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.