Items related to Elements of plane and spherical trigonometry with Logarithmi...
Elements of plane and spherical trigonometry with Logarithmic and other mathematical tables and examples of their use and hints on the art of computation Volume 1 - Softcover
Synopsis
This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1882 Excerpt: ...in the same way, the algebraic sum of their projections upon O Y is found to be a sin a--b sin /3--c sin y-f-etc., which sum is zero by Theorem I. Theokem IY. If the sum of the projections of a series of straight lines upon any two non-parallel lines be zero, the sum of their projections upon any third line will be zero. Note. This theorem follows immediately from Theorems T. and II., but we prove it algebraically in order to show an elegant application of the addition theorem. Proof. Put a, y, etc., the angles which the straight lines make with one of the lines of projection; it, the angle which the first two lines of projection make with each other. Then a--n, 0--it, y--it, etc., will be the angles which the lines make with the second line of projection. By hypothesis we have a cos a-f-b cos /3--c cos y--etc. = 0; (a) a cos (a--it)-(-b cos (/?--it)--c cos (y--it)--etc. = 0. The last equation, by the addition theorem, reduces to cos it (a cos a--b cos /3--c cos y-(-... )-(-sin n (a sin a--b sin /3-(-c sin y +. ) = 0. The first term of this equation vanishes by (a). Hence, the whole sum being zero, the second term must also vanish, which requires that we either have sin n = 0, which will give n = 0 or 180,--in which case the two lines would be parallel,--or a sin a-(-b sin /?-f-c sin y-f-etc. = 0. (b) Since, by hypothesis, the two lines are not parallel, the equation b must hold true. Now let 6 be the angle which any third line of projection forms with the first line. The angles which the lines a, b, c, etc., form with this third line will then be a--6, /3-6, y-6, etc. Therefore the sum of the projections upon this line will be a cos (a--6)--b cos (/?--6) + c cos (y--6)-j-etc., which reduces to cos 6 (a cos a--b cos /3-f-c cos y + etc.) + sin 6 (a sin a 5 s...
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