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“Cor. From this it appears, that two unequal solid angles may be contained by the same number of equal plane angles.".
For the solid angle at B, which is contained by the four plane angles EBA, EBC, GBA, GBC is not equal to the solid angle at the same point B, which is contained by the four plane angles FBA, FBC, GBA, GBC; for this last contains the other: And each of them is contained by four plane angles, which are equal to one another, each to each, or are the self-same, as has been proved: And indeed, there may be innumerable solid angles all unequal to one another, which are each of them contained by plane angles that are equal to one another, each to each: It is likewise manifest, that the before-mentioned solids are not similar, since their splid angles are not all equal."
Number to express the relations of the sides and angles of triangles. It depends, therefore, on the 17th of the first of Euclid, and ou the 7th of the first of the Supplement, the two propositions which do most immediately connect together the sciences of Arithmetic and Geometry.
The sine of an angle is defined above in the usual way, viz. the perpendicular drawo from one extremity of the arch, which measures the angle on the radius passing through the other; but in strictness the sine is not the perpendicular itself, but the ratio of that perpendicular to the radius, for it is this ratio which remains constant, while the angle continues the same, though the radius vary. It might be convenient, therefore, to define the sine to be the quotient which arises from dividing the perpendicular just described by the radius of the circle.
So also, if one of the sides of a right angled triangle about the right Angle be divided by the other, the quotient is the tangent of the angle opposite to the first-mentioned side, &c. But though this is certainly the rigorous way of conceiving the sines, tangents, &e. of angles, which are in reality not magnitudes, but the ratios of magnitudes; yet as this idea is a little more abstract than the common one, and would also in rolve some change in the lavguage of trigonometry, at the same time that it would in the end lead to nothing that is not attain, ed by making the radius equal to unity, 1 hayé adhered to the com, mon method, though I have thought it right to point out that which should in strictness be pursued.
A proposition is left out in the Plane Trigonometry, which the astronomers make use of in order, when two sides of a triangle, and the angle contained by them, are given,'to find the angles at the base, without making use of the sum or difference of the sides, which, in some cases, when only the Logarithms of the sides are given, cannot be conveniently found.
THEOREM. If, as the greater of any two sides of a triangle to the Jess, 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 he 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 to 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 :: tant (CDB+CBD): tan ž (CDB-CBD). (Prop. 5.); and also, BC+CA: BC-CA:: tan (CAB+CBA): tan(CAB-CBA). Therefore, since CD=CA,
} ), tan, I (CDB+CBD): tan i (CDB-CBD):: tani (CAB+CBA): tan i (CAB-CBA). But because the angles CDB+CBD=90°, tan (CDB+CBD): tan (CDB-CBD) :: R:tąn (45° -- CBD), (2 Cor. Prop. 3.); therefore, R:tan 45o-CBD); tan & (CAB+.CBA): tan $ (CAB-CBÀ); and CBD was already shown to be such an angle that BC:CA::R:tan CBD. Therefore, &c, Q. E. D.
Cor. If BC, CA, and the angle C are given to find the angles A and B; find an angle E such, that BC:CA::R: tąn E; then R: tan (45° -E):: tan (A+B): tan (A-B), 'Thus (A-B) is found, and I" (A+B) being given, A and B are each of them kuown, 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 propósi. tions 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.
PROP. V. THE angles at the base of an isosceles spherical triangle aré sym• metrical 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 'I'rigonometry, see Elementa Sphæricorum, Theor. 66. apud Wolfii Opera Math. tom. ii; Trigonometrie par Cagnoli, $ 463; Trigonometrie Spherique par Mauduit $ 165.