I 2 3 Given Two fides AC gle A oppofite to one of them Two fides AC, BC and an angle A oppofite to one of them! Two fides AC, Two fides AC, AB and the in-1 4cluded angle A Two fides AC, AB and the in5cluded angle Af Sought The angle B The included The other The other Either of the 'Solution. As fine BC fine A: : fine AC : fine B (by Cor. 1. to Theor. 1.) Notes This cafe is ambiguous when BC is lefs than AC;, fince it cannot be determined from the data whether B be acute or obtufe. Upon AB produced (if need be) let As rad. co-fine A tang. AC : fine AD :: tang. A Cafe. | Sought | The other Two angles A, Either of the ACB and the other fides, 7 fide AC be- fuppofe BC twixt them. Two angles A, Jone of them, Solution. As rad. co-fine AC ::tang A: co-tang. ACD (by Theor. 5.) whence BCD is alfo known; then (by Cor. to Theor. 3.) as fine ACD: fiue BCD co-fine A: co-fine B. As rad. : co-fine AC : : tang. A : co-tang. ACD (by Theor. 5.) whence BCD is alfo known; then as co-fine of BCD: co-fine ACD:: tan. AC : tang. BC (by Cor. 2. to Theor. 1.) BCIAS fine B: fine AC :: fine A: fine BC (by Cor, r. to Theor. 1.) The fide other Two angles A, The fide AB As rad.: co-fine A: tang. AC Note, In letting fall your perpendicular, let it always be from the end of a given fide and oppofite to a given angle. Of the nature and construction of Logarithme 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 fet of artificial numbers, fo proportioned among themfelves 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: 1 fhall here, for the fake of the young beginner (for whom this finall tract is chiefly intended) add a few pages upon this fübject. 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 fpe cies. Let, therefore, 1, a, a2, a3, a^, a3, a3, a", &c. be a geometrical progreffion whole first term is unity, and common ratio any given quantity a. Then it is manifest, 1. That, the fum 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 a'xa', or a'; and 3+ 4 (=7) is the index of a' X aa, or a'. 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 progreon is equal to the index of the quotient of one of them divided by the other. Thus 5-3 is the index of or a. Which is only a3 the converfe of the preceding article, 3. That, A 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 2X3 (6) is the index of a raised to the 3d power (or a). This is proved in p. 38, and alfo follows from article I. 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 (@1) the cube root of a. Which is only the converfe of the laft article. These 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, a3, a3, a', 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 them Jerves are equal. Thus, if "2, and a" = 3, then will m and n be logarithms of the numbers 2 and 3 respectively. Hence it is evident, that what has been above spe cified, 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 a 2, and a3, then will the logarithm of 6, (the product of 2 and 3) be equal to m+n (agreeable to article 1); because 2×3 (6)=a"Xa"a" D 4 m+n. But • 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 respect 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 propofed number, multiplied by the excess of the common ratio above unity. Thus, if e be an indefinitè fmall quantity, the hyperbolic logarithm of the natural number agreeing with any term +e" of the logarithmic progreffion 1, +e, I + el2, i + el', i + eft, &c. will be expreffed by ne. PROPOSITION I. The hyperbolic logarithm (L) of a number being given to find the number itself, anfwering thereto. n " Let be that term of the logarithmic progreffion 1, 1-+el', 'I+e", 1+e', 1+e", &c. which is equal to the required number (N). Then, becaule el is, univerfally, n ne + n. e' &c. we hall, also, 3 &c, N. But, because n (from the nature of logarithras) is here fuppofed indefinitely great, it is evident, fift, that the numbers connected to it by the fign, may be rejected, as far as any affigned number |