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
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1, 2, Bjornbo, loc. cit. p. 86.
3. Zeuthen, loc. cit. p. 27.