ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

126 GREEK AND HINDU SPHERICAL ASTRONOMY "the early spheric did not deal with the geometry of the sphere as such, still less did it contain anything of the nature of the spheri- cal trigonometry. (This deficiency was afterwards made good by Menelaus's Sphaerica).¹ Hence the Greek spherical trigonometry began with Menelaus (90 A.D.). His theorem in geometry is well-known—"If the sides of a plane triangle be cut by a trans- versal into six segments, the continued product of any three alternate segments, is equal to the continued product of the remaining three." From this proposition he deduced the so- called "regula sex quantitatum" or the theorem, if the sides of a spherical triangle be cut by an arc of a great circle into six segments, the continued product of the chords of the doubles of any three alternate segments is equal to the continued product of the chords of doubles of the remaining three segments." In plane geometry if the sides BC, CA, AB of a triangle be cut by any transversal at L, M, N, respectively, then we have (BL / LC) · (CM / MA) · (AN / NB) = 1. In spherics the theorem is : (Chord 2 BL / Chord 2 LC) · (Chord 2 CM / Chord 2 MA) · (Chord 2 AN / Chord 2 NB) = 1 Both these theorems are proved in Ptolemy's Syntaxis ( Karl Manitius's edi- Fig. 5 tion, Vol. I, pp. 45-51). If R be the radius of the sphere on which the spherical triangle ABC is constructed, then the chord of the arc 2 BL = 2 R sin BL. Hence Menelaus's theorem in spherics may be expres- sed as follows : (Sin BL / Sin LC) · (Sin CM / Sin MA) · (Sin AN / Sin NB) = 1 This theorem is true for any spherical triangle. If ∠B = AN = AM = 90° and L the pole of AB, then LMN is a secondary to the arc AB. There are four arcs of great circles; taking any three as forming a spherical triangle and the fourth as the transversal we readily get for the right-angled

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1. A.A. Bjornbo, "Studien über Menelaos' Sphärik" in Abhandlun- gen Zur Geschichte der Mathematischen Wissenschaften for 1902, pp. 89 et seq.; also Heath, Greek Mathematics, vol. II p. 261-73.