geography and astronomy, signifies the laying down upon paper those imaginary circles of the sphere by which the degrees of longitude and latitude are counted on celestial and terrestrial maps.

Projection of the sphere is either orthographic or stereographic. The orthographic projection supposes the eye placed at an infinite distance; whereas, in the stereographic projection, it is supposed to be only 90° distant from the primitive circle, or placed in its pole, and thence viewing the circles on the sphere. The primitive circle is that great circle which limits or bounds the representation or projection; and the place of the eye is called the projecting point. See Geography, n° 13, &c.

The laws of the orthographic projection are these:

1. The rays by which the eye, placed at an infinite distance, perceives any object are parallel. 2. A right-line, perpendicular to the plane of the projection, is represented by a point, where it cuts the plane of the projection. 3. A right-line, as AB, or CD, (fig. 3, n° 1.) not perpendicular, is projected into a right-line, as FE and GH, and is always comprehended between the extreme perpendiculars AF and BE, and CG and GH. 4. The projection of the right-line, AB, is the greatest when it is parallel to the plane of projection; being projected in a right-line equal to itself. 5. But an oblique line is always projected into one less than itself; and the more so, the nearer it approaches to a perpendicular, which, as already observed, is projected into a point. 6. A plane surface, as ABCD, (ibid. n° 2.) at right angles to the plane of the projection, is projected into the right line AB, in which it cuts the plane of the projection; and any arch, as BC, or CA, is projected into the corresponding lines BO, CO, or OA. 7. A circle parallel to the plane of projection, is represented by a circle equal to itself; and a circle oblique to the plane of projection, is represented by an ellipse.

As to the stereographic projection, its laws are these:

1. The representations of all circles, not passing thro' the projecting point, will be circles. Thus, let ACEDB (fig. 4, n° 1, 2, 3,) represent a sphere, cut by a plane RS, passing thro' the centre I, at right angles to the diameter EH, drawn from E the place of the eye; and let the section of the sphere by the plane RS; be the circle CFDL, whose poles are H and E. Suppose now AGB is a circle on the sphere to be projected, whose pole most remote from the eye is P; and the visual rays from the circle AGB meeting in E, form the cone AGBE, whereof the triangle AEB is a section thro' the vertex E, and diameter of the base AB; then will the figure agfb, which is the projection of the circle AGB, be itself a circle: for if the plane RS is supposed to revolve on the line CD, till it coincides with the plane of the circle ACEB; then will the circle CFDL coincide with the circle CEDH, and the projected circle agfb with the circle agfbk. Hence, the middle of the projected diameter is the centre of the projected circle, whether it be a great circle or a small one; the poles and centres of all circles, parallel to the plane of projection, fall in the centre of the projection; and all oblique great circles cut the primitive circle in two points diametrically opposite.

2. The Projection projected diameter of any circle subtends an angle at the eye equal to the distance of that circle from its nearest pole, taken on the sphere; and that angle is bisected by a right line, joining the eye and that pole. Thus let the plane RS (ibid. n° 4.) cut the sphere HFEG through its centre I; and let ABC be any oblique great circle, whose diameter AC is projected in ac, and KOL, any small circle parallel to ABC, whose diameter KL is projected in kl. The distances of those circles from their pole P, being the arches AHP, KHP; and the angles aEc, kEl, are the angles at the eye, subtended by their projected diameters, ac, kl. Then is the angle aEc measured by the arch AHP, and the angle kEl measured by the arch KHP, and those angles are bisected by EP. 3. Any point of a sphere is projected at the distance of the tangent of half the arch intercepted between that point and the pole opposite to the eye, from the centre of projection; the semi-diameter of the sphere being radius. Thus, let CEB (ibid. n° 5.) be a great circle of the sphere, whose centre is c, GH the plane of projection cutting the diameter of the sphere in b, B; E, C, the poles of the section by that plane; and a, the projection of A. Then is ca = the tangent of half the arch AC, as is evident by drawing CF = the tangent of half that arch, and joining cF. 4. The angle made by two projected circles, is equal to the angle which these circles make on the sphere. For let IACE and ABL (ibid. n° 6.) be two circles on a sphere intersecting in A; E, the projecting point; and RS, the plane of projection, wherein the point A is projected in a, in the line IC the diameter of the circle ACE. Also let DH, FA, be tangents to the circles ACE, ABL. Then will the projected angle daf be equal to the spheric angle BAC. 5. The distance between the poles of the primitive circle and an oblique circle, is equal to the tangent of half the inclination of those circles; and the distance of their centres is equal to the tangent of their inclination, the semi-diameter of the primitive being radius. For let AC (ibid. n° 7.) be the diameter of a circle, whose poles are P and Q, and inclined to the plane of projection in the angle AIF; and let a, c, p, be the projections of the points A, C, P; also let HaE be the projected oblique circle, whose centre is q. Now when the plane of projection becomes the primitive circle, whose pole is I; then is Ip = tangent of half the angle AIF, or of half the arch AF; and Ig = tangent of AF, or of the angle FHa = AIF. 6. If through any given point in the primitive circle, an oblique circle be described; then the centres of all other oblique circles passing through that point, will be in a right line drawn through the centre of the first oblique circle at right angles to a line passing through that centre, the given point, and the centre of the primitive. Thus let GACE (ibid. n° 8.) be the primitive circle, ADEI a great circle described through D, its centre being B. HK is a right line drawn through B perpendicular to a right line, Cl, passing through D, B, and the centre of the primitive circle. Then the centres of all other great circles, as FDG, passing through D, will fall into the line HK. 7. Equal arches of any two great circles of the sphere, will be intercepted between two other circles drawn on the sphere through the remotest poles of those great circles. For let PBEA. PROJECTURE PBEA (ibid. n° 9.) be a sphere, whereon AGB, CFD, are two great circles, whose remotest poles are E, P; and through these poles let the great circle PBEC, and the small circle PGE, be drawn, intersecting the great circles AGB, CFD, in the points B, G, and D, F. Then are the intercepted arches BG, and DF equal to one another. 8. If lines be drawn from the projected pole of any great circle, cutting the peripheries of the projected circle and plane of projection, the intercepted arches of those circumferences are equal; that is, the arch GB = f/d, (ibid.) 9. The radius of any small circle, whose plane is perpendicular to that of the primitive circle, is equal to the tangent of that lesser circle's distance from its pole; and the secant of that distance, is equal to the distance of the centres of the primitive and lesser circle. For let P (ibid. n° 10.) be the pole, and AB the diameter of a lesser circle, its plane being perpendicular to that of the primitive circle, whose centre is C; then d being the centre of the projected lesser circle, da is equal to the tangent of the arch PA, and dC = secant of PA.