Hence it follows that if from _D_, the dividing-point of _Ac_, we raise a perpendicular and make _DF_ equal to _EV_, a line _CF_, drawn from any point _C_ on the measuring-line to _F_, will mark the distance _AB_ on the inclined line, _AB_ being the portion of the given inclined line which forms the diagonal of the vertical rectangle of which _AC_ is the base. [27] The demonstration is in Appendix II., p. 104. PROBLEM XVIII. TO FIND THE SIGHT-LINE OF AN INCLINED PLANE IN WHICH TWO LINES ARE GIVEN IN POSITION.[28] As in order to fix the position of a line two points in it must be given, so in order to fix the position of a plane, two lines in it must be given. [Illustration: Fig. 48] Let the two lines be _AB_ and _CD_, Fig. 48. As they are given in position, the relative horizontals _AE_ and _CF_ must be given. Then by Problem XVI. the vanishing-point of _AB_ is _V_, and of _CD_, _V′_. Join _VV′_ and produce it to cut the sight-line in _X_. Then _VX_ is the sight-line of the inclined plane. Like the horizontal sight-line, it is of indefinite length; and may be produced in either direction as occasion requires, crossing the horizontal line of sight, if the plane continues downward in that direction. _X_ is the vanishing-point of all horizontal lines in the inclined plane. [28] Read the Article on this problem in the Appendix, p. 97, before investigating the problem itself. PROBLEM XIX. TO FIND THE VANISHING-POINT OF STEEPEST LINES IN AN INCLINED PLANE WHOSE SIGHT-LINE IS GIVEN. [Illustration: Fig. 49.] Let _VX_, Fig. 49., be the given sight-line. Produce it to cut the horizontal sight-line in _X_. Therefore _X_ is the vanishing-point of horizontal lines in the given inclined plane. (Problem XVIII.) Join _TX_, and draw _TY_ at right angles to _TX_. Therefore _Y_ is the rectangular vanishing-point corresponding to _X_.[29] From _Y_ erect the vertical _YP_, cutting the sight-line of the inclined plane in _P_. Then _P_ is the vanishing-point of steepest lines in the plane. All lines drawn to it, as _QP_, _RP_, _NP_, etc., are the steepest possible in the plane; and all lines drawn to _X_, as _QX_, _OX_, etc., are horizontal, and at right angles to the lines _PQ_, _PR_, etc. [29] That is to say, the vanishing-point of horizontal lines drawn at right angles to the lines whose vanishing-point is _X_. PROBLEM XX. TO FIND THE VANISHING-POINT OF LINES PERPENDICULAR TO THE SURFACE OF A GIVEN INCLINED PLANE. [Illustration: Fig. 50.] As the inclined plane is given, one of its steepest lines must be given, or may be ascertained. Let _AB_, Fig. 50., be a portion of a steepest line in the given plane, and _V_ the vanishing-point of its relative horizontal. Through _V_ draw the vertical _GF_ upwards and downwards. From _A_ set off any portion of the relative horizontal _AC_, and on _AC_ describe a semicircle in a vertical plane, _ADC_, cutting _AB_ in _E_.