[14] The student will observe, in practice, that, his paper lying flat on the table, he has only to draw the line _TV_ on its horizontal surface, parallel to the given horizontal line _AB_. In theory, the paper should be vertical, but the station-line _ST_ horizontal (see its definition above, page 5); in which case _TV_, being drawn parallel to _AB_, will be horizontal also, and still cut the sight-line in _V_. The construction will be seen to be founded on the second Corollary of the preceding problem. It is evident that if any other line, as _MN_ in Fig. 9., parallel to _AB_, occurs in the picture, the line _TV_, drawn from _T_, parallel to _MN_, to find the vanishing-point of _MN_, will coincide with the line drawn from _T_, parallel to _AB_, to find the vanishing-point of _AB_. Therefore _AB_ and _MN_ will have the same vanishing-point. Therefore all parallel horizontal lines have the same vanishing-point. It will be shown hereafter that all parallel _inclined_ lines also have the same vanishing-point; the student may here accept the general conclusion—“_All parallel lines have the same vanishing-point._” It is also evident that if _AB_ is parallel to the plane of the picture, _TV_ must be drawn parallel to _GH_, and will therefore never cut _GH_. The line _AB_ has in that case no vanishing-point: it is to be drawn by the construction given in Fig. 7. It is also evident that if _AB_ is at right angles with the plane of the picture, _TV_ will coincide with _TS_, and the vanishing-point of _AB_ will be the sight-point. [15] I spare the student the formality of the _reductio ad absurdum_, which would be necessary to prove this. [16] For definition of Sight-Magnitude, see Appendix I. It ought to have been read before the student comes to this problem; but I refer to it in case it has not. [17] The demonstration is in Appendix II. Article II. p. 101. PROBLEM IV. TO FIND THE DIVIDING-POINTS OF A GIVEN HORIZONTAL LINE. [Illustration: Fig. 15.] Let the horizontal line _AB_ (Fig. 15.) be given in position and magnitude. It is required to find its dividing-points. Find the vanishing-point _V_ of the line _AB_. With center _V_ and distance _VT_, describe circle cutting the sight-line in _M_ and _N_. Then _M_ and _N_ are the dividing-points required. In general, only one dividing-point is needed for use with any vanishing-point, namely, the one nearest _S_ (in this case the point _M_). But its opposite _N_, or both, may be needed under certain circumstances. PROBLEM V. TO DRAW A HORIZONTAL LINE, GIVEN IN POSITION AND MAGNITUDE, BY MEANS OF ITS SIGHT-MAGNITUDE AND DIVIDING-POINTS. [Illustration: Fig. 16.] Let _AB_ (Fig. 16.) be the given line. Find the position of the point _A_ in _a_.