CHAPTER X._A Short History of Astronomy

OBSERVATIONAL ASTRONOMY IN THE 18TH CENTURY.

“Through Newton theory had made a great advance and was ahead of

observation; the latter now made efforts to come once more level with

theory.”—BESSEL.

196. Newton virtually created a new department of astronomy,

=gravitational astronomy=, as it is often called, and bequeathed to his

successors the problem of deducing more fully than he had succeeded in

doing the motions of the celestial bodies from their mutual gravitation.

To the solution of this problem Newton’s own countrymen contributed

next to nothing throughout the 18th century, and his true successors

were a group of Continental mathematicians whose work began soon after

his death, though not till nearly half a century after the publication

of the _Principia_.

This failure of the British mathematicians to develop Newton’s

discoveries may be explained as due in part to the absence or scarcity

of men of real ability, but in part also to the peculiarity of the

mathematical form in which Newton presented his discoveries. The

_Principia_ is written almost entirely in the language of geometry,

modified in a special way to meet the requirements of the case; nearly

all subsequent progress in gravitational astronomy has been made by

mathematical methods known as =analysis=. Although the distinction

between the two methods cannot be fully appreciated except by those

who have used them both, it may perhaps convey some impression of the

differences between them to say that in the geometrical treatment

of an astronomical problem each step of the reasoning is expressed

in such a way as to be capable of being interpreted in terms of the

original problem, whereas in the analytical treatment the problem is

first expressed by means of algebraical symbols; these symbols are

manipulated according to certain purely formal rules, no regard being

paid to the interpretation of the intermediate steps, and the final

algebraical result, if it can be obtained, yields on interpretation

the solution of the original problem. The geometrical solution of

a problem, if it can be obtained, is frequently shorter, clearer,

and more elegant; but, on the other hand, each special problem has

to be considered separately, whereas the analytical solution can be

conducted to a great extent according to fixed rules applicable in a

larger number of cases. In Newton’s time modern analysis was only just

coming into being, some of the most important parts of it being in

fact the creation of Leibniz and himself, and although he sometimes

used analysis to solve an astronomical problem, it was his practice to

translate the result into geometrical language before publication; in

doing so he was probably influenced to a large extent by a personal

preference for the elegance of geometrical proofs, partly also by an

unwillingness to increase the numerous difficulties contained in the

_Principia_, by using mathematical methods which were comparatively

unfamiliar. But though in the hands of a master like Newton geometrical

methods were capable of producing astonishing results, the lesser men

who followed him were scarcely ever capable of using his methods to

obtain results beyond those which he himself had reached. Excessive

reverence for Newton and all his ways, combined with the estrangement

which long subsisted between British and foreign mathematicians, as the

result of the fluxional controversy (chapter IX., § 191), prevented

the former from using the analytical methods which were being rapidly

perfected by Leibniz’s pupils and other Continental mathematicians. Our

mathematicians remained, therefore, almost isolated during the whole

of the 18th century, and with the exception of some admirable work by

_Colin Maclaurin_ (1698-1746), which carried Newton’s theory of the

figure of the earth a stage further, nothing of importance was done in

our country for nearly a century after Newton’s death to develop the

theory of gravitation beyond the point at which it was left in the

_Principia_.

In other departments of astronomy, however, important progress was made

both during and after Newton’s lifetime, and by a curious inversion,

while Newton’s ideas were developed chiefly by French mathematicians,

the Observatory of Paris, at which Picard and others had done such

admirable work (chapter VIII., §§ 160-2), produced little of real

importance for nearly a century afterwards, and a large part of

the best observing work of the 18th century was done by Newton’s

countrymen. It will be convenient to separate these two departments of

astronomical work, and to deal in the next chapter with the development

of the theory of gravitation.

197. The first of the great English observers was Newton’s contemporary

_John Flamsteed_, who was born near Derby in 1646 and died at Greenwich

in 1720.[111] Unfortunately the character of his work was such that,

marked as it was by no brilliant discoveries, it is difficult to

present it in an attractive form or to give any adequate idea of its

real extent and importance. He was one of those laborious and careful

investigators, the results of whose work are invaluable as material for

subsequent research, but are not striking in themselves.

He made some astronomical observations while quite a boy, and wrote

several papers, of a technical character, on astronomical subjects,

which attracted some attention. In 1675 appointed a member of a

Committee to report on a method for finding the longitude at sea which

had been offered to the Government by a certain Frenchman of the name

of _St. Pierre_. The Committee, acting largely on Flamsteed’s advice,

reported unfavourably on the method in question, and memorialised

Charles II. in favour of founding a national observatory, in order that

better knowledge of the celestial bodies might lead to a satisfactory

method of finding the longitude, a problem which the rapid increase

of English shipping rendered of great practical importance. The King

having agreed, Flamsteed was in the same year appointed to the new

office of Astronomer Royal, with a salary of £100 a year, and the

warrant for building an Observatory at Greenwich was signed on June

12th, 1675. About a year was occupied in building it, and Flamsteed

took up his residence there and began work in July 1676, five years

after Cassini entered upon his duties at the Observatory of Paris

(chapter VIII., § 160). The Greenwich Observatory was, however, on

a very different scale from the magnificent sister institution. The

King had, it is true, provided Flamsteed with a building and a very

small salary, but furnished him neither with instruments nor with an

assistant. A few instruments he possessed already, a few more were

given to him by rich friends, and he gradually made at his own expense

some further instrumental additions of importance. Some years after

his appointment the Government provided him with “a silly, surly

labourer” to help him with some of the rough work, but he was compelled

to provide more skilled assistance out of his own pocket, and this

necessity in turn compelled him to devote some part of his valuable

time to taking pupils.

198. Flamsteed’s great work was the construction of a more accurate

and more extensive star catalogue than any that existed; he also

made a number of observations of the moon, of the sun, and to a less

extent of other bodies. Like Tycho, the author of the last great star

catalogue (chapter V., § 107), he found problems continually presenting

themselves in the course of his work which had to be solved before his

main object could be accomplished, and we accordingly owe to him the

invention of several improvements in practical astronomy, the best

known being his method of finding the position of the first point of

Aries (chapter II., § 42), one of the fundamental points with reference

to which all positions on the celestial sphere are defined. He was the

first astronomer to use a clock systematically for the determination of

one of the two fundamental quantities (the right ascension) necessary

to fix the position of a star, a method which was first suggested

and to some extent used by Picard (chapter VIII., § 157), and, as

soon as he could get the necessary instruments, he regularly used the

telescopic sights of Gascoigne and Auzout (chapter VIII., § 155),

instead of making naked-eye observations. Thus while Hevel (chapter

VIII., § 153) was the last and most accurate observer of the old

school, employing methods not differing essentially from those which

had been in use for centuries, Flamsteed belongs to the new school,

and his methods differ rather in detail than in principle from those

now in vogue for similar work at Greenwich, Paris, or Washington. This

adoption of new methods, together with the most scrupulous care in

details, rendered Flamsteed’s observations considerably more accurate

than any made in his time or earlier, the first definite advance

afterwards being made by Bradley (§ 218).

Flamsteed compared favourably with many observers by not merely taking

and recording observations, but by performing also the tedious process

known as reduction (§ 218), whereby the results of the observation are

put into a form suitable for use by other astronomers; this process

is usually performed in modern observatories by assistants, but in

Flamsteed’s case had to be done almost exclusively by the astronomer

himself. From this and other causes he was extremely slow in publishing

observations; we have already alluded (chapter IX., § 192) to the

difficulty which Newton had in extracting lunar observations from him,

and after a time a feeling that the object for which the Observatory

had been founded was not being fulfilled became pretty general among

astronomers. Flamsteed always suffered from bad health as well as

from the pecuniary and other difficulties which have been referred

to; moreover he was much more anxious that his observations should

be kept back till they were as accurate as possible, than that they

should be published in a less perfect form and used for the researches

which he once called “Mr. Newton’s crotchets”; consequently he took

remonstrances about the delay in the publication of his observations

in bad part. Some painful quarrels occurred between Flamsteed on the

one hand and Newton and Halley on the other. The last straw was the

unauthorised publication in 1712, under the editorship of Halley, of a

volume of Flamsteed’s observations, a proceeding to which Flamsteed not

unnaturally replied by calling Halley a “malicious thief.” Three years

later he succeeded in getting hold of all the unsold copies and in

destroying them, but fortunately he was also stimulated to prepare for

publication an authentic edition. The _Historia Coelestis Britannica_,

as he called the book, contained an immense series of observations made

both before and during his career at Greenwich, but the most important

and permanently valuable part was a catalogue of the places of nearly

3,000 stars.[112]

Flamsteed himself only lived just long enough to finish the second of

the three volumes; the third was edited by his assistants _Abraham

Sharp_ (1651-1742) and _Joseph Crosthwait_; and the whole was published

in 1725. Four years later still appeared his valuable Star-Atlas, which

long remained in common use.

The catalogue was not only three times as extensive as Tycho’s, which

it virtually succeeded, but was also very much more accurate. It

has been estimated[113] that, whereas Tycho’s determinations of the

positions of the stars were on the average about 1′ in error, the

corresponding errors in Flamsteed’s case were about 10″. This quantity

is the apparent diameter of a shilling seen from a distance of about

500 yards; so that if two marks were made at opposite points on the

edge of the coin, and it were placed at a distance of 500 yards, the

two marks might be taken to represent the true direction of an average

star and its direction as given in Flamsteed’s catalogue. In some cases

of course the error might be much greater and in others considerably

less.

Flamsteed contributed to astronomy no ideas of first-rate importance;

he had not the ingenuity of Picard and of Roemer in devising

instrumental improvements, and he took little interest in the

theoretical work of Newton;[114] but by unflagging industry and

scrupulous care he succeeded in bequeathing to his successors an

immense treasure of observations, executed with all the accuracy that

his instrumental means permitted.

199. Flamsteed was succeeded as Astronomer Royal by Edmund Halley, whom

we have already met with (chapter IX., § 176) as Newton’s friend and

helper.

Born in 1656, ten years after Flamsteed, he studied astronomy in his

schooldays, and published a paper on the orbits of the planets as

early as 1676. In the same year he set off for St. Helena (in latitude

16° S.) in order to make observations of stars which were too near

the south pole to be visible in Europe. The climate turned out to be

disappointing, and he was only able after his return to publish (1678)

a catalogue of the places of 341 southern stars, which constituted,

however, an important addition to precise knowledge of the stars. The

catalogue was also remarkable as being the first based on telescopic

observation, though the observations do not seem to have been taken

with all the accuracy which his instruments rendered attainable. During

his stay at St. Helena he also took a number of pendulum observations

which confirmed the results obtained a few years before by Richer at

Cayenne (chapter VIII., § 161), and also observed a transit of Mercury

across the sun, which occurred in November 1677.

After his return to England he took an active part in current

scientific questions, particularly in those connected with astronomy,

and made several small contributions to the subject. In 1684, as

we have seen, he first came effectively into contact with Newton,

and spent a good part of the next few years in helping him with the

_Principia_.

200. Of his numerous contributions to astronomy, which touched almost

every branch of the subject, his work on comets is the best known and

probably the most important. He observed the comets of 1680 and 1682;

he worked out the paths both of these and of a number of other recorded

comets in accordance with Newton’s principles, and contributed a good

deal of the material contained in the sections of the _Principia_

dealing with comets, particularly in the later editions. In 1705 he

published a _Synopsis of Cometary Astronomy_ in which no less than 24

cometary orbits were calculated. Struck by the resemblance between

the paths described by the comets of 1531, 1607, and 1682, and by

the approximate equality in the intervals between their respective

appearances and that of a fourth comet seen in 1456, he was shrewd

enough to conjecture that the three later comets, if not all four, were

really different appearances of the same comet, which revolved round

the sun in an elongated ellipse in a period of about 75 or 76 years.

He explained the difference between the 76 years which separate the

appearances of the comet in 1531 and 1607, and the slightly shorter

period which elapsed between 1607 and 1682, as probably due to the

perturbations caused by planets near which the comet had passed;

and finally predicted the probable reappearance of the same comet

(which now deservedly bears his name) about 76 years after its last

appearance, _i.e._ about 1758, though he was again aware that planetary

perturbation might alter the time of its appearance; and the actual

appearance of the comet about the predicted time (chapter XI., § 231)

marked an important era in the progress of our knowledge of these

extremely troublesome and erratic bodies.

201. In 1693 Halley read before the Royal Society a paper in which

he called attention to the difficulty of reconciling certain ancient

eclipses with the known motion of the moon, and referred to the

possibility of some slight increase in the moon’s average rate of

motion round the earth.

This irregularity, now known as the =secular acceleration of the moon’s

mean motion=, was subsequently more definitely established as a fact

of observation; and the difficulties met with in explaining it as a

result of gravitation have rendered it one of the most interesting of

the moon’s numerous irregularities (cf. chapter XI., § 240, and chapter

XIII., § 287).

202. Halley also rendered good service to astronomy by calling

attention to the importance of the expected transits of Venus across

the sun in 1761 and 1769 as a means of ascertaining the distance of

the sun. The method had been suggested rather vaguely by Kepler,

and more definitely by James Gregory in his _Optics_ published in

1663. The idea was first suggested to Halley by his observation of

the transit of Mercury in 1677. In three papers published by the

Royal Society he spoke warmly of the advantages of the method, and

discussed in some detail the places and means most suitable for

observing the transit of 1761. He pointed out that the desired result

could be deduced from a comparison of the durations of the transit of

Venus, as seen from different stations on the earth, _i.e._ of the

intervals between the first appearance of Venus on the sun’s disc and

the final disappearance, as seen at two or more different stations.

He estimated, moreover, that this interval of time, which would be

several hours in length, could be measured with an error of only about

two seconds, and that in consequence the method might be relied upon

to give the distance of the sun to within about 1∕500 part of its

true value. As the current estimates of the sun’s distance differed

among one another by 20 or 30 per cent., the new method, expounded

with Halley’s customary lucidity and enthusiasm, not unnaturally

stimulated astronomers to take great trouble to carry out Halley’s

recommendations. The results, as we shall see (§ 227), were, however,

by no means equal to Halley’s expectations.

203. In 1718 Halley called attention to the fact that three well-known

stars, Sirius, Procyon, and Arcturus, had changed their angular

distances from the ecliptic since Greek times, and that Sirius had

even changed its position perceptibly since the time of Tycho Brahe.

Moreover comparison of the places of other stars shewed that the

changes could not satisfactorily be attributed to any motion of the

ecliptic, and although he was well aware that the possible errors of

observation were such as to introduce a considerable uncertainty into

the amounts involved, he felt sure that such errors could not wholly

account for the discrepancies noticed, but that the stars in question

must have really shifted their positions in relation to the rest; and

he naturally inferred that it would be possible to detect similar

=proper motions= (as they are now called) in other so-called “fixed”

stars.

204. He also devoted a good deal of time to the standing astronomical

problem of improving the tables of the moon and planets, particularly

the former. He made observations of the moon as early as 1683, and by

means of them effected some improvement in the tables. In 1676 he had

already noted defects in the existing tables of Jupiter and Saturn, and

ultimately satisfied himself of the existence of certain irregularities

in the motion of these two planets, suspected long ago by Horrocks

(chapter VIII., § 156); these irregularities he attributed correctly

to the perturbations of the two planets by one another, though he was

not mathematician enough to work out the theory; from observation,

however, he was able to estimate the irregularities in question with

fair accuracy and to improve the planetary tables by making allowance

for them. But neither the lunar nor the planetary tables were ever

completed in a form which Halley thought satisfactory. By 1719 they

were printed, but kept back from publication, in hopes that subsequent

improvements might be effected. After his appointment as Astronomer

Royal in succession to Flamsteed (1720) he devoted special attention

to getting fresh observations for this purpose, but he found the

Observatory almost bare of instruments, those used by Flamsteed having

been his private property, and having been removed as such by his heirs

or creditors. Although Halley procured some instruments, and made with

them a number of observations, chiefly of the moon, the age (63) at

which he entered upon his office prevented him from initiating much, or

from carrying out his duties with great energy, and the observations

taken were in consequence only of secondary importance, while the

tables for the improvement of which they were specially designed were

only finally published in 1752, ten years after the death of their

author. Although they thus appeared many years after the time at which

they were virtually prepared and owed little to the progress of science

during the interval, they at once became and for some time remained the

standard tables for both the lunar and planetary motions (cf. § 226,

and chapter XI., § 247).

205. Halley’s remarkable versatility in scientific work is further

illustrated by the labour which he expended in editing the writings

of the great Greek geometer Apollonius (chapter II., § 38) and the

star catalogue of Ptolemy (chapter II., § 50). He was also one of the

first of modern astronomers to pay careful attention to the effects

to be observed during a total eclipse of the sun, and in the vivid

description which he wrote of the eclipse of 1715, besides referring

to the mysterious corona, which Kepler and others had noticed before

(chapter VII., § 145), he called attention also to “a very narrow

streak of a dusky but strong Red Light,” which was evidently a portion

of that remarkable envelope of the sun which has been so extensively

studied in modern times (chapter XIII., § 301) under the name of the

chromosphere.

It is worth while to notice, as an illustration of Halley’s unselfish

enthusiasm for science and of his power of looking to the future, that

two of his most important pieces of work, by which certainly he is

now best known, necessarily appeared during his lifetime as of little

value, and only bore their fruit after his death (1742), for his comet

only returned in 1759, when he had been dead 17 years, and the first of

the pair of transits of Venus, from which he had shewn how to deduce

the distance of the sun, took place two years later still (§ 227).

206. The third Astronomer Royal, _James Bradley_, is popularly known as

the author of two memorable discoveries, _viz._ the aberration of light

and the nutation of the earth’s axis. Remarkable as these are both in

themselves and on account of the ingenious and subtle reasoning and

minutely accurate observations by means of which they were made, they

were in fact incidents in a long and active astronomical career, which

resulted in the execution of a vast mass of work of great value.

The external events of Bradley’s life may be dealt with very briefly.

Born in 1693, he proceeded in due course to Oxford (B.A. 1714, M.A.

1717), but acquired his first knowledge of astronomy and his marked

taste for the subject from his uncle _James Pound_, for many years

rector of Wansted in Essex, who was one of the best observers of

the time. Bradley lived with his uncle for some years after leaving

Oxford, and carried out a number of observations in concert with him.

The first recorded observation of Bradley’s is dated 1715, and by

1718 he was sufficiently well thought of in the scientific world to

receive the honour of election as a Fellow of the Royal Society. But,

as his biographer[115] remarks, “it could not be foreseen that his

astronomical labours would lead to any establishment in life, and it

became necessary for him to embrace a profession.” He accordingly took

orders, and was fortunate enough to be presented almost at once to two

livings, the duties attached to which do not seem to have interfered

appreciably with the prosecution of his astronomical studies at Wansted.

In 1721 he was appointed Savilian Professor of Astronomy at Oxford,

and resigned his livings. The work of the professorship appears to

have been very light, and for more than ten years he continued to

reside chiefly at Wansted, even after his uncle’s death in 1724.

In 1732 he took a house in Oxford and set up there most of his

instruments, leaving, however, at Wansted the most important of all,

the “zenith-sector,” with which his two famous discoveries were made.

Ten years afterwards Halley’s death rendered the post of Astronomer

Royal vacant, and Bradley received the appointment.

The work of the Observatory had been a good deal neglected by Halley

during the last few years of his life, and Bradley’s first care was to

effect necessary repairs in the instruments. Although the equipment

of the Observatory with instruments worthy of its position and of the

state of science at the time was a work of years, Bradley had some

of the most important instruments in good working order within a few

months of his appointment, and observations were henceforward made

systematically. Although the 20 remaining years of his life (1742-1762)

were chiefly spent at Greenwich in the discharge of the duties of

his office and in researches connected with them, he retained his

professorship at Oxford, and continued to make observations at Wansted

at least up till 1747.

[Illustration: BRADLEY.]

207. The discovery of aberration resulted from an attempt to detect the

parallactic displacement of stars which should result from the annual

motion of the earth. Ever since the Coppernican controversy had called

attention to the importance of the problem (cf. chapter IV., § 92

and chapter VI., § 129), it had naturally exerted a fascination on

the minds of observing astronomers, many of whom had tried to detect

the motion in question, and some of whom (including the “universal

claimant” Hooke) professed to have succeeded. Actually, however, all

previous attempts had been failures, and Bradley was no more successful

than his predecessors in this particular undertaking, but was able to

deduce from his observations two results of great interest and of an

entirely unexpected character.

The problem which Bradley set himself was to examine whether any

star could be seen to have in the course of the year a slight motion

relative to others or relative to fixed points on the celestial sphere

such as the pole. It was known that such a motion, if it existed, must

be very small, and it was therefore evident that extreme delicacy in

instrumental adjustments and the greatest care in observation would

have to be employed. Bradley worked at first in conjunction with his

friend _Samuel Molyneux_ (1689-1728), who had erected a telescope at

Kew. In accordance with the method adopted in a similar investigation

by Hooke, whose results it was desired to test, the telescope was fixed

in a nearly vertical position, so chosen that a particular star in

the Dragon (γ Draconis) would be visible through it when it crossed

the meridian, and the telescope was mounted with great care so as to

maintain an invariable position throughout the year. If then the star

in question were to undergo any motion which altered its distance from

the pole, there would be a corresponding alteration in the position in

which it would be seen in the field of view of the telescope. The first

observations were taken on December 14th, 1725 (N.S.), and by December

28th Bradley believed that he had already noticed a slight displacement

of the star towards the south. This motion was clearly verified on

January 1st, and was then observed to continue; in the following March

the star reached its extreme southern position, and then began to move

northwards again. In September it once more altered its direction of

motion, and by the end of the year had completed the cycle of its

changes and returned to its original position, the greatest change in

position amounting to nearly 40′.

The star was thus observed to go through _some_ annual motion. It

was, however, at once evident to Bradley that this motion was not the

parallactic motion of which he was in search, for the position of

the star was such that parallax would have made it appear farthest

south in December and farthest north in June, or in each case three

months earlier than was the case in the actual observations. Another

explanation which suggested itself was that the earth’s axis might

have a to-and-fro oscillatory motion or nutation which would alter

the position of the celestial pole and hence produce a corresponding

alteration in the position of the star. Such a motion of the celestial

pole would evidently produce opposite effects on two stars situated on

opposite sides of it, as any motion which brought the pole nearer to

one star of such a pair would necessarily move it away from the other.

Within a fortnight of the decisive observation made on January 1st a

star[116] had already been selected for the application of this test,

with the result which can best be given in Bradley’s own words:—

“A nutation of the earth’s axis was one of the first things that

offered itself upon this occasion, but it was soon found to be

insufficient; for though it might have accounted for the change of

declination in γ Draconis, yet it would not at the same time agree

with the phaenomena in other stars; particularly in a small one

almost opposite in right ascension to γ Draconis, at about the same

distance from the north pole of the equator: for though this star

seemed to move the same way as a nutation of the earth’s axis would

have made it, yet, it changing its declination but about half as

much as γ Draconis in the same time, (as appeared upon comparing the

observations of both made upon the same days, at different seasons of

the year,) this plainly proved that the apparent motion of the stars

was not occasioned by a real nutation, since, if that had been the

cause, the alteration in both stars would have been near equal.”

One or two other explanations were tested and found insufficient, and

as the result of a series of observations extending over about two

years, the phenomenon in question, although amply established, still

remained quite unexplained.

By this time Bradley had mounted an instrument of his own at Wansted,

so arranged that it was possible to observe through it the motions of

stars other than γ Draconis.

Several stars were watched carefully throughout a year, and the

observations thus obtained gave Bradley a fairly complete knowledge of

the geometrical laws according to which the motions varied both from

star to star and in the course of the year.

208. The true explanation of =aberration=, as the phenomenon in

question was afterwards called, appears to have occurred to him about

September, 1728, and was published to the Royal Society, after some

further verification, early in the following year. According to a

well-known story,[117] he noticed, while sailing on the Thames, that a

vane on the masthead appeared to change its direction every time that

the boat altered its course, and was informed by the sailors that this

change was not due to any alteration in the wind’s direction, but to

that of the boat’s course. In fact the apparent direction of the wind,

as shewn by the vane, was not the true direction of the wind, but

resulted from a combination of the motions of the wind and of the boat,

being more precisely that of the motion of the wind _relative_ to the

boat. Replacing in imagination the wind by light coming from a star,

and the boat shifting its course by the earth moving round the sun

and continually changing its direction of motion, Bradley arrived at

an explanation which, when worked out in detail, was found to account

most satisfactorily for the apparent changes in the direction of a

star which he had been studying. His own account of the matter is as

follows:—

“At last I conjectured that all the phaenomena hitherto mentioned

proceeded from the progressive motion of light and the earth’s annual

motion in its orbit. For I perceived that, if light was propagated in

time, the apparent place of a fixed object would not be the same when

the eye is at rest, as when it is moving in any other direction than

that of the line passing through the eye and object; and that when the

eye is moving

in different directions, the apparent place of the object would be

different.

“I considered this matter in the following manner. I imagined C A to

be a ray of light, falling perpendicularly upon the line B D; then if

the eye is at rest at A, the object must appear in the direction A C,

whether light be propagated in time or in an instant. But if the eye

is moving from B towards A, and light is propagated in time, with a

velocity that is to the velocity of the eye, as C A to B A; then light

moving from C to A, whilst the eye moves from B to A, that particle of

it by which the object will be discerned when the eye in its motion

comes to A, is at C when the eye is at B. Joining the points B, C, I

supposed the line C B to be a tube (inclined to the line B D in the

angle D B C) of such a diameter as to admit of but one particle of

light; then it was easy to conceive that the particle of light at C (by

which the object must be seen when the eye, as it moves along, arrives

at A) would pass through the tube B C, if it is inclined to B D in the

angle D B C, and accompanies the eye in its motion from B to A; and

that it could not come to the eye, placed behind such a tube, if it had

any other inclination to the line B D....

[Illustration: FIG. 74.—The aberration of light. From Bradley’s paper

in the _Phil. Trans._]

“Although therefore the true or real place of an object is

perpendicular to the line in which the eye is moving, yet the visible

place will not be so, since that, no doubt, must be in the direction

of the tube; but the difference between the true and apparent place

will be (caeteris paribus) greater or less, according to the different

proportion between the velocity of light and that of the eye. So that

if we could suppose that light was propagated in an instant, then there

would be no difference between the real and visible place of an object,

although the eye were in motion; for in that case, A C being infinite

with respect to A B, the angle A C B (the difference between the true

and visible place) vanishes. But if light be propagated in time, (which

I presume will readily be allowed by most of the philosophers of this

age,) then it is evident from the foregoing considerations, that there

will be always a difference between the real and visible place of an

object, unless the eye is moving either directly towards or from the

object.”

Bradley’s explanation shews that the apparent position of a star is

determined by the motion of the star’s light _relative_ to the earth,

so that the star appears slightly nearer to the point on the celestial

sphere towards which the earth is moving than would otherwise be

the case. A familiar illustration of a precisely analogous effect

may perhaps be of service. Any one walking on a rainy but windless

day protects himself most effectually by holding his umbrella, not

immediately over his head, but a little in front, exactly as he would

do if he were at rest and there were a slight wind blowing in his face.

In fact, if he were to ignore his own motion and pay attention only to

the direction in which he found it advisable to point his umbrella,

he would believe that there was a slight head-wind blowing the rain

towards him.

[Illustration: FIG. 75.—The aberration of light.]

209. The passage quoted from Bradley’s paper deals only with the

simple case in which the star is at right angles to the direction of

the earth’s motion. He shews elsewhere that if the star is in any

other direction the effect is of the same kind but less in amount. In

Bradley’s figure (fig. 74) the amount of the star’s displacement from

its true position is represented by the angle B C A, which depends on

the proportion between the lines A C and A B; but if (as in fig. 75)

the earth is moving (without change of speed) in the direction A B′

instead of A B, so that the direction of the star is oblique to it, it

is evident from the figure that the star’s displacement, represented by

the angle A C B′, is less than before; and the amount varies according

to a simple mathematical law[118] with the angle between the two

directions. It follows therefore that the displacement in question is

different for different stars, as Bradley’s observations had already

shewn, and is, moreover, different for the same star in the course of

the year, so that a star appears to describe a curve which is very

nearly an ellipse (fig. 76), the centre (S) corresponding to the

position which the star would occupy if aberration did not exist. It

is not difficult to see that, wherever a star is situated, the earth’s

motion is twice a year, at intervals of six months, at right angles to

the direction of the star, and that at these times the star receives

the greatest possible displacement from its mean position, and is

consequently at the ends of the greatest axis of the ellipse which it

describes, as at A and A′, whereas at intermediate times it undergoes

its least displacement, as at B and B′. The greatest displacement S

A, or half of A A′, which is the same for all stars, is known as the

=constant of aberration=, and was fixed by Bradley at between 20″

and 20-1∕2″, the value at present accepted being 20″·47. The least

displacement, on the other hand, S B, or half of B B′, was shewn to

depend in a simple way upon the star’s distance from the ecliptic,

being greatest for stars farthest from the ecliptic.

[Illustration: FIG. 76.—The aberrational ellipse.]

210. The constant of aberration, which is represented by the angle A

C B in fig. 74, depends only on the ratio between A C and A B, which

are in turn proportional to the velocities of light and of the earth.

Observations of aberration give then the ratio of these two velocities.

From Bradley’s value of the constant of aberration it follows by an

easy calculation that the velocity of light is about 10,000 times

that of the earth; Bradley also put this result into the form that

light travels from the sun to the earth in 8 minutes 13 seconds. From

observations of the eclipses of Jupiter’s moons, Roemer and others had

estimated the same interval at from 8 to 11 minutes (chapter VIII., §

162); and Bradley was thus able to get a satisfactory confirmation of

the truth of his discovery. Aberration being once established, the same

calculation could be used to give the most accurate measure of the

velocity of light in terms of the dimensions of the earth’s orbit, the

determination of aberration being susceptible of considerably greater

accuracy than the corresponding measurements required for Roemer’s

method.

211. One difficulty in the theory of aberration deserves mention.

Bradley’s own explanation, quoted above, refers to light as a material

substance shot out from the star or other luminous body. This was in

accordance with the corpuscular theory of light, which was supported

by the great weight of Newton’s authority and was commonly accepted

in the 18th century. Modern physicists, however, have entirely

abandoned the corpuscular theory, and regard light as a particular

form of wave-motion transmitted through ether. From this point of

view Bradley’s explanation and the physical illustrations given are

far less convincing; the question becomes in fact one of considerable

difficulty, and the most careful and elaborate of modern investigations

cannot be said to be altogether satisfactory. The curious inference

may be drawn that, if the more correct modern notions of the nature

of light had prevailed in Bradley’s time, it must have been very much

more difficult, if not impracticable, for him to have thought of his

explanation of the stellar motions which he was studying; and thus an

erroneous theory led to a most important discovery.

212. Bradley had of course not forgotten the original object of his

investigation. He satisfied himself, however, that the agreement

between the observed positions of γ Draconis and those which resulted

from aberration was so close that any displacement of a star due

to parallax which might exist must certainly be less than 2″, and

probably not more than 1∕2″, so that the large parallax amounting to

nearly 30″, which Hooke claimed to have detected, must certainly be

rejected as erroneous.

From the point of view of the Coppernican controversy, however,

Bradley’s discovery was almost as good as the discovery of a

parallax; since if the earth were at rest no explanation of the least

plausibility could be given of aberration.

213. The close agreement thus obtained between theory and observation

would have satisfied an astronomer less accurate and careful than

Bradley. But in his paper on aberration (1729) we find him writing:—

“I have likewise met with some small varieties in the declination of

other stars in different years which do not seem to proceed from the

same cause.... But whether these small alterations proceed from a

regular cause, or are occasioned by any change in the materials, etc.,

of my instrument, I am not yet able fully to determine.”

The slender clue thus obtained was carefully followed up and led to

a second striking discovery, which affords one of the most beautiful

illustrations of the important results which can be deduced from the

study of “residual phenomena.” Aberration causes a star to go through

a cyclical series of changes in the course of a year; if therefore

at the end of a year a star is found not to have returned to its

original place, some other explanation of the motion has to be sought.

Precession was one known cause of such an alteration; but Bradley

found, at the end of his first year’s set of observations at Wansted,

that the alterations in the positions of various stars differed by a

minute amount (not exceeding 2″) from those which would have resulted

from the usual estimate of precession; and that, although an alteration

in the value of precession would account for the observed motions of

some of these stars, it would have increased the discrepancy in the

case of others. A nutation or nodding of the earth’s axis had, as we

have seen (§ 207), already presented itself to him as a possibility;

and although it had been shewn to be incapable of accounting for the

main phenomenon—due to aberration—it might prove to be a satisfactory

explanation of the much smaller residual motions. It soon occurred to

Bradley that such a nutation might be due to the action of the moon, as

both observation and the Newtonian explanation of precession indicated:—

“I suspected that the moon’s action upon the equatorial parts of

the earth might produce these effects: for if the precession of

the equinox be, according to Sir Isaac Newton’s principles, caused

by the actions of the sun and moon upon those parts, the plane of

the moon’s orbit being at one time above ten degrees more inclined

to the plane of the equator than at another, it was reasonable to

conclude, that the part of the whole annual precession, which arises

from her action, would in different years be varied in its quantity;

whereas the plane of the ecliptic, wherein the sun appears, keeping

always nearly the same inclination to the equator, that part of the

precession which is owing to the sun’s action may be the same every

year; and from hence it would follow, that although the mean annual

precession, proceeding from the joint actions of the sun and moon,

were 50″, yet the apparent annual precession might sometimes exceed

and sometimes fall short of that mean quantity, according to the

various situations of the nodes of the moon’s orbit.”

Newton in his discussion of precession (chapter IX., § 188;

_Principia_, Book III., proposition 21) had pointed out the existence

of a small irregularity with a period of six months. But it is evident,

on looking at this discussion of the effect of the solar and lunar

attractions on the protuberant parts of the earth, that the various

alterations in the positions of the sun and moon relative to the earth

might be expected to produce irregularities, and that the uniform

precessional motion known from observation and deduced from gravitation

by Newton was, as it were, only a smoothing out of a motion of a much

more complicated character. Except for the allusion referred to, Newton

made no attempt to discuss these irregularities, and none of them had

as yet been detected by observation.

Of the numerous irregularities of this class which are now known,

and which may be referred to generally as =nutation=, that indicated

by Bradley in the passage just quoted is by far the most important.

As soon as the idea of an irregularity depending on the position of

the moon’s nodes occurred to him, he saw that it would be desirable

to watch the motions of several stars during the whole period (about

19 years) occupied by the moon’s nodes in performing the circuit of

the ecliptic and returning to the same position. This inquiry was

successfully carried out between 1727 and 1747 with the telescope

mounted at Wansted. When the moon’s nodes had performed half their

revolution, _i.e._ after about nine years, the correspondence between

the displacements of the stars and the changes in the moon’s orbit was

so close that Bradley was satisfied with the general correctness of his

theory, and in 1737 he communicated the result privately to Maupertuis

(§ 221), with whom he had had some scientific correspondence.

Maupertuis appears to have told others, but Bradley himself waited

patiently for the completion of the period which he regarded as

necessary for the satisfactory verification of his theory, and only

published his results definitely at the beginning of 1748.

[Illustration: FIG. 77.—Precession and nutation.]

214. Bradley’s observations established the existence of certain

alterations in the positions of various stars, which could be accounted

for by supposing that, on the one hand, the distance of the pole from

the ecliptic fluctuated, and that, on the other, the precessional

motion of the pole was not uniform, but varied slightly in speed.

_John Machin_ (?-1751), one of the best English mathematicians of the

time, pointed out that these effects would be produced if the pole

were supposed to describe on the celestial sphere a minute circle in

a period of rather less than 19 years—being that of the revolution

of the nodes of the moon’s orbit—round the position which it would

occupy if there were no nutation, but a uniform precession. Bradley

found that this hypothesis fitted his observations, but that it would

be better to replace the circle by a slightly flattened ellipse, the

greatest and least axes of which he estimated at about 18″ and 16″

respectively.[119] This ellipse would be about as large as a shilling

placed in a slightly oblique position at a distance of 300 yards from

the eye. The motion of the pole was thus shewn to be a double one;

as the result of precession and nutation combined it describes round

the pole of the ecliptic “a gently undulated ring,” as represented in

the figure, in which, however, the undulations due to nutation are

enormously exaggerated.

215. Although Bradley was aware that nutation must be produced by the

action of the moon, he left the theoretical investigation of its cause

to more skilled mathematicians than himself.

In the following year (1749) the French mathematician D’Alembert

(chapter XI., § 232) published a treatise[120] in which not only

precession, but also a motion of nutation agreeing closely with that

observed by Bradley, were shewn by a rigorous process of analysis to be

due to the attraction of the moon on the protuberant parts of the earth

round the equator (cf. chapter IX., § 187), while Newton’s explanation

of precession was confirmed by the same piece of work. Euler (chapter

XI., § 236) published soon afterwards another investigation of the

same subject; and it has been studied afresh by many mathematical

astronomers since that time, with the result that Bradley’s nutation

is found to be only the most important of a long series of minute

irregularities in the motion of the earth’s axis.

216. Although aberration and nutation have been discussed first,

as being the most important of Bradley’s discoveries, other

investigations were carried out by him before or during the same time.

The earliest important piece of work which he accomplished was in

connection with Jupiter’s satellites. His uncle had devoted a good deal

of attention to this subject, and had drawn up some tables dealing

with the motion of the first satellite, which were based on those of

Domenico Cassini, but contained a good many improvements. Bradley seems

for some years to have made a practice of frequently observing the

eclipses of Jupiter’s satellites, and of noting discrepancies between

the observations and the tables; and he was thus able to detect several

hitherto unnoticed peculiarities in the motions, and thereby to form

improved tables. The most interesting discovery was that of a period

of 437 days, after which the motions of the three inner satellites

recurred with the same irregularities. Bradley, like Pound, made use of

Roemer’s suggestion (chapter VIII., § 162) that light occupied a finite

time in travelling from Jupiter to the earth, a theory which Cassini

and his school long rejected. Bradley’s tables of Jupiter’s satellites

were embodied in Halley’s planetary and lunar tables, printed in 1719,

but not published till more than 30 years afterwards (§ 204). Before

that date the Swedish astronomer _Pehr Vilhelm Wargentin_ (1717-1783)

had independently discovered the period of 437 days, which he utilised

for the construction of an extremely accurate set of tables for the

satellites published in 1746.

In this case as in that of nutation Bradley knew that his mathematical

powers were unequal to giving an explanation on gravitational

principles of the inequalities which observation had revealed to him,

though he was well aware of the importance of such an undertaking, and

definitely expressed the hope “that some geometer,[121] in imitation

of the great Newton, would apply himself to the investigation of these

irregularities, from the certain and demonstrative principles of

gravity.”

On the other hand, he made in 1726 an interesting practical application

of his superior knowledge of Jupiter’s satellites by determining,

in accordance with Galilei’s method (chapter VI., § 127), but with

remarkable accuracy, the longitudes of Lisbon and of New York.

217. Among Bradley’s minor pieces of work may be mentioned his

observations of several comets and his calculation of their respective

orbits according to Newton’s method; the construction of improved

tables of refraction, which remained in use for nearly a century;

a share in pendulum experiments carried out in England and Jamaica

with the object of verifying the variation of gravity in different

latitudes; a careful testing of Mayer’s lunar tables (§ 226), together

with improvements of them; and lastly, some work in connection with the

reform of the calendar made in 1752 (cf, chapter II., § 22).

218. It remains to give some account of the magnificent series of

observations carried out during Bradley’s administration of the

Greenwich Observatory.

These observations fall into two chief divisions of unequal merit,

those after 1749 having been made with some more accurate instruments

which a grant from the government enabled him at that time to procure.

The main work of the Observatory under Bradley consisted in taking

observations of fixed stars, and to a lesser extent of other bodies,

as they passed the meridian, the instruments used (the “mural

quadrant” and the “transit instrument”) being capable of motion only

in the meridian, and being therefore steadier and susceptible of

greater accuracy than those with more freedom of movement. The most

important observations taken during the years 1750-1762, amounting to

about 60,000, were published long after Bradley’s death in two large

volumes which appeared in 1798 and 1805. A selection of them had

been used earlier as the basis of a small star catalogue, published

in the _Nautical Almanac_ for 1773; but it was not till 1818 that

the publication of Bessel’s _Fundamenta Astronomiae_ (chapter XIII.,

§ 277), a catalogue of more than 3000 stars based on Bradley’s

observations, rendered these observations thoroughly available for

astronomical work. One reason for this apparently excessive delay is to

be found in Bradley’s way of working. Allusion has already been made

to a variety of causes which prevent the apparent place of a star, as

seen in the telescope and noted at the time, from being a satisfactory

permanent record of its position. There are various instrumental

errors, and errors due to refraction; again, if a star’s places at

two different times are to be compared, precession must be taken into

account; and Bradley himself unravelled in aberration and nutation

two fresh sources of error. In order therefore to put into a form

satisfactory for permanent reference a number of star observations,

it is necessary to make corrections which have the effect of allowing

for these various sources of error. This process of =reduction=, as

it is technically called, involves a certain amount of rather tedious

calculation, and though in modern observatories the process has been

so far systematised that it can be carried out almost according to

fixed rules by comparatively unskilled assistants, in Bradley’s time

it required more judgment, and it is doubtful if his assistants could

have performed the work satisfactorily, even if their time had not been

fully occupied with other duties. Bradley himself probably found the

necessary calculations tedious, and preferred devoting his energies to

work of a higher order. It is true that Delambre, the famous French

historian of astronomy, assures his readers that he had never found

the reduction of an observation tedious if performed the same day,

but a glance at any of his books is enough to shew his extraordinary

fondness for long calculations of a fairly elementary character, and

assuredly Bradley is not the only astronomer whose tastes have in this

respect differed fundamentally from Delambre’s. Moreover reducing

an observation is generally found to be a duty that, like answering

letters, grows harder to perform the longer it is neglected; and it

is not only less interesting but also much more difficult for an

astronomer to deal satisfactorily with some one else’s observations

than with his own. It is not therefore surprising that after Bradley’s

death a long interval should have elapsed before an astronomer appeared

with both the skill and the patience necessary for the complete

reduction of Bradley’s 60,000 observations.

A variety of circumstances combined to make Bradley’s observations

decidedly superior to those of his predecessors. He evidently

possessed in a marked degree the personal characteristics—of eye and

judgment—which make a first-rate observer; his instruments were mounted

in the best known way for securing accuracy, and were constructed by

the most skilful makers; he made a point of studying very carefully the

defects of his instruments, and of allowing for them; his discoveries

of aberration and nutation enabled him to avoid sources of error,

amounting to a considerable number of seconds, which his predecessors

could only have escaped imperfectly by taking the average of a number

of observations; and his improved tables of refraction still further

added to the correctness of his results.

Bessel estimates that the errors in Bradley’s observations of

the declination of stars were usually less than 4″, while the

corresponding errors in right ascension, a quantity which depends

ultimately on a time-observation, were less than 15″, or one second of

time. His observations thus shewed a considerable advance in accuracy

compared with those of Flamsteed (§ 198), which represented the best

that had hitherto been done.

219. The next Astronomer Royal was _Nathaniel Bliss_ (1700-1764), who

died after two years. He was in turn succeeded by _Nevil Maskelyne_

(1732-1811), who carried on for nearly half a century the tradition of

accurate observation which Bradley had established at Greenwich, and

made some improvements in methods.

To him is also due the first serious attempt to measure the density and

hence the mass of the earth. By comparing the attraction exerted by the

earth with that of the sun and other bodies, Newton, as we have seen

(chapter IX., § 185), had been able to connect the masses of several

of the celestial bodies with that of the earth. To connect the mass of

the whole earth with that of a given terrestrial body, and so express

it in pounds or tons, was a problem of quite a different kind. It is of

course possible to examine portions of the earth’s surface and compare

their density with that of, say, water; then to make some conjecture,

based on rough observations in mines, etc., as to the rate at which

density increases as we go from the surface towards the centre of the

earth, and hence to infer the average density of the earth. Thus the

mass of the whole earth is compared with that of a globe of water of

the same size, and, the size being known, is expressible in pounds or

tons.

By a process of this sort Newton had in fact, with extraordinary

insight, estimated that the density of the earth was between five and

six times as great as that of water.[122]

It was, however, clearly desirable to solve the problem in a less

conjectural manner, by a direct comparison of the gravitational

attraction exerted by the earth with that exerted by a known mass—a

method that would at the same time afford a valuable test of Newton’s

theory of the gravitating properties of portions of the earth, as

distinguished from the whole earth. In their Peruvian expedition (§

221), Bouguer and La Condamine had noticed certain small deflections

of the plumb-line, which indicated an attraction by Chimborazo, near

which they were working; but the observations were too uncertain to

be depended on. Maskelyne selected for his purpose Schehallien in

Perthshire, a narrow ridge running east and west. The direction of

the plumb-line was observed (1774) on each side of the ridge, and a

change in direction amounting to about 12″ was found to be caused by

the attraction of the mountain. As the direction of the plumb-line

depends on the attraction of the earth as a whole and on that of the

mountain, this deflection at once led to a comparison of the two

attractions. Hence an intricate calculation performed by _Charles

Hutton_ (1737-1823) led to a comparison of the average densities of the

earth and mountain, and hence to the final conclusion (published in

1778) that the earth’s density was about 4-1∕2 times that of water. As

Hutton’s estimate of the density of the mountain was avowedly almost

conjectural, this result was of course correspondingly uncertain.

A few years later _John Michell_ (1724-1793) suggested, and the famous

chemist and electrician _Henry Cavendish_ (1731-1810) carried out

(1798), an experiment in which the mountain was replaced by a pair of

heavy balls, and their attraction on another body was compared with

that of the earth, the result being that the density of the earth was

found to be about 5-1∕2 times that of water.

The =Cavendish experiment=, as it is often called, has since been

repeated by various other experimenters in modified forms, and one or

two other methods, too technical to be described here, have also been

devised. All the best modern experiments give for the density numbers

converging closely on 5-1∕2, thus verifying in a most striking way both

Newton’s conjecture and Cavendish’s original experiment.

With this value of the density the mass of the earth is a

little more than 13 billion billion pounds, or more precisely

13,136,000,000,000,000,000,000,000 lbs.

220. While Greenwich was furnishing the astronomical world with a

most valuable series of observations, the Paris Observatory had not

fulfilled its early promise. It was in fact suffering, like English

mathematics, from the evil effects of undue adherence to the methods

and opinions of a distinguished man. Domenico Cassini happened to hold

several erroneous opinions in important astronomical matters; he was

too good a Catholic to be a genuine Coppernican, he had no belief in

gravitation, he was firmly persuaded that the earth was flattened at

the equator instead of at the poles, and he rejected Roemer’s discovery

of the velocity of light. After his death in 1712 the directorship of

the Observatory passed in turn to three of his descendants, the last

of whom resigned office in 1793; and several members of the Maraldi

family, into which his sister had married, worked in co-operation with

their cousins. Unfortunately a good deal of their energy was expended,

first in defending, and afterwards in gradually withdrawing from, the

errors of their distinguished head. _Jacques Cassini_ for example, the

second of the family (1677-1756), although a Coppernican, was still a

timid one, and rejected Kepler’s law of areas; his son again, commonly

known as _Cassini de Thury_ (1714-1784), still defended the ancestral

errors as to the form of the earth; while the fourth member of the

family, _Count Cassini_ (1748-1845), was the first of the family to

accept the Newtonian idea of gravitation.

Some planetary and other observations of value were made by the

Cassini-Maraldi school, but little of this work was of first-rate

importance.

221. A series of important measurements of the earth, in which the

Cassinis had a considerable share, were made during the 18th century,

almost entirely by Frenchmen, and resulted in tolerably exact knowledge

of the earth’s size and shape.

The variation of the length of the seconds pendulum observed by Richer

in his Cayenne expedition (chapter VIII., § 161) had been the first

indication of a deviation of the earth from a spherical form. Newton

inferred, both from these pendulum experiments and from an independent

theoretical investigation (chapter IX., § 187), that the earth was

spheroidal, being flattened towards the poles; and this view was

strengthened by the satisfactory explanation of precession to which it

led (chapter IX., § 188).

On the other hand, a comparison of various measurements of arcs of the

meridian in different latitudes gave some support to the view that

the earth was elongated towards the poles and flattened towards the

equator, a view championed with great ardour by the Cassini school.

It was clearly important that the question should be settled by more

extensive and careful earth-measurements.

The essential part of an ordinary measurement of the earth consists

in ascertaining the distance in miles between two places on the same

meridian, the latitudes of which differ by a known amount. From

these two data the length of an arc of a meridian corresponding to a

difference of latitude of 1° at once follows. The =latitude= of a place

is the angle which the vertical at the place makes with the equator,

or, expressed in a slightly different form, is the angular distance of

the zenith from the celestial equator. The =vertical= at any place may

be defined as a direction perpendicular to the surface of still water

at the place in question, and may be regarded as perpendicular to the

true surface of the earth, accidental irregularities in its form such

as hills and valleys being ignored.[123]

The difference of latitude between two places, north and south of one

another, is consequently the angle between the verticals there. Fig.

78 shews the verticals, marked by the arrowheads, at places on the

same meridian in latitudes differing by 10°; so that two consecutive

verticals are inclined in every case at an angle of 10°.

[Illustration: FIG. 78.—The varying curvature of the earth.]

If, as in fig. 78, the shape of the earth is drawn in accordance with

Newton’s views, the figure shews at once that the arcs A A_{1}, A_{1}

A_{2}, etc., each of which corresponds to 10° of latitude, steadily

_increase_ as we pass from a point A on the equator to the pole B. If

the opposite hypothesis be adopted, which will be illustrated by the

same figure if we now regard A as the pole and B as a point on the

equator, then the successive arcs _decrease_ as we pass from equator

to pole. A comparison of the measurements made by Eratosthenes in

Egypt (chapter II., § 36) with some made in Europe (chapter VIII., §

159) seemed to indicate that a degree of the meridian near the equator

was longer than one in higher latitudes; and a similar conclusion was

indicated by a comparison of different portions of an extensive French

arc, about 9° in length, extending from Dunkirk to the Pyrenees, which

was measured under the superintendence of the Cassinis in continuation

of Picard’s arc, the result being published by J. Cassini in 1720.

In neither case, however, were the data sufficiently accurate to

justify the conclusion; and the first decisive evidence was obtained

by measurement of arcs in places differing far more widely in latitude

than any that had hitherto been available. The French Academy organised

an expedition to Peru, under the management of three Academicians,

_Pierre Bouguer_ (1698-1758), _Charles Marie de La Condamine_

(1701-1774), and _Louis Godin_ (1704-1760), with whom two Spanish naval

officers also co-operated.

The expedition started in 1735, and, owing to various difficulties, the

work was spread out over nearly ten years. The most important result

was the measurement, with very fair accuracy, of an arc of about 3° in

length, close to the equator; but a number of pendulum experiments of

value were also performed, and a good many miscellaneous additions to

knowledge were made.

But while the Peruvian party were still at their work a similar

expedition to Lapland, under the Academician _Pierre Louis Moreau de

Maupertuis_ (1698-1759), had much more rapidly (1736-7), if somewhat

carelessly, effected the measurement of an arc of nearly 1° close to

the arctic circle.

From these measurements it resulted that the lengths of a degree

of a meridian about latitude 2° S. (Peru), about latitude 47° N.

(France) and about latitude 66° N. (Lapland) were respectively 362,800

feet, 364,900 feet, and 367,100 feet.[124] There was therefore clear

evidence, from a comparison of any two of these arcs, of an increase

of the length of a degree of a meridian as the latitude increases; and

the general correctness of Newton’s views as against Cassini’s was thus

definitely established.

The extent to which the earth deviates from a sphere is usually

expressed by a fraction known as the =ellipticity=, which is the

difference between the lines C A, C B of fig. 78 divided by the greater

of them. From comparison of the three arcs just mentioned several very

different values of the ellipticity were deduced, the discrepancies

being partly due to different theoretical methods of interpreting the

results and partly to errors in the arcs.

A measurement, made by _Jöns Svanberg_ (1771-1851) in 1801-3, of an

arc near that of Maupertuis has in fact shewn that his estimate of the

length of a degree was about 1,000 feet too large.

A large number of other arcs have been measured in different parts

of the earth at various times during the 18th and 19th centuries.

The details of the measurements need not be given, but to prevent

recurrence to the subject it is convenient to give here the results,

obtained by a comparison of these different measurements, that the

ellipticity is very nearly 1∕292, and the greatest radius of the earth

(C A in fig. 78) a little less than 21,000,000 feet or 4,000 miles. It

follows from these figures that the length of a degree in the latitude

of London contains, to use Sir John Herschel’s ingenious mnemonic,

almost exactly as many thousand feet as the year contains days.

222. Reference has already been made to the supremacy of Greenwich

during the 18th century in the domain of exact observation. France,

however, produced during this period one great observing astronomer

who actually accomplished much, and under more favourable external

conditions might almost have rivalled Bradley.

_Nicholas Louis de Lacaille_ was born in 1713. After he had devoted

a good deal of time to theological studies with a view to an

ecclesiastical career, his interests were diverted to astronomy and

mathematics. He was introduced to Jacques Cassini, and appointed one of

the assistants at the Paris Observatory.

In 1738 and the two following years he took an active part in the

measurement of the French arc, then in process of verification.

While engaged in this work he was appointed (1739) to a poorly paid

professorship at the Mazarin College, at which a small observatory was

erected. Here it was his regular practice to spend the whole night, if

fine, in observation, while “to fill up usefully the hours of leisure

which bad weather gives to observers only too often” he undertook a

variety of extensive calculations and wrote innumerable scientific

memoirs. It is therefore not surprising that he died comparatively

early (1762) and that his death was generally attributed to overwork.

223. The monotony of Lacaille’s outward life was broken by the

scientific expedition to the Cape of Good Hope (1750-1754) organised by

the Academy of Sciences and placed under his direction.

The most striking piece of work undertaken during this expedition was

a systematic survey of the southern skies, in the course of which more

than 10,000 stars were observed.

These observations, together with a carefully executed catalogue

of nearly 2,000 of the stars[125] and a star-map, were published

posthumously in 1763 under the title _Coelum Australe Stelliferum_, and

entirely superseded Halley’s much smaller and less accurate catalogue

(§ 199). Lacaille found it necessary to make 14 new constellations

(some of which have since been generally abandoned), and to restore

to their original places the stars which the loyal Halley had made

into King Charles’s Oak. Incidentally Lacaille observed and described

42 nebulae, nebulous stars, and star-clusters, objects the systematic

study of which was one of Herschel’s great achievements (chapter XII.,

§§ 259-261).

He made a large number of pendulum experiments, at Mauritius as well

as at the Cape, with the usual object of determining in a new part

of the world the acceleration due to gravity, and measured an arc of

the meridian extending over rather more than a degree. He made also

careful observations of the positions of Mars and Venus, in order that

from comparison of them with simultaneous observations in northern

latitudes he might get the parallax of the sun (chapter VIII., §

161). These observations of Mars compared with some made in Europe by

Bradley and others, and a similar treatment of Venus, both pointed to

a solar parallax slightly in excess of 10″, a result less accurate

than Cassini’s (chapter VIII., § 161), though obtained by more reliable

processes.

A large number of observations of the moon, of which those made by

him at the Cape formed an important part, led, after an elaborate

discussion in which the spheroidal form of the earth was taken into

account, to an improved value of the moon’s distance, first published

in 1761.

Lacaille also used his observations of fixed stars to improve our

knowledge of refraction, and obtained a number of observations of the

sun in that part of its orbit which it traverses in our winter months

(the summer of the southern hemisphere), and in which it is therefore

too near the horizon to be observed satisfactorily in Europe.

The results of this—one of the most fruitful scientific expeditions

ever undertaken—were published in separate memoirs or embodied in

various books published after his return to Paris.

224. In 1757, under the title _Astronomiae Fundamenta_, appeared a

catalogue of 400 of the brightest stars, observed and reduced with

the most scrupulous care, so that, notwithstanding the poverty of

Lacaille’s instrumental outfit, the catalogue was far superior to any

of its predecessors, and was only surpassed by Bradley’s observations

as they were gradually published. It is characteristic of Lacaille’s

unselfish nature that he did not have the _Fundamenta_ sold in the

ordinary way, but distributed copies gratuitously to those interested

in the subject, and earned the money necessary to pay the expenses of

publication by calculating some astronomical almanacks.

Another catalogue, of rather more than 500 stars situated in the

zodiac, was published posthumously.

In the following year (1758) he published an excellent set of Solar

Tables, based on an immense series of observations and calculations.

These were remarkable as the first in which planetary perturbations

were taken into account.

Among Lacaille’s minor contributions to astronomy may be mentioned:

improved methods of calculating cometary orbits and the actual

calculation of the orbits of a large number of recorded comets, the

calculation of all eclipses visible in Europe since the year 1, a

warning that the transit of Venus would be capable of far less accurate

observation than Halley had expected (§ 202), observations of the

actual transit of 1761 (§ 227), and a number of improvements in

methods of calculation and of utilising observations.

In estimating the immense mass of work which Lacaille accomplished

during an astronomical career of about 22 years, it has also to be

borne in mind that he had only moderately good instruments at his

observatory, and _no assistant_, and that a considerable part of his

time had to be spent in earning the means of living and of working.

225. During the period under consideration Germany also produced one

astronomer, primarily an observer, of great merit, _Tobias Mayer_

(1723-1762). He was appointed professor of mathematics and political

economy at Göttingen in 1751, apparently on the understanding that he

need not lecture on the latter subject, of which indeed he seems to

have professed no knowledge; three years later he was put in charge

of the observatory, which had been erected 20 years before. He had at

least one fine instrument,[126] and following the example of Tycho,

Flamsteed, and Bradley, he made a careful study of its defects, and

carried further than any of his predecessors the theory of correcting

observations for instrumental errors.[127]

He improved Lacaille’s tables of the sun, and made a catalogue of

998 zodiacal stars, published posthumously in 1775; by a comparison

of star places recorded by Roemer (1706) with his own and Lacaille’s

observations he obtained evidence of a considerable number of proper

motions (§ 203); and he made a number of other less interesting

additions to astronomical knowledge.

226. But Mayer’s most important work was on the moon. At the beginning

of his career he made a careful study of the position of the craters

and other markings, and was thereby able to get a complete geometrical

explanation of the various librations of the moon (chapter VI., § 133),

and to fix with accuracy the position of the axis about which the moon

rotates. A map of the moon based on his observations was published with