distances from the point of contact; and those tangents form respectively the same angles as the arcs that mea sure the several distances from the principal point. In this projection too, a less circle will evidently be projected into an ellipse, a parabola, or a hyperbola, ac cording as the distance of its most remote point is less, equal to, or greater than 90°, from the centre of the plane of projection.

But for a farther developement of these properties, and for the geometrical constructions derived from them, such as want to enter more minutely into this subject may consult Emerson's Projection of the Sphere; the treatise in Bishop Horsley's Elementary Treatises on Practical Mathematics, or that in the Traité de Topographie, par Puissant.

CHAPTER IX.

On the Principles of Dialling.

1. DIALLING, or gnomonics, is the art of drawing on the surface of any given body, whether plane, angular, or curved, a sun-dial, that is, a figure, the dif ferent lines of which, when the sun shines, indicate by the shadow of a style or gnomon the time of the day.

2. The general principles which serve as a basis to the theory of dialling, cannot be more aptly illustrated than they have been by Ozanam and Ferguson, in the following contrivance.

Suppose a hollow transparent sphere DPBP, of glass, to represent the earth as transparent, and its equator divided into 24 equal parts by so many meridian semicircles a, b, c, d, e, &c. one of which is the geographical

meridian of any given place, as London, which it is

supposed is at the point a; then if the hour of 12 were marked at the equator, both upon that meridian and the opposite one, and all the rest of the hours in order on the other meridians, those meridians would be the hour circles of London. Because, as the sun appears to move

planes, or other surfaces, any way posited in this hollow terrestrial sphere, we should by the several intersections of these surfaces with the hour circles, obtain the hour lines for every variety of dial. And since the earth it

self compared with its distance from the sun, may, in reference to this branch of inquiry, be regarded as a point, if a small sphere of glass, or a small sphere constituted of a wire axis and wire hour circles, be placed upon any part of the earth's surface, so that its axis be parallel to the axis of the earth, and the sphere have lines upon it and planes within it, such as those above described, it will indicate the time of the day as accurately as if it were placed at the earth's centre, and the earth itself were as transparent as glass.

These general notions being premised, it will be proper to annex a few definitions.

3. The plane erected perpendicularly to the face of the dial, and the upper edge of which marks and bounds the shadow, is called the gnomon; the superior edge of the said plane is called the style of the dial, and it is always parallel to the earth's axis.

4. The line in which the plane of the gnomon intersects the plane of the dial is denominated the substyle.

5. The angle included between the style and the substyle is called the elevation of the style; in the following formulæ it will be denoted by E. In a horizontal dial this is, evidently, equal to the latitude of the place, or

EL.

6. While those dials whose planes are parallel to the plane of the horizon are called horizontal dials; such as have their planes perpendicular to the horizon are called erect dials.

7. Those erect dials whose planes are either parallel or perpendicular to the plane of the meridian, are called direct erect dials; they face one or other of the four cardinal points.

8. All other erect dials are called declining dials. 9. Those dials whose planes are neither parallel nor

perpendicular to the plane of the horizon, are called inclining or reclining dials. They may, at the same time, be either direct or declining, according as they present a sloping face to the cardinal points, or not.

10. The arch of the horizon which is intercepted between any given plane, and that of the prime vertical, is called the declination of the plane. It will be denoted by D, and will be regarded as positive when the declination is towards the north, negative when it is towards the south.

11. The inclination, 1, of a plane is the angle which it makes with a vertical plane.

12. The intersection of the plane of the dial and that of the meridian passing through the style is called the meridian of the dial, or the hour line of 12.

13. Those meridians passing through the style, which make angles of 15°, 30°, 45°, &c. with the meridian of the place (marking the hour line of 12) are called hour circles, and their intersections with the plane of the dial hour lines.

14. The angle formed by the substyle and the meridian is called the horary angle of the substyle: it will be denoted by M.

15. The angle included between the substyle and the horizon is termed the inclination of the substyle: it will be denoted by s.

GENERAL PROBLEM.

16. To determine the requisites in a dial of any proposed inclination to a vertical plane, and declination from the prime vertical.

Let HOR, in the marginal diagram, represent the horizon, HZR the meridian of the place, z the zenith, zo the prime vertical, p the elevated pole, H Pb, Po, portions of hour circles, zeb, zrb', corre

2

Z

R

sponding hour angles, and let the plane on which it in proposed to draw the dial be coincident with the plane of the great circle MoT. Also, let za perpendicular to MaT be drawn and produced to I. Then PRL, the latitude of the place; za 1, the inclination of the plane; OT HID, its declination; Ps drawn perpendicular to MT will be the position of the substyle; and мb is the inclination of the hour line to the meridian.

Now, we may regard the dial whose plane coincides with mar, as a vertical or erect dial at the place whose zenith is м, where м and z are on the same meridian, and, of course, reckon the hours alike. Let мz = l, and D' comp. of zмa. Then PZ = 90° L would become PMPZ + ZM = - 1 + 1 = 90° — (L — }). The right angled triangle мza, gives

90°

tan ma➡ sin za tan мza = sin Itan D..... (1.)

Here Ma is the angle between the meridian and ver⚫ tical, and is obviously evanescent when either 1 or D are. When I = 90°, мa becomes = HI = D.

(3.)

sinessin E=sin zma sin PM = cos D'cos (L).. (4.) where E is the height of the style, or elevation of the pole above the plane.

Then, Pb being any hour circle whatever, the spherical triangle Poм, gives us, from equa. (4) of spherical trigonometry [chap. vi. 23].

cot мb sin PM COS PM COS Pмb + sin Pмb cot P. But, cos Pмb= sin D', and sin Pмb = cos D ́;

tan мT=tan (ма + 90°) =

P

(5.)

cot Ma,

where MT is the inclination of the horizontal plane to

the meridian..