Napier's Rules for Right Angled Spherical Triangles

Theorem

 * NapiersRules.png

Let $\triangle ABC$ be a spherical triangle on the surface of a sphere whose center is $O$.

Let the sides $a, b, c$ of $\triangle ABC$ be measured by the angles subtended at $O$, where $a, b, c$ are opposite $A, B, C$ respectively.

Let $\angle C$ be a right angle.

Let the remaining parts of $\triangle ABC$ be arranged in the order as given above:
 * $a, b, A, c, B$

Let these quantities be arranged in a circle as above, where the prefix $\text{co-}$ meaning complement is added to $c, A, B$.

Let each one of the parts of this circle be called a middle part.

Let the two neighboring parts of the middle part be called adjacent parts.

Let the remaining two parts be called opposite parts.

Then: