496 Journal of the Asiatic Society of Bengal^ [July, 1907. 



hang some ideas. ^ Those who are really familiar with the works 

 in question must agree with Chasles, who wrote long ago : " L'ouv- 

 rage de Bhascara n'est qu'une imitation tres-imparfaite de celui 

 de Brahmagupta, qui y est commente et denature . • - . Les pro- 

 positions les plus importantes de Brahmagupta . . . - y sont onuses, 

 ou enoncees commes inexacfces . . • , Cette circonstance et les com- 

 mentaires de differens scoliastes, nous paraissent prouver que, 

 depuis Brahmagupta, les sciences, dans Tlnde, ont ete en decli- 

 nant" (p. 420). The significance of these statements, regarding 

 the verity of which there is not the slightest doubt, is great. We 

 are led to suspect, but not only by these considerations, that there 

 never was a school of Hindu mathematicians. Further, if Bbas* 

 kara and the other commentators were not competent enough to 

 appreciate Brahmagupta's work, it suggests the idea that, perhaps, 

 Brahmagupta himself was of the same type as his successors. 



Colebrooke says that Aryabhata was superior to any Hindu 

 who came after him and that deterioration rather than advance- 

 ment took place since the time of the more ancient author (p. 9). 



Wherein, then, does the reputation of Brahmagupta lie? In 

 the early part of the last century it was stated that his formula 

 (the correct one) for the area of the triangle was tfie earliest 

 known citation of it. Consequently it was assumed t)»at Brahma- 

 gupta was the discoverer of this useful formula. But, as was 

 bund out later on, the formula in question was known to Heron 

 the Elder (2nd century B.C.) and was demonstrated by him. Still 

 the reputation sticks. Moreover, Chasles thought that the pri- 

 ority of the statement of the same formula extended to quadri- 

 laterals rests with Brahmagupta, but it is even doubtful whether 

 the rule given was intetided to apply to quadrilaterals at all. 

 Certainly the commentators thought it did, but they did not un- 

 derstand its application. Krishna^s illustrations are ludicrous, 

 while Bhaskara did not understand that the formula applies to 

 cyclic quadrilaterals only and said* that anyone who believed 

 in it was a " blundering devil." 



Side by side with this correct formula for the triangle, Brah- 

 magupta states that the product of half the base and half the sum 

 of the other sides is the gross area of a triangle. That a mathe- 

 matician should state such a crude proposition is inconceivable. 

 It is, indeed, given by Boethius^ and Bede, but no one sets either 

 of these up as mathematicians : they aie recoj^nised as mere com- 

 pilers where mathematics is concerned, Accoi^ding to Chasles 

 this erroneous formula that is given by Brahmagupta (7th cen- 

 tury), Boethius (5th century), and Bede (7th century) must have 



a common origin. 



The next proposition by which Brahmagupta gains credit is 



i It must be remembered that the writer of this interesting book is not 

 to blame for the incorrect statements regarding Hindu mathematics. For 

 similar errors, see Cantor, Gow, etc., etc. 



^ Banerji's edition, p. 95. 



3 And also by Ahmes the Egyptian B.C. 1700, 



