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

136 GREEK AND INDIAN METHODS where Z is the zenith and Q is the point of intersection of the prime vertical and its secondary passing through the Sun and the north-south points. Zeuthen¹ points out that later in the same treatise Ptolemy finds the arc 2β described above the horizon by a star of given declination δ' by a procedure equivalent to the formula. (4) cos β=tan δ' tan ϕ. With regard to the 'Analemma' of Ptolemy. it may be noted, as Heath² says, that "the procedure amounts to a method of graphically constructing the arcs required as parts of an auxiliary circle in one plane." Many things may be, in practice, done graphically far more easily than by the theoreti- cal method. Besides, no theoretical calculations occur in the 'Analemma'. Zeuthen², following the method of this work, has deduced in the general case, the two equations. (5) cos Z=(cos δ, cos H+sin δ. tan ϕ) cos ϕ. cos δ.sin H (6) tan α=———————————————————————————————————————— sin δ ————— +(cos δ.cos H+sin δ.tan ϕ) sin ϕ cos δ These equations are suggested to a modern reader from a study of the figures in the 'Analemma.' But neither in this work nor in the 'Syntaxis' are they to be found. With regard to the first four formulae, it is possible that they were recognised by Ptolemy. With regard to the last two, Zeuthen³ remarks "mais le texte nen contient rien,' and they were certainly not recog- nised by Ptolemy. Besides the tangent function is wholly absent in Greek trigonometry. They are also different in form from those arrived at by the Indian method as explained before. Thus, it is clear that the Indian methods are in no way connected with the method of the 'Analemma.' Even taking for granted that the Indians followed a method of projection much allied to the method of the Analemma' there is no adequate reason for assuming that their method is derived from any Greek source. Analogy and precedence do not neces- sarily constitute originality—there is still the chance of a remoter origin from which both the systems drew their inspiration. The method of the 'Analemma,' as has been already stated, presents a —————————————————————————————————————— 1, 2, Bjornbo, loc. cit. p. 86. 3. Zeuthen, loc. cit. p. 27.