Def. 4. When the system of rays are all parallel to each other, and perpendicular to the horizon, and the projection is made on a plane perpendicular to the horizon, it is then called the orthography of the object.

The terms ichnography and orthography may be illustrated thus, Let GHIK, fig. 2, be a plane parallel to the horizon, and the rays Aa, Bb, Cc, &c, from the several points of the octahedron, ABCDEF being perpendicular to, and parallel to each other, the projection abcd made by them, is called the ichnography of the object.

Again, in fig. 2, let the rays Aa, Bb, Cc, &c. be parallel to each other, and perpendicular to the plane GHLM, on which the projection is made, and let the plane GHLM be perpendicular to the plane GHIK; then will the figure abcdef be the orthography of the object ABCDEF.

These are the common definitions of the terms ichnography and orthography, but we shall use them to signify any two projections that are made by systems of parallel rays, when the rays in those systems are perpendicular to each other, and to the plane on which the projections are made, as in the present figure, without any regard to their situation with respect to the horizon.

In this kind of projection, the projection of any particular point, or line, is usually called the seat of that point, or line, on the plane of projection. Thus, is the seat of the point A on the plane GHIK, and af is the seat of line AF the plane GHLM.

Def. 5. When the projection is made by a pyramid of rays, it is the schenography.

Thus, (fig. 1) the figure abcde projected on the plane FGHI by the rays AO, BO, CO, &c. coming from the several parts of the cube ABCDE to the point O, is the schenography of the object ABCDE.

We shall, hereafter, see that this projection is the true perspective representation of the object ABCDE, as it would appear to the eye of the spec

tator at O.

It is also evident, that the shadows of objects are thus generated, when the light is supposed to come from a single point. Though in case of the light coming from the Sun or Moon, they being at so great a distance, that the rays may be practically considered as parallel to each other.

Def. 6. The point of sight is where the spectator's eye should be placed to view the picture.

This point is the same with the vortex of the optic pyramid, as is evident from Theorem 2, where we will show, that the representation of any object is no other than the schenographic representation of the object on the plane of the picture.

Def. 7. If from the point of sight a line be drawn perpendicular to the picture, the point where that line intersects the picture is termed the centre of the picture, and the distance between this centre and the point of sight is termed the distance of the picture.

Def. 8. If a plane be imagined to pass through the point of sight parallel to the picture, this plane is termed the directing plane.

Def. 9. By original object is meant, whether it be the point, line, surface, or solid, the real object in its own situation, and which is represented in the picture.

Def. 10. By original planes, we mean the plane in which any original, point, line, or plane figure is situated.

Def. 11. The point where any original line cuts the picture, is

called simply the intersection of that line, or the intersecting point of that line.

Def. 12. The line produced by an original plane cutting the picture, is simply termed the intersection of the original plane.

Def. 13. The point where any original line cuts the directing plane, is termed the directing point of that original line. And a line drawn through that directing point, and the point of sight, is termed the director of that original line.

Def. 14. The line, whereby any original plane cuts the directing plane, is termed the directing line of that criginal plane.

Def. 15. A line drawn through the point sight parallel to any original line, is termed the parallel of that original line.

Def. 16. A plane passing through the point of sight parallel to any original plane, is termed the parallel of that original plane.

Def. 17. The point where the parallel of any original line cuts the picture, is called the vanishing point of that line, and the distance between the vanishing point and the point of sight, is called the distance of that vanishing point.

Cor. 1. Hence it is clear, that original lines, which are parallel to each other, have the same vanishing point, for a line passing through the spectator's eye parallel to them all, generates their common vanishing point by this definition.

Cor. 2. Those lines which are parallel to the picture have no vanishing point. Because the lines, which should produce the vanishing points, are, in this case, themselves parallel to the picture, and, therefore, never can cut or intersect it.

Cor. 3. The lines that generate the vanishing points of any two original lines, make the same angle at the spectator's eye, as the original lines do with each other.

Def. 18. The line, whereby the parallel of any original plane cuts the picture, is termed the vanishing line of that plane.

And if from the point of sight there be drawn a line cutting that vanishing line at right angles, the point where it is so cut is called the centre of the picture. And the distance between the centre and the point of sight is called the distance of the vanishing line.

Cor. 1. Hence, original planes, that are parallel, have the same vanishing line. For a plane parallel to them all, passing through the spectator's eye, produces that vanishing line.

Cor. 2. The vanishing points of lines in parallel planes, are in the vanishing line of those planes, for the lines that generate these vanishing points, are all in the plane that produces the vanishing line.

Cor. 3. The planes which produce the vanishing lines of two original planes, have their common intersection passing through the spectator's eye, parallel to the intersection of the original planes, and are inclined to each other in the same angle as the original planes are to each other.

Cor. 4. The vanishing point of the common intersection of two planes, is the intersection of the vanishing lines of those planes.

Cor. 5. The vanishing line, and intersection of the same original

plane are parallel to each other. Because they are generated by paralÏel planes.

Cor. 6. The distance of a vanishing line is the hypothenuse of a right-angled triangle, whose base being the principal distance, .ts perpendicular is the distance between the centre of the picture and the centre of that vanishing line.

Def. 19. The representation of any object is termed the projection of that object.

In order that the stude may comprehend more fully the true import of these definitions (see fig. 3), let the plane ABC, be supposed to be the surface of the picture, to be the point of sight (Def. 6), the plane ODE to be the directing plane parallel to the picture (Def. 8), FG to be an origina line (Def. 9) in the original plane FGH (Def. 10), cutting the picture in BI, and the directing plane in DE. And let OAC be a plane parallel to the original plane FGH, and cutting the picture in the line AC: aud let the line OV be parallel to the original line FG, and cut the picture in V. And let FG cut the picture in B, and the directing plane in D. These things being supposed, B is the intersection of the original line FG (Def. 11), D its directing point (Def. 13), and V its vanishing point (Def. 17), BI is the intersec tion (Def. 12), DE the directing line (Def. 14), OAC the parallel (Def. 16), and AC the vanishing line (Def. 18) of the original plane FGH, and if OS be drawn cutting the vanishing line AC at right angles in S, the point S will be the centre of the vanishing line (Ibid), and SO will be the distance (Ibid) of the vanishing line AC.

Axiom 1. The common intersection of two planes is a straight line.

Axiom 2. If two straight lines meet in a point, or are parallel to each other, a plane may pass through them both.

Axiom 3. If three straight lines cut one another, or if two of them are parallel, and cut by the third, they will all three be in the same plane, i. e. a plane passing through any two of them, it will also pass through the third.

Ariom 4. Every point in any straight line, is in the same plane that the said line is in,

Lemma 1. If (fig. 4) AEBD and BSO be two planes, cutting each other in ASB, and from any point O of one of the planes, be drawn two lines OS and OC, cutting the line AB, and the other plane AEBD at right angles in S and C, and there be drawn CS, that line CS will be perpendicular to ASB.

This follows from Prop. II. Lib. II. Elem.

Theo. 1. A line drawn from the centre of the picture to the centre of a vanishing line, is perpendicular to that vanishing line.

Imagine AEBD (fig. 4) to be the plane of the picture, O to be the point of sight, and OSB to be a parallel plane generating the vanishing ASB by its intersection with the picture; and let S be the centre of the vanishing line, and C be the centre of the picture. Then, having drawn OS and OC, draw CS, OS is perpendicular to AS (by Def. 18), and OC is perpendicular to the plane AEBD (by Def. 7). Therefore, CS is perpendicular to AS (by Lemma 1). Q. E. D.

Theo. 2. The perspective representation, or projection of any

object, is the same as the schenographic projection of it on the plane of the picture, the point of sight being the vertex of the optic cone.

For, by the explanation of the principle in Def. 1, since the light must come to the spectator's eye at O (fig. 1), in the same direction from any point a of the projection, as it would from the corresponding point A of the original object, it is evident that the rays aÒ and AO are in the same straight line. Whence it is evident, that the projection a is the intersection of the picture, which the ray AO and the whole projection abcde is the schenographic projection of the original object ABCDE, made by the optic pyramid OABCDE, whose vertex is the point of sight O. Q. E. D.

Cor. 1. The projection of a straight line is a straight line, for the optic pyramid OAB, which generates the projection de of any line DE, is the surface of a plane triangle ODE, all the rays going to O from the several points of the line DE, being in the plane passing through the lines DO and EO. Therefore, de is the intersection of the picture with the plane triangle ODE, and, consequently, is a straight line (by Axiom 1).

Cor. 2. The original of a projection may be any object that will produce the same pyramid of rays Thus, the original of the projection de, may be any line which produces the optic pyramid ODE, as well as the line DE.

This being the case, it may be asked, and with reason, whence it comes, that objects delineated on a picture appear to be what they are only intended to represent?

The cause of it is this, the mind has a habit of judging of objects as they are related to each other, whether it respects colour, light and slade, or form, or whatever be its situation. These circumstances are requisite to form a perfect picture, the mere outline is, sometimes, almost sufficient, because it expresses the due relation of the parts; as in a pavement, where all the stones appear to be square, though they are represented by very irregular figures, it is the due relation of the parts which produces this effect, for the represen tation of any one of the single squares, would hardly appear to be square, if it was not connected with the other squares in the pavement, to bias and determine the judgment.

Theo. 3. The projection of a straight line not parallel to the picture, passes through both its intersection and vanishing point.

All that has been said being clearly understood, and the definitions particularly, let fg, fig. 3, be the projection of the original line, FG, FO, and GO, being the rays which produce the projections ƒ and g of the points F and G (vide Theo. 2).

By this Theorem 2, fg is the intersection of the picture with the plane of the triangle OFG; but the whole of the line FGB is in that plane, and, consequently, the intersection B. Therefore, fg conti nued will pass through B.

This Theorem being the principal foundation of the whole practice of Perspective, the reader will do well to make it very familiar to him. To fix it the stronger on the mind and memory, I have repeated its substance in fig. 1, where the projection be meets the original line BC in its intersection K, and passes, also, through the vanishing point V, which is generated by its parallel OV

N.B. When the original line itself passes through its own vanishing point, the who.e projection of this said line will be in this point, so that in this case, the line may be said to vanish.

Cor. 1. The projections of all original lines that are parallel to one another, but not to the picture, pass through the same vanishing point; for they have but one parallel common to them all, and, consequently, but one vanishing point. This is represented in fig. 1, where the projection da and cb of the parallel lines DA and CB meet in their common vanishing point V.

Cor. 2. But if the original lines are parallel to the picture, as well as to each other, their representations will be parallel to each other, to their originals also. For the line passing through the eye of the spectator, which in other cases generates the vanishing point by its intersection with the picture, is, in this case, parallel to it, and, therefore, produces no vanishing point. So that the representations can never meet each other, nor that line passing through the eye of the spectator. And, consequently, they are parallel to each other to that line, and to the original lines.

Cor. 3. The centre of the picture is the vanishing point of lines perpendicular to the picture (vide Def. 7, 15, and 17).

Cor. 4. And hence it appears, that the representation of plane figures, which are parallel to the picture, are similar to their originals. For (fig. 5, No. 2,) the picture being AB, the original figure parallel to it being CDEF, and the representation being edef. If you resolve the original figure into triangles, by means of diagonal lines, as DF, the representation will be resolved into triangles by corresponding diagonals df. Whence, all the lines in the representation being parallel to all the lines in the original, every triangle edf will be like the corresponding triangle EDF, and, consequently, all the lines of the figure cdef will be in the same proportion to one another as the corresponding lines in the figure CDEF.

Theo. 4. The projection of a line parallel to the picture, is parallel to the original.

(Fig. 5.) Let the plane EF be the picture, AB the original line parallel to it, and ab its projection, O being the point of sight, and OAB the optic pyramid. By Theo. 2, ab is the intersection of the picture with the plane of the triangle OAB. Therefore, AB being parallel to the plane EF, ab is parallel to AB, for they are both in the plane of the triangle OAB, and do not meet; for if they did, their common intersection would be in the plane EF; and, consequently, AB would not, in that case, be parallel to the plane EF.

Cor. 1. Several original lines being parallel amongst themselves, and also to the picture, will, also, in their projection on the picture, be parallel to each other. Thus, ab and ed are parallel to each other, and to their originals AB and CD.

Cor. 2. The projection abcd of any plane figure ABCD parallel to the picture, is similar to the original. For, having drawn the diagonal AC, and its corresponding projection ac, the sides ab, be, ac, are parallel to their corresponding originals AB, BC, AC; wherefore the angles at a, b, c, are equal to the corresponding angles at A, B, and C; and, consequently, the triangle abc is similar to the triangle ABC. For the same reason, acd is similar to ACD; and, consequently, the figure abed is similar to ABCD.

Cor. 3. In the same case, the length of any line ab in the projection, is to the length of its original AB as the distance of the picture is to the distance between the point of sight and the plane of the original figure. Let OgG be perpendicular to those two planes, cutting them in g and G. Then it will be ab: AB :: OA : Oa ;: Og : OG (by Prop, XVII. Book II. Euc. Elem.) But g is the centre, Og is the distance of the picture, and Og is the distance between the point of sight O and the original plane ABCD; wherefore, ab is