A Short History of Astronomy, Arthur Berry [large screen ebook reader .TXT] 📗
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16 See, for example, the account of Galilei’s controversies, in chapter VI.
17 The poles of a great circle on a sphere are the ends of a diameter perpendicular to the plane of the great circle. Every point on the great circle is at the same distance, 90°, from each pole.
18 The word “zenith” is Arabic, not Greek: cf. chapter III., § 64.
19 Most of these names are not Greek, but of later origin.
20 That of M. Paul Tannery: Recherches sur l’Histoire de l’Astronomie Ancienne, chap. V.
21 Trigonometry.
22 The process may be worth illustrating by means of a simpler problem. A heavy body, falling freely under gravity, is found (the resistance of the air being allowed for) to fall about 16 feet in 1 second, 64 feet in 2 seconds, 144 feet in 3 seconds, 256 feet in 4 seconds, 400 feet in 5 seconds, and so on. This series of figures carried on as far as may be required would satisfy practical requirements, supplemented if desired by the corresponding figures for fractions of seconds; but the mathematician represents the same facts more simply and in a way more satisfactory to the mind by the formula s = 16 t2, where s denotes the number of feet fallen, and t the number of seconds. By giving t any assigned value, the corresponding space fallen through is at once obtained. Similarly the motion of the sun can be represented approximately by the more complicated formula l = nt + 2 e sin nt, where l is the distance from a fixed point in the orbit, t the time, and n, e certain numerical quantities.
23 At the present time there is still a small discrepancy between the observed and calculated places of the moon. See chapter XIII., § 290.
24 The name is interesting as a remnant of a very early superstition. Eclipses, which always occur near the nodes, were at one time supposed to be caused by a dragon which devoured the sun or moon. The symbols ☊ ☋ still used to denote the two nodes are supposed to represent the head and tail of the dragon.
25 In the figure, which is taken from the De Revolutionibus of Coppernicus (chapter IV., § 85), let D, K, M represent respectively the centres of the sun, earth, and moon, at the time of an eclipse of the moon, and let S Q G, S R E denote the boundaries of the shadow-cone cast by the earth; then Q R, drawn at right angles to the axis of the cone, is the breadth of the shadow at the distance of the moon. We have then at once from similar triangles
G K - Q M : A D - G K :: M K : K D.
Hence if K D = n. M K and ∴ also A D = n. (radius of moon), n being 19 according to Aristarchus,
G K-Q M: n. (radius of moon)-G K
:: 1 : n
n . (radius of moon) - G K
= n G K - n Q M
∴ radius of moon + radius of shadow
= (1 + 1∕n) (radius of earth).
By observation the angular radius of the shadow was found to be about 40′ and that of the moon to be 15′, so that
radius of shadow = 8∕3 radius of moon;
∴ radius of moon
= 3∕11 (1 + 1∕n) (radius of earth).
But the angular radius of the moon being 15′, its distance is necessarily about 220 times its radius,
and ∴ distance of the moon
= 60 (1 + 1∕n) (radius of the earth),
which is roughly Hipparchus’s result, if n be any fairly large number.
26 Histoire de l’Astronomie Ancienne, Vol. I., p. 185.
27 The chief MS. bears the title μεγάλη σύνταξις or great composition though the author refers to his book elsewhere as μαθηματικὴ σύνταξις (mathematical composition). The Arabian translators, either through admiration or carelessness, converted μεγάλη, great, into μεγίστη, greatest, and hence it became known by the Arabs as Al Magisti, whence the Latin Almagestum and our Almagest.
28 The better known apparent enlargement of the sun or moon when rising or setting has nothing to do with refraction. It is an optical illusion not very satisfactorily explained, but probably due to the lesser brilliancy of the sun at the time.
29 In spherical trigonometry.
30 A table of chords (or double sines of half-angles) for every 1∕2° from 0° to 180°.
31 His procedure may be compared with that of a political economist of the school of Ricardo, who, in order to establish some rough explanation of economic phenomena, starts with certain simple assumptions as to human nature, which at any rate are more plausible than any other equally simple set, and deduces from them a number of abstract conclusions, the applicability of which to real life has to be considered in individual cases. But the perfunctory discussion which such a writer gives of the qualities of the “economic man” cannot of course be regarded as his deliberate and final estimate of human nature.
32 The equation of the centre and the evection may be expressed trigonometrically by two terms in the expression for the moon’s longitude, a sinθ + b sin (2φ-θ), where a, b are two numerical quantities, in round numbers 6° and 1°, θ is the angular distance of the moon from perigee, and φ is the angular distance from the sun. At conjunction and opposition φ is 0° or 180°, and the two terms reduce to (a-b) 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 2 b sinφ cos (φ-θ),
which vanishes at conjunction or opposition, but reduces at the quadratures to 2 b 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 sin 2φ [cos (2φ + θ) + 2 cos (2φ - θ)],
which, it will be observed, vanishes at quadratures as well as at conjunction and opposition.
33 Here, as elsewhere, I have given no detailed account of astronomical instruments, believing such descriptions to be in general neither interesting nor intelligible to those who have not the actual instruments before them, and to be of little use to those who have.
34 The advantage derived from the use of the equant can be made clearer by a mathematical comparison with the elliptic motion introduced by Kepler. In elliptic motion the angular motion and distance are represented approximately by the formulae nt + 2e sin nt, a (1 - e cos nt) respectively; the corresponding formulæ given by the use of the simple eccentric are nt + e′ sin nt, a (1 - e′ cos nt). To make the angular motions agree we must therefore take e′ = 2e, but to make the distances agree we must take e′ = e; the two conditions are therefore inconsistent. But by the introduction of an equant the formulæ become nt + 2e′ sin nt, a (1 - e′ cos nt), and both agree if we take e′ = e. Ptolemy’s lunar theory could have been nearly freed from the serious difficulty already noticed (§ 48) if he had used an equant to represent the chief inequality of the moon; and his planetary theory would have been made accurate to the first order of small quantities by the use of an equant both for the deferent and the epicycle.
35 De Morgan classes him as a geometer with Archimedes, Euclid, and Apollonius, the three great geometers of antiquity.
36 The legend that the books in the library served for six months as fuel for the furnaces of the public baths is rejected by Gibbon and others. One good reason for not accepting it is that by this time there were probably very few books left to burn.
37 The data as to Indian astronomy are so uncertain, and the evidence of any important original contributions is so slight, that I have not thought it worth while to enter into the subject in any detail. The chief Indian treatises, including the one referred to in the text, bear strong marks of having been based on Greek writings.
38 He introduced into trigonometry the use of sines, and made also some little use of tangents, without apparently realising their importance: he also used some new formulæ for the solution of spherical triangles.
39 A prolonged but indecisive controversy has been carried on, chiefly by French scholars, with regard to the relations of Ptolemy, Abul Wafa, and Tycho in this matter.
40 For example, the practice of treating the trigonometrical functions as algebraic quantities to be manipulated by formulæ, not merely as geometrical lines.
41 Any one who has not realised this may do so by performing with Roman numerals the simple operation of multiplying by itself a number such as MDCCCXCVIII.
42 On trigonometry. He reintroduced the sine, which had been forgotten; and made some use of the tangent, but like Albategnius (§ 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°.
43 That of “lunar distances.”
44 He did not invent the measuring instrument called the vernier, often attributed to him, but something quite different and of very inferior value.
45 The name is spelled in a large number of different ways both by Coppernicus and by his contemporaries. He himself usually wrote his name Coppernic, and in learned productions commonly used the Latin form Coppernicus. The spelling Copernicus is so much less commonly used by him that I have thought it better to discard it, even at the risk of appearing pedantic.
46 Nullo demum loco ineptior est quam ... ubi nim’s pueriliter hallucinatur: Nowhere is he more foolish than ... where he suffers from delusions of too childish a character.
47 His real name was Georg Joachim, that by which
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