Let a b, Fig. 25., be the four-sided pyramid.As it is given in position and magnitude, the square base

Fig. 25.

on which it stands must be given in position and magnitude, and its vertical height, c D.1

Fig. 26.

Draw a square pillar, A B G E, Fig. 26., on the square base of the pyramid, and make the height of

1 If, instead of the vertical height, the length of A D is given, the vertical must be deduced from it. See the Exercises on this Problem in the Appendix, p. 283.The pillar A F equal to the vertical height of the pyramid c D (Problem IX.). Draw the diagonals G F, H I, on the top of the square pillar, cutting each

Fig. 27.other in c. Therefore c is the centre of the square F G h I.(Prob. VIII. Cor. II.)

Join C E, C A, C B.

Then A B c E is the pyramid required. If the base of the pyramid is above the eye, as when a square spire is seen on the top of a church-tower, the construction will be as in Fig. 27.

## Problem X

This is one of the most important foundational problems in perspective, and it is necessary that the student should entirely familiarise himself with its conditions.

In order to do so, he must first observe these general relations of magnitude in any pyramid on a square base.

Let A G H, Fig. 56., be any pyramid on a square base.

The best terms in which its magnitude can be given, are the length of one side of its base, A H, and its vertical altitude (C D in Fig. 25.); for, knowing these, we know all the other magnitudes. But these are not the terms in which its size will be usually ascertainable. Generally, we shall have given us, and be able to ascertain by measurement, one side of its base A H, and either A G the length of one of the lines of its angles, or B G (or b' g) the length of a line drawn from its vertex, G, to the middle of the side of its base. In measuring a real pyramid, A G will usually be the line most easily found; but in many architectural problems B G is given, or is most easily ascertainable.

Observe therefore this genera! construction.

Fig. 56.

Let A B D E, Fig. 57., be the square base of any pyramid. Draw its diagonals, a e, b d, cutting each other in its centre, C. Bisect any side, A B, in F. From F erect vertical F G. Produce F B to H, and make F H equal to A C.

Now if the vertical altitude of the pyramid (C D in Fig. 25.) be given, make F G equal to this vertical altitude. Join G B and G H. Then G B and G H are the true magnitudes of G B and G H in Figure 56.

If G B is given, and not the vertical altitude, with centre B, and distance G B, describe circle cutting F G in G, and F G is the vertical altitude.

If G H is given, describe the circle from H, with distance G H, and it will similarly cut F G in G.

It is especially necessary for the student to examine this construction thoroughly, because in many complicated forms of ornaments, capitals of columns, c, the lines B G and G H become the limits or bases of curves, which are elongated on the longer (or angle) profile G H, and shortened on the shorter (or lateral) profile B G. We will take a simple instance, but must previously note another construction.

It is often necessary, when pyramids are the roots of some ornamental form, to divide them horizontally at a given vertical height. The shortest way of doing so is in general the following.

Let A E c, Fig. 58., be any pyramid on a square base A B C. and A D C the square pillar used in its construction.

Then by construction (Problem X.) B D and A Fare both of the vertical height of the pyramid.

Of the diagonals, F E, D E, choose the shortest (in this case D e), and produce it to cut the sight-line in V. Therefore v is the vanishing-point of D E. Divide D B, as may be required, into the sight-magnitudes of the given vertical heights at which the pyramid is to be divided.

From the points of division, 1, 2, 3, c, draw to the vanishing-point v. The lines so drawn cut the angle line of

Fig. 57.

The pyramid, B E, at the required elevations. Thus, in the figure, it is required to draw a horizontal black band on the pyramid at three fifths of its height, and in breadth one twentieth of its height. The line B D is divided into five parts, of which three are counted from B upwards. Then the line drawn to v marks the base of the black band. Then one fourth of one of the five parts is measured, which similarly gives the breadth of the band. The terminal lines

Fig. 58 of the band are then drawn on the sides of the pyramid parallel to A B (or to its vanishing-point if it has one), and to the vanishing-point of B c.