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PERSPECTIVE. 



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Uw form* of crystal* can be drawn with the most perfect accuracy, 

 a mu*t dittinct conception obtained of them and of the relative posi 

 of their planes. And by analogous cumtnictioni diagrams of the 

 theorem* of solid geometry may be drawn, which would greatly facili- 

 tate the study of analytical geometry. 



ISO. It has been stated that perspective projection is principally 

 employed to furnish a pictorial outline of a building, machine, Ac., or 

 to convey an ides, of on object of that description to the spectator, but 

 to do this the perspective outline must excite in his mind the idea* of 

 the real forms of that object in their relative situation*, 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 geo- 

 metrical principle* 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. bince 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 con- 

 siderable 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 aa 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 out- 

 line 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 fol- 

 lowing principles must determine more accurately its situation. 



143. From the frequency of their occurrence under circumstances 

 favourable for the observation, the eye is accustomed to the apparent 

 convergence of long horizontal parallel lines, as in streets, aisles of 

 cathedrals, long avenues of trees, or walls, &c., but perpendicular 

 parallel lines are rarely if ever long enough to cause this optical effect. 

 Now we have proved that the projections of parallel lines never can 

 be parallel unless the originals are parallel to the plane of the picture ; 

 if therefore the draughtsman were to assume that plane not parallel to 

 the vertical lines of a building, &c., the convergence of the projections 

 of these lines would offend the eye of a person looking at his drawing, 

 as being at variance not only with his judgment of the real parallelism 

 of the lines in question, but even with his daily uncultured observa- 

 tion. But there i* another optical phenomenon regarding the appear- 

 ance of long parallel lines, which we must briefly allude to, because it 

 throws considerable light on the distinction between the apparent 

 forms of objects a* seen by the eye, which forms are functions of the 

 angles solely under which the original forms are seen, and the figures 

 on a plane, resulting from the section by that plane of the pyramids of 

 rays from those original forms, which sectional forms are functions of 

 the arcs subtending those angles. 



144. If a spectator stand opi>osite two or more long horizontal 

 parallel lines, as those of the facade 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 



offend the eye as being at variance with daily experience, and (till 

 more would any attempt to draw on a plane the apparent curvature of 

 the line* in question be reprehended a* being contrary to the verdict 

 of the judgment, which decides that the originals, being straight lines, 

 ought not to be represented by curve*. 



146. The draughtsman, consequently, must never assume his plane 

 of the picture parallel to the longest side of a building, lie., however 

 inurli he may be tempted to do so from the facility of making his con- 

 structions under this condition, when the projections of such a side 

 would subtend at the vertex an angle of more than IS" or 20. 



147. Keeping these conditions in view, the draughtsman may assume 

 the distance of his picture, or ita distance from the vertex, entirely 

 according to his own convenience, since it is only the absolute magni- 

 tude 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, ha will generally assume his plane 

 of the picture as coinciding with some principal vertical line of the 

 object or model. 



148. The ihadow of any object is obviously the projection of it on a 

 surface, by converging on parallel lines or rays, according as the lumi- 

 nary 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, there- 

 fore, we have obtained the projection of on 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 penpectirely 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 representa- 

 tion 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 c 8, ci, passing through the centre c, e, 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 H 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 i, M H ; 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. 



152. Draw cc', cc*, perpendicular to the projections of the ray, and 

 make them respectively equal to the distances of the centre of the 

 upbore from the co-ordinate planes; c', tf T, drawn through the points 

 in which the given ray cuts the co-ordinate planes, will represent that 

 ray brought into the co-ordinate planes by the turning round of its 

 projecting planes on its projections ; draw a b', d' if, perpendicular to 

 d T, tf i, making them equal to the diameter of the sphere ; then lines 

 drawn through a', b', parallel to C'T, will represent the two rays, 

 touching the surface of the solid and lying in the projecting plane of 

 the given ray brought into the co-ordinate plane along with that ray : 

 these lines will cut CT in Q R, the vertices of the major axis of the 

 elliptic outline of the shadow of the sphere on the co-ordinate plane. 

 The conjugate axis q r will be given by drawing lines parallel to c T 

 tangents to the projection of the sphere; for these last parallel tangents 

 will be the boundaries of the projections of the cylinder of rays. Lines 

 drawn through a', b', parallel to c c', will cut c T in the vertices A, B, 

 of the conjugate axis of the elliptic projection of the great circle, sepa- 

 rating the illuminated hemisphere from that in shade ; a diameter D E 

 to the circular projection of the sphere, drawn through e perpendicular 

 to c T, will be the major axis of this ellipse. 



163. For the plane of the great circle, of which A DDK is the pro- 

 jection, 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 CT in the original of AB, while the diameter o E is the projection of 

 the intersection with the plane of the same great circle, by a plane 

 passing through the given ray C 8, ci, and perpendicular to the plan- 



