LEGENDRE THEOREM ON SPHERICAL TRIANGLES

50 years of the Research Institute of Z. NádeníkGeodesy, Topography and CartographyThere immediately appearsProblem III: Is it possible to prove (N) or (N’) in a finite way modelled after Gauss’ proof inPart II?– – –1. For p = 0 formula (N’) reduces to the Legendre theorem in the form of (L’).2. For p =− 1/ 24 formula (N’) reduces tosin( α − ε / 4)= cycl.3aa − + (5)24The length of the chord joining the vertices B and C of our spherical triangle isa aa = 2sin = a− + (5) .2 24Hencesin( α −ε / 4) sin( β −ε / 4) sin( γ −ε/ 4)(Gr)= = .abcThis relation was established by J. Grunert in 1855 [3]. M. Burša called attention to it in [8] 1 , p. 35.3. For p =− 1/ 6 formula (N’) reduces tosinα sin β sin γ ,sin a sin b sin cbecause a− a 3 /6 + (5) sina. We thus arrive at the approximate sine law (of spherical trigonometry)which, of course, holds exactly.– – –That which I describe in the next two cases for side BC of spherical triangle ABC , is consideredalso applied to the other two sides CA and AB .Let a regular polygonal line with n sides be inscribed into side BC . The length of this polygonalline is3a a2nsin = a− + (5) .22n24nIf n = 1 , this polygonal line is reduced to chord BC , if n →∞, it becomes side BC . For2p=− 1/24nformula (N’) has the form⎡ ⎛1 1 ⎞ ⎤sin ⎢α−⎜−2 ⎟ε3⎥⎣ ⎝ 12n⎠ ⎦cycl.3 = aa − + (5)224nThe longitude of our inscribed polygonal line is in the denominator. If n = 1 , we arrive at Grunert’sformula (Gr) and if n →∞ at the Legendre theorem in the form (L’).Let us circumscribe polygonal line PPP 0 1 2… PP n n+1with P 0≡ B, Pn+ 1≡ C about side BC asfollows: 2PP 0 1= PP1 2= … Pn− 1Pn = 2PPn n+1; the segments PP 0 1and PPn n + 1touch side BC at point1 I reviewed this book in Zentralblatt für Mathematik 117 (1965), 177.346