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If, as the greater of any two sides of a triangle to the less, so the radius to the tangent of a certain angle; then will the radius be to the tangent of the difference between that angle and half a right angle, as the tangent of half the sum of the angles, at the base of the triangle to the tangent of half their difference.
Let ABC be a triangle, the sides of which are BC and CA, and the base AB, and let BC be greater than CA. Let DC be drawn at right angles to BC, and equal to AC ; join BD, and because (Prop. 1.) in the right angled triangle BCD, BC: CD::R tan CBD, CBD is the angle of which the tangent is Dto the radius as CD to BC, that is, as CA to BC, or as the least of the two sides of the triangle to the greatest.
But BC+CD: BC-CD :: tan tan (CDB-CBD). (Prop. 5.); and also, BC+CA: BC-CA:: tan
tan (CAB-CBA). Therefore, since CD=CA,
tan (CDB+CBD): tan (CDB-CBD): :
tan (CAB+CBA): tan (CAB-BBA). But because the angles CDB+CBD=90°, tan (CDB+CBD):
tan (CDB-CBD): : R: tan (45°-CBD), (2 Cor. Prop. 3.);
tan (CAB-CBA); and CBD was already shewn to be such an angle
Cor. If BC, CA, and the angle are given to find the angles A and B; find an angle E such, that BC: CA:: R: tan E; then R: tan (45°-E): tan (A+B): tan (A-B). Thus (A-B) is found, and (A+B) being given, A and B are each of them known, Lem. 2.
In reading the elements of Plane Trigonometry, it may be of use to observe, that the first five propositions contain all the rules absolutely necessary for solving the different cases of plane triangles. The learner, when he studies Trigonometry for the first time, may satisfy himself with these propositions, but should by no means neglect the others in a subsequent perusal.
PROP. VII. and VIII.
I have changed the demonstration which I gave of these propositions in the first edition, for two others considerably simpler and more concise, given me by Mr. JARDINE, teacher of the Mathematics in Edinburgh, formerly one of my pupils, to whose ingenuity and skill I am very glad to bear this public testimony.
HE angles at the base of an isosceles spherical triangle are symmetrical magnitudes, not admitting of being laid on one another, nor of coinciding, notwithstanding their equality. It might be considered as a sufficient proof that they are equal, to observe that they are each determined to be of a certain magnitude rather than any other, by conditions which are precisely the same, so that there is no reason why one of them should be greater than another. For the sake of those to whom this reasoning may not prove satisfactory, the demonstration in the text is given, which is strictly geometrical.
For the demonstrations of the two propositions that are given in the end of the Appendix to the Spherical Trigonometry, see Elementa Sphæricorum, Theor. 66. apud Wolfii Opera Math. tom. iii; Trigonometrie par Cagnoli, § 463; Trigonometrie Spherique par Mauduit $165.