loo OBSERVATIONS on the 



therefore, founded his calcvilations on another propofition, con- 

 taining a property of quadrilateral figures infcribed in a circle, 

 which he feems to have inveftigated on purpofe, and which is 

 ftill diflinguiflicd by his name. This propofition comprehends 

 in fadl Euclid's, and of coiirfe the Hindoo theorem, as a par- 

 ticular cafe ; and though this cafe would have been the mofl 

 ufeful to Ptolemy, of all others, it appears to have efcaped liis 

 obfervation ; on which account he did not perceive tliat every 

 number in liis tables might be calculated from the two preceding 

 numbers, by an operation extremely fimple, and every where the 

 fame; and therefore his method of con(lru6ting them is infinitely 

 more operofe and complicated than it needed to have been. 



Not only did this efcape Ptolemy, but it i-emained un- 

 noticed by the mathematicians, both Europeans and Arabians, 

 who came after him, though they applied the force of their 

 minds to nothing more than to trigonometry, and acfhually 

 enriched that fcience by a great number of valuable difcoveries. 

 They continued to conftrudl their tables by the fame methods 

 which Ptolemy had employed, till about the end of the fix- 

 teenth century, when the theorem in queftion, or that on which 

 the Hindoo rule is founded, was difcovered by Vieta. We are 

 however ignorant by what train of reafoning that excellent geo- 

 meter difcovered it ; for though it is publifhed in his Treatift 

 on A7igular SeB'tons^ it appears there not with his own demon- 

 ilration, but with one given by an ingenious mathematician, 

 of our own country, Alexander Anderson of Aberdeen. 

 It was then regarded as a theorem entirely new, and I know not 

 that any of the geometers of that age remarked its affinity to 

 the propofitions of Eiclid and Piolemy. It was foon after 

 applied in Europe, as it had been fo many ages before in Hin- 

 doftan, and cmickly gave to the conftrudHon of the trigonome- 

 trical canon all the fimphcity which it feems capable of attaining. 

 From all this, I think it might fairly be concluded, even if we 



bad 



