Editing History of geometry (section)

===Vedic India geometry===

[[Image:Rigveda MS2097.jpg|thumb|right|''[[Rigveda]]'' manuscript in [[Devanagari]]]]
The Indian [[Vedic period]] had a tradition of geometry, mostly expressed in the construction of elaborate altars.
Early Indian texts (1st millennium BC) on this topic include the ''[[Satapatha Brahmana]]'' and the ''[[Shulba Sutras|Śulba Sūtras]]''.<ref>A. Seidenberg, 1978. The origin of mathematics. Archive for the history of Exact Sciences, vol 18.</ref><ref>{{Harv|Staal|1999}}</ref><ref>Most mathematical problems considered in the ''Śulba Sūtras'' spring from "a single theological requirement,"  that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks. {{Harv|Hayashi|2003|p=118}}</ref>

The ''Śulba Sūtras'' has been described as "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians."{{Sfn|Hayashi|2005|p=363}} They make use of [[Pythagorean triples]],{{Sfn|Knudsen|2018|p=87}}<ref>Pythagorean triples are triples of integers <math> (a,b,c) </math> with the property: <math>a^2+b^2=c^2</math>. Thus, <math>3^2+4^2=5^2</math>, <math>8^2+15^2=17^2</math>, <math>12^2+35^2=37^2</math> etc.</ref> which are particular cases of [[Diophantine equations]].<ref name="cooke198">{{Harv|Cooke|2005|p=198}}: "The arithmetic content of the ''Śulva Sūtras'' consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others."</ref>

According to mathematician S. G. Dani, the Babylonian cuneiform tablet [[Plimpton 322]] written c. 1850 BC<ref>Mathematics Department, University of British Columbia, [http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html ''The Babylonian tabled Plimpton 322''].</ref> "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,<ref>Three positive integers <math>(a, b, c) </math> form a ''primitive'' Pythagorean triple if <math> c^2=a^2+b^2</math> and if the highest common factor of <math> a, b, c </math> is 1. In the particular Plimpton322 example, this means that <math> 13500^2+ 12709^2= 18541^2 </math> and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see Plimpton 322 for details.</ref> indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850&nbsp;BC.{{Sfn|Dani|2003|p=223}} "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India."{{Sfn|Dani|2003|p=223}} Dani goes on to say:{{Sfn|Dani|2003|p=223–224}}

<blockquote> As the main objective of the ''Sulvasutras'' was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the ''Sulvasutras''. The occurrence of the triples in the ''Sulvasutras'' is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily.</blockquote>