It is shown, by writers on spherical trigonometry, that in any right-angled spherical triangle, radius is to the cosine of either of the sides, as the cosine of the other side to the cosine of the hypothenuse. Suppose the plane angle F e L to be 27°, and the angle F c n 52°, to find the hypothenuse and angles of a right-angled spherical triangle, one of whose legs is 27° and the other 52°, it will therefore be— As radius, sine of 90° . . = 10.00000 This ascertains the angle which the jack-rafter makes with the hip. Since all the sides are now given, we shall have, by another well-known property, of the sines of the sides being as the sides of the opposite•angles, the following pro portion : As the sine of the hypothenuse 56° 44' . . = 9.92227 Is. to the sine of a right angle, or 90° . . . = 10.00000 Su is the sine of the side F c 11, 52° . . . = 9.89653 10 89753 0.92227 To the sine of the opposite angle 70° 28' = 9.97426 Therefore the backing is twice 70° 28' . . = 140° 56' In finding the angle opposite the side F c It, it was not necessary that the hypothenusal side should have first been found. It might have been found independently thus :—The sine of either of the sides about the right angle is to radius, as the tangent of the remaining side is to the tangent of the angle opposite to that side ; theretin'e, As the sine of the side F C L, 27° . . . . = 9.05705 Is to the tangent of the side F C 11 . = 10.10719 So is radius, sine of 90° = 10.00000 19 9.65705 To the tangent of the angle opposite the side 1' Cu, 70° 25' = 10.45014 In the same manner linty other bevels he found by trigo nometrical calculations; but as such extreme exactness is not necessary, the geometrical constructions ought to be well understood.
Ex AMPLE II.— The Aacn (Figure 2) of the wall plate ola hip span-roof and the height of the roof being !liven; to find the backing of the hips, the uni//es usvob upon the sides of the purlins by their limyitadinal arrises, and the angles made upon the sides of the jack-rafters; the roof briny rgltally inclined to the ti prod sides of the building, except of the oblique end, A n.
Figure 2.—Let the two sides, A n, n D, and n c of the wall plate he at right angles to each other, and the end c n at oblique angles to A it and c D; draw the sent, E F, of the ridge in the middle of the breadth, parallel to A 13 and n C ; make G and D it equal to halt' the breadth of the building ; join c:. n, which will be the seat of the common rafters adjoining the hips ; make E tequal to the height or the roof; and draw G and r u, which are the length of the common rafters. Draw E D and s A, the seats of the hips; make E K e1111111 to ; and draw a A, which gives the length of each hip. Through any pat, in the seat of the hip A E, draw m N perpendicular to A a, Caving the adjacent sides of the wall plate at M and a ; take the nearest distance from 1. to the rafter A a, and make r. o equal to it ; and draw o and o s ; and NI o N is the hacking of the hips, represented by their seats A K and D This operation is the same as having the two legs of a right-angled solid angle to tind the angle opposite to one of the legs; the angle m o N being exactly the double of the angle so found; for the hip angle of the roof consists of two equal solid angles.
Suppose the bevel end at c n to be inclined at a different angle to the other sides, and let F c and r 13 be the seats of the hips ; draw F Q perpendicular to F c, and F r perpcndieu lar to F a, each equal to the height of the roof; then draw Q C and p a, which are the lengths of the hip-rafters.
The backings s u T and v w x, are found in the same manner as above, and may be described in the same words.
From A, with the distance A li, describe an arc cutting c H at 3, and join A .1; then o a A will be the side bevel, which the jaek-rafters make with the hips; and if a right angle be added to c .1 A, it will form an obtuse angle, which is that made by the upper arris of the side of a pnrlin placed in the inclined side of the roof with the hip-rafter.
Let a be the position of a purlin in the rafter n 1; in G n take any point, b, and draw 6 c parallel to the inward direc tion of the purlin a; from b, with any distance, L c, describe an are, c d, cutting G n at d; draw L c, c f, and d 9, parallel to E F; the former two cutting E n at e and f; draw/ g parallel to Gn, and join e p; produce L e to h; and h e y, or L e y will be the angle required, according to which side it is applied: this will be found synonymous to one of the legs, and the adjacent angle of a right-angled solid angle being given, to find the hypothenuse. In the same manner, if neither side of the purlin should be parallel to the inclined side of the roof; as at k in the rafter G 1, the bevel or angle upon each side may be found, as shown.
P/ate I.—Fiyure 3, shows half the angle of the backing of the hips, the length of the common and hip-rafters, the bevel of the jack-rafters on their upper sides in an equal inclined roof, without laying down or drawing any more than the necessary seats; and this is all that is necessary when each side of the roof is alike; A u being the wall-plate between the hip and the rafter which joins the top of the hip, A c the seat of the rafter which joins the top of the hip, n C that of the hip, A 1' the length of the rafter %%bleb joins the hip 13 E the length of the hips, c it o half the hacking, A D 13 the angle which the jack-rafters form with the upper side of the hips, and, consequently, with the addition of a right angle, the side bevel of the purlin.
Figure 4, shows the same bevels, except that the side joint of the purlin is found by a different process, thus: from 13, with the distance n A, describe all are at D; fl•0111 e, with the distance A C, describe ani)ther arc, cutting the former at n; join 13 n, and the angle G a D will be the angle in the plane of the roof, made by the lower arris of the purlin and the joint against the hip-rafter.