23 , On (lie Hindoo Formulce 



proportional to the sine : thus the second difference opposite 

 30'' is 7-36, and that opposite 90" is 14-72, double of the 

 former : and to find the ratio of the second difference to the 

 sine they divided the radius 3'J37'75 by 14'72, and they 

 found 233*53 ft)r iheir quotient : dividing thus each sine by 

 its stcond difference they constantly found this same quo- 

 tient, whence ihev concluded that to obtain this second dif- 

 ference it was only necessary to divide the sine l)y 233*53. 



The rule for the first differences is not so simple ; for the 

 difference of ,9,A — 2s,\ Ac,{\+ ^ £i.K), and the 5,(A + 

 i A A), are not in the table. 



But the first of the first differences is at the same time the 

 first sine of the table ; whence they concluded, that with the 

 first sine, and the first of the first and second differences, 

 they were fully prepared for conqnuing all the rest : but in 

 the work the table was already computed throughout, when 

 it gave them their differential method; and the proof of this 

 is, that to make their table as they have given it they had 

 occT-ion to make the first sine 224*85 and not 225, which 

 would have given the first differences a little too great, and 

 the sines too small. 



It is true the Sonria SlJuGnta recommends to divide the 



number of minutes in a sign by 8 to obtain the first sine, 



which is the same as raakitig the sine equal to the arcj thus' 



30° 36c° 2 i6oo' , „ , . , ^ . . . 



-V- = --.^ = 7^ = 225' = 3° 43'; instead of which 



8 96 96 



the true value found by the above theorems is only 224-85. 



Let it be observed that there is nothing conjectural but the 

 reasoning which I have giyen them, for they really possessed 

 all the knowledge which I suppose them to have had. I do 

 not pretend, however, that they used decimal fractions ; it is 

 only to shorten the work that I have used them in recon- 

 structing their table of sines, for it is well known all their 

 calculations were made in sexagesimals. 



By taking proportional parts, the use of which was well 

 known to them, they might have extended their table to 

 every degree ; but those interpolated degrees would have had 

 their s'tgns less accurate, and tiiey have preferred giving those 

 \yhich result immediately from their formula;, to preserve in 



dl 



