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E. G. Ir it were propofcd to find the logarithm of 31, the 

 miihiple by means of which it would be moft conveniently found 

 by the application of produtls of two dimenfions, is according to 

 Brijgs, 341. Then, 17 x 20 = 340 = x— i, iiX3i = 34i=x' 

 and 18 X 19=342 = ^+ I. Whence the logarithms of 340 and 

 342 being known from the logarithms of their prime fadors, that 

 of 341 may be found by Dr. Hallcy's method — from which the 

 logarithm of 11 being fubtradcd we get the logarithm of 31. 

 Here ^ = 2 X 340 x 342 + i = 232561. So that ufing only the 

 firft term of the feries we c;et the logarithm true to the 17th place. 

 If we ufe the produds of three dimenfion?:, the advantage will be 

 confiderably greater. 



The logarithm of 31 maybe found from that of 899, which 



laft may be found by this method without knowing the logarithm 



of any prime number greater than 31. For 21' X 7 = 896 = 



X — 2 ; 3 X 13 X 23 = 897 = X — I ; 29 X 31 =^899 =x -\- I and 



'2x3x51* = 900 = x+ 2. Here we have, j' = -22'.^ — z. + i = 



-262074049. The firft term of the feries gives the logarithm of 



the ratio true to the 27th place — without ufing even a fingle term 



of the feries, we find the logarithm true to the ninth decimal 



place. 



If 



