Chapter XVII: Part 17

Kepler explained the double movement of the earth by the rotation of the sun. At one time the sun presented its friendly side, which attracted one planet, sometimes its adverse side, which repelled it. He also peopled the planets with souls and genii. He was led to his three great laws by musical analogies, just as William Herschel afterwards passed from music to astronomy. Kepler, who in his youth made almanacs, and once prophesied a hard winter, which came to pass, could not help putting an astrological interpretation on the disappearance of the brilliant star of 1572, which Tycho had observed. Theodore Beza thought that this star, which in December 1573 equalled Jupiter in brilliancy, predicted the second coming of Christ. Astronomers were only then beginning to study variable and periodic stars, and disturbances in that part of the heavens, which had till then, on the authority of Aristotle, been regarded as incorruptible, combined with the troubles of the times, must have given a new stimulus to belief in the signs in heaven. Montaigne (_Essais_, lib. i. chap, x.) relates a singular episode in the history of astrology. Charles V. and Francis I., who both bid for the friendship of the infamous Aretino, surnamed the divine, both likewise engaged astrologers to fight their battles. In Italy those who prophesied the ruin of France were sure to be listened to. These prophecies affected the public funds much as telegrams do nowadays. "At Rome," Montaigne tells us, "a large sum of money was lost on the Change by this prognostication of our ruin." The marquis of Saluces, notwithstanding his gratitude to Francis I. for the many favours he had received, including his marquisate, of which the brother was despoiled for his benefit, was led in 1536 to betray his country, being scared by the glorious prophecies of the ultimate success of Charles V. which were then rife. The influence of the Medici made astrologers popular in France. Richelieu, on whose council was Jacques Gaffarel (1601-1681), the last of the Kabbalists, did not despise astrology as an engine of government. At the birth of Louis XIV. a certain Morin de Villefranche was placed behind a curtain to cast the nativity of the future autocrat. A generation back the astrologer would not have been hidden behind a curtain, but have taken precedence of the doctor. La Bruyere dares not pronounce against such beliefs, "for there are perplexing facts affirmed by grave men who were eye-witnesses." In England William Lilly and Robert Fludd were both dressed in a little brief authority. The latter gives us elaborate rules for the detection of a thief, and tells us that he has had personal experience of their efficacy. "If the lord of the sixth house is found in the second house, or in company with the lord of the second house, the thief is one of the family. If Mercury is in the sign of the Scorpion he will be bald, &c." Francis Bacon abuses the astrologers of his day no less than the alchemists, but he does so because he has visions of a reformed astrology and a reformed alchemy. Sir Thomas Browne, too, while he denies the capacity of the astrologers of his day, does not venture to dispute the reality of the science. The idea of the souls of men passing at death to the stars, the blessedness of their particular sphere being assigned them according to their deserts (the metempsychosis of J. Reynaud), may be regarded as a survival of religious astrology, which, even as late as Descartes's day, assigned to the angels the task of moving the planets and the stars. Joseph de Maistre believed in comets as messengers of divine justice, and in animated planets, and declared that divination by astrology is not an absolutely chimerical science. Lastly, we may mention a few distinguished men who ran counter to their age in denying stellar influences. Aristarchus of Samos, Martianus Capella (the precursor of Copernicus), Cicero, Favorinus, Sextus Empiricus, Juvenal, and in a later age Savonarola and Pico della Mirandola, and La Fontaine, a contemporary of the neutral La Bruyere, were all pronounced opponents of astrology.

In England Swift may fairly claim the credit of having given the death-blow to astrology by his famous squib, entitled _Prediction for the Year 1708, by Isaac Bickerstaff, Esq._ He begins, by professing profound belief in the art, and next points out the vagueness and the absurdities of the philomaths. He then, in the happiest vein of parody, proceeds to show them a more excellent way:--"My first prediction is but a trifle, yet I mention it to show how ignorant these sottish pretenders to astrology are in their own concerns: it refers to Partridge the almanac-maker. I have consulted the star of his nativity by my own rules, and find he will infallibly die upon the 29th of March next about eleven at night of a raging fever. Therefore I advise him to consider of it and settle his affairs in time." Then followed a letter to a person of quality giving a full and particular account of the death of Partridge on the very day and nearly at the hour mentioned. In vain the wretched astrologer protested that he was alive, got a literary friend to write a pamphlet to prove it, and published his almanac for 1709. Swift, in his reply, abused him for his want of manners in giving a gentleman the lie, answered his arguments _seriatim_, and declared that the evidence of the publication of another almanac was wholly irrelevant, "for Gadbury, Poor Robin, Dove and Way do yearly publish their almanacs, though several of them have been dead since before the Revolution." Nevertheless a field is found even to this day for almanacs of a similar type, and for popular belief in them.

To astrological politics we owe the theory of heaven-sent rulers, instruments in the hands of Providence, and saviours of society. Napoleon, as well as Wallenstein, believed in his star. Many passages in the older English poets are unintelligible without some knowledge of astrology. Chaucer wrote a treatise on the astrolabe; Milton constantly refers to planetary influences; in Shakespeare's _King Lear_, Gloucester and Edmund represent respectively the old and the new faith. We still _contemplate_ and consider; we still speak of men as _jovial_, _saturnine_ or _mercurial_; we still talk of the _ascendancy_ of genius, or a _disastrous_ defeat. In French _heur_, _malheur_, _heureux_, _malheureux_, are all derived from the Latin _augurium_; the expression _ne sous une mauvaise etoile_, born under an evil star, corresponds (with the change of _etoile_ into _astre_) to the word _malotru_, in Provencal _malastrue_; and _son etoile palit_, his star grows pale, belongs to the same class of illusions. The Latia _ex augurio_ appears in the Italian _sciagura_, _sciagurato_, softened into _sciaura_, _sciaurato_, wretchedness, wretched. The influence of a particular planet has also left traces in various languages; but the French and English _jovial_ and the English _saturnine_ correspond rather to the gods who served as types in chiromancy than to the planets which bear the same names. In the case of the expressions _bien_ or _mal lune_, well or ill mooned, _avoir un quartier de lune dans la tete_, to have the quarter of the moon in one's head, the German _mondsuchtig_ and the English _moonstruck_ or _lunatic_, the fundamental idea lies in the strange opinions formerly held about the moon.

BIBLIOGRAPHY.--For the history of astrology with its affinities to
astronomy on the one hand, and to other forms of popular belief on the
other, the following works out of a large number that might be
mentioned are specially recommended:--A. Bouche-Leclercq,
_L'Astrologie grecque_ (Paris, 1899), with a full bibliography; Franz
Boll, _Sphaera_ (Leipzig, 1903); Franz Cumont, _Catalogus Codicum
Astrologorum Graecorum_ (Brussels, 1898; 7 parts published up to
1909); Franz Boll, "Die Erforschung der antiken Astrologie" (in _Neue
Jahrbucher fur das klassische Altertum_, Band xxi. Heft 2, pp.
103-126); Franz Cumont, _Les Religions orientates dans le paganisme
romain_ (Paris, 1907) (ch. vii. "L'Astrologie et la magie"); Alfred
Maury, _La Magie et l'astrologie a l'antiquite et au moyen age_ (4th
ed., Paris, 1877); R.C. Thompson, _Reports of the Magicians and
Astrologers of Nineveh and Babylon_ (2 vols., London, 1900); F.X.
Kugler, _Sternkunde und Sterndienst in Babel_ (Freiburg, 1907;--to be
completed in 4 vols.); Ch. Virolleaud, _L'Astrologie chaldeenne_
(Paris, 1905--to be completed in 8 parts--transliteration and
translations of cuneiform texts); Jastrow, _Religion Babyloniens und
Assyriens_ (Parts 13 and 14); also certain sections in
Bouche-Leclercq, _Histoire de la divination dans l'antiquite_ (Paris,
1879), vol. i. pp. 205-257; in Marcellin Berthelot, _Les Origines de
l'alchimie_ (Paris, 1885), pp. 1-56; Ferd. Hofer, _Histoire de
l'astronomie_ (Paris, 1846), pp. 1-90; in Rudolf Wolf, _Geschichte der
Astronomie_ (Munich, 1877), ch. i. See also the article by Ernst Riess
on Astrology in Pauly-Wissowa, _Realencyclopadie der klassischen
Altertumswissenschaft_, vol. ii. (Stuttgart, 1896). For modern and
practical astrology the following works may be found useful in
different ways: E.M. Bennett, _Astrology_ (New York, 1894); J.M.
Pfaff, _Astrologie_ (Bamberg, 1816); G. Wilde, _Chaldaean Astrology up
to date_ (1901); R. Garnett ("A.G. Trent"), "The Soul and the Stars,"
in the _University Magazine_, 1880 (reprinted in Dobson and Wilde,
_Natal Astrology_, 1893); Abel Haatan, _Traite d'astrologie
judiciaire_ (Paris, 1825); Fomalhaut, _Manuel d'astrologie spherique
el judiciaire_ (Paris, 1897). (M. Ja.)

ASTRONOMY (from Gr. [Greek: astron], a star, and [Greek: nemein], to classify or arrange). The subject matter of astronomical science, considered in its widest range, comprehends all the matter of the universe which lies outside the limit of the earth's atmosphere. The seeming anomaly of classifying as a single branch of science all that we know in a field so wide, while subdividing our knowledge of things on our own planet into an indefinite number of separate sciences, finds its explanation in the impossibility of subjecting the matter of the heavens to that experimental scrutiny which yields such rich results when applied to matter which we can handle at will. Astronomy is of necessity a science of observation in the pursuit of which experiment can directly play no part. It is the most ancient of the sciences because, before the era of experiment, it was the branch of knowledge which could be most easily systematized, while the relations of its phenomena to day and night, times and seasons, made some knowledge of the subject a necessity of social life. In recent times it is among the more progressive of the sciences, because the new and improved methods of research now at command have found in its cultivation a field of practically unlimited extent, in which the lines of research may ultimately lead to a comprehension of the universe impossible of attainment before our time.

The field we have defined is divisible into at least two parts, that of Astronomy proper, or "Astrometry," which treats of the motions, mutual relations and dimensions of the heavenly bodies; and that of Astrophysics (q.v.), which treats of their physical constitution. While it is true that the instruments and methods of research in these two branches are quite different in their details, there is so much in common in the fundamental principles which underlie their application, that it is unprofitable to consider them as completely distinct sciences.

Speaking in the most comprehensive way, and making an exception of the ethereal medium (see AETHER), which, being capable of experimental study, is not included in the subject of astronomy, we may say that the great masses of matter which make up the universe are of two kinds:--(1) incandescent bodies, made visible to us by their own light; (2) dark bodies, revolving round them or round each other. These dark bodies are known to us in two ways: (a) by becoming visible through reflecting the light from incandescent bodies in their neighbourhood, (b) by their attraction upon such bodies.

The incandescent bodies are of two classes: stars and nebulae. Among the stars our sun is to be included, as it has no properties which distinguish it from the great mass of stars except our proximity to it. The stars are supposed to be generally spherical, like the sun, in form, and to have fairly well-defined boundaries; while the nebulae are generally irregular in outline and have no well-defined limits. It is, however, probable that the one class runs into the other by imperceptible gradations. In the relation of the universe to us there is yet another separation of its bodies into two classes, one comprising the solar system, the other the remainder of the universe. The former consists of the sun and the bodies which move round it. Considered as a part of the universe, our solar system is insignificant in extent, though, for obvious reasons, great in practical importance to us, and in the facility with which we may gain knowledge relating to it.

Referring to special articles, SOLAR SYSTEM, STAR, SUN, MOON, &c. for a description of the various parts of the universe, we confine ourselves, at present, to setting forth a few of the most general modern conceptions of the universe. As to extent, it may be said, in a general way, that while no definite limits can be set to the possible extent of the universe, or the distance of its farthest bodies, it seems probable, for reasons which will be given under STAR, that the system to which the stars that we see belong, is of finite extent.

As the incandescent bodies of the universe are visible by their own light, the problem of ascertaining their existence and position is mainly one of seeing, and our facilities for attacking it have constantly increased with the improvement of our optical appliances. But such is not the case with the dark bodies. Such a body can be made known to us only when in the neighbourhood of an incandescent body; and even then, unless its mass or its dimensions are considerable, it will evade all the scrutiny of our science. The question of the possible number and magnitude of such bodies is therefore one that does not admit of accurate investigation. We can do no more than balance vague estimates of probability. What we do know is that these bodies vary widely in size. Those known to be revolving round certain of the stars are far larger in proportion to their central bodies than our planets are in respect to the sun; for were it otherwise we should never be able to detect their existence. At the other extreme we know that innumerable swarms of minute bodies, probably little more than particles, move round the sun in orbits of every degree of eccentricity, making themselves known to us only in the exceptional cases when they strike the earth's atmosphere. They then appear to us as "shooting stars" (see METEOR).

A general idea of the relation of the solar system to the universe may be gained by reflecting that the average distance between any two neighbouring stars is several thousand times the extent of the solar system. Between the orbit of Neptune and the nearest star known to us is an immense void in which no bodies are yet known to exist, except comets. But although these sometimes wander to distances considerably beyond the orbit of Neptune, it is probable that the extent of the void which separates our system from the nearest star is hundreds of times the distance of the farthest point to which a comet ever recedes.

We may conclude this brief characterization of astronomy with a statement and classification of the principal lines on which astronomical researches are now pursued. The most comprehensive problem before the investigator is that of the constitution of the universe. It is known that, while infinite diversity is found among the bodies of the universe, there are also common characteristics throughout its whole extent. In a certain sense we may say that the universe now presents itself to the thinking astronomer, not as a heterogeneous collection of bodies, but as a unified whole. The number of stars is so vast that statistical methods can be applied to many of the characters which they exhibit--their spectra, their apparent and absolute luminosity, and their arrangement in space. Thus has arisen in recent times what we may regard as a third branch of astronomical science, known as _Stellar Statistics_. The development of this branch has infused life and interest into what might a few years ago have been regarded as the most lifeless mass of figures possible, expressing merely the positions and motions of innumerable individual stars, as determined by generations of astronomical observers. The development of this new branch requires great additions to this mass, the product of perhaps centuries of work on the older lines of the science. To the statistician of the stars, catalogues of spectra, magnitude, position and proper motions are of the same importance that census tables are to the student of humanity. The measurement of the speed with which the individual stars are moving towards or from our system is a work of such magnitude that what has yet been done is scarcely more than a beginning. The discovery by improved optical means, and especially by photography, of new bodies of our system so small that they evaded all scrutiny in former times, is still going on, but does not at present promise any important generalization, unless we regard as such the conclusion that our solar system is a more complex organism than was formerly supposed.

One characteristic of astronomy which tends to make its progress slow and continuous arises out of the general fact that, except in the case of motions to or from us, which can be determined by a single observation with the spectroscope, the motion of a heavenly body can be determined only by comparing its position at two different epochs. The interval required between these two epochs depends upon the speed of the motion. In the case of the greater number of the fixed stars this is so slow that centuries may have to elapse before motion can be deduced. Even in the case of the planets, the variations in the form and position of the orbits are so slow that long periods of observation are required for their correct determination.

The process of development is also made slow and difficult by the great amount of labour involved in deriving the results of astronomical observations. When an astronomer has made an observation, it still has to be "reduced," and this commonly requires more labour than that involved in making it. But even this labour may be small compared with that of the theoretical astronomer, who, in the future, is to use the result as the raw material of his work. The computations required in such work are of extreme complexity, and the labour required is still further increased by the fact that cases are rather exceptional in which the results reached by one generation will not have to be revised and reconstructed by another; processes which may involve the repetition of the entire work. We may, in fact, regard the fabric of astronomical science as a building in the construction of which no stone can be added without a readjustment of some of the stones on which it has to rest. Thus it comes about that the observer, the computer, and the mathematician have in astronomical science a practically unlimited field for the exercise of their powers.

In treating so comprehensive a subject we may naturally distinguish between what we know of the universe and the methods and processes by which that knowledge is acquired. The former may be termed general, and the latter practical, astronomy. When we descend more minutely into details we find these two branches of the subject to be connected by certain principles, the application of which relates to both subjects. Considering as general or descriptive astronomy a description of the universe as we now understand it, the other branches of the subject generally recognized are as follows:--

_Geometrical_ or _Spherical Astronomy_, by the principles of which the positions and the motions of the heavenly bodies are defined.

_Theoretical Astronomy_, which may be considered as an extension of geometrical astronomy and includes the determination of the positions and motions of the heavenly bodies by combining mathematical theory with observation. Modern theoretical astronomy, taken in the most limited sense, is based upon _Celestial Mechanics_, the science by which, using purely deductive mechanical methods, the laws of motion of the heavenly bodies are derived by deductive methods from their mutual gravitation towards each other.

_Practical Astronomy_, which comprises a description of the instruments used in astronomical observation, and of the principles and methods underlying their application.

_Spherical or Geometrical Astronomy._

In astronomy, as in analytical geometry, the position of a point is defined by stating its distance and its direction from a point of reference taken as known. The numerical quantities by which the distance and direction, and therefore the position, are defined, are termed _co-ordinates_ of the point. The latter are measured or defined with regard to a fixed system of lines and planes, which form the basis of the system.

The following are the fundamental concepts of such a system.

(a) An origin or point of reference. The points most generally taken
for this purpose in astronomical practice are the following:--

(1) The position of a point of observation on the earth's surface. We
conceive its position to be that occupied by an observer. The position
of a heavenly body is then defined by its direction and distance from
the supposed observer.

(2) The centre of the earth. This point, though it can never be
occupied by an observer, is used because the positions of the heavenly
bodies in relation to it are more readily computed than they can be
from a point on the earth's surface.

(3) The centre of the sun.

(4) In addition to these three most usual points, we may, of course,
take the centre of a planet or that of a star in order to define the
position of bodies in their respective neighbourhoods.

Co-ordinates referred to a point of observation as the origin are
termed "apparent," those referred to the centre of the earth are
"geocentric," those referred to the centre of the sun, "heliocentric."

(b) The next concept of the system is a fundamental plane, regarded as
fixed, passing through the origin. In connexion with it is an axis
perpendicular to it, also passing through the origin. We may consider
the axis and the plane as a single concept, the axis determining the
plane, or the plane the axis. The fundamental concepts of this class
most in use are:--

(1) When a point on the earth's surface is taken as the origin, the
fundamental axis may be the direction of gravity at that point. This
direction defines the vertical line. The fundamental plane which it
determines is horizontal and is termed the plane of the horizon. Such
a plane is realized in the surface of a liquid, a basin of
quicksilver, for example.

(2) When the centre of the earth is taken as origin, the most natural
fundamental axis is that of the earth's rotation. This axis cuts the
earth's surface at the North and South Poles. The fundamental plane
perpendicular to it is the plane of the equator. This plane intersects
the earth's surface in the terrestrial equator. Co-ordinates referred
to this system are termed equatorial. A system of equatorial
co-ordinates may also be used when the origin is on the earth's
surface. The fundamental axis, instead of being the earth's axis
itself, is then a line parallel to it, and the fundamental plane is
the plane passing through the point, and parallel to the plane of the
equator.

(3) In the system of heliocentric co-ordinates, the plane in which the
earth moves round the sun, which is the plane of the ecliptic, is
taken as the fundamental one. The axis of the ecliptic is a line
perpendicular to this plane.

(c) The third concept necessary to complete the system is a fixed line
passing through the origin, and lying in the fundamental plane. This
line defines an initial direction from which other directions are
counted.

The geometrical concepts just defined are shown in fig. 1. Here O is
the origin, whatever point it may be; OZ is the fundamental axis
passing through it. In order to represent in the figure the position
of the fundamental plane, we conceive a circle to be drawn round O,
lying in that plane. This circle, projected in perspective as an
ellipse, is shown in the figure. OX is the fixed initial line by which
directions are to be defined.

Now let P be any point in space, say the centre of a heavenly body.
Conceive a perpendicular PQ to be dropped from this point on the
fundamental plane, meeting the latter in the point Q; PQ will then be
parallel to OZ. The co-ordinates of P will then be the following three
quantities:--

(1) The length of the line OP, or the distance of the body from the
origin, which distance is called the radius vector of the body.

(2) The angle XOQ which the projection of the radius vector upon the
fundamental plane makes with the initial line OX. This angle is called
the Longitude, Right Ascension or Azimuth of the body, in the various
systems of co-ordinates. We may term it in a general way the
longitudinal co-ordinate.

(3) The angle QOP, which the radius vector makes with the fundamental
plane. This we may call the latitudinal co-ordinate. Instead of it is
frequently used the complementary angle ZOP, known as the polar
distance of the body. Since ZOQ is a right angle, it follows that the
sum of the polar distance and the latitudinal co-ordinates is always
90 deg. Either may be used for astronomical purposes.

It is readily seen that the position of a heavenly body is completely
defined when these co-ordinates are given.

One of the systems of co-ordinates is familiar to every one, and may
be used as a general illustration of the method. It is our system of
defining the position of a point on the earth's surface by its
latitude and longitude. Regarding O (fig. 1) as the centre of the
earth, and P as a point on the earth's surface, a city for example, it
will be seen that OZ being the earth's axis, the circle MN will be the
equator. The initial line OX then passes through the foot of the
perpendicular dropped from Greenwich upon the plane of the equator,
and meets the surface at N. The angle QOP is the latitude of the place
and the angle NOQ its longitude. The longitudes and latitudes thus
defined are geocentric, and the latitude is slightly different from
that in ordinary use for geographic purposes. The difference arises
from the oblateness of the earth, and need not be considered here.

The conception of the co-ordinates we have defined is facilitated by
introducing that of the celestial sphere. This conception is embodied
in our idea of the vault of heaven, or of the sky. Taking as origin
the position of an observer, the direction of a heavenly body is
defined by the point in which he sees it in the sky; that is to say,
on the celestial sphere. Imagining, as we may well do, that the radius
of this sphere is infinite--then every direction, whatever the origin,
may be represented by a point on its surface. Take for example the
vertical line which is embodied in the direction of the plumb line.
This line, extended upwards, meets the celestial sphere in the zenith.
The earth's axis, continued indefinitely upwards, meets the sphere in
a point called the Celestial Pole. This point in our middle latitudes
is between the zenith and the north horizon, near a certain star of
the second magnitude familiarly known as the Pole Star. As the earth
revolves from west to east the celestial sphere appears to us to
revolve in the opposite direction, turning on the line joining the
Celestial Poles as on a pivot.

As we conceive of the sky, it does not consist of an entire sphere but
only as a hemisphere bounded by the horizon. But we have no difficulty
in extending the conception below the horizon, so that the earth with
everything upon it is in the centre of a complete sphere. The two
parts of this sphere are the visible hemisphere, which is above the
horizon, and the invisible, which is below it. Then the plumb line not
only defines the zenith as already shown, but in a downward direction
it defines the nadir, which is the point of the sphere directly below
our feet. On the side of this sphere opposite to the North Celestial
is the South Pole, invisible in the Northern Terrestrial Hemisphere
but visible in the Southern one.

The relation of geocentric to apparent co-ordinates depends upon the
latitude of the observer. The changes which the aspect of the heaven
undergoes, as we travel North and South, are so well known that they
need not be described in detail here; but a general statement of them
will give a luminous idea of the geometrical co-ordinates we have
described. Imagine an observer starting from the North Pole to travel
towards the equator, carrying his zenith with him. When at the pole
his zenith coincides with the celestial pole, and as the earth
revolves on its axis, the heavenly bodies perform their apparent
diurnal revolutions in horizontal circles round the zenith. As he
travels South, his zenith moves along the celestial sphere, and the
circles of diurnal rotation become oblique to the horizon. The
obliquity continually increases until the observer reaches the
equator. His zenith is then in the equator and the celestial poles are
in the North and South horizon respectively. The circles in which the
heavenly bodies appear to revolve are then vertical. Continuing his
journey towards the south, the north celestial pole sinks below the
horizon; the south celestial pole rises above it; or to speak more
exactly, the zenith of the observer approaches that pole. The circles
of diurnal revolution again become oblique. Finally, at the south pole
the circles of diurnal revolution are again apparently horizontal, but
are described in a direction apparently (but not really) the reverse
of that near the north pole. The reader who will trace out these
successive concepts and study the results of his changing positions
will readily acquire the notions which it is our subject to define.

We have next to point out the relation of the co-ordinates we have
described to the annual motion of the earth around the sun. In
consequence of this motion the sun appears to us to describe annually
a great circle, called the ecliptic, round the celestial sphere, among
the stars, with a nearly uniform motion, of somewhat less than 1 deg.
in a day. Were the stars visible in the daytime in the immediate
neighbourhood of the sun, this motion could be traced from day to day.
The ecliptic intersects the celestial equator at two opposite points,
the equinoxes, at an angle of 23 deg. 27'. The vernal equinox is taken
as the initial point on the sphere from which co-ordinates are
measured in the equatorial and ecliptic systems. Referring to fig. 1,
the initial line OX is defined as directed toward the vernal equinox,
at which point it intersects the celestial sphere.

The following is an enumeration of the co-ordinates which we have
described in the three systems:--

APPARENT SYSTEM.

Latitudinal Co-ordinate; Altitude or Zenith Distance.
Longitudinal " Azimuth.

EQUATORIAL SYSTEM.

Latitudinal Co-ordinate; Declination or Polar Distance.
Longitudinal " Right Ascension.

ECLIPTIC SYSTEM.

Latitudinal Co-ordinate; Latitude or Ecliptic Polar Distance.
Longitudinal " Longitude.

_Relation of the Diurnal Motion to Spherical Co-ordinates._--The
vertical line at any place being the fundamental axis of the apparent
system of co-ordinates, this system rotates with the earth, and so
seems to us as fixed. The other two systems, including the vernal
equinox, are fixed on the celestial sphere, and so seem to us to
perform a diurnal revolution from east towards west. Regarding the
period of the revolution as 24 hours, the apparent motion goes on at
the rate of 15 deg. per hour. Here we have to make a distinction of
fundamental importance between the diurnal motions of the sun and of
the stars. Owing to the unceasing apparent motion of the sun toward
the east, the interval between two passages of the same star over the
meridian is nearly four minutes less than the interval between
consecutive passages of the sun. The latter is the measure of the day
as used in civil life. In astronomical practice is introduced a day,
termed "sidereal," determined, not by the diurnal revolution of the
sun, but of the stars. The year, which comprises 365.25 solar days,
contains 366.25 sidereal days. The latter are divided into sidereal
hours, minutes and seconds as the solar day is. The conception of a
revolution through 360 deg. in 24 hours is applicable to each case.
The sun apparently moves at the rate of 15 deg. in a solar hour; the
stars at the rate of 15 deg. in a sidereal hour. The latter motion
leads to the use, in astronomical practice, of time instead of angle,
as the unit in which the right ascensions are to be expressed.
Considering the position of the vernal equinox, and also of a star on
the celestial sphere, it will be seen that the interval between the
transits of these two points across the meridian may be used to
measure the right ascension of a star, since the latter amounts to 15
deg. for every sidereal hour of this interval. For example, if the
right ascension of a star is exactly 15 deg., it will pass the
meridian one sidereal hour after the vernal equinox. For the relations
thus arising, and their practical applications, see TIME, MEASUREMENT
OF.

_Theoretical Astronomy._

Theoretical Astronomy is that branch of the science which, making use of the results of astronomical observations as they are supplied by the practical astronomer, investigates the motions of the heavenly bodies. In its most important features it is an offshoot of celestial mechanics, between which and theoretical astronomy no sharp dividing line can be drawn. While it is true that the one is concerned altogether with general theories, it is also true that these theories require developments and modifications to apply them to the numberless problems of astronomy, which we may place in either class.

Among the problems of theoretical astronomy we may assign the first
place to the determination of orbits (q.v.), which is auxiliary to the
prediction of the apparent motions of a planet, satellite or star. The
computations involved in the process, while simple in some cases, are
extremely complex in others. The orbit of a newly-discovered planet or
comet may be computed from three complete observations by well-known
methods in a single day. From the resulting elements of the orbit the
positions of the body from day to day may be computed and tabulated in
an ephemeris for the use of observers. But when definitive results as
to the orbits are required, it is necessary to compute the
perturbations produced by such of the major planets as have affected
the motions of the body. With this complicated process is associated
that of combining numerous observations with a view of obtaining the
best definitive result. Speaking in a general way, we may say that
computations pertaining to the orbital revolutions of double stars, as
well as the bodies of our solar system, are to a greater or less
extent of the classes we have described. The principal modification is
that, up to the present time, stellar astronomy has not advanced so
far that a computation of the perturbations in each case of a system
of stars is either necessary or possible, except in exceptional cases.

_Celestial Mechanics_.

Celestial Mechanics is, strictly speaking, that branch of applied mathematics which, by deductive processes, derives the laws of motion of the heavenly bodies from their gravitation towards each other, or from the mutual action of the parts which form them. The science had its origin in the demonstration by Sir Isaac Newton that Kepler's three laws of planetary motion, and the law of gravitation, in the case of two bodies, could be mutually derived from each other. A body can move round the sun in an elliptic orbit having the sun in its focus, and describing equal areas in equal times, only under the influence of a force directed towards the sun, and varying inversely as the square of the distance from it. Conversely, assuming this law of attraction, it can be shown that the planets will move according to Kepler's laws.

Thus celestial mechanics may be said to have begun with Newton's _Principia_. The development of the science by the successors of Newton, especially Laplace and Lagrange, may be classed among the most striking achievements of the human intellect. The precision with which the path of an eclipse is laid down years in advance cannot but imbue the minds of men with a high sense of the perfection reached by astronomical theories; and the discovery, by purely mathematical processes, of the changes which the orbits and motions of the planets are to undergo through future ages is more impressive the more fully one apprehends the nature of the problem. The purpose of the present article is to convey a general idea of the methods by which the results of celestial mechanics are reached, without entering into those technical details which can be followed only by a trained mathematician. It must be admitted that any intelligent comprehension of the subject requires at least a grasp of the fundamental conceptions of analytical geometry and the infinitesimal calculus, such as only one with some training in these subjects can be expected to have. This being assumed, the hope of the writer is that the exposition will afford the student an insight into the theory which may facilitate his orientation, and convey to the general reader with a certain amount of mathematical training a clear idea of the methods by which conclusions relating to it are drawn. The non-mathematical reader may possibly be able to gain some general idea, though vague, of the significance of the subject.

The fundamental hypothesis of the science assumes a system of bodies
in motion, of which the sun and planets may be taken as examples, and
of which each separate body is attracted toward all the others
according to the law of Newton. The motion of each body is then
expressed in the first place by Newton's three laws of motion (see
MOTION, LAWS OF, and MECHANICS). The first step in the process shows
in a striking way the perfection of the analytic method. The
conception of force is, so to speak, eliminated from the conditions of
the problem, which is reduced to one of pure kinematics. At the
outset, the position of each body, considered as a material particle,
is defined by reference to a system of co-ordinate axes, and not by
any verbal description. Differential equations which express the
changes of the co-ordinates are then constructed. The process of
discovering the laws of motion of the particle then consists in the
integration of these equations. Such equations can be formed for a
system of any number of bodies, but the process of integration in a
rigorous form is possible only to a limited extent or in special
cases.

The problems to be treated are of two classes. In one, the bodies are
regarded as material particles, no account being taken of their
dimensions. The earth, for example, may be regarded as a particle
attracted by another more massive particle, the sun. In the other
class of problems, the relative motion of the different parts of the
separate bodies is considered; for example, the rotation of the earth
on its axis, and the consequences of the fact that those parts of a
body which are nearer to another body are more strongly attracted by
it. Beginning with the first branch of the subject, the fundamental
ideas which it is our purpose to convey are embodied in the simple
case of only two bodies, which we may call the sun and a planet. In
this case the two bodies really revolve round their common centre of
gravity; but a very slight modification of the equations of motion
reduces them to the relative motion of the planet round the sun,
regarding the moving centre of the latter as the origin of
co-ordinates. The motion of this centre, which arises from the
attraction of the planet on the sun, need not be considered.

In the actual problems of celestial mechanics three co-ordinates
necessarily enter, leading to three differential equations and six
equations of solution. But the general principles of the problem are
completely exemplified with only two bodies, in which case the motion
takes place in a fixed plane. By taking this plane, which is that of
the orbit in which the planet performs its revolution, as the plane of
xy, we have only two co-ordinates to consider. Let us use the
following notation:

x, y, the co-ordinates of the planet relative to the sun as the origin.

M, m, the masses of the attracting bodies, sun and planet.

r, the distance apart of the two bodies, or the radius vector of m
relative to M. This last quantity is analytically defined by the
equation--

r^2 = x^2 + y^2

t, the time, reckoned from any epoch we choose.

The differential equations which completely determine the changes in
the co-ordinates x and y, or the motion of m relative to M, are:--

d^2x (M + m)x
---- = - --------
dt^2 r^3

d^2y (M + m)y
---- = - --------
dt^2 r^3

These formulae are worthy of special attention. They are the
expression in the language of mathematics of Newton's first two laws
of motion. Their statement in this language may be regarded as
perfect, because it completely and unambiguously expresses the naked
phenomena of the motion. The equations do this without expressing any
conception, such as that of force, not associated with the actual
phenomena. Moreover, as a third advantage, these expressions are
entirely free from those difficulties and ambiguities which are met
with in every attempt to express the laws of motion in ordinary
language. They afford yet another great advantage in that the
derivation of the results requires only the analytic operations of the
infinitesimal calculus.

The power and spirit of the analytic method will be appreciated by
showing how it expresses the relations of motion as they were
conceived geometrically by Newton and Kepler. It is quite evident that
Kepler's laws do not in themselves enable us to determine the actual
motion of the planets. We must have, in addition, in the case of each
special planet, certain specific facts, viz. the axes and eccentricity
of the ellipse, and the position of the plane in which it lies.
Besides these, we must have given the position of the planet in the
orbit at some specified moment. Having these data, the position of the
planet at any other time may be geometrically constructed by Kepler's
laws. The third law enables us to compute the time taken by the radius
vector to sweep over the entire area of the orbit, which is identical
with the time of revolution. The problem of constructing successive
radii vectores, the angles of which are measured off from the radius
vector of the body at the original given position, is then a geometric
one, known as Kepler's problem.

In the analytic process these specific data, called elements of the
orbit, appear as arbitrary constants, introduced by the process of
integration. In a case like the present one, where there are two
differential equations of the second order, there will be four such
constants. The result of the integration is that the co-ordinates x
and y and their derivatives as to the time, which express the
position, direction of motion and speed of the planet at any moment,
are found as functions of the four constants and of the time. Putting

a, b, c, d,

for the constants, the general form of the solution will be

x = f1(a, b, c, d, t)
y = f2(a, b, c, d, t) (2)

From these may be derived by differentiation as to t the velocities

dx/dt = f'1(a, b, c, d, t) = x'
dy/dt = f'2(a, b, c, d, t) = y' (3)

The symbols x' and y' are used for brevity to mean the velocities
expressed by the differential coefficients. The arbitrary constants,
a, b, c and d, are the elements of the orbit, or any quantities from
which these elements can be obtained. We note that, in the actual
process of integration, no geometric construction need enter.

Let us next consider the problem in another form. Conceive that
instead of the orbit of the planet, there is given a position P (fig.
2), through which the planet passed at an assigned moment, with a
given velocity, and in a given direction, represented by the
arrowhead. Logically these data completely determine the orbit in
which the planet shall move, because there is only one such orbit
passing through P, a planet moving in which would have the given
speed. It follows that the elements of the orbit admit of
determination when the co-ordinates of the planet at an assigned
moment and their derivatives as to time are given. Analytically the
elements are determined from these data by solving the four equations
just given, regarding a, b, c and d as unknown quantities, and x, y,
x', y' and t as given quantities. The solution of these equations
would lead to expressions of the form

a = [phi]1(x, y, x', y', t)
b = [phi]2(x, y, x', y', t) (4)
&c. &c.

one for each of the elements.

The general equations expressing the motion of a planet considered as
a material particle round a centre of attraction lead to theorems the
more interesting of which will now be enunciated.

(1) The motion of such a planet may take place not only in an ellipse
but in any curve of the second order; an ellipse, hyperbola, or
parabola, the latter being the bounding curve between the other two. A
body moving in a parabola or hyperbola would recede indefinitely from
its centre of motion and never return to it. The ellipse is therefore
the only closed orbit.

(2) The motion takes place in accord with Kepler's laws, enunciated
elsewhere.

(3) _Whewell's theorem_: if a point R be taken at a distance from the
sun equal to the major axis of the orbit of a planet and, therefore,
at double the mean distance of the planet, the speed of the latter at
any point is equal to the speed which a body would acquire by falling
from the point R to the actual position of the planet. The speed of
the latter may, therefore, be expressed as a function of its radius
vector at the moment and of the major axis of its orbit without
introducing any other elements into the expression. Another corollary
is that in the case of a body moving in a parabolic orbit the velocity
at any moment is that which would be acquired by the body in falling
from an infinite distance to the place it occupies at the moment.

(4) If a number of bodies are projected from any point in space with
the same velocity, but in various directions, and subjected only to
the attraction of the sun, they will all return to the point of
projection at the same moment, although the orbits in which they move
may be ever so different.

(5) At each distance from the sun there is a certain velocity which a
body would have if it moved in a circular orbit at that distance. If
projected with this velocity in any direction the point of projection
will be at the end of the minor axis of the orbit, because this is the
only point of an ellipse of which the distance from the focus is equal
to the semi-major axis of the curve, and therefore the only point at
which the distance of the body from the sun is equal to its mean
distance.

(6) The relation between the periodic time of a planet and its mean
distance, approximately expressed by Kepler's third law, follows very
simply from the laws of centrifugal force. It is an elementary
principle of mechanics that this force varies directly as the product
of the distance of the moving body from the centre of motion into the
square of its angular velocity. When bodies revolve at different
distances around a centre, their velocities must be such that the
centrifugal force of each shall be balanced by the attraction of the
central mass, and therefore vary inversely as the square of the
distance. If M is the central mass, n the angular velocity, and a the
distance, the balance of the two forces is expressed by the equation

an^2 = M/a^2,

whence a^3n^2 = M, a constant.

The periodic time varying inversely as n, this equation expresses
Kepler's third law. This reasoning tacitly supposes the orbit to be a
circle of radius a, and the mass of the planet to be negligible. The
rigorous relation is expressed by a slight modification of the law.
Putting M and m for the respective masses of the sun and planet, a for
the semi-major axis of the orbit, and n for the mean angular motion in
unit of time, the relation then is

a^3n^2 = M + m.

What is noteworthy in this theorem is that this relation depends only
on the sum of the masses. It follows, therefore, that were any portion
of the mass of the sun taken from it, and added to the planet, the
relation would be unchanged. Kepler's third law therefore expresses
the fact that the mass of the sun is the same for all the planets, and
deviates from the truth only to the extent that the masses of the
latter differ from each other by quantities which are only a small
fraction of the mass of the sun.

_Problem of Three Bodies._--As soon as the general law of gravitation
was fully apprehended, it became evident that, owing to the attraction
of each planet upon all the others, the actual motion of the planets
must deviate from their motion in an ellipse according to Kepler's
laws. In the _Principia_ Newton made several investigations to
determine the effects of these actions; but the geometrical method
which he employed could lead only to rude approximations. When the
subject was taken up by the continental mathematicians, using the
analytical method, the question naturally arose whether the motions of
three bodies under their mutual attraction could not be determined
with a degree of rigour approximating to that with which Newton had
solved the problem of two bodies. Thus arose the celebrated "problem
of three bodies." Investigation soon showed that certain integrals
expressing relations between the motions not only of three but of any
number of bodies could be found. These were:--

First, the law of the conservation of the centre of gravity. This
expresses the general fact that whatever be the number of the bodies
which act upon each other, their motions are so related that the
centre of gravity of the entire system moves in a straight line with a
constant velocity. This is expressed in three equations, one for each
of the three rectangular co-ordinates.

Secondly, the law of conservation of areas. This is an extension of
Kepler's second law. Taking as the radius vector of each body the line
from the body to the common centre of gravity of all, the sum of the
products formed by multiplying each area described, by the mass of the
body, remains a constant. In the language of theoretical mechanics,
the moment of momentum of the entire system is a constant quantity.
This law is also expressed in three equations, one for each of the
three planes on which the areas are projected.

Thirdly, the entire _vis viva_ of the system or, as it is now called,
the energy, which is obtained by multiplying the mass of each body
into half the square of its velocity, is equal to the sum of the
quotients formed by dividing the product of every pair of the masses,
taken two and two, by their distance apart, with the addition of a
constant depending on the original conditions of the system. In the
language of algebra putting m1, m2, m3, &c. for the masses of the
bodies, r_1.2, r_1.3, r_2.3, &c. for their mutual distances apart;
v1, v2, v3, &c., for the velocities with which they are moving at any
moment; these quantities will continually satisfy the equation

m1m2 m1m3 m2m3
1/2(m1[v1]^2 + m2[v2]^2 + ...) = ----- + ----- + ----- + ... + a constant.
r_1.2 r_1.3 r_2.3