" When a right line is parallel to one of the planes on which it is projected, its length is equal to that of its projec tion on the plane ; for the line and its projection, being both terminated at two point perpendicular at the plane of pro jection, are parallel to each other, and comprised between parallels. In this particular ease. therefore, the projection being given, the length of the right line, which is equal to it, is also given.
"A right line is always parallel to one of the two planes of projection, when its projection upon the second is parallel tc the first of its planes.
" It' the right line be at the same time oblique upon twc planes, its length will he greater than that of either of its projections ; but the true length may be obtained by a very simple operation.
" Figure !72.—Let A B be the right line ; a 1, a' 11, its given projections : to find its true length. From one of the extremities of the right line a, in the vertical plane falling from it, imagine a horizontal lino, A E. stretching out till it meet in E the vertical line falling from the other e.xt•emity at B ; this will give the rectangular triangle, A E 13, which must be constructed in order to obtain the length of the right line A 13, which is its hypothenuse. In this triangle, independently of the right angle. we know the side A E, which is equal to the given projection, a b. And if, in the vertical id we draw from the point a', the horizontal line a' e', which is the projection of A E, it will cut the vertical line 1/ n at the point e, which will he the projection ion of the point E. Thus e Mill be the vertical projection of 13 E, and con sequently of an equal length with it. Having ascertained, by these means, the two sides of the triangle. it will he easy to construct the triangle, whose 11‘ pothenuse will give the length of A B.
"Figure '2.—being drawn in perspective, has no affinity to constructions done in the manner of projections. We shall here give the construction of this first question in all its "Figure 3.—The right line L tst being supposed to be the intersection of the two planes of projection, and the lines a 1,, a" 6", the given projections of a right line; to find the knot]] of this line. Draw through the point a", the indefinite hori zontal n e, which will cut the line 1 b", at the point e, and upon it measure the length of It 1. from e to II. Draw the hypothenuse it b", and its length will be that of the right line required.
" As both the planes of projection are rectangular, this operation, which is performed upon one of the planes, may be also done upon the other, and will yield a similar result. From what has been said, the reader will perceive, that whenever we have the two projections of a body, terminated by plane surfaces, by rectilinear angles, and by the apices of solid angles (projections which are reducible to the system of rectilinear angles) it will be easy to ascertain the length of any of its dimensions : for this dimension will either be parallel to one of the two planes of projection, or it will lie at the same moment oblique to them both. In the first ease, the required length of the dimension will be equal to its pro jection; in the second, it may be reduced from the two pro jections, by the method just described.
" We come now to describe the mode by which the pro jections of solids, terminated by planes and rectilinear angles are constructed ; though there is no genera] rule for this operation : indeed, the construction of these projections will be more or less easy, according to the method in which the position of the apices of the angles of the solid is defined ; the nature of the operation being governed by that of the definition. The ease is precisely here as in algebra, in which there is no general method of reducing a problem to equa tions. In every particular instance, the process depends on the mode in which the relation between the given quan tities and those sought tbr, is expressed : and it is only by a variety of examples that young students can learn how to lay hold of these affinities, and to express them in equations. So likewise, in descriptive geometry, it is only by a multitude of examples, and by the use of the rule and compasses in our schools, that we can acquire the habit of forming construc tions, or accustom ourselves to make choice of the most familiar and elegant methods in each particular ease. We may farther observe, that, as in analyses, when a problem is reduced to equations, there are methods of treating those equations, and of deducting from them the value of each unknown quantity ; so also, in descriptive geometry, when the projections are made, there are certain general methods of constructing whatever may result from the terms, and respective positions of bodies.