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The sum of the three angles of any spherical triangle is greater than two right angles, and less than six.*
Hence a spherical triangle may have its three angles either right or obtuse ; and therefore, if two be given, the third cannot be thence determined.
If the three angles of a spherical triangle be each right or obtuse, the three sides are also each equal to, or greater than, 90°; and if each of the angles be acute, each of the sides is less than 90°; and the contrary:
A sides AB, AC, BC, of the spherical triangle ABC, be supposed indefinitely short, they will approach indefinitely sight lines, and the spherical surface indefinitely near to a plane surface ; therefore the triangle may, in such case, be considered as a plane triangle. Bat the sum of the angles of a plane triangle is only equal to two right angles ; therefore, while the sides of a spherical triangle are of a finite magnitude, the sum of its angles is greater than two right angles. Q. E. 1°D.
2. Every angle is less than two right angles; therefore the sum of the three angles of every spherical triangle is less than six right angles. Q. E. 2° D.
If from the three angles B, A and C, of a spherical triangle BAC, as poles, there be described on the surface of the sphere three arcs of great circles FD, DE and EF, which by their intersec
B tions form another spherical K
M triangle DEF; each side of
E the new triangle will be the
í supplement to the angle, which is at its pole, and cach of its angles the supplement to that side in the triangle BAC, to which it is opposite.*
Let the sides AB, AC and BC, of the triangle BAC, be produced till they meet those of the triangle DEF in the points G, L; I, N; K, M; then, since
1 the point A is the pole of the arc DILE, the distance of the points A, E, is 90° ; and, since C is the pole of the arc EF, the distance of the points C and E is also 90°; therefore the point E is the pole of the arc AC. We may prove in a similar manner, that F is the pole of BC, and D the pole of AB. Hence evidently DL=90°, and IE=90°; and therefore DL +IE=DL+EL+IL=DE+IL=180°. Consequently the arc DE is the supplement to the angle BAC, measured by the arc IL. We may prove in the same manner, that EF is the supplement to the angle BCA, measured by the arc MN, and that DF is the supplement to the angle ABC, measured by GK. Whence it follows, that each side of the triangle DEF is the supplement to the angle in the triangle BAC, which is at its pole.
Q. E. 1°D.
2. Since the arcs AL and BG are each 90°, AL+BG=GL +AB=180° ; but GL is the measure of the angle EDF, and consequently AB its supplement. We may prove in the same manner, that AC and BC are the supplements to the angles at E and F. Therefore the angles of the triangle DEF are supplemental to the sides of the triangle BAC, which are opposite to them. Q. E. 2°D.
Cor. If two right-angled spherical triangles have one common angle, the sines of their hypotenuses are as the sines of their legs, opposite to this angle.
at A. Since the planes GOP, GBP, are perpendicular to each other, if the radius be made equal to unity, the values, specified in the following table, may be easily obtained for all the parts of
sin. the triangle ; it being recollected, that tang. = and cot. =
BPXGQ_QPXBG BPXGQ 5. <BCA
BQXGP GPXBQ QPXBG Now, to shew how these sereral cxpressions are demonstrated, it is sufficient to.give merely the demonstration of the first line. For this purpose assunie any line GR, and regard it as the radius of the Tables ; let fall the perpendicular RS on GB. Then it is evident, that RS is the side of the arc BC, and GS its coside. In the similar triangles GRS, GQB, as QG : QB :: GR=1
; and QG :
GB GR=I : GS
: RS=sin. BC= OG
СВ cos, BCS
; whence is deduced tang. BC=
As radius is to the cosine of either angle, so is the tangent' of the hypotenuse to the tangent of the leg adjacent to this angle.
Cor. If two right-angled spherical triangles have one common leg, the tangents of their hypotenuses are in the inverse ratio of the cosines of the 'angles adjacent to this leg
The other expressions are demonstrated in the
The expressions for the cuiangents, being obtainable by merely inverting those for the tangents, are not inserted in the table.
The truth of the first six Theorems is proved by substituting the particular values of the terms of the proportions to be demonstrated, and then comparing the product of the extremes with that of the means. For the two products will always be found to be exactly equal. Thus, Theorem I. R: sin. BC :: sin. B sin. Ac.
PQ QP That is, I
во GQ OP
GQ Note. By this Theorer), the expressions for the sine, cosine, and tangent of the angle BCA were obtained.
Q. E. D.