CURB ROOF, a roof formed of four contiguous planes, of which each two have an external inclination, the ridge being the line of concourse of the two middle planes, and the highest of the three lines of concourse. This construction is frequently denominated a Jlansa•d roof, Mansard being the name of its inventor. It is very well adapted to a house surmounted by a parapet, so high as to cover the lower plane of the roof, as it gives a free or uninterrupted space from the level of its base to that of the summit of each lower plane, or, to the base of the two upper planes. In curb roofing, there is no danger of springing the walls by lateral pressure, for the distance between the base of the lower sides and the base of the upper sides, being sufficiently high for head-room, ties can always be fixed in these two situations, which %%ill prevent all danger. Indeed, if the four sides of the root' be properly balanced, the space may be made a complete void to the very ridge, or the upper part may be thrown into a cylindric arch.
A curb roof has great advantages over a common roof on account of the lower rafters pitching almost perpendicularly to their bases, and forming very nearly a continuation of the walls : whereas, in common roofs, the great inclination of the sides, and the quantity of head-•oom required, diminishes the space for lodging, in the breadth of the building ; in most cases there will be a loss of about 15 or 10 feet at least, and consequently, in small houses, no lodging-room whatever.
Curb roofs are generally lighted from dormer windows in the lower side.
The following contains the theory and practice of curb roofing, which is perhaps one of the most interesting parts in the science of carpentry : Proposition I. Figure 1. The position of several rafters, A 13, a c, c D, D E, &c., being given in a vertical plane, and movable about the angular points a, c, D, E, &C., while the points A and G remain stationary ; it is required to find the proportion of the forces at the angles, so that the rafters may be kept in equilibrio.
Through the points a, c, D, &e., draw the vertical lines c in, n p, &e., the direction of the forces ; make a i of any indefinite length, and complete the parallelogram a lc i k ; make c 1 equal a A-, and complete the parallelogram c l 711 a. Proceed in this manner with all the remaining parallelograms, making the two opposite forces in the direction of each rafter equal, and the diagonals a c n p, &c., will represent the forces required, as is evident from the construction. Then, to find the proportion of the weights upon any two angles, the sine of any angle is the same with the sine of its supple ment; therefore, the sine of the angle A B c is the same as the sine of a k i, and the sine of a c n the same as the sine of c m n ; likewise, the sine of the angle c an / is equal to the sine of the alternate angle in c a, and the sine of the angle D p o is equal to the sine of the angle p D q ; moreover, the sine of the angle i s A- is equal to the sine of the angle in c 1, and the sine of the angle in c n is equal to the sine of the angle p D o, and so on : then, because the sides of the triangle are the same as the sines of their opposite angles, it will be That is, the weights on any two angles are as the sines of these angles directly, and reciprocally as the product of the sines of the two parts of these angles formed by the vertical lines.
Corollary 1. Hence the weights on any two angles are as the sines of the angles directly, and as the produce of the co-sines of the angles of elevation reciprocally. For, draw a it perpendicular to a and produce i B and A B to land then will the angle K u 1, equal the angle It n i, be the idement of the angle n B K, VIZ.. the complement of the angle of elevation of the rafter A B above the horizon ; and because C a 1 is the supplement of c B the angles c B I and c n i have the simile sine, and the angle c B I is the complement of the angle n nc, viz, the angle of elevation of the rafter B C.
Corollary 2. Hence also the weights on any two angles are as the sines of the angles directly, and as the products ()Nile secants of elevation jointly ; for the secants of any two angles are reciprocally as the co-sines of these angles.
Corollary 3. The 11)ree which any rafter makes in its own direction, is as the secant of its elevation. For, make A equal ton b, draw the lines P L, &C., parallel to the vertical lines a C in, &c., and d ra w A N, B C L, &e., parallel to the horizon ; then because the angles N A r, II B L c a, &e., are the angles of elevation, and A B n, c &e., are all equal, if A N, B H, C &c., eonsidered as radii, A P, a h, c a, &e., are the secants of elevation, and also represent the forces on the rafters.
Corollary 4. Hence the horizontal pressures at A and c are equal; lbr all the perpendiculars drawn from the opposite angles of each parallelogram to meet the vertical diagonal, are all equal.
Corollary 5. Hence, if the position of any two rafters, and the proportion of the weights be given, the position of the remaining rafters may be determined.
Corollary 6. If the vertical line s D v be drawn, the hori zontal line A v o and the lines A S, A R, A 0, A T, &C. be drawn parallel to the rafters A B, B C, C D, D E, &C. meeting the vertical lines in s, R, Q, T; then will A 5, A R, A Q, A T, repre sent the forces, and s a, R 0, Q T, the forces upon the angles; for A s, A R, A Q, A T, &c. are the secants of the elevation, and the triangles A s R, A R Q, A Q T, are all similar to the triangles h a i, 1 c ?n, o 7), &C.
Corollary 7. In every roof kept in equilibrio by the weight of the rafters, if u, v, w, &c. be the centres of gravity of the rafters, and also represent their weights, then the weight pressing vertically on B, will be and the weight on