The back is the external surface. The face of the arch is the end surface. 565. Lines of the Arch. The springing lines are the intersections of the soffit with the abutment; as a', c', Fig. 121. In Fig. 115, B is the projection of a springing line. The span is the chord of the curve of right section, as DB, Fig. 115. A B Fig. 115-Represents an oval curve of three centres, the arcs of which are each 60°. DB, span of the curve. AC, rise. P, O, and R, centres of the arcs of 60°. The axis of the arch is the line passing through the centres of the span. The rise is the versed sine of the curve of right section, as AC, Fig. 115. The intrados is the intersection of the soffit with the face of the arch, as DCB. The extrados is the intersection of the back of the arch with the face. The intrados may be defined as the inner curve of a vertical right section, and the extrados as the outer one. The crown is the highest line of the soffit. The coursing joints are those lines which run lengthwise of the arch, and separate the several courses of the stones. The heading or ring joints are those lines which separate the stones, and are nearly or quite parallel to the face of the arch. 566. Volumes of the Arch. The blocks of stone which form the body of the arch are called voussoirs. The keystone is the highest stone of the arch. The impost stones are the highest stones of the abutment, and upon which the arch directly rests. 567. Cylindrical Arch. This is the most usual and the simplest form of arch. The soffit consists of a portion of a cylindrical surface. When the section of the cylinder perpendicular to the axis of the arch, termed a right section, cuts from the surface a semicircle, the arch is termed a full centre arch; when the section is an arc less than a semicircle, it is termed a segmental arch; when the section gives a semi-ellipse, it is termed an elliptical arch; when the section gives a curve resembling a semi-ellipse, formed of arcs of circles tangent to each other, the arch is termed an oval, (Fig. 115, or basket handle), and is called a curve of three, five, seven, etc., centres. In order to make the curve horizontal at the crown and symmetrical in reference to a vertical line through the centre, there must be an odd number of arcs. When the intrados is composed of two arcs meeting at the highest point of the curve, it is called a pointed, (Fig. 116,) or an obtuse or surbased arch, (Fig. 117.) 568. Oblique Arches. If the obliquity of the arch is small, it may be constructed like the right arch, but when the obliquity is considerable, or in other words when the angle between the axis and face is considerably less or greater than 90 degrees, the pressure upon the voussoirs near the end of the springing lines would be very oblique to the beds, and at the acute angles would tend to force the voussoirs out of place if the coursing joints are inade parallel to the axis. To obviate this defect the coursing joints are inclined to the cylindrical elements, as will now be explained. An ideal mode of determining the coursing joints is to conceive the arch to be intersected by an indefinite number of vertical planes parallel to the face, thus making an indefinite number of curves like the end ones. Then begin at any point, as d, Fig. 118, and pass a line along the soffit so as to cut all the former curves at right angles, and we have an ideal coursing joint. The line dc, Fig. 118, represents such a line. Other similar curves are also shown. The equation of these when developed is logarithmic. They are all asymptotes to the springing line. The plan of these curves is shown in Fig. 119. A suitable number of vertical intersections may be selected for determining the ring-joints, portions of which only are used, as b a, Fig. 118, and b', a', Fig. 119. This mode of determining the coursing joints is very objectionable in practice, because the voussoirs must constantly vary in width as we pass from one end to the other; and as the bed-surfaces are warped, it makes it exceedingly difficult to make the voussoirs of proper shape. The method of making the coursing joints nearly or quite parallel to each other, sometimes called the English method, is more simple, and gives as good results as the preceding method. the Fig. 120 is Fig. 121 is the elevation of an oblique arch, of which Fig. 120 is the plan. a co is the sof fit. a c' is the spring. co, spiral cours- Fig. 121. Fig. 121 is the elevation of such an oblique arch, and Fig. 120 is the plan. The system here shown is sometimes called "Buck's System." In order to construct this system graphically, we conceive that the soffit is developed, or rolled out about the springing line ac. Let mf be a right section (which is here supposed to be circular). Conceive that it is revolved down to coincide with the horizontal plane, and that the circumference is divided into a convenient number of equal parts, and through the points of division conceive. that cylindrical elements are drawn, as shown in the plan. In the development the circumference of the semicircle will become the line fb, and the cylindrical elements will be, as shown, parallel to the springing line a c. From the points where the horizontal projections of the cylindrical elements intersect the face a k, draw lines parallel to fb, and note their intersections with the developed position of the cylindrical elements, and the curve a db through these points will be the development of the intrados of oblique section. In a similar way find c A. Join ab with a straight line, and divide it into as many equal parts as there are to be voussoirs in the face. In the figure there are eight such parts. When there is an even number there will be a joint at the crown, but when an odd number there will be the appearance of a keystone at the crown. From cat the end of the springing-line ac draw a perpendicular cd to the line a b, and if it passes through one of the divisions previously determined on a b, we proceed with the construction; but if it does not, we make such a change in the data as will make it perpendicular. This may be done in several ways. We may erect a perpendicular to ab from the joint which is nearest the foot of the perpendicular previously drawn, and note where it intersects the springing-line, and change the length of the arch so that it will pass through that point. Or we may change the obliquity of the arch, or change the number of divisions of the line ab. If the foot of the perpendicular should fall near a division, the line may be changed so as to pass through the point and leave it slightly out of a perpendicular. We might also disregard the condition that the perpendicular de should pass through the end of the springing-line a c; but this is objectionable, because the opposite sides of the arch would then not be alike. Having fixed the position of cd, we proceed to draw lines through the several points of division of a b, parallel to c d. It should be observed that points through which these parallel lines are drawn are on the straight line a db, and not on the curved line a 1, 2, etc. The parallel lines thus drawn are the coursing joints. The development of the ring joints fn, etc., are perpendicular to the developed coursing joints, and hence will be normal to each other in their true position in the |