l6b QUESTIONS AND REMARKS ON 



Mr. Davis has obliged the world, point out some 

 very curious theorems, which must have been known 

 to the author of that ancient book. The rule, for 

 Instance, by which the trigonometrical canon of the 

 J] indie astronomers is constructed *, involves in it the 

 following theorem : "If there be three arches of a 

 <c circle in arithmetical progression, the sum of the 

 4C sines of the two extreme arches is to twice the sine 

 " of the middle arch, as the co-sine of the common 

 " difference of the arches to the radius of the circle. 

 Now this theorem, though not difficult to be demon- 

 strated, is yet so far from obvious, that it seems not 

 known to the mathematicians of Europe till the be- 

 ginning of the last century, when it was discovered 

 bv Viet a. It has ever since been used for the con- 

 struction of trigonometrical tables, as it affords a 

 method of calculating the sines and arches much 

 easier than that which depends on successive extrac- 

 lions of the square root. To find that this theorem 

 was known to the Brahnens many ages ago, is there- 

 fore extremely curious ; and the more so, because 

 there is some reason to think, that the commentator 

 on the Sidd/nm/a, quoted and translated by Mr. Da- 

 vis-f, did not understand the principle of this rule, 

 since the method which he lays down is entirely dif- 

 ferent, much less profound in theory, and much more 

 difficult in practice. If this be true, it indicates a re- 

 trograde order in the progrees of eastern science, which 

 must have had its origin in a very remote age. 



* 2 Asiatic Researches, 245. 



f P. 246, 2A-* 



II. Are 



