Because CP = a;, and CA = a, AP = PD=a: — a, and aP equal to x+a; therefore PN is a mean proportional between x + a and x - a, or between aP and PD; but, in the hyperbola, the transverse axis is to the conjugate as the mean proportion between x + a, and a:—a to the ordinate y or PM; but AR is divided in Q, in the same ratio as PN is in N. Therefore PN : PM :: AR : AQ. That is, as the mean proportional between x + a, and x—a is to the ordinate PM, so is the semi-transverse axis to the semi-conjugate axis. OF THE PARABOLA. DEFINITIONS RELATIVE TO THE PARABOLA. 321. That portion of the primary line which is within the curve, and which is terminated at one extremity by the vertex, is called the axis. 322. A straight line, drawn perpendicularly to the axis, between it and the curve, is called an ordinate. PARABOLA.—THEOREM 1. 323. The squares of the ordinates of the axis are to each other as their distances from the vertex. Let VRQ be a plane, passing through the axis of the cone, perpendicular to the cutting plane of the section AMII'M'; and let AH be their common section ; then AH will be the axis. Let QIRI' be a section of the cone parallel to the base: Then, because the base RIQI' is perpendicular to the plane VRQ, the two sections AMH'M', OMNM', are perpendicular to the plane VRQ; therefore their common sections, MM7, IP, are also perpendicular to VRQ and to the lines AH, RQ, ON; but, because the plane VRQ passes through the axis of the cone, it will divide every circle pa rallel to the base into two equal parts; therefore ON is a diameter of the circle OMNM'; and, because the chords MM', II', are perpendicular to the diameters RQ, ON, they will be bisected; let H be the point of bisection in IP, and P the point of bisection in MM'. 324. Let AP=ar, PM=y, AH=*, HI=y, HQ=», HR=OP=w, and PN=*. By similar triangles, .. APN, AHQ vx = tz and, by the circle, QIR y* = vw and, by the circle, NMO ttc = yi. By multiplying these equations, the result will be xy^—ztf. Therefore x : % :: y9 : y'. 325. Corollary 1.—Hence ?*=*-'. X Z 326. Corollary 2.—Hence, because j = PARABOLA. DEFINITIONS, CONTINUED. 327. The constant quantity p is called the parameter or latus rectum. 328. Corollary.—Hence the parameter is a third proportional to the distance of the ordinate from the vertex, and the ordinate itself; for 2x : 2y :: 2y :p, or px—ly1, that is, y*=ij»x. THE THREE CURVES OF THE CONIC SECTIONS.—PROBLEM. 329. The vertical section of a right cone being given, and the position of the axis of a conic section, to describe that section. Let AVB, {Jig. 2, In fig. 3, Draw any line, ai, cutting the assymtote IK at A, and the assymtote JH at g. Make hi equal to ag, and * will be a point in the curve. In the same manner we may find as many more points as we please. Let the axis be cf, Jig, 2, cutting the sides of the section AV, BV, at c and f. Draw cd, ef, parallel to AB, cutting AV at e and BV at d. In Jig. 4, draw AB equal to cf, fig. 2. Bisect AB in C. Make CF, Cf, each equal to the half of df, or the half of ce; then, with the transverse axis AB, and focii F,f, describe the ellipse ADBE. Again, in In the straight line AA', Jig. 5, take any point, D, and make DA, DA7 each equal to 105 CHAPTER II. CARPENTRY. Carpentry is the art of applying timber in the construction of buildings. The Cutting Of The Timbers, and adapting them to their various situations, so that one of the sides of every timber may be arranged according to some given surface, as indicated in the designs of the architect, requires profound skill in geometrical construction. For this purpose it is necessary, not only to be expert in the common problems, generally given in a course of practical geometry, but to have a thorough knowledge of the sections of solids and their coverings. Of these subjects, the first has already been explained in the series of Problems given in the geometrical part of this Work, and we are now about to treat on the other; that is, the Method of Covering them. As no line can be formed on the edge of a single piece of timber, so as to arrange with a given surface, nor in the intersection of two surfaces, (by workmen called a groin,) without a complete understanding of both, the reader is required not to pass them until the operations are perfectly familiar to his mind. For the more effectually rivetting the principles upon the mind of the student, it is requested that he should model them as he proceeds, and apply the sections and coverings found on the paper to the real sections and surfaces, by bending them around the solid. The Surfaces, which timbers are required to form, are those of cylinders, cylindroids, cones, The Formation of Arches, Groins, Niches, Angle-brackets, Lunettes, Roofs, &c depend entirely upon their Sections, or upon their This branch of carpentry, from its being subjected to geometrical rules, and described in schemes or diagrams upon a floor, sufficiently large for all the parts of the operation, has been called Descriptive Carpentry. In order to prepare the reader's mind for this subject, it will be necessary to point out the figures of the sections, as taken in certain positions. All The Sections Of A Cylinder, parallel to its base, are circles. All the sections of a cylinder, parallel to its axis, are parallelograms. And, if the axis of the cylinder be perpendicular to its base, all these parallelograms will be rectangles. If a cylinder be entirely cut through the curved surface, and if the section is not a circle, it is an ellipse. All The Sections Of A Cone, parallel to its base, are circles: all the sections of a cone, passing through its vertex, are triangles: all the sections of a cone, which pass entirely through the curved surface, and which are not circles, are ellipses: all the sections of a cone, which are parallel to one of its sides, are denominated parabolas; and all the sections of a cone, which are parallel to any line within the solid, passing through the vertex, are denominated hyperbolas. All The Sections Of A Sphere or Globe, made plane, are circles. The solid formed by a Semi-ellipse, revolving upon one of its axes, is termed an ellipsoid. All The Sections Of An Ellipsoid are similar figures: those sections, perpendicular to the fixed axis, are circles; and those parallel thereto are similar to the generating figure |