The Theory and Practice of Perspective Part 15

_ABCD_ is the given square, and only one vanis.h.i.+ng 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 vanis.h.i.+ng 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.

[Ill.u.s.tration: Fig. 134.]

This is to some extent the reverse of the previous problem. It also shows how the long vanis.h.i.+ng point can be dispensed with, and the perspective drawing brought within the picture.

LXXII

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.

[Ill.u.s.tration: Fig. 135.]

[Ill.u.s.tration: Fig. 136.]

LXXIII

OF PARALLELS AND DIAGONALS

[Ill.u.s.tration: Fig. 137 A.]

[Ill.u.s.tration: Fig. 137 B.]

[Ill.u.s.tration: Fig. 137 C.]

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 , 1/3, . 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 vanis.h.i.+ng 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_.

[Ill.u.s.tration: Fig. 137 D.]

LXXIV

THE SQUARE, THE OBLONG, AND THEIR DIAGONALS

[Ill.u.s.tration: Fig. 138.]

[Ill.u.s.tration: Fig. 139.]

Having found the centre of a square or oblong, such as Figs. 138 and 139, if we draw a third line through that centre at a given angle and then at each of its extremities draw perpendiculars _AB_, _DC_, we divide that square or oblong into three parts, the two outer portions being equal to each other, and the centre one either larger or smaller as desired; as, for instance, in the triumphal arch we make the centre portion larger than the two outer sides. When certain architectural details and s.p.a.ces are to be put into perspective, a scale such as that in Fig. 123 will be found of great convenience; but if only a ready division of the princ.i.p.al proportions is required, then these diagonals will be found of the greatest use.

LXXV

SHOWING THE USE OF THE SQUARE AND DIAGONALS IN DRAWING DOORWAYS, WINDOWS, AND OTHER ARCHITECTURAL FEATURES

This example is from Serlio's _Architecture_ (1663), showing what excellent proportion can be obtained by the square and diagonals. The width of the door is one-third of the base of square, the height two-thirds. As a further ill.u.s.tration we have drawn the same figure in perspective.

[Ill.u.s.tration: Fig. 140.]

[Ill.u.s.tration: Fig. 141.]

LXXVI

HOW TO MEASURE DEPTHS BY DIAGONALS

If we take any length on the base of a square, say from _A_ to _g_, and from _g_ raise a perpendicular till it cuts the diagonal _AB_ in _O_, then from _O_ draw horizontal _Og_, we form a square AgOg, and thus measure on one side of the square the distance or depth _Ag_. So can we measure any other length, such as _fg_, in like manner.

[Ill.u.s.tration: Fig. 142.]

[Ill.u.s.tration: Fig. 143.]

To do this in perspective we pursue precisely the same method, as shown in this figure (143).

To measure a length _Ag_ on the side of square _AC_, we draw a line from _g_ to the point of sight _S_, and where it crosses diagonal _AB_ at _O_ we draw horizontal _Og_, and thus find the required depth _Ag_ in the picture.

LXXVII

HOW TO MEASURE DISTANCES BY THE SQUARE AND DIAGONAL

It may sometimes be convenient to have a ready method by which to measure the width and length of objects standing against the wall of a gallery, without referring to distance-points, &c.

[Ill.u.s.tration: Fig. 144.]