In this final proposition, Lobachevski develops the trigonometric formulae of the imaginary plane. Although we can now replace the π-functions in any trigonometric equation with hyperbolic functions, Lobachevski chooses not to make these helpful translations. This lack, coupled with some awkward derivations and peculiar notation, make his work in this section appear particularly opaque. This is unfortunate, since the conclusions of this section are actually quite simple and admit easy proofs. To emphasize this fact, I shall derive Lobachevski's results in π-free notation and deviate from his unnecessarily convoluted proofs of the two laws of cosines.

The Need for a New Rectilinear Relation

We can interpret the five gems as statements about right spherical triangles or right rectilinear triangles. In their spherical interpretation (see TP 35 notes), they specify the following trigonometric relationships:

A relation among the triangle's three sides. (SG 5)

A relation among the two acute angles and a leg. (SG 3,4)

A relation among the hypotenuse, a leg, and the

acute angle they do not include. (SG 1,2)

In the notes to TP 35, we developed all of spherical trigonometry from these relationships. Naturally, as soon as we have equations that specify the same three relations for right rectilinear triangles, we will be able to develop all of plane trigonometry by an analogous procedure.