20 PHILOSOPHICAL TRANSACTIONS. [aNNO I695. 



ber : and seeing that in these continual proportionals, all the ratiunculae are 



equal, their sum, or the whole ratio, will be as the said momentum is directly ; 



that is, the logarithm of each ratio will be as the fluxion thereof. Wherefore, 



if the root of any infinite power be extracted out of any number, the difFe- 



rentiola of the said root from unity, will be as the logarithm of that number. 



So that logarithms thus produced may be of as many forms as you please to 



assume infinite indices of the power whose root you seek : as, if the index be 



supposed 100000 &c. infinitely, the roots will be the logarithms invented by the 



Lord Napier ; but if the said index were 2302585 &c. Mr. Briggs's logarithms 



would immediately be produced. And if you please to stop at any number of 



figures, and not to continue them on, it will suffice to assume an index of a 



figure or two more than your intended logarithm is to have, as Mr. Briggs's 



did, who to have his logarithms true to 14 places, by continual extraction of 



the square root, at last came to have the root of the 140737488355328th 



power; but how operose that extraction was, will be easily judged by those who 



shall undertake to examine his calculus. 



Now, though the notion of an infinite power may seem very strange, and 



to those that know the difficulty of the extraction of the roots of high 



powers, perhaps impracticable ; yet by the help of that admirable invention of 



Mr. Newton, whereby he determines the unciae or numbers prefixed to the 



members composing powers, on which chiefly depends the doctrine of series, 



the infinity of the index contributes to render the expression much more easy : 



for, if the infinite power to be resolved be put, after Mr. Newton's method, 

 j^ 1 



P+Pq, P+pq^'" or l~+~^"*, instead of 1 + -;^ 9 + \^ q q + 



\— 3m + 2mm 3, i— 6m+ 11 711 m— 6 mi.- u-u-i.i i. i 



g— 3 q H oT^i q ) o^c- which ]S the root when m is 



finite, becomes 1 -] — 9 — r— 9'? + :; — '7^ + :; — ?* + t-<7\ &c. m m 



being infiniti infinite, and consequently whatever is divided thereby vanishing. 



Hence it follows that - multiplied into 7 — -j-^^ + i??^ — ^-9*^--3-?^ 



&c. is the augment of the first of our mean proportionals between unity and 

 1+9, and is therefore the logarithm of the ratio of 1 to 1 + 9 ; and whereas 

 the infinite index m may be taken at pleasure, the several scales of logarithms 



to such indices will be as - or reciprocally as the indices. And if the index be 



taken 10000 &c. as in the case of Napier's logarithms, they will be simply 



