BE, these lines are corresponding dimensions of the two figures, and they are equal. Also, any two lines in one of the figures, as A'B', B'E', have the same inclination as the corresponding lines of the other figure AB, BE. COR.-The intersection of the primitive, with a plane perpendicular to it, is the projection of this plane's intersection with any other plane or surface or solid figure. PROPOSITION III. If a plane figure be inclined to the primitive, any of its dimensions parallel to the line of common section of its plane with the primitive, is equal to its projection; but any of its dimensions perpendicular to this line exceeds its projection in the ratio of radius to the cosine of obliquity. Let ON be the intersection of a plane figure with the primitive MR; A'B' a section of it by a projecting plane paral- M A D R A Then, since ON is perpendicular to the plane ODD', it is so to the lines OD, OD'; and therefore, the inclination of the plane of the figure with the primitive is measured by the angle DOD'. Now (Pr. III. 1) OD' : OD = Radius: cosine DOD'; but OD': OD = C'D': CD. Therefore C'D': CD Radius: cosine DOD'. Again, since A'B' is parallel to AB, AB' is a rectangle, and AB A'B'. = COR. 1.-The projection of a plane figure inclined to the primitive, is less than the original in the ratio of the radius to the cosine of obliquity. 1 For every line perpendicular to the line of common section exceeds its projection in this ratio; and the lines parallel to the line of common section are equal to their projections; hence the whole figure is greater than its projec tion in this ratio. ·『》 COR. 2. The projection of a circle inclined to the primi tive is an ellipse, unless it be a subcontrary section, in which case it is a circle. For, when its projection is a subcontrary section of the cylindrical projecting surface, it is a circle (Conic Sections, IV. 2). Again, when the circle is inclined, its diameter, which is parallel to the line of common section of its plane with the primitive, is projected into an equal line; and all the dimensions perpendicular to this line are projected into less lines in the ratio of radius to the cosine of obliquity. Hence the projection is an ellipse, of which the transverse axis is equal to the diameter of the circle, and its conjugate less in the preceding ratio. COR. 3.-The projection of a sphere is a circle of equal diameter. FOURTH BOOK. PERSPECTIVE. DEFINITIONS. 1. The theory of Linear Perspective treats of the method of projecting objects on a vertical plane from some given point of sight. 2. The plane of projection is also called the perspective plane, or the plane of the picture. 3. The point of sight is also called the point of view. 4. The centre of the perspective plane is also called the centre of the picture. 5. The distance of the point of view from the primitive, is called the distance of the picture. 6. A vertical plane passing through the axis of the primitive, is called the vertical plane; and a horizontal plane passing through it, is called the horizontal plane. 7. The intersection of the primitive with the horizontal plane, is called the horizontal line; and its intersection with the vertical plane is called the vertical line. 8. The intersection of the primitive with the ground plane, or that of the sensible horizon, is called the ground line. 9. The intersection of the primitive with a line from the point of view parallel to any original line, is called the vanishing point of that line. 10. Two points on the horizontal line, whose distances from the centre are equal to the distance of the picture, are called points of distance. PROPOSITION I. If the distance between the centre and the seat of any point be cut in the ratio of the distance of the picture to the distance of the point from the primitive, the point of section will be the perspective of the point. Let MR be the primitive, C its centre, S the point of sight, P' the given point, and N its seat. Then, if the point P in N For SC and NP' are parallel, and therefore in the same plane, and they are in the same plane with CN. Also, the angles C and N are equal, being right angles, and SC: CPP'N: NP; and hence the triangles SCP and P'NP are similar. Therefore the angles at P are equal, and therefore SP, PP', are in one straight line; and P is therefore the perspective of P'. PROPOSITION II. The sum of the distance of the picture, and the distance of any point from it, is to the distance of the picture, as the distance of that point from the vertical plane to the distance of its projection from the vertical line; and also as the distance of that point from the horizontal plane to the distance of its projection from the horizontal line. Let MR be the primitive, P' the given point, P its projection, and S the point of sight, and C the centre. Through P' let a plane P'C' pass parallel to MR, cutting the horizontal and vertical planes in the lines B'C' and C'A', and in that plane draw P'A' and P'B' parallel to the opposite sides of the figure P'C'; join S and the points B', P', A' and let these lines cut the primitive in B, P, A. M & Because the opposite sides of the figure P'C' are parallel, and the angle C a right angle, the figure is a rectangle; and PC being a section of the prism SA'B' by a plane parallel to the base, it is A milar triangles SC': SCA'C: AC=P'B': PB; and SC': SC=B'C': BC= P'A': PA; where P'B' and PB are the distances of the point and its perspective from the horizontal plane, and P'A', PA, their distances from the vertical plane. COR.-Let the distances of the perspective of the point from the horizontal and vertical planes be respectively v and h, and those of the point itself v' and h', the distance of the picture d, and that of the point from the primitive p, then d+p:d=v': v, and v = also, d+p: dh': h, and h= PROPOSITION III. dv' d + P dh' d + P ; To find the perspective of a point by means of a plane construction. K Let MR be the primitive, C its centre, CD the horizontal line, D the point of distance, and GR the ground line. Produce the side KR, and make RA = p, the point's distance beyond the plane, RG its M distance from the vertical plane passing through KR, and RH its distance above the ground plane. Make RBRA; and join C, R, P and D, B; through E draw EF pa- NG rallel to NR; join C, G; make P'G perpendicular to NR and RH; D 田 E join C, P'; and draw FP parallel to GP'; and P is the perspective of the point. For P' is evidently the seat of the point; and CP: PP'. CF: FG=CE: ER=CD: BR=d: p. Hence (Pr. IV. 1) P is the perspective of the point. COR. 1.-It is evident that the line EF is the locus of the perspective of all the points in the line drawn on the ground plane parallel to the ground line, and at a distance equal to RA from the primitive. COR. 2. The point F is evidently the perspective of the seat of the given point on the ground plane. Schol. The lines RN, RK, are sometimes called the scale of the front and the scale of heights, and CR is called the flying scale. M F A COR. 3.-The perspective of the point may be found more simply thus:-Let CA be the distance of the point from the horizontal plane, and AP' its distance from the vertical plane, D the point of distance, AP' parallel to DC, and CA perpendicular D P C R to it. Make P'F equal to the distance of the point from the primitive; join P', C, and D, F, then P is the perspective of P'. For P' is evidently the seat of the given point, and CP : PP' — CD : FP'; and therefore P is the perspective of the given point (Pr. IV. 1). PROPOSITION IV. The perspective of a plane figure parallel to the primitive is a similar figure, the dimensions of which are to the corresponding dimensions of the given figure in the ratio of the distance of the picture to the sum of the distance of the picture, and the distance of the point from the primitive. Let S be the point of sight, and let the figure A'B'C' be parallel to MR, then it is evident (So. Ge. II. 14, Cor. 1) that ABC is similar to A'B'C'; and that AC: A'C' = SC : SC'd:p+d. COR. 1. The projection of a straight line is a straight line, unless it be directed to the eye, in which case it is a point. The projecting surface for any straight line as A'B' is |