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

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 arc 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

M

S

R

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

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

2°.

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 can be drawn perpendicular to either side of the angle and included between the two sides?

a great circle that

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?

APPENDIX.

GRADED EXERCISES IN PLANE GEOMETRY.

ADDITIONAL DEFINITIONS.

1. The DISTANCE of a point from a line is measured on a perpendicular to that line.

2. The BISECTRIX of an angle is a line that divides the angle into two equal parts.

3. A MEDIAN is a line drawn from any vertex of a triangle to the middle of the opposite side.

4. The PROJECTION of a point, on a line, is the foot of a perpendicular drawn from the point to the line.

5. The PROJECTION of one straight line on another, is that part of the second line which is contained between the projections of the two extreme points of the first line, upon the second.

PROPOSITIONS.

I. THEOREM.-Show that the bisectrices of two adjacent angles are perpendicular to each other.

II. THEOREM.-Show that the perimeter of any triangle is greater than the sum of the distances from any point