finite, inclose the greatest surface possible, when placed so that the included angle may be equal to the sum of the other two angles of the triangle.

17. In a triangle, ABC, which has one of its angles, ABC, equal to the sum of the other two, the containing sides, AB, BC, are together less than a semicircumference.

18°. Of all spherical polygons, contained by the same given sides, that one contains the greatest portion of the spherical surface which has all its angles in the circumference of a circle.

19°. A circle includes a greater portion of the spherical surface than any spherical polygon of the same perimeter.

20°. The lunular surface, which is included by a spherical arc, and a small arc, is greater than any other surface which is included by the same perimeter, of which the same spherical arc is a part.

21°. Of all spherical polygons having the same number of sides and the same perimeter, the greatest is that which has all its sides equal and all its angles equal.

22°. Spherical pyramids, which stand upon equal bases, are equal to one another; so, likewise, are their solid angles.

23°. Any two spherical pyramids are to one another as their bases, and the solid angles of the pyramids are to one another in the same ratio.

24°. Every spherical pyramid is equal to the third part of the product of its base and the radius of the sphere.

25°. Every solid angle is measured by the spherical surface which is described with a given radius about the angular point, and intercepted between its planes.

26°. To find the diameter of a given sphere.*

27°. To find the quadrant of a great circle.

28°. Any point being given upon the surface of a sphere, to find the opposite extremity of the diameter which passes through that point.

29°. To join two given points upon the surface of a sphere.

30°. A spherical arc being given, to complete the great circle of which it is a part.

* In executing this and the following problems, it is not permitted to employ any thing like a flexible ruler or straightedge, but the student is supposed to be furnished simply with a pair of compasses of such construction as to be capable of embracing the extremities of any arc not greater than a quadrant. A Spherical Blackboard would be found useful.

EXERCISES.

31°. To bisect a given spherical arc.

279

32°. To draw a spherical arc which shall bisect a given spherical arc at right angles.

33°. To draw an arc which shall be perpendicular to a given spherical arc, from a given point in the same.

34°. To draw an arc which shall be perpendicular to a given spherical arc, from a given point without it.

35°. To bisect a given spherical angle.

36°. At a given point in a given arc, to make a spherical angle equal to a given spherical angle.

37°. To describe a circle through three given points upon the surface of a sphere.

38°. To find the poles of a given circle.

39°. Through two given points, A, B, and a third point, C, on the surface of a sphere, to describe two equal and parallel small circles; the points A, B, C, not lying in the circumference of the same great circle.

40°. To describe a triangle which shall be equal to a given spherical polygon, and shall have a side and adjacent angle the same with a given side and adjacent angle of the polygon.

41°. Given two spherical arcs together less than a semicircumference, to place them so that, with a third not given, they may contain the greatest surface possible.

42°. Through a given point to describe a great circle which shall touch two given equal and parallel small circles,

43°. To inscribe a circle in a given spherical triangle.

SECTION SECOND.

Projections of the Sphere.

PROPOSITION I.

The Orthographic Projection of every circle of the (573) sphere, as meridians and parallels of latitude, will be an ellipse, circle, or straight line, according as its plane shall be oblique, parallel, or perpendicular to the plane of projection.

10. For let M, N, P, Q, be the segments of any two chords, M+N, P+Q, of a circle, and m, n, p, q, their orthographic projections upon the plane of m+n, p+q; that is, projections made by perpendiculars let fall from the extremities of M, N; P, Q, we have,

which determines the nature of the curve of projection.

=

=

2°. If M+N pass through the centre, O, and P+Q be at right angles to M+N, then P will Q, and.. (574) p will q; whence it follows that m+n bisects a system of parallel chords p+q, and is itself bisected in o, the projection of O; therefore o is the centre of the curve. Assuming o as the origin of a system of oblique coordinates, (575) becomes,

R

n

N

m

Fig. 122.

M

} (574)

Fig. 1222.

(575)

putting m+ n = 24; but R being the radius of the circle and B

the projection of that

R which is parallel to P, we find

A Rcos(M,m),

=

B+Rcos(P,p);

3°. If R be assumed in such position that its projection, A, or, for the sake of distinction, a, shall be parallel to it and consequently b, the new value of B, perpendicular to a, there results

a=Rcos(M,m) = Rcos0° R,
b = Rcos(P,p) = Rcos I,

I being the inclination of the circle to the plane of projection; we have also

which is the equation of an ellipse of which the axes are 2a, 2b ; and the proposition is demonstrated, since when I = 0, b becomes parallel to and equal to R = a, and when I = 90°, b becoming = 0, the ellipse vanishes in a straight line.

Cor. 1. The ellipse may be referred to a system of ob- (579) lique coördinates such that its equation (576) shall be of the same form with that obtained for rectangular axes (578), and 2A, 2B, mutually bisecting each other and all chords drawn parallel to them, are denominated Conjugate Diameters, of which the axes 2a, 2b, are but particular values.

Cor. 2. If two systems of parallel chords intersect each (580) other in an ellipse, the products of their segments will be proportional (575); and this property may be extended to the case in which the points of intersection lie without the curve.

Cor. 3. An elliptical arc, MmApP, being given, the (581) centre, O, may be found by bisecting AOB drawn

through the middle points of any parallel chords,

MP, mp.

Fig. 1224.

M

Cor. 4. A conjugate, CD, to any diameter, AB, may (582) be found by drawing CD through the middle points of AB and PN a chord parallel to AB.

Cor. 5. A parallelogram, MPNQ, may be described in (583) an ellipse by drawing chords, MQ, PN; MP, QN, parallel to the conjugates AB, CD.

Cor. 6. The diagonals MN, PQ, mutually bisecting each (584) other in O, are obviously diameters, and P may be so chosen that MN shall be the conjugate axis = 2a, in which case PM, PN, are said to be Supplementary Chords.

Cor. 7. The tangents drawn through the extremities (585) A, B, C, D, of the conjugate diameters, are parallel to the supplementary chords, and to the corresponding diameters; and, by their intersections, constitute a parallelogram circumscribing the ellipse.

PROPOSITION II.

ZA

H
B

H

a

Fig. 123.

H

To make an Orthographic Projection of the Sphere. With the radius OA, equal to that of the required map, describe the meridian AZP,H,B P.H, passing through the north and south. poles, P„, P., and, consequently, perpendicular to the horizon seen edgewise in H„H ̧. Take the arc HP, measuring the elevation of the pole, equal to the latitude of the place Z over which the observer is supposed to be situated at a very great [infinite] distance. Having graduated the meridian and drawn the parallels of latitude ACB, which will be perpendicular to the axis POP, drop their extremities and centres, A, C, in the projections a, c, also project the elevated pole P in Pn Transfer the points a, c, P, to the central meridian H,OH, of the map, and through c draw a perpendicular to H,H,, intersecting the circumference in m, m,; cm and ca will be the semimajor and semiminor axes of the elliptical parallel of latitude mam,, which may be described by the first exercise under the ellipse. Next, for the meridians, let H,P,H,= h be the angle which any required meridian, P„H, makes with the vertical meridian P,H,, and HH, = H, the corresponding arc intercepted on the horizon, further PH, will = 1, the latitude of the place upon whose horizon the projection is made; therefore, by Napier's Rules, PH,H, being a right angle, we have

H

211

Pr

Fig. 1232.

H

Z

XII

Fig. 124.