Interior angles, such as E B C, Fig. 55. (suppose the corner of a room), are to be treated in the same way, each side of the room having its measurements separately carried to it from the

Find D,

measuring-line. It may sometimes happen in such cases that we have to carry the measurement up from the corner B, and that the sight-magnitudes are given us from the length of the line A B. For instance, suppose the room is eighteen feet high, and therefore A B is eighteen feet; and we have to lay off lengths of six feet on the top of the room-wall, в C. the dividing-point of в C. Draw a measuring-line, в F, from в; and another, gc, anywhere above. On B F lay off в G equal to one third of A B, or six feet; and draw from », through a and B, the lines G g, в b, to the upper measuring-line. Then gb is six feet on that measuring-line. Make bc, ch, &c., equal to bg; and draw ce, hf, &c., to D, cutting в c in e and f, which mark the required lengths of six feet each at the top of the wall

PROBLEM X.

THIS is one of the most important foundational problems in perspective, and it is necessary that the student should entirely familiarize 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.

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

G

A

B

H

Fig. 56.

Β'

The best terms in which its magnitude can be given, are the length of one side of its base, A II, 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 ▲ H, and either AG 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 general construction.

Let A B D E, Fig. 57., be the square base of any pyramid

Draw its diagonals, AE, BD, cutting each other in its centre,

с.

Bised any side, a в, in F.

G

From F erect vertical F G.

Produce F B to H, and make F H

equal to a c.

F B

A

H

C

Then G B and G H are the true magnitudes of G B and G 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 FG is the vertical altitude.

If GH is given, describe the circle from п, with distance GH, 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 GH become the limits or bases of curves, which are elongated on the longer (or angle) profile G н, and short

ened on the shorter (or lateral) profile в 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 o, Fig. 58., be any pyramid on a square base A B 0, and A Do the square pillar used in its construction.

Then by construction (Problem X.) в D and ▲ F are both of

the vertical height of the pyramid.

Of the diagonals, F E, D E, choose the shortest (in this case DE), 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 vanish ing-point v. The lines so drawn cut the angle line of the pyra mid, 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 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 в C.

If it happens that the vanishing-points of the diagonals are awkwardly placed for use, bisect the nearest base line of the pyramid in B, as in Fig. 59.

Erect the vertical D B and join G B and D G (G being the apex of pyramid).

Find the vanishing-point of D G, and use D B for division, carrying the measurements to the line G B.

In Fig. 59., if we join A D and D C, A DO is the vertical profile of the whole pyramid, and B DC of the half pyramid, cor responding to F G B in Fig. 57.

We may now proceed to an architectural example.

Let ▲ H, Fig. 60., be the vertical profile of the capital of a