Of the Nature and Conftruction of Logarithms, with their Application to the Doctrine of Triangles. A S the Business of Trigonometry is wonderfully facilitated by the Application of Logarithms; which are a Sett of artificial Numbers fo proportioned among themselves and adapted to the natural Numbers 2, 3, 4, 5, &c. as to perform the fame Things by Addition and Subtraction, only, as thefe do by Multiplication and Division: I fhall here, for the Sake of the young Beginner (for whom this fmall Tract is chiefly intended) add a few Pages upon this Subject. But, first of all, it will be neceffary to premife fomething, in general, with regard to the Indices of a geometrical Progreffion, whereof Logarithms. are a particular Species. Let, therefore, 1, a, a2, a3, aa, a3, ao, a2, &c. be a geometrical Progreffion whofe firft Term is Unity, and common Ratio any given Quantity a. Then it is manifeft, 1. That, the Sum of the Indices of any two Terms of the Progreffion is equal to the Index of the Product of thofe Terms. Thus 2+3 (5) is the Index of a2 x a3, or as; and 3 + 4 (7) is = the Index of a3 x at, or a2. This is univerfally demonstrated in p. 19. of my Book of Algebra. 2. That, the Difference of the Indices of any two Terms of the Progreffion is equal to the Index of the Quotient of one of them divided by the other. Thus 53 is the Index of or a2. Which is only a3 the Converse of the preceding Article. 3. That 3. That, the Product of the Index of any Term by a given Number (n) is equal to the Index of the Power whofe Exponent is the faid Number (n). Thus 2 x 3 (6) is the Index of a raised to the 3d Power (or a). This is proved in p. 36. and alfo follows from Article 1. 4. That, the Quotient of the Index of any Term of the Progreffion by a given Number (n) is equal to the Index of the Root of that Term defined by the fame Number (n). Thus (2) is the Index of (a) the Cube Root of ao. Which is only the Converfe of the last Article. Thefe are the Properties of the Indices of a geometrical Progreffion; which being univerfally true, let the common Ratio be now fuppofed indefinitely near to that of Equality, or the Excefs of a above Unity, indefinitely little; fo that fome Term, or other, of the Progreffion 1, a, a2, a3, aa, a3, &c. may be equal to, or coincide with, each Term of the Series of natural Numbers 2, 3, 4, 5, 6, 7, &c. Then are the Indices of thofe Terms called Logarithms of the Numbers to which the Terms themfelves are equal. Thus, if am = 2, and an = 3, then will m and n be Logarithms of the Numbers 2 and 3 respectively. Hence it is evident, that what has been above Specified, in relation to the Properties of the Indices of Powers, is equally true in the Logarithms of Numbers; fince Logarithms are nothing more than the Indices of fuch Powers as agree in Value with thofe Numbers. Thus, for Inftance, if the Logarithms of 2 and 3 be denoted by m and n; that is, if am = 2, and a" 3, then will the Logarithm of 6, (the Product of 2 and 3) be equal to m + n (agreeable to Article 1); because 2x 3 (6) a" × a" = am+". D 3 Bus But we must now observe, that there are various Forms or Species 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 most 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 propofed Number, multiplied by the Excess of the common Ratio above Unity. Thus, if e be an indefinite fmall Quantity, the hyperbolic Logarithm of the natural Number agreeing with any Term 1+el" of the logarithmic Progreffion 1, 1+e, I + el2, I tel3, I + elt, &c. will be expreffed by ne. PROPOSITION I. 2 The byperbolic Logarithm (L) of a Number being given, to find the Number itself, answering thereto. Lete" be that Term of the logarithmic Progreffion 1, 1+e', I tel2, it e3, I + el", &c. which is equal to the required Number (N). Then, becaufe eis, univerfally, I + ne + n. 11 I N2 e3 &c. we fhall, also, 72 = 2 3 &c. N. But, becaufen (from the Nature of Logarithms) is here fuppofed indefinitely great, it is evident, first, that the Numbers connected to it by the Sign, may be rejected, as far as any affigned Number Number of Terms, being indefinitely small in Comparifon of n: It is alfo evident, that they may be rejected in all the reft of the Terms of the Series; because these Terms (by reafon of the indefinite Smallness of e) bear no affignable Proportion to the preceding ones. Hence we have 1 + ne + 2 nt et 2.3.4 &c.N: But ne is (=L) the hyperbolic Logarithm of 1+e" (or N) by what has been already specified: Therefore 1 + L2 L3 L+ + + 2. E. I. L4 + &c. N. 2.3.4.5 2 2.3 2.3.4 PROP. II. To determine the hyperbolic Logarithm (L) of any given It Number (N). appears from the preceding Prop. that I + L L2 L3 + -+ &c. isN; Therefore, if x + 1 be 2 2.3 L2 L3 L4 put N, we shall have L+ + + &c.x; and, confequently, by reverting the Se garithms) we shall have 1+e=N” = 1 + x I by putting + x = N, and m = Therefore, being rejected in the Factors m-1, m—2, m—3 &c. as indefinitely small in Comparison of 1, 2, 3 &c. the Equation will become e = mx — mx2 &c. whence 2 + But this Series, tho' indeed the most easy and natural, is of little Use in determining the Logarithms of large Numbers; fince, in all fuch Cafes, it diverges, instead 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 Series that will always converge. First, then, let the Number whofe Logarithm you would find be denoted by I where it is manifeft (however great that Number may be) x will be always less than Unity: Moreover, let 1+el" (as before) be the Term of the logarithmic Progreffion agreeing with the propofed Number, or, which is the fame, let 1+el" I Then (by taking the Root on both Sides) -x |