the larger make it 4, and so on.

Or this form may be used: make _AB_ twice the length of _AC_ (Fig. 130),
or in any other proportion required. On _AC_ mark the points as in the
drawing you wish to enlarge. Make _AB_ the length that you wish to
enlarge to, draw _CB_, and then from each division on _AC_ draw lines
parallel to _CB_, and _AB_ will be divided in the same proportions, as I
have already shown (Fig. 117).

There is no doubt that it is easier to work direct from the vanishing
points themselves, especially in complicated architectural work, but at
the same time I will now show you how we can dispense with, at all
events, one of them, and that the farthest away.

[Illustration: Fig. 130.]

LXVIII

HOW TO DRAW A CUBE ON A GIVEN SQUARE, USING ONLY ONE VANISHING POINT

_ABCD_ is the given square (Fig. 131). At _A_ raise vertical _Aa_ equal
to side of square _AB'_, from _a_ draw _ab_ to the vanishing point.
Raise _Bb_. Produce _VD_ to _E_ to touch the base line. From _E_ raise
vertical _EF_, making it equal to _Aa_. From _F_ draw _FV_. Raise _Dd_
and _Cc_, their heights being determined by the line _FV_. Join _da_ and
the cube is complete. It will be seen that the verticals raised at each
corner of the square are equal perspectively, as they are drawn between
parallels which start from equal heights, namely, from _EF_ and _Aa_ to
the same point _V_, the vanishing point. Any other line, such as _OO'_,
can be directed to the inaccessible vanishing point in the same way as
_ad_, &c.

_Note._ This is only one of many original figures and problems in this
book which have been called up by the wish to facilitate the work of the
artist, and as it were by necessity.

[Illustration: Fig. 131.]

LXIX

A COURTYARD OR CLOISTER DRAWN WITH ONE VANISHING POINT

[Illustration: Fig. 132.]

In this figure I have first drawn the pavement by means of the diagonals
_GA_, _Go_, _Go_, &c., and the vanishing point _V_, the square at _A_
being given. From _A_ draw diagonal through opposite corner till it cuts
the horizon at _G_. From this same point _G_ draw lines through the
other corners of the square till they cut the ground line at _o_, _o_.
Take this measurement _Ao_ and mark it along the base right and left of
_A_, and the lines drawn from these points _o_ to point _G_ will give
the diagonals of all the squares on the pavement. Produce sides of
square _A_, and where these lines are intersected by t