chapter VI.

[17] The =poles= of a great circle on a sphere are the ends of a diameter perpendicular to the plane of the great circle. Every point on the great circle is at the same distance, 90°, from each pole.

[18] The _word_ “zenith” is Arabic, not Greek: cf. chapter III., § 64.

[19] Most of these names are not Greek, but of later origin.

[20] That of M. Paul Tannery: _Recherches sur l’Histoire de l’Astronomie Ancienne_, chap. V.

[21] Trigonometry.

[22] The process may be worth illustrating by means of a simpler problem. A heavy body, falling freely under gravity, is found (the resistance of the air being allowed for) to fall about 16 feet in 1 second, 64 feet in 2 seconds, 144 feet in 3 seconds, 256 feet in 4 seconds, 400 feet in 5 seconds, and so on. This series of figures carried on as far as may be required would satisfy practical requirements, supplemented if desired by the corresponding figures for fractions of seconds; but the mathematician represents the same facts more simply and in a way more satisfactory to the mind by the formula s = 16 t^2, where s denotes the number of feet fallen, and t the number of seconds. By giving t any assigned value, the corresponding space fallen through is at once obtained. Similarly the motion of the sun can be represented approximately by the more complicated formula l = nt + 2 e sin nt, where l is the distance from a fixed point in the orbit, t the time, and n, e certain numerical quantities.

[23] At the present time there is still a small discrepancy between the observed and calculated places of the moon. See chapter XIII., § 290.

[24] The name is interesting as a remnant of a very early superstition. Eclipses, which always occur near the nodes, were at one time supposed to be caused by a dragon which devoured the sun or moon. The symbols ☊ ☋ still used to denote the two nodes are supposed to represent the head and tail of the dragon.

[25] In the figure, which is taken from the _De Revolutionibus_ of Coppernicus (chapter IV., § 85), let D, K, M represent respectively the centres of the sun, earth, and moon, at the time of an eclipse of the moon, and let S Q G, S R E denote the boundaries of the shadow-cone cast by the earth; then Q R, drawn at right angles to the axis of the cone, is the breadth of the shadow at the distance of the moon. We have then at once from similar triangles

G K - Q M : A D - G K :: M K : K D.

Hence if K D = _n_. M K and ∴ also A D = _n_. (radius of moon), _n_ being 19 according to Aristarchus, G K-Q M: _n_. (radius of moon)-G K

:: 1 : _n_

_n_ . (radius of moon) - G K

= _n_ G K - _n_ Q M

∴ radius of moon + radius of shadow

= (1 + 1∕_n_) (radius of earth).

By observation the angular radius of the shadow was found to be about 40′ and that of the moon to be 15′, so that

radius of shadow = 8∕3 radius of moon;

∴ radius of moon

= 3∕11 (1 + 1∕_n_) (radius of earth).

But the angular radius of the moon being 15′, its distance is necessarily about 220 times its radius,

and ∴ distance of the moon

= 60 (1 + 1∕_n_) (radius of the earth),

which is roughly Hipparchus’s result, if _n_ be _any_ fairly large number.

[26] _Histoire de l’Astronomie Ancienne_, Vol. I., p. 185.

[27] The chief MS. bears the title μεγάλη σύνταξις or great composition though the author refers to his book elsewhere as μαθηματικὴ σύνταξις (mathematical composition). The Arabian translators, either through admiration or carelessness, converted μεγάλη, great, into μεγίστη, greatest, and hence it became known by the Arabs as _Al Magisti_, whence the Latin _Almagestum_ and our _Almagest_.

[28] The better known apparent enlargement of the sun or moon when rising or setting has nothing to do with refraction. It is an optical illusion not very satisfactorily explained, but probably due to the lesser brilliancy of the sun at the time.

[29] In spherical trigonometry.

[30] A table of chords (or double sines of half-angles) for every 1∕2° from 0° to 180°.

[31] His procedure may be compared with that of a political economist of the school of Ricardo, who, in order to establish some rough explanation of economic phenomena, starts with certain simple assumptions as to human nature, which at any rate are more plausible than any other equally simple set, and deduces from them a number of abstract conclusions, the applicability of which to real life has to be considered in individual cases. But the perfunctory discussion which such a writer gives of the qualities of the “economic man” cannot of course be regarded as his deliberate and final estimate of human nature.

[32] The equation of the centre and the evection may be expressed trigonometrically by two terms in the expression for the moon’s longitude, _a sin_θ + _b sin_ (2φ-θ), where _a_, _b_ are two numerical quantities, in round numbers 6° and 1°, θ is the angular distance of the moon from perigee, and φ is the angular distance from the sun. At conjunction and opposition φ is 0° or 180°, and the two terms reduce to (_a_-_b_) _sin_θ. This would be the form in which the equation of the centre would have presented itself to Hipparchus. Ptolemy’s correction is therefore equivalent to adding on

_b_ [_sin_θ + _sin_ (2φ - θ)], or 2 _b sin_φ _cos_ (φ-θ),

which vanishes at conjunction or opposition, but reduces at the quadratures to 2 _b sin_θ, which again vanishes if the moon is at apogee or perigee (θ = 0° or 180°), but has its greatest value half-way between, when θ = 90°. Ptolemy’s construction gave rise also to a still smaller term of the type,

_c sin_ 2φ [_cos_ (2φ + θ) + 2 _cos_ (2φ - θ)],

which, it will be observed, vanishes at quadratures as well as at conjunction and opposition.

[33] Here, as elsewhere, I have given no detailed account of astronomical instruments, believing such descriptions to be in general neither interesting nor intelligible to those who have not the actual instruments before them, and to be of little use to those who have.

[34] The advantage derived from the use of the equant can be made clearer by a mathematical comparison with the elliptic motion introduced by Kepler. In elliptic motion the angular motion and distance are represented approximately by the formulae _nt_ + 2_e sin nt_, _a_ (1 - _e cos nt_) respectively; the corresponding formulæ given by the use of the simple eccentric are _nt + e′ sin nt_, _a_ (1 - _e′ cos nt_). To make the angular motions agree we must therefore take _e′_ = 2_e_, but to make the distances agree we must take _e′ = e_; the two conditions are therefore inconsistent. But by the introduction of an equant the formulæ become _nt_ + 2_e′ sin nt_, _a_ (1 - _e′ cos nt_), and _both_ agree if we take _e′ = e_. Ptolemy’s lunar theory could have been nearly freed from the serious difficulty already noticed (§ 48) if he had used an equant to represent the chief inequality of the moon; and his planetary theory would have been made accurate to the first order of small quantities by the use of an equant both for the deferent and the epicycle.

[35] De Morgan classes him as a geometer with Archimedes, Euclid, and Apollonius, the three great geometers of antiquity.

[36] The legend that the books in the library served for six months as fuel for the furnaces of the public baths is rejected by Gibbon and others. One good reason for not accepting it is that by this time there were probably very few books left to burn.

[37] The data as to Indian astronomy are so uncertain, and the evidence of any important original contributions is so slight, that I have not thought it worth while to enter into the subject in any detail. The chief Indian treatises, including the one referred to in the text, bear strong marks of having been based on Greek writings.

[38] He introduced into trigonometry the use of _sines_, and made also some little use of _tangents_, without apparently realising their importance: he also used some new formulæ for the solution of spherical triangles.

[39] A prolonged but indecisive controversy has been carried on, chiefly by French scholars, with regard to the relations of Ptolemy, Abul Wafa, and Tycho in this matter.

[40] For example, the practice of treating the trigonometrical functions as _algebraic_ quantities to be manipulated by formulæ, not merely as geometrical lines.

[41] Any one who has not realised this may do so by performing with Roman numerals the simple operation of multiplying by itself a number such as MDCCCXCVIII.

[42] On trigonometry. He reintroduced the _sine_, which had been forgotten; and made some use of the _tangent_, but like Albategnius (§ 59 _n._) did not realise its importance, and thus remained behind Ibn Yunos and Abul Wafa. An important contribution to mathematics was a table of sines calculated for every minute from 0° to 90°.

[43] That of “lunar distances.”

[44] He did not invent the measuring instrument called the _vernier_, often attributed to him, but something quite different and of very inferior value.

[45] The name is spelled in a large number of different ways both by Coppernicus and by his contemporaries. He himself usually wrote his name Coppernic, and in learned productions commonly used the Latin form Coppernicus. The spelling Copernicus is so much less commonly used by him that I have thought it better to discard it, even at the risk of appearing pedantic.

[46] _Nullo demum loco ineptior est quam ... ubi nim’s pueriliter hallucinatur_: Nowhere is he more foolish than ... where he suffers from delusions of too childish a character.

[47] His real name was Georg Joachim, that by which he is known having been made up by himself from the Latin name of the district where he was born (Rhætia).

[48] The _Commentariolus_ and the _Prima Narratio_ give most readers a better idea of what Coppernicus did than his larger book, in which it is comparatively difficult to disentangle his leading ideas from the mass of calculations based on them.

[49] _Omnis enim quæ videtur secundum locum mutatio, aut est propter locum mutatio, aut est propter spectatæ rei motum, aut videntis, aut certe disparem utriusque mutationem. Nam inter mota æqualiter ad eadem non percipitur motus, inter rem visam dico, et videntem_ (De Rev., I. v.).

I have tried to remove some of the crabbedness of the original passage by translating freely.

[50] To Coppernicus, as to many of his contemporaries, as well as to the Greeks, the simplest form of a revolution of one body round another was a motion in which the revolving body moved as if rigidly attached to the central body. Thus in the case of the earth the second motion was such that the axis of the earth remained inclined at a constant angle to the line joining earth and sun, and therefore changed its direction in space. In order then to make the axis retain a (nearly) fixed direction in space, it was necessary to add a _third_ motion.

[51] In this preliminary discussion, as in fig. 40, Coppernicus gives 80 days; but in the more detailed treatment given in Book V. he corrects this to 88 days.

[52] Fig. 42 has been slightly altered, so as to make it agree with fig. 41.

[53] Coppernicus, instead of giving longitudes as measured from the first point of Aries (or vernal equinoctial point, chapter I., §§ 11, 13), which moves on account of precession, measured the longitudes from a standard fixed star (α _Arietis_) not far from this point.

[54] According to the theory of Coppernicus, the diameter of the moon when greatest was about 1∕8 greater than its average amount; modern observations make this fraction about 1∕13. Or, to put it otherwise, the diameter of the moon when greatest ought to exceed its value when least by about 8′ according to Coppernicus, and by about 5′ according to modern observations.

[55] Euclid, I. 33.

[56] If P be the synodic period of a planet (in years), and S the sidereal period, then we evidently have (1∕P) + 1 = 1∕S for an inferior planet, and 1 - (1∕P) = 1∕S for a superior planet.

[57] Recent biographers have called attention to a cancelled passage in the manuscript of the _De Revolutionibus_ in which Coppernicus shews that an ellipse can be generated by a combination of circular motions. The proposition is, however, only a piece of pure mathematics, and has no relation to the motions of the planets round the sun. It cannot, therefore, fairly be regarded as in any way an anticipation of the ideas of Kepler (chapter VII.).

[58] It may be noticed that the differential method of parallax (chapter VI., § 129), by which such a quantity as 12′ could have been noticed, was put out of court by the general supposition, shared by Coppernicus, that the stars were all at the same distance from us.

[59] There is little doubt that he invented what were substantially logarithms independently of Napier, but, with characteristic inability or unwillingness to proclaim his discoveries, allowed the invention to die with him.

[60] A similar discovery was in fact made twice again, by Galilei (chapter VI., § 114) and by Huygens (chapter VIII., § 157).

[61] He obtained leave of absence to pay a visit to Tycho Brahe and never returned to Cassel. He must have died between 1599 and 1608.

[62] He even did not forget to provide one of the most necessary parts of a mediæval castle, a prison!

[63] It would be interesting to know what use he assigned to the (presumably) still vaster space _beyond_ the stars.

[64] Tycho makes in this connection the delightful remark that Moses must have been a skilled astronomer, because he refers to the moon as “the lesser light,” notwithstanding the fact that the apparent diameters of sun and moon are very nearly equal!

[65] By transversals.

[66] On an instrument which he had invented, called the _hydrostatic balance_.

[67] A fair idea of mediaeval views on the subject may be derived from one of the most tedious Cantos in Dante’s great poem (_Paradiso_, II.), in which the poet and Beatrice expound two different “explanations” of the spots on the moon.

[68] _Ludovico delle Colombe_ in a tract _Contra Il Moto della Terra_, which is reprinted in the national edition of Galilei’s works, Vol. III.

[69] In a letter of May 4th, 1612, he says that he has seen them for eighteen months; in the _Dialogue on the Two Systems_ (III., p. 312, in Salusbury’s translation) he says that he saw them while he still lectured at Padua, _i.e._ presumably by September 1610, as he moved to Florence in that month.

[70] _Historia e Dimostrazioni intorno alle Macchie Solari._

[71] Acts i. 11. The pun is not quite so bad in its Latin form: _Viri Galilaci_, etc.

[72] _Spiritui sancto mentem fuisse nos docere, quo modo ad Coelum eatur, non autem quomodo Coelum gradiatur._

[73] From the translation by Salusbury, in Vol. I. of his _Mathematical Collections_.

[74] The only point of any importance in connection with Galilei’s relations with the Inquisition on which there seems to be room for any serious doubt is as to the stringency of this warning. It is probable that Galilei was at the same time specifically forbidden to “hold, teach, or defend in any way, whether verbally or in writing,” the obnoxious doctrine.

[75] This is illustrated by the well-known optical illusion whereby a white circle on a black background appears larger than an equal black one on a white background. The apparent size of the hot filament in a modern incandescent electric lamp is another good illustration.

[76] Actually, since the top of the tower is describing a slightly larger circle than its foot, the stone is at first moving eastward slightly faster than the foot of the tower, and therefore should reach the ground slightly to the _east_ of it. This displacement is, however, very minute, and can only be detected by more delicate experiments than any devised by Galilei.

[77] From the translation by Salusbury, in Vol. I. of his _Mathematical Collections_.

[78] The official minute is: _Et ei dicto quod dicat veritatem, alias devenietur ad torturam_.

[79] The three days June 21-24 the only ones which Galilei _could_ have spent in an actual prison, and there seems no reason to suppose that they were spent elsewhere than in the comfortable rooms in which it is known that he lived during most of April.

[80] Equivalent to portions of the subject now called _dynamics_ or (more correctly) _kinematics_ and _kinetics_.

[81] He estimates that a body falls in a second a distance of 4 “bracchia,” equivalent to about 8 feet, the true distance being slightly over 16.

[82] _Two New Sciences_, translated by Weston, p. 255.

[83] The astronomer appears to have used both spellings of his name almost indifferently. For example, the title-page of his most important book, the _Commentaries on the Motions of Mars_ (§ 141), has the form Kepler, while the dedication of the same book is signed Keppler.

[84] The regular solids being taken in the order: cube, tetrahedron, dodecahedron, icosahedron, octohedron, and of such magnitude that a sphere can be circumscribed to each and at the same time inscribed in the preceding solid of the series, then the radii of the six spheres so obtained were shewn by Kepler to be approximately proportional to the distances from the sun of the six planets Saturn, Jupiter, Mars, Earth, Venus, and Mercury.

[85] Two stars 4′ apart only just appear distinct to the naked eye of a person with average keenness of sight.

[86] _Commentaries on the Motions of Mars_, Part II., end of chapter XIX.

[87] An ellipse is one of several curves, known as =conic sections=, which can be formed by taking a section of a cone, and may also be defined as a curve the sum of the distances of any point on which from two fixed points inside it, known as the =foci=, is always the same.

Thus if, in the figure, S and H are the foci, and P, Q are _any_ two points on the curve, then the distances S P, H P added together are equal to the distances S Q, Q H added together, and each sum is equal to the length A A′ of the ellipse. The ratio of the distance S H to the length A A′ is known as the =eccentricity=, and is a convenient measure of the extent to which the ellipse differs from a circle.

[88] The ellipse is _more_ elongated than the actual path of Mars, an accurate drawing of which would be undistinguishable to the eye from a circle. The eccentricity is 1∕3 in the figure, that of Mars being 1∕10.

[89] _Astronomia Nova_ αἰτιολογητος _seu Physica Coelestis, tradita Commentariis de Motibus Stellae Martis._ _Ex Observationibus G. V. Tychonis Brahe._

[90] It contains the germs of the method of infinitesimals.

[91] _Harmonices Mundi Libri V._

[92] There may be some interest in Kepler’s own statement of the law: “Res est certissima exactissimaque, quod proportionis quae est inter binorum quorumque planetarum tempora periodica, sit praecise sesquialtera proportionis mediarum distantiarum, id est orbium ipsorum.”—_Harmony of the World_, Book V., chapter III.

[93] _Epitome_, Book IV., Part 2.

[94] Introduction to the _Commentaries on the Motions of Mars_.

[95] Substantially the _filar micrometer_ of modern astronomy.

[96] Galilei, at the end of his life, appears to have thought of contriving a pendulum with clockwork, but there is no satisfactory evidence that he ever carried out the idea.

[97] In modern notation: time oπf oscillation = 2π√(_l_∕_g_).

[98] _I.e._ he obtained the familiar formula (_v^2_)∕_r_, and several equivalent forms for _centrifugal force_.

[99] Also frequently referred to by the Latin name _Cartesius_.

[100] According to the unreformed calendar (O.S.) then in use in England, the date was Christmas Day, 1642. To facilitate comparison with events occurring out of England, I have used throughout this and the following chapters the Gregorian Calendar (N.S.), which was at this time adopted in a large part of the Continent (cf. chapter II., § 22).

[101] From a MS. among the Portsmouth Papers, quoted in the Preface to the Catalogue of the Portsmouth Papers.

[102] W. K. Clifford, _Aims and Instruments of Scientific Thought_.

[103] It is interesting to read that Wren offered a prize of 40_s._ to whichever of the other two should solve this the central problem of the solar system.

[104] The familiar _parallelogram of forces_, of which earlier writers had had indistinct ideas, was clearly stated and proved in the introduction to the _Principia_, and was, by a curious coincidence, published also in the same year by _Varignon_ and _Lami_.

[105] It is between 13 and 14 billion billion pounds. See chapter X. § 219.

[106] As far as I know Newton gives no short statement of the law in a perfectly complete and general form; separate parts of it are given in different passages of the _Principia_.

[107] It is commonly stated that Newton’s value of the motion of the moon’s apses was only about half the true value. In a scholium of the _Principia_ to prop. 35 of the third book, given in the first edition but afterwards omitted, he estimated the annual motion at 40°, the observed value being about 41°. In one of his unpublished papers, contained in the Portsmouth collection, he arrived at 39° by a process which he evidently regarded as not altogether satisfactory.

[108] Throughout the Coppernican controversy up to Newton’s time it had been generally assumed, both by Coppernicans and by their opponents, that there was some meaning in speaking of a body simply as being “at rest” or “in motion,” without any reference to any other body. But all that we can really observe is the motion of one body relative to one or more others. Astronomical observation tells us, for example, of a certain motion relative to one another of the earth and sun; and this motion was expressed in two quite different ways by Ptolemy and by Coppernicus. From a modern standpoint the question ultimately involved was whether the motions of the various bodies of the solar system relatively to the earth or relatively to the sun were the simpler to express. If it is found convenient to express them—as Coppernicus and Galilei did—in relation to the sun, some simplicity of statement is gained by speaking of the sun as “fixed” and omitting the qualification “relative to the sun” in speaking of any other body. The same motions might have been expressed relatively to any other body chosen at will: _e.g._ to one of the hands of a watch carried by a man walking up and down on the deck of a ship on a rough sea; in this case it is clear that the motions of the other bodies of the solar system relative to this body would be excessively complicated; and it would therefore be highly inconvenient though still possible to treat this particular body as “fixed.”

A new aspect of the problem presents itself, however, when an attempt—like Newton’s—is made to explain the motions of bodies of the solar system as the result of forces exerted on one another by those bodies. If, for example, we look at Newton’s First Law of Motion (chapter VI., § 130), we see that it has no meaning, unless we know what are the body or bodies relative to which the motion is being expressed; a body at rest relatively to the earth is moving relatively to the sun or to the fixed stars, and the applicability of the First Law to it depends therefore on whether we are dealing with its motion relatively to the earth or not. For most terrestrial motions it is sufficient to regard the Laws of Motion as referring to motion relative to the earth; or, in other words, we may for this purpose treat the earth as “fixed.” But if we examine certain terrestrial motions more exactly, we find that the Laws of Motion thus interpreted are not quite true; but that we get a more accurate explanation of the observed phenomena if we regard the Laws of Motion as referring to motion relative to the centre of the sun and to lines drawn from it to the stars; or, in other words, we treat the centre of the sun as a “fixed” point and these lines as “fixed” directions. But again when we are dealing with the solar system generally this interpretation is slightly inaccurate, and we have to treat the centre of gravity of the solar system instead of the sun as “fixed.”

From this point of view we may say that Newton’s object in the _Principia_ was to shew that it was possible to choose a certain point (the centre of gravity of the solar system) and certain directions (lines joining this point to the fixed stars), as a base of reference, such that all motions being treated as relative to this base, the Laws of Motion and the law of gravitation afford a consistent explanation of the observed motions of the bodies of the solar system.

[109] He estimated the annual precession due to the sun to be about 9″, and that due to the moon to be about four and a half times as great, so that the total amount due to the two bodies came out about 50″, which agrees within a fraction of a second with the amount shewn by observation; but we know now that the moon’s share is not much more than twice that of the sun.

[110] He once told Halley in despair that the lunar theory “made his head ache and kept him awake so often that he would think of it no more.”

[111] December 31st, 1719, according to the unreformed calendar (O.S.) then in use in England.

[112] The apparent number is 2,935, but 12 of these are duplicates.

[113] By Bessel (chapter XIII., § 277).

[114] The relation between the work of Flamsteed and that of Newton was expressed with more correctness than good taste by the two astronomers themselves, in the course of some quarrel about the lunar theory: “Sir Isaac worked with the ore I had dug.” “If he dug the ore, I made the gold ring.”

[115] Rigaud, in the memoirs prefixed to Bradley’s _Miscellaneous Works_.

[116] A telescopic star named 37 Camelopardi in Flamsteed’s catalogue.

[117] The story is given in T. Thomson’s _History of the Royal Society_, published more than 80 years afterwards (1812), but I have not been able to find any earlier authority for it. Bradley’s own account of his discovery gives a number of details, but has no allusion to this incident.

[118] It is _k sin_ C A B, where _k_ is the constant of aberration.

[119] His observations as a matter of fact point to a value rather greater than 18″, but he preferred to use round numbers. The figures at present accepted are 18″·42 and 13″·75, so that his ellipse was decidedly less flat than it should have been.

[120] _Recherches sur la précession des équinoxes et sur la nutation de l’axe de la terre._

[121] The word “geometer” was formerly used, as “géomètre” still is in French, in the wider sense in which “mathematician” is now customary.

[122] _Principia_, Book III., proposition 10.

[123] It is important for the purposes of this discussion to notice that the vertical is _not_ the line drawn from the centre of the earth to the place of observation.

[124] 69 miles is 364,320 feet, so that the two northern degrees were a little more and the Peruvian are a little less than 69 miles.

[125] The remaining 8,000 stars were not “reduced” by Lacaille. The whole number were first published in the “reduced” form by the British Association in 1845.

[126] A mural quadrant.

[127] The ordinary approximate theory of the _collimation error_, _level error_, and _deviation error_ of a transit, as given in textbooks of spherical and practical astronomy, is substantially his.

[128] The title-page is dated 1767; but it is known not to have been actually published till three years later.

[129] For a more detailed discussion of the transit of Venus, see Airy’s _Popular Astronomy_ and Newcomb’s _Popular Astronomy_.

[130] _Some_ other influences are known—_e.g._ the sun’s heat causes various motions of our air and water, and has a certain minute effect on the earth’s rate of rotation, and presumably produces similar effects on other bodies.

[131] The arithmetical processes of working out, figure by figure, a non-terminating decimal or a square root are simple cases of successive approximation.

[132] “C’est que je viens d’un pays où, quand on parle, on est pendu.”

[133] Longevity has been a remarkable characteristic of the great mathematical astronomers: Newton died in his 85th year; Euler, Lagrange, and Laplace lived to be more than 75, and D’Alembert was almost 66 at his death.

[134] This body, which is primarily literary, has to be distinguished from the much less famous Paris Academy of Sciences, constantly referred to (often simply as the Academy) in this chapter and the preceding.

[135] E.g. _Mélanges de Philosophie, de l’Histoire, et de Littérature_; _Éléments de Philosophie_; _Sur la Destruction des Jésuites_.

[136] _I.e._ he assumed a law of attraction represented by μ∕_r^2_ + ν∕_r^3_.

[137] This appendix is memorable as giving for the first time the method of _variation of parameters_ which Lagrange afterwards developed and used with such success.

[138] That of the distinguished American astronomer Dr. G. W. Hill (chapter XIII., § 286).

[139] They give about ·78 for the mass of Venus compared to that of the earth.

[140] The orbit might be a parabola or hyperbola, though this does not occur in the case of any known planet.

[141] On the _Calculus of Variations_.

[142] The establishment of the general equations of motion by a combination of _virtual velocities_ and _D’Alembert’s principle_.

[143] _Théorie des Fonctions Analytiques_ (1797); _Resolution des Équations Numériques_ (1798); _Leçons sur le Calcul des Fonctions_ (1805).

[144] _Théorie Analytique des Probabilités._

[145] The fact that the post was then given by Napoleon to his brother Lucien suggests some doubts as to the unprejudiced character of the verdict of incompetence pronounced by Napoleon against Laplace.

[146] _Outlines of Astronomy_, § 656.

[147] Laplace, _Système du Monde_.

[148] If _n_, _n′_ are the mean motions of the two planets, the expression for the disturbing force contains terms of the type = _sin_(_np_ ± _n′p′_) _t_, _cos_ where _p_, _p′_ are integers, and the coefficient is of the order _p_⁓_p′_ in the eccentricities and inclinations. If now _p_ and _p′_ are such that _np_⁓_n′p′_ is small, the corresponding inequality has a period 2π∕(_np_⁓_n′p′_), and though its coefficient is of order _p_⁓_p′_, it has the small factor _np_⁓_np′_ (or its square) in the denominator and may therefore be considerable. In the case of Jupiter and Saturn, for example, _n_ = 109,257 in seconds of arc per annum, _n′_ = 43,996; 5_n′_ - 2_n_ = 1,466; there is therefore an inequality of the _third_ order, with a period (in years) = 360°∕1,466″ = 900.

[149] This statement requires some qualification when perturbations are taken into account. But the point is not very important, and is too technical to be discussed.

[150] ∑_e^2m_√_a_ = _c_, ∑_tan^2im_√_a_ = _c′_, where _m_ is the mass of any planet, _a_, _e_, _i_ are the semi-major axis, eccentricity, and inclination of the orbit. The equation is true as far as squares of small quantities, and therefore it is indifferent whether or not _tan i_ is replaced as in the text by _i_.

[151] Nearly the whole of the “eccentricity fund” and of the “inclination fund” of the solar system is shared between Jupiter and Saturn. If Jupiter were to absorb the whole of each fund, the eccentricity of its orbit would only be increased by about 25 per cent., and the inclination to the ecliptic would not be doubled.

[152] Of tables based on Laplace’s work and published up to the time of his death, the chief solar ones were those of _von Zach_ (1804) and _Delambre_ (1806); and the chief planetary ones were those of _Lalande_ (1771), of _Lindenau_ for Venus, Mars, and Mercury (1810-13), and of _Bouvard_ for Jupiter, Saturn, and Uranus (1808 and 1821).

[153] The motion of the satellites of Uranus (chapter XII., § 253, 255) is in the opposite direction. When Laplace first published his theory their motion was doubtful, and he does not appear to have thought it worth while to notice the exception in later editions of his book.

[154] This statement again has to be modified in consequence of the discoveries, beginning on January 1st, 1801, of the minor planets (chapter XIII., § 294), many of which have orbits that are far more eccentric than those of the other planets and are inclined to the ecliptic at considerable angles.

[155] _Système du Monde_, Book V., chapter VI.

[156] In his paper of 1817 Herschel gives the number as 863, but a reference to the original paper of 1785 shews that this must be a printer’s error.

[157] The motion of Castor has become slower since Herschel’s time, and the present estimate of the period is about 1,000 years, but it is by no means certain.

[158] More precisely, counting motions in right ascension and in declination separately, he had 27 observed motions to deal with (one of the stars having no motion in declination); 22 agreed in sign with those which would result from the assumed motion of the sun.

[159] The method was published by Legendre in 1806 and by Gauss in 1809, but it was invented and used by the latter more than 20 years earlier.

[160] The figure has to be enormously exaggerated, the angle SσE as shewn there being about 10°, and therefore about 100,000 times too great.

[161] Sir R. S. Ball and the late Professor Pritchard (§ 279) have obtained respectively ·47″ and ·43″; the mean of these, ·45″, may be provisionally accepted as not very far from the truth.

[162] An average star of the 14th magnitude is 10,000 times fainter than one of the 4th magnitude, which again is about 150 times less bright than Sirius. See § 316.

[163] Newcomb’s velocity of light and Nyrén’s constant of aberration (20″·4921) give 8″·794; Struve’s constant of aberration (20″·445), Loewy’s (20″·447), and Hall’s (20″·454) each give 8″·81.

[164] _Fundamenta Nova Investigationis Orbitae Verae quam Luna perlustrat._

[165] _Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Störungen._

[166] _E.g._ in Grant’s _History of Physical Astronomy_, Herschel’s _Outlines of Astronomy_, Miss Clerke’s _History of Astronomy in the Nineteenth Century_, and the memoir by Dr. Glaisher prefixed to the first volume of Adams’s _Collected Papers_.

[167] This had been suggested as a possibility by several earlier writers.

[168] The discovery of a terrestrial substance with this line in its spectrum has been announced while this book has been passing through the press.

[169] Observations made on Mont Blanc under the direction of M. Janssen in 1897 indicate a slightly larger number than Dr. Langley’s.

[170] _Catalogus novus stellarum duplicium_, _Stellarum duplicium et multiplicium mensurae micrometricae_, and _Stellarum fixarum imprimis duplicium et multiplicium positiones mediae pro epocha 1830_.

[171] _I.e._ 2·512... is chosen as being the number the logarithm of which is ·4, so that (2·512...)^{5∕2} = 10.

[172] If L be the ratio of the light received from a star to that received from a standard first magnitude star, such as Aldebaran or Altair, then its magnitude _m_ is given by the formula

L = (1∕2·512)^{m - 1} = (1∕100)^{(m - 1)∕5}, whence m - 1 = -5∕2log L.

A star brighter than Aldebaran has a magnitude less than 1, while the magnitude of Sirius, which is about nine times as bright as Aldebaran, is a _negative_ quantity,-1·4, according to the Harvard photometry.

End of Project Gutenberg's A Short History of Astronomy, by Arthur Berry