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Copernican Cosine Theorems for Spherical Triangles

Chapter XIV

Theorems III and XII

We consider three points A, B, and C on a sphere of radius R. If we connect them by arcs (along great circles), we obtain a spherical triangleABC.
We consider right-angled spherical triangles with sides shorter than a semicircle, as shown in the
figure.

III Copernicus’ /b>Third Theorem<:

In a right-angled spherical triangle ABC on a sphere of radius R (where angle C is a right angle), the following proportion between the sides holds:

AB / BC = R / BC

That is, the ratio of the hypotenuse to one leg equals the ratio of the radius to the adjacent leg.
This means that if we know two sides, we can determine the third.

[Copernicus, Nicolaus (1473-1543), “De revolutionibus orbium coelestium”, Kujawsko-Pomorska Digital library, UMK, 1854, Chapter XIII, pp. 63-64]

In other words, Copernicus proved a special case of the spherical law of cosines. Now, we may assume that the sphere has radius R =1.

We measure in radians the length of the side (arc) opposite to a given angle as an arc on the sphere (from the sphere’s center).
We have:

AB = c BC = a AC = b
(here a, b, c are the measures of angles AOB, BOC, AOC in radians)

If angle C is right, we can write Copernicus’ theorem as:

cos c / cos b = cos a

[Copernicus, Nicolaus (1473-1543), “De revolutionibus orbium coelestium”, Kujawsko-Pomorska Digital library, UMK, 1854, Chapter XIII, pp. 63-64]

Theorem of Pythagoras for spherical triangles

That is, if we know two sides, we can find the third.This is the spherical version of the Pythagorean theorem, which can be written as:

cos c = cos b cos a

We also have the general spherical law of cosines for triangles on a sphere of radius R=1, where α, β, γ are the spherical angles of triangle ABC.

Copernicus’ Twelfth Theorem:
cos c = cos a cos b + sin a sin b cos γ

That is, if we know two sides and at least one angle, we can determine the third, which is the spherical version of the law of cosines for all plane triangles.

[Copernicus, Nicolaus (1473-1543), “De revolutionibus orbium coelestium”, Kujawsko-Pomorska Digital library, UMK, 1854, Chapter XIII, pp. 63-64]