(Contributed by Walter Hooker)

The nature of the various forms of arches and the technical terms used in connection therewith have already been explained under the head of Arches in Volume I. of this work, and need not therefore be repeated here.

A Segmental Arch is struck from a centre at some convenient point below the springing line (or junction of the arch with the supports), and the voussoirs are struck from this centre, the intrados being divided into a number of equal parts to mark the position of the joints as shown in Fig. 92. Fig. 92.

A Semicircular Arch is somewhat similar to the above, with the exception that the centre is on the line of the springing and the radius is equal to half the opening. The curve of the intrados thus dies into the vertical faces of the supports, as shown in Fig. 93.

It is customary, though not universal, to table the upper surfaces of the voussoirs so that the arch ring may bond in with the coursing of the plain wall, thus adding stability to the work. Sometimes the extrados is cut to the same curve as the intrados, as in the case of the segmented arch shown in Fig. 92, but this renders the work less homogeneous than if tabled surfaces are used. On the right-hand side of Fig. 93 the tabling is shown set out from the semicircular dotted line, but this method makes the courses of the wall unequal in height. The method of setting out shown on the left-hand side allows greater latitude of adjustment of the courses. Fig. 93.

## Three-Centred Arches

One method of setting out a three-centred arch is shown in Fig. 94. Fig. 94.

Take XW as the span and YZ as the rise.

Let YP = the difference between XY and YZ, and on YP construct an equilateral triangle Ypk. Bisect the angle Xkp by a line PN cutting the line XY in N. Mark off NM on the line YX = KN, and from YW cut

Arches - Plane off a part YQ equal to YM; and with MQ as base describe an equilateral triangle Mqo.

M, Q, and O are the centres for the arch. If OM and OQ are produced they will meet the arcs at their junctions. The joints above the intersections of the greater and lesser curves are struck from O, and the others (or those approaching the springing) from M and Q.

It is an obvious fact that an infinite number of different three-centred arches can be constructed; so long as the centres of the side segments are on the springing line and the centre of the middle segment is on the centre line of the arch.

It should be noted, however, that the rise should be at least equal to one-third the span, otherwise the arch will be weak.

## Elliptical Arches

Take the distance between the abutments, as at AB (Fig. 95), and equal to the span and bisect it at C. Draw CD perpendicularly to AB and equal to the proposed rise of the arch.

The curve of the intrados is struck by means of a trammel, or a number of points are found upon the curve by means of a lath of wood used as a trammel. Fig. 95.

A lath is cut whose length AC (see Fig. 96) is equal to half the major axis AB of the ellipse, and a point J is marked on it so that AJ = CD = half the minor axis DE. The lath may then be placed in any oblique direction, so long as the marked points, as C1 and J1 lie on the minor and major axes respectively, when the extremity A1 will mark a point on the circumference of the ellipse. A succession of points thus obtained will outline the ellipse.

Another good method of drawing ellipses is as follows (see Fig. 97) -

Draw two semicircles, one with radius equal to the rise, the other with radius equal to half the span.

Divide the circumference of each semicircle into the same number of equal parts, as in Fig. 97. From each divisional point in the larger circumference draw vertical lines. From the divisional points of the smaller circumference draw horizontal lines to cut the vertical lines. The crossings of these lines mark points on an ellipse.

To obtain the joints, the foci E and F (Fig. 95) must first be found. This is done by striking an arc from the centre D with a radius equal to half the major axis of the intrados, cutting the springing line in E and F. Divide the intrados into a convenient odd number of equal parts. Join each of these points to both foci, and bisect the angle between each pair of lines. The bisecting lines give the direction of the joints. The operation is shown in Fig. 95. All the other joints have been found in a similar manner. The joints will be found to be normal to the curve of the intrados. Fig. 96.

The elliptic arch has been described in previous paragraphs For estimating the length of its semi-circumference, the following computation may prove useful: the circumference, and consequently half of this gives the desired dimension π = 3.4I59.

Let A be the transverse diameter and B the conjugate; then Pointed Arches are such as form a point at their apex, as shown in Figs. 98, 99, and 100. The method of setting out the various forms of pointed arches is as follows -  Fig. 97.

## Lancet Arches

Let AB (Fig. 98) be the opening between the abutments and also the springing line, and CD (greater than AB) the apex height. Join AC and bisect in E.

From E draw EG at right angles to AC, cutting AB in G. G is the centre of the arc AC. The other centre is found in the same way, as shown at H.

In this case it is seen that the centres fall without the abutments, causing the apex of the arch to be acutely pointed; for which reason this type of arch is termed a Lancet Arch.