Therefore _ad_ is the sight-magnitude of _AB_, as _aR_ is of _AB_ in Fig. 11. [Illustration: Fig. 78.] Remove again from the figure all lines except _PV_, _VT_, _PT_, _ab_, _ad_, and the measuring-line. Set off on the measuring-line _am_ equal to _ad_. Draw _PQ_ parallel to _am_, and through _b_ draw _mQ_, cutting _PQ_ in _Q_. Then, by the proof already given in page 20, _PQ_ = _PT_. Therefore if _P_ is the vanishing-point of an inclined line _AB_, and _QP_ is a horizontal line drawn through it, make _PQ_ equal to _PT_, and _am_ on the measuring-line equal to the sight-magnitude of the line _AB_ _in the diagram_, and the line joining _mQ_ will cut _aP_ in _b_. We have now, therefore, to consider what relation the length of the line _AB_ in this diagram, Fig. 77., has to the length of the line _AB_ in reality. Now the line _AE_ in Fig. 77. represents the length of _AE_ in reality. But the angle _AEB_, Fig. 77., and the corresponding angle in all the constructions of the earlier problems, is in reality a right angle, though in the diagram necessarily represented as obtuse. [Illustration: Fig. 79.] Therefore, if from _E_ we draw _EC_, as in Fig. 79., at right angles to _AE_, make _EC_ = _EB_, and join _AC_, _AC_ will be the real length of the line _AB_. Now, therefore, if instead of _am_ in Fig. 78., we take the real length of _AB_, that real length will be to _am_ as _AC_ to _AB_ in Fig. 79. And then, if the line drawn to the measuring-line _PQ_ is still to cut _aP_ in _b_, it is evident that the line _PQ_ must be shortened in the same ratio that _am_ was shortened; and the true dividing-point will be _Q′_ in Fig. 80., fixed so that _Q′P′_ shall be to _QP_ as _am′_ is to _am_; _am′_ representing the real length of _AB_. But _am′_is therefore to _am_ as _AC_ is to _AB_ in Fig. 79. Therefore _PQ′_ must be to _PQ_ as _AC_ is to _AB_. But _PQ_ equals _PT_ (Fig. 78.); and _PV_ is to _VT_ (in Fig. 78.) as _BE_ is to _AE_ (Fig. 79.). Hence we have only to substitute _PV_ for _EC_, and _VT_ for _AE_, in Fig. 79., and the resulting diagonal _AC_ will be the required length of _PQ′_. [Illustration: Fig. 80.] It will be seen that the construction given in the text (Fig. 46.) is the simplest means of obtaining this magnitude, for _VD_ in Fig. 46. (or _VM_ in Fig. 15.) = _VT_ by construction in Problem IV. It should, however, be observed, that the distance _PQ′_ or _PX_, in Fig. 46., may be laid on the sight-line of the inclined plane itself, if the measuring-line be drawn parallel to that sight-line. And thus any form may be drawn on an inclined plane as conveniently as on a horizontal one, with the single exception of the radiation of the verticals, which have a vanishing-point, as shown in Problem XX. THE END. Transcriber’s Note A handful of unequivocal typographical errors has been corrected. For increased clarity, a few diagrams have been shifted from their original position in the text.