"TRACT 21.

LOGARITHMS.

431

of his age. His method was first printed in the Philosophical Transactions for the year 1695, and it is entitled “ A mostcompendious and facile method for constructing the loga-rithms, exemplified and demonstrated from the nature ofnumbers, without any regard to the hyperbola, with a speedymethod tor finding the number from the given logarithm.”

Instead of the more ordinary definition ol logarithms, asmimerorum proportwnalium eequidifferentes comites, in thistract our learned author adopts this other, numeri ralioncmexponentes , as being better adapted to the principle on whichlogarithms are here constructed, where those quantities arenot considered as the logarithms of the numbers, for example,of 2, or of 3, or of 10, but as the logarithms of the ratios of1 to 2, or 1 to 3, or 1 to 10, In this consideration he firstpursues the idea of Kepler and Mercator , remarking that anysuch ratio is proportional to, and is measured by, the numberof equal ratiunculse contained in each ; which ratiunculse areto be understood as in a continued scale of proportionals, in-finite in number, between the two terms of the ratio ; whichinfinite number of mean proportionals, is to that infinitenumber of the like and equal ratiunculse between any othertwo terms, as the logarithm of the one ratio, is to the loga-rithm of the other : thus, if there be supposed between 1 and10 an infinite scale of mean proportionals, whose number is100000 &c in infinitum ; then between 1 and 2 there wili be30102 &c of such proportionals ; and between 1 and 3 therewill be 47712 &c of them ; which numbers therefore are thelogarithms of the ratios of 1 to 10, 1 to 2, and 1 to 3. Butfor the sake of Aw mode of constructing logarithms, he changesthis idea of equal ratiunculse, for that of other ratiunculse, soconstituted, as that the same infinite number of them shall becontained in the ratio of 1 to every other number whatever;and that therefore these latter ratiunculse will be of unequalor different magnitudes in all the different ratios, and, in suchsort, that in any one ratio, the magnitude of each of the ra-tiunculas in this latter case, will be as the number of them inthe former. And therefore, if between 1 and any number