113. By one or other of these principles, the images of any definite right lines, and therefore of any rectilinear figure, may be obtained. For one or more original lines may be always assumed as passing through one or more points, the images of which are required; so that the indefinite images of the assumed lines will give those of the points sought, by its intersections with the images of other lines, in which those points lie: and these assumed lines may be so taken as to define the images with more precision, or to obviate the necessity of drawing radials of lines but little inclined to the intersecting lines of the planes in which they lie. Z R is the perpendicular distance of the point in which the auxiliary radial cuts the original plane from its intersecting line; R therefore is the centre of the circular section of the conical surface before alluded to (102). Make z R' in z v' equal to zR: draw vs to make at V, with v B, the complement of the angle at which the faces of the tetrahedron are inclined to each other. From R', as a centre, with RS for a radius, describe a circle. Draw lines to touch this circle, parallel; respectively to A B, BD, A D. Through the point p, in which the tangent parallel to AB cuts Y Z, and through P1, the vanishing point of AB, draw P, P, the vanishing line of the face of the solid meeting the face ABD in AB: and on the same principles P2 Po, P, P, the vanishing lines of the two remaining faces are found; then PP, P, in which these vanishing lines intersect each other, will be the vanishing points (94) of the edges of the solid, and lines accordingly drawn from a,b,d to these points will complete the image of the tetrahedron. 114. Simple as is the construction above described, for finding the vanishing lines of planes making any proposed angle with a given plane, it may frequently be avoided by availing ourselves properly of the symmetry of the solid to be delineated. Thus, in the example before us, after finding the image, a, b, d, of one face of the tetrahedron, we might have determined the image of the centre of that face by drawing those of the perpendiculars on each side of the triangle from the opposite angles; a line drawn through this centre and through would be the image of one, perpendicular to the plane of the triangle (97); this line would pass through the vertex of the pyramid, or through the angular point in which the other three faces meet, by finding the image of this point, which can be easily done by first determining the intersecting point of the perpendicular, and the intersecting line of any plane in which it lies; then lines drawn from a, b, and d to this image e would complete the figure. 115. When a vanishing line is obtained, it is frequently requisite to determine its centre and distance, or its principal radial; this is done by the construction employed to determine the vanishing line P, P. Thus, to determine the centre, &c., of vanishing line P, P., draw a parallel to it through c, making cv" equal o v, the distance of the picture; also draw cv" perpendicular to the vanishing line for its Make c" v" equal auxiliary one, cutting the former in c" its centre. o'v", the principal radial; then v"" P, VP, VP, being drawn, they will be the radials of the three sides, a b, eb, ea, of the face of the solid, and will be found, accordingly, to make angles of 60° with each other (83). The radial v" P, will also be found equal to VP,, these lines representing one and the same line, only brought into the plane of the picture by the rotation of two different vanishing planes on their vanishing lines. 116. The perspective projection of a curve may always be found by means of the images of a sufficient number of points in the original, or by the projection of some inscribed or circumscribed polygon; if the curve be a plane one: in this case the image of a tangent to the original curve will be a tangent to the image of that curve. For if the image of the tangent meet that of the curve in more than one point, these points must be the images of points in the original curve through which the original of the tangent must pass: which is contrary to the supposition. But there are some theorems regarding the perspective projection of a circle, and constructions founded on them, which ought to be well understood by the draughtsman. 117. The rays from the circumference of a circle, obviously, form a conical surface, the section of which, by the plane of the picture, will be one of the conic sections. If the original circle, or base of the cone of rays, be parallel to the plane of the picture, the image will be a circle, the radius of which will be to that of the original in the ratio of the distance of the picture (70) to the distance of the plane of the original circle from the vertex (89). 118. If an original circle do not touch, or cut, the station line of its equal to the distance of a point a in the original from its intersecting point, and PV equal to the length of the radial of the line; then A, v being joined, P V plane, its image will be an ellipse wherever the plane of the picture may be; unless the section by the plane of the picture happen to be a subcontrary one, an exception to which we shall recur on a subsequent occasion. If the station line be a tangent to the circle, its image will be a parabola; and if that line cut the circle, the image will be the opposite branches of an hyperbola, lying on contrary sides of the vanishing line of the original plane (80). A R DF M B 119. Let KNLM be an original circle, AB being the station line (81); Draw the the image of the circle will in this instance, be an ellipse. diameter o D to the circle, perpendicular to AB; and let & be the point in C D through which the chords of the tangents from all points in a B pass, according to the well known property of the circle. Let v represent the vertex, the vertical plane being supposed to be turned round on the station line AB, till it coincide with the plane of the circle; v v' being the director perpendicular to the station line. Make DE, in D R, equal to the tangent to the circle drawn from D; bisect V E by a perpendicular, cutting A B in F; on F as a centre, with F V or FE for a radius, intersect A B in A and B, and draw lines through these points and through G; KL, MN will be the originals of the axis of the elliptic image of the given circle, wherever the plane of the picture may be assumed, and at whatever angle that plane and the vertical one be inclined to the plane of the circle. 120. If A, B be two points in A B, such that each is in the chord of the tangents from the other point produced, then, from the properties of the circle, A E, B E will be equal respectively to the tangents AN, BL, drawn from those points; and the square on A B is equal to the E therefore lies in the sum of the squares on AN, B L, or on A E, BE. circumference of a circle described on A B as a diameter. Since the angle A V B, made by the directors of AL, BN is a right angle by construction; the images of AL, BN will be perpendicular to each other, Again, and parallel, respectively, to those of the tangents AN, AM; BL, BK having the same station points with the chords K L, M N. since AL is harmonically divided in K and G, and BN in м and G, the image of KL will be bisected by that of G, and the image of MN will be also bisected by the image of a (87): hence those images being diameters to the ellipse, mutually bisecting each other, and parallel reciprocally to the tangents which are the images (86) of AN, AM, BK, B L, the images of KL, MN must be conjugate diameters, and since those diameters are perpendicular to each other, they must be the axes, 121. If v', the foot of the director v v', coincided with D, or if v v were in the auxiliary vanishing plane, the perpendicular to v E would be parallel to AB, and PQ, SR would be the originals of the axes, which accordingly would be parallel and perpendicular to the intersecting line. But in every other position of VV', with reference to the circle, these axes must be oblique to that intersecting line, while the angles they form with it will vary according to the distance of v' from D, and according to the length of the director v v'. 122. The points G and E will not be common to two or more concentric circles, the originals of the axes of the elliptic projections of concentric circles will not be in the same straight lines, nor will they have the same station points, except in the case of v' and D coinciding, when the originals of the axes will be parallel and perpendicular to AB. 123. If A B touched or cut the original circle, the originals of the axcs, &c., of the parabolic or hyperbolic projections might be found on the same principles: but as these curves do not often occur in practical perspective drawing, we shall not dwell on the subject. 124. The only solids with curved surfaces that need be considered are, the cylinder, the cone, and the sphere. 125. If a line be conceived to pass through the vertex, parallel to the axis of a cylinder, whether right or oblique, two planes passing through this parallel will touch the cylinder in two lines of its surface, also parallel to its axis, which will be the originals of the straight outline of the perspective projection, or image, of that cylinder. 126. These two tangential planes will cut the plane of the base of the solid, or that of any section of it whatsover, in two lines, which will be tangents to the curve of that section. And the parallel to the axis through the vertex is obviously the radial of that axis, which, by its intersection with the plane of the picture, will determine the vanishing point of that axis; and this vanishing point, remembered, is the image of the point, in any original plane, cutting the cylinder in which the two tangents to the curve of the section in that plane meet, which have been shown to be the originals of the outline of the solid. must be AV will cut YP in a, the image of the point a. For however the lines YA, PV may be drawn, the triangles YAa, avP will be similar; the antecedents YA, PV being constant, the consequents ya, ar must be so likewise. When this principle is applied, the two parallels may be so taken that the line VA joining their extremities may cut YP at nearly right angles, and so define the point with precision. 127. If therefore the image of the base or of any section of the cylinder by a plane be obtained, lines drawn tangents to this image the section will be the intersections with that plane of two others through the vanishing point of the axis will give the straight parts of passing through the vertex and tangential to the surface of the solid, the outline of the solid; these outlines must also be tangents to every and these two planes will touch the cone in lines which will be the other curve which is the image of any section of the original cylinder. originals of the outline of its image. 128. If a line pass through the vertex and the apex of a cone, and 129. The ray just mentioned passing through the apex of a cone is meet the plane of its base, or any other plane cutting the cone, two analogous to the radial of a cylinder passing through the vertex, the lines drawn through the point of intersection tangents to the curve of cylinder being considered as a cone, with its apex infinitely distant. N 130. If the line through the vertex and the apex of a cone, or the the square circumscribing the circular base, and the inscribed ellipse ray of that apex, be parallel to the plane of its base, or of any section, that of the circle itself, or this ellipse is the outline of the sphere. the tangents to the base lying in its plane, or in that of such section, 135. If the distance of the vertex (70) be supposed to be indefinitely must be drawn parallel to that ray, and the image of the apex will be great, compared to the magnitude of the object to be represented, the the vanishing point of these parallel tangents. pyramid of rays may be conceived to become a prism, or the rays to be 131. Let c be the centre of the picture; ab, bisected in e, being parallel. On this supposition the vanishing points of the lines of the given as the image of a diameter, parallel to the plane of the picture, original object would be indefinitely distant from the centre of the of a sphere* e, therefore being the image of its centre (88). Draw an picture, and the images of parallel original lines would be parallels. indefinite line through c and e, and cv perpendicular to it, equal to The isometric projection of a parallelopiped (57) is obviously a limited the assumed distance of the picture; take any point e at pleasure in case of this kind, the limitation being necessary from the object in ec, but as far from e as convenient; draw a B through e perpendicular view, which induces us to adopt that kind of projection. But there to ce, making e a, eß, equal to ea, eb. Join ve and set off its length are occasions on which it is desirable to delineate rectilinear objects each way from e to l and m along a line perpendicular to ce. pictorially, which from their small relative size, and from other con132. By this construction l m is a vanishing line, of which e is the siderations, do not require the application of perspective projection, centre, ve equal to its principal radial, and ve its auxiliary vanishing and which would not be adequately represented by an isometric one. line (95); I and m will obviously be the vanishing points of the In such cases the draughtsman may readily accomplish his purpose by diagonals of every square, lying in original planes having Im for their combining the principles of projection on co-ordinate planes with pervanishing line, the sides of that square being parallel and perpendicular spective, as in the following example. to 'the intersecting line of its plane ; accordingly the quadrilateral fghi is the image of such a square, lying in such a plane, and the line a B being made equal to the given image of a diameter of the sphere, a B and ab are the images of equal original lines parallel to the picture and equally distant from it, or both lying in a plane parallel to that of the picture. If therefore an ellipse be described in f ghi, touching the sides in the points a Bgd, and having its transverse axis in ce, this ellipse will be the image of an original circle equal to a great one of the sphere, and having its plane parallel to that passing through the vertex and the centre of the sphere, or this original circle may be regarded as the oblique plan, on a plane parallel to it, of the section of the sphere by the vanishing plane, the projecting lines being parallel to the plane of the picturc. 133. Draw v n perpendicular to ve, cutting ec in n, and through n draw a vanishing line perpendieular to en, or having en also for its auxiliary vanishing line; make no, np, each equal to the auxiliary radial vn; make er, e 8,t in lm, each equal to the semi-conjugate axis of the ellipse last drawn, and complete the trapezium w x y z as the image of a square having op for its vanishing line, and its sides 136. Let a hexagonal figure, abcdefg, be drawn, with the condition parallel and perpendicular to the intersecting line of its plane. An that each pair of opposite sides shall be parallel, and consequently ellipse described in w x y z, having its transverse axis in en, will be the equal; from the angles a, c, f draw lines parallel to the alternate sides, outline of the sphere. 134. For n being the auxiliary vanishing point of the plane of the lines parallel to the remaining sides respectively, and meeting in h. and meeting in a point d, and from the intermediate angles b, e,g draw original of fghi, o p is the vanishing line of all planes perpendicular to The figure thus formed will be the orthographic or orthogonal projection that original plane, and intersecting it in lines parallel to the plane of of a cube, under certain unknown conditions of inclination of the plane the picture. The original square of the quadrilateral w x y z is there. of projection to the projecting lines, and of these to the original planes fore perpendicular to the plane of the original of fghi, or to the of the solid. vanishing plane passing through the vertex and centre of the sphere. 137. The projections of the centres of each face of the cube, as q, Now it will be seen that the conjugate axis of the ellipse in fghi is may be found by drawing the diagonals, as ac, bd, and if lines be the oblique plan (59) of the chord of the tangents from the vertex to drawn through the centres of each pair of opposite faces, as pr, which the section of the sphere by the vanishing plane, which chord of the lines will obviously be parallel to the edges of the solid, and perpentangents must be a diameter of the small circle of the solid, consti- dicular to the planes of the faces, they will pass through the vertices tuting the original of its apparent outline; this small circle being the of right pyramids placed on each face. By making the altitude of base of the cone of rays tangential to its surface (62), and having its these pyramids, as pq, equal to half the projection of the parallel edges plane perpendicular to that of the vanishing plane passing through the bf, &c., of the solid, we obtain the remaining angles, l, m, n, o, p, r, of vertex and centre of the sphere ; w x y z is consequently the image of the solid termed a rhomboidal dodecahedron, one diagonal of each ab need not be perpendicular to the line ce ; it is shown so in the figure, face of which is one edge of the original cube. to avoid unnecessary lines; but as every diameter of the sphere which lies in a 138. By previously constructing the projection of a cube in the plane parallel to that of the picture is also parallel to that plane, ab may make ratio of the projections of any lines parallel to the edges of that cabe, manner just described, the sides of which will furnish a scale of the any angle whatever with ce. The points ra, sb, are not the same, though they cannot be distinguished in the projection of any parallelopiped may be obtained, and from this the figure. again the image of any symmetrical solid deduced. In this manner 1 the forms of crystals can be drawn with the most perfect accuracy, and a most distinct conception obtained of them and of the relative position of their planes. And by analogous constructions diagrams of the theorems of solid geometry may be drawn, which would greatly facilitate the study of analytical geometry. 139. It has been stated that perspective projection is principally employed to furnish a pictorial outline of a building, machine, &c., or to convey an idea of an object of that description to the spectator, but to do this the perspective outline must excite in his mind the ideas of the real forms of that object in their relative situations, such as would be excited by the object itself, when viewed from a given point. But there are limitations to the apparent forms of objects, arising from the structure of the eye and the laws of vision, which the draughtsman must never lose sight of, when he practically applies the purely geometrical principles we have deduced, or otherwise he may produce an accurate projection of an object which would be perfectly unintelligible to an ordinary spectator; as the outline of the sphere, deduced in the preceding example, would be to an uninitiated eye. 140. Since the eye can only embrace at one time a very limited field of view, in order to see the whole of an object without changing the place of the eye, the spectator must not be nearer to it than a certain distance, for otherwise he would have to turn his head to see the successive parts, and at each such change of position the apparent forms of those parts just escaping from his view would undergo a considerable modification, arising from the structure of the eye itself. Few persons are aware of these modifications, owing to the effects of habit and the result of the judgment, which induce us unconsciously to assign the real and constant forms we know the parts of the object to possess to the apparent forms under which those parts are seen. Indeed it requires a considerable degree of abstraction and education of the eye to make the mind cognisant of the fact, that it is never the real form of an object that presents itself, a truth familiar to artists, who know that persons when first attempting to draw an object before them by eye, invariably draw it as they know it to be, and not as they really see it. 141. We have stated that the perspective projection of an object is rarely viewed from the precise point from which alone it ought to be viewed, so that the forms in the projection may suggest the ideas of the original forms whence they were deduced; consequently the outline should not in any part deviate greatly from what we may call the average form under which the true one would present itself to the eye. To effect this accordance the draughtsman must assume his point of view, or vertex, at such a proportional distance from the object itself, or from the imaginary model of it, that the rays from the points of it farthest apart, may not contain an angle greater than 60° at most, and, if circumstances allow of it, of not more than 45°. In short the pyramid of rays from an object to the vertex should be included within a cone the angle at the apex of which is not greater than that above named. 142. The distance of the vertex from the object being determined from these considerations, and its position with respect to the various parts of the original object decided on, by the conditions of the kind of view of that object it is proposed to delineate, the position of the plane of the picture should, generally speaking, be perpendicular to the axis of the cone or pyramid of rays before alluded to; but the following principles must determine more accurately its situation. offend the eye as being at variance with daily experience, and still more would any attempt to draw on a plane the apparent curvature of the lines in question be reprehended as being contrary to the verdict of the judgment, which decides that the originals, being straight lines, ought not to be represented by curves. 146. The draughtsman, consequently, must never assume his plane of the picture parallel to the longest side of a building, &c., however much he may be tempted to do so from the facility of making his constructions under this condition, when the projections of such a side would subtend at the vertex an angle of more than 15° or 20°. 147. Keeping these conditions in view, the draughtsman may assume the distance of his picture, or its distance from the vertex, entirely according to his own convenience, since it is only the absolute magnitude of the image or projection which is altered by the different distances of the picture, the figure of the image being similar on all parallel planes, as long as the vertex and object remain the same. For the sake of facility of construction, he will generally assume his plane of the picture as coinciding with some principal vertical line of the object or model. 148. The shadow of any object is obviously the projection of it on a surface, by converging on parallel lines or rays, according as the luminary is supposed to be at a finite or at an infinite distance, as the sun may be considered to be as regards terrestrial objects. When, therefore, we have obtained the projection of an object by the principles just explained, they will also enable us to obtain the projection of its shadow on one or more planes or surfaces, as supposed to be cast by an artificial light or by the sun; the problem being simply to deter mine the projection of the intersection of a pyramid or prism of rays passing from a given or assumed point through the points of a projected object. 149. If the object be perspectively projected, and the luminary be the sun, the vanishing point of the parallel rays, whose direction must be given or assumed, will represent the sun, since that vanishing point is the image of a point infinitely distant. 150. Although our power of forming correct conceptions of the true form of an object, as derived from a projection or pictorial representation of it, is much increased by the addition of light and shade, and of shadows of the object correctly projected by rules identical with those by which its outline was obtained, yet as soon as we thus approach the domain of a higher art, that of painting, the mathematical precision of the shadows we should obtain by our rules must yield to more important considerations connected with the art alluded to. Hence it is that the draughtsman seldom applies the geometrical principles for finding the true shadows of the engine, building, or analogous object, the outline of which he has delineated; for at an early stage of his practice in drawing he ought to have acquired sufficient knowledge of art to be able to add to his outline the effect of light and shade without any gross violation of truth of nature, and with a better pictorial effect than he could ensure by geometrical rules. We shall consequently only give two simple examples relating to the projection of shadows, rather as affording additional illustrations of the prin ciples of projections, than for any practical utility as regards the specific subject of shadows. 151. Let the line cs, es, passing through the centre c, c, of a sphere, be given as the direction of the solar rays; it is proposed to determine the shadow of that sphere on the given plane L M n. It is obvious that the problem is to determine the section of the right cylindrical surface, formed by the system of parallel rays, which are tangential to the spherical surface, by the plane LMn; and that the great circle of the sphere passing through the points in which these rays touch it will be the base of the cylinder, and the boundary between the illuminated hemisphere and that in shadow. 143. From the frequency of their occurrence under circumstances 144. If a spectator stand opposite two or more long horizontal parallel lines, as those of the façade of a long building, or of a garden wall, for example, he very palpably perceives the apparent convergence of these parallels in both directions, as they recede from him to the right and left; on reflection, he is therefore convinced that the apparent form of the really parallel straight lines are curves, produced by the varying angles under which the equal ordinates between the parallels are seen, as they become more and more distant from the eye. 145. The parallel projections of such long horizontal lines, which would result from the plane of the picture being assumed parallel to the originals, would reassume their natural apparent curvature, if viewed from the correct vertex; but if not, their parallelism would 153. For the plane of the great circle, of which ADBE is the projection, is obviously by the construction perpendicular to the given ray, and the plane of this circle is cut by the projecting plane of the given ray or in the original of AB, while the diameter D E is the projection of the intersection with the plane of the same great circle, by a plane passing through the given ray cs, cs, and perpendicular to the plan 155. It is clear that in this example the two elliptic outlines of the shadows of the sphere on the co-ordinate planes, must cut y z in two common points; because the segments of the ellipse on either side of YZ of each outline is the projection on the one co-ordinate plane of that portion of the cylinder of rays which forms on the other co-ordinate plane the portion of the outline of the shadow on the same side of Y Z. L, perpendicular to Yz, is the trace of the elevation projecting plane of cs, cs; o, o, is the point in which this same plane cuts the trace of the given plane, consequently Lo is the plan of the intersection of those two planes, and r', in which this line is cut by the plan of the ray cs is the intersection of that ray, and the given plane; the elevation t' of the same intersection may be obtained by applying the same constructions to the other traces and projections. 156. The two pair of parallel planes, which are respectively perpendicular to the co-ordinate planes, and therefore to each other, and which are parallel to the given ray, touch the sphere in the points a, a; B,b; D, d; E, c. These four planes will be cut by the plane L M n in a parallelogram, the sides of the projections of which must be parallel to those of the ray c s, c, 8, and to the lines Lo, wn. Draw t' perpendicular to ot, and make t' equal to tL; join oL', which will represent the intersection of the projecting plane with LMn; draw lines through d', e', parallel to c's, and from the points in which these parallels cut o' draw parallels to Lt to cut ot; again lines drawn through these last intersections parallel to wn will be the two sides of the elevation of the rectangle above mentioned; the parallel tangents at a and b will complete the figure; and ot, wn, will cut the opposite sides in the points in which the elliptic outline of the shadow of the sphere will touch those sides, or the points which represent the shadows of d, e, a, and b. 157. The plan of this parallelogram may be determined in the same manner, or by the other constructions explained for determining the projections on the other co-ordinate plane from those already determined on the first, and which are sufficiently indicated in the figure to render further description of them unnecessary. 158. If L represent a luminous body, and P a point, then, by P ※し imagining a plane to pass through them, the intersection of that plane with the plane on which the shadow is cast will cut the ray LP in Q, the shadow of the point. To determine this intersection, we have only to draw two parallel lines through L and P, in any direction, and deter L mine the points and p, or l', p', in which these parallels meet the plane of the shadow: then lp, LP being drawn, they will cut each other in 9, the shadow of the point. This is the principle employed in the following construction. 159. Let abcdefg be the perspective projection of a cube, c being the centre of the picture, cv the distance of the picture, P, X the vanishing line of the face abcd, and y z its intersecting line; while Y'z' is that of the face efg, parallel to the former. Let Y Z and w z be given as the vanishing and intersecting lines of a plane, on which the shadow of the cube, as cast by the luminous body given in position, is to be determined. 160. xz, xw, being drawn, will represent the lines in which the plane of the shadow intersects those of the parallel faces of the solid (94). If we suppose planes parallel to that of the picture to pass through the various points of the cube, as a, these will intersect the two original planes in lines, as a a, a a', parallel to Yz, zw; and a line, a a', through the point of the cube, parallel to the auxiliary vanishing line, will meet a a' in the point a', which will be the oblique projection of the point a on the plane of the shadow. Therefore, by drawing lines through the points a, b, c, d, parallel to yz, to cut x z in a, B,.. 8, then lines parallel to w z, through a, 8, ... 8, will cut lines parallel to w c', drawn through a, b, c, d, in the oblique projections of those points on the plane of the shadow, and by referring e, f, g ... to wx, in the same way, we obtain the oblique projections of the other angles of the cube. ་ 161. Since the sides of the cube a b, cd, ef, &c., are parallels, their oblique projections will be parallels (59), consequently the images of these parallels a' b', c' d', e' f', &c., will have a common vanishing point P' in the vanishing line of the plane in which the oblique projections lie; for the same reason, a'd', b'c, fg, &c., will have a common vanishing point P', in Y X. Now it is obvious that the vanishing points P1, P2, are, by an extension of the principle, the oblique projections on the plane of the shadow of the vanishing points P, P, of the original sides of the cube; consequently the former may be determined from the last-named vanishing points by simply drawing lines through them parallel to w c to cut YX in P',, P2. 2 * 162. If I had been given as the image of the point in which a line through the luminary perpendicular to the plane Y z met that plane, the image of the luminary would be determined by making k*, drawn to the auxiliary vanishing point Q, the image of the given perpendicular height of the luminary above the original plane. A line through parallel to wo will meet c'k produced in 7, the oblique projection of the luminary on the plane Yz.* Its oblique projection l' on the plane of the shadow may be either determined as those of a, b, c, d, &c., were, or by drawing a line, as al, at pleasure, to cut the vanishing line X P in some vanishing point; this vanishing point may be transferred to XY by a parallel to ow; then a line drawn through a', the oblique That is to say, a' is the perspective image of the oblique projection of the original point of which a is the perspective image. projection of a, to this transferred vanishing point, will cut * l pro- 163. Draw lines through l' and through the oblique projection duced in 1', the oblique projection of the luminary on the plane of the a', u', c, d, e, &c., intersecting each such line by the luminous ray shadow. &c., in the shadows a", 6", c", &c., of the angles of the cube, *a, *b, * C, a and these points being joined, the figure thus produced will be the | the table-land of the Desaguadero, is made of enormous masses of image of the shadow of the cube on the plane as proposed. porphyry, and it is still nearly perfect in several parts of the Montaña. 164. The oblique projecting lines a a', &c., were assumed parallel to Humboldt obtained an ancient Peruvian cutting instrument, which the picture and its auxiliary plane, simply for facility of construction, had been found in a mine not far from Cuzco : the material consisted or else the points a, b, c, e, f, &c., might, as well as the luminary, have of 94 parts of copper and 6 of tin, a composition which rendered it been projected on the plane of the shadow by lines in any direction, hard enough to be used nearly like steel. With instruments made of provided these lines were parallels according to the above principle this material the Peruvians cut the enormous masses of which their (158). buildings were composed. Some of the buildings near Cuzco contain 165. Draw lines through * and Þy, Ps, and Q, the vanishing points of stones 40 feet long, 20 feet wide, and nearly 7 feet thick. These the sides of the cube ; then the images, as a"", c" d", "f", of the stones are fitted together with great skill, and, as it was supposed, shadows of the parallel lines of the original solid will meet in a point without cement. But Humboldt discovered in some ruins a thin layer in the corresponding line drawn through * and the vanishing point of of cement, consisting of gravel and an argillaceous earth; in other those originals. For the planes passing through * and the parallels edifices, he says, it is composed of bitumen. These stones are all ab,cd, ef, &c., must intersect in a common line, passing through parallelopipedons, and worked with such exactness that it would be and parallel to those originals : this common intersection will there impossible to perceive the joinings if their exterior surface were quite fore have the same vanishing point, P, as those originals; the line * P, level ; but being a little convex, the junctures form slight depressions, represents that common intersection. Now the shadows a" l", c"d", e"für which constitute the only exterior ornament of the buildings. The are the intersections of the before-mentioned planes by another, namely, doors of the buildings are from 7 to 84 feet high. The sides of the by the plane of the shadow, and these intersections, a"b", "d", &c., doors are not parallel, but approach each other towards the top. The must meet each other in a point in *P, the common intersection of the niches, of which several occur in the inner side of the walls, have the planes passing through * and the original lines. The same reasoning form of the doors. applies to the other shadows of the corresponding parallel sides of The walls of Cuzco are formed of huge polygonal blocks of limestone, the cube. from 8 to 10 feet in length, and the same in width, and admirably PERSULPHOHYDROCYANIC ACID. Synonymous with hydro- fitted to each other without cement, precisely like those shown in the persulphocyanic acid. [CYANOGEN.) cut of the Walls of Epirus in PELASGIAN ARCHITECTURE ; but it is PERSULPHOMOLYBDENIC ACID. [MOLYBDENUM, Sulphides of.] remarkable that not only is the masonry almost perfect of its kind, PERTURBATIONS. (GRAVITATION.] but that the walls are planned as fortifications with a degree of skill PERTUSSIS. (HOOPING-Cough.] that excites the highest admiration of the military engineer. PERUVIAN ARCHITECTURE. Remains of ancient Peruvian The oldest known building in Peru is that called the house of Manco buildings are dispersed over the western parts of South America, from Capac, which stands on an island in the lake of Titicaca. It is built of the equator to 15o S. lat., especially over the Montaña. Nothing rather small irregular polygonal blocks of stones; is curvilinear in plan certain is known of their date, but the oldest is attributed to Manco and has small rude towers. The interior is divided into small square Capac, the traditional founder of ancient Peruvian civilisation, who rooms, which are lighted only from the doorways. Not far from it is is said to have flourished about three centuries before the conquest of a later and less rude building of two stories, known as the House Peru by the Spaniards, or 1200 A.D. These remains are characterised of the Virgins of the Sun. It is nearly square in plan, and is by simplicity, symmetry, and solidity. There are no columns, pilasters, divided into twelve small square rooms on each story, those on or arches, and the buildings exhibit a singular uniformity and a com- the ground-foor being lighted by the doorways, and those above plete want of all exterior ornaments. The structures, whether mere mostly by very narrow windows, but some are without any opening walls or buildings, are all of stone; the blocks in some instances being or light. squared and laid in horizontal courses, in others consisting of huge The most extensive Peruvian buildings occur in the table-land of polygonal masses ; while in both the doorways and openings for Cuzco, which was the most ancient seat of the monarchy of the Incas. light are formed of jambs inclined towards the top, and covered with a There are also ancient remains within the boundaries of the present large stone as a lintel; being, in fact, in all respects almost the exact republic of Ecuador. Near the ridge called Chisinche, not far from counterpart of the Pelasgian masonry of Greece and Italy (PELASGIAN the volcano Cotopaxi, are the ruins of a large building called the ARCHITECTURE]—a circumstance the more remarkable since, if Peruvian Palace of the Incas. It was a square, of which each side was about tradition be not altogether at fault, some eighteen centuries must have 30 yards long, and it had four doors. The interior was divided into elapsed between the latest example of Pelasgic construction in Europe eight apartments, three of which are still in tolerable preservation. and the earliest in Peru. Not far from the mountain-pass of Assuay is a building called IngapThe great road of the Incas, which runs from Quito to Cuzco and pilca, or the Fortress of Cañar, consisting of a wall of very large stones, |