shall therefore show you another method equally
correct, but much simpler than either, which I have invented for our
use, and which indeed forms one of the chief features of this book.

LXXVIII

HOW BY MEANS OF THE SQUARE AND DIAGONAL WE CAN DETERMINE
THE POSITION OF POINTS IN SPACE

Apart from the aid that perspective affords the draughtsman, there is a
further value in it, in that it teaches us almost a new science, which
we might call the mystery of aspect, and how it is that the objects
around us take so many different forms, or rather appearances, although
they themselves remain the same. And also that it enables us, with,
I think, great pleasure to ourselves, to fathom space, to work out
difficult problems by simple reasoning, and to exercise those inventive
and critical faculties which give strength and enjoyment to mental life.

And now, after this brief excursion into philosophy, let us come down to
the simple question of the perspective of a point.

[Illustration: Fig. 145.]

[Illustration: Fig. 146.]

Here, for instance, are two aspects of the same thing: the geometrical
square _A_, which is facing us, and the perspective square _B_, which we
suppose to lie flat on the table, or rather on the perspective plane.
Line _A'C'_ is the perspective of line _AC_. On the geometrical square
we can make what measurements we please with the compasses, but on the
perspective square _B'_ the only line we can actually measure is the
base line. In both figures this base line is the same length. Suppose we
want to find the perspective of point _P_ (Fig. 146), we make use of the
diagonal _CA_. From _P_ in the geometrical square draw _PO_ to meet the
diagonal in _O_; through _O_ draw perpendicular _fe_; transfer length
_fB_, so found, to the base of the perspective square; from _f_ draw
_fS_ to point of sight; where it cuts the diagonal in _O_, draw
horizontal _OP'_, which gives us the point required. In the same way we
can find the perspective of any number of points on any side of the
square.

LXXIX

PERSPECTIVE OF A POINT PLACED IN ANY POSITION WITHIN THE SQUARE

Let the point _P_ be the one we wish to put into perspective. We have
but to repeat the process of the previous problem, making use of our
measurements on the base, the diagonals, &c.

[Illustration: Fig. 147.]

Indeed these figures are so plain and evident that further description
of them is hardly necessary, so I will here giv