PROJECTION

, in Mechanics, the act of giving a projectile its motion.

If the direction of the force, by which the projectile is put in motion, be perpendicular to the horizon, the Projection is said to be perpendicular; if parallel to the apparent horizon, it is said to be an horizontal Projection; and if it make an oblique angle with the horizon, the Projection is oblique. In all cases the angle which the line of direction makes with the horizontal line, is called the angle of Elevation of the projectile, or of Depression when the line of direction points below the horizontal line.

, in Perspective, denotes the appearance or representation of an object on the perspective plane. So, the Projection of a point, is a point, where the optic ray passes from the objective point through the plane to the eye; or it is the point where the plane cuts the optic ray. — And hence it is easy to conceive what is meant by the projection of a line, a plane, or a solid.

Projection of the Sphere in Plano, is a representation of the several points or places of the surface of the sphere, and of the circles described upon it, upon a transparent plane placed between the eye and the sphere, or such as they appear to the eye placed at a given distance. For the laws of this Projection, see PERSPECTIVF; the Projection of the sphere being only a particular case of perspective.

The chief use of the Projection of the sphere, is in the construction of planispheres, maps, and charts; which are said to be of this or that Projection, according to the several situations of the eye, and the perspective plane, with regard to the meridians, parallels, and other points or places to be represented.

The most usual Projection of maps of the world, is that on the plane of the meridian, which exhibits a right sphere; the first meridian being the horizon. The next is that on the plane of the equator, which has the pole in the centre, and the meridians the radii of a circle, &c; and this represents a parallel sphere. See Map.—The primitive circle is that great circle.

The Projection of the sphere is usually divided into Orthographic and Stereographic; to which may be added Gnomonic.

Orthographic Projection, is that in which the surface of the sphere is drawn upon a plane, cutting it in the middle; the eye being placed at an infinite distance vertically to one of the hemispheres. And

Stereographic Projection of the sphere, is that in which the surface and circles of the sphere are drawn upon the plane of a great circle, the eye being in the pole of that circle.

Gnomonical Projection of the Sphere, is that in which the surface of the sphere is drawn upon a plane without side of it, commonly touching it, the eye being at the centre of the sphere. See Ggnomonical Projection.

1. The rays coming from the eye, being at an infinite distance, and making the Projection, are parallel to each other, and perpendicular to the plane of Projection.

2. A right line perpendicular to the plane of Projection, is projected into a point, where that line meets the said plane.

3. A right line, as AB, or CD, not perpendicular, but either parallel or oblique to the plane of the Projection, is projected into a right line, as EF or GH, and is always comprehended between the extreme perpendiculars AE and BF, or CG and DH.

4. The Projection of the right line AB is the greatest, when AB is parallel to the plane of the Projection.

5. Hence it is evident, that a line parallel to the plane of the Projection, is projected into a right line equal to itself; but a line that is oblique to the plane of Projection, is projected into one that is less than itself.

6. A plane surface, as ACBD, perpendicular to the plane of the Projection, is projected into the right line, as AB, in which it cuts that plane —Hence it is evident, that the circle ACBD perpendicular to the plane of Projection, passing through its centre, is pro jected into that diameter AB in which it cuts the plane of the Projection. Also any arch as Cc is projected into Oo, equal to ca, the right sine of that arch; and the complemental arc cB is projected into oB, the versed sine of the same arc cB.

7. A circle parallel to the plane of the Projection, is projected into a circle equal to itself, having its centre the same with the centre of the Projection, and its radius equal to the cosine of its distance from the plane. And a circle oblique to the plane of the Projection, is projected into an ellipsis, whose greater axis is equal to the diameter of the circle, and its less axis equal to double the cosine of the obliquity of the circle, to a radius equal to half the greater axis.

1. In this Projection a right circle, or one perpendicular to the plane of Projection, and passing through the eye, is projected into a line of half tangents.

2. The Projection of all other circles, not passing through the projecting point, whether parallel or oblique, are projected into circles.|

Thus, let ACEDB represent a sphere, cut by a plane RS, passing through the centre I, perpendicular 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, of which the triangle AEB is a section through the vertex E, and diameter of the base AB: then will the figure agbf, which is the Projection of the circle AGB, be itself a circle. Hence, the middle of the projected diameter is the centre of the projected circle, whether it be a great circle or a small one: Also 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 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 cut the sphere HFEG through its centre I; and let ABC be any oblique great circle, whose diameter AC is projected into 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 arcs AHP, KHP; and the angles aEc, kEl, are the angles at the eye, subtended by their projected diameters, ac and kl. Then is the angle aEc measured by the arc AHP, and the angle kEl measured by the arc KHP; and those angles are bisected by EP.

3. Any point of a sphere is projected at such a distance from the centre of Projection, as is equal to the tangent of half the arc intercepted between that point and the pole opposite to the eye, the semidiameter of the sphere being radius. Thus, let CbEB be a great circle of the sphere, whose centre is c, GH the plane of Projection cutting the diameter of the sphere in b and B; also E and C the poles of the section by that plane; and a the projection of A. Then ca is equal the tangent of half the arc AC, as is evident by drawing CF = the tangent of half that arc, 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 be two circles on a sphere intersecting in A; E the projecting point; and RS the plane of Projection, in which the point A is projected in a, in the line IC, the diameter of the circle ACE. Also let DH and FA be tangents to the circles ACE and ABL. Then will the projected angle daf be equal to the spherical 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 semidiameter of the primitive being radius. For let AC 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 Iq = 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, and perpendicular to a line passing through that centre, the given point, and the centre of the pri- mitive circle. Thus, let GACE 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 CI passing through D and B and the centre of the primitive circle. Then the centres of all other great circles, as FDG, passing through D, will fall in the line HK.

7. Equal arcs 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 be a sphere, on which AGB and CFD are two great circles, whose remotest poles are E and P; and through these poles let the great circle PBEC and the small circle PGE be drawn, cutting the great circles AGB and CFD in the points B, G, D, F. Then are the intercepted arcs BG and DF equal to one another.

8. If lines be drawn from the projected pole of any great circle, cutting the peripherics of the projected circle and plane of Projection; the intercepted arcs of those peripherics are equal; that is, the arc BG = df.

9. The radius of any lesser 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 be the pole, and AB the diameter of a lesser circle, its plane being perpendicular to that of the primi- tive circle, whose centre is C: then d being the centre of the projected lesser circle, da is equal to the tangent of the arc PA, and dC = the secant of PA. See Stereographic Projection.

Mercator's Projection. See Mercator and Chart.

Projection of Globes, &c. See Globe, &c.

Polar Projection. See Polar.

Projection of Sbadows. See Shadow.

, or Projecture, in Building, the outjetting or prominency which the mouldings and members have, beyond the plane or naked of the wall, column, &c.

Monstrous Projection. See Anamorphosis.

PROJECTIVE Dialling, a manner of drawing the hour lines, the furniture &c of dials, by a method of projection on any kind of furface whatever, without regard to the situation of those surfaces, either as to declination, reclination, or inclination. See DIALLING.