Editing History of geometry (section)

==Classical Indian geometry==
{{see also|Indian mathematics}}

In the [[Bakhshali manuscript]], there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."<ref name=hayashi2005-371>{{Harv|Hayashi|2005|p=371}}</ref> [[Aryabhata]]'s ''[[Aryabhatiya]]'' (499) includes the computation of areas and volumes.

[[Brahmagupta]] wrote his astronomical work ''[[Brahmasphutasiddhanta|{{IAST|Brāhma Sphuṭa Siddhānta}}]]'' in 628. Chapter 12, containing 66 [[Sanskrit]] verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).<ref name=hayashi2003-p121-122>{{Harv|Hayashi|2003|pp=121–122}}</ref> In the latter section, he stated his famous theorem on the diagonals of a [[cyclic quadrilateral]]:<ref name=hayashi2003-p121-122/>

'''Brahmagupta's theorem:''' If a cyclic quadrilateral has diagonals that are [[perpendicular]] to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side.

Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of [[Heron's formula]]), as well as a complete description of [[rational triangle]]s (''i.e.'' triangles with rational sides and rational areas).

'''Brahmagupta's formula:''' The area, ''A'', of a cyclic quadrilateral with sides of lengths ''a'', ''b'', ''c'', ''d'', respectively, is given by

: <math> A = \sqrt{(s-a)(s-b)(s-c)(s-d)}</math>

where ''s'', the [[semiperimeter]], given by: <math> s=\frac{a+b+c+d}{2}.</math>

'''Brahmagupta's Theorem on rational triangles:''' A triangle with rational sides <math>a, b, c </math> and rational area is of the form:

:<math>a = \frac{u^2}{v}+v, \ \ b=\frac{u^2}{w}+w, \ \ c=\frac{u^2}{v}+\frac{u^2}{w} - (v+w) </math>
for some rational numbers <math>u, v, </math> and <math> w </math>.<ref>{{Harv|Stillwell|2004|p=77}}</ref>

[[Parameshvara Nambudiri]] was the first mathematician to give a formula for the [[radius]] of the [[circle]] circumscribing a cyclic quadrilateral.<ref>Radha Charan Gupta [1977] "Parameshvara's rule for the circumradius of a cyclic quadrilateral", ''Historia Mathematica'' 4: 67–74</ref> The expression is sometimes attributed to [[Lhuilier]] [1782], 350 years later. With the sides of the [[cyclic quadrilateral]] being ''a, b, c,'' and ''d'', the radius ''R'' of the circumscribed circle is:
:<math> R = \sqrt {\frac{(ab + cd)(ac + bd)(ad + bc)}{(- a + b + c + d)(a - b + c + d)(a + b - c + d)(a + b + c - d)}}.</math>