100 OBSERVATIONS on the 



therefore, founded his calculations 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 

 flill diftinguifhed by his name. This propofition comprehends 

 in fa(5l Euclid's, and of courfe the Hindoo theorem, as a par- 

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

 tifeful to Ptolemy, of all others, it appears to have efcaped his 

 obfervation ; on which account he did not perceive that every 

 number in his tables might be calculated from the two preceding 

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

 fame; and therefore his method of conllrudling them is infinitely 

 more operofe and complicated than it needed to have been. 



Not only did this efcape Ptolemy, but it remained 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 acftually 

 enriched that fcience by a great number of valuable difcoveries. 

 They continued to conflru(5l their tables by the fame methods 

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

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

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

 however ignorant by what train of reafoning that excellent geo- 

 meter difcovered it ; for though it is publilhed in his T?'eatife 

 on Angular Se6iions, 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 quickly gave to the conflru(5lion of the trigonome- 

 trical canon ail the fimplicity which it feems capable of attaining. 

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



had 



