GNOMONIC; GLOBULAR; ORTHOGONAL, ORTHOGRAPHIC STER EOC RAPITIC ; &c. The present article is merely intended to point out the general principle of all projections, and also to note the theoretical importance of the subject.
Imagine a surface of any kind, through every point of which passes a curve the character of which depends upon that point, insomuch that, given a point of the surface, the curve which passes through that point is given in !character and position. If any second surface be taken, which is cut by all the curves emanating from the points of the first, every point of the first surface has a point corresponding to it on the second. Thus if the curve passing through A on the first surface cut the second surface in a, the point A is said to be projected on the second surface at a by means of the projecting curve A a. Similarly any line on the first surface is projected into a line on the second, which last contains the projections of all the points on the first ; and the projections of the several boundaries of a figure on the first surface are boundariee of a figure on the second, which is the projection • of the first figure.
It is perhaps not usual to make so wide a definition of projection in since the only caeca which aro commonly considered are those in which the projecting lines are all straight, and either parellel to one another, as in the orthographic projection, or all peening through the none point, as in common perspective. But such a conception of projection is necessary : in :Mercator's projection, for example, [Mar) the points of a sphere are projected on a circumscribing cylinder, not by straight lines passing through a point, but either by straight lines disposed according to a complicated law, or else by curves. If a relation between any point and its be given, so that either can be found from Cho other, the passage from one to the other maybe made either on a straight line or on an infinite variety of curves; but it may happen that the law which the disposition of the projecting straight linos follows may be of a more difficult character than that which would be required if a curve, not in itself so simple as a straight tine, were substituted.
When the foundations of plane geometry were fixed, and the first principles of solid geometry were superaddel, it was natural that the eery simple idea of the perspective projection should excite attention.
In a country in which the first principles at least of drawing were practically known, the following problem must have suggested itself to 'the geometers : If through a given point lines be drawn through all ,the points of the boundary of a plane figure, until they are stopped by another plane, required the figure traced out upon the second plane. 'A straight lino was known thus to give a straight line : a moment's consideration of the circle, the only other line then considered, would show that a projection of a circle and a plane section of a cone are the came things. Hence probably the first idea of a conic section ; and thus, if the conjecture be correct, the attention was turned front that point, which would, if properly kept in view, have led to the theory of projections in place of one isolated branch of it. The properties of the Iconic sections, as deduced in the ancient manner from the cone, are neither so general nor so easy as they might be made ; and it may be confidently expected, considering the progress which the doctrine of projections has made of late years, that the method of considering the ellipse, hyperbola, and parabola as projections of the circle, will become established in elementary teaching, in preference to the detached geometrical and algebraical methods now in use.
We have already spoken of the geometry of projections [Geosternv): unfortunately there is no elementary work which gives a general view of its first principles supported by sufficient application ; and until such a work shall appear, the student must search for himself the writings of Mongo, CarnOt, Chasles, Poncelet, &c. The 'history of Geometry,' by M. Chasles, referred to in the article cited, will furnish many more references ; and the Propriates Projectives dos Figures,' by M. Poneelet, perhaps the work in which the student may most easily make an advantageous beginning of the subject. Much has been done of late years to introduce some elements into works on the conic sections : but projection has net yet been properly treated as a distinct subject.