Law of Species/Examples/Arbitrary Example 1

Example of Use of Law of Species

Consider the right spherical triangle with:

side $c$ equal to $30 \degrees$

angle $B$ equal to $30 \degrees$.

The other sides are found by using Napier's Rules of Circular Parts.

Hence:

$\sin b = \sin c \sin B$

and so:

$\sin b = \dfrac 1 2 \cdot \dfrac 1 2 = \dfrac 1 4$

But from the Law of Species, $b$ and $B$ are of the same species.

We have that $B$ is an acute angle.

So $b$ must also be an acute angle.

Hence $b = \inv \sin {\dfrac 1 4}$ and less than $90 \degrees$.

So:

$b = 14 \degrees \, 29 \minutes$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): species
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): species