HOW TO DRAW LINES WHICH SHALL MEET AT A DISTANT POINT, BY MEANS OF DIAGONALS
This is in a great measure a repetition of the foregoing figure, and therefore needs no further explanation.
I must, however, point out the importance of the point _G_. In angular perspective it in a measure takes the place of the point of distance in parallel perspective, since it is the vanishing point of diagonals at 45° drawn between parallels such as _AV_, _DV_, drawn to a vanishing point _V_. The method of dividing line _AV_ into a number of parts equal to _AB_, the side of the square, is also shown in a previous figure (Fig. 120).
HOW TO DIVIDE A SQUARE PLACED AT AN ANGLE INTO A GIVEN NUMBER OF SMALL SQUARES
_ABCD_ is the given square, and only one vanishing point is accessible. Let us divide it into sixteen small squares. Produce side _CD_ to base at _E_. Divide _EA_ into four equal parts. From each division draw lines to vanishing point _V_. Draw diagonals _BD_ and _AC_, and produce the latter till it cuts the horizon in _G_. Draw the three cross-lines through the intersections made by the diagonals and the lines drawn to _V_, and thus divide the square into sixteen.
This is to some extent the reverse of the previous problem. It also shows how the long vanishing point can be dispensed with, and the perspective drawing brought within the picture.
FURTHER EXAMPLE OF HOW TO DIVIDE A GIVEN OBLIQUE SQUARE INTO A GIVEN NUMBER OF EQUAL SQUARES, SAY TWENTY-FIVE
Having drawn the square _ABCD_, which is enclosed, as will be seen, in a dotted square in parallel perspective, I divide the line _EA_ into five equal parts instead of four (Fig. 135), and have made use of the device for that purpose by measuring off the required number on line _EF_, &c. Fig. 136 is introduced here simply to show that the square can be divided into any number of smaller squares. Nor need the figure be necessarily a square; it is just as easy to make it an oblong, as _ABEF_ (Fig. 136); for although we begin with a square we can extend it in any direction we please, as here shown.
To find the centre of a square or other rectangular figure we have but to draw its two diagonals, and their intersection will give us the centre of the figure (see 137 A). We do the same with perspective figures, as at B. In Fig. C is shown how a diagonal, drawn from one angle of a square _B_ through the centre _O_ of the opposite side of the square, will enable us to find a second square lying between the same parallels, then a third, a fourth, and so on. At figure _K_ lying on the ground, I have divided the farther side of the square _mn_ into ¼, 1/3, ½. If I draw a diagonal from _G_ (at the base) through the half of this line I cut off on _FS_ the lengths or sides of two squares; if through the quarter I cut off the length of four squares on the vanishing line _FS_, and so on. In Fig. 137 D is shown how easily any number of objects at any equal distances apart, such as posts, trees, columns, &c., can be drawn by means of diagonals between parallels, guided by a central line _GS_.