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Extra resources for A Catalog of Special Plane Curves

Sample text

C x/ D cos b: (B) Comparing the values of sin p given in (A) and (B) gives sin a sin B D sin b sin A: (6) Dividing both sides of the last equation in Group (B) by both sides of the last equation in Group (A) gives cos b tan x D cot c: cos a sin c The combination of the second, third and last equations in (A) gives tan x D tan a cos B: Comparing these two values of tan x leads us to the equation cos b cos a cos c D sin a sin c cos B: 20 (7) Pangeometry 21 Likewise, if the perpendicular p falls outside the triangle and adds the arc x to the arc c, and the angle D to the angle C , there arise two right spherical triangles.

Parallel to a. b/. ˛/; which is the equation that we found above, and which is therefore proven in all cases. x/ D for x D 1, which agrees with what we have adopted and proved above. ˇ/ and 2 . c/: 44 There is an equivalent formula in Lobachevsky’s first published memoir, the Elements of geometry x x (1829) ([95], p. x/ D e 2e . a/: (12) Now let a, b, c be the three edges of an arbitrary rectilinear triangle and let A, B, C be the angles opposite to these edges. Let us drop from the vertex of the angle C a perpendicular p on the edge c.

Parallel to a. b/. ˛/; which is the equation that we found above, and which is therefore proven in all cases. x/ D for x D 1, which agrees with what we have adopted and proved above. ˇ/ and 2 . c/: 44 There is an equivalent formula in Lobachevsky’s first published memoir, the Elements of geometry x x (1829) ([95], p. x/ D e 2e . a/: (12) Now let a, b, c be the three edges of an arbitrary rectilinear triangle and let A, B, C be the angles opposite to these edges. Let us drop from the vertex of the angle C a perpendicular p on the edge c.

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