Fig. 40. tical planes such as OO'P'P, (fig. 40) passing through the eye; construct upon the vertical plane the projection P′′X” of each of the sections, and from the point O" draw O′′P" tangent to this curve. Having the projections of the visual ray, we can find as in the preceding problem, the perspective of the point P situated upon the visible limit of the proposed object. 100. If we suppose the eye to be situated at an infinite distance from the object, so that the visual rays may be considered as parallel; having designated by a straight line the direction in which the body must be seen, nothing more remains to place the points in perspective but to draw from these points, lines parallel to the given line, and to find their intersection with the plane of delineation. It is readily seen that in this hypothesis, the apparent contour of the body is determined by tangents to its surface which are parallel to each other, the whole of which, taken together, form a cylindrical surface. To determine these tangents we choose cutting planes vertical and parallel to the given line; and the tangents to the vertical projections of the sections must be drawn parallel to the given line which marks the direction of the visual ray. This perspective has a great analogy to the orthographic projection discussed in the former part of this introductory treatise; and may be used in constructing problems of the same kind with those which were solved in the second section. There is no necessity for making these projections by perpendicular lines; in many instances the solutions would be as simple if the projections were made by oblique lines. 1 shall proceed to give some propositions which may serve as the foundation of another method of perspective which may be applied with great facility to bodies terminated by planes and straight lines. 101. If we draw through the eye a straight line parallel to a straight line situated in any manner whatever with respect to the plane of delineation, the point where this parallel meets this plane belongs to the perspective of the straight line proposed. In fact, all the lines drawn from the eye to the dif ferent points of any proposed straight line, form a plane which, by its intersection with the plane of the picture, determines the perspective of this straight line; but the line OO' (fig. 43) being parallel to the proposed line and Fig. 43. passing through the eye which we suppose in O, is situated necessarily in this plane; therefore, the point O' where it meets the plane of delineation TA, belongs to the perspective of the straight line proposed. It is evident that the point P' where the proposed straight line meets the picture itself, makes also a part of its own perspective; therefore, to trace this perspective, it is sufficient to know the points where the proposed line and a line drawn parallel to it through the eye, will meet the plane of the picture. 102. It follows from the preceding article, that the perspectives of any number of lines parallel to each other, will all meet in a single point in the plane of the picture; this point is called in treatises on perspective, their vanishing point. Only one line, indeed, can be drawn. through the eye, which will be parallel to all these others; their perspectives must therefore pass through the point where this line meets the plane of the picture. 103. If the proposed lines were at the same time parallel to the plane of the picture, the straight line OO' drawn through the eye, would not meet the picture ; and consequently their perspectives would be parallel among themselves. We may be convinced, a priori, of the truth of this proposition, by the following reasoning: The two proposed straight lines being parallel, the planes formed by the assemblage of rays drawn from the eye to different points of these lines, and containing their perspectives, have necessarily their intersection parallel to these same lines, (El. 214) and consequently, to the plane of the picture. The perspectives can meet only in the points common to the intersection and to the plane of the picture; they will therefore be parallel to each other. Hence we derive a very simple method of putting lines and points in perspective. We draw a perpendicular OO" from the eye to the plane of delineation (fig. 44); the point O" where it meets Fig. 44. plane is called the centre of the picture, sometimes call Fig. 44. ed the point of sight. It follows from what has been said, that the perspectives of all lines perpendicular to the plane of the picture, must meet in this point. We project the proposed point P upon the plane of the picture, which we suppose to be vertical; the point P′′ where this projection falls, will be that in which the perpendicular drawn from the proposed point to the plane of the picture, meets it; and P′′O" will be the perspective of this line. Next draw P'M, making with AB an angle equal to half a right angle; this will be the projection of a horizontal line drawn from the point P to the plane of the picture and making the same angle with it; and its meeting with this plane will be at the point M" placed at a height MM" equal to P'P. But if we take, upon O′′D” parallel to AB, a magnitude O"D" equal to the distance OO" of the eye from the picture, it is evident that the line OD" will be parallel to all the horizontal lines which can be drawn to the picture, at an inclination of 45°, in the direction MP'; consequently the perspectives of these lines must all meet in the point D", which is called the point of distance. Having drawn M"D", this straight line must contain the perspective of the point P; but this perspective must also be found under O"P"; it is therefore in R". From the three preceding articles we derive the following rules. (1). When an original straight line is parallel to the base of the picture, the perspective of this line is also parallel to the base of the picture (103). (2). The perspectives of all straight lines perpendicular to the picture, are directed towards the centre of the picture (102). (3). The perspectives of all horizontal straight lines which make with the picture an angle of 45°, will meet in the point of distance (103). (4). The perspective of a horizontal straight line making any angle whatever with the picture, will have its vanishing point in the horizon, at the intersection with the picture of a straight line drawn parallel to the proposed line, through the eye. (5). The horizon of the picture is the vanishing line of the perspectives of all horizontal planes. (6). And by analogy-The vertical line passing through the centre of the picture, is the vanishing line of all vertical planes perpendicular to the plane of the pic ture. (7). The perspectives of planes parallel to the picture, can have no vanishing lines, and are always figures similar to their originals. 104. We can find the perspective of objects by means of the receding scale, which dispenses with tracing their ground plan, and elevation; it is constructed in the following manner. We refer the objects to three rectangular co-ordinate planes, the first horizontal, and passing through the ground line AB (fig. 45); the second vertical, perpendicular to Fig. 45. the plane of delineation, and passing through the edge BT the third, the plane of delineation itself, ABT, which we here suppose to be vertical. A point will then be given when we know its respective distances from these three planes (30). The distance of this point from the plane of the picture will be measured upon BC; its distance from the vertical plane passing through BC and BT, will be measured upon AB; and its distance from the horizontal plane, or the height of the point, will be measured upon BT. Now, the two lines AB and BT being in the plane of the picture, it is sufficient to transfer to this plane the divisions of the third line, BC; this is done by drawing to the centre of the picture the line BO" which will be the perspective of the line BC, and by drawing to the point of distance D", the straight lines a D", 1D", 2 D" &c. which will cut BO" in the points c, 1, 2, 3, &c. corresponding to the parts Bb, b1, 12, &c. of the line BC. The line BO", thus divided, is the receding scale which marks the apparent sinking of objects in the picture; and if we draw through the points of division, of this scale, lines parallel to AB, they may be considered as the ground lines of several planes drawn parallel to the plane of the picture at the depths marked by the corresponding divisions of the scale; they would contain the perspectives of horizontal projections, or the bases of objects situated in these planes. If we then take upon the straight line AB, which is called the front scale, a part Be equal to the distance of the proposed point from the vertical plane passing through Fig. 45. BC and BT, and draw to the centre of the picture the straight line e O", the meeting of this line with g 2 parallel to AB, will give the perspective of the horizontal projection, or of the base of the proposed object. Finally, if upon BT, the scale of heights, we take the part ef equal to the height of the proposed point, and draw fO", this last straight line will meet g h perpendicnlar to g 2 at the point h which will be the perspective sought. We see that this process gives, by operating directly upon the plane of delineation, the perspective of all the objects which we may wish to represent, when we have constructed the receding scale. 105. When the picture is so large as to render the construction inconvenient, we may calculate the divisions of the receding scale, by considering the similar triangles O" c D" and ac B; from which we have O"D" a B+O′′D′′ a B + O′′D′′ a B a B B c' BO" a B Bc lated by a similar proportion, since e O" ef and the h g g0" straight lines e O′′ and g O′′ are evidently to each other as the distances BO" and O′′ 2. a B O"D" The proportion will give the distance 0′′ c 105. General Remark. We have, in what precedes, the general means of putting into perspective the apparent outlines and the remarkable points of objects; but these processes, which constitute linear perspective, are by no means sufficient to give a complete representation of a body. Light and shade, and the gradations of tint, all concur to represent the prominences, depressions, and distances of objects. All these circumstances may be rigorously determined by methods analogous to those which have been given. To do this, it is only necessary to make such an analysis of the enunciation of |