Case. 1 The solution of the cases of oblique spherical triangles. Two sides AC, The included BC, and an an gle A opposite 2 to one of them Two sides AC, BC, and an an. gle opposite to one of them Two sides AC, AB, and the in. 4 cluded angle A Solution. As sine BC: sine A: sine AC sine B (by Cor. 1. to Theor. 1.) Note. This case is ambiguous when BC is less than AC; since it cannot be determined from the data whether B be acute or obtuse. Upon AB produced (if need be) let fall angle ACB the perpendicular CD; then (by Theor. 5.) rad. co-sine AC tang. A cotang. ACD; but (by Cor. 2. to Theor. 1.) as tang BC: tang. AC:: co-sine ACD : co sine BCD. Whence ACB = ACDBCD is known. · The other The other As rad. : co-sine A:: tang. AC: tang. AD (by Theor. 1.); and (by Cor. to Theor. 2.) as co-sine AC co-sine BC: cosine AD: co-sine BD. Note. This and the last case are both ambiguous when the first is so. As rad. : co-sine A:: tang AC: tang. AD (by Theor. 1.); whence BD is also known; then (by Cor. to Theor. 2.) as co-sine AD: co-sine BD :: co-sine AC : co-sine BC. Two sides AC, Either of the As rad. : co-sine A:: tang. AC: tang. AB,and the in- other angles, AD (by Theor. 1.); whence BD is 5 cluded angle A suppose B known; then (by Cor. to Theor. 4) as sine BD: sine AD: : tang. A tang. B. Case. Given Sought Solution. 8 9 Two angles A, The side BC As sine B: sine AC B, and a side opposite the BC (by Cor. 1. to Theor. 1.) one of them other Two angles A, The side ABAs rad, co-sine A sine A: sine :: tang. AC tang.! B, and a side betwixt them AD (by Theor. 1.); and as tang. B AC opposite to tang. A: : sine AD sine BD (by Cor. one of them to Theor. 4.); whence AB is also known. As rad. : co-sine AC tang. A: cotang. ACD (by Theor. 5.); and as cosine A co-sine B: sine ACD: sine BCD (by Cor. to Theor. 3.); whence ACB is also known. The other angle ACB : Two angles A, B, and a side 10 AC opposite to one of them Note. In letting fall your perpendicular, let it always be from the end of a given side and opposite to a given angle. Of the Nature and Construction of Logarithms, with their application to the doctrine of Triangles. AS the business of trigonometry is wonderfully facilitated by the application of logarithms; which are a set of artificial numbers, so proportioned among themselves and adapted to the natural numbers 2, 3, 4, 5, &c. as to perform the same things by addition and subtraction, only, as these do by multiplication and division: I shall here, for the sake of the young beginner (for whom this small tract is chiefly intended), add a few pages upon this subject. But, first of all, it will be necessary to premise something, in general, with regard to the indices of a geometrical progression, whereof logarithms are a particular species. Let, therefore, 1, a, a2, a3, aa, a3, ao, a7, &c, be a geometrical progression, whose first term is unity, and common ratio any given quantity a. Then it is manifest, 1. That the sum of the indices of any two terms of the progression is equal to the index of the product of those terms. Thus 2+3 (5) is = the index of a2 × a3, or a3; and 3 + 4 (7) is the index of a3 x a1, or a. This is universally demonstrated in p. 19 of my book of Algebra. = 2. That the difference of the indices of any two terms of the progression is equal to the index of the quotient of one of them divided by the other. Thus 5 3 is the index of or a2. ticle. Which is only the converse of the preceding ar 3. That the product of the index of any term by a given number (n) is equal to the index of the power whose exponent is the said number (n). Thus 2 x 3 (6) is = the index of a2 raised to the third power (or a"). This is proved in p. 43, and also follows from article 1. 4. That the quotient of the index of any term of the progression by a given number (n) is equal to the index of the root of that term defined by the same number (n). Thus (2) is the index of (a) the cube root of ao. Which is only the converse of the last article. These are the properties of the indices of a geometrical progression; which being universally true, let the common ratio be now supposed indefinitely near to that of equality, or the excess of a aboye unity, indefinitely little; so that 'some term or other of the progression 1, a, a2, a3, aa, a3, &e. may be equal to, or eoincide with each term of the series of natural numbers, 2, 3, 4, 5, 6, 7, &c. Then are the indices of those terms called logarithms of the numbers to which the terms themselves are equal. Thus, if am = 2, and 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; since logarithms are Hothing more than the indices of such powers as agree in value with those numbers. Thus, for instance, if the logarithms of 2 and 3 be denoted by m and n; that is, if um = 2,. and an 3, then will the logarithm of 6 (the product of 2 and 3) be equal to m+n (agreable to article 1.); because m+". = 2 × 3(6)= am × a" = a′′ But we must now observe, that there are various forms or species of logarithms; because it is evident that what has been hitherto said, in respect to the properties of indices, holds equally true in relation to any equimultiples, or like parts, of them; which have, manifestly, the same properties and proportions, with regard to each other, as the indices themselves. But the most simple kind of all is Napier's, otherwise called the hyperbolical. The hyperbolical logarithm of any number is the index of that term of the logarithmic progression agreeing with the proposed number, multiplied by the excess of the common ratio above unity. Thus, if e be an indefinitely small quantity, the hyperbolic logarithm of the natural number agreeing with any term 1+e]" of the logarithmic progression 1, 1+e, 1+e 3 1+e]3, 1+e]*, &c. will be expressed by ne. PROPOSITION I. The hyperbolic logarithm (L) of a number being given, to find the number itself, answering thereto. |