DEMONSTRATION. Let ABC be the triangle proposed, and let o be the centre of the sphere; join AO, BO, and co. Take any point D in oc, and in the planes AOC, BOC, draw DE and DF each at right an- O< gles to oc, and join EF. Then because the EDF is the · measure of the inclination of the planes AOC, BOC, it is also the measure of the spherical ACB. (D. 133.) In the plane triangle EDF, -(N. 92.) EO2+ OF2- EF2 also in the plane triangle EOF, From the second of these equations EF2E02+OF2 — 2EO X OF X COS EOF which substituted in the first equation gives (ED2+DF2-E02-OF2) + (2EO XOF X COS EOF) 2DE X DF ED2 OD2 (Euclid 47 of I.) and or2 consequently coS EDF = (EO X OF X COS EOF)-(OD2 x rad) Now, EDF = c; 4 EOF=AB;; sine AC' DF OE DE X DF = DE sine DOE :(O. 44.) = rad OF rad sine / DOF (A) Since the preceding conclusion does not depend on any peculiar relation which the c has to the other angles, a similar equation will be equally true for the angles A and B. Hence (rad. cos a)-(rad. cos b. cos c) (rad2. cos c)-(rad. cos b. cos a) These are the formulæ from which Lagrange and Legendre* begin their investigations, and of the four quantities involved, any three being given the fourth may be found. They are applicable to every species of spherical triangles, whether rightangled, quadrantal, or oblique-angled, but the formula for right-angled and quadrantal triangles have already been given. (O. 169, and P. 169.) (B) If A, B, C represent the three angles of a spherical triangle, the opposite sides may be represented by 180°-A, 180°-B, 180°-c; and if a, b, c represent the three sides, their opposite angles may be represented by 180°-a, 180°—b, 180°-e (U. 137.) Hence rad2. cos (180°—▲)—rad. cos (180°-B). cos (180°-c) sine (180°-B). sine (180° —c) Cos (180°-a)= But cos (180°-a) = cos a; cos (180°— a) = COS A, &c. (N. 101.) consequently (rad3. cos a)+(rad. cos B. cos c) sine B. sine c may In the same manner the cosines of the other sides be determined. (C) It has been shewn (G. 176.) that the sines of the sides of any spherical triangle have the same ratio to each other as the sines of their opposite angles; hence by using the notation of Legendre, we shall have sine a. I. Sine A= sine B sine a. sine c sine c sine b. sine c Sine B sine a Sine c=- sine a sine b. sine A sine c sine b sine c II. Sine a= sine B sine c . sine B Sine b= (D) The general expressions for the cosines, which have been obtained by this proposition, may be arranged thus: *The former in the Journal de L'Ecole Polytechnique, and the latter in his Éléments de Géométrie. And by reducing these last equations. IV. Cos a= (cos A. sine b. sine c)+(rad. cos b. cos c) rad2 Cos b(Cos B. sine a . sine c) + (rad. cos a. cos c) rad2 (Cos c. sine b. sine a)+(rad. cos b. cos a) rad (rade. cos a)+(rad. cos B. cos c) Cos c= V. Cos a= (B. 183.) Cos b Cos c= (rad2. cos c)+(rad . cos a COS B) And by reducing these equations, we shall have VI. Cos A= Cos B= Cos c= (cos b. sine a. sine c) — (rad. cos a. cos c) (E) The six preceding articles afford solutions to all the different cases of oblique-angled spherical triangles. The Ist. The IId. Finds the angles, when two sides and an angle opposite to one of them are given. Finds a side, when two angles and a side opposite to one of them are given. The IIId. Finds the angles, from the three sides being given. The IVth. Finds the third side, when two sides and their contained angle are given. The Vth. Finds the sides, from the three angles being given. The VIth. Finds the third angle, when two angles and the side adjacent to both of them are given. (F) But none of the foregoing formulæ are conveniently adapted to logarithmical calculation. Let the value of the cosine of c(D. 183.) be substituted in the formula rad2-rad. cos c-2 sine2c (2d equation I. 117.) we COS C 2 sine c (rad cosc)-(rad.cosa.cosb) shall have 1. (sine a. sine b)+(cos a. cos b)-(rad.cos c) sine a. sine b sine a. sine b. rad ;but(sine a. sineb) 2 sine2c rad2 +(cos a. cos b)=rad. cos (a—b) (D.115.), hence rad. cos (a-b)-(rad. cos c) sine a. sine b From the 4th equation (F. 116.) rad. cos o-rad. cos P= 2 sine (P+Q). sine (P-Q), which, by putting q=(a—b) and PC, becomes rad. cos (a-b)-rad. cos c 2 sine (c+a-b). sine (c-a+b), hence we obtain 2 sinec 2. sine (c+a-b). sine (c-a+b) sine a sine b sine crad sine (c+a-b). sine (c—a+b). But 1/2 (c+a−b)=1/(a+b+c)−b, and (c+b−a)=(a+b+c)—a, sine (a+b+c)—b.sine (a+b+c)—a, and it is evident that the same formula will be obtained, with only a change of the letters, for the angles A and B. When c is near 90° this will not be a convenient rule for producing an accurate result, because the difference of the logarithmical sines for 1" is then very small (see the note page 54); if c be less than 45°, it will be proper to use this rule. 2 (G) Again, if the value of the cosine of c (D. 183.) be substituted in the formula rad2+ rad. cos c=2 cos'c (1st Equation I. 117. we shall have 1+ COS C 2 cos2 c (rad. cos c)—(cos b. cos a) rad rad2 :1+ sine a. sine b ; but (cos a. cos b) (sine a. sine b)+(rad. cosc) — (cos b. cos a) sine a. sine b (sine a. sine 6)=rad. cos (a+b), (D. 115.) Hence (sine a. sine b)-(cos a. cos b)=-rad. cos (a+b), therefore 2 cosc (rad. cos c)-rad. cos (a+b) From the 4th equation (F. 116.) (rad. cos g)-rad. cos P= 2 sine (P+Q). sine (PQ), which, by putting 2=c and P= (a+b),becomes (rad.cosc)-rad. cos(a+b)=2 sine (a+b+c) .sine (a+b-c), hence we obtain 2 cosc 2 sine (a+b+c). sine 1 (a+b−c) ; that is cos &c=rad / sine (a+b+c). sine ‡ (a+b−c). But sine a. sine b * (a+b−c)=3(a+b+c)—c hence cos + c=rad \/ sine ¦ (a+b+c). sine ÷ (a+b+c) —c sine a sine b and it is obvious that the same formula will be obtained, with only a change of the letters, for the angles A and B. Α When c is very small, this rule should not be used where a very accurate result is wanted, because the logarithmical cosines of very small arcs, in a table carried to seven places of figures, differ but little from each other (see the note page 54); if c be between 45°, and 90°, this rule may be used with advantage. (H) Also, because C rad sine c cos c tangc (N. 104), and rad. cos cot c(O. 104.) we shall obtain by division, sinec Tang+c=rad \/ sine (a+b+c)—b. sine¿(a+b+c)− a -9 sine (a+b+c). sine (a+b+c)—c sine (a+b+c). sine (a+b+c)—c Cot &c=rad sine and ne (a+b+c)—b.sines (a+b+c)—a (I) Any of the three preceding formulæ (F. 184, G. 185, or H. 186.) will determine an angle when the three sides are given, and by continuing Legendre's* mode of investigation, formulæ for determining a side in terms of the three angles may be obtained; thus, let the value of cos a (B. 183.) be substituted in the formula rad2-rad. cos a=2 sine 2 a (2d Equation I. 117.) we have 1 cos a 2 sine2a (rad2. cos a)+(rad. cos B. cos c) sine B. sine C rad (sine B. sine c)-(rad. cos a)—(cos B. cos c) ̧ rad. cos (B+c)=(cos B . cos c)-(sine B. sine c), (D.115.) or, (sine B. sine c)-(cos B. cos c)=-rad. cos (B+c), therefore * Eléments de Géométrie, 6th edition, page 391, et seq. |