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Brahmagupta (ad 628) was the first mathematician to provide the formula for the area of a cyclic quadrilateral. His contributions to geometry are significant. He is the first person to discuss the method of finding a cyclic quadrilateral with rational sides.
Feb 1, 2010 · Chasles [1837, 420–447] sees in Brahmagupta’s Propositions XII.21–38 an outline of a general theory of quadrilaterals in which all the lines considered by Brahmagupta are rational. 25 Indeed, XII.33–38 explain how to obtain, in terms of arbitrary quantities (r a ¯ śi), the sides of distinguished quadrilaterals. 26 Chasles considers that Brahmagupta’s propositions constitute steps leading up to the construction of such rational quadrilaterals.
Brahmagupta was very well acquainted with the mathematics and astronomy of Aryabhata I. He seems to criticize Aryabhata I severely in his work “Brahma Sphuta Siddhanta” but seems to be compelled to bow to Aryabhata’s popularity by referring to Aryabhata with tremendous reverence in his later work, “Khandakadyaka.”
Feb 1, 1985 · Aryabhata II (xv, 69) pointed out that the rule is accurate in the case of a triangle, but not for a quadrilateral. 8ripati followed Brahmagupta here, completely disregarding the warning of Aryabhata Il, that "the mathematician who wishes to tell the area or the altitudes of a quadrilateral without knowing a diagonal is either a fool or a blunderer" (xv, 70). d:ripati's second rule for the area of a quadrilateral is remarkable for its contradic, Lion of the corresponding rules of gridhara ...
Brahmagupta solved completely to find the rational solutions of the equation Nx 2 +k=y 2. He also exhibited how to find the integral solutions of the equation Nx 2 +1= y 2 , provided an integral solution ( x y )=( α β ) can be found for the equation Nx 2 + k = y 2 , when k =−1, ±2, ±4.
Description. Mathematics in India has a long and impressive history. Presented in chronological order, this book discusses mathematical contributions of Pre-Modern Indian Mathematicians from the Vedic period (800 B.C.) to the 17 th Century of the Christian era. These contributions range across the fields of Algebra, Geometry and Trigonometry.
Aug 1, 2016 · Chasles then turned to Colebrooke's translation of the mathematical tracts of Brahmagupta's Brāhma-sphuṭa-siddhānta, and mainly focused on that part registered by his British predecessor as chapter XII, “Arithmetic (Ganita)”, section IV, “Plane figure”. 115 Therein, he highlighted some most valuable propositions which Colebrooke had overlooked, “probably because they stood as so many enigmas, due to their laconism and the obscurity of their statement, in each of which certain ...
May 1, 1986 · In 1879 another mathematician, Li Liu [am], wrote Haidao suanjing wei bi [an] [Notes on the Sea island mathematical manual], in which he used the tian yuan [ao] ["celestial element"] method to solve the sea island problems. As the tian yuan method was a product of the Song-Yuan period (A.D. 960-1368), its use by Liu Hui is indeed doubtful.
Jan 1, 2008 · Any Brahmagupta quadrilateral with four integer sides has integer area and even perimeter. Proof. Since the sides are integral and the area is rational, then (4K 4 ) 2 = 8abcd + 2 parenleftbig a 2 b 2 + a 2 c 2 + a 2 d 2 + b 2 c 2 + b 2 d 2 + c 2 d 2 parenrightbig − a 4 − b 4 − c 4 − d 4 implies that 4K 4 ∈ Z.
Nov 1, 2012 · Propositions XII.21–27 of Brahmagupta’s Brāhma-sphuṭa-siddhānta (BSS, India, 628 A.D.) constitute a coherent mathematical discourse leading to the derivation of the expression of the area of an arbitrary cyclic quadrilateral; Proposition XII.28, giving the expressions of its diagonals, follows from it. 1 The purpose of this paper is to extend this analysis to XII.29–32, that show how to determine the lengths of all the portions of diagonals and perpendiculars defined by their ...
