o 1099. Throar:M IV. The radius vector is equal to the sum of the distances between the focus Coroll. 1. If through the point G (fig. 447.) the line GQ be drawn perpendicular to the axis, it is called the directrix of the parabola. By the property shown in this theorem, it appears that if any line QM be drawn parallel to the axis, and if FM be joined, the straight line FM is equal to QM; for QM is equal to GP. Coroll. 2. Hence, also, the curve is easily described by points. Take AG equal to AF, (fig. 447.), and draw a number of lines M, M perpendicular to the axis AP; then with the distances GP, GP, &c. as radii, and from F T as a centre, describe arcs on each side of AP, cutting the lines MM, MM, &c. at MM, &c.; then through all the points M, M, M., &c. draw a curve, which will be a parabola. 1 100. TheoreM. V. If a tangent be drawn . *—so from the verter of an ordinate to meet the aris M.V. T —SM produced, the subtangent PT (fig. 448.) will P he equal to twice the distance of the ordinate Fig. 447. from the verter. If MT be a tangent at M, the extremity of the ordinate PM; then the sub-tangent PT is equal to twice AP. For draw MK parallel to AH, But when the ordinates HI and PM coincide, MT will become a tangent, and GK will From this property is obtained an easy and accurate method of drawing a tangent to any point of the curve of a parabola. Thus, let it be re- T T quired to draw a tangent to any point M in the curve. Produce PA to T (fig. 449.), and draw MP perpendicular to PT, meeting AP in the point P. Make AP equal to AP, and join MT, which will be the tangent A But, by Theorem III., FM = AF + AP; Therefore, by equality, FM = FT. Coroll. 1. If MN be drawn perpendicular to MIT to meet the axis in N, then will FN = FM = FT. For draw FH perpendicular to MT, and it also bisects MT, because FM = FT ; and since HF and MN are parallel, and MT is bisected in H, the line TN will also be bisected in F. It therefore follows that FN = FM = FT. Coroll. 2. The subnormal PN is a constant quantity, and it is equal to half the parameter, or to 2AF. For since TMN is a right angle, Coroll. 3. The tangent of the vertex AH is a mean proportional between A F and AP. For since FHT is a right angle, therefore AH is a mean proportional between A.F and AT; and since AT = AP, A H is a mean proportional between A F and AP. Also FH is a mean proportional between FA and FT, or between FA and FM. Coroll. 4. The tangent makes equal angles with FM and the axis AP, as well as with FC and CI. 1 102. Theor EM VII. A line parallel to the aris, intercepted by a double ordinate and a tangent at the rerter of that ordinate, will be divided by the curre in the same ratio us the line itself dirides the double ordinate. T Let QM (fig. 451.) be the double ordinate, MT the tangent, AP the axis, GK the intercepted line divided by the curve in the point I; then will GI : IK :: MK : KQ. 1 103. ProbleM. I. To describe a parabola. If a thread, equal in length to the leg BC (fig. 452.) of a right angle or square, be fixed to the end C, and the other end of the thread be fixed to a point F in a plane, then if the square be moved in that plane so that the leg AB may slide along the straight line GH, and the point D be always kept close to the edge BC of the square, and the two parts F D and DC of the string kept stretched, the point D will describe a curve on the plane, which will be a parabola. 1 104. Prob. II. Given the double ordinate DE and the abscissa BC in position and magnitude, to describe a parabola. Through B (figs. 453, 454.) draw FG parallel to DE and DF, and EG parallel to CD. f rt a Divide DC and g d parts. From Fig. 452. the points of Fig. 454. 2 division in DE draw lines to B. Through the points of division in DC draw lines parallel to BC, and through the l points of intersection of the corresponding lines draw a curve, and complete the other half in the same manner; then will DBE be the complete curve of the parabola. The less BC U-T- so o * is in proportion to CD, the nearer the curve will approach to ig. 455. the arc of a circle, as in fig. 422. ; and hence we may describe the curve for diminishing the shaft of a column, or draw a flat segment of a circle. 1105. PRob. III. The same parts being given, to describe the parabola by the intersection of straight lines. Produce CB to F (fig. 455.), and make BF equal to BC. Join FD and F.E. Divide DF and FE in the same proportion, or into the same number of equal parts. Let the divisions be numbered from D to F, and from F to E, and join every two corresponding points by a straight line; then the intersection of all the straight lines will form the parabola required. Fig. 455. 1106. PRob. IV. To draw a straight line from a given point in the curve of a parabola, which shall be a tangent to the curve at that point. Let DC (fig. 456.) be the double ordinate, CB the abscissa to the parabolic curve DBC, and let it be required to draw a tangent from the point e in the 1, C C curve. Draw ef parallel to DC, cutting Fig. 456. BC in f: produce CB to g, and make Bø equal to Bs, and join ge, then will gebe the tangent required. In the same manner DH will be found to be a tangent at D. If ek be drawn perpendicular to the tangent ge, then will e K be also perpendicular to the curve, and in the proper direction for a joint in the masonry of a parabolic arch. 1107. The uses of the parabolic curve in architecture are many. The theorists say that it is the curve of equilibrium for an arch which has to sustain a load uniformly diffused over its length, and that therefore it should be included in the depth of lintels and flat arches; and that it is nearly the best form for suspension and other bridges, and for roofs. It is also considered the best form for beams of equal strength. It may be here also remarked, that it is the curve described by a projectile, and that it is the form in which a jet of water is delivered from an orifice made in the side of a reservoir. So is it the best curve for the reflection of light to be thrown to a distance. In construction it occurs in the intersection of conic surfaces by planes parallel to the side of the cone, and is a form of great beauty for the profiles of mouldings, in which manner it was much used in Grecian buildings. GENERAL METHop of DETERMINING The species AND OF DESCRIBING CONIC SECTIONS. 1108. In a conic section, let there be given the abscissa AB (fig. 457.), an ordinate BC, and a tangent CD to the curve at the extremity of the ordinate to determine the species of the conic section, and to describe the figure. Draw AD parallel to BC, and join AC (Nos. 1. and 2.). Bisect AC in E, and produce DE and AB, so as to meet in F when DE is not parallel to AB; then in the case where DE will meet AB or AB produced in F, the point F will be the centre of an ellipsis or hyperbola. In this case produce AF to G, and make FG equal to FA ; then if the ordinate BC and the centre be upon the same side of the apex A, the curve to which the given parts belong is an ellipsis; but if they be on different sides of it, the curve is an hyperbola. When the line DE (No. 3.) is parallel to AB, the figure is a parabola. 1 109. In a conic section, the abscissa AB (fig. 458.), an ordinate BC, and a point D in the curve being given, to determine the species of the curve, and thence to describe it. Draw CG parallel to AB (Nos. 1. and 2.), and AG parallel to BC. Join AD, and produce it to meet CG in e. Divide the ordinate CB in f in the same proportion as CG is divided, then will Cf:f B:: Ce: eG. Join Df, and produce it or f D to meet AB or BA in h; then if the points D and h fall upon opposite sides of the ordinate BC, the curve is an ellipsis ; but if D and h fall upon the same side of the ordinate BC, the curve will be an hyperbola. If Df (No. 3.) be parallel to AB, the curve will be a parabola. In the case of the ellipsis and hyperbola, Ah is a diameter; and therefore we have a diameter and ordinate to describe the curve. SECT. VI. DEsc Riptive groMETRY. 1110. The term Descriptive Geometry, first used by Monge and other French geometers to express that part of the science of geometry which consists in the application of geometrical rules to the representation of the figures and the various relations of the forms of bodies, according to certain conventional methods, differs from common perspective by the design or representation being so made that the exact distance between the different points of the body represented can always be found; and thus the mathematical relations arising from its form and position may be deduced from the representation. Among the English writers on practical architecture, it has usually received the name of projection, from the circumstance of the different points and lines of the body being projected on the plane of representation; for, in descriptive geometry, points in space are represented by their orthographical projection on two planes at right angles to each other, called the planes of projection, one of which planes is usually supposed to be horizontal, in which case the other is vertical, the projections being called horizontal or vertical, according as they are on one or other of these planes. 1111. In this system, a point in space is represented by drawing a perpendicular from it to each of the planes of projection; the point whereon the perpendicular falls is the projection of the proposed point. Then, as points in space are the boundaries of lines, so their projections similarly form lines, by whose means their projection is obtained; and by the projections of points lying in curves of any description, the projections of those curves are obtained. 1119. For obvious reasons, surfaces cannot be similarly represented ; but if we suppose the surface to be represented, covered by a system (4 lines, according to some determinate law, then these lines projected on each of the two planes will, by their boundaries, enable us to project the surface in a rigorous and satisfactory manner. 1113. There are, however, some surfaces which may be more simply represented; for a plane is completely defined by the straight lines in which it intersects the two planes of projection, which lines are called the traces of the plane. So a sphere is completely defined by the two projections of its centre and the great circle which limits the projections of its points. So also a cylinder is defined by its intersection (or trace) with one of the planes of projection and by the two projections of one of its ends; and a cone by its intersection with one of the planes of projection and the two projections of its summit. 1114. Monge, before mentioned, Hachette, Vallée, and Leroi, are the most systematic writers on this subject, whose immediate application to architecture, and to the mechanical arts, and most especially to engineering, is very extensive; in consequence, indeed, of which it is considered of so much importance in France, as to form one of the principal departments of study in the Polytechnic School of Paris. A sufficient general idea of it for the architectural student may be obtained in a small work of Le Croix, entitled, Complément des Elemens de Geometrie. In the following pages, and occasionally in other parts of this work, we shall detail all those points of it which are connected more immediately with our subject, inasmuch as we do not think it necessary to involve the reader in a mass of scientific matter connected therewith, which we are certain he would never find necessary in the practice of the art whereon we are engaged. 1 115. In order to comprehend the method of tracing geometrically the projections of all sorts of objects, we must observe, -I. That the visible faces only of solids are to be expressed. II. That the surfaces which enclose solids are of two sorts, rectilinear and curved. These, however, may be divided into three classes, – Ist. Those included by plane surfaces, as prisms, pyramids, and, generally, similar sorts of figures used in building. 2d. Those included by surfaces whereof some are plane and others with a simple curvature, as cylinders, cones, or parts of them, and the voussoirs of arches. 3d. Solids enclosed by one or several surfaces of double flexure, as the sphere, spheroids, and the voussoirs of arches on circular planes. 1116. First class, or solids with plane surfaces. – The plane surfaces by which these solids are bounded form at their junction edges or arrisses, which may be represented by right lines. 1 117. And it is useful to observe in respect of solids that there are three sorts of angles formed by them. First, those arising from the meeting of the lines which bound the faces of a solid. Second, those which result from the concurrence of several faces whose edges unite and form the summit of an angle: thus a solid angle is composed of as many plane angles as there are planes uniting at the point, recollecting however that their number must be at least three. Third, the angles of the planes, which is that formed by two of the faces of a solid. A cube enclosed by six square equal planes comprises twelve rectilineal edges or arrisses and eight solid angles. in 18. Pyramids are solids standing on any polygonal bases, their planes or faces being triangular and meeting in a point at the top, where they form a solid angle. 1119. Prisms, like pyramids, may be placed on all sorts of polygonal bases, but they rise on every side of the base in parallelograms instead of triangles, thus having throughout similar form and thickness. 1120. Though, strictly speaking, pyramids and prisms are polyhedrons, the latter term is only applied to those solids whose faces forming polygons may each be considered as the base of a separate pyramid. 1121. In all solids with plane surfaces the arrisses terminate in solid angles formed by several of these surfaces, which unite with one another; whence, in order to find the projection of the right lines which represent those arrisses, all that we require to know is the position of the solid angles where they meet; and as a solid angle is generally composed of several plane angles, a single solid angle will determine the extremity of all the arrisses by which it is formed. 1122. Second class : solids terminated by plane and curved surfaces. – Some of these, as cones for instance, exhibit merely a point and two surfaces, one curved and the other flat. The meeting of these surfaces forms a circular or elliptical arris common to both. The projection of an entire cone requires several points for the curvature which forms its base; but a single point only is necessary to determine its summit. This solid may be considered as a pyramid with an elliptic or circular base; and to facilitate its projection a polygon is inscribed in the ellipsis or circle, which serves as its base. 1123. If the cone is truncated or cut off, polygons may in like manner be inscribed in the curves which produce the sections. 1124. Cylinders may be considered as prisms whose bases are formed by circles, ellipses, or other curves, and their projections may be obtained in a similar manner: that is, by inscribing polygons in the curves which form their bases. 1125. Third class : solids whose surfaces have a double curvature.—A solid of this sort may be enclosed in a single surface, as a sphere or spheroid. 1126. As these bodies present neither angles nor lines, they can only be represented by the apparent curve which seems to bound their superficies. This curve may be determined by tangents parallel to a line drawn from the centre of the solid perpendicularly to the plane of projection. 1127. If these solids are truncated or cut by planes, we must, after having traced the curves which represent them entire, inscribe polygons in each curve produced by the sections, in order to proceed as directed for cones and cylinders. 1128. To obtain a clear notion of the combination of several pieces, as, for instance, of a vault, we must imagine the bodies themselves annihilated, and that nothing remains but the arrisses or edges which form the extremes of the surfaces of the voussoirs. The whole assemblage of material lines which would result from this consideration being considered transparent would project upon a plane perpendicular to the rays of light, traces defining all these edges that we have supposed material, some foreshortened, and others of the same size. These will form the outlines of the vault, whence follow the subjoined remarks. I. That in order, on a plane, to obtain the projection of a right line representing the arris of any solid body, we must on such plane let fall verticals from each of its extremities. II. That if the arris be parallel to the plane of the drawing, the line which represents its projection is the same size as the original. III. That if it be oblique, its representation will be shorter than the original line. IV. That perpendiculars by means of which the projection is made being parallel to each other, the line projected cannot be longer than the line it represents. V. That in order to represent an arris or edge perpendicular to the plane of projection, a mere point marks it because it coincides in the length with the perpendiculars of projection. VI. That the measure of the obliquity of an arris or edge will be found by verticals let fall from its extremities. 1129. In conducting all the operations relative to projections, they are referable to two planes, whereof one is horizontal and the other vertical. Pivoj Ection of Right Lines. 1130. The projection of a line AB (fig. 459.) perpendicular to a horizontal plane is ex Fig. 459. Fig. 460. Fig. 461. Fig. 462. pressed on such plane by a point K, and by the lines ab, a'b', equal to the original on vertical planes, whatever their direction. 1131. An inclined line CD (fig. 460.) is represented on an horizontal or a vertical plane by ca, c'd, shorter than the line itself, except on a vertical plane, parallel to its projection, and on the horizontal plane c”d", where it is equal to the original CD. 1132. An inclined line EF (fig. 461.) moveable on its extremity E, may, by preserving the same inclination in respect of the plane on which it lies, have its projection successively in all the radii of the circle Ef, determined by the perpendicular let fall from the point F. 1133. Two lines GH, IK (fig. 462.), whereof one is parallel to an horizontal plane and the other inclined, may have the same projection m, n, upon such plane. Upon a vertical |