since a/d is a small fraction, we have, to a first approximation, 



376 



ASTRONOMY. 



(ii.) At A the Moon's attraction per unit mass is km/A'M' 2 , 
and is less than that at (7, since AM > CM. Hence the Moon 
tends to accelerate C more than A, and thus to draw the 
Earth away from A with relative acceleration F', where 



= Jan 




To a first approximation, therefore, 



(l+a/d 



Thus a body either at A or A tends to separate from the 
Earth, as if acted on by a force away from C, of magnitude 
approximately = 2kina/d* per unit mass. 




FIG. 147. 



(iii.) Consider now the effect of the Moon's attraction on a 
body at B. This produces a force per unit mass of 
which may be resolved into components 



and 



Since we have taken 7>3f = CM, the first component is 
equal to km/ CM 3 ; that is, to the force at C. This component 
therefore tends to make a body at B move with the rest of 
the Earth, and produces no relative acceleration. Therefore 
the Moon tends to draw a body at B towards the Earth with 
relative acceleration /, represented by the second component ; 

thus 



FUKTHER APPLICATIONS OF THE LAW OP GRAVITATION. 377 

The point B is approximately the end of the diameter BCB 
perpendicular to AC (since BM, CM, B'M are nearly parallel 
in the neighbourhood of the Earth). 

Hence the relative acceleration at B is approximately per- 
pendicular to CM, and its magnitude 

f=km=km . 
d it' 

Similarly at B' the Moon tends to draw a body towards C, 
with relative acceleration /= kma/d*. 

At either of these points, B, B', therefore, a body tends to 
approach the Earth, as if acted on by a force towards the Earth's 
centre, of magnitude kma/d 3 per unit mass. Generally, the 
Moon's attraction at any point tends to accelerate a body, 
relatively to the Earth, as if it were acted on by a force depend- 
ing on the difference in magnitude and direction between the 
Moon's attractions at that point and at the Earth's centre. 

This apparent force is called the Moon's disturbing 
force or tide-generating force. AVc sec that the dis- 
turbingforce produces a pull along^L4' and a squeeze along////. 

A similar consequence arises from the attraction of the Sun. 
The Sun's actual attraction on the Earth as a whole keeps the 
Earth in its annual orbit, but the variations in the attraction 
at different points give rise to an apparent distribution of 
force on the Earth which is the Sun's disturbing force or 
tide-generating force. 

441. To find approximately the Moon's or Sun's 
Disturbing Force at any point. 

Let be any point of the Earth. Draw ON perpen- 
dicular on CM. " Then the difference of the Moon's attractions 
at and N tends to accelerate towards JV, with a relative 
acceleration 1cm . NO/d* [by 440 (iii.)]- Also, the difference 
of the attractions at N, C tends to accelerate TV away from C 
with a relative acceleration 2km. CN/d 3 [by 440 (i.)]. 

The whole acceleration of 0, relative to C, is compounded 
of these two relative accelerations. Therefore, if X. Foe the 
components of the disturbing force at in the directions 
CN, NO, 

ON v ^ 
.- , Y= K 



378 ASTRONOMY. 

442. Hence the following geometrical construction ; 
On CN produced take a point IT such that 



Tlicn the line OH represents the disturbing force at in 
direction, and its magnitude is 

v 7 on 

F = 1cm. -. 
d* 

The Sun's tide-raising force may be found exactly in the 
same way. The force is everywhere directed towards a point 
on the diameter of the Earth through the Sun, found by a 
similar construction to the above. And if r, S denote the 
Sun's distance and mass, the force is proportional to S/r s 
instead of mj&. 

In all these investigations we see that the tide-raising force 
due to an attracting body is proportional directly to its mass 
and inversely to the cube (not the square) of its distance. 

From this it is easy to compare the tide-raising forces due 
to different bodies acting at different distances. 

EXAMPLES. 

1. To compare the tide-raising forces due to the Sun and Moon. 
The masses of the Sun and Moon are respectively 331,000 and 
gL times the Earth's mass. Also, the Sun's distance is about 390 
times the Moon's. 

.*. Sun's tide-raising force : Moon's tide-raising force 



= 33 : 73 nearly = 3:7 nearly. 

Thus the Sun's tide-raising force is about'three-sevenths of that 
of the Moon. 

2. To find what would be the change in the Moon's tide-raising 
force if the Moon's distance were doubled and its mass were in- 
creased sixfold. 

If /,/'betheold and new tide-raising forces at corresponding points, 

/'/= f = 4 f 
J ' J 2 3 ' l a> 4 

Therefore the tide-raising force would have three-quarters of its 
present value. 

3. To compare the Moon's tide-raising forces at perigee and 
apogee. 

The greatest and least distances of the Moon being in the ratio of 



FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 379 

l + TT to ! Te> or 19 to 17 ( 270), the tide-raising power at perigee 
is greater than at apogee in the ratio of 19 3 : 17 3 or 6859 : 4913, or 
roughly 7 : 5. 

4. To compare the maximum and minimum values of the Sun's 
tide-raising force. 

The eccentricity of the Earth's orbit being ^ these are in the 
ratio of (1 + sV) 3 : (1 g^) 3 , or approximately 1 + ^ : I--&, or 
21 : 19. As before, the force is greatest at perigee and least at 
apogee. 



Moon 




443. The Equilibrium Theory of the Tides. Let 

us imagine the Earth to be a solid sphere covered with an 
ocean of uniform depth. If we plot out the disturbing forces 
at different points of the Earth by the construction of 442, 
we shall find the distribution represented in Eig. 148, the 
lines representing the forces both in magnitude and direction. 
Here the disturbing force tends to raise the ocean at the 
sub-lunar point A and at the opposite point A, and to de- 
press it at the points B, B'. At intermediate points it tends 
to draw the water away from B and B\ towards A and A'. 

Hence the surface of the ocean will assume an oval form, 
as represented by the thick line in Eig. 148, and there will 
be high water at the sublunar point A and the opposite point 
A, low water along the circle of the Earth BB', distant 90 
from the sublunar point. Thus we have the same tides 
occurring simultaneously at opposite sides of the Earth. 

It may be shown that the oval curve aba'b' is an ellipse 
whose major axis is aa'. The surface of the ocean, therefore, 
assumes the form of the figure produced by revolving this 
ellipse about its major axis. This figure is called a prolate 
spheroid, and is thus distinguished from an oblate spheroid, 
which is formed by revolution about the minor axis. 

2c 



ASTRON. 



380 ASTEONOMT, 

But though this is the form which the ocean would assume 
if it were at rest, a stricter mathematical investigation shows 
that the Earth's rotation would cause the surface of the sea 
to assume a very different form. 

In fact, if the Earth were covered over with a sufficiently 
shallow ocean of uniform depth, and rotating, we should really 
have low tide very near the sublunar point A and its anti- 
podal point A', and high tide at the two points on the Earth's 
equator distant 90 from the Moon (Fig. 149). 

If the Moon were to move in the equator, the equilibrium 
theory would always give low ^ water at the poles. This 
phenomenon is uninnue'nced by the Earth's rotation, and since 
the Moon is never more than about 28 from the equator, we 
see that the Moon's tide-raising force has the general effect 
of drawing some of the ocean from the poles towards the 
equator. 

*444. A few other consequences of the equilibrium theory may 
also be enumerated. (1) According to it the height of the tides, or 
the difference of height between high and low water at any place, is 
directly proportional to the tide-generating force, and consequently, 
with the results of Example 1 of 442, the heights of the solar and 
lunar tides are in the proportion of 3 to 7. (2) Since the distortion 
of the mass of liquid is resisted by gravity, the height of the tide 
depends on the ratio of the tide-producing force to gravity, and 
therefore is inversely proportional to the intensity of gravity, and 
therefore to the density of the Earth ; if the density were halved, 
the height of the tides would be doubled. (3) If the diameter of 
the Earth were doubled, its density remaining the same, the inten- 
sity of gravity and the tide-producing force would both be doubled, 
since both are proportional to the Earth's radius. This would cause 
the ocean to assume the same shape as before, only all its dimensions 
would be doubled. f Consequently the height of the tide would also 
be doubled, and it thus appears that the height of the tide is pro- 
portional to the Earth's radius. 

We thus have the means of comparing the tides which would be 
produced on different celestial bodies, for the above properties show 
that the height of tide is proportional to ma/Dd?, where a and D are 
the radius and density of the body under consideration, TO, d the 
mass and distance of the disturbing body. 

*445. Canal Theory of the Tides. As an illustration, 
let us consider what would happen in a circular canal, not 
extremely deep, supposed to extend round the equator of a re- 

f Of course this is not a very strict proof. 



FTTRTHEB APPLICATIONS OF THE LAW OF GBAYITATION. 381 

volving globe. Then, in Fig. 149, it is clear that the direction 
of the disturbing force would, if it acted alone, cause the 
water in the quadrants AB and AB' to flow towards A ; 
and, in the quadrants A'B and AB', towards A. Hence this 
force acts in the same direction as the Earth's rotation in the 
quadrants B'A and BA, and in the opposite direction in AB 
and AB'. Hence, as the water is carried from A to B, it is 
constantly being retarded, from B to A it is accelerated, 
from A to B' it is retarded, and from B' to A it is again 
accelerated, the average velocity being, of course, that of 
the Earth's rotation. Hence the velocity is least at B and 
', and greatest at A and A'. 



Moon 




Now, it is easy to see that when water moves steadily 
in a uniform canal it must be shallow where it is swift and 
deep where it is slow. For, if we consider any portion 
of the canal, say AB, the quantity that flows in at one end 
A is equal to the quantity that flows out at the other 
end B. But it is evident that if the depth of the canal 
were doubled at any point without altering the velocity of 
the liquid, twice as much liquid would flow through the 
canal ; consequently, in order that the amount which flows 
through might be the same as before, we should have to halve 
the velocity of the liquid. This shows that where the canal 
is deepest the water must be travelling most slowly. Con- 
versely, where the velocity is least the depth must be greatest, 
and where the velocity is greatest the depth must be least. 
Hence the depth is least at A and A', and greatest at B and 
J?, just the opposite to what we should have expected from 
the equilibrium theory. 



882 ASTRONOMY. 

In a canal constructed round any parallel of latitude the 
same would be the case ; and hence, if we could imagine a 
uniform ocean replaced by a series of such parallel canals, 
low tide would occur at every place when the Moon was in 
the meridian. 

This theory (due to Newton), though sounder than Laplace's 
equilibrium theory, is still not quite mathematically correct. 
The true explanation of the tides, even in an ocean of uniform 
depth, is far more complicated, and quite beyond the scope 
of this book. 




446. Lunar Day and Lunar Time. According to 
either hypothesis, the recurrence of high and low water depends 
on the Moon's motion relative to the meridian ; hence, in 
investigating this, it is convenient to introduce another kind 
of time, depending on the Moon's diurnal motion. 

The lunar day is the interval between two consecutive 
upper transits of the Moon across the meridian. 

In a lunation, or 29 mean solar days, the Moon performs 
one direct revolution relative to the Sun, and therefore per- 
forms one retrograde revolution less relative to the meridian. 
Thus 29J mean days = 28J lunar days ; whence the mean 
length of a lunar day 

= O + BT) mean 8 lar days = 24h. 50m. 32s. nearly. 
The lunar time is measured by the Moon's hour an^le, 
converted into hours, minutes, and seconds, at the rate of ^15 
to the hour. 



ttjItTHEB, APPLICATIONS OP THE LAW OF GRAVITATION. 383 

*447. Semi-diurnal, Diurnal and Fortnightly Tides. 

It has been found convenient to regard the tides produced 
by the Moon's disturbing force as divided into three parts, 
whose periods .are half a day, a day and a fortnight, the 
" day " being the lunar day of the last paragraph. 

If we adopt the equilibrium theory as a working hypothesis, 
the lunar tide must be highest when the Moon is nearest to 
the zenith or nadir. Hence high tide takes place at the 
Moon's upper and lower transits, when its zenith distance and 
nadir distance are least respectively. But, for a place in N. 
lat. (Fig. 150) when the Moon's declination is K, it describes a 
small circle Q,'R\ and its least zenith distance ZQ 'is less than its 
least nadir distance NR ; hence the two tides are unequal 
in height. This phenomenon can be represented by supposing 
a diurnal tide, high only once in a lunar clay, combined 
with a semi-diurnal tide, high twice in this period. 

Again, the Moon's meridian Z.D. and N.D. go through a 
complete cycle of changes, owing to the change of the Moon's 
declination, whose period is a month. But after half a month, 
the Moon's declination will have the same value but opposite 
sign, and hence the diurnal circles Q[R' , Q,"R" ', equidistant 
from the equator Q,R, are described at intervals of a fortnight. 
But NJR"= ZQ', ZQ"=: NR' ; hence the two tides have the 
same heights. This can be represented by supposing a fort- 
nightly tide of the proper height combined with the 
diurnal and semi-diurnal ones. 

In just the same way the smaller tides caused by the Sun 
may be artificially represented by combining a diurnal 
and semi-diurnal tide (the solar day being used) and a 
six-monthly tide. 

448. Spring and Neap Tides. Priming andLagging 

We have hitherto considered chiefly the tides due to the 
action of the Moon. In reality, however, the tides are due 
to the combined action of the Sun and Moon, the tide-raising 
forces due to these bodies being in the proportion of about 
3 to 7 (Ex. 1, 442). We shall make the assumption that the 
height of the tide at any place is the algebraic sum of the 
heights of the tides which would be produced at that place by 
the Sun and Moon separately. 



384 ASTRONOMY. 

/ At new or full Moon the Sun is nearly in the line AA\ 

(, and the tide-raising powers of the Sun and Moon both act in 

)the same direction, and tend to draw the water from B^B' to 

i A^^AL^ hence the whole tide is that due to the sum of the 

) separate disturbing forces of the Sun and Moon. The tides 

/ are then most marked, the height of high water and.depth_pf 

I low water being at their maximum. Such tides are called 

Spring Tides. We notice that the height of the spring 

tide = 1 +f or \- ^ * na * f tnc lunar tide alone. 




Moon 



At the Moon's first or last quarter the Sun is in a 

line BB' perpendicular to A A'. Hence the Sun tends to draw 
the water away from A, A' to B, B >', while the Moon tends to 
draw the water in the opposite direction. The Moon's action 
being the greater, preponderates, but the Sun's action 
diminishes the tides as much as possible. The variations 
are therefore at their minimum, although high water still 
occurs at the same time as it would if the Sun were absent. 
These tides are called Neap Tides. The^ndght of the 
neap, _ tide is the difference of the heights of the lunar and 
solar tides, and is therefore f of that of the lunar tide. 

Hence spring tides and neap tides are in the ratio of 
(roughly) 10 to 4. 

For any intermediate phase of the Moon, the Sun's action 
is somewhat different. 

Between new Moon and first quarter, the Sun is over 
a point S l behind A. Here the Moon tends to draw the 
water towards A, A', and the Sun tends to draw the water 
towards S l and the antipodal point $ s . Therefore the com- 
bined action tends to draw the water towards two points Q, Q' 



APPLICATIONS OF THE LAW OF GRAVITATION". 385 

between A and S l and between A and S. 6 respectively, whose 
longitudes are rather less than those of A and A respectively. 
The resulting position of high water is therefore displaced to 
the west, and the high water occurs earlier than it would if 
due to the Moon's influence alone. The tides are then said 
to prime. 

Between first quarter and full Moon the Sun is over 
a point $ 2 between ' and A, and the combined action of the 
Sun and Moon tends to draw the water towards two points 
jR, R', whose longitudes are slightly greater than those of 
A, A. The resulting high tides are therefore displaced east- 
wards, and occur later than they would if the Sun were 
absent. The tides are then said to lag. 

Between full Moon and last quarter the Sun is over 
some point $ 8 between and A', but the antipodal point S l 
is between A and B' ; hence the tide primes. 

Between last quarter and new Moon, when the Sun 
is at a point S between B and A, it is evident in like manner 
that the tide lags. 

Hence Spring Tides occur at the syzygies (conjunction 
and opposition). 

Neap Tides occur at the quadratures. 
From syzygy to quadrature, the tide primes. 
From quadrature to syzygy, the tide lags. 

The heights of the spring and neap tides vary with the varying 
distances of the Sun and Moon from the Earth. Spring tides are 
the highest possible when both the Sun and Moon are in perigee, 
while neap tides are the most marked when the Moon is in apogee 
but the Sun is in perigee (because the Sun then pulls against the 
Moon with the greatest power, as far as the Sun's action is con- 
cerned). Both the spring and neap tides, and also the priming and 
lagging, are on the whole most marked when the Sun is near per ^ee, 
i.e.. about January. 

It may be here stated, without proof, that, taking the Sun s am 
Moon's tide-raising forces to be in the proportion ot o 7, ti 
maximum interval of priming or lagging is found \ 
61 minutes. 



386 ASTRONOMY. 

449. Establishment of the Port. Both the equilibrium 
and canal theories completely fail to represent the actual tides 
on the sea, owing to the irregular distribution of land and 
water on the Earth, combined with the varying depth of the 
ocean. These circumstances render the prediction of tides 
by calculation one of the most complicated problems of prac- 
tical astronomy, and the computations have to be based largely 
on previous observations. In consequence of the barriers 
offered to the passage of tidal waves by large continents, 
lunar high tide does not occur either when the Me on crosses 
the meridian, as it would on the equilibrium theory, or 
when the Moon's hour angle is 90, as it would on the canal 
theory. But this continental retardation causes the high 
tide to occur later than it would on the equilibrium theory, 
by an interval which is constant for any given place. This 
interval, reckoned inlunar hours, is called the Establishment 
of the Port for the place considered. Thus the establish- 
ment of the port at London Bridge is Ih. 58m., so that lunar 
high water occurs Ih. 58m. after the Moon's transit, i.e., 
when the Moon's hour angle, reckoned in time, is Ih. 58m. 

The same causes affect the solar tide as the lunar, hence 
the Sun's hour angle (or the local apparent time) at the solar 
high tide is also equal to the establishment of the port. 

The actual high tide, being due to the Sun and Moon con- 
jointly, is earlier or later than the lunar tide by the amount 
of priming or lagging. By adding a correction' for this to 
the establishment of the port, the lunar time of high water 
may be found for any phase of the Moon ; and we notice in 
particular that at the Moon's four quarters (syzygies and 
quadratures), the lunar time of high water is equal to the 
establishment of the port. And, knowing the lunar time of 
high water, the corresponding mean time can be found, for 

(mean solar time) (lunar time) 

= (mean 0's hour angle) ( ([ 's hour angle) 
= ( d 's R.A.) -(mean Q's R.A.) 
[since R.A. and hour angle are measured in opposite directions]. 

Now the Moon's R.A. is given in the Nautical Almanack 
for every hour of every day in the year. Also the mean Sun's 
R.A. at noon is the sidereal time of mean noon, and is given 



FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 387 

in the Nautical Almanack. Hence the mean Sun's R.A. 
[which = (sidereal time) - (mean time)] is easily found for 
any intermediate time. 

Hence the mean time of high water can be readily found. 

The establishments of different ports, and the times of high 
water at London Bridge, are given in the Nautical Almanack. 

*450. If only a very rough calculation is required, we may proceed 
as in 35, 40. We assume the Moon's R.A. to increase uniformly ; 
we shall then have 

( <[ 's R.A.) - ( 0's R.A.) = ( O elongation) ; 
.'. (solar time) = (lunar time) + ( <t 's elongation). 
Knowing the Moon's age, its elongation may be found, as in 40, 
and this must be converted into time, at the rate of Ih. to 15. We 
then have (time of high water) 

= (establishment) + (amount of lag.) + ( C 's elongation in time) 

EXAMPLE. To find, roughly, the time of high water at the Moon's 
first quarter, at London Bridge. 

Here there is no priming or lagging. Hence the lunar time, or 
's hour angle at high water, is equal to the establishment, or Ih. 
58m. Also the Moon's elongation is 90. Hence the Sun's hour 
angle, in time, = Ih. 58m. + 6h ., and high water occurs about 7h. 58m. 

*451. Tidal Constants. The excess of the establishment of the 
port at any place, over that at London Bridge, expressed in mean 
time, is sometimes called the Tidal Constant of that place. 

If we assume the amount of priming or lagging to be the same 
at both places, the tidal constant is the difference between the times 
of high water at London Bridge and the given place. Hence, 
knowing the tidal constant and the time of high water at London 
Bridge, the time at any other place can be found. 

Tables of tidal constants, and of the heights of the spring and neap 
tides at different places, are given in Whitakers Almanack. 

EXAMPLE. To find the times of high water at Cardiff and Ports- 
mouth on January 25, 1892, the tide intervals from London Bridge 
being +4h. 58m. and 2h. 17m. From the Almanack we find 
times of high water at London Bridge are 

Jan. 24. . Jan. 25. 

9h. 15m. aft. 9h. 53m. morn., lOh. 31m. aft. 
Add 4h. 58m. 4h. 58m. 4h. 58m. 

/. Times at Cardiff are 

(Jan. 25) 2h. 13m. morn. 2h. 51m. aft. 
Again, subtract from first line 2h. 17m. 2h. 17m. 



times at Portsmouth are (Jan. 25) 7h. 36m. morn., 8h. 14m. aft. 



388 



ASTROXOJltf. 



452. The Masses of the Sun and Moon can be com- 
pared by observing the relative heights of the solar and lunar 
tide, the relative distances of the Sun and Moon being known. 
Or, if the ratio of the masses be supposed known, the dis- 
tances could be compared by this method. In this manner 
Newton (A.D. 1687) found the masses of the Moon and Earth 
to be in the proportion of 1 : 40 nearly. D. Bernouilli 
(1738) found 1 : 70, and Lubbock (1862) found 1 : 67-3. 
The two last make the Moon's mass a little too great. Newton 
makes it double what it ought to be. 




FIG. 152. 



453. Effects of Tidal Friction. Retardation of 
Earth's Rotation. Acceleration of Moon's Orbital 
Motion. All liquids possess a certain kind of friction, 
known as u viscosity," which tends to resist their motion 
when they are changing their form, and to convert part of 
their kinetic energy into heat. Owing to this friction between 
the Earth and the oceans, the Earth, in its diurnal rotation, 
tends to carry the tidal wave round slightly in front of the 
point underneath the Moon, taking the positions of high water 
forward from the line JI'CMto AC A. The Moon, on the 
contrary, tends to draw the water back from A : A', the dis- 
turbing forces AH, A' II' forming a couple, which is resisted 
only by the Earth's friction. Hence the ocean exerts an 
equal frictional couple on the Earth, and this couple tends to 
diminish the angular velocity of the Earth's diurnal rotation, 
and thus increase its period. 

Therefore tidal friction tends to gradually lengthen the day. 



APPLICATIONS OF THE JAW OF GBAVlTATlOtf. 389 

But if the Moon exerts a couple on the Earth, tending to 
retard it, the Earth must exert an equal and opposite couple 
on the Moon, tending to accelerate it. That it really does so 
is manifest from Fig. 152. The portion of the ocean heaped 
up at A, being nearer the Moon, exerts a greater attraction than 
that at A', in addition to which the angle CM A is very 
slightly greater than GNA '. Hence the resultant of the 
attractions of equal masses of water at A and A acts on M 
in a direction slightly in front of M C, and tends to pull the 
Moon forward. This tends to increase the Moon's areal 
velocity. (Compare 413.) Since the areal velocity of a 
body revolving in a circle varies as the square root of the 
radius ( 411, Cor.), the Moon's distance must be gradually 
increased by this means, and hence also its periodic time.* 

Therefore tidal friction tends to increase the Moon's distance 
and to lengthen the month. 

Still the final effect of tidal friction must be to equalize 
the lengths of the day and lunar month. The angular 
velocities of the Earth and Moon "both decrease, but the 
effect of the couple, in producing retardation, is far more 
considerable on the Earth than on the Moon. 

The student who has not read Rigid Dynamics may illustrate this 
statement by the comparative ease with which a small top can be 
spun with the fingers, and the great difficulty of imparting an equal 
angular velocity to the same body by whirling it round in a circle 
at the end of a string of considerable length. The top represents 
the Earth, and the body on the long string the Moon. 

In Rigid Dynamics it is shown that when a system of bodies are 
revolving under their mutual reactions, their angular momentum, or 
moment of momentum about their centre of mass, remains constant. 
Hence the decrease in the Earth's angular momentum is equal to 
the increase in that of the Moon. Now the angular momentum 
of a particle revolving in an orbit is twice the product of its mass 
into its areal velocity, and this is also approximately true of the 
Moon. Hence, since the Moon's distance from the common centre 
of mass is far greater (about sixty times as great) than the distance 
of any point on the Earth from its axis of rotation, it is evident 
that the same change in angular momentum produces far more effect 
on the angular velocity of the Earth than on that of the Moon. 

* This increase of the distance more than counterbalances the 
tendency to increase the Moon's actual velocity. For the actnai 
velocity is inversely proportional to the square root of the distance 
( 409), and therefore diminishes as the distance increases, 
Similarly, the angular velocity is decreased. 



390 ASTfcONOMt. 

It thus appears that, after the lapse of probably many 
millions of years, tidal friction will equalize the periods of 
rotation of the Earth and Moon, and the day and month will be 
of equal length, each being probably about 1,400 hours long. 
The Earth will then always turn the same face towards the 
Moon, just as the Moon now does towards the Earth ; hence 
there will be no lunar tides, and the retardation due to lunar 
tidal friction will no longer exist. 

The solar tides will, however, still continue to exist, pro- 
vided that there is any water left on the Earth. The effect 
of solar tidal friction will be to retard the Earth's rotation, 
thus further lengthening the day ; and this again will retard 
the Moon's orbital motion, and diminish its areal velocity. 
The Moon will, therefore, approach the Earth, and will 
ultimately fall into the Earth ; and finally, the Earth will 
always turn the same face towards the Sun, so that there 
will always be day over one hemisphere and night over the 
other. 

This theory of the probable future history of the Earth is 
due to Professor G. H. Darwin. It is certain that the effect 
of tidal friction on the Earth's rotation must be very small ; 
hence a very long period must necessarily elapse before any 
perceptible increase in the length of the day can be detected. 
The records of history afford no data sufficiently accurate to 
furnish conclusive evidence of such a lengthening, but there 
are some grounds for believing that the sidereal day is in- 
creasing in length by about *006 of a second in 1,000 years. 

Moreover, the Earth is gradually cooling, and consequently 
is shrinking ; and this shrinkage, by bringing the particles of 
the Earth nearer to the axis, causes an increase of the angular 
velocity of rotation.* It is quite possible that an increase of 
this nature is at the present time either wholly or partially 
counteracting the retardation due to tidal friction. 



* For, according to the principles of Rigid Dynamics, the angular 
momentum of the Earth = (its angular velocity) x (its moment of 
inertia). And if the angular momentum remains constant, and the 
moment of inertia decreases through shrinkage, the angular velocity 
must increase. 



FUHTHER APPLICATIONS OF THE LAW OF GEAVITATIOfl . 391 

454. The Moon's Form and Rotation. The theory of 
tidal friction affords a simple explanation of how it is that the 
Moon always turns the same face to the Earth. Remember- 
ing that the Earth's mass is 81 times the Moon's, but that its 
radius is about four times as great, the Earth's tide-raising 
force at a point on the Moon would be about 81/4, or over 
twenty times as great as the Moon's on the Earth. Although 
there are now no oceans on the Moon, still we have some 
evidence that water may once have existed on its surface. 
Furthermore, the large volcanic craters with which its sur- 
face is dotted prove that the Moon was at one time filled 
with molten lava, and that it was probably wholly in a liquid 
or viscous state at an earlier period of its history. At that 
time the huge tides on the Moon, ever following the Earth, 
must, by their friction, have gradually equalized the Moon's 
period of rotation with its period of revolution about the 
Earth, in just the same way as if the Moon were surrounded 
by a friction belt attached to the Earth. This continued till 
the Moon always turned the same face to the Earth. 

If the Moon was then not quite solid, the Earth's tide- 
raising force, which had then become constant, must have 
drawn it out into the form demanded by the equilibrium 
theory, namely, to a first approximation, a prolate spheroid, 
with its longest diameter pointed towards the Earth. 

It may easily be seen, from the expressions in 440, that th- 
tide-raising force of a body is slightly greater at the poinl 
just under it than at the opposite point (when we do not 
only consider approximate values). Hence the Moon is not 
quite spheroidal, but is more drawn out on the side toward^ 
the Earth than on the remote side. Its form is, therefore, 
that of an egg, the small end being towards the Earth. This 
result of theory cannot, of course, be confirmed by direct 
observation, the remote side being invisible ; butHansen, by 
the theory of perturbations, has shown that _ the Moon's 
centre of mass is further from the Earth than its centre of 
figure, thus furnishing independent evidence in favour of 
the theory. 







392 ASTBONOMY. 

*455. Application to Solar System. Since the Sun's 
tide-raising force on different planets varies inversely as the 
cube of their distance, the solar tides are far greater on the 
nearer planets than on those more remote. It is, therefore, 
quite natural to suppose that the effects of tidal friction may 
have produced such a great retardation in the rotations of 
Mercury, and possibly also Venus, that one or both of these 
bodies already turn the same face towards the Sun, while the 
Earth, and the remoter planets, must necessarily take a much 
longer time to undergo the necessary retardation, and it 
would be very unnatural to expect JS~eptune, for example, 
always to turn the same face to the Sun. Thus Professor 
Schiaparelli's recent researches on the rotations of Mercury 
and Yenus are in support of the theory of tidal friction. 



SECTION III. Precession and Nutation. 

456. In 141 we stated that the plane of the Earth's 
equator is not fixed in space, but that its intersections with 
the ecliptic have a slow retrograde motion. This phenome- 
non, which is known as Precession, is due to the fact that the 
Earth is not quite spherical, and that, in consequence of its 
spheroidal form, the Sun's and Moon's attractions exert a dis- 
turbing couple on it. 

457. The Sun's and Moon's Disturbing Couples 
on the Earth. 

Let the plane of the paper in Fig. 153 contain the Earth's 
polar axis PP', and the Moon's centre M, say at the time 
when the Moon's south declination is greatest. 

Inside the Earth inscribe a sphere PAP' A', touching its 
surface at the poles. Then we may (for the sake of illustra- 
tion) regard the protuberant portion of the Earth outside this 
sphere as a kind of tide firmly fixed to the Earth, and the argu- 
ments of the last section ( 453) show that the variations in the 
Moon's attraction at different points give rise to a distribution 
of disturbing force identical with the tide -raising force, tending 
to draw this protuberant part with its longest diameter QR 
pointing towards the Moon. The Moon's attraction on the 
matter inside the inscribed sphere- passes exactly through the 



FTJBTHEB APPLICATIONS OP THE LAW OP GRAVITATION. 393 

Earth's centre 0, and produces no such couple ; but the dis- 
turbing forces at A, A', which are represented by AH, A' IT, 
form a couple on the protuberant parts, A Q, A'R, tending 
to turn the diameter A A towards CM. The same is true of 
the disturbing forces at any other pair of opposite points of 
the Earth in the quadrants HCK, H'CK'. Of course there 
are couples in the two other quadrants tending in the reverse 
direction, but they have less matter to act on, and are there- 
fore insufficient to balance the former couples. 




FIG. 153. 



When the Moon is at the opposite point of its orbit, i.e., at 
its greatest N. declination, it is again in the line CH', and 
again tends to draw the Earth's equatorial plane towards the 
line HH'. For any intermediate position of the Moon the 
couple is smaller, and it vanishes when the Moon is on the 
equator ; still, on the whole, the Moon's disturbing force always 
tends to draw the plane of the Earth? s equator towards the plane 
of the Moon's orbit. 

Similarly, the Sun's disturbing force always tends to draw 
the plane of the Earth's equator towards the ecliptic. 

Since the Moon's nodes are rotating ( 273), the plane of the 
Moon's orbit is not fixed ; but it is inclined to the ecliptic at 
a small angle (5), while the plane of the equator is inclined 
to the ecliptic at a much larger angle (23|). The average 
effect of the Moon's disturbing couple is thus to pull the 
Earth's equator towards the plane of the ecliptic. This ten- 
dency is increased by the Sun's disturbing couple ; and the 
two are proportional to the Sun's and Moon's tide-producing 
forces, i.e., as 3 : 7 roughly. For this reason, the resulting 
phenomenon is sometimes called luni-solar precession. 



394 ASTBONOMT. 

*458. Effect of the Couple on the Earth's Axis. 

If the Earth were without rotation, the tendency of this 
couple would be to bring the plane of the equator into coinci- 
dence with the ecliptic, with the result that the equator 
would oscillate from side to side of the ecliptic, like a pendu- 
lum under gravity. But the rapid diurnal motion of the 
Earth entirely alters the phenomena. 

Let CR be a semi-diameter of the Earth, perpendicular to 
CP and CM. The precessional couple would, alone, produce a 
slow rotation in the direction PQM; i.e., about CR. If 
now the Earth's rotation be represented in magnitude and 
direction by CP, measured along the Earth's axis, this addi- 
tional rotation must be represented by a very short length 
CR', measured along CR. 




FIG. 154. 

Take PP', equal and parallel to CR' ; then, since PP' is 
very small, CP' is of almost exactly the same length as CP. 
But angular velocities, and momenta about lines which repre- 
sent them in magnitude, are compounded by the same law as 
forces, velocities, &c. \ef. 387 (iii.)] along the same lines of 
corresponding magnitudes. 

Hence, the resultant axis of rotation is shifted from CP to 
CP', in a direction perpendicular to the plane of the acting 
couple. 

A full explanation of what follows would be impossible 
without a close acquaintance with Rigid Dynamics. But it is 
evident that a body flattened at the poles will spin more 
readily about the line CP than about any other line drawn in 
its substance. Hence it is easy to understand that the polar 
axis CPis itself deflected towards CP', and thus moves per- 
pendicular to the acting couple. 



FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 395 

This motion can be illustrated by that of a rapidly spinning 
top, or of a gyroscope, the phenomena of which can readily be 
investigated by experiment. 

459. Precession of a Spinning Toy. Experiment 1. 

Let a top be set spinning rapidly about its extremity, in the oppo- 
site direction to the hands of a watch, as seen from above, the top 
being supported at a point on its axis below its centre of gravity. 
The weight of the top, acting vertically through the centre of gravity, 
tends to upset the top by pulling its axis out of the vertical. But if 
the top is spinning sufficiently rapidly, we know that it will not fall, 
the only effect of gravity being to make it " reel," i.e., to cause its 
axis of rotation to describe a cone about the vertical through the 
point of support, revolving slowly in the counter-clockwise direction. 
This slow revolution may be called the precession of the top, and 
the experiment shows that when a top is acted on by a couple (such 
as that due to its weight) tending to pull its axis away from the 
vertical, it processes in the same direction in which it is spinning. 

Experiment 2. Now suppose the top suspended from its upper 
extremity, being thus supported above its centre of gravity. The 
couple due to the weight and the reaction of the support, now tends 
to draw the axis of the top towards the vertical. In this case the 
axis of the top will be found to slowly describe a cone in the opposite 
direction ; that is, the top now precesses in the opposite direction to 
that in which it is spinning. 

Experiment 3. Suppose the top supported as in Experiment 1. 
If we give the top a push away from the vertical, its axis will not 
move in this direction, but its processional motion will increase. If 
we give a push in the direction of precession, its axis will approach 
the vertical. If we push the axis in the direction of the vertical, it 
will not move towards the vertical, but its rate of processional motion 
will be increased, i.e., the top will acquire an additional increased 
precessional motion. If we push it in the direction opposite to 
that of precession, the axis will begin to move away from the vertical. 
In every case the axis of the top moves in a direction perpendicular 
to the direction of the force acting on it, and therefore a couple 
acting on a very rapidly spinning top produces displacement of the 
axis in a plane perpendicular to the plane of the couple.* 

[If we push the top by pressing the side of a pencil against its 
axis, it thus always moves in the direction in which the axis would 
roll along the side of the pencil. Of course the displacement of the 
axis is not due to rolling, as may easily be shown by repeating tl 
same experiment with a gyroscope, this time pushing one of the 
hoops carrying the top instead of touching the top itself ; here no 
such rolling is possible.] 

* These experiments may easily be performed by the reader with 
any good-sized top. 

ASTBOX. 2 D 



396 ASTB-ONOMt. 




r'r 
FIG. 155. 

460. Precession of the Earth's Axis. On the celestial 
sphere, let P, K he the poles of the equator and ecliptic 
respectively. The Sun's disturbing couple and the mean 
couple due to the Moon tend to pull the Earth's equator 
towards the ecliptic, or to pull the polar axis P towards the 
axis of the ecliptic K. Hence the Earth "behaves like a top 
suspended from above its centre of gravity, and the polar 
axis slowly describes a cone about the axis of the ecliptic, 
revolving in the opposite direction to that of the Earth's 
rotation, i.e., in the retrograde direction.* The pole P there- 
fore slowly describes a small circle PP' about It, the pole of 
the ecliptic, with angular radius P7T, equal to the obliquity 
of the ecliptic, i.e., 23 27'. As the pole revolves from P to 
P' it carries the equator from r Q^ to f '$=', thus carry- 
ing the equinoctial points T and b slowly backwards along 
the ecliptic. The average angle T T ', or P/fP'f , described in 
a year, is 50-2", and P therefore performs a complete revolu- 
tion about -5Tin 25,800 years ( 141). 

* Sec also Fig. 154. If K be pole of ecliptic (CK nearly perpen- 
dicular to CM) it is evident that as P travels towards P' it moves 
in the retrograde direction about It". 

t PT andETT are each 90; .'. T is. pole of arc KP-, .'. LrKP is 
a right angle. Similarly, T 'KP' is a right angle ; 

/. PKP' = L TKr' = arc T T 7 , 
since T T'^ is a great circle, whose pole is K. 






FURTHER APPLICATIONS OP THE LAW of GRAVITATION. 397 

The position of the ecliptic is not affected by precession. 
Hence the celestial latitude xH of any star x remains constant, 
and its celestial longitude T Jf increases by the amount of pre- 
cession r T', that is, at the rate of '50-2" per year. 

A star's declination and right ascension are, however, con- 
tinually changing. This change is, of course, due to the 
motion of the equator, and not of the star. Thus, as P moves 
to P', the KP.D. of the star x decreases from Px to P'x, and 
its 11. A. changes from TPx to T'P'x. (The circles 
T-P, T'F, %P, %P' are not represented, in order not to com- 
plicate the figure unnecessarily. The reader should draw a 
figure, inserting them.) 

The declinations of some stars are increasing, of others 
decreasing. 

461. To apply the Corrections for Precession. 

The changes in the decl. and R.A. of a star in one year are 
always small, except in the case of the Pole Star, which is 
so near the pole that a slight displacement of the pole pro- 
duces a great change in the R.A. With this exception, the 
rates of change of the decl. and R.A. of a star remain sensibly 
constant for a considerable period. Hence, if the coordi- 
nates are observed on any given date, and their rates of 
variation are known, their values at any other date may be 
found by adding or subtracting corrections obtained by mul- 
tiplying these rates of variation by the elapsed time. 

The rates of variation may be regarded as constant so long 
as the interval of time is small compared with the period of 
rotation of the pole. They are therefore sensibly uniform for 
several years. 

The most convenient plan, in correcting for precession, is to 
calculate the right ascensions and declinations of all stars for 
the same date or epoch. 

For this purpose, the time of the vernal equinox in the year 
1900 is now frequently chosen as the standard epoch of refer- 
ence. "When the R.A. and decl. of a star are known, their 
rates of variation can be calculated by Spherical Trigonometry 
in terms of the known rate of precession, and the correction 
can then be applied. 



398 A.STBONOMY. 

It would, of course, be possible to proceed somewhat differ- 
ently, namely, from the decl. and R.A. to find tbe star's lat. 
and long. Tbe long, could tben be increased by tbe amount 
of tbe precession, namely, 50 -2" x (tbe number of years 
elapsed) ; and from tbe new lat. and long, tbe new decl. and 
R.A. could be found ; but tbe calculations would be longer. 

For the purpose of facilitating observations of time, latitude 
and longitude, and instrumental errors, tbe declinations and 
right ascensions of certain bright stars are calculated at 
intervals of ten days in tbe Nautical Almanack ; these stars 
are the clock stars of 54. 

The effects of aberration, as well as of precession and nuta- 
tion, are taken into account, the tabulated coordinates being 
those of tbe apparent and not tbe true positions of the star. 
Such stars can therefore be used to determine clock error and 
other errors, without applying any further correction. 



462. Various Effects of Precession. 

Since the R.A. and decl. of a star depend only on tbe 
relative positions of the star and equator, their variations due 
to precession are just the same as they would be if the equator 
and ecliptic were fixed, and tbe stars bad a direct motion of 
rotation, of 50*2" per annum, about the pole of the ecliptic. 

If we make this supposition, tbe stars will describe circles 
about JTin a period of 25,800 years. 

(i.) If a star's distance Ex from the pole of the ecliptic is 
less than the obliquity t, or its latitude (I) greater than 
90 t, it will describe a circle ax^'x^ (Fig. 156), of radius 
90 Z, not enclosing the pole P, and its greatest and least 
N.P.D. will be 

Pa' = f+(90-f), Pa = t-(90-Z). 

Also the star's R.A. will fluctuate between the values 
rPx, and rP# 2 . Now r is the pole of PK; hence EPr 
is a right angle, and rPK= 270; therefore the maximum 
and minimum R.A. are 270 + JTPx v and 270 KPx r 



FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 399 

(ii.) If, on the other hand, the star's latitude is <90' *', 
it will describe a circle byb', enclosing the pole P. Its 
greatest and least N.P.D. will be 

PV = (90 - /) + , Pb = (90- /) - i. 
The star's E.A. will continually increase from to 360. 

In either case the star's N.P.D. will increase as its longitude 
increases from 90 (at a or b) to 270 (at a' or J'), and will 
decrease over the other half of the path. 




The Pole Star will, after a time, move away from the pole, 
and its place will be then occupied in succession by other 
stars whose latitude is very nearly = 90 i = 66 33'. If 
I, L be the latitude and longitude of such a star, it will be 
nearest the pole in an interval of (90 -Z)-^ 50 -2" years, and 
its N.P.D. will then be (90- J)~- 

That precession has shifted the equinoctial points from the 
constellations Aries and Libra, into Pisces and Virgo, has 
already been mentioned. Since there are twelve signs ot the 
zodiac, the equinoctial points shift from one " sign ' into the 
next in 25,800/12 years, *.*., about 2,150 years, 



400 ISTfiONOMT. 

463. Effects on the Climate of the Earth's Hemi- 
spheres. We have seen ( 132) that the fact of the Earth 
being in perihelion near the winter solstices renders the 
climate of the Earth's northern hemisphere more equable, but 
makes the seasons more marked in southern hemisphere. 
Owing, however, to precession, combined with the progres- 
sive motion of the apse line ( 153), the reverse will be the 

case in 180x60x6 or 10,545 years. The summer in the 

50-22 + 11-25' 

northern hemisphere will then be hotter, but shorter, and the 
winter colder and longer. On the whole, the climate will be 
colder, as the Earth's radiation will be more rapid during the 
heat of summer, and therefore a larger proportion of the heat 
received from the Sun will be lost before the winter. 

In a recent paper, Sir Robert Ball has shown that the ice 
ages, of which we have geological evidence, can probably be 
accounted for in this manner. The eccentricity of the Earth's 
orbit is not constant, but is changing very slowly, and is 
decreasing at the present time. "When the orbit had its 
greatest eccentricity and the winter solstice coincided with 
aphelion, the autumn and winter were 199 days long, spring 
and summer being only 166 days long. At this time the 
climate of the northern hemisphere must have been so exceed- 
ingly cold that the whole of northern Europe, including 
Germany and Switzerland, was ice-bound. When aphelion 
coincided with the summer solstice a similar effect took place 
in the southern hemisphere, but the northern hemisphere was 
warmer and more genial than it is now, spring and summer 
being 199 days long, and autumn and winter only 166 days 
long. Thus, at the time of greatest eccentricity 'there must 
have been long ages of arctic climate, oscillating from one 
hemisphere to the other and bask in a period of 10,500 years, 
alternating with more equable, and, perhaps, almost tropical 
climates. 



464. Nutation of the Earth's Axis. In treating of 
precession, we have supposed the Earth's poles to describe 
small circles uniformly about the poles of the ecliptic. This 



PT7ETHEE APPLICATIONS OF THE LAW OP GRAVITATION. 401 

they would do if the Sun's and Moon's disturbing couples on 
the Earth were always constant in magnitude, and always 
tended to pull the Earth's poles directly towards the poles of 
the ecliptic. But the couples, so far from being constant 
are subject to periodic variations, in consequence of which 
the Earth's poles really describe a wavy curve (shown in 
-Fig. 157), threading alternately in and out of the small 
circle which would be described under precession alone if the 
couple were constant. This- phenomenon is called Nutation, 
because it causes the Earth's poles to nod to and from the 
pole of the ecliptic. 




Nutation is really compounded of several independent 
periodic motions of the Earth's axis; the most important of 
these is known as Lunar Nutation, and has for its period 
the time of a sidereal revolution of the Moon's nodes, i.e., 
about 18 years 220 days. The effect of lunar nutation may 
be represented by imagining the pole P to revolve in a small 
ellipse about its mean position jp as centre, in the above period, 
in the retrograde direction, while p revolves about JT, the 
pole of the ecliptic, with the uniform angular velocity of 
precession of 50'2" per annum. The major and minor axes 
of the little ellipse are along and perpendicular to Kp re- 
spectively, their semi-lengths being pa 9" and pb = 6 -8" 
respectively. The angle pKb = bp/smJp (Sph. Georn. 17) 
^ 6-8"cosec 23 27' = 1 7' 1" nearly. 



402 ASTRONOMY. 

465. General Effects of Lunar Nutation. In con- 
sequence of lunar nutation, the obliquity of the ecliptic is 
subject to periodic variations. For this obliquity is equal 
to the arc KP, and as P revolves about its mean position 
from one end to the other of the major axis of the little 
ellipse, the arc KP becomes alternately greater and less than 
its mean value p, by 9". Thus the greatest and least values 
of the obliquity of the ecliptic differ by 18", and the obliquity 
fluctuates between the values 23 27' 20" and 23 27' 2" once 
in about 18f years. 




FIG. 158. 

Again, when the pole is at an extremity of the minor axis 
i, it has regreded further than its mean position p by the 
angle pKb, which we have seen is about 17-1". Hence, also, 
the first point of Aries has regreded 17'1" further than it 
would have gone had its motion been uniform. Similarly, at 
V it has regreded 17-1" less than it wt'uld have done if 
moving uniformly. Hence the first point of Aries oscillates 
to and fro about its mean position through an arc of 34' 2" in 
the period of 18-| years, while its mean position moves through 
an angle 18f X 50-2", or about 15' 37". 

The angular distance between the true and mean positions 
of the first point of Aries is called the Equation of the 
Equinoxes. It is, of course, equal to the angle pKP. 

Nutation does not affect the position of the ecliptic ; hence 
the latitudes of stars are unaltered by it. Their apparent 
longitudes are, however, increased by the equation of the 
equinoxes. Both this cause and the varying obliquity of 
the ecliptic produce variations in a star's R.A. and decl. 



FURTHER APPLICATIONS OP TUB LAW OP GRAVITATION. 403 

466. Discovery of Nutation. Nutation was discovered 
by Bradley soon after his discovery of aberration, while con- 
tinuing his observations on the star y Draconis and on a small 
star in the constellation Camelopardus, by its effect on the 
declinations of these stars. The peculiarity which led him 
to separate nutation from aberration was their difference of 
period. The period of the former phenomenon is about 19 
years, while that of the aberration displacement is only a 
year. Had the observed variations in declination been due 
to aberration alone, the declination would always have had 
the same apparent value at the same time of year, but such 
was not the case. 

Newton had, sixty years previously (1687), proved the 
existence of nutation from theory, but had supposed that its 
effects would be inappreciable. 

467. To correct for Nutation, the coordinates of a star 
are always referred to. the mean position of the ecliptic, i.e., 
the position which the ecliptic would occupy if its pole were 
at p, the centre of the little ellipse. Hence, since the 
apparent decl. and RA. of a star x are measured by 90 Px 
andr^ (= 270 + 7TP^), the corrected decl. and K.A. are 
90 jfcc and 270 + JTpx. If the star's position is specified 
by its celestial latitude and longitude, the only correction 
required is to increase the longitude by the equation of the 
equinoxes. 

*468. Bessel's Day Numbers. If the declinations and right ascen- 
sion of stars have been tabulated for a certain date, their apparent 
values for any other date, as affected by precession, nutation, and 
aberration, can be found by adding certain small corrections to the 
tabulated values, and it is found that these may be put into the 
form Change of R.A. = Aa + Bb + Cc + Dd, 

Change of decl. - Aa' + Bb' + Cc' -I- Dd', 

where A, B, 0, D are constants, whose values depend only on the 
date, and are the same for all stars ; while a, 6, c, d, a', &, c, d 
depend only on the coordinates of the star, being alway* constant for 
the same star, and independent of the time of observation. 

The four quantities A, B, C, D are called Bessel's Day Numbers, 
and their logarithms are given in the Nautical Almanack for every 
day of the year. The logarithms of the eight constants a, 6, c,d, 
a', b', c', d', have been tabulated for many thousands of stars m tt 
star catalogues of the Boyal Astronomical Society. 



404 ASTRONOMY. 

469. Physical Cause of Nutation. If the Moon were 
to move exactly in the ecliptic, the average couples exerted 
by the Moon as well as the Sun would both tend to pull the 
Earth's pole directly towards K, the pole of the ecliptic. 
But the Moon's orbit is inclined to the ecliptic at an angle 
of 5 ; hence, if L be its pole, KL = 5, and the Moon's 
average disturbing couple tends to pull the pole P towards 
L instead of K. When we consider the Sun's action also, 
the resultant of the two couples tends to pull the pole towards 
a point H which is intermediate between K and Z, but 
nearer to L (because the Moon's disturbing couple is about 
2J times the Sun's). Hence the pole P moves off in a direc- 
tion perpendicular to HP, and not to KP. In consequence 
of the rotation of the Moon's nodes, Z, and therefore also ZT, 
revolves in a small circle about P in the period of 18| years 
(see Fig. 159). 

Let Z t , Z 2 , Z 3 , Z 4 , Z 6 be the positions of Z, and P v P 2 , P 3 , 
P 4 , P 5 the positions of P, when the angle PKL is 0, 90, 
180, 270, 360 respectively, H v H^ the positions of ZT cor- 
responding to Z 2 , Z 4 . Then at P l and P 8 the couple is 
directed towards JT, and therefore P is then moving perpen- 
dicular to KP. At P 2 the couple is directed towards J7 2 , an ^ 
the pole P 2 moves perpendicularly to JToPg, thuspassingfromthe 
inside to the outside of the small circle described by its mean 
position. Similarly, at P 4 the pole, by moving perpendicularly 
to II fv passes back from the outside to the inside of the 
small circle which it would describe if the couple were 
always directed towards K. Thus the wavy form of the 
curve described by P is accounted for. And since the whole 
space Pj-ffPg or Zj J5fZ 5 , traversed in a revolution of Z, is very 
small, the period of oscillation is almost exactly that of 
revolution of the Moon's nodes. 

Again, the Moon's couple depends on the angular distance 
PZ, and is greater the greater this distance (as may easily be 
seen by 457). Hence the resultant couple, and therefore 
also the precessional motion, is least at P l and greatest at P 3 . 
This accounts for the variable rate of motion of P, which 
gives rise to the equation of the equinoxes. 



FUTITHEE APPLICATIONS OF THE LAW OF GRAVITATION. 405 





FIG. 159. 



FIG. 160. 



* 470. Solar and Monthly Nutations. The variations in the inten- 
sity of the Sun's and Moon's disturbing couples during their orbital 
revolutions give rise to two other kinds of nutation. Let us first 
consider the variations in the Sun's disturbing couple, which pro- 
duce Solar Nutation. It appears from 457, that the couple vanishes 
when the Sun is on the equator, and that it is greater the greater 
the Sun's declination. Also it is readily evident from Fig. 153 that 
the couple in general acts in a plane through the Sun and the 
Earth's poles, tending to turn the poles more nearly perpendicular 
to the direction of the Sun. This shows that the couple is not 
really directed towards the pole of the ecliptic (though this is its 
average direction for the year) except at the solstices (Fig. 160). 

Now at the vernal equinox, when the Sun is at T , the couple 
vanishes, and therefore the Earth's tendency to precession, due to 
the Sun, vanishes. Between the vernal equinox and the summer 
solstice, when the Sun is at Sj, the couple is along SjP away from 
Si, and this tends to make the pole precess along PG' perpendicu- 
larly to SiP. At the summer solstice the couple along CP is a 
maximum, and tends to produce precession along PG 5 perpendicular 
to KP. At 8-2 the couple along S 2 P tends to make the pole precess 
in the direction PG". At the autumnal equinox, ^, the couple, and 
therefore the velocity of solar precession, vanishes. AtS a the Sun's 
declination is negative, and the couple tends to draw P towards S 3 ; 
hence the Earth again tends to precess along PG'. At the winter 
solstice the direction of precession is again along PG, and the pro- 
cessional velocity again a maximum. Finally, at 84 the direction 
of precession is again along PG". 



406 ASTBONOMY. 

Hence the variations in the Sun'a declination cause the pole to 
thread its way in and out of the circle it would describe under 
uniform precession once every six months, and to cause the velocity 
of revolution about K to fluctuate in the same period. This gives 
rise to the nutation known as Solar Nutation, whose period is half a 
tropical year. In the case of the Moon the corresponding 
phenomenon is known as Monthly Nutation, and its period is half 
a month ; the explanation is exactly the same. 

The variations in the obliquity of the ecliptic due to these two 
causes are small, because, owing to the comparatively small period in 
which they recur, the pole has not time to oscillate to and from K 
to any great extent. Moreover, the couple, and therefore the rate of 
motion of P, decreases as the inclination of PG' to PO increases. 
When the Sun is at T or ^ the displacement, if it existed, would be 
along PK, in the most advantageous direction for producing nutation, 
but at this instant the couple vanishes. 

The solar nutation only displaces the pole about 1'2" to or from 
K, and the displacement due to monthly nutation is imperceptible. 
The effects on the equation of the equinoxes are more apparent. 
Under the Sun's action alone, the pole would come to rest twice a 
year, viz., at the equinoxes, and under the Moon's action its rate of 
motion would vanish twice a month, viz., when the Moon crossed 
the equator. At all other times the couples tend to produce retro- 
grade never direct motion of the pole about K. Hence the pre- 
cessional motion can never vanish unless the Sun and Moon should 
happen to cross the equator simultaneously. 



SECTION IY. Lunar and Planetary Perturbations. 

471. In consequence of the universality of gravitation, 
every body in the solar system has its motion more or less 
disturbed by the attraction of every other body. Kepler's 
Laws (with the modification of the Third Law given in 421 ) 
would only be strictly true if each planet were attracted 
solely by the Sun, and each satellite described its relative 
orbit solely under the attraction of its primary. Hence the 
fact that these laws very nearly agree with the results of 
observation shows that the mutual attractions of the planets 
are small compared with that which the Sun exerts on each 
of them, and that, in the orbital motion of a satellite, by far 
the greater part of the relative acceleration is due to the 
attraction of the primary. 



FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 407 

472. Lunar Perturbations- "We have seen, in Section 
I., that tEe Moon's motion consists of two component parts, a 
monthly orbital motion relative to the Earth or, more strictly, 
relative to the centre of mass of the Earth and Moon and 
the annual orbital motion of this centre of mass in an ellipse 
about the Sun. If the acceleration of the Sun's attraction 
were the same in magnitude and direction at the Moon as at 
the Earth, it would be exactly the acceleration required to 
produce the latter component, and the relative orbit about 
the Earth would be determined by the Earth's attraction 
alone. This is very nearly the case, owing to the great dis- 
tance of the Sun. But the small differences of the 
accelerations caused by the Sun's attraction on the Earth and 
Moon tend to modify the relative motion of these two bodies, 
by giving rise to perturbations ( 272). The relative 
accelerations thus produced may be represented by a distri- 
bution of disturbing force due to the Sun, just in the same 
way that the relative accelerations of the oceans, which cause 
the tides, are determined by distributions of disturbing force 
due to the Sun and Moon. And since the Sun's distance is 
nearly 400 times the Moon's, the expressions for ^the dis- 
turbing force, corresponding to those investigated in 441, 
are sufficiently approximate to account for the more impor- 
tant lunar perturbations. 




FIG. 161. 



Let S, E, M denote the centres of the Sun, Earth, and 
Moon. Drop M K perpendicular on ES, and on JSf produced 
take KH 2EK. Then, if S denote the mass and r the dis- 
tance of the Sun, the Sun's disturbing force produces at M a 
relative acceleration along M H of magnitude JcS . MH/r*, 
its components being Jc.S. MR ft* along MK and 2k . S . EK/r* 
parallel to EK. 

This force tends to accelerate the Moon towards the Earth 
at quadrature (3f,), and away from the Earth at conjunction 
and opposition (M w 3f 4 ) . At any other position it accelerates 



408 ASTRONOMY. 

the Moon towards a point (.fi^) in the line JZS, and thus 
makes the Moon tend to approach the Sun, if its elongation 
(Af^ZftS) is less than 90; but it accelerates the Moon 
towards a point (7T 3 ) away from the Sun if its angle of 
elongation from the Sun be obtuse. 

473. The Rotation of the Moon's Nodes. Let CL 

represent the ecliptic, JV^J/jJVi' the great circle which the 
Moon would appear to describe on the celestial sphere if there 
were no disturbing force acting upon it, and let 77", between 
JVi and JV/on the ecliptic, represent either the Sun's position 
on the celestial sphere or that of the point antipodal to it. 
Then the reasoning of the last paragraph shows that the dis- 
turbing force acts in the plane HEM V and therefore has a 
component at M l directed along the tangent to the great 
circle 




Now let us suppose that the Moon is revolving under the 
Earth's attraction alone, but that on arriving at J/j it is 
acted on by a sudden impulse or blow directed towards H. 
Clearly the effect of such an impulse is to bend the direction 
of motion inward, from Mffl to J^JV^and the Moon will then 
begin to describe a great circle Jfj.$i', which, if produced both 
ways, will intercept the ecliptic at 'points N v N behind JVj, 
JV/. The inclination of the orbit to the ecliptic will also be 
diminished slightly if M l is within 90 of ^ ; for the exterior 
angle MN^H > MN^ since the sides of the triangle 
J/iJVyV 2 are each less than 90. But when the Moon comes 
to My let another impulse act towards H. This will deflect 
the direction of motion from M^N^ to M^, and the Moon 
will now begin to describe the great circle NJil^N^ whose 
nodes Ny N^ are still further behind their initial positions. 
The inclination of the orbit to the ecliptic will, however, be 
increased this time. 

It is easy to see that the same general effect takes place 
when the Moon is acted on by a continuous force, always 



FURTHER AT-PLICATIOKS OP THE LAW OP GRAVITATION. 409 

tending towards the ecliptic, instead of a series of impulses. 
Such a force continuously deflects the Moon's direction of 
motion, and draws the Moon down so that it returns to 
the ecliptic more quickly than it would otherwise. Hence 
the Moon, after leaving one node, arrives at the next hefore 
is has quite described 180, and the result is an apparent 
retrograde (never direct] motion of the nodes, combined with 
periodic, hut small, fluctuations in the inclination of the 
orbit. 

*474. The retrograde motion of the Moon's nodes is, in some respects, 
analogous to the precession of the equinoxes, and, although the analogy 
is somewhat imperfect, the former phenomenon gives an illustration 
of the way in which the latter is produced. If the Earth had a 
string of satellites, like Saturn's rings, chosely packed together in a 
circle in the plane of the equator, the Sun's disturbing force, ever 
ac derating them towards the ecliptic, would, as in the case of the 
Moon, cause a retrograde motion of the points of intersection of all 
of their paths with the ecliptic, and this would give the appearance 
of a kind of retrograde precession of the plane of the rings. If the 
particles, instead of being separate, were united into a solid ring, the 
general phenomena would be the same. And it is not unnatural to 
expect that what occurs in a simple ring should also occur, to a 
greater or less degree, in the case of other bodies that are somewhat 
flattened out perpendicularly to their axis of rotation, such as the 
Earth, thus accounting for the precession of the equinoxes. (Of 
course this is only an illustration, not a rigorous proof; in fact, if 
the Earth were qiiite spherical it would behave very differently.) 




FIG. 163. 



*475 Perturbations due to Average Value of Eadial Disturbing 
Force -Let d be the Moon's distance. Then, when the Moon is m 
conjunction or opposition, the Sun's disturbing force acts away from 
the Earth, and is of magnitude 2k8d/* (Fig 163) When the 
Moon is in quadrature the disturbing force acts towards the B. 
but is only half as great. Hence, on the average, the disturbing 
force tends to pull the Moon away from the Earth. 

In consequence, the Moon's average centrifugal force must be 
rather less than it would be at the same distance from the Eartl ,f 
there were no disturbing force, and the effect of this is^ to make the 
month a little longer than it would be otherwise for the same c 
tance of the Moon. 



410 ASTRONOMY. 

Moreover, the disturbing force increases as the Moon's distance 
increases, but the Earth's attraction diminishes, being proportional 
to the inverse square of the distance ; this has the effect of making 
the whole average acceleration along the radius vector decrease more 
rapidly as the distance increases than it would according to the law 
of inverse squares. The result of this cause is the progressive 
motion of the apse line. It is difficult to explain this in a simple 
manner, but the following arguments may give some idea of how 
the effect takes place. At apogee the Moon's average acceleration 
is less, and at perigee it is greater than if it followed the law of 
inverse squares and had the same mean value. Hence, when the 
Moon's distance is greatest, as at apogee, the Earth does not pull 
the Moon back so quickly, and it takes longer to come back to its 
least distance, so that it does not reach perigee till it has revolved 
through a little more than 180. Similarly, at perigee the greater 
average acceleration to the Earth does not allow the Moon to fly 
out again quite so quickly, and it does not reach apogee till it has 
described rather more than 180. Hence, in each case, the line of 
apsides moves forward on the whole. 

*476. Variation, Evection, Annual Equation, Parallactic 
Inequality. When the Moon is nearer than the Earth to the Sun 
(M } , Fig. 162), the Moon is more attracted than the Earth, and 
therefore the disturbing force is towards the Sun ( 472). Its 
effect is, therefore, to accelerate the Moon from last quarter to con- 
junction, and to retard it from conjunction to first quarter. When 
the Moon is more distant than the Earth from the Sun (A/ 3 , Fig. 163), 
it is less attracted than the Earth, and therefore the disturbing 
force is away from the Sun. Thus the Moon is accelerated from 
first quarter to full Moon, and retarded from full Moon to last 
quarter.f Hence we see that the Moon's motion in each case 
must be swiftest at conjunction and opposition, and slowest at the 
quadratures. This phenomenon is known as the Variation. 

The force towards the Earth is greatest at the quadratures, and 
least at the conjunction and opposition, since at the former the Sun 
pulls the Moon towards, and at the latter away from the Earth. 
Either cause tends to make the orbit more curved at the quadratures 
and less curved at the syzygies. For, if v is the velocity, R the 
radius of curvature, then v*]R = normal acceleration. Hence R is 
greatest, and the orbit therefore least curved, when v is greatest, 
and the normal acceleration is least. The effect of this cause would 
be to distort the orbit, if it were a circle, into a slightly oval curve, 
which would be most flattened, and therefore narrowest (compare 



f These retardations and accelerations are closely analogous to 
those of the water in an equatorial canal ( 445). 



-PUHTHEB APPLICATIONS OF THE LAW OF GRAVITATION. 411 

arguments of 114, 115), at the points towards and opposite the 
Sun ; most rounded, and therefore broadest, at the points distant 
90 from the Sun. 

Of course the Moon's undisturbed orbit is not really circular, but 
elliptic, and far more elliptic than the oval into which a circular 
orbit would be thus distorted. But a distortion still takes place, and 
gives rise to periodical changes in the eccentricity, depending on the 
position of the apse line, and known as evection. 

The Sun's disturbing force is greatest when the Sun is nearest, 
and least when the Sun is furthest. These fluctuations, between 
perihelion and aphelion, give rise to another perturbation, called the 
whose most noticeable effect consists in the con- 



sequent variations in the length of the month ( 475). 

If, instead of resorting to a first approximation, we employ more 
accurate expressions for the Sun's disturbing force on the Moon, it 
is evident that this force is greater when the Moon is near con- 
junction than at the corresponding position near opposition ; just 
as the disturbing force which produces the tides is really greater 
under the Moon than at the opposite point. Hence the Moon is 
mere disturbed from last quarter through new Moon to first quarter 
than from first quarter through full Moon to last quarter. Hence 
the time of first quarter is slightly accelerated, and that of last 
quarter retarded. This is called the Moon's Parallactic Inequality. 
Its amount is proportional to TcScP/r*, instead of fcSeF/r^like the 
other perturbations). For many reasons this perturbation is of 
considerable use in determinations of the Sun's mass and distance. 

477. Planetary Perturbations. The Sun's mass is so 
great, compared with the masses of the planets, that the 
orbital motion of one planet about the Sun is but slightly 
affected by the attraction of any other planet. The mutual 
attractions of the planets, and their actions on the Sun, give 
rise to small planetary perturbations, which cause each 
planet to diverge slowly from its elliptical orbit, besides 
accelerating or retarding its motion. 

Since the orbital motions of the planets are all usually 
referred to the Sun as their common centre or " origin," nnd 
not to the centre of mass of the solar system, the perturba- 
tions of one planet, due to a second, depend, not on the actual 
acceleration produced by the latter, but on the differences of 
the accelerations which it produces on the former planet and on 
the Sun. 

As in the case of the Moon, the force which produces this 
difference of accelerations is called the disturbing force. 

ASTRON. 2 E 



412 ASTRONOMt. 

*478. Geometrical Construction for the Disturbing Force. The 
approximate expressions, investigated in 472, for _the Sun s dis- 
turbing force on the Moon, are inapplicable to the disturbing force 
of one planet on another, because the distance of the distorting body 
from the Sun is no longer very large, compared with that of the 
disturbed body. We must, therefore, adopt the following con- 
struction (Fig. 164) : 

Let P, Q be two planets, of masses Jf, If ; 8 the Sun. Then the 
planet P produces an acceleration fcM/PQ 2 on Q along QP, and an 
acceleration IcM/PS* on 8 along SP. To find the acceleration of Q, 
relative to 8, due to this cause, take a point T on PQ such that 
PT : P3 = PS 2 : PQ 2 . Then the accelerations of 8, Q, due to P, are 
fcjf . SP//SP 3 and TcM . TP/SP 3 respectively. Hence, by the triangle of 
accelerations, the acceleration of Q, relative to 8, is represented^ in 
magnitude and direction by fclf .TS/SP 3 . Therefore the disturbing 
force per unit mass on Q, due to P, is parallel to T/8, and of magni- 
tude TeM. TS/SP*. 




FIG. 164. 



Similarly, if we take a point 2* on QP such that QT : Q8 
= QS 2 : QP 2 , the disturbing force per unit mass on P, due to Q, is 
parallel to T'8, and is of magnitude JcM' . T'S/SQ 3 . 

The disturbing force on Q, due to P, and that on P, due to Q, are 
not equal and opposite, because they depend on the planets' attrac- 
tions on 8, as well as on their mutual attractions. 

When PQ = P/8, the points Q, T evidently coincide, and the dis- 
turbing force on Q is along the radius vector QS. When PQ < P8, 
PT>PQ, so that the disturbing force on Q tends to pull Q about 8 
(as in Fig. 164) towards P, and when PQ>PS, the disturbing force 
tends to push Q about 8 away from P. 



JTTETHER APPLICATIONS OF THE LAW OP GBAVITATION. 413 



f 7 f ' * h i? distlu * bin S f on P is along PS. 

1*2 PUU P about 5 toward Q. an * when 
push P about fl owa^/ from Q. 

*479. Periodic Perturbations on an Interior Planet.-Let us con- 
sider, m the first place, the perturbations produced by one planet 
E on another planet F, whose orbit is nearer the Sun- as for 
example, the perturbations produced by the Earth on Venus by 
Jupiter or Mars on the Earth, or by Neptune on Uranus. 

Let A, B be the positions of the planet, relative to E, when in 
heliocentri^ conjunction and opposition respectively; U, IT points 
on the relative orbit such that EU = EU' = E8. (These points are 
near but not quite coincident with the positions of greatest elonga- 
tion. ) Then, if we only consider the component relative acceleration 
of F perpendicular to the radius vector Fflf, this vanishes when the 
planet is at U or 17, as shown in the last paragraph. 




FIG. 165. 

The tangential acceleration also vanishes at A and B. Over the 
arc U'AU the relative acceleration is towards E, therefore the planet's 
orbital velocity is accelerated from U' to A j similarly it is retarded 
from A to U. 

Again, at a point F 2 on the arc UBU', the relative acceleration is 
away from the Earth, and this accelerates the planet's orbital 
velocity between U and B, and retards it between B and U'. 

It follows that V is moving most swiftly at A and B, and most 
slowly at U and U'. Hence, if we neglect the eccentricity of the 
orbit, we see that the planet, after passing A, will shoot ahead of the 
position it would occupy if moving uniformly; thus the disturbing 
force displaces the planet forwards during its path from A to near 
U. Somewhere near U, when the planet is moving with its least 
velocity, it begins to lag behind the position it would occupy if 
moving uniformly; thus from near Uto B the disturbing force dis- 
places the planet backwards. Similarly, it may be seen that from B 
to near U' the planet is displaced forwards, and from near U ' to A 
it is displaced backwards. 



414 



ASTRONOMY. 



The principal effect of the component of the disturbing force 
along the radius vector, is to cause rotation of the planet's apsides, 
as in the case of the Moon. The direction of their rotation depends 
on the direction of the force, and is not always direct. The eccen- 
tricity of the orbit is also affected by this cause, as in the phenome- 
non of lunar evection, and the periodic time is slightly changed. 

Owing to the inclination of the planes of the orbits of E, V, the 
attraction of E, in general, gives rise to a small component perpen- 
dicular to the plane of F's orbit, which is always directed towards 
the plane of E's orbit. This component produces rotation of the line 
of nodes, or line of intersection of the planes of the two orbits. This 
rotation is always in the retrograde direction, and is to be explained 
in exactly the same way as the rotation of the Moon's nodes. 

It is thus a remarkable fact that since all the bodies in the solar 
system (except the satellites of Uranus and Neptune) rotate in the 
direct direction, all the planes of rotation and revolution, and all 
their lines of intersection (i.e., the lines of nodes, and the lines of 
equinoxes) in the whole solar system, with the above exceptions, 
have a retrograde motion. 




A 

FIG. 166. 

*480. Periodic Perturbations of an Exterior Planet. The accele- 
rations and retardations produced by a planet E on one J, whose 
orbit is more remote from the Sun, during the course of a synodic 
period, may be investigated in a similar manner to the corre- 
sponding perturbations of an interior planet, assuming the orbits to 
be nearly circular. 

If SJ is less than 2SE there are two points if, N on the relative 
orbit at which EM = EN = E8. At these points the disturbing 
torce is purely radial, and it appears, as before, that the planet J is 
lerated from heliocentric conjunction A to M, and from helio- 
Ce ^ r i C T PPOSition B to N ' retarded from N to A, and from M to B. 
c, ' then E8<EA > hence the attraction of E is greater on 

the hun than on J, and the disturbing force therefore always accele- 
rates the planet J towards B. Thus the planet's orbital velocity 
increases from A to B, and decreases from B to A, and it is greatest at 
a and least at A. Therefore from B to A the planet is displaced in 



FURTHER APPLICATIONS OF THE LAW OF GRAVITATION. 415 

advance of its mean position, and from A to B falls behind its mean 
position. 

The effects of the radial and orthogonal components of the dis- 
turbing force in altering the period and causing rotation of the apse 
line, and regression of the nodes, can be investigated in the same 
way for a superior as for an inferior planet. 

*481. Inequalities of Long Period. If the orbits of the planets 
were circular (except for the effects of perturbations), and in the 
same plane, their mutual perturbations would be strictly periodic, 
and would recur once in every synodic period. Owing, however, to 
the inclinations and eccentricities of the orbits, this is not the case. 
The mutual attractions of the planets produce small changes in the 
eccentricities and inclinations, and even in their periodic times, 
which depend on the positions of conjunction and opposition relative 
to the lines of nodes and apses. Neglecting the motion of these 
latter lines, the perturbations would only be strictly periodic if the 
periodic times of two planets were commensurable ; the period of 
recurrence being the least common multiple of the periods of the 
two planets. But when the periodic times of two planets are nearly 
but not quite in the proportion of two small whole numbers, inequali- 
ties of long period are produced, whose effects may, in the course 
of time, become considerable. 

Thus, for example, the periodic times of Jupiter and Saturn are 
very nearly but not quite in the proportion of 2 to 5. If the propor- 
tionality were exact, then 5 revolutions of Jupiter would take the 
same time as 2 revolutions of Saturn 5 and, since Jupiter would thus 
gain three revolutions on Saturn, the interval would contain 3 
synodic periods. Thus, after 3 synodic periods had elapsed from 
conjunction, another conjunction would occur at exactly the same 
place in the two orbits, and the perturbations would be strictly 
periodical. 

But, in reality, the proportionality of periods is not exact; the 
positions of every third conjunction are very slowly revolving in 
the direct direction. They perform a complete revolution in 
2,640 years. But there are three points on the orbits at^which con- 
junctions occur, and these are distant very nearly 120 from one 
another. It follows that when the positions of conjunction have 
revolved through 120, they will again occur at the same points on 
the orbits, and the perturbations will again be of the same kind as 
initially. The time required for this is one-third the above period, 
or 880 years, and consequently Jupiter and Saturn are subject to 
lono-- period inequalities which recur only once in 880 years. 

Again, the periodic times of Venus and the Earth are nearly in 
the proportion of 8 to 13 ; consequently 5 conjunctions of Venus occur 
in almost exactly 8 years, thus giving rise to perturbations havinj 
a period of 8 years. But the proportion is not exact, and, consequently, 
there are other mutual perturbations having a very long period. 



416 ASTBOtfOMY. 

One of the most important secular perturbations is the alternate 
increase and decrease in the eccentricity of the Earth's orbit. This, 
at the present time, is becoming gradually more and more circular, 
but in about 24,000 years the eccentricity will be a minimum, and 
will then once more begin to increase. The effects of this cause on 
the climate of the Earth's two hemispheres have already been 
considered ( 463). 

482. Gravitational Methods of Finding the Sun's 
Distance. The Earth's perturbations on Mars and Venus 
furnish a good method of finding the Sun's distance. For 
the magnitude of these perturbations depends on the ratio of 
the Earth's mass, or rather the sum of the masses of the 
Earth and Moon (since both are instrumental in producing 
the perturbations), to the Sun's mass. Hence, if S, M, m 
denote the masses of the Sun, Earth, and Moon, it is possible, 
from observations of these perturbations, to find the ratio of 
(M+m) : S. 

But, if r, d be the distances of the Sun and Moon from the 
Earth, T and Fthe length of the sidereal lunar month and 
year, we have, by Kepler's corrected Third Law, 

(M+m) T* : (S+M+m) Y z = d 3 : r 8 ; 
whence the ratio of r to d is known. If, now, the Moon's 
distance d be determined by observation in any of the ways 
described in Chapter VIII., or by the gravitational method 
of 423, the Sun's distance r may be immediately found. 

This method was used by Leverrier in 1872. From obser- 
vations of certain perturbations of Venus he found the values 
8-853" and 8-859" for the Sun's parallax, while the rotation 
of the apse line of Mars gave the value 8'866". 

The perturbations of Encke's comet were used in a similar 
way by Von Asten, in 1876, to find the Sun's parallax, the 
value thus obtained being rather greater, viz., 9-009". 

The lunar perturbations also furnish data for determining 
the Sun's distance, the principal of these being the parallactic 
inequality of the Moon ( 476). Several computations of the 
Sun's parallax have thus been made, the results being 8 -6" 
by Laplace in 1804, 8-95" by Leverrier in 1858, 8-838" by 
Newcomb in 1867. See also 437 for the determination of 
the parallax from the apparent monthly displacement of the 
Sun. 



FUETHEB APPLICATIONS OF THE LAW OF GRAVITATION. 417 

483. Determination of Masses. The mass of any 
planet which is not furnished with a satellite can be deter 
mined in terms of the Sun's mass by means of the perturba- 
tions it produces on the orbits of other planets. The 
amount of these perturbations is always proportional to the 
disturbing force, and this again is proportional to the mass 
of the disturbing planet. In this manner the mass of Yenus 
has been found to be about 1/400,000 of the Sun's mass, and 
that of Mercury about 1/5,000,000. 

484. The Discovery of Neptune. The narrative of the 
discovery of Neptune is one of the most striking and remark- 
able in the annals of theoretical astronomy, and forms a fitting 
conclusion to this chapter. 

In 1795, or about 14 years after its discovery, the planet 
Uranus was observed to deviate slightly from its predicted 
position, the_ observed longitude becoming slightly greater 
than that given by theory. The_ discrepancy increased till 
1822, when Uranus appeared to undergo a retardation, and 
to again approach its predicted position. About 1830 the 
observed and computed longitudes of the planet were equal, 
but the retardation still continued, and by 1845 Uranus had 
fallen behind its computed position by nearly 2'. 

As early as 1821, Alexis Bouvard pointed out that these 
discrepancies indicated the existence of a planet exterior to 
Uranus, but the matter remained in abeyance until 1846, 
when the late Mr. (afterwards Prof.) Adams, in Cambridge, 
and M. Leverrier, in Paris, independently and almost simul- 
taneously, undertook the problem of determining the position, 
orfcit, and mass oi_an unknown planet which would give rise 
to the observed perturbations. Adams was undoubtedly the 
first by a few months in performing the computations, but 
the actual search for the planet at the observatory of Cam- 
bridge was delayed from pressure of other work. Meanwhile 
Leverrier sent the results of his calculations to Dr. Galle, 
of Berlin, who, within a few hours of receiving them, turned 
his telescope towards the place predicted for the planet, and 
found it within about 52' of that place. Subsequent exami- 
nation of star charts showed that the planet had been pre- 
viously observed on several occasions, but had always been 
mistaken for a fixed star. 



418 ASTRONOMY. 

It will be seen from 479 that the acceleration of Uranus 
up to 1822, and its subsequent retardation, are ^at once 
accounted for by supposing an exterior planet to be in helio- 
centric conjunction with the Sun about the year 1822. But 
Adams and Leverricr sought for far more accurate details 
concerning the planet. At the same time the data afforded 
by the observed perturbations of Uranus were insufficient to 
determine all the unknown elements of the new planet's 
orbit, and therefore the problem admitted of any number of 
possible solutions. In other words, any number of different 
planets could have produced the observed perturbations. 
To render the problem less indeterminate, however, both 
astronomers assumed that the disturbing body moved nearly 
in the plane of the ecliptic and in a nearly circular orbit, 
that its distance and period were connected by Kepler's 
Third Law, and that its distance from the Sun followed 
Bode's Law. The latter assumption led to considerable 
errors, including an erroneous estimation of the planet's 
period by Kepler's Third Law. For when Neptune was 
observed, its distance was found to be only 30 '04 times the 
Earth's distance, instead of 38-8 times, as it would have been 
according to Bode's Law. Nevertheless, the actual planet 
was subsequently found to fully account for all the observed 
perturbations of Uranus. 

The discovery of Neptune affords most powerful evidence 
of the truth of the Law of Gravitation, and so indeed does 
the theory of perturbations generally. The fact that the 
planetary motions are observed to agree closely with theory, 
that computations of astronomical constants (such as the 
Sun's and Moon's distances), based upon gravitational methods, 
agree so closely with those obtained by other methods, when 
possible errors of observation are taken into account, affords 
an indisputable proof that the resultant acceleration of any 
body in the solar system can always be resolved into com- 
ponents directed to the various other bodies, each__cpmponent 
boingju:oportional directly to the mass and inversely to the 
sijuare of the distance of the corresponding body. Such a 
truth cannot be regarded as a fortuitous coincidence ; it can 
only be explained by supposing every body in the universe 
to attract every other body in accordance with Newton's Law 
of Universal Gravitation. 



FUKTHEK APPLICATIONS OF THE LAW Qlf OKAV1TATION. 



EXAMPLES. XIY. 

1. If the Sun's parallax be 8'80", and the Sun's displacement at 
first quarter of Moon 6'52", calculate the mass of the Moon, the 
Earth's radius being taken as 3,963 miles. 

2. Supposing the Moon's distance to be 60 of the Earth's radii, 
and the Sun's distance to be 400 times that of the Moon, while his 
mass is 25,600,000 times the Moon's mass, compare the effects of 
the Sun and Moon in creating a tide at the equator, in the event of 
a total eclipse occurring at the equinox. 

3. If the Earth and Moon were only half their present distance 
from the Sun, what difference would this make to the tides ? Cal- 
culate roughly what the proportion between the Sun's tide-raisng 
power and the Moon's would then be, assuming the Moon's distance 
from -the Earth remained the same as at present. 

4. Taking the Moon's mass as -^ of the Earth's, and its distance 
as 60 times the Earth's radius, show that the Moon's tide-raising 
force increases the intensity of gravity by 1/17,280,000 when the 
Moon is on the horizon, and that it decreases the intensity of gravity 
by 1/8,640,000 when the Moon is in the zenith. 

5. Compare the heights of the solar tides on the Earth and on 
Mercury, taking the density of Mercury to be twice that of the 
Earth, its diameter "38 of the Earth's diameter, and its solar distance 
38 of the Earth's solar distance. 

6. Explain how the pushing forward of the Moon by the tidal 
wave enlarges the Moon's orbit. 

7. Show that, owing to precession, the right ascension of a star 
at a greater distance than 23| from the pole of the ecliptic will 
undergo all possible changes, but that a star at a less distance than 
23^ will always have a right ascension greater than twelve hours. 

8. Prove that for a short time precession does not alter the decli- 
nations of stars whose right ascensions are 6h., or 18h. 

9. Exhibit in a diagram the position of the pole star (R.A. 
= Ih. 20m., decl. = 88 40 7 ) relative to the poles of the equator and 
ecliptic, and hence show that owing to precession its R.A. is increas- 
ing rapidly, but that its polar distance is decreasing. 

10. Describe the disturbing effects of Neptune on Uranus for a 
short time before and after heliocentric conjunction, pointing out 
when Uranus is displaced in the direct, and when in the retrograde 
direction. 



420 A.STRONOMr. 



EXAMINATION PAPER. XIV. 

1. Show that the Moon's orbit is everywhere concave to the Sun. 

2. Show that the tide-raising force of a heavenly body is nearly 
proportional to its (mass) -f- (distance) 3 . 

3. How is it that we have tides on opposite sides of the Earth at 
once ? 

4. Explain the production of the tides on the equilibrium theory. 

5. Define the terms spring tide, neap tide, priming and lagging , 
establishment of the port, lunar time. 

6. What is meant by the expression " Luni-solar Precession" ? 
Describe the action of the Sun and of the Moon in causing the 
Precession. 

7. Give a general description of Precession. Does precession 
change the position of (a) the equator, (6) the ecliptic among the 
stars ? 

8. Describe nutation. What is the cause of Lunar Nutation? 
What is meant by the equation of the equinoxes ? 

9. Give a brief account of the discovery of Neptune. 

10. Explain how the retrograde motion of the Moon's nodes is 
caused bv the Sun's attraction on the Earth and Moon. 



NOTE I. 
DIAGRAM FOE SOUTH LATITUDES. 

In order to familiarize the student with astronomical 
diagrams drawn under different conditions, we subjoina/ftgure 
showing the principal circles of the celestial splndre of an 
observer in South latitude 45 at about 19h. of JSereal time 
(QWRr = 270+15 = 19h.). The figure Jnows also the 
Sun's daily paths at t^e" solstices ; also the arcs T^R^= QM, 
and MX, which measure the E.A. and N. decl. of the star x. 




R" 



fi 



NADIR 
FIG. 169. 



N.POLE 



NOTE II. 

THE PHOTOCHEONOGRAPH. 

Quite recently photography has been applied to recording 
transits, as an alternative for the methods explained in 
Chap. II., 49, 50. The image of the observed star is 



422 ASTRONOMY. 

jjf ejected on a sensitized plate placed in front of the transit 
circle, and, owing to the diurnal motion, it moves horizontally 
across the plate. The plate is made to oscillate slightly in a 
vertical direction, by means of clockwork, say once in a 
second, and this motion, combined with the horizontal motion 
of the image, causes it to describe a zigzag or wavy streak on 
the plate. The star's position at each second is indicated by 
the undulations, and the position of these is capable of being 
measured with great exactness. 



NOTE III. 
NOTE ON 104. 
It may be proved, by Spherical Trigonometry, that 

sin nP = sin xP sin nxP, or sin I = cos d sin nxP , 
cos 2 d cos 2 nxP = cos 2 d cos 2 d sin 2 nxP = cos 2 d sin 2 1 
= cos (d + 1} cos (d I) ; 



acceleration t = - 



15 

U" 



15 </{ cos (<M-Z) cos (<?/)} 
The same formula is applicable to 135, 190. 



sees. 




APPENDIX. 



PROPERTIES OF THE ELLIPSE. 

For the benefit of'those readers who have not studied Conic 
Sections, we subjoin a list of those properties of the ellipse which 
are of astronomical importance. The proofs are given in books on 
Conic Sections. 



B 



APSE 




FIG. 168. 

1. DEFINITION. A conic section is a curve such that the distance 
of every point on it from a certain fixed point is proportional to its 
perpendicular distance from a certain fixed straight line. 

The fixed point is called the focus, the fixed line is called the 
directrix, and the constant ratio of distances is called the eccen- 
tricity. 

If this constant ratio or eccentricity is less than unity, the curve 
is called an ellipse. In this case the curve assumes the form of a 
closed oval, as shown in the figure. 

If 8 is the focus, and if from A, P, L, P r , A', &c., any points on 
the curve, perpendiculars AX, PM, &c., be drawn on the directrix, 
and if the eccentricity be e, the definition requires that 
= 8 A _ 8P_ = SL_ = SP' = -8 A' = . 

AX ~ PM LK P'M' A'X 
and that e is less than unity. 

The other conic sections, the parabola and hyperbola, are defined 
by the same property, save that in the former e = 1, and in the 
latter e > 1 ; but they are of little astronomical importance, except 
as representing the paths described by non-periodic comets. 



424 



ASTRONOMY. 



2. An ellipse has two foci (each focus having a corresponding 
directrix), and the sum of the distances of any point from the two 
foci is constant. 

Thus in Fig 169, 8, Hare the two foci, and the sum 8P + PH is 
the same for all positions of P on the curve. 

From this property an ellipse may easily be drawn. For, let two 
small pins be fixed at 8 and H, and let a loop of string SPH be passed 
over them and round a pencil-point P ; then, if the pencil be moved so 
as to keep the string tight, its point P will trace out an ellipse. 
For SP + PH +H8 = constant, and /. SP + PR = constant. 

3. For all positions of P on the ellipse, SP is inversely propor- 
tional to 1 -t e cos ASP, so that 

SP(l + ecos^4SP) = I = constant, 

e being the eccentricity and 8A the line through 8 perpendicular 
to the directrix. 

>r 




4. The line joining the two foci is perpendicular to the directrices. 
The portion of this line (AA'), bounded by the curve, is called the 

major axis or axis major. Its middle point C is called the centre, 
and the^ curve is symmetrical about this point. 

The line BCB', drawn through the centre perpendicular to AC A' 
and terminated by the curve, is called the minor axis or axis minor. 
The lengths of the major and minor axes are usually denoted by 
2a and 26 respectively. 

5. The extremities A, A' of the major axis are called the apses 
or apsides. Since, by (2), 8P + HP is constant, therefore, taking 
P at A or A', SP + HP = SA + HA = 8 A' + HA' 

= $(SA + HA + SA' + HA') evidently 
= AA' = 2a. 
Taking P at B, SB + HB = 2a ; 

.'. SB (evidently) = HB = a = CA. 



PBOPERTIES OF THE ELLIPSE. 425 

6. The eccentricity e = 08/CA ; /. CS = e . CA, and 

52 = 02 = SB 2 - CiS 2 (Euc. I. 47) = a*-a 2 e 2 = a 2 (1-e 2 ) j 



Hence also 
S4= (L4-CS = a(l-e) and 8 A' = CA' 

7. The latus rectum is the chord LSL' drawn through the focus 
perpendicular to the major axis AA'. Its length is 21, where 
I = a (Ift-e 2 ). Also I is the constant of (3), for when P coincides 
with L, ASP = 90; .'. cos ASL = 0, and 8L = I. [Fig. 168.] 

8. The tangent fPT and normal PGg, at P, bisect respectively 
the exterior and interior angles (SPI, 8PH) formed by the lines 
JSP, HP. 

9. If the normal meets the major and minor axes in G, g, 

PO : Pg = OB 2 : CA 2 (= 6 2 : a 2 ). 

10. If ST, drawn perpendicular on the tangent at P, meets HP 
produced in I, then evidently SP IP ; 

.'. HI = SP + HP = 2a [by (2)]. 

If HT' is the other focal perpendicular on the tangent, it is known 
that rectangle 8T . HT' = constant = 6 2 . 

11. Relation between the focal radius SP and the focal perpen- 
dicular on the tangent ST. 

Let SP = r, 8T = p. 

Then cos TIP = cos T8P = pfr. 

By Trigonometry, 

= I8* + IH--2. 18. IH. cos SIHi 

4aV = 4j> 2 + 4a 2 - 8pa x p/r ; 

2 (1-e 2 ) 2a 







This may also be proved from the similarity of the triangles 
8PT } HPT', which gives 8T : HT' = SP : HP ; 

/. ST 2 : ST.HT = )S(P : .HP and ST.HF = b 2 (10) j 
.\ p 2 : b 2 =r : 2a-r. 

12. If a circular cone (i.e., either a right or oblique cone on a 
circular base) is cut in two by a plane not intersecting its base, the 
curve of section is an ellipse. More generally, the form of a circle 
represented in perspective, or the oval shadow cast by a spherical 
globe or a circular disc on any plane, are ellipses. A circle is a 
particular form of ellipse for the case where 6 = a and /. e = 0. 

13. The area of the ellipse is -nab. 



426 ASTRONOMY. 



TABLE OF ASTRONOMICAL CONSTANTS. 

(Approximate values, calculated, when variable, for the Spring 
Equinox, A.D. 1900.) 



THE CELESTIAL SPHERE. 

Latitude of London (Greenwich Observatory), 51 28' 31", 

Cambridge Observatory, 52 12' 51". 

Obliquity of Ecliptic, 23 27' 8", 

OPTICAL CONSTANTS. 

Coefficient of Astronomical Eefraction, 57". 

Horizontal Eefraction, 33'. 

Coefficient of Aberration, 20'493". 

Velocity of Light in miles per second, 186,330. 

metres 299,860,000. 

Equation of Light, 8m. 18s 

TIME CONSTANTS. 

Sidereal Day in mean solar units = 1 l/366days = 23h. 56m.4'ls. 
Mean Solar Day in sidereal units = 1 + 1/365| days = 24h. 3m. 56'5s. 
Year, Tropical, in mean time, 365d. 5h. 48m. 45'51s. 

Sidereal, 365d. 6h. 9m. 8'97s. 

Anomalistic, 365d. 6h. 13m. 48'09s. 

Civil, if the number of the year is not divisible by 4, 

or if it be divisible by 100, bnt not by 400, 365 days. 

In other cases, 366 

Month, Sidereal, 27'32166d. = 27d. 7h. 43m. 11 '4s. 

Synodic, 29'53059d. = 29d. 12h. 44m. 30s. 

Metonic Cycle, 235 Synodic Months = 6939'69d 

= 19 tropical years (all but 2 hours). 

Period of Botatiou of Moon's Nodes (Sidereal), 6793'391d. = 18'60yr. 
(Synodic), 346'644d. 

= 346d. 14Jh. 
Apsides (Sidereal), 3232'575d. = 8'85yr, 

(Synodic), 411'74d. 
Saros 223 Synodic Months = 6585'29d. = 18*0906 yr, 

= 18 yr. 10 or 11 days. 

= 19 Synodic periods of Moon's Nodes (very nearly/ 
= 16 Apsides (nearly). 

Equation of Time, Maximum due to Eccentricity, 7m. 

Obliquity, 10m, 



TABLE OF ASTRONOMICAL CONSTANTS. 



427 



Equatorial Circumference, 



THE EARTH. 

Equatorial Eadius, 3963-296 miles. 

Polar 3949-791 

Mean 3959"! 

22,902 
360 x 60 = 21,600 geographical miles. 

4 x 1Q7 = 40,000,000 metres. 
Ellipticity or Compression, l-f-293. 

Eccentricity, '0826. 

Density (Water = 1), 5'58. 

Mass, 6067 x 10 18 tons. 

Mean Acceleration of Gravity in ft. per sec. per sec., 32-18. 
Eatio of C entrif ugal Force to Gravity at Equator, 1 -. 289. 

Eccentricity of its Orbit, l-s-60. 

Annual Progressive Motion of Apse Line, H'25". 

Eetrograde Motion of Equinoxes (Precession), 50-22". 
Period of Precession, 25,695 years. 

Nutation, 18'6 

Greatest change in Obliquity due to Nutation, 9'23". 

Equation of Equinoxes, 15' 37". 

THE SUN. 

Mean Parallax, 8'80". 

Angular Semi-diameter, 16' 1". 

Distance in miles, 92,800,000. 

Diameter in miles, 866,400. 

in Earth's radii, 109. 

Density in terms of Earth's, ^. 

(taking water a& lj, 1 J 4. 

Mass in terms' of Earth's, 324,439. 

Period of Axial Eotation, 25d. 5h. 37m. 

THE MOON. 

Mean Parallax, 57' 2707". 

Angular Semi-diameter, 15' 34". 

Distance in miles, 238,840, 

in Earth's radii, 60'27. 

in terms of Sun's distance, 1/389. 

Diameter in miles, 2,162. 

in terms of Earth's, 3/11. 

Density in terms of Earth's, '61. 

(taking water as 1), 3'4. 

Mass, in terms of Earth's, 1/81. 

Eccentricity of Orbit, 1/18. 

Inclination of Orbit to Ecliptic, 5 8'. 

Ecliptic Limits, Lunar, 12 5' and 9 30'. 

Solar, 18 31' and 15 21'. 

Tide-raising force in terms of Sun's, 7/3. 

ASTRON. 2 F 




ANSWERS. 



NOTE. Where only rough values of the astronomical data are 
given in the questions, the answers can only be regarded as rough 
approximations, not as highly accurate results. It is impossible to 
calculate results correctly to a greater number of significant figures 
than are given in the data employed, and any extra figures so 
calculated will necessarily be incorrect. As the use of working 
examples is to learn astronomy rather than arithmetic, it is ad- 
visable to supply from memory the rough values of such astronomi- 
cal constants as are not given in the questions. These values will 
thus be remembered more easily than if the more accurate values 
were taken from the tables on pages 426, 427, though reference to 
the latter should be made until the student is familiar with them. 



I. EXAMPLES (p. 33). 

1. Only their relative positions are stated; these do not completely 

fix them. 

2. 6 P.M., 6 A.M.; on the meridian. 8. On September 19. 

9. (i.) Early in July ; (ii.) middle of June the Sun passes it about 

June 26. 

10. 304 = 20h. 16m.; at 8h. 13m. P.M. 

11. Near the S. horizon about 10 P.M. early in October. 

12. 38 27', 51 33', 28 5', or if Sun transits N. of zenith 8 27', 

81 33', 58 5'. 

I. EXAMINATION PAPER (p. 34). 

7. 30. 8. 61 58' 37", 15 4' 21". 9. 6h. 43m. 16s. (roughly). 
10. The figure should make Capella slightly W. of N., altitude about 

15; o Lyras a little S.E. of zenith, altitude about 75; 

a Scorpii slightly W. of S., altitude about 12; o Ursse Hajoris 

N.W., altitude about 60. 



ANSWERS. 429 

II. EXAMPLES (p. 61). 

. Direct. 7. Interval = 12 sidereal hours. 9. 2 3 29' 58'5". 

11. 12 39' 9". 12. I7h. 29m. 52'42s. 

II. EXAMINATION PAPER (p. 62). 

6. Positive. 1O. lrn.2'52s., + 0718. 

III. EXAMPLES (p. 84). 

2. 4,267ft. 

3. aN., L-90W. and o S., L + 90W., if L = W. longitude 

given place. 
5. 13m. 6. 39-8 miles. 7. 3960. 

8. 6084ft. 10. 49' 6" per hour. 

MISCELLANEOUS QUESTIONS (p. 85). 

2. N.P.D. = 85, hour angle = 30 W. 

3. Because declination circle has not been defined. 

5. 22h. 40m., 9h. 20m., 14h. Om., 19h. 36m.. 10. 52". 

' V 

III. 1 EXAMINATION PAPER (p. 86). 

1. 24,840 miles, 3,953 miles. 

2. 3-285 ft., 6,084 ft., T69ft. per second. 3. 507 ft. 
5. 3,437,700 fathoms, 6,366,200 metres (roughly), 1,851-851 metres. 

9. See 97, cor. 

IV. EXAMPLES (p. 113). 
5. 45. 7. Star, 6h. 15m. 26'35s. ; Sun, Oh. 13m. 51'90s. 

10. 3481 : 3721, or 29 : 31 nearly. 

IV. EXAMINATION PAPER (p. 114). 

3. See 130, 151. 

3. Oh. 36m. 21'26s. (Note that the clock has a losing rate of 
3m. 22'05s. on sidereal time ; it gives solar time approxi- 
mately.) 

V. EXAMPLES (p. 137). 

1. Retrograde. 3. 3'9m. 6. 347 centuries exactly. 

7. Star's hour angle = 4h. llm. 3s., N.P.D. = 53. 

8. October 28, 15h. 39m. 27'32s. 

1O. 12h. 27m. 13'26s. at Louisville = 18h. 9m. 13'26 at Greenwich. 



430 ASTRONOMY. 

MISCELLANEOUS QUESTIONS (p. 138). 

3. Eastward. 5. Use Figs. 47, 50. 6. See 439. 

7. See 161. 8. llh. 59m. 15'9s. ; - 1m. 7'4s. 

9. 366-25 : 365'25 or 1465 : 1461. 

V. EXAMINATION PAPER (p. 139). 

4. - 10m. ; morning 20m. longer. 5. See 172. 

8. (i.) 7h. 13m. 5s. ; (ii.) 7h. 12m. 48s. 9. June 26. 

10. 1824, 1852, 1880, 1920. 

VI. EXAMPLES (p. 151). 

3. 3,963 miles. 

4. From 50 9' 47" to 49 59' 55" (refraction at altitude 5 - 9' 47" 

by tables). 

5. 44 53' 28". 8. 84 33' ; 377 miles or 327 nautical miles. 

VI. EXAMINATION PAPER (p. 152). 

4. 462". 7. 44 58' 54". 1O. Ih. 12m. 

VII. EXAMPLES (p. 188). 

1. 37 49'. 2. 51 44' 26-09*. 

4. 50 54' 58'6" or 60 43' 23'6" according aa star transits N. or S. 

of zenith. 

5. 44 55', or, if corrected for refraction (cf. Ex. 2, p. 168), 

44 53' 54". 

6. 51 33', 38 27', 61 54'. 8. - 10m. } i.e., 10m. fast. 

9. 12 30'. 1O. Ih.Om. 11. 2 32'. 12. 27'. 
13. See 237. 18. Lat. = cos- l -fr = 87 54' nearly. 

VIII. EXAMPLES (p. 217). 

2. 92,819,000 (see Ex. 2, p. 195). 

3. At 6 p.m. ; about same length as Midsummer Sun, i.e., 16|h. 

4. See 261. 5. 8' 48". 6. Use 266. 

7. lOd. 4h. at noon. 

8. Gibbous, bright limb turned slightly below direction of W. 

Hour angle = 30, decl. = 0. 

10. (i.) No harvest moon ; (ii.) Phenomena practically unaltered. 



ANSWERS, 431 

VIIL EXAMINATION PAI^ER (p. 218). 
4. See 260. 7. 71 33". 9. When we have a solar eclipse. 

IX. EXAMPLES (p. 236). 

1. 23|S. 

2. Favourable if moon passes from N. to S. at ecliptic on March 21. 

4. 4m. 38s. 5. Length = (Earth's radius) -~ sin (8 P). 

7. 6h. 32m. if month unaltered; or, by 329, a lunation = about 

10 days, and then time = 2h. 10m. 

8. 40 Earth's radii = 158,000 miles (roughly). 

9. Total Solar. 1O. 128' (cf. 291). 

IX. EXAMINATION PAPER (p. 237). 

6. 850,000, 230,000, and 5,800 miles (roughly). 

7. See 292, 295-297. 9. No. 

10. In Fig. 93 take M on xm produced, such that sin xM = #m/(p - P). 

X. EXAMPLES (p. 265). 

1. 291'96 days, or, if conjunctions are of the same kind, 583'92 days. 

2. 40. 3. 19 : 6, or nearly 3:1. 4. IQi^h., 120h. 

5. p + P s with notation of 290. 6. 888 million miles, 164 yrs. 
7. 6 months ; Vi or '63 of Earth's mean distance. 8. 398 days. 

9. f of a year = 137 days. 

10. Stationary at heliocentric conjunction only, never retrograde. 

X. EXAMINATION PAPER (p. 266). 

3. i_i_ years = 378 days. 

4. See 323, 324. The alterations in Venus's brightening are 

really not inconsiderable (see Ex. 3, p. 205). 

6. Most rapid approach at quadrature j velocity that with which the 

Earth would describe its orbit in synodic period. 

9. 287 days. 

10. Draw the circular orbits about , radii 4, 7, 10, 16, 52 ( 304). 

The heliocentric longitudes (measured from Q T ) are roughly 
as follows: $153, ? 175, 0220, <? 20, ^211. The 
C should be drawn close to at an elongation C 90 
at first quarter. 



432 

XI. EXAMPLES (p. 311). 

2. 432,000 miles. 3. 2,250 miles. 

6. 9,282,000 and 92,820,000 million miles respectively. 

7. 37'8 billion miles - 378 x 10 n miles. 

8. 5 : TT = 1'6 : 1 roughly. 

10. It will always appear half-way between its actual direction and 

a point on the ecliptic 90 behind Sun. Path is roughly a 
small circle of angular radius 45. 

11. 4 35'. 

13. (i.) On ecliptic 90 from Sun. (ii.) In same or opposite direc- 

tion to Sun. Effects greatest along great circles distant 90 
from these points. 

14. (i.) At either pole of ecliptic, (ii.) In ecliptic. 

16. Jan. 21, 10'25" Eastwards; Feb., 17-75" E. ; Mar., 20'50" E. ; 
April, 17-75" E. ; May, 10'25" E. ; June, 0" ; July, 10'25" 
Westwards; Aug.,l7'75"W.j Sept.,20'50"W.; Oct., 1775" W. ; 
Nov., 10-25" VV. ; Dec., 0". 

18. 973,800 miles. 

MISCELLANEOUS QUESTIONS (p. 313). 

5. 15 E. 6. In the autumn. 

7. 17d. 5h. ; star is on equator, hour angle 60 E. 8. 1 : v7 : 7. 

9. 24h. 50m. 30s. mean units = 24h. 54m. 35s. sidereal units. 

10. At the equinoxes. 11. See 376. 

XII. EXAMPLES (p. 335). 

1. 12 Ve sidereal hours = 16h. 58m. 5s. sidereal time. 

2. Pendulum revolving in direction of hands of watch will have 

less velocity in S. hemisphere. 
7. Increased (i.) 59 54' 51"; (ii.) 60 15' 27". 12. 109 Ibs. 

XII. EXAMINATION PAPER (p. 336). 

3. By observing deviation of a projectile ( 390), or by 387 or 389. 

4. 16 V3 = 27-7157 sidereal hours = Id. 3h. 33m. mean time. 

5. 3*368 cm. per sec. per sec. ; ,$ T . 9. See 3PO. 



ANSWERS. 433 

XIII. EXAMPLES (p. 368). 

1. 2-97 miles per sec. 2. 15 ft., or, if g = 32'2, 15'576ft. 

3. 5h.35m. 4. 5-39 days. 5.2,959,000. 11. 8'98". 

13. The distances from the centre of the Sun are 457,579 miles, 

457,579 -H 278 milesj and 457,579 - 281 miles ; but these results 
can only be considered as approximate. 

14. 32'155 greater, owing to attraction of mountain. 

17. '253 of Earth's density ; T415, taking water = 1. 

18. 894 poundals. 

20. At first a hyperbola under the Earth's attraction. After going 
some distance this attraction would become insensible, and 
the Moon would describe an ellipse about the Sun rather 
more eccentric than the Earth's present orbit. 

XIV. EXAMPLES (p. 419). 



3. 24 : 7, by Ex. 1, 442 Cor., or 16 : 5, using result of last example. 
5. Tide on Mercury is higher in proportion 1 : '2888, or 45 : 13, or 

7 : 2 nearly. 
1O. Direct shortly before, retrograde shortly after. 

XIV. EXAMINATION PAPER (p. 420), 
7. (a) Yes; (6) No. 







UNIVERSITY 



INDEX. 



(The numbers refer to the pages throughout.) 



Aberration of Light, 295; cor- 
rection for aberration deter- 
mined, 298 ; its general effect 
on the celestial sphere, 299; 
jomparison with annual paral- 
lax, 300 ; to show that the 
aberration curve of a star is an 
ellipse, 301 ; its discovery by 
Bradley, 302 ; the constant de- 
termined by observation, 302 ; 
relation between the coefficient 
of aberration and the equation 
of light, 304 ; relation between 
the coefficient of aberration 
and the Sun's parallax, 310. 

diurnal, 308 ; its effect on 

meridian observations, 309 ; 
determination of its coefficient 
by observations of the azimuths 
of stars on the horizon, 310. 

planetary, 306. 

Altazimuth, 54. 

Altitude, 8. 

Angular diameter, 8. 

distance, 3. 

measure, its conversion to 

time, 14. 

velocities of planets, to com- 
pare, 342. 

Annual equation, 411. 

Anomalistic year, 127. 

Aphelion, 111. 

Apogee, 106, 210. 

Apparent area, 105, 109; 
Moon's phase, 204. 

midnight, 24. 

motion of a planet, 258. 

noon, 24. 

solar day, 24. 

solar time, its disadvantages 

115. 



Apparent Sun, 117. 
.pse, 106. 

Moon's, 210, 410. 
line, 106, 111, 210; deter- 
mination of its position, 109; 
its progressive motion, 109, 211, 
414. 

Arctic and Antarctic circles, 88. 

Areal velocity, 343 ; relation be- 
tween areal velocity and actual 
(linear) velocity, 344. 

Aries, first point of, 7j to find, 
99, 100; retrograde motion of 
(see Precession}. 

ARISTARCHUS : his method of 
finding the Sun's distance, 205. 

Asteroids, 240. 

Astronomical clock, 13, 36. 

diagrams : their practical 
application, 28. 

telescope, 36. 

terms, table of, 12. 
Astronomy defined, 1 ; its prac- 
tical uses, 153. 

Descriptive, Gravitational, 

and Physical, defined, 1. 
Autumnal equinox, 21. 
Azimuth, 8. 

Bar, double, 78. 

Base line, measurement of, 78. ^ 

BESSEL : his method of determin- 
ing the annual parallax of a star, 
290 ; his day numbers, 403. 

Binary stars, 292. 

Black drop, 279. 

BODE'S Law, 239. 

BRADLEY : his discovery of aber- 
ration, 302 ; his discovery of 
nutation, 403; his determina- 
tion of refraction, 146. 



INDEX. 



Calendar, Julian, 128; Gregorian 
correction, 128. 

month, 200. 

ardinal points, 7. 

CASSINI: his formula of refrac 

tion, 145. 
CAVENDISH: his experiment for 

finding the Earth's mass, 362. 
Celestial equator, 6. 

horizon, 5. 

latitude, 10. 

longitude, 10. 

meridian, 6. 

poles, G. 

sphere, 2. 

entre of mass, 355. 
'Centrifugal force, 324; its effects 
on the acceleration of falling 
bodies, 325 ; loss of weight of 
a body due to it, 326 
'Ceres, 241. 

Chronograph, 43 ; photo-, 421 
Chronometer, 160; its error and 

rate, 161. 

Circle, of position, 187 ; transit, 38. 
Circumpolar stars, 16; determi 

nation of latitude by, 167. 
Civil Year, 128. 
Clock, astronomical, 13, 36. 

error and rate, 44, 45. 

stars, 45, 398. 

'Colatitude, 11. 
Collimating Eyepiece, 49. 
Collimation, error, 46. 

line of, 39. 

Colures, 23. 
Compass, points of, 9. 
Conjunctions, 200, 245. 
Coordinates : their use explained, 
8 ; advantages of the different 
systems, 11 ; table of, 12 
transformation of, 16. 
Culmination, 16. 



485 



Day and night, relative lengths, 
89 92. 

lunar, 382. 
mean, 117. 

numbers, Bessel's, 403. 
perpetual, 92. 
sidereal, 13. 

Declination, 9, 10; name of, 9- 
expressed in terms of latitude 
and meridian Z. D., 15 ; deter- 
mination of the Sun's, 23- 
method of observing, 51. 
Declination Circle, 9, 56. 
DELISLE : his method of deter- 
mining the Sun's parallax, 271 
Density of a heavenly body its 

determination, 366. 
Dip of horizon; defined, 73- its 
determination, 74, 75; its effect 
on tho times of rising and 
setting, 76, 422. 
Direct motion, 22. 
Disappearance of a ship at sea 75 
Diurnal motion of the stars 5. 

aberration, 308 

Double bar, 78. 



Day, apparent solar, 24; explana- 
tion of gam or loss of a clay in 
going round the world, 72. 



Earth : early observations of its 
form, 63; general effects of 
change of position on it 64- 
its rotation, 64; measurement 
of its radius, 67; A. R. Wai- 
Jace's method of finding its 
radius, 77; ordinary methods 
of finding its radius, 78; its 
exact form, 81 ; determination 
of its equatorial and polar 
n,^; its exact dimensions, 
; its mean radius, 83; its 
ellipticity or compression, 83 
its eccentricity, 88 , its zones) 
88 ; determination of the eccen- 
tricity of its orbit, 107- its 
phases, 206; its place in the 
solar system, 240; its rotation, 
315; arguments in favour of 
its rotation, 316; dynamical 
proofs of its rotation, 317 



436 



INDEX. 



Earth (continued) : general 
effects of its centrifugal force, 
324; its figure, 328; evidence 
in favour of its annual motion 
round the Sun, 337; verification 
of the law of gravitation, 356 ; 
its so-called "weight," 362 ; the 
Cavendish experiment, 362 ; 
the mountain method of finding 
its mass, 366 ; its mass deter- 
mined by the common balance, 
364; its mass determined by 
observations of the attraction 
of tides in estuaries, 365 ; the 
pendulum method of finding its 
mass, 365 ; its displacement 
due to the Moon, 371 ; its rota- 
tion retarded by tidal friction, 
388 ; precession of its axis, 396 ; 
nutation of its axis, 400. 

Earth's way, 299. 

Eclipses, 219 et seqq. ; different 
kinds of lunar E., 220 ; effects 
of refraction on lunar E., 150, 
221 ; different kinds of solar E., 
222 ; determination of greatest 
or least number possible in a 
year, 229 ; of Jupiter's satellites, 
241 ; their retardation, 293. 

Ecliptic, 7, 20, 99, 111 ; its obli- 
quity, 11 ; determination of its 
obliquity, 26, 104. 

Ecliptic limits, 226, 228. 

Ellipse, properties of, 423. 

Elongation, 200, 244 ; changes of 
E. of planet, 244, 246. 

Equation, Annual, 411. 

of equinoxes, 402. 

of light, 293 ; its relation to 

the coefficient of aberration, 
304. 

of time, 117 ; due to un- 
equal motion, 118 ; due to ob- 
liquity, 119; its graphical 
representation, 121 ; it vanishes 

1 four times a year, 122 ; its 
maximum values. 123 ; its de- 
termination, ]24. 



Equation, personal, 46. 
Equator, celestial, 6. 

terrestrial, 64. 
Equatorial, 56 ; its use, 57. 
Equinoctial colure, 23. 

points, 7, 20, 23. 

time, 134. 

Equinoxes, 20, 21, 23 ; precession 

of, 103. 
Evection, 411. 

Fathom, 67. 

First point of Aries, 7, 20; its 
determination, 100. 

First point of Libra, 7, 20. 

FLAMSTEED : his method of deter- 
mining the first point of Aries, 
100; advantages of the method, 
102. 

FOUCAULT : his pendulum experi- 
ment, 318 ; his gyroscope, 321 ; 
his determination of the velo- 
city of light, 293. 

Full Moon, 203. 

Geocentric latitude, 83, 112. 

longitude, 112. 

lunar distances, 180. 

parallax : its general effects > 

192; correction for, 192. 
Geodesy, 77. 
Geographical latitude, 83. 

mile, 67. 

Gibbosity of Mars, 252. 
Gibbous Moon, 203. 
Globes : their use, 3. 
Gnomon, 25, 125. 
Golden Number, 215. 
Gravitation : Newton's law of> 

352 ; remarks, 353 ; verification 

for the Earth and Moon, 356. 
Gravity: to compare its intensity 

at different places, 329, 334; 

to find its value, 334. 
GREGORY, Pope : his correction 

of the Julian Calendar, 128. 
Gyroscope or Gyrostat, 321, 395. 



INDEX. 



43? 



HALLET : his method of deter- 
mining the Sun's parallax by ob- 
serving a transit of Venus, 271. 

Harvest Moon, 216. 

Heliocentric latitude, 112. 
longitude, 112. 

Heliometer, 59. 

Horizon, celestial, 5; artificial,159, 

visible, 5, 73-76. 

- dip of, 73. 

Horizontal parallax, 191. 

point, 50. 

Hour angle, 9 ; expressed in time, 
13; its connection with right 
ascension, 15. 

circle, 56. 

Instruments for meridian obser- 
vations, 35 ; for ex-meridian 
observations, 54; for geodesy, 
78-80 ; for navigation, 153. 

Introductory Chapter on Spheri- 
cal Geometry, i.-vi. 

JULIUS C^SAR : his calendar, 128. 

Juno, 241,269. 

Jupiter, 241 ; its satellites, 241. 

KATER'S reversible pendulum, 329. 

KEPLER : his laws of planetary 
motion, 106, 111, 253 ; verifica- 
tion of his first law, 107, 254 ; 
verification of the second law, 
108, 254 ; deductions from the 
second law, 109; verification 
of the third law, 256 ; Newton's 
deductions from his laws, 339, 
345, 346, 348 ; his third law for 
circular orbits, 340 ; correction 
of the third law, 354. 

Knot, 68. 

Known star, 15, 45. 

Lagging of the tides, 383-5. 

Latitude of a place defined, 10; 
phenomena depending on 
change of latitude, 65 ; change 
due to ship's motion, 72. 



Latitude (continued) : determi- 
nation by meridian observa- 
tions, 162 ; determination by 
ex-meridian observations, 169. 

celestial, 10. 

geocentric, 83, 112. 

geographical, 83. 

heliocentric, 112. 

parallel of, 71 ; length of 

any arc of a given parallel, 71- 

Leap year, 128. 

Libra, first point of, 7. 

Light, refraction of, 140 ; its velo- 
city, 293; aberration of, 295 
to find the time taken by the 
light from a star to the Earth,. 
305. 

Light-year, 305. 

Local time: its determination,. 
171. 

Log-line : its use in navigation,. 
68. 

Longitude, celestial, 10. 

geocentric, 112. 

heliocentric, 112. 

terrestrial, 69 ; phenomena 

depending on change of terres- 
trial longitude, 70 ; change due- 
to ship's motion, 72; its deter- 
mination at sea, 177; the 
method of lunar distances, 179;. 
clearing the distance, 179 ; its 
determination by celestial 
signals, 181 ; its determination, 
on land, 182 ; its determination 
by transmission of chronome- 
ters, 182; by chronograph, 184^ 
by terrestrial signals, 185; by 
Moon-culminating stars, 186;. 
bv Captain Sumner's method,. 
187, 

Loop of retrogression, 261. 

Lunar distances, determination of 
longitude by, 179. 

geocentric, 180. 

mountains: determination of 

their height, 207. 

Lunation. 27. 



488 



INDEX. 



Mars, 240 ; Kepler's observations 
on Mars, 254 ; its parallax used 
to determine that of the Sun, 
268. 

Mass, astronomical unit of, 352. 

Mean noon, 117. 

solar day, 117. 

solar time, 117 ; its deter- 
mination at a given instant of 
sidereal time, 132. 

Sun, 116, 117. 

time, 116. 

Mercury, 239 ; its period of rota- 
tion, 264 ; frequency of its 
transits, 282; its mass, 360, 417. 

Meridian, celestial, 6. 

line : its determination, 175. 

prime, 69. 

terrestrial, 64. 
it Meteors : their motion, 4. 

Metonic cycle, 215. 

Metre, 67. 

Micrometers, 58. 

Midnight, apparent, 24. 

Mile, geographical, 67. 

nautical, 67. 

Moon : its motion, 27 ; its age, 27 ; 
itsposition denned by its centre, 
53; illusory variations in its 
size, 149 ; method of taking its 
altitude by the sextant, 158; 
determination of its parallax, 
196 ; its distance, 197 ; its dia- 
meter determined, 199; its 
elongation, 200 ; determination 
of its synodic period, 201 ; its 
phases, 202; relation between 
phase and elongation, 204 ; its 
use in finding the Sun's dis- 
tance, 205; its appearance 
i-elative to the horizon, 206; 
determination of the height of 
lunar mountains, 207 ; its orbit 
about the Earth, 209; eccen- 
tricity of its orbit, 210; its 
nodes, 210; its perturbations, 
210, 407 ; retrograde motion 
of its nodes, 211, 408, 409. 



Moon (continued) : progressive 
motion of its apse line, 211, 410 ; 
its rotation, 212 ; its librations, 
213; general effects of libra- 
tion, 214 ; its eclipses, 219-221 ; 
determination of its geocentric 
distance consistent with an 
eclipse, 224; its greatest lati- 
tude at syzygy consistent with 
an eclipse, 226; synodic revo- 
lution of its nodes, 228; its 
occupations, 232 ; verification 
of the law of gravitation, 356; 
effect of its attraction, 357 ; its 
mass, 357 ; concavity of its path 
about the Sun, 374 ; its disturb- 
ing or tide-generating force, 
375, 377 ; its orbital motion 
accelerated by tidal friction, 
388 ; its form and rotation, 
391 ; its disturbing couple on 
the Earth, 392 ; the rotation of 
its nodes, 408; its other in- 
equalities, 410, 411. 

Nadir, 5. 

point, determination of, 49. 

Nautical mile, 67. 

Neptune, 243 ; its discovery, 417. 

New Moon, 27. 

NEWTON, Sir ISAAC : his deduc- 
tions from Kepler's laws, 339, 
345, 346, 348 ; his law of uni- 
versal gravitation, 352. 

Nodes, 27, 210 ; their retrograde 
motion, 211. 

North polar distance of a circuin- 
polar star, 17. 

Number of eclipses in year, 229. 

Nutation, lunar, 401 ; its general 
effects, 402 ; its discovery, 403 ; 
to correct for, 403 ; its physical 
causes, 404. 

monthly, 406, 

solar, 405. 

Obliquity of ecliptic, 11 j its de 
termination, 26. 



INDEX. 



439 



Observatory, 35. 
Occultafcions, 232. 
Offing, 73. 
Opposition, 200. 



Parallactic inequality, 411. 

Parallax, 179, 191; geocentric 
parallax, 191; horizontal paral- 
lax, 191 ; general effects of 
and correction for geocentric 
parallax, 192 ; relation between 
horizontal parallax and dis- 
tance of celestial body, 194; 
compared with refraction, 195; 
parallax of Moon determined, 
196 ; parallax of planet deter- 
mined, 198 ; relation between 
parallax and angular diameter, 
199 ; determination of the 
Sun's parallax, 268 et seqq.; 
annual parallax denned, 283; 
to find the correction for 
annual parallax, 284; relation 
between the parallax and dis- 
tance of a star, 285 ; its general 
effects on the position of a star, 
286; determination of the an- 
nual parallax of a star, 290. 

Pendulum, Foucault's, 318 ; Cap- 
tain Kater's reversible, 329 j 
oscillations of a simple pen- 
dulum, 330; to find the change 
in the time of oscillation due 
to a variation in its length or 
in the intensity of gravity, 
330; to compare the times of 
oscillation of two pendulums of 
nearly equal periods, 333 ; pen- 
dulum method of finding the 
Earth's mass, 365. 

Perigee, 106, 210. 

Perihelion, 111. 

Perpetual day : determination of 
its length, 97. 

Personal equation, 46. 

Phases of Moon, 202 ; of planet, 
251, 252. 



Perturbations, lunar, 210, 407; 
rotation of nodes, 408 ; due to 
average value of radial disturb- 
ing force, 409; variation, evec- 
tion, annual equation and 
parallactic inequality, 410, 411. 

planetary, 411 ; periodical, 

413, 414 ; inequalities of long 
period, 415 ; secular, 416. 

Photography, stellar, 60, 421. 

Planet : its position defined by 
centre, 53; determination of 
its parallax, 198 ; its occulta- 
tion, 235 ; definition, 238 ; in- 
ferior and superior planets, 
244 ; changes in elongation of 
a inferior planet, 244 ; to find 
the ratio of the distance from 
the Sun of an inferior planet 
to that of the Earth, 246; 
changes in elongation of a 
superior planet, 247; to com- 
pare the distance from the Sun 
of a superior planet with that 
of the Earth, 248 ; determina- 
tion of the synodic period of an 
inferior planet, 249; relation 
between the synodic and side- 
real periods of a planet, 250 ; 
phases of the planets, 251, 252 ; 
motions relative to stars, 258 ; 
transits of inferior planets, 
271; its aberration, 306, 307; 
to compare the velocities and 
angular velocities of two planets 
moving in circular orbits, 342 ; 
having given the velocity of a 
planet at any point of its orbit, 

^ to construct the ellipse de- 
scribed under the Sun's attrac- 
tion, 350 ; to find the mass of a 
planet which has one or more 
satellites, 359; its perturba- 
tions, 411 j masses determined, 
417. 

Points of the compass, 9* 

Polar distance, 9- 

point : its determination, 51. 



440 



INDEX. 



Pole, celestial, 6. 

terrestrial, 64. 

Port, establishment of the, 386. 

Precession of the equinoxes, 103, 
392. 

Earth's axis, 396. 

a spinning-top, 395. 

luni-solar, 393 ; to apply the 

corrections for, 397 ; various 
effects of, 398; its effects on 
the climate of the Earth's 
hemispheres, 400. 

Prime vertical, 7. 

Prime vertical instrument : deter- 
mination of latitude by its use, 
170. 

Priming of the tides, 383-5. 

-Quadrature, 200. 

Hadiant, 4. 

Eeading microscope, 40. 

Refraction, 140; laws of R., 
140; relative index of E., 140; 
general description of atmo- 
spherical R., 141 ; its effect 
on the apparent altitude of a 
star, 141 ; law of successive 
R., 142; formula for astro- 
nomical R., 142 ; Cassini's for- 
mula, 145; coefficient found 
by meridian observations, 146 ; 
other methods of determination, 
147 ; its effects on rising and 
setting, 148; effects on dip and 
distance of horizon, 149 ; effects 
on lunar eclipses and occulta- 
tions, 150, 221 ; comparison of 
R. with parallax, 195. 

Retrograde motion, 22, 258. 

Right ascension, 10 ; expressed in 
time, 14 ; connection with hour 
angle, 15. 

ROEMER : his method of finding 
the velocity of light, 293. 

dotation of "Earth, 64, 315 ; of 
Moon, 212; of Moon'snodes,211, 
408 ; of Sun and planets, 264. 



Saros of the Chaldeans, 231. 

Satellite, defined, 238 ; their obe- 
dience to Kepler's laws, 257. 

Saturn, 242 ; phases of its rings, 
252. 

Seasons, 94 ; effect of the length 
of day on temperature, 94 ; 
other causes affecting tempera- 
ture, 94; unequal length of, 109. 

Secondary, iii., 238. 

Sextant, 154 ; its errors, 157 ; 
determination of theindex error, 
157 ; method of taking altitudes 
at sea, 158 ; method of taking 
altitudes of Sun or Moon, 158. 

Sidereal day, 13. 

month, 200 ; its relation to 

the synodic month, 200. 

noon, 13. 

period, 200, 250. 

time, 13, 25 ; its disadvan- 
tages, 115 ; its determination 
at a given instant of mean solar 
time, 131 ; its determination at 
Greenwich or in any longitude, 
133. 

year, 127. 



Solar day, apparent, 24. 

system, tabular view of, 243 

its centre of mass, 361. 

time, 24 ; its disadvantages, 

115. 

Solstices, 21, 23. 

Solstitial colure, 23. 

points, 23. 

Southing of stars, 16. 

Spectrum analysis, 60.' 

Stars : independence of their di- 
rections relative to observer's 
position on the Earth, 4 ; their 
diurnal motion, 5, 13; culmi- 
nation, 16 ; southing, 16 ; cir- 
cumpolar stars, 16 ; rising and 
setting, 18 ; time of transit, 19 ; 
to show that a star appears to 
describe an ellipse, owing to 
parallax, 287 ; owing to aber- 
ration, 301. 



INDEX. 



441 



Stars, morning and evening, 25. 

Stationary points, 258 ; their de- 
termination, 262, 263. 

Sub-solar point, 187. 

Summer solstice, 21. 

and winter, causes of, 94. 

SUMNER, Captain : his method of 
finding longitude, 187. 

Sundial, 125 ; geometrical method 
of graduation, 126. 

Sun : its annual motion, 7 ; its 
annual motion in the ecliptic, 
20 ; its motion in longitude, 
right ascension and declination, 
20, 21 ; its variable motion in 
right ascension, 22 ; determi- 
nation of its right ascension and 
declination, 23, 24 ; its position 
defined by its centre, 53 ; its 
diurnal path at different sea- 
sons and places, 88 ; to find 
length of time of sunrise or 
sunset, 98 ; observations of its 
relative orbit, 105 ; its apparent 
area, 105, 109 ; its apparent 
annual motion accounted for, 
110; illusory variations in size, 
149 ; method of finding its alti- 
tude by the sextant, 158 ; diffi- 
culty of finding its parallax, 197; 
its distance determined by 
Aristarchus, 205 ; solar eclipses, 
219, 222, 234; description, 238 ; 
its period of rotation, 264 ; de- 
termination of its distance from 
the Earth, 268 et seqq. ; its paral- 
lax determined by observation 
of the parallax of Mars, 268; 
parallax by observations on the 
asteroids and Venus, 269; paral- 
lax determined by observations 
-of the transit of Venus, 271 et 
$eqq.; advantages and disadvan- 
tages of Halley's and Delisle's 
methods, 280 ; relation between 
coefficient of aberration, Sun's 
parallax, and velocity of light, 
306. 



Sun (continued) : to find the ratio 
of its mass to the Earth's, 358 : 
gravity on its surface, 366 ; its 
parallax determined by observa- 
tions of lunar and solar displace- 
ments of the Earth, 373; its 
disturbing or tide-generating 
force, 375, 377 ; its mass com- 
pared with that of the Moon, 
from observations of the rela- 
tive heights of the solar and 
lunar tides, 388 ; its disturbing 
couple on the Earth, 392; gravi- 
tational methods of finding its 
distance, 416. 

Synodic month, 200. 
period, 200, 250. 



Syzygy, 200. 

Telescope, astronomical, 37. 

Terrestrial equator, 64. 

longitude, 69. 

meridian, 64. 

pole, 64. 

Theodolite, 79. 

Tidal constants, 387. 

friction, 388 ; application to 

the solar system, 392. 

Tides, 375 ; equilibrium theory of 
their formation, 379; canal 
theory, 380; semi - diurnal, 
diurnal, and fortnightly tides 
due to the Moon, 383; semi-diur- 
nal, diurnal, and six-monthly 
tides due to the Sun, 383; spring 
and neap tides, 383_j their 
priming and lagging, 383- 
385 ; establishments of ports 
386. 

Time: its reduction to circular 
measure, 14 ; relation between 
the different units, 129, 134. 
equinoctial, 134. 

local : its determination by 

method of equal altitudes, 171, 
172. 

lunar, 382. 



Trade winds, 323. 



442 



LtfDEX. 



Transit, 14 ; eye and ear method 
of taking transits, 42; of Venus, 
271-282"; of Mercury, 282. 

circle, 38 ; corrections re- 
quired for right ascension, 44 ; 
corrections required for decli- 
nation, 49. 

Triangulation, 79. 

Tropics, 88. 

Tropical year, 127. 

True Sun, 117. 

Uranus, 242. 

Variation, 410. 

Venus, 240; its period of rota- 
tion, 264; observations of its 
transit used to determine the 
Sun's parallax, 271 ; determi- 
nation of the frequency of its 
transits, 281; its mass, 360, 
417. 

Velocity, angular, 342. 

area!, 343. 

of light, 293. 

Velocities of planets compared, 
342. 



Vernier, 157. 
Vernal equinox, 20, 
Vertical, 7. 

circle, 7. 

prime, 7. 

Vesta, 2-40. 

WALLACE, ALFRED RUSSELL : hi 
method of finding the Earth's- 
radius, 77. 

Waning and waxing Moons, 203. 

Winter solstice, 21. 

Year, 20. 

anomalistic, 127. 

civil, 128. 

' leap, 128. 

sidereal, 127. 

synodic, 128. 

tropical, 127. 

Zenith, 5. 

distance, 8. 

point, 51. 

sector, 8& 



Zodiac, 25. 



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