The Angular Excess of a Spherical Triangle

As we have said, the parallel axiom is equivalent to the fact that the angles in a triangle sum to $\pi$ (i.e. $180^0$ ). It follows that both the spherical axiom and the hyperbolic axiom must lead to geometries in which the angles do not sum to $\pi$ , To quantify this departure from Euclidean Geometry, we introduce the angular excess, defined by the amount $\epsilon$ by which the angle sum of a triangle exceeds $\pi$ .

$\epsilon\equiv\text{(angle sum of triangle)}-\pi$

See the spherical triangle shown below:

The angle sum for this triangle is $(\pi+\alpha)$ and therefore, the angular excess is $\epsilon=(\pi+\alpha)-\pi=\alpha$

A crucial insight now arises if we compare the triangle’s angular excess with its area $A$ . Let the radius of the sphere be $R$ . Since the triangle occupies a fraction $\Large$\frac{\alpha}{2\pi}$$ , so the area of the triangle,
$\mathcal A=\frac{\alpha}{2\pi}\times 2\pi R^2=\alpha R^2$
Thus we get,
$\boxed{\epsilon=\frac{1}{R^2}\mathcal A}$

In 1603, the English mathematician Thomas Harriot made the remarkable discovery that this relationship holds for any triangle Δ on the sphere (This discovery is most often attributed to Albert Girard, who discovered it about 25 years later and was the one to publish it for the first time, we will thus call this result by Harriot-Girard’s Theorem).

***Thomas Harriot (1560-1621),*** an *English astronomer and mathematician*

Harriot‘s elementary but ingenious argument is shown in the following short animation video (done for sphere of unit radius) taken from the YouTube channel 2maniac. This argument was later rediscovered by Euler in 1781.