which the contiguous sides of a cube assume to the vertical plane, when it is resting upon one of its corners and the diagonal line joining the corners is horizontal. This position is shown by the cube drawn in Fig. 3, page 83, and the lozenge representing the upper surface of the cube differs considerably from the square, which is its actual or orthographic projection, as shown in juxtaposition in dotted lines. This figure will explain why a true isometric drawing is smaller than the object it represents, apart from any scale it may be drawn at. If the square c- -a-b- -d, shown in dotted lines, were revolved on the diagonal c-b, until the corner a reached point a', the lullTine lozenge c-a'-b-d' would be its appearance to an observer immediately in front; it would also be, as we shall see presently, the true isometric projection of the square, which is also one side of a cube. It will be obvious that although the position of the sides of the Square are altered during its movement, their actual lengths are not; nevertheless it will be found on measuring the projection that the full lines are shorter than the dotted lines, which are the real dimensions, whilst the diagonal c-b remains the same throughout. The proportion of an isometric line to the real line it represents is as the square root of 2 is to the square root of 3.
An isometric scale constructed upon the above principle for the purpose of making an isometric projection in due proportion to a given scale is shown in Fig. 1.
To construct an Isometric Scale.â€”Draw two lines, A-B and B-C, perpendicular to each other and equal in length. Join A-C, then A-B: A-C as 1 : ^'2. Next make A-B equal in length to A-C, and D-E parallel to and equal in height to B-C. Join A-E. Then A-E is to A-D as the ^'3 is to ^'2. We have now two lines in the desired proportion or ratio to each other, and if we construct a common or regular scale upon the longer line, A--E, as shown, and project its divisions perpendicularly to the base line A-D, the said divisions will be proportionately reduced thereon, and an isometric scale constructed,
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