Find the vanishing-point _V_, and most convenient dividing-point _M_, of the line _AB_. Join _aV_. Through _a_ draw a horizontal line _ab′_ and make _ab′_ equal to the sight-magnitude of _AB_. Join _b′M_, cutting _aV_ in _b_. Then _ab_ is the line required. COROLLARY I. [Illustration: Fig. 17.] Supposing it were now required to draw a line _AC_ (Fig. 17.) twice as long as _AB_, it is evident that the sight-magnitude _ac′_ must be twice as long as the sight-magnitude _ab′_; we have, therefore, merely to continue the horizontal line _ab′_, make _b′c′_ equal to _ab′_, join _cM′_, cutting _aV_ in _c_, and _ac_ will be the line required. Similarly, if we have to draw a line _AD_, three times the length of _AB_, _ad′_ must be three times the length of _ab′_, and, joining _d′M_, _ad_ will be the line required. The student will observe that the nearer the portions cut off, _bc_, _cd_, etc., approach the point _V_, the smaller they become; and, whatever lengths may be added to the line _AD_, and successively cut off from _aV_, the line _aV_ will never be cut off entirely, but the portions cut off will become infinitely small, and apparently “vanish” as they approach the point _V_; hence this point is called the “vanishing” point. COROLLARY II. It is evident that if the line _AD_ had been given originally, and we had been required to draw it, and divide it into three equal parts, we should have had only to divide its sight-magnitude, _ad′_, into the three equal parts, _ab′_, _b′c′_, and _c′d′_, and then, drawing to _M_ from _b′_ and _c′_, the line _ad_ would have been divided as required in _b_ and _c_. And supposing the original line _AD_ be divided _irregularly into any number_ of parts, if the line _ad′_ be divided into a similar number in the same proportions (by the construction given in Appendix I.), and, from these points of division, lines are drawn to _M_, they will divide the line _ad_ in true perspective into a similar number of proportionate parts. The horizontal line drawn through _a_, on which the sight-magnitudes are measured, is called the “MEASURING-LINE.” And the line _ad_, when properly divided in _b_ and _c_, or any other required points, is said to be divided “IN PERSPECTIVE RATIO” to the divisions of the original line _AD_. If the line _aV_ is above the sight-line instead of beneath it, the measuring-line is to be drawn above also: and the lines _b′M_, _c′M_, etc., drawn _down_ to the dividing-point. Turn Fig. 17. upside down, and it will show the construction. PROBLEM VI. TO DRAW ANY TRIANGLE, GIVEN IN POSITION AND MAGNITUDE, IN A HORIZONTAL PLANE. [Illustration: Fig. 18.] Let _ABC_ (Fig. 18.) be the triangle. As it is given in position and magnitude, one of its sides, at least, must be given in position and magnitude, and the directions of the two other sides. Let _AB_ be the side given in position and magnitude. Then _AB_ is a horizontal line, in a given position, and of a given length.