EXERCISES.

31°. To bisect a given spherical arc.

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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.

1°. 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,

=

cos (M,m) cos2(P,p)

=

R

which determines the nature of the curve of projection.
2o. 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. As-
suming o as the origin of a system of oblique co-
ordinates, (575) becomes,

B

P

n

N

M

m

Fig. 122.

} (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, ɑ, shall be parallel to it and consequently b, the new value of B, perpendicular to a, there results

B

R

(576)

} (577)

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

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,

C

MP, mp.

N

B

M

Fig. 1224.

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.

H

B

Fig. 123.

Hn

SH

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 PH, passing through the north and south. poles, P, P., and, consequently, perpendicular to the horizon seen edgewise in HH. 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 P2 in Pr• Transfer the points a, c, p,, to the central meridian H2OH, of the map, and through c draw a perpendicular to H,H,, intersecting the circumference in m, m2,; 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 PH, and H,H1 = H, the corresponding arc intercepted on the horizon, further P„H, will = 7, the latitude of the place upon whose horizon the projection is made; therefore, by Napier's Rules, P„HH, being a right angle, we have

=

In

Pr

Hs

Fig. 1232.

XII
Fig. 124.

or

GNOMONIC PROJECTION.

cosl coth cot He,

=

tanH = sin tanh.

H found by calculation,

Having laid off HH, and drawn the diameter H,O, and Or perpendicular to it, transfer the semimajor axis H,O to the straight edge of a thin ruler or slip of paper, and, having applied this line in pt, and marked its point, y, of intersection with OH,, proceed to describe the meridian Hip, according to the first exercise under the ellipse.*

The work now described is to be combined in a single figure, when the features of the spherical surface will be laid down according to the projections of the meridians and parallels of latitude.

PROPOSITION III.

To make the Gnomonic Projection when the dial is horizontal. Let H,H,H, be the horizontal face of the dial, O its (fig. 124) centre and OP, the style elevated according to the latitude of the place; the shadow at noon will fall upon the north point, H2, and its positions, OH,, for all other hours will be determined by (586), where h = the hour angle.

PROPOSITION IV.

To make the Gnomonic Projection of the Style upon a vertical south plane.

The construction will be similar to the preceding, (fig. 124) observing that,

tan(XII)T = cosl tanh

(587)

* An instrument very convenient for this purpose may be constructed, consisting of a slender ruler carrying a pen or pencil in its extremity and furnished with two moveable pins, one to glide in a groove cut in the stem of a wooden T and the other along its top.