[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.