By T. A. Saraswati Amma

This booklet is a geometric survey of the Sanskrit and Prakrt clinical and quasi-scientific literature and finishing with the early a part of the seventeenth century. The paintings seeks to blow up the speculation that the Indian mathematical genius used to be predominantly genius used to be predominantly algebraic and computational and that's eschewed proofs and rationales.

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If PO*, z) = \xz\ and p ( y , z) = \yz\, th en j) (x, y) < \x z \ + Ia iJ = P (*. z) + p (y, z ). If p (x, z) — I x z I then p (x, y) < \ x z \ = p (x, z) and, bgnce, p (x, y) p (*, z) + p (y, z). Similarly, if p(y, z) — \ y z \ , then p(*. y) < p(y, z) and, hence, P {x, P (x, У Х P (x, z) + p (y, z). In the case where the three points X 9 y 9 z are not collinear, the triangle inequality follows from the Axiom of Pasch and the ad ditivity of the set function o. In fact, let x ÿ z be a projective triangle DESARGUESIAN METRIZATIONS OF A PROJECTIVE SPACE 23 and let its side xz, y z be the shorter of the intervals of the corresponding lines xz and yz.

If O denotes the set of parameters of the trans formation A, then the metric pô is defined by the formula Iû |< ô The average pô of the metric p makes sense in the whole space if it is interpreted in this way. Moreover, pa is a Desarguesian metric and, REGULAR APPROXIMATION OF DESARGUESIAN METRICS 45 outside the plane a, it has the regularity properties established in Section 5. In the case of a metric of elliptic type, the metric pô converges to the metric p in the whole space as Ô-> 0 . The transition from the metric pa to the metric pô8also represents an averaging and, hence, again reduces to a Desarguesian metric, converging uniformly to the metric pô in the whole space as e 0.

The quantity A is equal to the value of the set function о on the set of lines intersecting the lateral sides of the tirangle. Hence where the integration is carried out over the set of lines g intersecting both lateral sides uw and vw of the triangle. Because y is continuous and a and ô are small, we have the approximation For sufficiently small a and ô the integral on the right is proportional to the altitude of the triangle ^ a ô /2 (the integration with respect to p ) and to the base angle a (the integration with respect to £).