SOLID ANGLES.
Of the Construction of Solid Jingles, consisting of any three Plane ?ingles.
In these, besides the three plane angles, there are also to be considered the three inclinations of the planes ; so that in a solid angle there are six parts, any three of which being given, the other three may be found. Solid angles are either right angled or oblique : when they arc right angled, two of the planes form a right angle with each other ; and this is the only case necessary to be con sidered, as all the cases of oblique solid angles can be solved by the help of those which are right angled. The planes containing the right angle are called legs ; the plane subtending the right angle, the hypothenuse ; and the three inclinations, angles.
Piton. H. In a right angled solid angle are given the two planes ABC and ADD, containing the right angle to find the hypothenuse, and the angle contained by the hypothenuse and the leg MID. (Fig. 4.) In AB take any point A ; draw AC at right angles with A13, and ADE at right angles with 13D ; make BE equal to BC, and 13E will be the hypothenuse ; from AB cut oil* AF equal to Al), and join EC, and AFC will be the angle required.
For if the plane A BD be raised upon AB, at a right angle with the plane ABC, and the plane ACE turned up on AC until AF coincide with AD, and, lastly, the plane DBE turned round the line DB until BE fall upon VC ; then BE will fall upon BC, FA and EC will he at right angles to Ill), and consequently AFC will be the inclination of the plains.
Piton. III. Given one of the legs, and the angle op posite, to lind the other leg. (Fig. 5.) Let ABC be the given leg ; make ADC equal to the given angle; front the point C draw CA at right with All; from A, with the radius AD, describe the arc DE, and draw BE a tangent to the arc at E; then will ABE be the other leg required.
By these problems, the various levels in roofing are ascertained, as the backing of the hips, and the side joints of 'indicts and jack rafters; and in hand-railing, the spring of the plank, and the intersection of the plane of the plank with a horizontal plane.
Piton. IV. In a right angled solid angle, consisting of three plane angles, are given one of the legs and the ad jacent angle to lind the hypothenuse. (Fig. 6.) Let ABC be the given leg ; from any point A draw Al) perpendicular to All ; make the angle DA E equal to the adjacent angle ; from A, with any radius AD, de scribe an arc DE, cutting AE at E; draw DV and EC parallel to All; and draw CI' parallel to Al), and join F, Il, and F13.1 will be the hypothenuse.
Piton. V. In an oblique angled solid angle, consist ing of three plane angles, are given one of the sides, and the two adjacent angles, to find the other side. (Fig. 7.) Let ABC be the given side ; in AB take any point A, and draw Al) perpendicular to AB; make the angle 1) 1E equal to one of the given angles ; draw Eli and DI, parallel to AB, the former cutting All at :11, any where in CB, or in CB produced take any point I', draw VG perpendicular to CI', and make the angle GFI equal to the other given angle ; produce CF to II, and make Flt equal to AlE ; draw III parallel to VG, Ili parallel to BC, and KL parallel to All ; join BL, and ABL is the side required.
The applications of these problems are numerous, in cutting timbers to it at any anuh• .s<=inst another ; like wise in ohliquo stone arches, which have cylindric or cylind•oidic intradoscs, in cutting the sides of the bevels of the stone to the given angles. In these, a series of bevels may be found, which will have one side common, while the other side of each is only varied in the most easy manner. If the arch startd in a vertical wall, the eonstant angle will be a right angle ; but if otherwise, it will be acute or obtuse : Thus, suppose AB, Fig. 8. to be the line of an erect wall upon the ground, and CD the direction of the arch ; or, let AB be the intersection of any plane, and CD the direction of a prismatic piece of timber, consisting of several sides, to he cut so as to fit against the plane : Suppose EC', ECG, ECU, &c. to be the angles which the several beds make with each other, in a plane perpendicular to the sides of the arch, or the angles w hich a series of planes would make at the same intersection, parallel to the sides of the same prism ; then CDf, CDg, CD/i, &c. are respectively the angles which two adjoining sides of the beds make with each other, or the angles which the several arrises of the prism make with the sides of the end to be cut. In like manner, Fig. 9. shows a series of angles when the plane Inclines or reclines; each is to be found in the same man ner as in Fig. 7. the angles DCf, DCg, DC/i, &e. corre sponding with EDF, EDG, EDIT, &e. In Fig. 8. it will be only necessary to find the angles on one side ; but in Fig. 9. they must be found un both sides.