Chapter XVIII: Part 18

The theorems of motion just cited are expressed by seven integrals, or
equations expressing a law that certain functions of the variables and
of the time remain constant. It is remarkable that although the seven
integrals were found almost from the beginning of the investigation,
no others have since been added; and indeed it has recently been shown
that no others exist that can be expressed in an algebraic form. In
the case of three bodies these do not suffice completely to define the
motion. In this case, the problem can be attacked only by methods of
approximation, devised so as to meet the special conditions of each
case. The special conditions which obtain in the solar system are such
as to make the necessary approximation theoretically possible however
complex the process may be. These conditions are:--(1) The smallness
of the masses of the planets in comparison with that of the sun, in
consequence of which the orbit of each planet deviates but slightly
from an ellipse during any one revolution; (2) the fact that the
orbits of the planets are nearly circular, and the planes of their
orbits but slightly inclined to each other. The result of these
conditions is that all the quantities required admit of development in
series proceeding according to the powers of the eccentricities and
inclinations of the orbits, and the ratio of the masses of the several
planets to the mass of the sun.

_Perturbations of the Planets._--Kepler's laws do not completely
express the motion of a planet around a central body, except when no
force but the mutual attraction of the two bodies comes into play.
When one or more other bodies form a part of the system, their action
produces deviations from the elliptic motion, which are called
_perturbations_. The problem of determining the perturbations of the
heavenly bodies is perhaps the most complicated with which the
mathematical astronomer has to grapple; and the forms under which it
has to be studied are so numerous that they cannot be easily arranged
under any one head. But there is one conception of perturbations of
such generality and elegance that it forms the common base of all
those methods of determining these deviations which have high
scientific interest. This conception is embodied in the method of
"variation of elements," originally due to J.L. Lagrange. The simplest
method of presenting it starts with the second view of the elliptic
motion already set forth.

We have shown that, when the position of a planet and the direction
and speed of its motion at a certain instant are given, the elements
of the orbit can be determined. We have supposed this to be done at a
certain point P of the orbit, the direction and speed being expressed
by the variables x, y, x' and y'. Now, consider the values of these
same variables expressing the position of the planet at a second point
Q, and the speed with which it passes that point. With this position
and speed the elements of the orbit can again be determined. Since the
orbit is unchanged so long as no disturbing force acts, it follows
that the elements determined by means of the two sets of values of the
variables are in this case the same. In a word, although the position
and speed of the planet and the direction of its motion are constantly
changing, the values of the elements determined from these variables
remain constant. This fact is fully expressed by the equations (4)
where we have constants on one side of the equation equal to functions
of the variables on the other. Functions of the variables possessing
this property of remaining constant are termed _integrals_.

Now let the planet be subjected to any force additional to that of the
sun's attraction,--say to the attraction of another planet. To fix the
ideas let us suppose that the additional attraction is only an impulse
received at the moment of passing the point P. The first effect will
evidently be to change either the velocity or the direction in which
the planet is moving at the moment, or both. If, with the changed
velocity we again compute the elements they will be different from the
former elements. But, if the impulse is not repeated, these new
elements will again remain invariable. If repeated, the second impulse
will again change the elements, and so on indefinitely. It follows
that, if we go on computing the elements a, b, c, d from the actual
values of x, y, x' and y', at each moment when the planet is subject
to the attraction of another body, they will no longer be invariable,
but will slowly vary from day to day and year to year. These ever
varying elements represent an ever varying elliptic orbit,--not an
orbit which the planet actually describes through its whole course,
but an ideal one in which it is moving at each instant, and which
continually adjusts itself to the actual motion of the planet at the
instant. This is called the _osculating_ orbit.

The essential principle of Lagrange's elegant method consists in
determining the variations of this osculating ellipse, the
co-ordinates and velocities of the planet being ignored in the
determination. This may be done because, since the elements and
co-ordinates completely determine each other, we may concentrate our
attention on either, ignoring the other. The reason for taking the
elements as the variables is that they vary very slowly, a property
which facilitates their determination, since the variations may be
treated as small quantities, of which the squares and products may be
neglected in a first solution. In a second solution the squares and
products may be taken account of, and so on as far as necessary.

If the problem is viewed from a synthetic point of view, the stages of
its solution are as follows. We first conceive of the planets as
moving in invariable elliptic orbits, and thus obtain approximate
expressions for their positions at any moment. With these expressions
we express their mutual action, or their pull upon each other at any
and every moment. This pull determines the variations of the ideal
elements. Knowing these variations it becomes possible to represent by
integration the value of the elements as algebraic expressions
containing the time, and the elements with which we started. But the
variations thus determined will not be rigorously exact, because the
pull from which they arise has been determined on the supposition that
the planets are moving in unvarying orbits, whereas the actual pull
depends on the actual position of the planets. Another approximation
is, therefore, to be made, when necessary, by correcting the
expression of the pull through taking account of the variations of the
elements already determined, which will give a yet nearer
approximation to the truth. In theory these successive approximations
may be carried as far as we please, but in practice the labour of
executing each approximation is so great that we are obliged to stop
when the solution is so near the truth that the outstanding error is
less than that of the best observations. Even this degree of precision
may be impracticable in the more complex cases.

The results which are required to compare with observations are not
merely the elements, but the co-ordinates. When the varying elements
are known these are computed by the equations (2) because, from the
nature of the algebraic relations, the slowly varying elements are
continuously determined by the equations (4), which express the same
relations between the elements and the variables as do the equations
(2) and (3). This method is, therefore, in form at least, completely
rigorous. There are some cases in which it may be applied unchanged.
But commonly it proves to be extremely long and cumbrous, and
modifications have to be resorted to. Of these modifications the most
valuable is one conceived by P.A. Hansen. A certain mean elliptic
orbit, as near as possible to the actual varying orbit of the planet,
is taken. In this orbit a certain fictitious planet is supposed to
move according to the law of elliptic motion. Comparing the longitudes
of the actual and the fictitious planet the former will sometimes be
ahead of the latter and sometimes behind it. But in every case, if at
a certain time t, the actual planet has a certain longitude, it is
certain that at a very short interval dt before or after t, the
fictitious planet will have this same longitude. What Hansen's method
does is to determine a correction dt such that, being applied to the
actual time t, the longitude of the fictitious planet computed for the
time t + dt, will give the longitude of the true planet at the time t.
By a number of ingenious devices Hansen developed methods by which dt
could be determined. The computations are, as a general rule, simpler,
and the algebraic expressions less complex, than when the computations
of the longitude itself are calculated. Although the longitude of the
fictitious planet at the fictitious time is then equal to that of the
true planet at the true time, their radii vectores will not be
strictly equal. Hansen, therefore, shows how the radius vector is
corrected so as to give that of the true planet.

In all that precedes we have considered only two variables as
determining the position of the planet, the latter being supposed to
move in a plane. Although this is true when there are any number of
bodies moving in the same plane, the fact is that the planets move in
slightly different planes. Hence the position of the plane of the
orbit of each planet is continually changing in consequence of their
mutual action. The problem of determining the changes is, however,
simpler than others in perturbations. The method is again that of the
variation of elements. The position and velocity being given in all
three co-ordinates, a certain osculating plane is determined for each
instant in which the planet is moving at that instant. This plane
remains invariable so long as no third body acts; when it does act the
position of the plane changes very slowly, continually rotating round
the radius vector of the planet as an instantaneous axis of rotation.

_Secular and Periodic Variations._--When, following the preceding
method, the variations of the elements are expressed in terms of the
time, they are found to be of two classes, _periodic_ and _secular_.
The first depend on the mean longitudes of the planets, and always
tend back to their original values when the planets return to their
original positions in their orbits. The others are, at least through
long periods of time, continually progressive.

A luminous idea of the nature of these two classes of variation may be
gained by conceiving of the motion of a ship, floating on an ocean
affected by a long ground swell. In consequence of the swell, the ship
is continually pitching in a somewhat irregular way, the oscillations
up and down being sometimes great and sometimes small. An observer on
board of her would notice no motion except this. But, suppose the tide
to be rising. Then, by continued observation, extended over an hour or
more, it will be found that, in the general average, the ship is
gradually rising, so that two different kinds of motion are
superimposed on each other. The effect of the rising tide is in the
nature of a secular variation, while the pitching is periodic.

But the analogy does not end here. If the progressive rise of the ship
be watched for six hours or more, it will be found gradually to cease
and reverse its direction. That is to say, making abstraction of the
pitching, the ship is slowly rising and falling in a total period of
nearly twelve hours, while superimposed upon this slow motion is a
more rapid motion due to the waves. It is thus with the motions of the
planets going through their revolutions. Each orbit continually
changes its form and position, sometimes in one direction and
sometimes in another. But when these changes are averaged through
years and centuries it is found that the average orbit has a secular
variation which, for a number of centuries, may appear as a very slow
progressive change in one direction only. But when this change is more
fully investigated, it is found to be really periodic, so that after
thousands, tens of thousands, or hundreds of thousands of years, its
direction will be reversed and so on continually, like the rising and
falling tide. The orbits thus present themselves to us in the words of
a distinguished writer as "Great clocks of eternity which beat ages as
ours beat seconds."

The periodic variations can be represented algebraically as the
resultant of a series of harmonic motions in the following way: Let L
be an angle which is increasing uniformly with the time, and let n be
its rate of increase. We put L0 for its value at the moment from which
the time is reckoned. The general expression for the angle will then
be

L = nt + L0.

Such an angle continually goes through the round of 360 deg. in a
definite period. For example, if the daily motion is 5 deg., and we
take the day as the unit of time, the round will be completed in 72
days, and the angle will continually go through the value which it had
72 days before. Let us now consider an equation of the form

U = a sin (nt + L0).

The value of U will continually oscillate between the extreme values
+a and -a, going through a series of changes in the same period in
which the angle nt + L0 goes through a revolution. In this case the
variation will be simply periodic.

The value of any element of the planet's motion will generally be
represented by the sum of an infinite series of such periodic
quantities, having different periods. For example

U = a sin (nt + L0) + b sin (mt + L1) + c sin (kt + L2) &c.

In this case the motion of U, while still periodic, is seemingly
irregular, being much like that of a pitching ship, which has no one
unvarying period.

In the problems of celestial mechanics the angles within the
parentheses are represented by sums or differences of multiples of the
mean longitudes of the planets as they move round their orbits. If l
be the mean longitude of the planet whose motion we are considering,
and l' that of the attracting planet affecting it, the periodic
inequalities of the elements as well as of the co-ordinates of the
attracted planet, may be represented by an infinite series of terms
like the following:--

a sin (l' - l) + b sin (2l' - l) + c sin (l' - 2l) + &c.

Here the coefficients of l and l' may separately take all integral
values, though as a general rule the coefficients a, b, c, &c.
diminish rapidly when these coefficients become large, so that only
small values have to be considered.

The most interesting kind of periodic inequalities are those known as
"terms of long period." A general idea both of their nature and of
their cause will be gained by taking as a special case one celebrated
in the history of the subject--the great inequality between Jupiter
and Saturn. We begin by showing what the actual fact is in the case of
these two planets. Let fig. 3 represent the two orbits, the sun being
at C. We know that the period of Jupiter is nearly twelve years, and
that of Saturn a little less than thirty years. It will be seen that
these numbers are nearly in the ratio of 2 to 5. It follows that the
motions of the mean longitudes are nearly in the same proportion
reversed. The annual motion of Jupiter is nearly 30 deg., that of
Saturn a little more than 12 deg. Let us now consider the effect of
this relation upon the configurations and relations of the two
planets. Let the line CJ represent the common direction of the two
planets from the sun when they are in conjunction, and let us follow
the motions until they again come into conjunction. This will occur
along a line CR1, making an angle of nearly 240 deg. with CJ. At this
point Saturn will have moved 240 deg. and Jupiter an entire revolution
+ 240 deg., making 600 deg. These two motions, it will be seen, are in
the proportion 5:2. The next conjunction will take place along CS1,
and the third after the initial one will again take place near the
original position JQ, Jupiter having made five revolutions and Saturn
two.

The result of these repetitions is that, during a number of
revolutions, the special mutual actions of the two planets at these
three points of their orbits repeat themselves, while the actions
corresponding to the three intermediate arcs are wanting. Thus it
happens that if the mutual actions are balanced through a period of a
few revolutions only there is a small residuum of forces corresponding
to the three regions in question, which repeats itself in the same
way, and which, if it continued indefinitely, would entirely change
the forms of the two orbits. But the actual mean motions deviate
slightly from the ratio 2:5, and we have next to show how this
deviation results in an ultimate balancing of the forces. The annual
mean motions, with the corresponding combinations, are as follows:--

Jupiter:--n = 30 deg. .349043
Saturn:--n' = 12 deg. .221133
2n = 60 deg. .69809
5n' = 61 deg. .10567
5n' - 2n = 0 deg. .40758

If we make a more accurate computation of the conjunctions from these
data, we shall find that, in the general mean, the consecutive
conjunctions take place when each planet has moved through an entire
number of revolutions + 242.7 deg. It follows that the third
conjunction instead of occurring exactly along the line CQ1 occurs
along CQ2, making an angle of nearly 8 deg. with CQ1. The successive
conjunctions following will be along CR2, CS2, CQ3, &c., the law of
progression being obvious.

The balancing of the series of forces will not be complete until the
respective triplets of conjunctions have filled up the entire space
between them. This will occur when the angle whose annual motion is
5n' - 2n has gone through 360 deg. From the preceding value of 5n' -
2n we see that this will require a little more than 883 years. The
result of the continued action of the two planets upon each other is
that during half of this period the motion of one planet is constantly
retarded and of the other constantly accelerated, while during the
other half the effects are reversed. There is thus in the case of each
planet an oscillation of the mean longitude which increases it and
then diminishes it to its original value at the end of the period of
883 years.

The longitudes, latitudes and radii vectores of a planet, being
algebraically expressed as the sum of an infinite periodic series of
the kind we have been describing, it follows that the problem of
finding their co-ordinates at any moment is solved by computing these
expressions. This is facilitated by the construction of tables by
means of which the co-ordinates can be computed at any time. Such
tables are used in the offices of the national Ephemerides to
construct ephemerides of the several planets, showing their exact
positions in the sky from day to day.

We pass now to the second branch of celestial mechanics viz. that in
which the planets are no longer considered as particles, but as
rotating bodies of which the dimensions are to be taken into account.
Such a body, in free space, not acted on by any force except the
attraction of its several parts, will go on rotating for ever in an
invariable direction. But, in consequence of the centrifugal force
generated by the rotation, it assumes a spheroidal form, the
equatorial regions bulging out. Such a form we all know to be that of
the earth and of the planets rotating on their axes. Let us study the
effect of this deviation from the spherical form upon the attraction
exercised by a distant body.

We begin with the special case of the earth as acted upon by the sun
and moon. Let fig. 4 represent a section of the earth through its axis
AB, ECQ being a diameter of the equator. Let the dotted lines show the
direction of the distant attracting body. The point E, being more
distant than C, will be attracted with less force, while Q will be
attracted with a greater force than will the centre C. Were the force
equal on every point of the earth it would have no influence on its
rotation, but would simply draw its whole mass toward the attracting
body. It is therefore only the _difference_ of the forces on different
parts of the earth that affects the rotation.

Let us, therefore, divide the attracting forces at each point into two
parts, one the average force, which we may call F, and which for our
purpose may be regarded as equal to the force acting at C; the others
the residual forces which we must superimpose upon the average force F
in order that the combination may be equal to the actual force. It is
clear that at Q this residual force as represented by the arrow will
be in the same direction as the actual force. But at E, since the
actual force is less than F, the residual force must tend to diminish
F, and must, therefore, act toward the right, as shown by the arrow.
These residual forces tend to make the whole earth turn round the
centre C in a clockwise direction. If nothing modified this tendency
the result would be to bring the points E and Q into the dotted lines
of the attraction. In other words the equator would be drawn into
coincidence with the ecliptic. Here, however, the same action comes
into play, which keeps a rotating top from falling over. (See
GYROSCOPE and MECHANICS.) For the same reason as in the case of the
gyroscope the actual motion of the earth's axis is at right angles to
the line joining the earth and the attracting centre, and without
going into the details of the mathematical processes involved, we may
say that the ultimate mean effect will be to cause the pole P of the
earth to move at right angles to the circle joining it to the pole of
the ecliptic. Were the position of the latter invariable, the
celestial pole would move round it in a circle. Actually the curve in
which it moves is nearly a circle; but the distance varies slightly
owing to the minute secular variation in the position of the ecliptic,
caused by the action of the planets. This motion of the celestial pole
results in a corresponding revolution of the equinox around the
celestial sphere. The rate of motion is slightly variable from century
to century owing to the secular motion of the plane of the ecliptic.
Its period, with the present rate of motion, would be about 26,000
years, but the actual period is slightly indeterminate from the cause
just mentioned.

The residual force just described is not limited to the case of an
ellipsoidal body. It will be seen that the reasoning applies to the
case of any one body or system of bodies, the dimensions of which are
not regarded as infinitely small compared with the distance of the
attracting body. In all such cases the residual forces virtually tend
to draw those portions of the body nearest the attracting centre
toward the latter, and those opposite the attracting centre away from
it. Thus we have a tide-producing force tending to deform the body,
the action of which is of the same nature as the force producing
precession. It is of interest to note that, very approximately, this
deforming force varies inversely as the cube of the distance of the
attracting body.

The action of the sun upon the satellites of the several planets and
the effects of this action are of the same general nature. For the
same reason that the residual forces virtually act in opposite
directions upon the nearer and more distant portions of a planet they
will virtually act in the case of a satellite. When the latter is
between its primary and the sun, the attraction of the latter tends to
draw the satellite away from the primary. When the satellite is in the
opposite direction from the sun, the same action tends to draw the
primary away from the satellite. In both cases, relative to the
primary, the action is the same. When the satellite is in quadrature
the convergence of the lines of attraction toward the centre of the
sun tends to bring the two bodies together. When the orbit of the
satellite is inclined to that of the primary planet round the sun, the
action brings about a change in the plane of the orbit represented by
a rotation round an axis perpendicular to the plane of the orbit of
the primary. If we conceive a pole to each of these orbits, determined
by the points in which lines perpendicular to their planes intersect
the celestial sphere, the pole of the satellite orbit will revolve
around the pole of the planetary orbit precisely as the pole of the
earth does around the pole of the ecliptic, the inclination of the two
orbits remaining unchanged.

If a planet rotates on its axis so rapidly as to have a considerable
ellipticity, and if it has satellites revolving very near the plane of
the equator, the combined actions of the sun and of the equatorial
protuberances may be such that the whole system will rotate almost as
if the planes of revolution of the satellites were solidly fixed to
the plane of the equator. This is the case with the seven inner
satellites of Saturn. The orbits of these bodies have a large
inclination, nearly 27 deg., to the plane of the planet's orbit. The
action of the sun alone would completely throw them out of these
planes as each satellite orbit would rotate independently; but the
effect of the mutual action is to keep all of the planes in close
coincidence with the plane of the planet's equator.

_Literature._--The modern methods of celestial mechanics may be
considered to begin with Joseph Louis Lagrange, whose theory of the
variation of elements is developed in his _Mecanique analytique_. The
practical methods of computing perturbations of the planets and
satellites were first exhaustively developed by Pierre Simon Laplace
in his _Mecanique celeste_. The only attempt since the publication of
this great work to develop the various theories involved on a uniform
plan and mould them into a consistent whole is that of de Pontecoulant
in _Theorie analytique du systeme du monde_ (1829-46, Paris). An
approximation to such an attempt is that of F.F. Tisserand in his
_Traite de mecanique celeste_ (4 vols., Paris). This work contains a
clear and excellent resume of the methods which have been devised by
the leading investigators from the time of Lagrange until the present,
and thus forms the most encyclopaedic treatise to which the student
can refer.

Works less comprehensive than this are necessarily confined to the
elements of the subject, to the development of fundamental principles
and general methods, or to details of special branches. An elementary
treatise on the subject is F.R. Moulton's _Introduction to Celestial
Mechanics_ (London, 1902). Other works with the same general object
are H.A. Resal, _Mecanique celeste_; and O.F. Dziobek, _Theorie der
Planetenbewegungen_. The most complete and systematic development of
the general principles of the subject, from the point of view of the
modern mathematician, is found in J.H. Poincare, _Les Methodes
nouvelles de la mecanique celeste_ (3 vols., Paris, 1899, 1892, 1893).
Of another work of Poincare, _Lecons de mecanique celeste_, the first
volume appeared in 1905.

_Practical Astronomy._

Practical Astronomy, taken in its widest sense, treats of the instruments by which our knowledge of the heavenly bodies is acquired, the principles underlying their use, and the methods by which these principles are practically applied. Our knowledge of these bodies is of necessity derived through the medium of the light which they emit; and it is the development and applications of the laws of light which have made possible the additions to our stock of such knowledge since the middle of the 19th century.

At the base of every system of astronomical observation is the law
that, in the voids of space, a ray of light moves in a right line. The
fundamental problem of practical astronomy is that of determining by
measurement the co-ordinates of the heavenly bodies as already
defined. Of the three co-ordinates, the radius vector does not admit
of direct measurement, and must be inferred by a combination of
indirect measurements and physical theories. The other two
co-ordinates, which define the direction of a body, admit of direct
measurement on principles applied in the construction and use of
astronomical instruments.

In the first system of co-ordinates already described the fundamental
axis is the vertical line or direction of gravity at the point of
observation. This is not the direction of gravity proper, or of the
earth's attraction, but the resultant of this attraction combined with
the centrifugal force due to the earth's rotation on its axis. The
most obvious method of realizing this direction is by the plumb-line.
In our time, however, this appliance is replaced by either of two
others, which admit of much more precise application. These are the
basin of mercury and the spirit-level. The surface of a liquid at rest
is necessarily perpendicular to the direction of gravity, and
therefore horizontal. Considered as a curved surface, concentric with
the earth, a tangent plane to such a surface is the plane of the
horizon. The problem of measuring from an axis perpendicular to this
plane is solved on the principle that the incident and reflected rays
of light make equal angles with the perpendicular to a reflecting
surface. It follows that if PO (fig. 5) is the direction of a ray,
either from a heavenly body or from a terrestrial point, impinging at
O upon the surface of quicksilver, and reflected in the direction OR,
the vertical line is the bisector OZ, of the angle POR. If the point P
is so adjusted over the quicksilver that the ray is reflected back on
its own path, P and R lying on the same line above O, then we know
that the line PO is truly vertical. The zenith-distance of an object
is the angle which the ray of light from it makes with the vertical
direction thus defined.

To show the principle involved in the spirit-level let MN (fig. 6) be
the tube of such a level, fixed to an axis OZ on which it may revolve.
If this axis is so adjusted that in the course of a revolution around
it the bubble of the level undergoes no change of position, we know
that the axis is truly vertical. Any slight deviation from verticality
is shown by the motion of the bubble during the revolution, which can
be measured and allowed for. The level may not be actually attached to
an axis, a revolution of 180 deg. being effected round an imaginary
vertical axis by turning the level end for end. The motion of the
bubble then measures double the inclination of this imaginary axis, or
the deviation of a cylinder on which the level may rest from
horizontality.

The problem of determining the zenith distance of a celestial object
now reduces itself to that of measuring the angle between the
direction of the object and the direction of the vertical line
realized in one of these ways. This measurement is effected by a
combination of two instruments, the telescope and the graduated
circle. Let OF (fig. 7) be a section of the telescope, MN being its
object glass. Let the parallel dotted lines represent rays of light
emanating from the object to be observed, which, for our purpose, we
regard as infinitely distant, a star for example. These rays come to a
focus at a point F lying in the focal plane of the telescope. In this
plane are a pair of cross threads or spider lines which, as the
observer looks into the telescope, are seen as AB and CD (fig. 8). If
the telescope is so pointed that the image of the star is seen in
coincidence with the cross threads, as represented in fig. 8, then we
know that the star is exactly in the line of sight of the telescope,
defined as the line joining the centre of the object glass, and the
point of intersection of the cross threads. If the telescope is moved
around so that the images of two distant points are successively
brought into coincidence with the cross threads, we know that the
angle between the directions of these points is equal to that through
which the telescope has been turned. This angle is measured by means
of a graduated circle, rigidly attached to the tube of the telescope
in a plane parallel to the line of sight. When the telescope is turned
in this plane, the angular motion of the line of sight is equal to
that through which the circle has turned.

Stripped of all unnecessary adjuncts, and reduced to a geometric form,
the ideal method by which the zenith distance of a heavenly body is
determined by the combination which we have described is as
follows:--Let OP (fig. 9) be the direction of a celestial body at
which a telescope, supplied with a graduating circle, is pointed. Let
OZ be an axis, as nearly vertical as it can easily be set, round which
the entire instrument may revolve through 180 deg. After the image of
the body is brought into coincidence with the cross threads, the
instrument is turned through 180 deg. on the axis, which results in
the line of sight of the telescope pointing in a certain direction OQ,
determined by the condition QOZ = ZOP. The telescope is then a second
time pointed at the object by being moved through the angle QOP.
Either of the angles QOZ and ZOP is then one half that through which
the telescope has been turned, which may be measured by a graduated
circle, and which is the zenith distance of the object measured from
the direction of the axis OZ. This axis may not be exactly vertical.
Its deviation from the vertical line is determined by the motion of
the bubble of a spirit-level rigidly attached either to the axis, or
to the telescope. Applying this deviation to the measured arc, the
true zenith distance of the body is found.

When the basin of quicksilver is used, the telescope, either before or
after being directed toward P, is pointed directly downwards, so that
the observer mounting above it looks through it into the reflecting
surface. He then adjusts the instrument so that the cross threads
coincide with their images reflected from the surface of the
quicksilver. The angular motion of the telescope in passing from this
position to that when the celestial object is in the line of sight is
the distance (ND) of the body from the nadir. Subtracting 90 deg. from
(ND) gives the altitude; and subtracting (ND) from 180 deg. gives the
zenith distance.

In the measurement of equatorial co-ordinates, the polar distance is
determined in an analogous way. We determine the apparent position of
an object near the pole on the celestial sphere at any moment, and
again at another moment, twelve hours later, when, by the diurnal
motion, it has made half a revolution. The angle through the celestial
pole, between these two positions, is double the polar distance. The
pole is the point midway between them. This being ascertained by one
or more stars near it, may be used to determine by direct measurements
the polar distances of other bodies.

The preceding methods apply mainly to the latitudinal co-ordinate. To
measure the difference between the longitudinal co-ordinates of two
objects by means of a graduated circle the instruments must turn on an
axis parallel to the principal axis of the system of co-ordinates, and
the plane of the graduated circle must be at right angles to that
axis, and, therefore, parallel to the principal co-ordinate plane. The
telescope, in order that it may be pointed in any direction, must
admit of two motions, one round the principal axis, and the other
round an axis at right angles to it. By these two motions the
instrument may be pointed first at one of the objects and then at the
other. The motion of the graduated circle in passing from one pointing
to the other is the measure of the difference between the longitudinal
co-ordinates of the two objects.

In the equatorial system this co-ordinate (the right ascension) is
measured in a different way, by making the rotating earth perform the
function of a graduated circle. The unceasing diurnal motion of the
image of any heavenly body relative to the cross threads of a
telescope makes a direct accurate measure of any co-ordinate except
the declination almost impossible. Before the position of a star can
be noted, it has passed away from the cross threads. This troublesome
result is utilized and made a means of measurement. Right ascensions
are now determined, not by measuring the angle between one star and
another, but, by noting the time between the transits of successive
stars over the meridian. The difference between these times, when
reduced to an angle, is the difference of the right ascensions of the
stars. The principle is the same as that by which the distance between
two stations may be determined by the time required for a train moving
at a uniform known speed to pass from one station to the other. The
uniform speed of the diurnal motion is 15 deg. per hour. We have
already mentioned that in astronomical practice right ascensions are
expressed in time, so that no multiplication by 15 is necessary.

Measures made on the various systems which we have described give the
apparent direction of a celestial object as seen by the observer. But
this is not the true direction, because the ray of light from the
object undergoes refraction in passing through the atmosphere. It is
therefore necessary to correct the observation for this effect. This
is one of the most troublesome problems in astronomy because, owing to
the ever varying density of the atmosphere, arising from differences
of temperature, and owing to the impossibility of determining the
temperature with entire precision at any other point than that
occupied by the observer, the amount of refraction must always be more
or less uncertain. The complexity of the problem will be seen by
reflecting that the temperature of the air inside the telescope is not
without its effect. This temperature may be and commonly is somewhat
different from that of the observing room, which, again, is commonly
higher than the temperature of the air outside. The uncertainty thus
arising in the amount of the refraction is least near the zenith, but
increases more and more as the horizon is approached.

The result of astronomical observations which is ordinarily wanted is
not the direction of an object from the observer, but from the centre
of the earth. Thus a reduction for parallax is required. Having
effected this reduction, and computed the correction to be applied to
the observation in order to eliminate all known errors to which the
instrument is liable, the work of the practical astronomer is
completed.

The instruments used in astronomical research are described under
their several names. The following are those most used in
astrometry:--

The equatorial telescope (q.v.) is an instrument which can be directed
to any point in the sky, and which derives its appellation from its
being mounted on an axis parallel to that of the earth. By revolving
on this axis it follows a star in its diurnal motion, so that the star
is kept in the field of view notwithstanding that motion.

Next in extent of use are the transit instrument and the meridian
circle, which are commonly united in a single instrument, the transit
circle (q.v.), known also as the meridian circle. This instrument
moves only in the plane of the meridian on a horizontal east and west
axis, and is used to determine the right ascensions and declinations
of stars. These two instruments or combinations are a necessary part
of the outfit of every important observatory. An adjunct of prime
importance, which is necessary to their use, is an accurate clock,
beating seconds.

_Use of Photography._--Before the development of photography, there
was no possible way of making observations upon the heavenly bodies
except by the eye. Since the middle of the 19th century the system of
photographing the heavenly bodies has been introduced, step by step,
so that it bids fair to supersede eye observations in many of the
determinations of astronomy. (See PHOTOGRAPHY: _Celestial_.)

The field of practical astronomy includes an extension which may be
regarded as making astronomical science in a certain sense universal.
The science is concerned with the heavenly bodies. The earth on which
we live is, to all intents and purposes, one of these bodies, and, so
far as its relations to the heavens are concerned, must be included in
astronomy. The processes of measuring great portions of the earth, and
of determining geographical positions, require both astronomical
observations proper, and determinations made with instruments similar
to those of astronomy. Hence geodesy may be regarded as a branch of
practical astronomy. (S. N.)

_History of Astronomy._

Origin of the science.

A practical acquaintance with the elements of astronomy is indispensable to the conduct of human life. Hence it is most widely diffused among uncivilized peoples, whose existence depends upon immediate and unvarying submission to the dictates of external nature. Having no clocks, they regard instead the face of the sky; the stars serve them for almanacs; they hunt and fish, they sow and reap in correspondence with the recurrent order of celestial appearances. But these, to the untutored imagination, present a mystical, as well as a mechanical aspect; and barbaric familiarity with the heavens developed at an early age, through the promptings of superstition, into a fixed system of observation. In China, Egypt and Babylonia, strength and continuity were lent to this native tendency by the influence of a centralized authority; considerable proficiency was attained in the arts of observation; and from millennial stores of accumulated data, empirical rules were deduced by which the scope of prediction was widened and its accuracy enhanced. But no genuine science of astronomy was founded until the Greeks sublimed experience into theory.

Chinese astronomy.

Already, in the third millennium B.C., equinoxes and solstices were determined in China by means of culminating stars. This is known from the orders promulgated by the emperor Yao about 2300 B.C., as recorded in the _Shu Chung_, a collection of documents antique in the time of Confucius (550-478 B.C.). And Yao was merely the renovator of a system long previously established. The _Shu Chung_ further relates the tragic fate of the official astronomers, Hsi and Ho, put to death for neglecting to perform the rites customary during an eclipse of the sun, identified by Professor S.E. Russell[1] with a partial obscuration visible in northern China 2136 B.C. The date cannot be far wrong, and it is by far the earliest assignable to an event of the kind. There is, however, no certainty that the Chinese were then capable of predicting eclipses. They were, on the other hand, probably acquainted, a couple of millenniums before Meton gave it his name, with the nineteen-year cycle, by which solar and lunar years were harmonized;[2] they immemorially made observations in the meridian; regulated time by water-clocks, and used measuring instruments of the nature of armillary spheres and quadrants. In or near 1100 B.C., Chou Kung, an able mathematician, determined with surprising accuracy the obliquity of the ecliptic; but his attempts to estimate the sun's distance failed hopelessly as being grounded on belief in the flatness of the earth. From of old, in China, circles were divided into 365-1/4 parts, so that the sun described daily one Chinese degree; and the equator began to be employed as a line of reference, concurrently with the ecliptic, probably in the second century B.C. Both circles, too, were marked by star-groups more or less clearly designated and defined. Cometary records of a vague kind go back in China to 2296 B.C.; they are intelligible and trustworthy from 611 B.C. onward. Two instruments constructed at the time of Kublai Khan's accession in 1280 were still extant at Peking in 1881. They were provided with large graduated circles adapted for measurements of declination and right ascension, and prove the Chinese to have anticipated by at least three centuries some of Tycho Brahe's most important inventions.[3] The native astronomy was finally superseded in the 17th century by the scientific teachings of Jesuit missionaries from Europe.

Egyptian astronomy.

Astrolatry was, in Egypt, the prelude to astronomy. The stars were observed that they might be duly worshipped. The importance of their heliacal risings, or first visible appearances at dawn, for the purposes both of practical life and of ritual observance, caused them to be systematically noted; the length of the year was accurately fixed in connexion with the annually recurring Nile-flood; while the curiously precise orientation of the Pyramids affords a lasting demonstration of the high degree of technical skill in watching the heavens attained in the third millennium B.C. The constellational system in vogue among the Egyptians appears to have been essentially of native origin; but they contributed little or nothing to the genuine progress of astronomy.

Babylonian astronomy.

With the Babylonians the case was different, although their science lacked the vital principle of growth imparted to it by their successors. From them the Greeks derived their first notions of astronomy. They copied the Babylonian asterisms, appropriated Babylonian knowledge of the planets and their courses, and learned to predict eclipses by means of the "Saros." This is a cycle of 18 years 11 days, or 223 lunations, discovered at an unknown epoch in Chaldaea, at the end of which the moon very nearly returns to her original position with regard as well to the sun as to her own nodes and perigee. There is no getting back to the beginning of astronomy by the shores of the Euphrates. Records dating from the reign of Sargon of Akkad (3800 B.C.) imply that even then the varying aspects of the sky had been long under expert observation. Thus early, there is reason to suppose, the star-groups with which we are now familiar began to be formed. They took shape most likely, not through one stroke of invention, but incidentally, as legends developed and astrological persuasions became defined.[4] The zodiacal series in particular seem to have been reformed and reconstructed at wide intervals of time (see ZODIAC). Virgo, for example, is referred by P. Jensen, on the ground of its harvesting associations, to the fourth millennium B.C., while Aries (according to F.K. Ginzel) was interpolated at a comparatively recent time. In the main, however, the constellations transmitted to the West from Babylonia by Aratus and Eudoxus must have been arranged very much in their present order about 2800 B.C. E.W. Maunder's argument to this effect is unanswerable.[5] For the space of the southern sky left blank of stellar emblazonments was necessarily centred on the pole; and since the pole shifts among the stars through the effects of precession by a known annual amount, the ascertainment of any former place for it virtually fixes the epoch. It may then be taken as certain that the heavens described by Aratus in 270 B.C. represented approximately observations made some 2500 years earlier in or near north latitude 40 deg.

In the course of ages, Babylonian astronomy, purified from the astrological taint, adapted itself to meet the most refined needs of civil life. The decipherment and interpretation by the learned Jesuits, Fathers Epping and Strassmeier, of a number of clay tablets preserved in the British Museum, have supplied detailed knowledge of the methods practised in Mesopotamia in the 2nd century B.C.[6] They show no trace of Greek influence, and were doubtless the improved outcome of an unbroken tradition. How protracted it had been, can be in a measure estimated from the length of the revolutionary cycles found for the planets. The Babylonian computers were not only aware that Venus returns in almost exactly eight years to a given starting-point in the sky, but they had established similar periodic relations in 46, 59, 70 and 83 years severally for Mercury, Saturn, Mars and Jupiter. They were accordingly able to fix in advance the approximate positions of these objects with reference to ecliptical stars which served as fiducial points for their determination. In the Ephemerides published year by year, the times of new moon were given, together with the calculated intervals to the first visibility of the crescent, from which the beginning of each month was reckoned; the dates and circumstances of solar and lunar eclipses were predicted; and due information was supplied as to the forthcoming heliacal risings and settings, conjunctions and oppositions of the planets. The Babylonians knew of the inequality in the daily motion of the sun, but misplaced by 10 deg. the perigee of his orbit. Their sidereal year was (4-1/2)^m too long,[7] and they kept the ecliptic stationary among the stars, making no allowance for the shifting of the equinoxes. The striking discovery, on the other hand, has been made by the Rev. F.X. Kugler[8] that the various periods underlying their lunar predictions were identical with those heretofore believed to have been independently arrived at by Hipparchus, who accordingly must be held to have borrowed from Chaldaea the lengths of the synodic, sidereal, anomalistic and draconitic months.

Greek astronomy. Thales.

Pythagoras.

Heraclides.

A steady flow of knowledge from East to West began in the 7th century B.C. A Babylonian sage named Berossus founded a school about 640 B.C. in the island of Cos, and perhaps counted Thales of Miletus (c. 639-548) among his pupils. The famous "eclipse of Thales" in 585 B.C. has not, it is true, been authenticated by modern research;[9] yet the story told by Herodotus appears to intimate that a knowledge of the Saros, and of the forecasting facilities connected with it, was possessed by the Ionian sage. Pythagoras of Samos (fl. 540-510 B.C.) learned on his travels in Egypt and the East to identify the morning and evening stars, to recognize the obliquity of the ecliptic, and to regard the earth as a sphere freely poised in space. The tenet of its axial movement was held by many of his followers--in an obscure form by Philolaus of Crotona after the middle of the 5th century B.C., and more explicitly by Ecphantus and Hicetas of Syracuse (4th century B.C.), and by Heraclides of Pontus. Heraclides, who became a disciple of Plato in 360 B.C., taught in addition that the sun, while circulating round the earth, was the centre of revolution to Venus and Mercury.[10] A genuine heliocentric system, developed by Aristarchus of Samos (fl. 280-264 B.C.), was described by Archimedes in his _Arenarius_, only to be set aside with disapproval. The long-lived conception of a series of crystal spheres, acting as the vehicles of the heavenly bodies, and attuned to divine harmonies, seems to have originated with Pythagoras himself.

Eudoxus.

The first mathematical theory of celestial appearances was devised by Eudoxus of Cnidus (408-355 B.C.).[11] The problem he attempted to solve was so to combine uniform circular movements as to produce the resultant effects actually observed. The sun and moon and the five planets were, with this end in view, accommodated each with a set of variously revolving spheres, to the total number of 27. The Eudoxian or "homocentric" system, after it had been further elaborated by Callippus and Aristotle, was modified by Apollonius of Perga (fl. 250-220 B.C.) into the hypothesis of deferents and epicycles, which held the field for 1800 years as the characteristic embodiment of Greek ideas in astronomy. Eudoxus further wrote two works descriptive of the heavens, the _Enoptron_ and _Phaenomena_, which, substantially preserved in the _Phaenomena_ of Aratus (fl. 270 B.C.), provided all the leading features of modern stellar nomenclature.

School of Alexandria.

Aristarchus.

Greek astronomy culminated in the school of Alexandria. It was, soon after its foundation, illustrated by the labours of Aristyllus and Timocharis (c. 320-260 B.C.), who constructed the first catalogue giving star-positions as measured from a reference-point in the sky. This fundamental advance rendered inevitable the detection of precessional effects. Aristarchus of Samos observed at Alexandria 280-264 B.C. His treatise on the magnitudes and distances of the sun and moon, edited by John Wallis in 1688, describes a theoretically valid method for determining the relative distances of the sun and moon by measuring the angle between their centres when half the lunar disk is illuminated; but the time of dichotomy being widely indeterminate, no useful result was thus obtainable. Aristarchus in fact concluded the sun to be not more than twenty times, while it is really four hundred times farther off than our satellite. His general conception of the universe was comprehensive beyond that of any of his predecessors.

Eratosthenes.