proportional to AC, DB, HE. Then, with the axis major, EF, and the semi-axis minor, HG, describe a semi-ellipse, and it will be the section of the ellipsoid required. If AC be the axis major, BD will be the axis minor. In this case, join DC, and draw EG parallel to DC; then HG will be the height found geometrically. PROBLEM 43. 226. To find the section of a cylindric ring, perpendicular to the plane passing through the axis of the ring, the line of section being given. Let ABED, (fig. 10, pl. VI,) be the section of the ring, passing through its axis, and let AB be a straight line, passing or tending to the centre of the two concentric circles, AD and BE; also, let DE be the line of section. On AB describe a semi-circle, and take a, b, c, &c., any number of points in its circumference; draw the ordinates, ae, bf, cg, &c. Through the points, e, f, g, &c., in the diameter AB, draw the concentric circles, ei, fk, gl, &c., cutting the sectional line DE in the points i, k, l, &c. Through the points, i, k, l, &c., draw in, ko, lp, &c., perpendicular to DE; transfer the ordinates ea, fb, gc,&c., of the semi-circle, to in, ko, lp, &c.: and, through the points D, n, o, p, &c. draw the curve, DnopqE, which is the section required. PLANE TRIGONOMETRY. DEFINITIONS OF TERMS IN TRIGONOMETRY.* 227. THE COMPLEMENT OF AN ARC is the difference between that arc and a quadrant or quarter of a circle. Thus, the arc BC, which is the difference between AC and AB, is the complement of AB; and AB is, in like manner, the complement of BC. * Trigonometry is that branch of Geometry which treats exclusively on the properties, relations, 228. The SUPPLEMENT OF AN ARC is the remainder between that arc and a semi-circle. Thus, the arc given being AB, its supplement is BC. 229. The SINE OF AN ARC is a straight line, drawn from one extremity of the arc, upon and perpendicular to a radius or diameter. Thus, BM is the sine of the arc AB; and here it is evident that an arc and its supplement have the same sine. B 230. The Co-SINE OF AN ARC is the sine of the complement of that arc. Hence, BO or IM is the co-sine of the arc AB; and, therefore, the sine of the complement BC. 231. The TANGENT OF AN ARC is a straight line, drawn from one extremity of the arc, where it touches it, to meet the prolongation of the radius through the other extremity. The line AK, touching the arc at A, and extended to meet the radius IB produced, is the tangent of the arc AB. 232. The Co-TANGENT OF AN ARC is the tangent of c the complement of that arc. Thus, CL is the co-tangent of the arc AB, or the tangent of the arc BC, In the annexed diagram let AB, AC, AD, AE, AF, AG, AH, to A, be the several portions of the circumference, by supposing the point B to revolve round the circumference from A to B, C, D, E, F, G, H, the sine of any arc, in the first quadrant, increases from A to C, where it is the greatest possible, and then decreases to E, where B K A it becomes zero; the sine will, therefore, be positive for the first semi-circumference, and in the other half it will be negative. The co-sine will be positive in the first quarter, negative in the second and third, and again positive in the fourth. The tangent will be positive in the first quarter, negative in the second, positive in the third, and negative in the fourth. TRIGONOMETRY.-THEOREM 1. 233. If a perpendicular be drawn from an angle of a triangle, to the opposite side, which is the base; then, as the base is to the sum of the two sides, so is the difference of the sides to the difference of the segments of the base. For, (theorem 62, page 56) AC2-CD2-AD2 .... and, again, (theorem 62).. ....... BC2-CD2=BD2. Subtract the second equation from the first, and the result is ... AC2-BC2=AD2 - BD2: but, since the difference of the squares of any two quantities is equal to a rectangle contained by their sum and difference; therefore..... (AC+BC) (AC—BC)=(AD+BD) (AD--BD) Whence, (theorem 40, page 41) AD+BD: AC+BC :: AC-BC : AD-BD. TRIGONOMETRY.—THEOREM 2. 234. The sum of the two sides of a triangle is to their difference as the tangent of half the sum of the angles at the base is to the tangent of half their difference. D Let ABC be a triangle; then, of the two sides, CA and CB, let CB be the greater. Produce CA to E, and make CE = CB, and join BE. Produce BC to F; and, through A, draw FD, perpendicular to EB, meeting it in D; then FBD will be half the E sum of the angles at the base, and ABD half their difference. Likewise, DF is the tangent of the angle FBD, and AD the tangent of the angle ABD : moreover BF is the sum of the two sides BC, CA, and AE is their difference. Then, by similar triangles, BFD, EAD,....BF × AD = FD × EA. Wherefore BF: AE :: FD: AD; which is the proposition to be demonstrated. B CONIC SECTIONS. DEFINITIONS OF CONIC SECTIONS. 235. A CONE is a solid body, terminating in a point, called its vertex, and having a circle for its base, connected to the vertex by a curved surface, which every where coincides with a straight line passing through its vertex, and through any point in the circumference of the base. If a cone be cut by an imaginary plane, the figure of the section so formed acquires its name according to the inclination or direction of the cutting plane. 236. A plane passing through the vertex of a cone, and meeting the plane of the base, is called a directing plane, and the line of common section is called a directing line. 237. If a cone be cut by a plane parallel to the directing plane, the section is denominated a conic section. 238. If the directing line fall without the base of the cone, the section is called an ellipse. 239. If the directing line touch the circumference of the base, the section is called a parabola. 240. If the directing line fall within the base, the section is called an hyperbola. 241. Equal opposite cones are those which have their axes in the same straight line; and, if cut by a plane through their common line of axis, the sides of the section will be two straight lines cutting each other. Hence the two equal and opposite cones join each other at their vertices, and have their vertical angles equal. 242. If the plane which produces the section of an hyperbola be extended so as to cut the opposite cone, the two sections are denominated opposite hyperbolas. 243. If the plane of a conic section be cut perpendicularly by another plane, which passes through the axis of the cone, the line of common section, in the plane of the figure, is called the primary line. 244. A point where the primary line cuts a conic section is called a vertex of that conic section. Hence the ellipse has two vertices, opposite hyperbolas have each one, and the parabola has one. OF THE ELLIPSE. 245. That portion of the primary line terminated at each extremity by the vertices of the curve, is called the axis major, or transverse axis. 246. A straight line, drawn perpendicularly to the axis major, from any point in it, to meet the curve, is called an ordinate. 247 The middle of the axis major is called the centre of the figure. In the figure here annexed, Aa is the axis major, PM an ordinate to it, and the point C, in the middle of Aa, is the centre of the ellipse, AMa. THE ELLIPSE. THEOREM 1. 248. The squares of the ordinates of the axis are to each other as the rectangles of the segments of the axis, from each ordinate to each of the two vertices of the curve. Let VDF be a plane passing through the axis of the cone, perpendicular to the cutting plane of the section A1Mami, and let Aa be their common section, meeting the conic surface in the points A, a; (then Aa will be the axis major,) and let Q1Ri, NMOm, be sections of the cone, parallel to the base. D M 鐵 |