VOL. XXX.] PHILOSOPHICAL TRANSACTIONS. 305 



table, against tliern in the second column I set a and b for their logarithms, ex- 

 pressing by an equation the manner how they are compounded of the logarithms 

 of 1 and 10, for which I write /2 and /lO. Then multiplying the two num- 

 bers in the first column together, I have a third number 1.024, against which I 

 write c for its logarithm, expressing likewise by an equation in what manner c 

 is formed of the foregoing logarithms a and b. And in the same manner the 

 calculation is continued ; only observing this compendium, that before multi- 

 plying the last two numbers already got in the table, I consider what power of 

 one of them must be used to bring the product the nearest to unit that can be. 

 This is found, after we have gone a little way in the table, only by dividing the 

 differences of the numbers from unit one by the other, and taking the quotient 

 with the nearest, for the index of the power wanted. Thus the last two num-' 

 bers in the table being 0.8 and 1 .024, their differences from unit are 0.200 and 

 0.024 ; therefore r— ^ gives 9 for the index ; therefore multiplying the gth 

 power of 1.024 by 0.8, I have the next number 0-990352031429, whose 

 logarithm is d = 9c + b* In seeking the index in this manner by division of 

 the differences, the quotient ought generally to be taken with the least: but in 

 the present case it happens to be the most, because instead of the difference 

 between 0.8 and 1, we ought strictly to have taken the difference between the 

 reciprocal 1.25 and 1, which would have given the index 10; and that would 

 be too great, because the product by that means would have been larger than 1, 

 as 1 .024 is. Whereas this approximation requires that the numbers in the first 

 column be alternately greater and less than 1, as may be seen in the table. 



When I have in this manner continued the calculation, till I have got the 

 numbers small enough, I suppose the last logarithm to be equal to nothing. 

 Which gives me an equation, from which having got away the letters by means 

 of the foregoing equations, I have the relation of the logarithms proposed. In 

 this manner if I suppose g = O, I have 2136/2 — 643 / 10 = O : which gives 

 the logarithm of 2 true in 7 figures, and too great in the 8th ; which happens 

 because the number corresponding with g is greater than unit. 



There is another expedient which renders this calculation still shorter. It is 

 founded on this consideration, that when x is very small, 1 -f- x^ is very nearly 

 1 + ^^' Hence if 1 + ^j ^"d 1 — 2 be the last two numbers already gotten 

 in the first column of the t able, an d the ir pow ers 1 + a^p and 1 — z|" be such 

 as will make the produc t 1 + ^j*" X 1 + J|" very near to unit, m and n may 

 be fou nd t hus : 1 -f •^i"' = f + ^^'^s and 1 — zj" = 1 — wz, consequently 

 1 + '^r X 1 — zj" = 1 -f- mx — nz — mnzx^ or (neglecting mnzx) 1 -j- mx 



VOL. VI. Rr 



