Law of Species

Theorem

Let $T$ be a right spherical triangle whose angles are $A$, $B$ and $C$ and whose respective sides opposite those angles are $a$, $b$ and $c$.

Let $C$ be the right angle of $T$.

Then:

$(1): \quad$ $A$ and $a$ are of the same species, and $B$ and $b$ are of the same species

$(2): \quad$ if $c < 90 \degrees$, then $a$ and $b$ are of the same species, as are $A$ and $B$

$(3): \quad$ if $c > 90 \degrees$, then $a$ and $b$ are of the opposite species, as are $A$ and $B$.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Examples

Arbitrary Example

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:

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

Also known as

The law of species is also known as the law of quadrants.

Sources

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