for computing Eclipses, Tables of Sines, &'c. ] 9 



admit : but I cannot forbear saying a few words on the In- 

 dian tables of sines, and of the two methods by which they 

 have been con)piUed ; for, since they have been printed, I 

 have found that in the note put at the bottom of this table 

 I have not sufficiently appreciated the excellence of the me- 

 thod, from having been drawn into an error by a constant 

 number, which does not appear to have been given accu- 

 rately enough in the memoir. 



In this table the sines are expressed in minutes; they pro- 

 ceed by intervals of 3|- degrees^ and suppose the radius 343S, 

 or rather 343775. On the side of the right sines it gives the 

 versed sines. 



If we examine the process recommended by the Indian 

 author, we readily perceive thai his method is reduced to the 

 computing beforehand a first difference, which is at the same 

 time the first sine of the table j after which, to obtain the 

 second sine, he calculates the second difference, which he 

 subtracts from the first difference ; this gives him the first 

 difference between the first and second sine, and then the 

 second sine : after this he computes another second differ- 

 ence, to obtain a new first difference and a new sine, and so 

 on to the end of the table. This process is exactly that 

 which I have pointed out in the preface to Borda's Decimal 

 Tables, without knowing that the method which appeared 

 unknown even to the moderns, had been practised so long 

 ago in India. 



My*formula is A'c>>^,A = — .ls%iAA5,A = (— chord)^ 

 A A $, A. (See page 48 of the Decimal Tables.) 



A A being a constant quantity in a table of sines, it fol- 

 lows, that to obtain the second difference of any sine what- 

 ever, it is necessary to multiply this sine by a constant num- 

 ber: now A A in the Indian tables is 3" 45', therefore 4 s'-, 



i- A A = 4i^ l** 52' 30" = 0-004 12821 = ^ ; whence 



2J.3-53 

 it follows, that the constant factor to find the second diflc- 



rencc is • that is to say, the last found sine must be 



^33' Si 



• Delambrc usci A to denote Jiinl differ nice, and A?) to (Jciiotc second dif- 

 ftrcrue. 



B 2 divided 



