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GENERAL PROPOSITION. In an oblique angled triangle, of the three sides and three angles, any three being given, the other three may be found, except when the three angles are given; in which case the ratios of the sides are only giver, being the same with the ratios of the sines of the angles opposite to them.
Given. Sought. 1 A,B, and there- BC, AC. S, C:SA :: AB : BC Fig. 16. 17. fore C, and the
and also S, C: S, B :: AB side AB..
1: AC. (2.)
2 AB, AC, and The angles AC: AB :: S, B: S, C. B, two sides and A and C. K2.) This case admits of two an angle oppo
solutions; for C may be site to one of
greater or less than a quathem.;
drant. (Cor, to def. 4.)
8 AB, AC, and The angles AB+AO: AB-AC::T. A, two sides, B and C.
C+B C-B and the in
(3). cluded angle.
the sum and difference of
BA: AC::R:T, ABD,
therefore B and C are given
2 ACX CB : ACq+CB, - AB :: R: Cos, c. If ABq+CBq be greater than ABq. Fig. 16.
2 AC xCB : ABq-AC9
-CBq :: R: Cos, c. 11 AB, BC, CA, A,B,C, the ABq be greater than ACq+ the three sides. three angles. CBq. Fic. 17. (4.)
2. If ABq be greater than ACq+CBq. Fig. 17. BC : BA+AC:: BA-AC:BD +DC; and BC the difference of BD, DC, is given, therefore each of them is given. (7.)
And CA : CD::R: Cd s, C. (1.) and C being found, A and B are found by case 2 or 3.
A Trigonometrical Canon is a Table, which, beginning
from one second or one minute, orderly expresses the lengths that every sine, tangent, and secant have, in respect of the radius, which is supposed unity; and is conceived to be divided into 10000000 or more decimal parts. And so the sine, tangent, or secant of an arc, may be had by help of this table; and, contrariwise, a sine, tangent, or secant being given, we may find the arc it expresses. Take notice, that in the following tract, R signifies the radius, S a sine, Cos, a cosine, T a tangent, and Cot. a cotangent; also ACq signifies the square of the right line AC; and the marks or characters, +, ,, :, ::, and V, are severally used to signify addition, subtraction, equality, proportionality, and the extraction of the square root. Again, when a line is drawn over the sum or difference of two quantities, then that sum or difference is to be considered as one quantity,
Constructions of the Trigonometrical Canon.
PROP. I. THEOR. The two sides of any right angled triangle being given, the other side is also given.
For (by 47. 1.) ACq=ABq+BCg, and ACq-BCq= Fig. 28. ABq, and interchangeably ACq-ABq=BCq. Whence, by the extraction of the square root, there is given AC= VABq + BCq; and AB = V AC - BCq; and BC =
PROP. II. PROB. The sine DE of the arc BD, and the radius CD, Fig. 29. being given, to find the cosine DF.