PERSPECTIVE, a term popularly given to an application of geometrical principles, by means of which a pictorial outline of a certain class of objects may be delineated on a plane surface. Thus understood it has been defined to be the art of representing on a plane surface the outlines of objects in the same forms, positions, and relative proportions which they bear to each other in nature as seen from a given point. A perspective representation corresponds in fact to a section of a cone of rays, the apex of which is in,the eye of the observer (62, 64). In other words, if a pane of glass were interposed between us and any object or objects which we were viewing through a small hole (or sight), and we were to trace on the glass an accurate outline of the objects as they appeared on it, that outline would be a perspective representation : and it is the purpose of the art of per spective to show how this may be done if the necessary data be furnished. Perspective constitutes however only a specific case of a more general application of the geometrical principles above alluded to, which enable us to make constructions relating to geometrical solids, bearing the same relation to geometry of three dimensions that practical bears to theoretical plane geometry. In the present article these principles will be explained, and their application to perspective be shown (66 ct ae2),—specifically with reference to geometrical solids, but capable of extension to the objects delineated in pictorial repre sentations.
The analyst, in his investigations of symbolical expressions for the relations of geometrical magnitudes, refers these, according to the species of magnitudes under conaiderstion, either to co-ordinate lines on a plane, or to co-ordinate planes, assumed at pleasure in space. [Co-ountnivEs.] The draughtsman, or practical geometrician, makes his constructions on the lines and figures themselves, when they lie wholly in one plane ; and when he has to make constructions on geometrical solids, be is compelled to refer the various points, lines, and figures connected with or constituting those solids to one or more planes, to effect his object ; and from constructions on these planes he can determine the unknown quantities of the original solids by means of their projections, as they are termed, knowing the conditions under which these projections were obtained.
1. The series of points of any geometrical solid are most simply supposed to be referred to a plane by parallel right lines, passing through them perpendicular to the plane ; the intersections of these lines with the plane are termed the projections of the original points on that plane.
2. Let us conventionally designate the original points by Italic capital letters, and their projections by small letters ; thus p means a point in space, p its projection on a plane.
3. The points 1, ni, n, on a plane A Y Z, are therefore understood as referring to the points in space, situated in right lines passing through 1, no, n respectively, perpendicular to that plane ; but these projections alone do not define the relations between the original points ; for 1, no., n are each the projections of an infinite number of original points, of all in short through which each projecting line may pass. To define the specific points a, M, N, we must consequently not only have 1, no, n, but the lengths respectively of the projecting lines al, mot, we, or the distances at which a, N are respectively situated from their projections.
4. This second series of essential data is furnished by the projections t, at, sr of the original points on a second co-ordinate plane B Y z, per pendicular to the first, and therefore parallel to the former projecting lines, by which 1, no, n were determined ; while, conversely, the first plane must be parallel to the projecting lines by which L, si, x are determined. For if a third plane be conceived to pass through the two projecting lines at, al, of any point c, and therefore necessarily perpendicular to the two co-ordinate planes, the intersections of this third plane with the two latter will, together with the two projecting lines themselves, form a rectangle ; consequently the distance of any projection L, from the common intersection Y Z of the co-ordinate planes, is equal to the length of the projecting line which is parallel to it; while, conversely, the distance of the other projection 1 of the same point a from the same common intersection is equal to the length of the projecting line L a.