If, in Fig. 43. or Fig. 44., the lines _AY_ and _A′Y′_ be produced,
the student will find that they meet.
Let _P_, Fig. 45., be the point at which they meet.
From _P_ let fall the vertical _PV_ on the sight-line, cutting the
sight-line in _V_.
Then the student will find experimentally that _V_ is the
vanishing-point of the line _AC_.[26]
Complete the rectangle of the base _AC′_, by drawing _A′C′_ to _V_,
and _CC′_ to the vanishing-point of _AA′_.
Join _Y′C′_.
Now if _YC_ and _Y′C′_ be produced downwards, the student will find
that they meet.
Let them be produced, and meet in _P′_.
Produce _PV_, and it will be found to pass through the point _P′_.
Therefore if _AY_ (or _CY_), Fig. 45., be any inclined line drawn in
perspective by Problem XV., and _AC_ the relative horizontal (_AC_ in
Figs. 39, 40.), also drawn in perspective.
Through _V_, the vanishing-point of _AV_, draw the vertical _PP′_
upwards and downwards.
Produce _AY_ (or _CY_), cutting _PP′_ in _P_ (or _P′_).
Then _P_ is the vanishing-point of _AY_ (or _P′_ of _CY_).
[Illustration: Fig. 45.]
The student will observe that, in order to find the point _P_ by this
method, it is necessary first to draw a portion of the given inclined
line by Problem XV. Practically, it is always necessary to do so, and,
therefore, I give the problem in this form.
Theoretically, as will be shown in the analysis of the problem, the
point _P_ should be found by drawing a line from the station-point
parallel to the given inclined line: but there is no practical means
of drawing such a line; so that in whatever terms the problem may be
given, a portion of the inclined line (_AY_ or _CY_) must always be
drawn in perspective before P can be found.
[26] The demonstration is in Appendix II. Article III.