CHAPTER XI._A Short History of Astronomy

GRAVITATIONAL ASTRONOMY IN THE 18TH CENTURY.

“Astronomy, considered in the most general way, is a great problem

of mechanics, the arbitrary data of which are the elements of the

celestial movements; its solution depends both on the accuracy of

observations and on the perfection of analysis.”

LAPLACE, Preface to the _Mécanique Céleste_.

228. The solar system, as it was known at the beginning of the 18th

century, contained 18 recognised members: the sun, six planets, ten

satellites (one belonging to the earth, four to Jupiter, and five to

Saturn), and Saturn’s ring.

Comets were known to have come on many occasions into the region of

space occupied by the solar system, and there were reasons to believe

that one of them at least (chapter X., § 200) was a regular visitor;

they were, however, scarcely regarded as belonging to the solar system,

and their action (if any) on its members was ignored, a neglect which

subsequent investigation has completely justified. Many thousands of

fixed stars had also been observed, and their places on the celestial

sphere determined; they were known to be at very great though unknown

distances from the solar system, and their influence on it was regarded

as insensible.

The motions of the 18 members of the solar system were tolerably

well known; their actual distances from one another had been roughly

estimated, while the _proportions_ between most of the distances were

known with considerable accuracy. Apart from the entirely anomalous

ring of Saturn, which may for the present be left out of consideration,

most of the bodies of the system were known from observation to be

nearly spherical in form, and the rest were generally supposed to be so

also.

Newton had shewn, with a considerable degree of probability, that these

bodies attracted one another according to the law of gravitation; and

there was no reason to suppose that they exerted any other important

influence on one another’s motions.[130]

The problem which presented itself, and which may conveniently be

called Newton’s problem, was therefore:—

_Given these 18 bodies, and their positions and motions at any time,

to deduce from their mutual gravitation by a process of mathematical

calculation their positions and motions at any other time; and to shew

that these agree with those actually observed._

Such a calculation would necessarily involve, among other quantities,

the masses of the several bodies; it was evidently legitimate to assume

these at will in such a way as to make the results of calculation agree

with those of observation. If this were done successfully the masses

would thereby be determined. In the same way the commonly accepted

estimates of the dimensions of the solar system and of the shapes of

its members might be modified in any way not actually inconsistent with

direct observation.

The general problem thus formulated can fortunately be reduced to

somewhat simpler ones.

Newton had shewn (chapter IX., § 182) that an ordinary sphere

attracted other bodies and was attracted by them, as if its mass were

concentrated at its centre; and that the effects of deviation from

a spherical form became very small at a considerable distance from

the body. Hence, except in special cases, the bodies of the solar

system could be treated as spheres, which could again be regarded as

concentrated at their respective centres. It will be convenient for the

sake of brevity to assume for the future that all “bodies” referred to

are of this sort, unless the contrary is stated or implied. The effects

of deviations from spherical form could then be treated separately

when required, as in the cases of precession and of other motions of a

planet or satellite about its centre, and of the corresponding action

of a non-spherical planet on its satellites; to this group of problems

belongs also that of the tides and other cases of the motion of parts

of a body of any form relative to the rest.

Again, the solar system happens to be so constituted that each body’s

motion can be treated as determined primarily by one other body only.

A planet, for example, moves nearly as if no other body but the sun

existed, and the moon’s motion relative to the earth is roughly the

same as if the other bodies of the solar system were non-existent.

The problem of the motion of two mutually gravitating spheres was

completely solved by Newton, and was shewn to lead to Kepler’s first

two laws. Hence each body of the solar system could be regarded as

moving nearly in an ellipse round some one body, but as slightly

disturbed by the action of others. Moreover, by a general mathematical

principle applicable in problems of motion, the effect of a number of

small disturbing causes acting conjointly is nearly the same as that

which results from adding together their separate effects. Hence each

body could, without great error, be regarded as disturbed by one body

at a time; the several disturbing effects could then be added together,

and a fresh calculation could be made to further diminish the error.

The kernel of Newton’s problem is thus seen to be a special case of the

so-called =problem of three bodies=, viz.:—

_Given at any time the positions and motions of three mutually

gravitating bodies, to determine their positions and motions at any

other time._

Even this apparently simple problem in its general form entirely

transcends the powers, not only of the mathematical methods of the

early 18th century, but also of those that have been devised since.

Certain special cases have been solved, so that it has been shewn to be

possible to suppose three bodies initially moving in such a way that

their future motion can be completely determined. But these cases do

not occur in nature.

In the case of the solar system the problem is simplified, not only by

the consideration already mentioned that one of the three bodies can

always be regarded as exercising only a small influence on the relative

motion of the other two, but also by the facts that the orbits of

the planets and satellites do not differ much from circles, and that

the planes of their orbits are in no case inclined at large angles

to any one of them, such as the ecliptic; in other words, that the

eccentricities and inclinations are small quantities.

Thus simplified, the problem has been found to admit of solutions of

considerable accuracy by methods of approximation.[131]

In the case of the system formed by the sun, earth, and moon, the

characteristic feature is the great distance of the sun, which is

the disturbing body, from the other two bodies; in the case of the

sun and two planets, the enormous mass of the sun as compared with

the disturbing planet is the important factor. Hence the methods of

treatment suitable for the two cases differ, and two substantially

distinct branches of the subject, =lunar theory= and =planetary

theory=, have developed. The problems presented by the motions of the

satellites of Jupiter and Saturn, though allied to those of the lunar

theory, differ in some important respects, and are usually treated

separately.

229. As we have seen, Newton made a number of important steps towards

the solution of his problem, but little was done by his successors

in his own country. On the Continent also progress was at first very

slow. The _Principia_ was read and admired by most of the leading

mathematicians of the time, but its principles were not accepted,

and Cartesianism remained the prevailing philosophy. A forward step

is marked by the publication by the Paris Academy of Sciences in

1720 of a memoir written by the _Chevalier de Louville_ (1671-1732)

on the basis of Newton’s principles; ten years later the Academy

awarded a prize to an essay on the planetary motions written by _John

Bernouilli_ (1667-1748) on Cartesian principles, a Newtonian essay

being put second. In 1732 Maupertuis (chapter X., § 221) published

a treatise on the figure of the earth on Newtonian lines, and the

appearance six years later of Voltaire’s extremely readable _Éléments

de la Philosophie de Newton_ had a great effect in popularising the

new ideas. The last official recognition of Cartesianism in France

seems to have been in 1740, when the prize offered by the Academy for

an essay on the tides was shared between a Cartesian and three eminent

Newtonians (§ 230).

The rapid development of gravitational astronomy that ensued between

this time and the beginning of the 19th century was almost entirely

the work of five great Continental mathematicians, Euler, Clairaut,

D’Alembert, Lagrange, and Laplace, of whom the eldest was born in 1707

and the youngest died in 1827, within a month of the centenary of

Newton’s death. Euler was a Swiss, Lagrange was of Italian birth but

French by extraction and to a great extent by adoption, and the other

three were entirely French. France therefore during nearly the whole of

the 18th century reigned supreme in gravitational astronomy, and has

not lost her supremacy even to-day, though during the present century

America, England, Germany, Italy, and other countries have all made

substantial contributions to the subject.

It is convenient to consider first the work of the three first-named

astronomers, and to treat later Lagrange and Laplace, who carried

gravitational astronomy to a decidedly higher stage of development than

their predecessors.

230. _Leonhard Euler_ was born at Basle in 1707, 14 years later

than Bradley and six years earlier than Lacaille. He was the

son of a Protestant minister who had studied mathematics under

_James Bernouilli_ (1654-1705), the first of a famous family of

mathematicians. Leonhard Euler himself was a favourite pupil of John

Bernouilli (the younger brother of James), and was an intimate friend

of his two sons, one of whom, _Daniel_ (1700-1782), was not only a

distinguished mathematician like his father and uncle, but was also

the first important Newtonian outside Great Britain. Like so many

other astronomers, Euler began by studying theology, but was induced

both by his natural tastes and by the influence of the Bernouillis to

turn his attention to mathematics. Through the influence of Daniel

Bernouilli, who had recently been appointed to a professorship at

St. Petersburg, Euler received and accepted an invitation to join the

newly created Academy of Sciences there (1727). This first appointment

carried with it a stipend, and the duties were the general promotion

of science; subsequently Euler undertook more definite professorial

work, but most of his energy during the whole of his career was devoted

to writing mathematical papers, the majority of which were published

by the St. Petersburg Academy. Though he took no part in politics,

Russian autocracy appears to have been oppressive to him, reared as

he had been among Swiss and Protestant surroundings; and in 1741 he

accepted an invitation from Frederick the Great, a despot of a less

pronounced type, to come to Berlin, and assist in reorganising the

Academy of Sciences there. On being reproached one day by the Queen for

his taciturn and melancholy demeanour, he justified his silence on the

ground that he had just come from a country where speech was liable

to lead to hanging;[132] but notwithstanding this frank criticism he

remained on good terms with the Russian court, and continued to draw

his stipend as a member of the St. Petersburg Academy and to contribute

to its Transactions. Moreover, after 25 years spent at Berlin, he

accepted a pressing invitation from the Empress Catherine II. and

returned to Russia (1766).

He had lost the use of one eye in 1735, a disaster which called from

him the remark that he would henceforward have less to distract him

from his mathematics; the second eye went soon after his return

to Russia, and with the exception of a short time during which an

operation restored the partial use of one eye he remained blind till

the end of his life. But this disability made little difference to his

astounding scientific activity; and it was only after nearly 17 years

of blindness that as a result of a fit of apoplexy “he ceased to live

and to calculate” (1783).

Euler was probably the most versatile as well as the most prolific of

mathematicians of all time. There is scarcely any branch of modern

analysis to which he was not a large contributor, and his extraordinary

powers of devising and applying methods of calculation were employed

by him with great success in each of the existing branches of applied

mathematics; problems of abstract dynamics, of optics, of the motion

of fluids, and of astronomy were all in turn subjected to his analysis

and solved. The extent of his writings is shewn by the fact that, in

addition to several books, he wrote about 800 papers on mathematical

and physical subjects; it is estimated that a complete edition of his

works would occupy 25 quarto volumes of about 600 pages each.

Euler’s first contribution to astronomy was an essay on the tides which

obtained a share of the Academy prize for 1740 already referred to,

Daniel Bernouilli and Maclaurin (chapter X., § 196) being the other two

Newtonians. The problem of the tides was, however, by no means solved

by any of the three writers.

He gave two distinct solutions of the problem of three bodies in a form

suitable for the lunar theory, and made a number of extremely important

and suggestive though incomplete contributions to planetary theory. In

both subjects his work was so closely connected with that of Clairaut

and D’Alembert that it is more convenient to discuss it in connection

with theirs.

231. _Alexis Claude Clairaut_, born at Paris in 1713, belongs to the

class of precocious geniuses. He read the Infinitesimal Calculus and

Conic Sections at the age of ten, presented a scientific memoir to the

Academy of Sciences before he was 13, and published a book containing

some important contributions to geometry when he was 18, thereby

winning his admission to the Academy.

Shortly afterwards he took part in Maupertuis’ expedition to Lapland

(chapter X., § 221), and after publishing several papers of minor

importance produced in 1743 his classical work on the figure of the

earth. In this he discussed in a far more complete form than either

Newton or Maclaurin the form which a rotating body like the earth

assumes under the influence of the mutual gravitation of its parts,

certain hypotheses of a very general nature being made as to the

variations of density in the interior; and deduced formulae for the

changes in different latitudes of the acceleration due to gravity,

which are in satisfactory agreement with the results of pendulum

experiments.

Although the subject has since been more elaborately and more

generally treated by later writers, and a good many additions have been

made, few if any results of fundamental importance have been added to

those contained in Clairaut’s book.

He next turned his attention to the problem of three bodies, obtained

a solution suitable for the moon, and made some progress in planetary

theory.

[Illustration: FIG. 80.—The path of Halley’s comet.]

Halley’s comet (chapter X., § 200) was “due” about 1758; as the time

approached Clairaut took up the task of computing the perturbations

which it would probably have experienced since its last appearance,

owing to the influence of the two great planets, Jupiter and Saturn,

close to both of which it would have passed. An extremely laborious

calculation shewed that the comet would have been retarded about 100

days by Saturn and about 518 days by Jupiter, and he accordingly

announced to the Academy towards the end of 1758 that the comet might

be expected to pass its perihelion (the point of its orbit nearest

the sun, P in fig. 80) about April 13th of the following year, though

owing to various defects in his calculation there might be an error

of a month either way. The comet was anxiously watched for by the

astronomical world, and was actually discovered by an amateur, _George

Palitzsch_ (1723-1788) of Saxony, on Christmas Day, 1758; it passed its

perihelion just a month and a day before the time assigned by Clairaut.

Halley’s brilliant conjecture was thus justified; a new member was

added to the solar system, and hopes were raised—to be afterwards

amply fulfilled—that in other cases also the motions of comets might

be reduced to rule, and calculated according to the same principles as

those of less erratic bodies. The superstitions attached to comets were

of course at the same time still further shaken.

Clairaut appears to have had great personal charm and to have been a

conspicuous figure in Paris society. Unfortunately his strength was not

equal to the combined claims of social and scientific labours, and he

died in 1765 at an age when much might still have been hoped from his

extraordinary abilities.[133]

232. _Jean-le-Rond D’Alembert_ was found in 1717 as an infant on the

steps of the church of St. Jean-le-Rond in Paris, but was afterwards

recognised, and to some extent provided for, by his father, though

his home was with his foster parents. After receiving a fair school

education, he studied law and medicine, but then turned his attention

to mathematics. He first attracted notice in mathematical circles by

a paper written in 1738, and was admitted to the Academy of Sciences

two years afterwards. His earliest important work was the _Traité de

Dynamique_ (1743), which contained, among other contributions to the

subject, the first statement of a dynamical principle which bears his

name, and which, though in one sense only a corollary from Newton’s

Third Law of Motion, has proved to be of immense service in nearly all

general dynamical problems, astronomical or otherwise. During the next

few years he made a number of contributions to mathematical physics,

as well as to the problem of three bodies; and published in 1749 his

work on precession and nutation, already referred to (chapter X., §

215). From this time onwards he began to give an increasing part of his

energies to work outside mathematics. For some years he collaborated

with Diderot in producing the famous French Encyclopaedia, which began

to appear in 1751, and exercised so great an influence on contemporary

political and philosophic thought. D’Alembert wrote the introduction,

which was read to the _Académie Française_[134] in 1754 on the occasion

of his admission to that distinguished body, as well as a variety of

scientific and other articles. In the later part of his life, which

ended in 1783, he wrote little on mathematics, but published a number

of books on philosophical, literary, and political subjects;[135] as

secretary of the Academy he also wrote obituary notices (_éloges_)

of some 70 of its members. He was thus, in Carlyle’s words, “of

great faculty, especially of great clearness and method; famous in

Mathematics; no less so, to the wonder of some, in the intellectual

provinces of Literature.”

D’Alembert and Clairaut were great rivals, and almost every work of the

latter was severely criticised by the former, while Clairaut retaliated

though with much less zeal and vehemence. The great popular reputation

acquired by Clairaut through his work on Halley’s comet appears to have

particularly excited D’Alembert’s jealousy. The rivalry, though not a

pleasant spectacle, was, however, useful in leading to the detection

and subsequent improvement of various weak points in the work of each.

In other respects D’Alembert’s personal characteristics appear to have

been extremely pleasant. He was always a poor man, but nevertheless

declined magnificent offers made to him by both Catherine II. of

Russia and Frederick the Great of Prussia, and preferred to keep his

independence, though he retained the friendship of both sovereigns and

accepted a small pension from the latter. He lived extremely simply,

and notwithstanding his poverty was very generous to his foster-mother,

to various young students, and to many others with whom he came into

contact.

233. Euler, Clairaut, and D’Alembert all succeeded in obtaining

independently and nearly simultaneously solutions of the problem of

three bodies in a form suitable for lunar theory. Euler published in

1746 some rather imperfect Tables of the Moon, which shewed that he

must have already obtained his solution. Both Clairaut and D’Alembert

presented to the Academy in 1747 memoirs containing their respective

solutions, with applications to the moon as well as to some planetary

problems. In each of these memoirs occurred the same difficulty which

Newton had met with: the calculated motion of the moon’s apogee was

only about half the observed result. Clairaut at first met this

difficulty by assuming an alteration in the law of gravitation, and

got a result which seemed to him satisfactory by assuming gravitation

to vary partly as the inverse square and partly as the inverse cube of

the distance.[136] Euler also had doubts as to the correctness of the

inverse square. Two years later, however (1749), on going through his

original calculation again, Clairaut discovered that certain terms,

which had appeared unimportant at the beginning of the calculation and

had therefore been omitted, became important later on. When these were

taken into account, the motion of the apogee as deduced from theory

agreed very nearly with that observed. This was the first of several

cases in which a serious discrepancy between theory and observation

has at first discredited the law of gravitation, but has subsequently

been explained away, and has thereby given a new verification of its

accuracy. When Clairaut had announced his discovery, Euler arrived by a

fresh calculation at substantially the same result, while D’Alembert by

carrying the approximation further obtained one that was slightly more

accurate. A fresh calculation of the motion of the moon by Clairaut won

the prize on the subject offered by the St. Petersburg Academy, and

was published in 1752, with the title _Théorie de la Lune_. Two years

later he published a set of lunar tables, and just before his death

(1765) he brought out a revised edition of the _Théorie de la Lune_ in

which he embodied a new set of tables.

D’Alembert followed his paper of 1747 by a complete lunar theory

(with a moderately good set of tables), which, though substantially

finished in 1751, was only published in 1754 as the first volume of his

_Recherches sur différens points importans du système du Monde_. In

1756 he published an improved set of tables, and a few months afterward

a third volume of _Recherches_ with some fresh developments of the

theory. The second volume of his _Opuscules Mathématiques_ (1762)

contained another memoir on the subject with a third set of tables,

which were a slight improvement on the earlier ones.

Euler’s first lunar theory (_Theoria Motuum Lunae_) was published in

1753, though it had been sent to the St. Petersburg Academy a year or

two earlier. In an appendix[137] he points out with characteristic

frankness the defects from which his treatment seems to him to suffer,

and suggests a new method of dealing with the subject. It was on this

theory that Tobias Mayer based his tables, referred to in the preceding

chapter (§ 226). Many years later Euler devised an entirely new way

of attacking the subject, and after some preliminary papers dealing

generally with the method and with special parts of the problem, he

worked out the lunar theory in great detail, with the help of one of

his sons and two other assistants, and published the whole, together

with tables, in 1772. He attempted, but without success, to deal in

this theory with the secular acceleration of the mean motion which

Halley had detected (chapter X., § 201).

In any mathematical treatment of an astronomical problem some data

have to be borrowed from observation, and of the three astronomers

Clairaut seems to have been the most skilful in utilising observations,

many of which he obtained from Lacaille. Hence his tables represented

the actual motions of the moon far more accurately than those of

D’Alembert, and were even superior in some points to those based

on Euler’s very much more elaborate second theory; Clairaut’s last

tables were seldom in error more than 1-1∕2′, and would hence serve

to determine the longitude to within about 3∕4°. Clairaut’s tables

were, however, never much used, since Tobias Mayer’s as improved by

Bradley were found in practice to be a good deal more accurate; but

Mayer borrowed so extensively from observation that his formulae cannot

be regarded as true deductions from gravitation in the same sense in

which Clairaut’s were. Mathematically Euler’s second theory is the most

interesting and was of the greatest importance as a basis for later

developments. The most modern lunar theory[138] is in some sense a

return to Euler’s methods.

234. Newton’s lunar theory may be said to have given a _qualitative_

account of the lunar inequalities known by observation at the time when

the _Principia_ was published, and to have indicated others which had

not yet been observed. But his attempts to explain these irregularities

_quantitatively_ were only partially successful.

Euler, Clairaut, and D’Alembert threw the lunar theory into an

entirely new form by using analytical methods instead of geometrical;

one advantage of this was that by the expenditure of the necessary

labour calculations could in general be carried further when required

and lead to a higher degree of accuracy. The result of their more

elaborate development was that—with one exception—the inequalities

known from observation were explained with a considerable degree of

accuracy quantitatively as well as qualitatively; and thus tables, such

as those of Clairaut, based on theory, represented the lunar motions

very closely. The one exception was the secular acceleration: we have

just seen that Euler failed to explain it; D’Alembert was equally

unsuccessful, and Clairaut does not appear to have considered the

question.

235. The chief inequalities in planetary motion which observation had

revealed up to Newton’s time were the forward motion of the apses of

the earth’s orbit and a very slow diminution in the obliquity of the

ecliptic. To these may be added the alterations in the rates of motion

of Jupiter and Saturn discovered by Halley (chapter X., § 204).

Newton had shewn generally that the perturbing effect of another planet

would cause displacements in the apses of any planetary orbit, and

an alteration in the relative positions of the planes in which the

disturbing and disturbed planet moved; but he had made no detailed

calculations. Some effects of this general nature, in addition to those

already known, were, however, indicated with more or less distinctness

as the result of observation in various planetary tables published

between the date of the _Principia_ and the middle of the 18th century.

The irregularities in the motion of the earth, shewing themselves as

irregularities in the apparent motion of the sun, and those of Jupiter

and Saturn, were the most interesting and important of the planetary

inequalities, and prizes for essays on one or another subject were

offered several times by the Paris Academy.

The perturbations of the moon necessarily involved—by the principle of

action and reaction—corresponding though smaller perturbations of the

earth; these were discussed on various occasions by Clairaut and Euler,

and still more fully by D’Alembert.

In Clairaut’s paper of 1747 (§ 233) he made some attempt to apply his

solution of the problem of three bodies to the case of the sun, earth,

and Saturn, which on account of Saturn’s great distance from the sun

(nearly ten times that of the earth) is the planetary case most like

that of the earth, moon, and sun (cf. § 228).

Ten years later he discussed in some detail the perturbations of the

earth due to Venus and to the moon. This paper was remarkable as

containing the first attempt to estimate masses of celestial bodies

by observation of perturbations due to them. Clairaut applied this

method to the moon and to Venus, by calculating perturbations in the

earth’s motion due to their action (which necessarily depended on their

masses), and then comparing the results with Lacaille’s observations

of the sun. The mass of the moon was thus found to be about 1∕67

and that of Venus 2∕3 that of the earth; the first result was a

considerable improvement on Newton’s estimate from tides (chapter IX.,

§ 189), and the second, which was entirely new, previous estimates

having been merely conjectural, is in tolerable agreement with modern

measurements.[139] It is worth noticing as a good illustration of the

reciprocal influence of observation and mathematical theory that, while

Clairaut used Lacaille’s observations for his theory, Lacaille in turn

used Clairaut’s calculations of the perturbations of the earth to

improve his tables of the sun published in 1758.

Clairaut’s method of solving the problem of three bodies was also

applied by _Joseph Jérôme Le François Lalande_ (1732-1807), who is

chiefly known as an admirable populariser of astronomy but was also an

indefatigable calculator and observer, to the perturbations of Mars by

Jupiter, of Venus by the earth, and of the earth by Mars, but with only

moderate success.

D’Alembert made some progress with the general treatment of planetary

perturbations in the second volume of his _Recherches_, and applied his

methods to Jupiter and Saturn.

236. Euler carried the general theory a good deal further in a series

of papers beginning in 1747. He made several attempts to explain

the irregularities of Jupiter and Saturn, but never succeeded in

representing the observations satisfactorily. He shewed, however, that

the perturbations due to the other planets would cause the earth’s

apse line to advance about 13″ annually, and the obliquity of the

ecliptic to diminish by about 48″ annually, both results being in fair

accordance both with observations and with more elaborate calculations

made subsequently. He indicated also the existence of various other

planetary irregularities, which for the most part had not previously

been observed.

In an essay to which the Academy awarded a prize in 1756, but which was

first published in 1771, he developed with some completeness a method

of dealing with perturbations which he had indicated in his lunar

theory of 1753. As this method, known as that of the =variation of the

elements= or =parameters=, played a very important part in subsequent

researches, it may be worth while to attempt to give a sketch of it.

If perturbations are ignored, a planet can be regarded as moving in an

ellipse with the sun in one focus. The size and shape of the ellipse

can be defined by the length of its axis and by the eccentricity; the

plane in which the ellipse is situated is determined by the position of

the line, called the line of =nodes=, in which it cuts a fixed plane,

usually taken to be the ecliptic, and by the inclination of the two

planes. When these four quantities are fixed, the ellipse may still

turn about its focus in its own plane, but if the direction of the apse

line is also fixed the ellipse is completely determined. If, further,

the position of the planet in its ellipse at any one time is known, the

motion is completely determined and its position at any other time can

be calculated. There are thus six quantities known as =elements= which

completely determine the motion of a planet not subject to perturbation.

When perturbations are taken into account, the path described by

a planet in any one revolution is no longer an ellipse, though it

differs very slightly from one; while in the case of the moon the

deviations are a good deal greater. But if the motions of a planet at

two widely different epochs are compared, though on each occasion the

path described is very nearly an ellipse, the ellipses differ in some

respects. For example, between the time of Ptolemy (A.D. 150) and that

of Euler the direction of the apse line of the earth’s orbit altered by

about 5°, and some of the other elements also varied slightly. Hence

in dealing with the motion of a planet through a long period of time

it is convenient to introduce the idea of an elliptic path which is

gradually changing its position and possibly also its size and shape.

One consequence is that the actual path described in the course of a

considerable number of revolutions is a curve no longer bearing much

resemblance to an ellipse. If, for example, the apse line turns round

uniformly while the other elements remain unchanged, the path described

is like that shewn in the figure.

[Illustration: FIG. 81.—A varying ellipse.]

Euler extended this idea so as to represent any perturbation of a

planet, whether experienced in the course of one revolution or in a

longer time, by means of changes in an elliptic orbit. For wherever a

planet may be and whatever (within certain limits[140]) be its speed

or direction of motion some ellipse can be found, having the sun in

one focus, such that the planet can be regarded as moving in it for a

short time. Hence as the planet describes a perturbed orbit it can be

regarded as moving at any instant in an ellipse, which, however, is

continually altering its position or other characteristics. Thus the

problem of discussing the planet’s motion becomes that of determining

the elements of the ellipse which represents its motion at any time.

Euler shewed further how, when the position of the perturbing planet

was known, the corresponding rates of change of the elements of the

varying ellipse could be calculated, and made some progress towards

deducing from these data the actual elements; but he found the

mathematical difficulties too great to be overcome except in some of

the simpler cases, and it was reserved for the next generation of

mathematicians, notably Lagrange, to shew the full power of the method.

237. _Joseph Louis Lagrange_ was born at Turin in 1736, when Clairaut

was just starting for Lapland and D’Alembert was still a child; he was

descended from a French family three generations of which had lived in

Italy. He shewed extraordinary mathematical talent, and when still a

mere boy was appointed professor at the Artillery School of his native

town, his pupils being older than himself. A few years afterwards

he was the chief mover in the foundation of a scientific society,

afterwards the Turin Academy of Sciences, which published in 1759 its

first volume of Transactions, containing several mathematical articles

by Lagrange, which had been written during the last few years. One of

these[141] so impressed Euler, who had made a special study of the

subject dealt with, that he at once obtained for Lagrange the honour of

admission to the Berlin Academy.

In 1764 Lagrange won the prize offered by the Paris Academy for an

essay on the libration of the moon. In this essay he not only gave

the first satisfactory, though still incomplete, discussion of the

librations (chapter vi., § 133) of the moon due to the non-spherical

forms of both the earth and moon, but also introduced an extremely

general method of treating dynamical problems,[142] which is the basis

of nearly all the higher branches of dynamics which have been developed

up to the present day.

Two years later (1766) Frederick II., at the suggestion of D’Alembert,

asked Lagrange to succeed Euler (who had just returned to St.

Petersburg) as the head of the mathematical section of the Berlin

Academy, giving as a reason that the greatest king in Europe wished to

have the greatest mathematician in Europe at his court. Lagrange

accepted this magnificently expressed invitation and spent the next 21

years at Berlin.

[Illustration: LAGRANGE.]

During this period he produced an extraordinary series of papers on

astronomy, on general dynamics, and on a variety of subjects in pure

mathematics. Several of the most important of the astronomical papers

were sent to Paris and obtained prizes offered by the Academy; most of

the other papers—about 60 in all—were published by the Berlin Academy.

During this period he wrote also his great _Mécanique Analytique_, one

of the most beautiful of all mathematical books, in which he developed

fully the general dynamical ideas contained in the earlier paper on

libration. Curiously enough he had great difficulty in finding a

publisher for his masterpiece, and it only appeared in 1788 in Paris.

A year earlier he had left Berlin in consequence of the death of

Frederick, and accepted an invitation from Louis XVI. to join the Paris

Academy. About this time he suffered from one of the fits of melancholy

with which he was periodically seized and which are generally supposed

to have been due to overwork during his career at Turin. It is said

that he never looked at the _Mécanique Analytique_ for two years

after its publication, and spent most of the time over chemistry

and other branches of natural science as well as in non-scientific

pursuits. In 1790 he was made president of the Commission appointed to

draw up a new system of weights and measures, which resulted in the

establishment of the metric system; and the scientific work connected

with this undertaking gradually restored his interest in mathematics

and astronomy. He always avoided politics, and passed through the

Revolution uninjured, unlike his friend Lavoisier the great chemist

and Bailly the historian of astronomy, both of whom were guillotined

during the Terror. He was in fact held in great honour by the various

governments which ruled France up to the time of his death; in 1793 he

was specially exempted from a decree of banishment directed against

all foreigners; subsequently he was made professor of mathematics,

first at the École Normale (1795), the École Polytechnique (1797),

the last appointment being retained till his death in 1813. During

this period of his life he published, in addition to a large number

of papers on astronomy and mathematics, three important books on pure

mathematics,[143] and at the time of his death had not quite finished

a second edition of the _Mécanique Analytique_, the second volume

appearing posthumously.

238. _Pierre Simon Laplace_, the son of a small farmer, was born at

Beaumont in Normandy in 1749, being thus 13 years younger than his

great rival Lagrange. Thanks to the help of well-to-do neighbours,

he was first a pupil and afterwards a teacher at the Military School

of his native town. When he was 18 he went to Paris with a letter of

introduction to D’Alembert, and, when no notice was taken of it, wrote

him a letter on the principles of mechanics which impressed D’Alembert

so much that he at once took interest in the young mathematician and

procured him an appointment at the Military School at Paris. From this

time onwards Laplace lived continuously at Paris, holding various

official positions. His first paper (on pure mathematics) was published

in the Transactions of the Turin Academy for the years 1766-69, and

from this time to the end of his life he produced an uninterrupted

series of papers and books on astronomy and allied departments of

mathematics.

Laplace’s work on astronomy was to a great extent incorporated in his

_Mécanique Céleste_, the five volumes of which appeared at intervals

between 1799 and 1825. In this great treatise he aimed at summing up

all that had been done in developing gravitational astronomy since the

time of Newton. The only other astronomical book which he published

was the _Exposition du Système du Monde_ (1796), one of the most

perfect and charmingly written popular treatises on astronomy ever

published, in which the great mathematician never uses either an

algebraical formula or a geometrical diagram. He published also in 1812

an elaborate treatise on the theory of probability or chance,[144] on

which nearly all later developments of the subject have been based, and

in 1819 a more popular _Essai Philosophique_ on the same subject.

[Illustration: P. S. LAPLACE.]

Laplace’s personality seems to have been less attractive than that of

Lagrange. He was vain of his reputation as a mathematician and not

always generous to rival discoverers. To Lagrange, however, he was

always friendly, and he was also kind in helping young mathematicians

of promise. While he was perfectly honest and courageous in upholding

his scientific and philosophical opinions, his politics bore an

undoubted resemblance to those of the Vicar of Bray, and were professed

by him with great success. He was appointed a member of the Commission

for Weights and Measures, and afterwards of the Bureau des Longitudes,

and was made professor at the École Normale when it was founded. When

Napoleon became First Consul, Laplace asked for and obtained the

post of Home Secretary, but—fortunately for science—was considered

quite incompetent, and had to retire after six weeks (1799)[145]; as

a compensation he was made a member of the newly created Senate. The

third volume of the _Mécanique Céleste_, published in 1802, contained

a dedication to the “Heroic Pacificator of Europe,” at whose hand he

subsequently received various other distinctions, and by whom he was

created a Count when the Empire was formed. On the restoration of the

Bourbons in 1814 he tendered his services to them, and was subsequently

made a Marquis. In 1816 he also received a very unusual honour for a

mathematician (shared, however, by D’Alembert) by being elected one of

the Forty “Immortals” of the _Académie Française_; this distinction

he seems to have owed in great part to the literary excellence of the

_Système du Monde_.

Notwithstanding these distractions he worked steadily at mathematics

and astronomy, and even after the completion of the _Mécanique Céleste_

wrote a supplement to it which was published after his death (1827).

His last words, “_Ce que nous connaissons est peu de chose, ce que nous

ignorons est immense_,” coming as they did from one who had added so

much to knowledge, shew his character in a pleasanter aspect than it

sometimes presented during his career.

239. With the exception of Lagrange’s paper on libration, nearly all

his and Laplace’s important contributions to astronomy were made when

Clairaut’s and D’Alembert’s work was nearly finished, though Euler’s

activity continued for nearly 20 years more. Lagrange, however,

survived him by 30 years and Laplace by more than 40; and together they

carried astronomical science to a far higher stage of development than

their three predecessors.

240. To the lunar theory Lagrange contributed comparatively little

except general methods, applicable to this as to other problems of

astronomy; but Laplace devoted great attention to it. Of his special

discoveries in the subject the most notable was his explanation of

the secular acceleration of the moon’s mean motion (chapter X., §

201), which had puzzled so many astronomers. Lagrange had attempted to

explain it (1774), and had failed so completely that he was inclined

to discredit the early observations on which the existence of the

phenomenon was based. Laplace, after trying ordinary methods without

success, attempted to explain it by supposing that gravitation was

an effect not transmitted instantaneously, but that, like light, it

took time to travel from the attracting body to the attracted one; but

this also failed. Finally he traced it (1787) to an indirect planetary

effect. For, as it happens, certain perturbations which the moon

experiences owing to the action of the sun depend among other things

on the eccentricity of the earth’s orbit; this is one of the elements

(§ 236) which is being altered by the action of the planets, and has

for many centuries been very slowly decreasing; the perturbation in

question is therefore being very slightly altered, and the moon’s

average rate of motion is in consequence very slowly increasing, or

the length of the month decreasing. The whole effect is excessively

minute, and only becomes perceptible in the course of a long time.

Laplace’s calculation shewed that the moon would, in the course of a

century, or in about 1,300 complete revolutions, gain about 10″ (more

exactly 10″·2) owing to this cause, so that her place in the sky would

differ by that amount from what it would be if this disturbing cause

did not exist; in two centuries the angle gained would be 40″, in

three centuries 90″, and so on. This may be otherwise expressed by

saying that the length of the month diminishes by about one-thirtieth

of a second in the course of a century. Moreover, as Laplace shewed (§

245), the eccentricity of the earth’s orbit will not go on diminishing

indefinitely, but after an immense period to be reckoned in thousands

of years will begin to increase, and the moon’s motion will again

become slower in consequence.

Laplace’s result agreed almost exactly with that indicated by

observation; and thus the last known discrepancy of importance in the

solar system between theory and observation appeared to be explained

away; and by a curious coincidence this was effected just a hundred

years after the publication of the _Principia_.

Many years afterwards, however, Laplace’s explanation was shewn to be

far less complete than it appeared at the time (chapter XIII., § 287).

The same investigation revealed to Laplace the existence of alterations

of a similar character, and due to the same cause, of other elements in

the moon’s orbit, which, though not previously noticed, were found to

be indicated by ancient eclipse observations.

241. The third volume of the _Mécanique Céleste_ contains a general

treatment of the lunar theory, based on a method entirely different

from any that had been employed before, and worked out in great

detail. “My object,” says Laplace, “in this book is to exhibit in the

one law of universal gravitation the source of all the inequalities

of the motion of the moon, and then to employ this law as a means of

discovery, to perfect the theory of this motion and to deduce from it

several important elements in the system of the moon.” Laplace himself

calculated no lunar tables, but the Viennese astronomer _John Tobias

Bürg_ (1766-1834) made considerable use of his formulae, together with

an immense number of Greenwich observations, for the construction

of lunar tables, which were sent to the Institute of France in 1801

(before the publication of Laplace’s complete lunar theory), and

published in a slightly amended form in 1806. A few years later (1812)

_John Charles Burckhardt_ (1773-1825), a German who had settled in

Paris and worked under Laplace and Lalande, produced a new set of

tables based directly on the formulae of the _Mécanique Céleste_.

These were generally accepted in lieu of Bürg’s, which had been in

their turn an improvement on Mason’s and Mayer’s.

Later work on lunar theory may conveniently be regarded as belonging to

a new period of astronomy (chapter XIII., § 286).

242. Observation had shewn the existence of inequalities in the

planetary and lunar motions which seemed to belong to two different

classes. On the one hand were inequalities, such as most of those

of the moon, which went through their cycle of changes in a single

revolution or a few revolutions of the disturbing body; and on the

other such inequalities as the secular acceleration of the moon’s

mean motion or the motion of the earth’s apses, in which a continuous

disturbance was observed always acting in the same direction, and

shewing no signs of going through a periodic cycle of changes.

The mathematical treatment of perturbations soon shewed the

desirability of adopting different methods of treatment for two classes

of inequalities, which corresponded roughly, though not exactly,

to those just mentioned, and to which the names of =periodic= and

=secular= gradually came to be attached. The distinction plays a

considerable part in Euler’s work (§ 236), but it was Lagrange who

first recognised its full importance, particularly for planetary

theory, and who made a special study of secular inequalities.

When the perturbations of one planet by another are being studied,

it becomes necessary to obtain a mathematical expression for the

disturbing force which the second planet exerts. This expression

depends in general both on the elements of the two orbits, and on the

positions of the planets at the time considered. It can, however, be

divided up into two parts, one of which depends on the positions of the

planets (as well as on the elements), while the other depends only on

the elements of the two orbits, and is independent of the positions in

their paths which the planets may happen to be occupying at the time.

Since the positions of planets in their orbits change rapidly, the

former part of the disturbing force changes rapidly, and produces in

general, at short intervals of time, effects in opposite directions,

first, for example, accelerating and then retarding the motion of the

disturbed planet; and the corresponding inequalities of motion are the

periodic inequalities, which for the most part go through a complete

cycle of changes in the course of a few revolutions of the planets,

or even more rapidly. The other part of the disturbing force remains

nearly unchanged for a considerable period, and gives rise to changes

in the elements which, though in general very small, remain for a long

time without sensible alteration, and therefore continually accumulate,

becoming considerable with the lapse of time: these are the secular

inequalities.

Speaking generally, we may say that the periodical inequalities are

temporary and the secular inequalities permanent in their effects, or

as Sir John Herschel expresses it:—

“The secular inequalities are, in fact, nothing but what remains after

the mutual destruction of a much larger amount (as it very often is)

of periodical. But these are in their nature transient and temporary;

they disappear in short periods, and leave no trace. The planet is

temporarily withdrawn from its orbit (its slowly varying orbit), but

forthwith returns to it, to deviate presently as much the other way,

while the varied orbit accommodates and adjusts itself to the average

of these excursions on either side of it.”[146]

“Temporary” and “short” are, however, relative terms. Some periodical

inequalities, notably in the case of the moon, have periods of only a

few days, and the majority which are of importance extend only over

a few years; but some are known which last for centuries or even

thousands of years, and can often be treated as secular when we only

want to consider an interval of a few years. On the other hand, most of

the known secular inequalities are not really permanent, but fluctuate

like the periodical ones, though only in the course of immense periods

of time to be reckoned usually by tens of thousands of years.

One distinction between the lunar and planetary theories is that in the

former periodic inequalities are comparatively large and, especially

for practical purposes such as computing the position of the moon a few

months hence, of great importance; whereas the periodic inequalities

of the planets are generally small and the secular inequalities are the

most interesting.

The method of treating the elements of the elliptic orbits as variable

is specially suitable for secular inequalities; but for periodic

inequalities it is generally better to treat the body as being

disturbed from an elliptic path, and to study these deviations.

“The simplest way of regarding these various perturbations consists

in imagining a planet moving in accordance with the laws of elliptic

motion, on an ellipse the elements of which vary by insensible

degrees; and to conceive at the same time that the true planet

oscillates round this fictitious planet in a very small orbit the

nature of which depends on its periodic perturbations.”[147]

The former method, due as we have seen in great measure to Euler, was

perfected and very generally used by Lagrange, and often bears his name.

243. It was at first naturally supposed that the slow alteration in

the rates of the motions of Jupiter and Saturn (§§ 235, 236, and

chapter X., § 204) was a secular inequality; Lagrange in 1766 made an

attempt to explain it on this basis which, though still unsuccessful,

represented the observations better than Euler’s work. Laplace in his

first paper on secular inequalities (1773) found by the use of a more

complete analysis that the secular alterations in the rates of motions

of Jupiter and Saturn appeared to vanish entirely, and attempted to

explain the motions by the hypothesis, so often used by astronomers

when in difficulties, that a comet had been the cause.

In 1773 _John Henry Lambert_ (1728-1777) discovered from a study of

observations that, whereas Halley had found Saturn to be moving more

slowly than in ancient times, it was now moving faster than in Halley’s

time—a conclusion which pointed to a fluctuating or periodic cause of

some kind.

Finally in 1784 Laplace arrived at the true explanation. Lagrange had

observed in 1776 that if the times of revolution of two planets are

exactly proportional to two whole numbers, then part of the periodic

disturbing force produces a secular change in their motions, acting

continually in the same direction; though he pointed out that such

a case did not occur in the solar system. If moreover the times of

revolution are _nearly_ proportional to two whole numbers (neither

of which is very large), then part of the periodic disturbing force

produces an irregularity that is not strictly secular, but has a very

long period; and a disturbing force so small as to be capable of

being ordinarily overlooked may, if it is of this kind, be capable of

producing a considerable effect.[148] Now Jupiter and Saturn revolve

round the sun in about 4,333 days and 10,759 days respectively; five

times the former number is 21,665, twice the latter is 21,518, which

is very little less. Consequently the exceptional case occurs; and on

working it out Laplace found an appreciable inequality with a period of

about 900 years, which explained the observations satisfactorily.

The inequalities of this class, of which several others have been

discovered, are known as =long inequalities=, and may be regarded

as connecting links between secular inequalities and periodical

inequalities of the usual kind.

244. The discovery that the observed inequality of Jupiter and Saturn

was not secular may be regarded as the first step in a remarkable

series of investigations on secular inequalities carried out by

Lagrange and Laplace, for the most part between 1773 and 1784,

leading to some of the most interesting and general results in the

whole of gravitational astronomy. The two astronomers, though living

respectively in Berlin and Paris, were in constant communication, and

scarcely any important advance was made by the one which was not at

once utilised and developed by the other.

The central problem was that of the secular alterations in the

elements of a planet’s orbit regarded as a varying ellipse. Three of

these elements, the axis of the ellipse, its eccentricity, and the

inclination of its plane to a fixed plane (usually the ecliptic), are

of much greater importance than the other three. The first two are the

elements on which the size and shape of the orbit depend, and the first

also determines (by Kepler’s Third Law) the period of revolution and

average rate of motion of the planet;[149] the third has an important

influence on the mutual relations of the two planets. The other three

elements are chiefly of importance for periodical inequalities.

It should be noted moreover that the eccentricities and inclinations

were in all cases (except those specially mentioned) considered as

small quantities; and thus all the investigations were approximate,

these quantities and the disturbing forces themselves being treated as

small.

245. The basis of the whole series of investigations was a long paper

published by Lagrange in 1766, in which he explained the method of

variation of elements, and gave formulae connecting their rates of

change with the disturbing forces.

In his paper of 1773 Laplace found that what was true of Jupiter and

Saturn had a more general application, and proved that in the case of

any planet, disturbed by any other, the axis was not only undergoing

no secular change at the present time, but could not have altered

appreciably since “the time when astronomy began to be cultivated.”

In the next year Lagrange obtained an expression for the secular change

in the inclination, _valid for all time_. When this was applied to

the case of Jupiter and Saturn, which on account of their superiority

in size and great distance from the other planets could be reasonably

treated as forming with the sun a separate system, it appeared that

the changes in the inclinations would always be of a periodic nature,

so that they could never pass beyond certain fixed limits, not

differing much from the existing values. The like result held for the

system formed by the sun, Venus, the earth, and Mars. Lagrange noticed

moreover that there were cases, which, as he said, fortunately did not

appear to exist in the system of the world, in which, on the contrary,

the inclinations might increase indefinitely. The distinction depended

on the masses of the bodies in question; and although all the planetary

masses were somewhat uncertain, and those assumed by Lagrange for

Venus and Mars almost wholly conjectural, it did not appear that any

reasonable alteration in the estimated masses would affect the general

conclusion arrived at.

Two years later (1775) Laplace, much struck by the method which

Lagrange had used, applied it to the discussion of the secular

variations of the eccentricity, and found that these were also of a

periodic nature, so that the eccentricity also could not increase or

decrease indefinitely.

In the next year Lagrange, in a remarkable paper of only 14 pages,

proved that whether the eccentricities and inclinations were treated

as small or not, and whatever the masses of the planets might be,

the changes in the length of the axis of any planetary orbit were

necessarily all periodic, so that for all time the length of the axis

could only fluctuate between certain definite limits. This result was,

however, still based on the assumption that the disturbing forces could

be treated as small.

Next came a series of five papers published between 1781 and 1784 in

which Lagrange summed up his earlier work, revised and improved his

methods, and applied them to periodical inequalities and to various