But we must now observe, that there are various forms or fpecies of logarithms; because it is evident that what has been hitherto faid, in refpect to the properties of indices, holds equally true in relation to any equimultiples, or like parts, of them; which have, manifeftly, the fame properties and proportions, with regard to each other, as the indices themselves. But the moft fimple kind of all, is Neiper's, otherwife called the hyperbolical. The hyperbolical logarithm of any number is the index, of that term of the logarithmic progreffion agreeing with the proposed number, multiplied by the excess of the common ratio above unity. n Thus, if e be an indefinite fmall quantity, the hyperbolic logarithm of the natural number agreeing with any term +el of the logarithmic progreffion 1, 1+e, I + e2, I + e3, i + eft, &fe will be expreffed by ne, 2 PROPOSITION I. The hyperbolic logarithm (L) of a number being given, to find the number itself, answering thereto. 2 3 Lete" be that term of the logarithmic progreffion 1, 1+e › I + e › I + el', I + eft, &c, which is equal to the required number (N). Then, because el is, universally, = 1 + ne + n 3 .e3 &c. N. But, because n (from the nature of logarithms) is here fuppofed indefinitely great, it is evident, first, that the numbers connected to it by the fign, may be rejected, as far as any affigned number number of terms being indefinitely fmall in comparifon of n it is alfo evident, that they may be rejected in all the rest of the terms of the feries; because these terms (by reafon of the indefinite fmallness of e) bear no affignable proportion to the preceding ones. Hence we have i + n4e4 &c. N: but ne is n2e2 ne + + + 2 2.3 2.3.4 (L) the hyperbolic logarithm of 1+el" (or N) by what has been already specified: therefore 1 + To determine the hyperbolic logarithm (L) of any given + number (N). It appears from the preceding Prop. that 1+L L2 2 L3 + &c. is N: therefore, if x + 1 be 2.3 = L2 L3 L+ put = N, we shall have L+ + + 2 2.3 2.3.4 &c. x; and confequently, by reverting the fe Because el I m garithms) we shall have e=N"=1+x"; I by putting + x = N, and m = -. m ท +* being = = 1 + mx + m. n 2 x &c. we have emx+m. 2 2 3 m being rejected in the factors m-1, M2, m—3 &c. as indefinitely fmall in comparison of 1, 2, 3 &c. the equation will become e = mx mx2 2 + But this feries, tho' indeed the most easy and natural, is of little ufe in determining the logarithms of large numbers; fince, in all such cases, it diverges, inftead of converging. It will be proper, therefore, to give, here, the invention of other methods, which authors have had recourfe to, in order to obtain a feries that will always converge. First, then, let the number whose logarithm ; where it is you would find be denoted by manifest (however great that I I number may be) x will be always lefs than unity: moreover, let n +el (as before) be the term of the logarithmic progreffion agreeing with the propofed, number, or, which is the fame, let 1 + "= I - X then, by taking the root on both fides) we jected in the factors m - 1, m2, &c. (as before) our equation will become =ne the hyperbolic logarithm of I I Which feries, it is manifeft, will always converge, will be always lefs than unity. But it is further obfervable that this feries has exactly the fame form (except in its figns) with that above for the logarithm of 1 + x; and that, if both of them be added together, the feries 2x + 2x5 2x7 7 2x3 + + &c. thence arifing, will be more 3 5 fimple than either of them; fince one half of the terms will be intirely deftroyed thereby. Therefore, because the fum of the logarithms of any two numbers is equal to the logarithm of the product of those numbers, (fee Article 1.) it is mani2x3 2x5 feft that 2x + + &c. will truly exprefs not only because the even powers are here deftroyed, but because x, in finding the logarithm of any given number (N), will have a lefs value. 3 But But now, to determine what this value must be, make N-I I + x Ι x = N, and then x will be found N+1; but if the quantity proposed be a fraction Example. Let it be propofed to find the hyperbolic logarithm of the number 2. Which values being respectively divided by the numbers, 1, 3, 5, 7, 9, &c. and the feveral quotients added together, (fee the general feries) we fhall have ,346573590 &c. whofe double, being ,693147180 &c. is the hyperbolical logarithm of the number 2. After |