Chapter XIII: The Nineteenth Century (1)

“The greater the sphere of our knowledge, the larger is the surface of its contact with the infinity of our ignorance.”

272. The last three chapters have contained some account of progress made in three branches of astronomy which, though they overlap and exercise an important influence on one another, are to a large extent studied by different men and by different methods, and have different aims. The difference is perhaps best realised by thinking of the work of a great master in each department, Bradley, Laplace, and Herschel. So great is the difference that Delambre in his standard history of astronomy all but ignores the work of the great school of mathematical astronomers who were his contemporaries and immediate predecessors, not from any want of appreciation of their importance, but because he regards their work as belonging rather to mathematics than to astronomy; while Bessel (§ 277), in saying that the function of astronomy is “to assign the places on the sky where sun, moon, planets, comets, and stars have been, are, and will be,” excludes from its scope nearly everything towards which Herschel’s energies were directed.

Current modern practice is, however, more liberal in its use of language than either Delambre or Bessel, and finds it convenient to recognise all three of the subjects or groups of subjects referred to as integral parts of one science.

The mutual relation of gravitational astronomy and what has been for convenience called observational astronomy has been already referred to (chapter X., § 196). It should, however, be noticed that the latter term has in this book hitherto been used chiefly for only one part of the astronomical work which concerns itself primarily with observation. Observing played at least as large a part in Herschel’s work as in Bradley’s, but the aims of the two men were in many ways different. Bradley was interested chiefly in ascertaining as accurately as possible the apparent positions of the fixed stars on the celestial sphere, and the positions and motions of the bodies of the solar system, the former undertaking being in great part subsidiary to the latter. Herschel, on the other hand, though certain of his researches, _e.g._ into the parallax of the fixed stars and into the motions of the satellites of Uranus, were precisely like some of Bradley’s, was far more concerned with questions of the appearances, mutual relations, and structure of the celestial bodies in themselves. This latter branch of astronomy may conveniently be called =descriptive astronomy=, though the name is not altogether appropriate to inquiries into the physical structure and chemical constitution of celestial bodies which are often put under this head, and which play an important part in the astronomy of the present day.

273. Gravitational astronomy and exact observational astronomy have made steady progress during the nineteenth century, but neither has been revolutionised, and the advances made have been to a great extent of such a nature as to be barely intelligible, still less interesting, to those who are not experts. The account of them to be given in this chapter must therefore necessarily be of the slightest character, and deal either with general tendencies or with isolated results of a less technical character than the rest.

Descriptive astronomy, on the other hand, which can be regarded as being almost as much the creation of Herschel as gravitational astronomy is of Newton, has not only been greatly developed on the lines laid down by its founder, but has received—chiefly through the invention of spectrum analysis (§ 299)—extensions into regions not only unthought of but barely imaginable a century ago. Most of the results of descriptive astronomy—unlike those of the older branches of the subject—are readily intelligible and fairly interesting to those who have but little knowledge of the subject; in particular they are as yet to a considerable extent independent of the mathematical ideas and language which dominate so much of astronomy and render it unattractive or inaccessible to many. Moreover, not only can descriptive astronomy be appreciated and studied, but its progress can materially be assisted, by observers who have neither knowledge of higher mathematics nor any elaborate instrumental equipment.

Accordingly, while the successors of Laplace and Bradley have been for the most part astronomers by profession, attached to public observatories or to universities, an immense mass of valuable descriptive work has been done by amateurs who, like Herschel in the earlier part of his career, have had to devote a large part of their energies to professional work of other kinds, and who, though in some cases provided with the best of instruments, have in many others been furnished with only a slender instrumental outfit. For these and other reasons one of the most notable features of nineteenth century astronomy has been a great development, particularly in this country and in the United States, of general interest in the subject, and the establishment of a large number of private observatories devoted almost entirely to the study of special branches of descriptive astronomy. The nineteenth century has accordingly witnessed the acquisition of an unprecedented amount of detailed astronomical knowledge. But the wealth of material thus accumulated has outrun our powers of interpretation, and in a number of cases our knowledge of some particular department of descriptive astronomy consists, on the one hand of an immense series of careful observations, and on the other of one or more highly speculative theories, seldom capable of explaining more than a small portion of the observed facts.

In dealing with the progress of modern descriptive astronomy the proverbial difficulty of seeing the wood on account of the trees is therefore unusually great. To give an account within the limits of a single chapter of even the most important facts added to our knowledge would be a hopeless endeavour; fortunately it would also be superfluous, as they are to be found in many easily accessible textbooks on astronomy, or in treatises on special parts of the subject. All that can be attempted is to give some account of the chief lines on which progress has been made, and to indicate some general conclusions which seem to be established on a tolerably secure basis.

274. The progress of exact observation has of course been based very largely on instrumental advances. Not only have great improvements been made in the extremely delicate work of making large lenses, but the graduated circles and other parts of the mounting of a telescope upon which accuracy of measurement depends can also be constructed with far greater exactitude and certainty than at the beginning of the century. New methods of mounting telescopes and of making and recording observations have also been introduced, all contributing to greater accuracy. For certain special problems photography is found to present great advantages as compared with eye-observations, though its most important applications have so far been to descriptive astronomy.

275. The necessity for making allowance for various known sources of errors in observation, and for diminishing as far as possible the effect of errors due to unknown causes, had been recognised even by Tycho Brahe (chapter V., § 110), and had played an important part in the work of Flamsteed and Bradley (chapter X., §§ 198, 218). Some further important steps in this direction were taken in the earlier part of this century. The method of =least squares=, established independently by two great mathematicians, _Adrien Marie Legendre_ (1752-1833) of Paris and _Carl Friedrich Gauss_ (1777-1855) of Göttingen,[159] was a systematic method of combining observations, which gave slightly different results, in such a way as to be as near the truth as possible. Any ordinary physical measurement, _e.g._ of a length, however carefully executed, is necessarily imperfect; if the same measurement is made several times, even under almost identical conditions, the results will in general differ slightly; and the question arises of combining these so as to get the most satisfactory result. The common practice in this simple case has long been to take the arithmetical mean or average of the different results. But astronomers have constantly to deal with more complicated cases in which _two_ or more unknown quantities have to be determined from observations of different quantities, as, for example, when the elements of the orbit of a planet (chapter XI., § 236) have to be found from observations of the planet’s position at different times. The method of least squares gives a rule for dealing with such cases, which was a generalisation of the ordinary rule of averages for the case of a single unknown quantity; and it was elaborated in such a way as to provide for combining observations of different value, such as observations taken by observers of unequal skill or with different instruments, or under more or less favourable conditions as to weather, etc. It also gives a simple means of testing, by means of their mutual consistency, the value of a series of observations, and comparing their probable accuracy with that of some other series executed under different conditions. The method of least squares and the special case of the “average” can be deduced from a certain assumption as to the general character of the causes which produce the error in question; but the assumption itself cannot be justified _a priori_; on the other hand, the satisfactory results obtained from the application of the rule to a great variety of problems in astronomy and in physics has shewn that in a large number of cases unknown causes of error must be approximately of the type considered. The method is therefore very widely used in astronomy and physics wherever it is worth while to take trouble to secure the utmost attainable accuracy.

276. Legendre’s other contributions to science were almost entirely to branches of mathematics scarcely affecting astronomy. Gauss, on the other hand, was for nearly half a century head of the observatory of Göttingen, and though his most brilliant and important work was in pure mathematics, while he carried out some researches of first-rate importance in magnetism and other branches of physics, he also made some further contributions of importance to astronomy. These were for the most part processes of calculation of various kinds required for utilising astronomical observations, the best known being a method of calculating the orbit of a planet from three complete observations of its position, which was published in his _Theoria Motus_ (1809). As we have seen (chapter XI., § 236), the complete determination of a planet’s orbit depends on six independent elements: any complete observation of the planet’s position in the sky, at any time, gives two quantities, _e.g._ the right ascension and declination (chapter II., § 33); hence three complete observations give six equations and are theoretically adequate to determine the elements of the orbit; but it had not hitherto been found necessary to deal with the problem in this form. The orbits of all the planets but Uranus had been worked out gradually by the use of a series of observations extending over centuries; and it was feasible to use observations taken at particular times so chosen that certain elements could be determined without any accurate knowledge of the others; even Uranus had been under observation for a considerable time before its path was determined with anything like accuracy; and in the case of comets not only was a considerable series of observations generally available, but the problem was simplified by the fact that the orbit could be taken to be nearly or quite a parabola instead of an ellipse (chapter IX., § 190). The discovery of the new planet Ceres on January 1st, 1801 (§ 294), and its loss when it had only been observed for a few weeks, presented virtually a new problem in the calculation of an orbit. Gauss applied his new methods—including that of least squares—to the observations available, and with complete success, the planet being rediscovered at the end of the year nearly in the position indicated by his calculations.

277. The theory of the “reduction” of observations (chapter X., § 218) was first systematised and very much improved by _Friedrich Wilhelm Bessel_ (1784-1846), who was for more than thirty years the director of the new Prussian observatory at Königsberg. His first great work was the reduction and publication of Bradley’s Greenwich observations (chapter X., § 218). This undertaking involved an elaborate study of such disturbing causes as precession, aberration, and refraction, as well as of the errors of Bradley’s instruments. Allowance was made for these on a uniform and systematic plan, and the result was the publication in 1818, under the title _Fundamenta Astronomiae_, of a catalogue of the places of 3,222 stars as they were in 1755. A special problem dealt with in the course of the work was that of refraction. Although the complete theoretical solution was then as now unattainable, Bessel succeeded in constructing a table of refractions which agreed very closely with observation and was presented in such a form that the necessary correction for a star in almost any position could be obtained with very little trouble. His general methods of reduction—published finally in his _Tabulae Regiomontanae_ (1830)—also had the great advantage of arranging the necessary calculations in such a way that they could be performed with very little labour and by an almost mechanical process, such as could easily be carried out by a moderately skilled assistant. In addition to editing Bradley’s observations, Bessel undertook a fresh series of observations of his own, executed between the years 1821 and 1833, upon which were based two new catalogues, containing about 62,000 stars, which appeared after his death.

278. The most memorable of Bessel’s special pieces of work was the first definite detection of the parallax of a fixed star. He abandoned the test of brightness as an indication of nearness, and selected a star (_61 Cygni_) which was barely visible to the naked eye but was remarkable for its large proper motion (about 5″ per annum); evidently if a star is moving at an assigned rate (in miles per hour) through space, the nearer to the observer it is the more rapid does its motion appear to be, so that apparent rapidity of motion, like brightness, is a probable but by no means infallible indication of nearness. A modification of Galilei’s differential method (chapter VI., § 129, and chapter XII., § 263) being adopted, the angular distance of _61 Cygni_ from two neighbouring stars, the faintness and immovability of which suggested their great distance in space, was measured at frequent intervals during a year. From the changes in these distances σ _a_, σ _b_ (in fig. 85), the size of the small ellipse described by σ could be calculated. The result, announced at the end of 1838, was that the star had an annual parallax of about 1∕3″ (chapter VIII., § 161), _i.e._ that the star was at such distance that the greatest angular distance of the earth from the sun viewed from the star (the angle S σ E in fig. 86, where S is the sun and E the earth) was this insignificant angle.[160] The result was confirmed, with slight alterations, by a fresh investigation of Bessel’s in 1839-40, but later work seems to shew that the parallax is a little less than 1∕2″.[161] With this latter estimate, the apparent size of the earth’s path round the sun as seen from the star is the same as that of a halfpenny at a distance of rather more than three miles. In other words, the distance of the star is about 400,000 times the distance of the sun, which is itself about 93,000,000 miles. A mile is evidently a very small unit by which to measure such a vast distance; and the practice of expressing such distances by means of the time required by light to perform the journey is often convenient. Travelling at the rate of 186,000 miles _per second_ (§ 283), light takes rather more than six years to reach us from _61 Cygni_.

279. Bessel’s solution of the great problem which had baffled astronomers ever since the time of Coppernicus was immediately followed by two others. Early in 1839 _Thomas Henderson_ (1798-1844) announced a parallax of nearly 1″ for the bright star α _Centauri_ which he had observed at the Cape, and in the following year _Friedrich Georg Wilhelm Struve_ (1793-1864) obtained from observations made at Pulkowa a parallax of 1∕4″ for _Vega_; later work has reduced these numbers to 3∕4″ and 1∕10″ respectively.

A number of other parallax determinations have subsequently been made. An interesting variation in method was made by the late Professor _Charles Pritchard_ (1808-1893) of Oxford by _photographing_ the star to be examined and its companions, and subsequently measuring the distances on the photograph, instead of measuring the angular distances directly with a micrometer.

At the present time some 50 stars have been ascertained with some reasonable degree of probability to have measurable, if rather uncertain, parallaxes; α _Centauri_ still holds its own as the nearest star, the light-journey from it being about four years. A considerable number of other stars have been examined with negative or highly uncertain results, indicating that their parallaxes are too small to be measured with our present means, and that their distances are correspondingly great.

280. A number of star catalogues and star maps—too numerous to mention separately—have been constructed during this century, marking steady progress in our knowledge of the position of the stars, and providing fresh materials for ascertaining, by comparison of the state of the sky at different epochs, such quantities as the proper motions of the stars and the amount of precession. Among the most important is the great catalogue of 324,198 stars in the northern hemisphere known as the Bonn _Durchmusterung_, published in 1859-62 by Bessel’s pupil _Friedrich Wilhelm August Argelander_ (1799-1875); this was extended (1875-85) so as to include 133,659 stars in a portion of the southern hemisphere by _Eduard Schönfeld_ (1828-1891); and more recently Dr. _Gill_ has executed at the Cape photographic observations of the remainder of the southern hemisphere, the reduction to the form of a catalogue (the first instalment of which was published in 1896) having been performed by Professor _Kapteyn_ of Groningen. The star places determined in these catalogues do not profess to be the most accurate attainable, and for many purposes it is important to know with the utmost accuracy the positions of a smaller number of stars. The greatest undertaking of this kind, set on foot by the German Astronomical Society in 1867, aims at the construction, by the co-operation of a number of observatories, of catalogues of about 130,000 of the stars contained in the “approximate” catalogues of Argelander and Schönfeld; nearly half of the work has now been published.

The greatest scheme for a survey of the sky yet attempted is the photographic chart, together with a less extensive catalogue to be based on it, the construction of which was decided on at an international congress held at Paris in 1887. The whole sky has been divided between 18 observatories in all parts of the world, from Helsingfors in the north to Melbourne in the south, and each of these is now taking photographs with virtually identical instruments. It is estimated that the complete chart, which is intended to include stars of the 14th magnitude,[162] will contain about 20,000,000 stars, 2,000,000 of which will be catalogued also.

281. One other great problem—that of the distance of the sun—may conveniently be discussed under the head of observational astronomy.

The transits of Venus (chapter X., §§ 202, 227) which occurred in 1874 and 1882 were both extensively observed, the old methods of time-observation being supplemented by photography and by direct micrometric measurements of the positions of Venus while transiting.

The method of finding the distance of the sun by means of observation of Mars in opposition (chapter VIII., § 161) has been employed on several occasions with considerable success, notably by Dr. Gill at Ascension in 1877. A method originally used by Flamsteed, but revived in 1857 by _Sir George Biddell Airy_ (1801-1892), the late Astronomer Royal, was adopted on this occasion. For the determination of the parallax of a planet observations have to be made from two different positions at a known distance apart; commonly these are taken to be at two different observatories, as far as possible removed from one another in latitude. Airy pointed out that the same object could be attained if only one observatory were used, but observations taken at an interval of some hours, as the rotation of the earth on its axis would in that time produce a known displacement of the observer’s position and so provide the necessary base line. The apparent shift of the planet’s position could be most easily ascertained by measuring (with the micrometer) its distances from neighbouring fixed stars. This method (known as the =diurnal method=) has the great advantage, among others, of being simple in application, a single observer and instrument being all that is needed.

The diurnal method has also been applied with great success to certain of the minor planets (§ 294). Revolving as they do between Mars and Jupiter, they are all farther off from us than the former; but there is the compensating advantage that as a minor planet, unlike Mars, is, as a rule, too small to shew any appreciable disc, its angular distance from a neighbouring star is more easily measured. The employment of the minor planets in this way was first suggested by Professor _Galle_ of Berlin in 1872, and recent observations of the minor planets _Victoria_, _Sappho_, and _Iris_ in 1888-89, made at a number of observatories under the general direction of Dr. Gill, have led to some of the most satisfactory determinations of the sun’s distance.

282. It was known to the mathematical astronomers of the 18th century that the distance of the sun could be obtained from a knowledge of various perturbations of members of the solar system; and Laplace had deduced a value of the solar parallax from lunar theory. Improvements in gravitational astronomy and in observation of the planets and moon during the present century have added considerably to the value of these methods. A certain irregularity in the moon’s motion known as the =parallactic inequality=, and another in the motion of the sun, called the =lunar equation=, due to the displacement of the earth by the attraction of the moon, alike depend on the ratio of the distances of the sun and moon from the earth; if the amount of either of these inequalities can be observed, the distance of the sun can therefore be deduced, that of the moon being known with great accuracy. It was by a virtual application of the first of these methods that Hansen (§ 286) in 1854, in the course of an elaborate investigation of the lunar theory, ascertained that the current value of the sun’s distance was decidedly too large, and Leverrier (§ 288) confirmed the correction by the second method in 1858.

Again, certain changes in the orbits of our two neighbours, Venus and Mars, are known to depend upon the ratio of the masses of the sun and earth, and can hence be connected, by gravitational principles, with the quantity sought. Leverrier pointed out in 1861 that the motions of Venus and of Mars, like that of the moon, were inconsistent with the received estimate of the sun’s distance, and he subsequently worked out the method more completely and deduced (1872) values of the parallax. The displacements to be observed are very minute, and their accurate determination is by no means easy, but they are both secular (chapter XI., § 242), so that in the course of time they will be capable of very exact measurement. Leverrier’s method, which is even now a valuable one, must therefore almost inevitably outstrip all the others which are at present known; it is difficult to imagine, for example, that the transits of Venus due in 2004 and 2012 will have any value for the purpose of the determination of the sun’s distance.

283. One other method, in two slightly different forms, has become available during this century. The displacement of a star by aberration (chapter X., § 210) depends upon the ratio of the velocity of light to that of the earth in its orbit round the sun; and observations of Jupiter’s satellites after the manner of Roemer (chapter VIII., § 162) give the =light-equation=, or time occupied by light in travelling from the sun to the earth. Either of these astronomical quantities—of which aberration is the more accurately known—can be used to determine the velocity of light when the dimensions of the solar system are known, or _vice versa_. No independent method of determining the velocity of light was known until 1849, when _Hippolyte Fizeau_ (1819-1896) invented and successfully carried out a laboratory method.

New methods have been devised since, and three comparatively recent series of experiments, by M. _Cornu_ in France (1874 and 1876) and by Dr. _Michelson_ (1879) and Professor _Newcomb_ (1880-82) in the United States, agreeing closely with one another, combine to fix the velocity of light at very nearly 186,300 miles (299,800 kilometres) per second; the solar parallax resulting from this by means of aberration is very nearly 8″·8.[163]

284. Encke’s value of the sun’s parallax, 8″·571, deduced from the transits of Venus (chapter X., § 227) in 1761 and 1769, and published in 1835, corresponding to a distance of about 95,000,000 miles, was generally accepted till past the middle of the century. Then the gravitational methods of Hansen and Leverrier, the earlier determinations of the velocity of light, and the observations made at the opposition of Mars in 1862, all pointed to a considerably larger value of the parallax; a fresh examination of the 18th century observations shewed that larger values than Encke’s could easily be deduced from them; and for some time—from about 1860 onwards—a parallax of nearly 8″·95, corresponding to a distance of rather more than 91,000,000 miles, was in common use. Various small errors in the new methods were, however, detected, and the most probable value of the parallax has again increased. Three of the most reliable methods, the diurnal method as applied to Mars in 1877, the same applied to the minor planets in 1888-89, and aberration, unite in giving values not differing from 8″·80 by more than two or three hundredths of a second. The results of the last transits of Venus, the publication and discussion of which have been spread over a good many years, point to a somewhat larger value of the parallax. Most astronomers appear to agree that a parallax of 8″·8, corresponding to a distance of rather less than 93,000,000 miles, represents fairly the available data.

285. The minute accuracy of modern observations is well illustrated by the recent discovery of a variation in the latitude of several observatories. Observations taken at Berlin in 1884-85 indicated a minute variation in the latitude; special series of observations to verify this were set on foot in several European observatories, and subsequently at Honolulu and at Cordoba. A periodic alteration in latitude amounting to about 1∕2″ emerged as the result. Latitude being defined (chapter X., § 221) as the angle which the vertical at any place makes with the equator, which is the same as the elevation of the pole above the horizon, is consequently altered by any change in the equator, and therefore by an alteration in the position of the earth’s poles or the ends of the axis about which it rotates.

Dr. _S. C. Chandler_ succeeded (1891 and subsequently) in shewing that the observations in question could be in great part explained by supposing the earth’s axis to undergo a minute change of position in such a way that either pole of the earth describes a circuit round its mean position in about 427 days, never deviating more than some 30 feet from it. It is well known from dynamical theory that a rotating body such as the earth can be displaced in this manner, but that if the earth were perfectly rigid the period should be 306 days instead of 427. The discrepancy between the two numbers has been ingeniously used as a test of the extent to which the earth is capable of yielding—like an elastic solid—to the various forces which tend to strain it.

286. All the great problems of gravitational astronomy have been rediscussed since Laplace’s time, and further steps taken towards their solution.

Laplace’s treatment of the lunar theory was first developed by _Marie Charles Theodore Damoiseau_ (1768-1846), whose _Tables de la Lune_ (1824 and 1828) were for some time in general use.

Some special problems of both lunar and planetary theory were dealt with by _Siméon Denis Poisson_ (1781-1840), who is, however, better known as a writer on other branches of mathematical physics than as an astronomer. A very elaborate and detailed theory of the moon, investigated by the general methods of Laplace, was published by _Giovanni Antonio Amadeo Plana_ (1781-1869) in 1832, but unaccompanied by tables. A general treatment of both lunar and planetary theories, the most complete that had appeared up to that time, by _Philippe Gustave Doulcet de Pontécoulant_ (1795-1874), appeared in 1846, with the title _Théorie Analytique du Système du Monde_; and an incomplete lunar theory similar to his was published by _John William Lubbock_ (1803-1865) in 1830-34.

A great advance in lunar theory was made by _Peter Andreas Hansen_ (1795-1874) of Gotha, who published in 1838 and 1862-64 the treatises commonly known respectively as the _Fundamenta_[164] and the _Darlegung_,[165] and produced in 1857 tables of the moon’s motion of such accuracy that the discrepancies between the tables and observations in the century 1750-1850 were never greater than 1″ or 2″. These tables were at once used for the calculation of the _Nautical Almanac_ and other periodicals of the same kind, and with some modifications have remained in use up to the present day.

A completely new lunar theory—of great mathematical interest and of equal complexity—was published by _Charles Delaunay_ (1816-1872) in 1860 and 1867. Unfortunately the author died before he was able to work out the corresponding tables.

Professor Newcomb of Washington (§ 283) has rendered valuable services to lunar theory—as to other branches of astronomy—by a number of delicate and intricate calculations, the best known being his comparison of Hansen’s tables with observation and consequent corrections of the tables.

New methods of dealing with lunar theory were devised by the late Professor _John Couch Adams_ of Cambridge (1819-1892), and similar methods have been developed by Dr. _G. W. Hill_ of Washington; so far they have not been worked out in detail in such a way as to be available for the calculation of tables, and their interest seems to be at present mathematical rather than practical; but the necessary detailed work is now in progress, and these and allied methods may be expected to lead to a considerable diminution of the present excessive intricacy of lunar theory.

287. One special point in lunar theory may be worth mentioning. The secular acceleration of the moon’s mean motion which had perplexed astronomers since its first discovery by Halley (chapter X., § 201) had, as we have seen (chapter XI., § 240), received an explanation in 1787 at the hands of Laplace. Adams, on going through the calculation, found that some quantities omitted by Laplace as unimportant had in reality a very sensible effect on the result, so that a certain quantity expressing the rate of increase of the moon’s motion came out to be between 5″ and 6″, instead of being about 10″, as Laplace had found and as observation required. The correction was disputed at first by several of the leading experts, but was confirmed independently by Delaunay and is now accepted. The moon appears in consequence to have a certain very minute increase in speed for which the theory of gravitation affords no explanation. An ingenious though by no means certain explanation was suggested by Delaunay in 1865. It had been noticed by Kant that =tidal friction=—that is, the friction set up between the solid earth and the ocean as the result of the tidal motion of the latter—would have the effect of checking to some extent the rotation of the earth; but as the effect seemed to be excessively minute and incapable of precise calculation it was generally ignored. An attempt to calculate its amount was, however, made in 1853 by _William Ferrel_, who also pointed out that, as the period of the earth’s rotation—the day—is our fundamental unit of time, a reduction of the earth’s rate of rotation involves the lengthening of our unit of time, and consequently produces an apparent increase of speed in all other motions measured in terms of this unit. Delaunay, working independently, arrived at like conclusions, and shewed that tidal friction might thus be capable of producing just such an alteration in the moon’s motion as had to be explained; if this explanation were accepted the observed motion of the moon would give a measure of the effect of tidal friction. The minuteness of the quantities involved is shewn by the fact that an alteration in the earth’s rotation equivalent to the lengthening of the day by 1∕10 second in 10,000 years is sufficient to explain the acceleration in question. Moreover it is by no means certain that the usual estimate of the amount of this acceleration—based as it is in part on ancient eclipse observations—is correct, and even then a part of it may conceivably be due to some indirect effect of gravitation even more obscure than that detected by Laplace, or to some other cause hitherto unsuspected.

288. Most of the writers on lunar theory already mentioned have also made contributions to various parts of planetary theory, but some of the most important advances in planetary theory made since the death of Laplace have been due to the French mathematician _Urbain Jean Joseph Leverrier_ (1811-1877), whose methods of determining the distance of the sun have been already referred to (§ 282). His first important astronomical paper (1839) was a discussion of the stability (chapter XI., § 245) of the system formed by the sun and the three largest and most distant planets then known, Jupiter, Saturn, and Uranus. Subsequently he worked out afresh the theory of the motion of the sun and of each of the principal planets, and constructed tables of them, which at once superseded earlier ones, and are now used as the basis of the chief planetary calculations in the _Nautical Almanac_ and most other astronomical almanacs. Leverrier failed to obtain a satisfactory agreement between observation and theory in the case of Mercury, a planet which has always given great trouble to astronomers, and was inclined to explain the discrepancies as due to the influence either of a planet revolving between Mercury and the sun or of a number of smaller bodies analogous to the minor planets (§ 294).

Researches of a more abstract character, connecting planetary theory with some of the most recent advances in pure mathematics, have been carried out by _Hugo Gyldén_ (1841-1896), while one of the most eminent pure mathematicians of the day, M. _Henri Poincaré_ of Paris, has recently turned his attention to astronomy, and is engaged in investigations which, though they have at present but little bearing on practical astronomy, seem likely to throw important light on some of the general problems of celestial mechanics.

289. One memorable triumph of gravitational astronomy, the discovery of Neptune, has been described so often and so fully elsewhere[166] that a very brief account will suffice here. Soon after the discovery of Uranus (chapter XII., § 253) it was found that the planet had evidently been observed, though not recognised as a planet, as early as 1690, and on several occasions afterwards.

When the first attempts were made to compute its orbit carefully, it was found impossible satisfactorily to reconcile the earlier with the later observations, and in Bouvard’s tables (chapter XI., § 247, note) published in 1821 the earlier observations were rejected. But even this drastic measure did not cure the evil; discrepancies between the observed and calculated places soon appeared and increased year by year. Several explanations were proposed, and more than one astronomer threw out the suggestion that the irregularities might be due to the attraction of a hitherto unknown planet. The first serious attempt to deduce from the irregularities in the motion of Uranus the position of this hypothetical body was made by Adams immediately after taking his degree (1843). By October 1845 he had succeeded in constructing an orbit for the new planet, and in assigning for it a position differing (as we now know) by less than 2° (four times the diameter of the full moon) from its actual position. No telescopic search for it was, however, undertaken. Meanwhile, Leverrier had independently taken up the inquiry, and by August 31st, 1846, he, like Adams, had succeeded in determining the orbit and the position of the disturbing body. On the 23rd of the following month Dr. Galle of the Berlin Observatory received from Leverrier a request to search for it, and on the same evening found close to the position given by Leverrier a strange body shewing a small planetary disc, which was soon recognised as a new planet, known now as Neptune.

It may be worth while noticing that the error in the motion of Uranus which led to this remarkable discovery never exceeded 2′, a quantity imperceptible to the ordinary eye; so that if two stars were side by side in the sky, one in the true position of Uranus and one in the calculated position as given by Bouvard’s tables, an observer of ordinary eyesight would see one star only.

290. The lunar tables of Hansen and Professor Newcomb, and the planetary and solar tables of Leverrier, Professor Newcomb, and Dr. Hill, represent the motions of the bodies dealt with much more accurately than the corresponding tables based on Laplace’s work, just as these were in turn much more accurate than those of Euler, Clairaut, and Halley. But the agreement between theory and observation is by no means perfect, and the discrepancies are in many cases greater than can be explained as being due to the necessary imperfections in our observations.

The two most striking cases are perhaps those of Mercury and the moon. Leverrier’s explanation of the irregularities of the former (§ 288) has never been fully justified or generally accepted; and the position of the moon as given in the _Nautical Almanac_ and in similar publications is calculated by means of certain corrections to Hansen’s tables which were deduced by Professor Newcomb from observation and have no justification in the theory of gravitation.

291. The calculation of the paths of comets has become of some importance during this century owing to the discovery of a number of comets revolving round the sun in comparatively short periods. Halley’s comet (chapter XI., § 231) reappeared duly in 1835, passing through its perihelion within a few days of the times predicted by three independent calculators; and it may be confidently expected again about 1910. Four other comets are now known which, like Halley’s, revolve in elongated elliptic orbits, completing a revolution in between 70 and 80 years; two of these have been seen at two returns, that known as Olbers’s comet in 1815 and 1887, and the Pons-Brooks comet in 1812 and 1884. Fourteen other comets with periods varying between 3-1∕3 years (Encke’s) and 14 years (Tuttle’s), have been seen at more than one return; about a dozen more have periods estimated at less than a century; and 20 or 30 others move in orbits that are decidedly elliptic, though their periods are longer and consequently not known with much certainty. Altogether the paths of about 230 or 240 comets have been computed, though many are highly uncertain.

292. In the theory of the tides the first important advance made after the publication of the _Mécanique Céleste_ was the collection of actual tidal observations on a large scale, their interpretation, and their comparison with the results of theory. The pioneers in this direction were Lubbock (§ 286), who presented a series of papers on the subject to the Royal Society in 1830-37, and _William Whewell_ (1794-1866), whose papers on the subject appeared between 1833 and 1851. Airy (§ 281), then Astronomer Royal, also published in 1845 an important treatise dealing with the whole subject, and discussing in detail the theory of tides in bodies of water of limited extent and special form. The analysis of tidal observations, a large number of which taken from all parts of the world are now available, has subsequently been carried much further by new methods due to Lord _Kelvin_ and Professor _G. H. Darwin_. A large quantity of information is thus available as to the way in which tides actually vary in different places and according to different positions of the sun and moon.

Of late years a good deal of attention has been paid to the effect of the attraction of the sun and moon in producing alterations—analogous to oceanic tides—in the earth itself. No body is perfectly rigid, and the forces in question must therefore produce some tidal effect. The problem was first investigated by Lord Kelvin in 1863, subsequently by Professor Darwin and others. Although definite numerical results are hardly attainable as yet, the work so far carried out points to the comparative smallness of these bodily tides and the consequent great rigidity of the earth, a result of interest in connection with geological inquiries into the nature of the interior of the earth.

Some speculations connected with tidal friction are referred to elsewhere (§ 320).

293. The series of propositions as to the stability of the solar system established by Lagrange and Laplace (chapter XI., §§ 244, 245), regarded as abstract propositions mathematically deducible from certain definite assumptions, have been confirmed and extended by later mathematicians such as Poisson and Leverrier; but their claim to give information as to the condition of the actual solar system at an indefinitely distant future time receives much less assent now than formerly. The general trend of scientific thought has been towards the fuller recognition of the merely approximate and probable character of even the best ascertained portions of our knowledge; “exact,” “always,” and “certain” are words which are disappearing from the scientific vocabulary, except as convenient abbreviations. Propositions which profess to be—or are commonly interpreted as being—“exact” and valid throughout all future time are consequently regarded with considerable distrust, unless they are clearly mere abstractions.

In the case of the particular propositions in question the progress of astronomy and physics has thrown a good deal of emphasis on some of the points in which the assumptions required by Lagrange and Laplace are not satisfied by the actual solar system.

It was assumed for the purposes of the stability theorems that the bodies of the solar system are perfectly rigid; in other words, the motions relative to one another of the parts of any one body were ignored. Both the ordinary tides of the ocean and the bodily tides to which modern research has called attention were therefore left out of account. Tidal friction, though at present very minute in amount (§ 287), differs essentially from the perturbations which form the main subject-matter of gravitational astronomy, inasmuch as its action is irreversible. The stability theorems shewed in effect that the ordinary perturbations produced effects which sooner or later compensated one another, so that if a particular motion was accelerated at one time it would be retarded at another; but this is not the case with tidal friction. Tidal action between the earth and the moon, for example, gradually lengthens both the day and the month, and increases the distance between the earth and the moon. Solar tidal action has a similar though smaller effect on the sun and earth. The effect in each case—as far as we can measure it at all—seems to be minute almost beyond imagination, but there is no compensating action tending at any time to reverse the process. And on the whole the energy of the bodies concerned is thereby lessened. Again, modern theories of light and electricity require space to be filled with an “ether” capable of transmitting certain waves; and although there is no direct evidence that it in any way affects the motions of earth or planets, it is difficult to imagine a medium so different from all known forms of ordinary matter as to offer _no_ resistance to a body moving through it. Such resistance would have the effect of slowly bringing the members of the solar system nearer to the sun, and gradually diminishing their times of revolution round it. This is again an irreversible tendency for which we know of no compensation.

In fact, from the point of view which Lagrange and Laplace occupied, the solar system appeared like a clock which, though not going quite regularly, but occasionally gaining and occasionally losing, nevertheless required no winding up; whereas modern research emphasises the analogy to a clock which after all is running down, though at an excessively slow rate. Modern study of the sun’s heat (§ 319) also indicates an irreversible tendency towards the “running down” of the solar system in another way.

294. Our account of modern descriptive astronomy may conveniently begin with planetary discoveries.

The first day of the 19th century was marked by the discovery of a new planet, known as Ceres. It was seen by _Giuseppe Piazzi_ (1746-1826) as a strange star in a region of the sky which he was engaged in mapping, and soon recognised by its motion as a planet. Its orbit—first calculated by Gauss (§ 276)—shewed it to belong to the space between Mars and Jupiter, which had been noted since the time of Kepler as abnormally large. That a planet should be found in this region was therefore no great surprise; but the discovery by _Heinrich Olbers_ (1758-1840), scarcely a year later (March 1802), of a second body (_Pallas_), revolving at nearly the same distance from the sun, was wholly unexpected, and revealed an entirely new planetary arrangement. It was an obvious conjecture that if there was room for two planets there was room for more, and two fresh discoveries (_Juno_ in 1804, _Vesta_ in 1807) soon followed.

The new bodies were very much smaller than any of the other planets, and, so far from readily shewing a planetary disc like their neighbours Mars and Jupiter, were barely distinguishable in appearance from fixed stars, except in the most powerful telescopes of the time; hence the name =asteroid= (suggested by William Herschel) or =minor planet= has been generally employed to distinguish them from the other planets. Herschel attempted to measure their size, and estimated the diameter of the largest at under 200 miles (that of Mercury, the smallest of the ordinary planets, being 3000), but the problem was in reality too difficult even for his unrivalled powers of observation. The minor planets were also found to be remarkable for the great inclination and eccentricity of some of the orbits; the path of Pallas, for example, makes an angle of 35° with the ecliptic, and its eccentricity is 1∕4, so that its least distance from the sun is not much more than half its greatest distance. These characteristics suggested to Olbers that the minor planets were in reality fragments of a primeval planet of moderate dimensions which had been blown to pieces, and the theory, which fitted most of the facts then known, was received with great favour in an age when “catastrophes” were still in fashion as scientific explanations.

The four minor planets named were for nearly 40 years the only ones known; then a fifth was discovered in 1845 by _Karl Ludwig Hencke_ (1793-1866) after 15 years, of search. Two more were found in 1847, another in 1848, and the number has gone on steadily increasing ever since. The process of discovery has been very much facilitated by improvements in star maps, and latterly by the introduction of photography. In this last method, first used by Dr. _Max Wolf_ of Heidelberg in 1891, a photographic plate is exposed for some hours; any planet present in the region of the sky photographed, having moved sensibly relatively to the stars in this period, is thus detected by the trail which its image leaves on the plate. The annexed figure shews (near the centre) the trail of the minor planet _Svea_, discovered by Dr. Wolf on March 21st, 1892.

At the end of 1897 no less than 432 minor planets were known, of which 92 had been discovered by a single observer, M. _Charlois_ of Nice, and only nine less by Professor _Palisa_ of Vienna.

The paths of the minor planets practically occupy the whole region between the paths of Mars and Jupiter, though few are near the boundaries; no orbit is more inclined to the ecliptic than that of Pallas, and the eccentricities range from almost zero up to about 1∕3.

Fig. 89 shews the orbits of the first two minor planets discovered, as well as of No. 323 (_Brucia_), which comes nearest to the sun, and of No. 361 (not yet named), which goes farthest from it. All the orbits are described in the standard, or west to east, direction. The most interesting characteristic in the distribution of the minor planets, first noted in 1866 by _Daniel Kirkwood_ (1815-1895) is the existence of comparatively clear spaces in the regions where the disturbing action of Jupiter would by Lagrange’s principle (chapter XI., § 243) be most effective: for instance, at a distance from the sun about five-eighths that of Jupiter, a planet would by Kepler’s law revolve exactly _twice_ as fast as Jupiter; and accordingly there is a gap among the minor planets at about this distance.

Estimates of the sizes and masses of the minor planets are still very uncertain. The first direct measurement of any of the discs which seem reliable are those of Professor _E. E. Barnard_, made at the Lick Observatory in 1894 and 1895; according to these the three largest minor planets, Ceres, Pallas, and Vesta, have diameters of nearly 500 miles, about 300 and about 250 miles respectively. Their sizes compared with the moon are shewn on the diagram (fig. 90). An alternative method—the only one available except for a few of the very largest of the minor planets—is to measure the amount of light received, and hence to deduce the size, on the assumption that the reflective power is the same as that of some known planet. This method gives diameters of about 300 miles for the brightest and of about a dozen miles for the faintest known.

Leverrier calculated from the perturbations of Mars that the total mass of all known or unknown bodies between Mars and Jupiter could not exceed a fourth that of the earth; but such knowledge of the sizes as we can derive from light-observations seems to indicate that the total mass of those at present known is many hundred times less than this limit.

295. Neptune and the minor planets are the only planets which have been discovered during this century, but several satellites have been added to our system.

Barely a fortnight after the discovery of Neptune (1846) a satellite was detected by _William Lassell_ (1799-1880) at Liverpool. Like the satellites of Uranus, this revolves round its primary from east to west—that is, in the direction contrary to that of all the other known motions of the solar system (certain long-period comets not being counted).

Two years later (September 16th, 1848) _William Cranch Bond_ (1789-1859) discovered, at the Harvard College Observatory, an, eighth satellite of Saturn, called _Hyperion_, which was detected independently by Lassell two days afterwards. In the following year Bond discovered that Saturn was accompanied by a third comparatively dark ring-now commonly known as the =crape ring=—lying immediately inside the bright rings (see fig. 95); and the discovery was made independently a fortnight later by _William Rutter Dawes_ (1799-1868) in England. Lassell discovered in 1851 two new satellites of Uranus, making a total of four belonging to that planet. The next discoveries were those of two satellites of Mars, known as _Deimos_ and _Phobos_, by Professor _Asaph Hall_ of Washington on August 11th and 17th, 1877. These are remarkable chiefly for their close proximity to Mars and their extremely rapid motion, the nearer one revolving more rapidly than Mars rotates, so that to the Martians it must rise in the west and set in the east. Lastly, Jupiter’s system received an addition after nearly three centuries by Professor Barnard’s discovery at the Lick Observatory (September 9th, 1892) of an extremely faint fifth satellite, a good deal nearer to Jupiter than the nearest of Galilei’s satellites (chapter VI., § 121).

296. The surfaces of the various planets and satellites have been watched with the utmost care by an army of observers, but the observations have to a large extent remained without satisfactory interpretation, and little is known of the structure or physical condition of the bodies concerned.