APPLIED MECHANICS. [ M K. -H AXICA L DR AWIW a SHADOWS. 



I..-. I.' 



coming from above downwards. Further, as our draw- 

 ings are intended to represent the visible surfaces of 

 object*, we must suppose those sides that are exposed 

 to be illuminated ; and therefore select rays, coming 

 from holiind u, and striking on and past the objects. 

 As to whether these rays shall come in a direction from 

 K-ft towards right, or from right towards left, is a matter 

 of indifference. The only circumstance which can affect 

 our choice in this last respect is, that draughtsmen 

 generally sit with their left side towards the light, and 

 therefore are more likely to select rays coming in the 

 direction to which they are accustomed : that is, from 

 loft towards right Now, in order to see clearly the 

 effect of rays supposed to come in the chosen direction, 

 in illuminating certain surfaces, and in producing shades 

 and shadows, we shall suppose a piece of 'paper, cut to 

 the form of Fig. 42, consisting of three equal squares, 

 and that portions of the diagonals of these squares are 

 drawn, lying at mitre-angles, 

 or angles of 45, with the side* 

 of the squares. Now, lot us 

 suppose that the paper is 

 folded along the two lines 

 a b and o c, so as to turn up 

 the two squares perpendicular 

 to the third, aud to each 

 other, as in Fig. 43, forming 

 three sides of a square box, 

 with the diagonal lines drawn 



upon them. If the lines of 



G projection upwards, towards 



the right and towards the front from the extremities of 

 these three diagonal lines respectively, be drawn till they 



. as indicated by the dotted rtf. . 



lines, we get a line de lying ob- 

 liquely to each of three sides, of 

 which line the diagonal lines are 

 correct projections. Now, this 

 line d e would represent a portion 

 of one of the rays of light which, 

 as we suppose, illuminate the ob- c 

 jects, and all the other rays would bo parallel to it. 

 The shades and shadows, therefore, of plans, elevations, 

 and sections, projected on the three planes, will all be 

 determined by lines lying at angles of 45 to the boun- 

 daries of those planes. AM an example of the mode of 

 correctly delineating shadows according to this law, we 

 may explain the mode of defining the shadows in Fig. 44. 

 1. The line a b drawn at 45 from the edge of the 

 section, gives the breadth c 6 of the bottom, over which 

 Fig. M. 



PU 1, looking from bor. PUn J, looking from below. 

 the shadow of the side extends ; we therefore set off m n 

 and p o in the Plan 1, each equal to c 6 ; and drawing 

 linen from n and o parallel to the edges, we have the 

 outline of that shadow. 



2. Drawing d e and fg at 45 from two lower edges of 

 the section, we get the extent of the shadows observable 

 in 1'1,-in 2, the inner shadow * wv of the projecting part 

 of tin- bottom being limited, because the line d t falls 

 within the edge of the bottom in section, while the 

 shadow n ( y extends as far as the edges of the Plan 2, 

 and would extend further if there were any surface to 



catch the shadow, because the line/;/ in section extend* 

 beyond the edge of the body. 



3. Again, in the elevation, h i gives the point t as the 

 limit of the shadow of the upper projection, while k (, 

 extending beyond the projecting part of the bottom, 

 shows that all its surface must be within the shadow. 



For drawings of bodies, whose boundaries are plane 

 surfaces, and whose edges are straight lines, the shadow- 

 ing is very simple ; and the proper inclinations and 

 boundaries of the shadows almost suggest themselves, 

 without the necessity for their actual projection. But 

 when the bodies are bounded by curved surfaces, such as 

 cylinders, cones, spheres, and the like, the projection of 

 the shadows is somewhat more difficult. A little con- 

 sideration, however, will in most of those cases enable 

 the draughtsman to give a sufficiently faithful represen- 

 tation of the desired effect. It must not be supposed 

 that the shadowing of mechanical drawings is intended to 

 give them any merit in an artistic point of view ; for, as 

 they are not real perspective representations of objects, 

 but only imaginary projections? so the shadows tinted on 

 them are not the representations of shadows actually seen 

 on the objects themselves, but geometrical projections of 

 shadows, that would occur on certain suppositions as to 

 the direction of illuminating rays. The only purpose of 

 the shadowing is to give a clearer conception of the solid 

 form intended to be shown ; and the draughtsman should, 

 therefore, take care to make these shadows geometrically 

 accurate. After having solved a few problems by actual 

 jir.ijection, he will find it easy to give tolerably faithful 

 views of shadows by the eye. We will work out two 

 problems in circular shadowing as examples of projection ; 

 and we have selected two kinds of shadows most com. 

 monly occurring in mechanical drawing for this purpose. 

 First, in Fig. 45, we have an elevation, and a plan 

 Fig.il. looking from below, of a 



cylindrical body, with a 

 flange or projecting rim 

 round its upper end. The 

 plan and elevation have a 

 common centre line, corre- 

 sponding with the axis. It 

 is evident that the shadow 

 of the lower edge of the 

 flange is the boundary of 

 the shadow of the flange on 

 the elevation. If, then, we 

 take any point, such as a 

 on the plan, tracing it up 

 to , on the elevation, and 

 drawing a b at 45, tracing 

 up from 6 a line in eleva- 

 tion till it meets a line 

 drawn at 45 from a,, we 

 get 6, in the elevation as 

 the shadow of a,. Taking 

 a number of points, such as 

 a, and finding their shadows in a similar manner, \\<> 

 should be able to trace through them a curved lino 

 /, 6, d, as the boundary of the shadow of the flange. 

 The extreme point/, is found at once by taking /on the 

 plan, drawing /, tracing e up to t on the elevation, 

 and drawing , /, . The line c d on the plan is supposed 

 to touch the inner circle in d ; therefore, d g, the re- 

 mainder of the visible part of that circle, must be all in 

 shadow, as the body itself intercepts the rays. The line 

 Fig. 46. traced up from d to the 



elevation, and meeting the 



1 proper shadow line r, il t in 

 I d,, is therefore the boun- 

 dary of the illuminated 

 part of the elevation ; and 

 if the shadow part be filled 

 in with a Muck tint, wehavo 

 the appearance presented 

 in Fig. 40 of the oblique 

 projection or shadow of 

 the flange on the cylin- 

 der. It is manifest that this shadow does not couvry 



