FOOTNOTES:

FOOTNOTES:1In our climate 2,000 is about the greatest number ever visible at once, even to a keen-sighted person.2Owing to the greater brightness of the stars overhead they usually seem a little nearer than those near the horizon, and consequently the visible portion of the celestial sphere appears to be rather less than a half of a complete sphere. This is, however, of no importance, and will for the future be ignored.3A right angle is divided into ninety degrees (90°), a degree into sixty minutes (60′), and a minute into sixty seconds (60″).4I have made no attempt either here or elsewhere to describe the constellations and their positions, as I believe such verbal descriptions to be almost useless. For a beginner who wishes to become familiar with them the best plan is to get some better informed, friend to point out a few of the more conspicuous ones, in different parts of the sky. Others can then be readily added by means of a star-atlas, or of the star-maps given in many textbooks.5The names, in the customary Latin forms, are: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces; they are easily remembered by the doggerel verses:—The Ram, the Bull, the Heavenly Twins,And next the Crab, the Lion shines,The Virgin and the Scales,The Scorpion, Archer, and He-Goat,The Man that bears the Watering-pot,And Fish with glittering tails.6This statement leaves out of account small motions nearly or quite invisible to the naked eye, some of which are among the most interesting discoveries of telescopic astronomy; see, for example, chapterX.,§§ 207-215.7The custom of calling the sun and moon planets has now died out, and the modern usage will be adopted henceforward in this book.8It may be noted that our word “day” (and the corresponding word in other languages) is commonly used in two senses, either for the time between sunrise and sunset (day as distinguished from night), or for the whole period of 24 hours or day-and-night. The Greeks, however, used for the latter a special word, νυχθήμερον.9Compare the French: Mardi, Mercredi, Jeudi, Vendredi; or better still the Italian: Martedi, Mercoledi, Giovedi, Venerdi.10See, for example,Old Moore’sorZadkiel’s Almanack.11We have little definite knowledge of his life. He was born in the earlier part of the 6th centuryB.C., and died at the end of the same century or beginning of the next.12Theophrastus was born about half a century, Plutarch nearly five centuries, later than Plato.13Republic, VII. 529, 530.14Confused, because the mechanical knowledge of the time was quite unequal to giving any explanation of the way in which these spheres acted on one another.15I have introduced here the familiar explanation of the phases of the moon, and the argument based on it for the spherical shape of the moon, because, although probably known before Aristotle, there is, as far as I know, no clear and definite statement of the matter in any earlier writer, and after his time it becomes an accepted part of Greek elementary astronomy. It may be noticed that the explanation is unaffected either by the question of the rotation of the earth or by that of its motion round the sun.16See, for example, the account of Galilei’s controversies, in chapterVI.17Thepolesof 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.18Theword“zenith” is Arabic, not Greek: cf. chapterIII., § 64.19Most of these names are not Greek, but of later origin.20That of M. Paul Tannery:Recherches sur l’Histoire de l’Astronomie Ancienne, chap.V.21Trigonometry.22The 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 formulas= 16t2, wheresdenotes the number of feet fallen, andtthe number of seconds. By givingtany 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 formulal=nt+ 2esinnt, wherelis the distance from a fixed point in the orbit,tthe time, andn,ecertain numerical quantities.23At the present time there is still a small discrepancy between the observed and calculated places of the moon. See chapterXIII.,§ 290.24The 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.25In the figure, which is taken from theDe Revolutionibusof Coppernicus (chapterIV.,§ 85), letD,K,Mrepresent respectively the centres of the sun, earth, and moon, at the time of an eclipse of the moon, and letS Q G,S R Edenote the boundaries of the shadow-cone cast by the earth; thenQ 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 trianglesG K-Q M:A D-G K::M K:K D.Hence ifK D=n.M Kand ∴ alsoA D=n. (radius of moon),nbeing 19 according to Aristarchus,G K-Q M:n. (radius of moon)-G K:: 1 :nn. (radius of moon) -G K=nG K-nQ 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 thatradius 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, ifnbeanyfairly large number.26Histoire de l’Astronomie Ancienne, Vol. I., p. 185.27The 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 asAl Magisti, whence the LatinAlmagestumand ourAlmagest.28The 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.29In spherical trigonometry.30A table of chords (or double sines of half-angles) for every 1∕2° from 0° to 180°.31His 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.32The equation of the centre and the evection may be expressed trigonometrically by two terms in the expression for the moon’s longitude,a sinθ +b sin(2φ-θ), wherea,bare 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)sinθ. 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 onb[sinθ +sin(2φ - θ)], or 2b sinφcos(φ-θ),which vanishes at conjunction or opposition, but reduces at the quadratures to 2b sinθ, 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 sin2φ [cos(2φ + θ) + 2cos(2φ - θ)],which, it will be observed, vanishes at quadratures as well as at conjunction and opposition.33Here, 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.34The 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 formulaent+ 2e sin nt,a(1 -e cos nt) respectively; the corresponding formulæ given by the use of the simple eccentric arent + e′ sin nt,a(1 -e′ cos nt). To make the angular motions agree we must therefore takee′= 2e, but to make the distances agree we must takee′ = e; the two conditions are therefore inconsistent. But by the introduction of an equant the formulæ becoment+ 2e′ sin nt,a(1 -e′ cos nt), andbothagree if we takee′ = 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.35De Morgan classes him as a geometer with Archimedes, Euclid, and Apollonius, the three great geometers of antiquity.36The 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.37The 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.38He introduced into trigonometry the use ofsines, and made also some little use oftangents, without apparently realising their importance: he also used some new formulæ for the solution of spherical triangles.39A 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.40For example, the practice of treating the trigonometrical functions asalgebraicquantities to be manipulated by formulæ, not merely as geometrical lines.41Any one who has not realised this may do so by performing with Roman numerals the simple operation of multiplying by itself a number such asMDCCCXCVIII.42On trigonometry. He reintroduced thesine, which had been forgotten; and made some use of thetangent, but like Albategnius (§ 59n.) 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°.43That of “lunar distances.”44He did not invent the measuring instrument called thevernier, often attributed to him, but something quite different and of very inferior value.45The 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.46Nullo 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.47His 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).48TheCommentariolusand thePrima Narratiogive 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.49Omnis 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.50To 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 athirdmotion.51In 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.52Fig. 42 has been slightly altered, so as to make it agree with fig. 41.53Coppernicus, instead of giving longitudes as measured from the first point of Aries (or vernal equinoctial point, chapterI.,§§ 11, 13), which moves on account of precession, measured the longitudes from a standard fixed star (αArietis) not far from this point.54According 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.55Euclid, I. 33.56IfPbe the synodic period of a planet (in years), andSthe sidereal period, then we evidently have (1∕P) + 1 = 1∕Sfor an inferior planet, and 1 - (1∕P) = 1∕Sfor a superior planet.57Recent biographers have called attention to a cancelled passage in the manuscript of theDe Revolutionibusin 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 (chapterVII.).58It may be noticed that the differential method of parallax (chapterVI.,§ 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.59There 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.60A similar discovery was in fact made twice again, by Galilei (chapterVI.,§ 114) and by Huygens (chapterVIII.,§ 157).61He obtained leave of absence to pay a visit to Tycho Brahe and never returned to Cassel. He must have died between 1599 and 1608.62He even did not forget to provide one of the most necessary parts of a mediæval castle, a prison!63It would be interesting to know what use he assigned to the (presumably) still vaster spacebeyondthe stars.64Tycho 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!65By transversals.66On an instrument which he had invented, called thehydrostatic balance.67A 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.68Ludovico delle Colombein a tractContra Il Moto della Terra, which is reprinted in the national edition of Galilei’s works, Vol. III.69In a letter of May 4th, 1612, he says that he has seen them for eighteen months; in theDialogue 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.70Historia e Dimostrazioni intorno alle Macchie Solari.71Acts i. 11. The pun is not quite so bad in its Latin form:Viri Galilaci, etc.72Spiritui sancto mentem fuisse nos docere, quo modo ad Coelum eatur, non autem quomodo Coelum gradiatur.73From the translation by Salusbury, in Vol. I. of hisMathematical Collections.74The 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.75This 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.76Actually, 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 theeastof it. This displacement is, however, very minute, and can only be detected by more delicate experiments than any devised by Galilei.77From the translation by Salusbury, in Vol. I. of hisMathematical Collections.78The official minute is:Et ei dicto quod dicat veritatem, alias devenietur ad torturam.79The three days June 21-24 the only ones which Galileicouldhave 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.80Equivalent to portions of the subject now calleddynamicsor (more correctly)kinematicsandkinetics.81He estimates that a body falls in a second a distance of 4 “bracchia,” equivalent to about 8 feet, the true distance being slightly over 16.82Two New Sciences, translated by Weston, p. 255.83The astronomer appears to have used both spellings of his name almost indifferently. For example, the title-page of his most important book, theCommentaries on the Motions of Mars(§ 141), has the form Kepler, while the dedication of the same book is signed Keppler.84The 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.85Two stars 4′ apart only just appear distinct to the naked eye of a person with average keenness of sight.86Commentaries on the Motions of Mars, Part II., end of chapterXIX.87An ellipse is one of several curves, known asconic 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 thefoci, is always the same.Fig. 59.—An ellipse.Thus if, in the figure,SandHare the foci, andP,Qareanytwo points on the curve, then the distancesS P,H Padded together are equal to the distancesS Q,Q Hadded together, and each sum is equal to the lengthA A′of the ellipse. The ratio of the distanceS Hto the lengthA A′is known as theeccentricity, and is a convenient measure of the extent to which the ellipse differs from a circle.88The ellipse ismoreelongated 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∕10.89Astronomia Novaαἰτιολογητοςseu Physica Coelestis, tradita Commentariis de Motibus Stellae Martis.Ex Observationibus G. V. Tychonis Brahe.90It contains the germs of the method of infinitesimals.91Harmonices Mundi Libri V.92There 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., chapterIII.93Epitome, Book IV., Part 2.94Introduction to theCommentaries on the Motions of Mars.95Substantially thefilar micrometerof modern astronomy.96Galilei, 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.97In modern notation: time oπf oscillation = 2π√(l∕g).98I.e.he obtained the familiar formula (v2)∕r, and several equivalent forms forcentrifugal force.99Also frequently referred to by the Latin nameCartesius.100According 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. chapterII.,§ 22).101From a MS. among the Portsmouth Papers, quoted in the Preface to the Catalogue of the Portsmouth Papers.102W. K. Clifford,Aims and Instruments of Scientific Thought.103It is interesting to read that Wren offered a prize of 40s.to whichever of the other two should solve this the central problem of the solar system.104The familiarparallelogram of forces, of which earlier writers had had indistinct ideas, was clearly stated and proved in the introduction to thePrincipia, and was, by a curious coincidence, published also in the same year byVarignonandLami.105It is between 13 and 14 billion billion pounds. See chapterX.§ 219.106As 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 thePrincipia.107It 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 thePrincipiato 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.108Throughout 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 (chapterVI.,§ 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 thePrincipiawas 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.109He 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.110He 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.”111December 31st, 1719, according to the unreformed calendar (O.S.) then in use in England.112The apparent number is 2,935, but 12 of these are duplicates.113By Bessel (chapterXIII.,§ 277).114The 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.”115Rigaud, in the memoirs prefixed to Bradley’sMiscellaneous Works.116A telescopic star named 37 Camelopardi in Flamsteed’s catalogue.117The story is given in T. Thomson’sHistory 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.118It isk sinC A B, wherekis the constant of aberration.119His 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.120Recherches sur la précession des équinoxes et sur la nutation de l’axe de la terre.121The word “geometer” was formerly used, as “géomètre” still is in French, in the wider sense in which “mathematician” is now customary.122Principia, Book III., proposition 10.123It is important for the purposes of this discussion to notice that the vertical isnotthe line drawn from the centre of the earth to the place of observation.12469 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.125The remaining 8,000 stars were not “reduced” by Lacaille. The whole number were first published in the “reduced” form by the British Association in 1845.126A mural quadrant.127The ordinary approximate theory of thecollimation error,level error, anddeviation errorof a transit, as given in textbooks of spherical and practical astronomy, is substantially his.128The title-page is dated 1767; but it is known not to have been actually published till three years later.129For a more detailed discussion of the transit of Venus, see Airy’sPopular Astronomyand Newcomb’sPopular Astronomy.130Someother 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.131The 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.”133Longevity 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.134This 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.135E.g.Mélanges de Philosophie, de l’Histoire, et de Littérature;Éléments de Philosophie;Sur la Destruction des Jésuites.136I.e.he assumed a law of attraction represented by μ∕r2+ ν∕r3.137This appendix is memorable as giving for the first time the method ofvariation of parameterswhich Lagrange afterwards developed and used with such success.138That of the distinguished American astronomer Dr. G. W. Hill (chapterXIII.,§ 286).139They give about ·78 for the mass of Venus compared to that of the earth.140The orbit might be a parabola or hyperbola, though this does not occur in the case of any known planet.141On theCalculus of Variations.142The establishment of the general equations of motion by a combination ofvirtual velocitiesandD’Alembert’s principle.143Théorie des Fonctions Analytiques(1797);Resolution des Équations Numériques(1798);Leçons sur le Calcul des Fonctions(1805).144Théorie Analytique des Probabilités.145The 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.146Outlines of Astronomy, § 656.147Laplace,Système du Monde.148Ifn,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,coswherep,p′are integers, and the coefficient is of the orderp⁓p′in the eccentricities and inclinations. If nowpandp′are such thatnp⁓n′p′is small, the corresponding inequality has a period 2π∕(np⁓n′p′), and though its coefficient is of orderp⁓p′, it has the small factornp⁓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; 5n′- 2n= 1,466; there is therefore an inequality of thethirdorder, with a period (in years) = 360°∕1,466″ = 900.149This 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∑e2m√a=c, ∑tan2im√a=c′, wheremis the mass of any planet,a,e,iare 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 nottan iis replaced as in the text byi.151Nearly 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.152Of tables based on Laplace’s work and published up to the time of his death, the chief solar ones were those ofvon Zach(1804) andDelambre(1806); and the chief planetary ones were those ofLalande(1771), ofLindenaufor Venus, Mars, and Mercury (1810-13), and ofBouvardfor Jupiter, Saturn, and Uranus (1808 and 1821).153, The motion of the satellites of Uranus (chapterXII.,§ 253255) 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.154This statement again has to be modified in consequence of the discoveries, beginning on January 1st, 1801, of the minor planets (chapterXIII.,§ 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.155Système du Monde, Book V., chapterVI.156In 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.157The 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.158More 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.159The 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.160The figure has to be enormously exaggerated, the angleSσEas shewn there being about 10°, and therefore about 100,000 times too great.161Sir 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.162An 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 § 316.163Newcomb’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″·81.164Fundamenta Nova Investigationis Orbitae Verae quam Luna perlustrat.165Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Störungen.166E.g.in Grant’sHistory of Physical Astronomy, Herschel’sOutlines of Astronomy, Miss Clerke’sHistory of Astronomy in the Nineteenth Century, and the memoir by Dr. Glaisher prefixed to the first volume of Adams’sCollected Papers.167This had been suggested as a possibility by several earlier writers.168The discovery of a terrestrial substance with this line in its spectrum has been announced while this book has been passing through the press.169Observations made on Mont Blanc under the direction of M. Janssen in 1897 indicate a slightly larger number than Dr. Langley’s.170Catalogus novus stellarum duplicium,Stellarum duplicium et multiplicium mensurae micrometricae, andStellarum fixarum imprimis duplicium et multiplicium positiones mediae pro epocha 1830.171I.e.2·512... is chosen as being the number the logarithm of which is ·4, so that (2·512...)5∕2= 10.172If 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 magnitudemis given by the formulaL = (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 anegativequantity,-1·4, according to the Harvard photometry.

1In our climate 2,000 is about the greatest number ever visible at once, even to a keen-sighted person.

1In our climate 2,000 is about the greatest number ever visible at once, even to a keen-sighted person.

2Owing to the greater brightness of the stars overhead they usually seem a little nearer than those near the horizon, and consequently the visible portion of the celestial sphere appears to be rather less than a half of a complete sphere. This is, however, of no importance, and will for the future be ignored.

2Owing to the greater brightness of the stars overhead they usually seem a little nearer than those near the horizon, and consequently the visible portion of the celestial sphere appears to be rather less than a half of a complete sphere. This is, however, of no importance, and will for the future be ignored.

3A right angle is divided into ninety degrees (90°), a degree into sixty minutes (60′), and a minute into sixty seconds (60″).

3A right angle is divided into ninety degrees (90°), a degree into sixty minutes (60′), and a minute into sixty seconds (60″).

4I have made no attempt either here or elsewhere to describe the constellations and their positions, as I believe such verbal descriptions to be almost useless. For a beginner who wishes to become familiar with them the best plan is to get some better informed, friend to point out a few of the more conspicuous ones, in different parts of the sky. Others can then be readily added by means of a star-atlas, or of the star-maps given in many textbooks.

4I have made no attempt either here or elsewhere to describe the constellations and their positions, as I believe such verbal descriptions to be almost useless. For a beginner who wishes to become familiar with them the best plan is to get some better informed, friend to point out a few of the more conspicuous ones, in different parts of the sky. Others can then be readily added by means of a star-atlas, or of the star-maps given in many textbooks.

5The names, in the customary Latin forms, are: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces; they are easily remembered by the doggerel verses:—The Ram, the Bull, the Heavenly Twins,And next the Crab, the Lion shines,The Virgin and the Scales,The Scorpion, Archer, and He-Goat,The Man that bears the Watering-pot,And Fish with glittering tails.

5The names, in the customary Latin forms, are: Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricornus, Aquarius, and Pisces; they are easily remembered by the doggerel verses:—

The Ram, the Bull, the Heavenly Twins,And next the Crab, the Lion shines,The Virgin and the Scales,The Scorpion, Archer, and He-Goat,The Man that bears the Watering-pot,And Fish with glittering tails.

The Ram, the Bull, the Heavenly Twins,And next the Crab, the Lion shines,The Virgin and the Scales,The Scorpion, Archer, and He-Goat,The Man that bears the Watering-pot,And Fish with glittering tails.

The Ram, the Bull, the Heavenly Twins,And next the Crab, the Lion shines,The Virgin and the Scales,The Scorpion, Archer, and He-Goat,The Man that bears the Watering-pot,And Fish with glittering tails.

The Ram, the Bull, the Heavenly Twins,

And next the Crab, the Lion shines,

The Virgin and the Scales,

The Scorpion, Archer, and He-Goat,

The Man that bears the Watering-pot,

And Fish with glittering tails.

6This statement leaves out of account small motions nearly or quite invisible to the naked eye, some of which are among the most interesting discoveries of telescopic astronomy; see, for example, chapterX.,§§ 207-215.

6This statement leaves out of account small motions nearly or quite invisible to the naked eye, some of which are among the most interesting discoveries of telescopic astronomy; see, for example, chapterX.,§§ 207-215.

7The custom of calling the sun and moon planets has now died out, and the modern usage will be adopted henceforward in this book.

7The custom of calling the sun and moon planets has now died out, and the modern usage will be adopted henceforward in this book.

8It may be noted that our word “day” (and the corresponding word in other languages) is commonly used in two senses, either for the time between sunrise and sunset (day as distinguished from night), or for the whole period of 24 hours or day-and-night. The Greeks, however, used for the latter a special word, νυχθήμερον.

8It may be noted that our word “day” (and the corresponding word in other languages) is commonly used in two senses, either for the time between sunrise and sunset (day as distinguished from night), or for the whole period of 24 hours or day-and-night. The Greeks, however, used for the latter a special word, νυχθήμερον.

9Compare the French: Mardi, Mercredi, Jeudi, Vendredi; or better still the Italian: Martedi, Mercoledi, Giovedi, Venerdi.

9Compare the French: Mardi, Mercredi, Jeudi, Vendredi; or better still the Italian: Martedi, Mercoledi, Giovedi, Venerdi.

10See, for example,Old Moore’sorZadkiel’s Almanack.

10See, for example,Old Moore’sorZadkiel’s Almanack.

11We have little definite knowledge of his life. He was born in the earlier part of the 6th centuryB.C., and died at the end of the same century or beginning of the next.

11We have little definite knowledge of his life. He was born in the earlier part of the 6th centuryB.C., and died at the end of the same century or beginning of the next.

12Theophrastus was born about half a century, Plutarch nearly five centuries, later than Plato.

12Theophrastus was born about half a century, Plutarch nearly five centuries, later than Plato.

13Republic, VII. 529, 530.

13Republic, VII. 529, 530.

14Confused, because the mechanical knowledge of the time was quite unequal to giving any explanation of the way in which these spheres acted on one another.

14Confused, because the mechanical knowledge of the time was quite unequal to giving any explanation of the way in which these spheres acted on one another.

15I have introduced here the familiar explanation of the phases of the moon, and the argument based on it for the spherical shape of the moon, because, although probably known before Aristotle, there is, as far as I know, no clear and definite statement of the matter in any earlier writer, and after his time it becomes an accepted part of Greek elementary astronomy. It may be noticed that the explanation is unaffected either by the question of the rotation of the earth or by that of its motion round the sun.

15I have introduced here the familiar explanation of the phases of the moon, and the argument based on it for the spherical shape of the moon, because, although probably known before Aristotle, there is, as far as I know, no clear and definite statement of the matter in any earlier writer, and after his time it becomes an accepted part of Greek elementary astronomy. It may be noticed that the explanation is unaffected either by the question of the rotation of the earth or by that of its motion round the sun.

16See, for example, the account of Galilei’s controversies, in chapterVI.

16See, for example, the account of Galilei’s controversies, in chapterVI.

17Thepolesof 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.

17Thepolesof 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.

18Theword“zenith” is Arabic, not Greek: cf. chapterIII., § 64.

18Theword“zenith” is Arabic, not Greek: cf. chapterIII., § 64.

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

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

20That of M. Paul Tannery:Recherches sur l’Histoire de l’Astronomie Ancienne, chap.V.

20That of M. Paul Tannery:Recherches sur l’Histoire de l’Astronomie Ancienne, chap.V.

21Trigonometry.

21Trigonometry.

22The 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 formulas= 16t2, wheresdenotes the number of feet fallen, andtthe number of seconds. By givingtany 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 formulal=nt+ 2esinnt, wherelis the distance from a fixed point in the orbit,tthe time, andn,ecertain numerical quantities.

22The 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 formulas= 16t2, wheresdenotes the number of feet fallen, andtthe number of seconds. By givingtany 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 formulal=nt+ 2esinnt, wherelis the distance from a fixed point in the orbit,tthe time, andn,ecertain numerical quantities.

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

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

24The 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.

24The 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.

25In the figure, which is taken from theDe Revolutionibusof Coppernicus (chapterIV.,§ 85), letD,K,Mrepresent respectively the centres of the sun, earth, and moon, at the time of an eclipse of the moon, and letS Q G,S R Edenote the boundaries of the shadow-cone cast by the earth; thenQ 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 trianglesG K-Q M:A D-G K::M K:K D.Hence ifK D=n.M Kand ∴ alsoA D=n. (radius of moon),nbeing 19 according to Aristarchus,G K-Q M:n. (radius of moon)-G K:: 1 :nn. (radius of moon) -G K=nG K-nQ 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 thatradius 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, ifnbeanyfairly large number.

25In the figure, which is taken from theDe Revolutionibusof Coppernicus (chapterIV.,§ 85), letD,K,Mrepresent respectively the centres of the sun, earth, and moon, at the time of an eclipse of the moon, and letS Q G,S R Edenote the boundaries of the shadow-cone cast by the earth; thenQ 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 ifK D=n.M Kand ∴ alsoA D=n. (radius of moon),nbeing 19 according to Aristarchus,

G K-Q M:n. (radius of moon)-G K:: 1 :nn. (radius of moon) -G K=nG K-nQ 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, ifnbeanyfairly large number.

26Histoire de l’Astronomie Ancienne, Vol. I., p. 185.

26Histoire de l’Astronomie Ancienne, Vol. I., p. 185.

27The 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 asAl Magisti, whence the LatinAlmagestumand ourAlmagest.

27The 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 asAl Magisti, whence the LatinAlmagestumand ourAlmagest.

28The 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.

28The 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.

29In spherical trigonometry.

29In spherical trigonometry.

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

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

31His 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.

31His 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.

32The equation of the centre and the evection may be expressed trigonometrically by two terms in the expression for the moon’s longitude,a sinθ +b sin(2φ-θ), wherea,bare 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)sinθ. 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 onb[sinθ +sin(2φ - θ)], or 2b sinφcos(φ-θ),which vanishes at conjunction or opposition, but reduces at the quadratures to 2b sinθ, 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 sin2φ [cos(2φ + θ) + 2cos(2φ - θ)],which, it will be observed, vanishes at quadratures as well as at conjunction and opposition.

32The equation of the centre and the evection may be expressed trigonometrically by two terms in the expression for the moon’s longitude,a sinθ +b sin(2φ-θ), wherea,bare 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)sinθ. 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[sinθ +sin(2φ - θ)], or 2b sinφcos(φ-θ),

which vanishes at conjunction or opposition, but reduces at the quadratures to 2b sinθ, 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 sin2φ [cos(2φ + θ) + 2cos(2φ - θ)],

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

33Here, 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.

33Here, 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.

34The 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 formulaent+ 2e sin nt,a(1 -e cos nt) respectively; the corresponding formulæ given by the use of the simple eccentric arent + e′ sin nt,a(1 -e′ cos nt). To make the angular motions agree we must therefore takee′= 2e, but to make the distances agree we must takee′ = e; the two conditions are therefore inconsistent. But by the introduction of an equant the formulæ becoment+ 2e′ sin nt,a(1 -e′ cos nt), andbothagree if we takee′ = 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.

34The 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 formulaent+ 2e sin nt,a(1 -e cos nt) respectively; the corresponding formulæ given by the use of the simple eccentric arent + e′ sin nt,a(1 -e′ cos nt). To make the angular motions agree we must therefore takee′= 2e, but to make the distances agree we must takee′ = e; the two conditions are therefore inconsistent. But by the introduction of an equant the formulæ becoment+ 2e′ sin nt,a(1 -e′ cos nt), andbothagree if we takee′ = 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.

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

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

36The 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.

36The 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.

37The 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.

37The 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.

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

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

39A 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.

39A 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.

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

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

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

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

42On trigonometry. He reintroduced thesine, which had been forgotten; and made some use of thetangent, but like Albategnius (§ 59n.) 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°.

42On trigonometry. He reintroduced thesine, which had been forgotten; and made some use of thetangent, but like Albategnius (§ 59n.) 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°.

43That of “lunar distances.”

43That of “lunar distances.”

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

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

45The 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.

45The 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.

46Nullo 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.

46Nullo 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.

47His 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).

47His 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).

48TheCommentariolusand thePrima Narratiogive 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.

48TheCommentariolusand thePrima Narratiogive 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.

49Omnis 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.

49Omnis 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.

50To 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 athirdmotion.

50To 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 athirdmotion.

51In 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.

51In 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.

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

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

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

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

54According 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.

54According 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.

55Euclid, I. 33.

55Euclid, I. 33.

56IfPbe the synodic period of a planet (in years), andSthe sidereal period, then we evidently have (1∕P) + 1 = 1∕Sfor an inferior planet, and 1 - (1∕P) = 1∕Sfor a superior planet.

56IfPbe the synodic period of a planet (in years), andSthe sidereal period, then we evidently have (1∕P) + 1 = 1∕Sfor an inferior planet, and 1 - (1∕P) = 1∕Sfor a superior planet.

57Recent biographers have called attention to a cancelled passage in the manuscript of theDe Revolutionibusin 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 (chapterVII.).

57Recent biographers have called attention to a cancelled passage in the manuscript of theDe Revolutionibusin 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 (chapterVII.).

58It may be noticed that the differential method of parallax (chapterVI.,§ 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.

58It may be noticed that the differential method of parallax (chapterVI.,§ 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.

59There 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.

59There 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.

60A similar discovery was in fact made twice again, by Galilei (chapterVI.,§ 114) and by Huygens (chapterVIII.,§ 157).

60A similar discovery was in fact made twice again, by Galilei (chapterVI.,§ 114) and by Huygens (chapterVIII.,§ 157).

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

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

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

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

63It would be interesting to know what use he assigned to the (presumably) still vaster spacebeyondthe stars.

63It would be interesting to know what use he assigned to the (presumably) still vaster spacebeyondthe stars.

64Tycho 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!

64Tycho 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!

65By transversals.

65By transversals.

66On an instrument which he had invented, called thehydrostatic balance.

66On an instrument which he had invented, called thehydrostatic balance.

67A 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.

67A 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.

68Ludovico delle Colombein a tractContra Il Moto della Terra, which is reprinted in the national edition of Galilei’s works, Vol. III.

68Ludovico delle Colombein a tractContra Il Moto della Terra, which is reprinted in the national edition of Galilei’s works, Vol. III.

69In a letter of May 4th, 1612, he says that he has seen them for eighteen months; in theDialogue 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.

69In a letter of May 4th, 1612, he says that he has seen them for eighteen months; in theDialogue 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.

70Historia e Dimostrazioni intorno alle Macchie Solari.

70Historia e Dimostrazioni intorno alle Macchie Solari.

71Acts i. 11. The pun is not quite so bad in its Latin form:Viri Galilaci, etc.

71Acts i. 11. The pun is not quite so bad in its Latin form:Viri Galilaci, etc.

72Spiritui sancto mentem fuisse nos docere, quo modo ad Coelum eatur, non autem quomodo Coelum gradiatur.

72Spiritui sancto mentem fuisse nos docere, quo modo ad Coelum eatur, non autem quomodo Coelum gradiatur.

73From the translation by Salusbury, in Vol. I. of hisMathematical Collections.

73From the translation by Salusbury, in Vol. I. of hisMathematical Collections.

74The 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.

74The 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.

75This 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.

75This 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.

76Actually, 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 theeastof it. This displacement is, however, very minute, and can only be detected by more delicate experiments than any devised by Galilei.

76Actually, 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 theeastof it. This displacement is, however, very minute, and can only be detected by more delicate experiments than any devised by Galilei.

77From the translation by Salusbury, in Vol. I. of hisMathematical Collections.

77From the translation by Salusbury, in Vol. I. of hisMathematical Collections.

78The official minute is:Et ei dicto quod dicat veritatem, alias devenietur ad torturam.

78The official minute is:Et ei dicto quod dicat veritatem, alias devenietur ad torturam.

79The three days June 21-24 the only ones which Galileicouldhave 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.

79The three days June 21-24 the only ones which Galileicouldhave 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.

80Equivalent to portions of the subject now calleddynamicsor (more correctly)kinematicsandkinetics.

80Equivalent to portions of the subject now calleddynamicsor (more correctly)kinematicsandkinetics.

81He 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.

81He 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.

82Two New Sciences, translated by Weston, p. 255.

82Two New Sciences, translated by Weston, p. 255.

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

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

84The 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.

84The 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.

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

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

86Commentaries on the Motions of Mars, Part II., end of chapterXIX.

86Commentaries on the Motions of Mars, Part II., end of chapterXIX.

87An ellipse is one of several curves, known asconic 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 thefoci, is always the same.Fig. 59.—An ellipse.Thus if, in the figure,SandHare the foci, andP,Qareanytwo points on the curve, then the distancesS P,H Padded together are equal to the distancesS Q,Q Hadded together, and each sum is equal to the lengthA A′of the ellipse. The ratio of the distanceS Hto the lengthA A′is known as theeccentricity, and is a convenient measure of the extent to which the ellipse differs from a circle.

87An ellipse is one of several curves, known asconic 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 thefoci, is always the same.

Fig. 59.—An ellipse.

Fig. 59.—An ellipse.

Thus if, in the figure,SandHare the foci, andP,Qareanytwo points on the curve, then the distancesS P,H Padded together are equal to the distancesS Q,Q Hadded together, and each sum is equal to the lengthA A′of the ellipse. The ratio of the distanceS Hto the lengthA A′is known as theeccentricity, and is a convenient measure of the extent to which the ellipse differs from a circle.

88The ellipse ismoreelongated 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.

88The ellipse ismoreelongated 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.

89Astronomia Novaαἰτιολογητοςseu Physica Coelestis, tradita Commentariis de Motibus Stellae Martis.Ex Observationibus G. V. Tychonis Brahe.

89Astronomia Novaαἰτιολογητοςseu Physica Coelestis, tradita Commentariis de Motibus Stellae Martis.Ex Observationibus G. V. Tychonis Brahe.

90It contains the germs of the method of infinitesimals.

90It contains the germs of the method of infinitesimals.

91Harmonices Mundi Libri V.

91Harmonices Mundi Libri V.

92There 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., chapterIII.

92There 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., chapterIII.

93Epitome, Book IV., Part 2.

93Epitome, Book IV., Part 2.

94Introduction to theCommentaries on the Motions of Mars.

94Introduction to theCommentaries on the Motions of Mars.

95Substantially thefilar micrometerof modern astronomy.

95Substantially thefilar micrometerof modern astronomy.

96Galilei, 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.

96Galilei, 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.

97In modern notation: time oπf oscillation = 2π√(l∕g).

97In modern notation: time oπf oscillation = 2π√(l∕g).

98I.e.he obtained the familiar formula (v2)∕r, and several equivalent forms forcentrifugal force.

98I.e.he obtained the familiar formula (v2)∕r, and several equivalent forms forcentrifugal force.

99Also frequently referred to by the Latin nameCartesius.

99Also frequently referred to by the Latin nameCartesius.

100According 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. chapterII.,§ 22).

100According 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. chapterII.,§ 22).

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

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

102W. K. Clifford,Aims and Instruments of Scientific Thought.

102W. K. Clifford,Aims and Instruments of Scientific Thought.

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

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

104The familiarparallelogram of forces, of which earlier writers had had indistinct ideas, was clearly stated and proved in the introduction to thePrincipia, and was, by a curious coincidence, published also in the same year byVarignonandLami.

104The familiarparallelogram of forces, of which earlier writers had had indistinct ideas, was clearly stated and proved in the introduction to thePrincipia, and was, by a curious coincidence, published also in the same year byVarignonandLami.

105It is between 13 and 14 billion billion pounds. See chapterX.§ 219.

105It is between 13 and 14 billion billion pounds. See chapterX.§ 219.

106As 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 thePrincipia.

106As 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 thePrincipia.

107It 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 thePrincipiato 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.

107It 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 thePrincipiato 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.

108Throughout 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 (chapterVI.,§ 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 thePrincipiawas 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.

108Throughout 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 (chapterVI.,§ 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 thePrincipiawas 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.

109He 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.

109He 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.

110He 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.”

110He 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.”

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

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

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

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

113By Bessel (chapterXIII.,§ 277).

113By Bessel (chapterXIII.,§ 277).

114The 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.”

114The 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.”

115Rigaud, in the memoirs prefixed to Bradley’sMiscellaneous Works.

115Rigaud, in the memoirs prefixed to Bradley’sMiscellaneous Works.

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

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

117The story is given in T. Thomson’sHistory 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.

117The story is given in T. Thomson’sHistory 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.

118It isk sinC A B, wherekis the constant of aberration.

118It isk sinC A B, wherekis the constant of aberration.

119His 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.

119His 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.

120Recherches sur la précession des équinoxes et sur la nutation de l’axe de la terre.

120Recherches sur la précession des équinoxes et sur la nutation de l’axe de la terre.

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

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

122Principia, Book III., proposition 10.

122Principia, Book III., proposition 10.

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

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

12469 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.

12469 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.

125The 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.

125The 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.

126A mural quadrant.

126A mural quadrant.

127The ordinary approximate theory of thecollimation error,level error, anddeviation errorof a transit, as given in textbooks of spherical and practical astronomy, is substantially his.

127The ordinary approximate theory of thecollimation error,level error, anddeviation errorof a transit, as given in textbooks of spherical and practical astronomy, is substantially his.

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

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

129For a more detailed discussion of the transit of Venus, see Airy’sPopular Astronomyand Newcomb’sPopular Astronomy.

129For a more detailed discussion of the transit of Venus, see Airy’sPopular Astronomyand Newcomb’sPopular Astronomy.

130Someother 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.

130Someother 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.

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

131The 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.”

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

133Longevity 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.

133Longevity 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.

134This 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.

134This 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.

135E.g.Mélanges de Philosophie, de l’Histoire, et de Littérature;Éléments de Philosophie;Sur la Destruction des Jésuites.

135E.g.Mélanges de Philosophie, de l’Histoire, et de Littérature;Éléments de Philosophie;Sur la Destruction des Jésuites.

136I.e.he assumed a law of attraction represented by μ∕r2+ ν∕r3.

136I.e.he assumed a law of attraction represented by μ∕r2+ ν∕r3.

137This appendix is memorable as giving for the first time the method ofvariation of parameterswhich Lagrange afterwards developed and used with such success.

137This appendix is memorable as giving for the first time the method ofvariation of parameterswhich Lagrange afterwards developed and used with such success.

138That of the distinguished American astronomer Dr. G. W. Hill (chapterXIII.,§ 286).

138That of the distinguished American astronomer Dr. G. W. Hill (chapterXIII.,§ 286).

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

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

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

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

141On theCalculus of Variations.

141On theCalculus of Variations.

142The establishment of the general equations of motion by a combination ofvirtual velocitiesandD’Alembert’s principle.

142The establishment of the general equations of motion by a combination ofvirtual velocitiesandD’Alembert’s principle.

143Théorie des Fonctions Analytiques(1797);Resolution des Équations Numériques(1798);Leçons sur le Calcul des Fonctions(1805).

143Théorie des Fonctions Analytiques(1797);Resolution des Équations Numériques(1798);Leçons sur le Calcul des Fonctions(1805).

144Théorie Analytique des Probabilités.

144Théorie Analytique des Probabilités.

145The 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.

145The 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.

146Outlines of Astronomy, § 656.

146Outlines of Astronomy, § 656.

147Laplace,Système du Monde.

147Laplace,Système du Monde.

148Ifn,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,coswherep,p′are integers, and the coefficient is of the orderp⁓p′in the eccentricities and inclinations. If nowpandp′are such thatnp⁓n′p′is small, the corresponding inequality has a period 2π∕(np⁓n′p′), and though its coefficient is of orderp⁓p′, it has the small factornp⁓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; 5n′- 2n= 1,466; there is therefore an inequality of thethirdorder, with a period (in years) = 360°∕1,466″ = 900.

148Ifn,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

wherep,p′are integers, and the coefficient is of the orderp⁓p′in the eccentricities and inclinations. If nowpandp′are such thatnp⁓n′p′is small, the corresponding inequality has a period 2π∕(np⁓n′p′), and though its coefficient is of orderp⁓p′, it has the small factornp⁓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; 5n′- 2n= 1,466; there is therefore an inequality of thethirdorder, with a period (in years) = 360°∕1,466″ = 900.

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

149This 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∑e2m√a=c, ∑tan2im√a=c′, wheremis the mass of any planet,a,e,iare 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 nottan iis replaced as in the text byi.

150∑e2m√a=c, ∑tan2im√a=c′, wheremis the mass of any planet,a,e,iare 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 nottan iis replaced as in the text byi.

151Nearly 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.

151Nearly 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.

152Of tables based on Laplace’s work and published up to the time of his death, the chief solar ones were those ofvon Zach(1804) andDelambre(1806); and the chief planetary ones were those ofLalande(1771), ofLindenaufor Venus, Mars, and Mercury (1810-13), and ofBouvardfor Jupiter, Saturn, and Uranus (1808 and 1821).

152Of tables based on Laplace’s work and published up to the time of his death, the chief solar ones were those ofvon Zach(1804) andDelambre(1806); and the chief planetary ones were those ofLalande(1771), ofLindenaufor Venus, Mars, and Mercury (1810-13), and ofBouvardfor Jupiter, Saturn, and Uranus (1808 and 1821).

153, The motion of the satellites of Uranus (chapterXII.,§ 253255) 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.

153, The motion of the satellites of Uranus (chapterXII.,§ 253255) 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.

154This statement again has to be modified in consequence of the discoveries, beginning on January 1st, 1801, of the minor planets (chapterXIII.,§ 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.

154This statement again has to be modified in consequence of the discoveries, beginning on January 1st, 1801, of the minor planets (chapterXIII.,§ 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.

155Système du Monde, Book V., chapterVI.

155Système du Monde, Book V., chapterVI.

156In 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.

156In 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.

157The 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.

157The 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.

158More 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.

158More 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.

159The 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.

159The 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.

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

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

161Sir 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.

161Sir 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.

162An 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.

162An 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.

163Newcomb’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.

163Newcomb’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.

164Fundamenta Nova Investigationis Orbitae Verae quam Luna perlustrat.

164Fundamenta Nova Investigationis Orbitae Verae quam Luna perlustrat.

165Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Störungen.

165Darlegung der theoretischen Berechnung der in den Mondtafeln angewandten Störungen.

166E.g.in Grant’sHistory of Physical Astronomy, Herschel’sOutlines of Astronomy, Miss Clerke’sHistory of Astronomy in the Nineteenth Century, and the memoir by Dr. Glaisher prefixed to the first volume of Adams’sCollected Papers.

166E.g.in Grant’sHistory of Physical Astronomy, Herschel’sOutlines of Astronomy, Miss Clerke’sHistory of Astronomy in the Nineteenth Century, and the memoir by Dr. Glaisher prefixed to the first volume of Adams’sCollected Papers.

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

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

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

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

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

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

170Catalogus novus stellarum duplicium,Stellarum duplicium et multiplicium mensurae micrometricae, andStellarum fixarum imprimis duplicium et multiplicium positiones mediae pro epocha 1830.

170Catalogus novus stellarum duplicium,Stellarum duplicium et multiplicium mensurae micrometricae, andStellarum fixarum imprimis duplicium et multiplicium positiones mediae pro epocha 1830.

171I.e.2·512... is chosen as being the number the logarithm of which is ·4, so that (2·512...)5∕2= 10.

171I.e.2·512... is chosen as being the number the logarithm of which is ·4, so that (2·512...)5∕2= 10.

172If 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 magnitudemis given by the formulaL = (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 anegativequantity,-1·4, according to the Harvard photometry.

172If 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 magnitudemis 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 anegativequantity,-1·4, according to the Harvard photometry.

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