angle is taken as the unit of polyedral angles, and the tri-rectangular spherical triangle is taken as its measure. If the vertex of a polyedral angle is taken as the centre of a sphere, the portion of the surface intercepted by its faces is the measure of the polyedral angle, a tri-rectangular triangle of the same sphere being the unit.

The area of a spherical polygon is equal to its spherical excess multiplied by the tri-rectangular triangle.

Let ABCDE be a spherical polygon on a sphere whose centre is O, the sum of whose angles is S, and the number of whose sides is n: then is its area equal to

For, draw the diagonals AC, AD, dividing the polygon into spherical triangles: there are n 2 such triangles. Now, the area of each triangle is equal to its spherical excess into the tri-rectangular triangle:

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hence, the sum of the areas of all the triangles, or the area of the polygon, is equal to the sum of all the angles of the triangles, or the sum of the angles of the polygon diminished by 2 (n-2), into the tri-rectangular triangle; or,

GENERAL SCHOLIUM 1.

From any point P on a hemisphere, two arcs of a great circle, PC and PD, can always be drawn, which shall be perpendicular to the circumfer

ence of the base of the hemisphere, and they will in general be unequal. Now, it may be proved, by a course of reasoning analogous to that employed in Book I., Proposition XV.:

C

R

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1°. That the shorter of the two arcs, PC, is the shortest arc that can be drawn from the given point to the circumference; and, therefore, that the longer of the two, PED, is the longest arc that can be drawn from the given point to the circumference:

2. That two oblique arcs, PQ and PR, drawn from the same point, to points of the circumference at equal distances from the foot of the perpendicular, are equal:

3. That of two oblique arcs, PR and PS, drawn from the same point, that is the longer which meets the circumference at the greater distance from the foot of the perpendicular.

GENERAL SCHOLIUM 2.

The arc of a great circle drawn perpendicular to an are of a second great circle of a sphere, passes through the poles of the second arc (P. III., C. 3). The measure of a spherical angle is the arc of a great circle included between the sides of the angle and at the distance of a quadrant from its vertex (P. IV.). It is evident, therefore,

that the pole of either side of an acute spherical angle lies without the sides of the angle; and that the pole of either side of an obtuse spherical angle lies within the sides of the angle.

Now, let A be an acute spherical angle, ST its measure, MN any arc of a great circle, other than ST, drawn perpendicular to the side AQ, and included between the two sides AQ and AR, and P the pole of the side AQ: and

P

M

S

T

Let B be an obtuse spherical angle, CD its measure, EF any arc of a great circle, other than CD, drawn perpendicular to the side BH, and in

cluded between the two sides BH

C

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and, hence, is the longest arc of a great circle that can be drawn perpendicular to the side AQ and included between the two sides AQ and AR: and

2o. That CD is shorter than EF, and, hence, is the shortest arc of a great circle that can be drawn perpendicular to the side BH and included between the two sides BH and BG.

EXERCISES.

1. The sides of a spherical triangle are 80°, 100°, and 110°; find the angles of its supplemental triangle, and the angles of each of its polar triangles.

2. Find the area of a tri-rectangular triangle, on a sphere whose diameter is 8 feet.

3. Find the area of a tri-rectangular triangle, on a sphere whose surface and volume may be expressed by the same number.

4. The angle of a lune, on a sphere whose radius is 5 feet, is 50°; find the area of the lune and the volume of the corresponding wedge.

5. The area of a lune is 33.5104 square feet and the angle of the lune is 60°; find the surface and the volume of the sphere.

6. Show that if two spherical triangles on unequal spheres are mutually equiangular, they are similar.

7. Show how to circumscribe a circle about a given spherical triangle.

8. Show how to inscribe a circle in a given spherical triangle.

9. Show that the intersection of the surfaces of two spheres is a circle, and that the line which joins the centres of two intersecting spheres is perpendicular to the circle in which their surfaces intersect.

10. Show that two spherical pyramids of the same or equal spheres, which have symmetrical triangles for bases, are equal in volume. [Proof analogous to that in P. XVI.] 11. The circumferences of two great circles intersect on the surface of a hemisphere whose diameter is 10 feet, and the acute angle formed by them is 40°; find the sum of the opposite triangles thus formed and the sum of the corresponding spherical pyramids.

12. Show that the volume of a triangular spherical pyramid is equal to its base multiplied by one third the radius of the sphere.

13. Show that the volume of any spherical pyramid is equal to its base multiplied by one third the radius of the sphere.

14. Find the volume of a spherical pyramid whose base is a tri-rectangular triangle, the diameter of the sphere being 8 feet.

15. The angles of a triangle, on a sphere whose radius is 9 feet, are 100°, 115°, and 120°; find the area of the triangle and the volume of the corresponding spherical pyramid.

16. A spherical pyramid, of a sphere whose diameter is 10 feet, has for its base a triangle of which the angles are 60°, 80°, and 85°; what is its ratio to a pyramid whose base is a tri-rectangular triangle of the same sphere?

17. The sum of the angles of a regular spherical octagon is 1140°, and the radius of the sphere is 12 feet; find the area of the octagon.

18. The volume of a spherical pyramid, whose base is an equiangular triangle, is 84.8232 cubic feet, and the radius of the sphere is 6 feet; find one of the angles of the base. 19. Given a spherical angle of 40°; what is the number of degrees in the longest arc of a great circle that can be drawn perpendicular to either side of the angle and included between the two sides?

20. Given a spherical angle of 115°; what is the number of degrees in the shortest arc of a great circle that can be drawn perpendicular to either side of the angle and included between the two sides?