11 PHILOSOPHICAL TRANSACTIONS. [anNO ISQS. 



After the same manner may the difference of the said two logarithms be very 

 fitly applied to find the logarithms of prime numbers, having the logarithms 

 of the next two numbers above and below them : for the difference of the ratio 

 of c to -L z and of -i- z to h, is the ratio oi ab \.o \ zz, and the half of that ratio 

 is that of V ah io \ z, or of the geometrical mean to the arithmetical. And 

 consequently the logarithm thereof will be the half difference of the logarithms 

 of those rationes, viz. 



— mto - — + — - + ^-7 + -r-ft occ. 



Which is a theorem of good dispatch to find the logarithm of 4- z. But the 

 same is yet much more advantageously performed by a rule derived from the 

 foregoing, and beyond which in my opinion nothing better can be hoped. For 

 the ratio o{abio^zzor\aa-\-^ab-\-^bb, has the difference of its terms 

 ■^a a — \ a b -\- \ b b, or the square of-^a — :l. b z= \ x x, which in the pre- 

 sent case of finding the logarithms of prime numbers is always unity, and calling 

 the sum of the terms ^zz-\-ab=iyy, the logarithm of the ratio of \/ a b 

 to 4- a + 4- ^ or -'.- z will be found to be 



— into — + r— ^ + + — + 5 &C. 



which converges much faster than any theorem hitherto published for this 

 purpose. ^ , -. ,; ■ , , ^. . ^ 



Here note that — is all along applied to adapt these rules to all sorts of 



logarithms. If m be 10000 Sec. it may be neglected, and you will have Napier's 

 logarithms, as was hinted before ; but if you desire Briggs's logarithms, which 

 are now generally received, yon inust divide the series by 



2,30258509299404568/i017991454684364207601 101488628772976033328, 



or multiply it by the reciprocal thereof, viz. 

 0,434294481903251827651128918916605082294397005803666566114454. 



But to save so operose a multiplication, which is more than all the rest of the 

 work, it is expedient to divide this multiplicator by the powers of z or _y con- 

 tinually, according to the direction of the theorem, especially where x is small 

 and integral, reserving the proper quotes to be added together, when you have 

 produced the logarithm to as many figures as you desire, of which method I 

 will give a specimen. 



If the curiosity of any gentleman, who has leisure, would prompt him to 

 undertake to do the logarithms of all prime numbers under 1000(H), to 25 or 

 30 figures, I dare assure him that the facility of this method will invite him 



