A treatise on spherical trigonometry, and its application to geodesy and astronomy, with numerous examples. By John Casey.

Spherical Triangles. 17
This demonstration is taken from the works of WALLIS, Vol. n.,
p. 875. The theorem is due to ALBERT GIRARD, a Flemish
Mathematician of the 17th century. In 1787, more than 150
years after its discovery, an important application of it was
made by General Roy in correcting the spherical angles, observed in the Trigonometrical Survey of Britain, Phil. Trans.,
Vol. vIII., p. 163, year 1790. See also IMem. Acad., Paris, 1787,
p. 358, and Mem. Inst., Vol. vi., p. 511.
Cor. 1.-The area of a great circle: area of the spherical
triangle:: r: 2E. (9)
Cor. 2.-If Z denote the sum of the angles of a spherical
polygon of n sides, its area is
{( + (2- n)r} r2. (10)
EXERCISES.-II.
1. If a triangle coincides with its supplemental triangle, prove that all its
sides are quadrants and all its angles right.
2. The sum of two opposite angles of a spherical quadrilateral inscribed
in a small circle is equal to the sum of the two others, and each sum is
greater than two right angles.
3. The spherical excess of a spherical triangle is equal to the circumference of a great circle diminished by the perimeter of the supplemental
triangle.
4. The sum of two opposite sides of a spherical quadrilateral, circumscribed to a small circle, is equal to the sum of the remaining sides.
5. If A, B, C, D be four concyclic points on a sphere, prove that
sin ~ BC. s in AD + sin CA. sin i BD + sin AB. sin C CD = 0.
(11)
This follows from Ptolemy's theorem, since chord BC = 2 sin ~ BC, &c.
6. In the same case, if
AB = a, BC=b, CD =c, DA = d, AC=e, BD =f;
prove that
sin e sin a a. sin d + sin b. sini e
(12)
sin Sf sin a. sin i b + sin c. sin d()
O