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One leg The oppo- As radius : fine A :: 008
AB and the site angle fine of AB:co-fine of C (by
One leg The hyp. As co-fine of A: radius ::
tang. AB : tang. AC (by 9 adjacent
dius : sine AB (by Theor. 4.) IO
opposite angle A One leg | The adja- As co-line BC : radius : : BCand the cent angle co-fine of A : fine C (by opposite
с angle A
As fin. A: fin. BC:: radius BC and the AC : fine AC (by Theor. 1.) 12
opposite angle A Both legs
As radius : co-fine AB :: 13 AB and BC AC co-fine BC : co-fine AC
(by Theor. 2. Both legs An angle, As fine AB: radius :: tang. 14 AB and BC suppole A BC : tang. A (by Theor. 4.)
As fin. A:co - fine C :: ra• 15 A and C suppole dius: co-fine AB (by Theor.
3.) Both angles The hyp. As tang. A:co-tang. C:: 161 A and C AC". radius : co-fine AC (by
Note, The 10th, uth and 12th cases are ambiguous ; since it cannot be determined, by the data, whether AB, C, and AC; be greater or less than go degrees each.
The solution of the cases of oblique spherical tri
Two sides AC, Either of the As rad. : co-fine A :: tang. AC :
AB and the in- other angles, tang. AD (by Tbeor. 1.) whence BD 5cluded angle A suppose B is known; then (by Cor. to Tbeor.4.)
as fine BD : : fine AD :: tang. A:
Note, In letting fall your perpendicular, let it always be from the end of a given fide and oppofte to a given angle,
Of the nature and construction of logarithms,
with their application to the doctrine of triangles.
S the business of trigonometry is wonder
fully facilitated by the application of logarithms; which are a fett of artificial numbers fo 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 divifion : 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 spe. cies.
Let, therefore, i, a, a, a, a, a, a, a, &c. be a geometrical progression whose first 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 progression is equal to the index of the produa of those terms. Thus 2 +3 (5) is = the index of a’ xa', or as ; and 3 + 41=7) is = the index of a} * a*, 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 progreffion is equal to the index of the quotient of one of them divided by the other. Thus
as 5-3 is = the index of or a”. Which is only the converse of the preceding article.
3. That, the produet of the index of any term by a given number (n) is equal to the index of the power wbose exponent is the faid number (n). Thus 2 * 3 (6) is = the index of a' raised to the 3d power (or ao). This is proved in p. 38, and also follows from article 1.
4. That, the quotient of the index of any term of the progression by a given nnmber (n) is equal to the index of the root of that term defined by the same number (n). Thus § (2) is = the index of (aa) the cube root of a'. Which is only the converse of the last article.
These are the properties of the indices of a geo: metrical progression, which being universally true, let the common ratio be now supposed indefinitely near to that of equality, or the excess of a above unity, indefinitely little ; so that some term, or other, of the progression 1, a, a, a', at, as, &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 those terms called logarithms of the numbers to which the terms thenselves 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 fpecified, in relation to the properties of the indices of powers, is equally true in the logarithms of numbers ; since logarithms are nothing 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 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 2x3 (6)=am xa" = am+n.