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light, and the remaining parts of such body are said to be in shade. But when one body stands on or before another, and intercepts the sun's rays from the latter, which is thereby deprived of the action upon it of the rays of light, the part so deprived of the immediate action of the light is said to be in shadow. It seems hardly necessary to observe, that the parts of any body nearest the source of light will be the brightest in appearance, whilst those furthest removed from it will, unless under the action of reflected light, be the darkest.
2460. It has been the practice, in architectural drawings, to represent the shadows of their objects at an angle of forty-five degrees with the horizon, as well on the elevations as on the plans. The praetice has this great convenience, namely, that the breadth of the shadow cast will then actually measure the depth of each projecting member which casts it, and the shadowed elevation may be thus made to supply a plan of the external parts of the building. Now, if in the elevation the shadows be cast at an angle of forty-five degrees, it will on a little consideration be manifest, that, being only projections of a more lengthened shadow (for those on the plan are at an angle of forty-five degrees), the actual shadow seen diagonally must be at such an angle as will make its projection equal to forty-five degrees upon the elevation ; because all elevations, sections, and plans, being themselves nothing more than projections of the objects they represent, are determined by perpendicular, horizontal, or inclined parallel lines drawn from the points which bound them to the plane of projection, and simi. larly, a shadow in vertical projection, which forms an angle of forty-five degrees with the horizon, can only be the representation on such projection of an angle, whose measure it is our business now to determine.
2461. In the cube ABCDEFGH (fig. 838.) the line BD, forming an angle of forty-five degrees with the horizon, is a projection or representation of the diagonal BH on the vertical plane ABD; and our object being to find the actual angle AHB, whereof the angle ADB is the projection, we have the following method. Let each side of the cube, for example,
Fig. 838. =10. Then (by 907.) AD? + DH?= AH2,
That is, 10 x 10+ 10 x 10=200= AH, consequently AH=14•142100.
As AH (=14:14142100) or Ar. Co. Log. 9.8494850
10000000 To tangent of angle FHB=35° 16' . =9.8494850
The angle ABH is therefore 54° 44. Hence it follows, that when shaduws are projected on the plan as well as on the elevation, at an angle of forty-five degrees, the height of the sun which projects them must be 35° 16'.
2462. It is of the utmost importance to the student to recollect this fact, because it will be hereafter seen that it will give him great facility in obviating difficulty where confusion of lines may lead him astray, being, in fact, not only a check, but an assistance in proving the accuracy of his work.
2463. We now proceed to submit to the student a series of examples, containing the most common cases of shadowing, and which, once well understood, will enable him to execute any other case that may be presented to his notice.
2464. In fig. 839. we have on the left-hand side of the diagram the common astragal fillet and cavetto occurring in the Tuscan and other pilasters, above in elevation and below in plan. The right-hand part shows the same connected with a wall, whereon a shadow is cast by the several parts. LL is a line showing the direction of the light in projection at an angle of forty-five degrees. It will on experiment be found, by a continuation of the line, or by one parallel to it, to touch the side of the asti agal at a, whence an
Fig. 839. horizontal line drawn along it will determine its line of shade. We here again repeat, to prevent misunderstanding, that in the matter we are now attempting to explain we are not dealing with reflected light, nor with the softening off of shadows apparent in convex objeots, but are about to
determine the mere boundaries of shade and shadow of those under co rest must be learned from observation, for the circumstances under w must constantly vary. This, however, we think, we may safely state, aries of shade and shadow only be accurately given in a drawing (h the satisfaction they will afford to the spectator will be sufficient, with ment. But it is not to be understood from this that we discounte ment of finish in architectural subjects; all that we mean to say is, that To return to the diagram : it is manifest that if the boundary of shade point parallel to the direction of the light a line ab will determine the be on the fillet at b, and that from the lower edge of such fillet at f a line the direction of the light will give at c the boundary of the shadow shaft S. As, in the foregoing explanation, a was the upper boundary of ducing the horizontal line which it gave to a on the right-hand side o obtain there a corresponding point whence a line aa' parallel to the direct to be drawn indefinitely ; and on the plan a line aa, also parallel to the dir cutting the wall WW whereon the shadow is cast at a. From the point la line from a, where the shadow cuts the wall on the plan, cutting aa' in a', point a' in the shadow. The point e, by a line therefrom parallel to the di will determine similarly the situation e' by obtaining its relative seat o which perhaps will be at once seen by taking the extreme point d of the astragal, and therefrom drawing dd' parallel to the direction of the light dd, drawn similarly parallel to the direction of the light, and cutting WW boundary of the shadow on the plan, and from that point a vertical dd boundary of shadow of the extreme projection of the astragal is thu boundary of shadow of the fillet on the right-hand side at b, similar and by the vertical bb', gives the boundary point of the shadow fro operation in respect of cc gives the boundary of shadow from c to c'i We have not described this process in a strictly mathematical manner, is rather to lead the student to think for himself a little in conducting suppose the matter will not be perfectly understood by him even on a si the diagram.
2465. In the diagram (fig. 840.) is represented a moulding of common occurrence in architectural subjects, and, as before, the right-hand side is the appearance of its shadow on the wall WW on the plan. It will be immediately seen that LL being the projected representation of the rays of light, the line aa determines the boundary of shadow on the ovolo, and that at b, the boundary of its shade, is also given by a line touching that point parallel to the rays, or rather projected rays, of light. On the right-hand side of the figure oo', drawn indefinitely parallel to the direction of the light, and determined by a vertical from a", the intersection by a"a" with the w the line of shadow of oa'. The line aa determines the shadow on thcontinued to a' horizontally gives also a like termination to a" in the sha dary upwards of the ovolo's shade, is represented to the right by b', and to the right on the plan by b, whence by a vertical cutting the line b'b" in b”, the boundary of shadow which b' will cast is obtained. cc on the plan is in projection the distance of the line of shade c' from the wall whereon the shadow is cast, and its place in the shadow is at c", ee"b" being the length of horizontal shadow produced by the circumstances.
In fig. 841., which, it will be seen. is a common fillet and cavetto, LL is, as before, the direction of the
light, and aa gives the boundary of shadow, as well of the fillet's lower edge as of the lower edge of the cavetto itself. In respect of the right-hand side of the figure, a'a' is a line showing in profile the extent of projection of the fillet before the wall line WW, and froin a' a line drawn indefinitely parallel to the direction of the light, and terminated by the intersection of a vertical from a' in a', will give the point a' in the shadow. So is bb found through a vertical from b on the wall, by a line drawn parallel to the direction of the light from b on the plan. The several points being connected by lines, we gain the boundaries of the shadow, wherein a'a'' is represented by a"a”.
2466. Fig. 842. exhibits a fillet and cyma reversa or ogee, wherein, as before, LL is the direction of the light at a similar angle to that used on the plan. Froin the lower edge of the fillet, parallel to the direction of the light, is obtained the point a on the ogee, and from b a similarly parallel line gives the boundary of shadow in c. A line from o in direction of the light, drawn indefinitely, intercepted by a vertical line from d', its projection on the plan in d determines o'd, the boundary of the shadow of the fillet on the wall WW. CC"" is the line of profile of the projecting boundary in elevation, of the shade of the ogee before the wall, whereon its shadow is terminated from c and c'" by a vertical c'" c'". bb', the boundary of shade of the ogee itself, is found in shadow by the line b b'' drawn indefinitely parallel to the direction of the light, and terminated by a vertical from b', the point on the wall correspondent to b on the plan, the place of the shade's point in the elevation. By the junction of the lines so found, we shall have the outline of the shades and shadows cast. It is here to be observed, that the portion of light a'b' which the moulding retains is represented in the shadow by a'b'", all the other parts of its curved form being hidden, first by the projection of the fillet, and secondly by the line of shade bb", which acts in the same way as the fillet itself in producing the line aa', for the moment the light is intercepted, whether by a straight or curved profile, shadow must follow the shade of the moulding, whatever it be; and this is by the student to be especially observed.
2467. Fig. 843. exhibits the mode of obtaining the shadows and shade in the cyma recta. LL is the direction of the light, parallel whereto the line ab determines the line of horizontal shadow cast by the lower edge of the fillet upon the cyma, and cd that of the under part of the cyma itself upg the fillet at d. cc' is the upper boundary of the shade of the cyma, and e the point for determining the shadow of the lower fillet, the points abcd corresponding with abcd on the plan. WW on the right hand is the face of the wall, whereto the lines e'e", ďd", c'c", ", and a'a" are drawn parallel to the direction of the light. From e" d'c''B'a' vertical being drawn, cutting the indefinite lines oo', a'a', &c. parallel to the direction of the light in e", d'", c", b', and a", we have the form of the shadow in elevation. The part from b' to c' of the cyma being in light, its shadow will be the curve c'b", wherein, if it be required on a large scale, any number of points may be taken to determine its form by means of correspondent points on the plan as for the parts already described.
2468. Fig. 844. is the plan and elevation of some steps, surrounded by a wall, and Pin the plan is a square pillar standing in front of them. It will be seen that the line AB
Fig. 816. of a corona, so as to show the shadows of the modillions, independent with other parts of the assemblage. FG, HI, and AB parallel to the dir determine, by means of verticals from d and i, the points of shadows fro ent points c, 1, the points D, L, and I, whereof L is the point of shadow
2470. In fig. 846. we approach a little nearer to the form of a modill line EF determines the shadow of the corona, and AB by means of the li verticals dD, KK, the boundary of the side HL of the modillions. horizontally from B will give the under sides of their shadows. FG is the shadow of the eorona.
2471. Fig. 847. gives the finished modillion, and the lines Aa, Bb, mine, by horizontal lines drawn from them, the shadows which we are seeking. The auxiliary lines, to which no letters are attached, cannot fail of being understood ; but if difficulty arise in comprehending them, it will be removed by planning the several points, and therefrom drawing on the plan, to meet what may be oalled the frieze, vertical lines to intercept those from the correspondent points in the elevation, and the operation will be facilitated, perhaps, by projecting the form of the curved lines (as seen in the figure) whereof
Fig. 847. the modillion is formed.
2472. Fig. 848. will scarcely require a description. It is a geometri
Doric triglyph and frieze, with the usual acces-
Fig. 849. of a three-quarter column, forming part of an arcade. The abacus is the mere block of material AK. In the plan ab shows the length of the line of shadow AB, and is determined by the vertical b B.
In the same way, CD is found by cd and the vertical dD. KG is the representation of kg on the plan, and by a vertical from g the line GH is also determined ; H giving also by the horizontal line FH, in which H is already found, the situation of shadow of the point E of the abacus, as also by a vertical from f. LMN are places of the shadow of the column on the impost moulding of the arch, whereof two correspondent points are seen in 1 and n.
2474. The form of shadow of the console in fig. 850. will be seen on inspection to have been found from the lines aa, cc, dd, &c. on the elevation, corresponding with aa, cc, dd, &c. on the section, all which are parallel to the direction of the light, and sufficiently explain themselves.
2475. Fig. 851. is the elevation and section of a hemispherical niche, wherein are shown the shadows cast thereon by the vertical wall in which it is placed. Through the centre O draw DD at right angles to the direction of the light, and from O draw OA parallel to the direction of the light: A will be found the point in the wall casting the longest shadow. Produce AO indefi. nitely; and from a, the corresponding point in the section to A on the elevation, draw aa', parallel to it, which will cut the surface of the niche in a'. Draw the horizontal line a' al cutting AO produced in a'", and all will represent in the shadow the point A in the circumference. Take any other
Fig. 851. point B in the edge of the niche, and by means of a line drawn therefrom horizontally we have the correspondent point b of B in the section From B draw in the direction of the light the line BW" b'', cutting DD on the diameter in b'; transfer the point b'" in the elevation to b in the section, and draw bb' in the direction of the light indefinitely, Then with Bb!" as a radius from b as a centre, describe an arc cutting bb' in b'; and from b' draw the horizontal line b' b', cutting Bb" produced in b', and bh will be the point in the shadow corresponding to B in the elevation. To avoid the confusion which