BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUN^D THE GIFT OF Henrg W, Sag* 1891 AMipbl. ' u. arY992 Cornell University Library Oescri 11 ptive eometry-pure and applied 3 1924 032 184 248 olin,anx The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924032184248 DESCRIPTIVE GEOMETRY AND MECHANICAL DRAWING SERIES By Frederick Newton Willson, C.E.. A.M. The specialized treatises constituting this series are uniform in size (nine inches by twelve inches, with text-page six and three-quarters by nine), printed on plate paper, and elaborately illustrated with photo-engravings, wood-cuts, cero- graphic process blocks, half-tone plates, etc. They are adapted both to class-room use and self-instruction. 1. ^Stote"Taking, Dimensioning and Ltettefing. A text-book on Free-hand and Mechanical Lettering in general, and on the lettering and dimensioning of " working drawings ; " also con- taining full instructions as to the sketching of bridge and machine details, for inspection or design. Various conventional methods of represent- ing materials are also given. Roman and Gothic letters, vertical and inclined, together with the Soennecken Round Writing, Reinhardt Gothic, and other alphabets much employed by engineers and architects, receive ample illustration ; while the total of sixty-five complete alphabets affords an unusual range of choice among serviceable fonrns. Full instructions are given as to the proportioning of titles, spacing, mechanical *' short-cuts,' ' etc. ; also a large number of designs for fancy comers and borders. $r-zS ^^^- 2, The Third Angle ^VTethod of making Working Dramings, A practical treatise on the American draught jng-office system of applying the principles of projection in the making of ' ' shop ' ' drawings. It contains a large number of general problems on projection, intersections, and the development of surfaces, together with such special problems as an upper-chord post-connection of a bridge ; standard screws, bolts and nuts, with tables of proportions ; helical springs ; rail and valve sections ; spur gear. Illustrated with eighty-five cerographic-process blocks, and eight wood engravings. $1.25 7iet. 3. Some ^Vlathematical Curves and their Graphical Construction. This work presents in compact form the more interesting and important properties, methods of construction, and practical applications of the curves with which it is essential that the architect and engineer should be familiar. It is also adapted to class-room use in mathematical courses. Among its special features are sections on homologous plane and space figures, given in connection with the conic sections, and laying a sound foundation for work in projective geometry ; link-motion curves and centroids, as an introduction to kinematic geometry ; historic notes and problems. Among other topics treated are the Helix ; Common Cycloid and its Companion ; Curtate and Prolate Ortho-cycloids; Hypo, Epi, and Peri-trochoids ; Special Trochoids, as the Ellipse, Straight Line, Lima^on, Cardioid, Trisectrix, Involute, Spiral of Archimedes ; Parallel Curves ; Conchoid ; Quadratrix ; Cissoid ; Tractrix ; Witch of Agnesi ; Cartesian Ovals ; Cassian Ovals ; Catenary ; Logarithmic Spiral; Hyperbolic Spiral. Lituus, and the Ionic Volute. The work concludes with a chapter on the nomenclature and double generation of cycloidal curves. Illustrated with fifty-four ceiographic blocks and one half-tone. $1-50 «i'^- 4. Practical Engineering Drauiing and Third Angle Projection, A practical course for students in scientific, technical and manual training schools, and for engineering or architectural draughtsmen. It includes not only the contents of the Brst three volumes of this series, but also full instructions as to the choice and use of drawing instruments and materials ; line tinting and shading ; conventional methods of representation; plane problems of frequent recurrence; blue-printing and other methods of graphic reproduction and illustration; isometric drawing; and cavalier perspective (oblique projection). One hundred and seventy-eight pages, two hundred and seventy illustrations, and sixty-five alphabets. 2.80 net. B, Shades, Shadouus and liinear Perspective. A short course for students of engineering or architecture, and for professional draughtsmen. For its reading a knowledge of elementary projection drawing is assumed. The methods employed in the best American practice receive especial emphasis. Illustrated by twenty-one cerographic blocks and three half-tones. 1.00 tiet. 6. Descriptive Geometry — Pure and Applied, uaith a chapter on fligher Plane Curves and the flelix. This work contains in logical sequence not only the matter constituting volumes 2, 3 and 5 of this series, but also a chapter on the pure descriptive geometry of Monge, with felaborate illustration of the mathematical surfaces of most importance to the graphicist, and with applica- tions to Trihedrals, Spherical Projections, Axonometric (including Isometric) Projection, One-plane Descriptive Geometry, Oblique Projection, etc., the whole constituting a broad, educational course. One hundred and ninety pages, illustrated with two hundred and eighty-four cerographic blocks, six half-tones, and twenty-one wood engravings. -^--.oo nvt. 7. Theoretical and Practical Graphics. This work embodies the entire contents of the six preceding treatises, in a volume of three hundred pages, with four hundred and seventy eight illustrations, sixty-five alphabets, and thirty-eight border designs. It constitutes a progressive course in graphical science. $4.00 ft,-t Published by THE MACMILLAN COMPANY, 66 Fifth Ave., New York. Loudon: MACMILLAN & CO., Limited. DESCRIPTIVE GEOMETRY-PURE AND APPLIED. DESCRIPTIVE GEOMETRY— PURE AND APPLIED WITH A CHAPTER ON HIGHER PLANE CURVES AND THE HELIX A THEORETICAL AND PRACTICAL TREATISE INCLUDING PROBLEMS OF INTERSECTION, DEVELOPMENT, TANGENCY, ETC., WOEKING DRAWINGS BY BOTH THE FIRST AND THIRD ANGLE SYSTEMS, TRIHEDRALS, MAP PROJECTION, SHADOWS AND PERSPECTIVE, AXONOMETRIC (including isometric) PROJECTION, OBLIQUE PROJECTION, KINEMATIC GEOMETRY, PROJECTIVE GEOMETRY AND RELIEF PERSPECTIVE, THE CONIC SECTIONS, TROCHOIDS, SPIRALS, ETC. PREPARED FOR COURSES IN GENERAL SCIENCE, ENGINEERING AND ARCHITECTURE FREDERICK NEWTON WILLSON C.E. (rensselaer) ) A.M. (princeton) Pi-ofesso7' of Descriptive Geometry^ Stereotomy and Technical Drawing in the John C. Ch'een School of Science, Princeton University ; Mem. Am. Soc. Mechanical Engineers; Associate Am,. Soc. Civil Engineers; Mem. Am,. Mathematical Society; Fellow American Association for the Advancement of Science. NEW YORK THE MACMILLAN COMPANY LONDON : MACMILLAN & CO., Ltd. 1898 ALL RIGHTS RESERVED COPYRIGHT 1897 BY FREDERICK N. WII,LSON. PREFACE URING the latter part of the eighteenth century, and in the time of her greatest need, France ^^y was laid under peculiar obligations by one of her most loyal sons — Gaspard Monge — who gave her, almost simultaneously, himself and a new science. In his Gmmetrie Descriptive the government instantly recognized a system which would give to its possessors a great advantage over others, and the new invention was therefore most carefully guarded as a state secret, the officers who were instructed in it being under strict orders to conceal it even from those who were employed in other departments of the public service. This monopoly was quite effectively maintained for some time, but in 1794 Monge was permitted to lecture on the subject in the newly established Ecole Normale of Paris. His geometry made its first visible public appearance in the form of short-hand notes, taken by his pupils. It was soon after issued in print under the title Geometrie Descriptive : Legons donn'ees aux Ecoles Kormnlcs, Van 3 de la Repuhlique. It was but a small work, yet marked an era in the his- tory of geometry, and for the first time placed both the Fine and the Constructive Arts on a thoroughly scientific foundation. Dealing with pure theory, it was to have been followed by a number of works by Monge on its practical applications, a series which he was precluded from preparing by arduous and prolonged services to his country, under Napoleon. Since Monge's day the extensions and modifications of the science have resulted in a very con- siderable literature. The purpose of this addition thereto is to furnish in concise form a broad view of the developments and applications of the subject, at the same time laying a sound foundation not only for all constructive work in engineering or architecture, but also for advanced mathematical work along projective lines. The author has aimed to illustrate the subject in a manner not only in keeping with its importance but also calculated to obviate as far as possible the difficulties usually met in its study by those who are somewhat deficient in the power of mental visualizing. The Third Angle method of applying orthographic projection, now almost universally employed in machine drawing and design, is treated fully, but its relation to the original method so indicated that the student should have no difficulty in transferring from one system to the other at will. Except as projectively obtained, the curves in Chapter V can hardly be considered as descriptive geometry matter, yet their inclusion is amply justified by their constant occurrence in projection draw- ing, and is an essential preliminary to the study of the algebraic surfaces of Chapter IX. Although the page numbers indicate that the following matter is selected from the author's larger work — Theoretical and Practical Graphics — yet having been originally prepared with a view to its separate issue in this form, it will be found that with the exception of two or three unimportant allusions it constitutes an entirely independent treatise. F. N. W. ''The study of Descriptive Geometry possesses an important philosophical peculiarity^ quite independent of its high industrial utility. This is the advantage which it so pre-eminently offers in habituating the mind to consider very complicated geometrical combinations in space, and to follow with precision their continual correspondence with the figures which are actually traced — of thus exercisiiig to the utmost, in the most certain and precise manner, that important faculty of the human mind which is properly called * imagination,^ and which consists, in its elementary and positive acceptation, in representing to ourselves, clearly and easily, a vast and variable collection of ideal objects, as if they were really before us. While it belongs to the geometry of the ancients by the character of its solutions, on the other hand it approaches the geometry of the m.oderns by the nature of the questions which compose it. These questions are in fact eminently remarkable for that generality which constitutes the true fundamental character of modern geo?netry ; for the methods used are always conceived as applicable to any figures whatever, the peculiarity of each havi7ig only a purely secondary influence." Auguste Comte: Cours de Philosophic Positive. "y? mathematical problem may usually be attacked by what is termed in military parlance the method of * systematic approach;' that is to say, its solution may be gradually felt for, even though the successive steps leading to that solution cannot be clearly foreseen. But a Descriptive Geometry problem must be seen through and through before it can be^ attempted. The entire scope of its conditions as well as each step toward its solution must be grasped by the Imagination. It must be 'taken by assault.'" George Sydenham Clarke, Captain, Royal Engineers. THEORETICAL AND PRACTICAL GRAPHICS. CHAPTER I. FIRST PRINCIPLES, WITH GENERAL SURVEY OP THE FIELD OF GRAPHIC SCIENCE. E"ig-- i. 1. Geometrically considered, any combination of points, lines and surfaces is called a figure. A figure lying wholly in one plane is called a plane figure; otherwise a space figure. 2. Among the methods of investigating and demonstrating the mathematical properties of figures, and of solving problems relating to them, that called projection is at once one of the most valuable and interesting, constituting, as it does, the common basis of nearly all graphic representations, whether of artist, architect or engineer. When using this method figures are always considered in connection with a certain point called a centre of prelection. In Fig. 1 let S be an assumed centre of projection and A any point in space. The straight line SA, joining S with A, is called a prelecting line or ray, or simply a projector, and its intersection, a, with any line CD, is its projection upon that line. It is otherwise expressed by saying that A is pro- jected upon CD at a. In the same way the point B is projected^ from S upon the plane MN at b; or, in other words, b is — for the assumed position of S — the projection' of B upon the plane. It is with projection upon a plane that we are principally concerned. The word "projection" is used not only to indicate the method of representation but also the representation itself. In certain other branches of mathematics it has a yet more extended significance, being employed to denote the represen- tation of any curve or surface upon any other. 3. A figure, as ABC (Fig. 2), is projected upon a plane, MN, by drawing projectors, SA, SB, SC, through its vertices and prolonging them, if necessary^, to meet the plane. The figure abc, formed by joining the points in Avhich the projectors intersect the plane, is then the projection of the first, or original figure. The plane upon which the jjrojection is made is called the plane of projection. 4. Were abc (Fig. 3) the original figure and MN the plane of projection, then would ^ -B C be the projection desired. Each figure may thus be considered a projection of the other for a given position of S, and when so related figures are said to correspond to each other. Points that are collinear (or in line) with the centre of projection, as a and A, are called corresponding points. 1 Were 5 the muzzle of a gun, and B a huUet speeding from it toward the plane, it would he projected against or through the plane at b. The appropriateness of the term "projection" Is obvious. 2 In Fig. 3 the projectors meet the plane between the centre £^ and the given figure. E'ig'. 3. // \n w/ 7 «4ii| f f 2 THEORETICAL AND PRACTICAL GRAPHICS. 5. Having indicated what projections are and how obtained, it will be well, before giving their grand divisions and sub - divisions, to state the nature and extent of the field in which they may be employed. The mathematical properties of geometrical figures, as also the propositions and problems involving: them, are divided into two classes, metrical and descriptive. In the first class the idea of quantity necessarily enters, either directly — as in measurement, or indirectly — as in ratio.' In the second or descriptive class, however, we find involved only those properties dependent upon relative position.'^ Descriptive properties are unaltered by projection, while, as ordinarily regarded, but few metrical properties are projective.' The main province of projection is obvious. 6. Descriptive Geometry is that branch of mathematics in which figures are represented and their descriptive properties investigated and demonstrated by means of projection. -tos^ tos„ to So, DIVISIONS OF PROJECTION. 7. All projections may be divided into two general classes. Central and Parallel. If the centre of projection be at a finite distance, as in Figs. 2 and 3, the projection obtained' is called a central projection ; but if we suppose it to be at ^ infinity, as in Fig. 4, projectors from it will then evidently be parallel, and the resulting figure is called a parallel projection of the original figure. Parallel projection is thus seen to be merely a special case of central projection, yet each has been independently developed to a high degree and has an extensive literature. 8. The terms Conical and Cylindrical are employed by many writers synonymously with central and parallel respectively. Central projections are also occasionally called Radial or Polar. ^^s- B. Eemaek. — A straight line is said to generate a conical surface (see Fig. 5) when it constantly passes through a fixed point (the vertex), and is guided in its motion hy a given fixed curve (the- ^/ directrix). The moving straight line is a generatrix of the surface, and its various positions are called _/ elements of the surface. If the vertex of a conical surface be removed to infinity the elements will become parallel, and we shall have a cylindrical surface, which may be also defined as the surface (see Fig. 6V generated by a straight line that is guided in its motion by a given fixed curve, and is in any position parallel to a given, fixed, straight line. The origin of the terms conical and cylindrical as applied to projection is obvious. We have now to mention the more important sub - divisions of projections, with the sciences based upon them. The names depend in certain cases upon the nature of the centre of projection, while- in others they are due to some particular application. Under Central (or Conicdl) projectio)) we have: — 9. Projective Geometry {Geometry of Position). ^Vhile in its most general sense this science includes aU projections, yet in its ordinary acceptation it may ])g defined as that l)raneh of mathematics in which— with the centre of projection considered as a mathematical point at a finite di.'n of a circle from a point in the plane of the latter, we would recjuire simply a secant vanishing line, M N (Fig. 94), and an axis of homology parallel to it. Take any ]>(.)int P on the vanishing line and join it with any point A' of the circle. PK meets the axis at // ; hence whatever line corrcqjoniJs to PA" must also meet the axis at y. OP is analogous to 5' .4 of Fig. 93, in that it meets its corresponding line at infinity, i.e., is ]>arallel to it. Therefore y k, parallel to OP, corresponds to Py, and meets the ray OK at l\ corresponding to A'. Then A' joined with any other point R gives A';. Join 2 with k and prolong R to intersect /.■ 2, olitaining r, another point of the hyperbola. 151. In Fig. 93, were a tangent drawn to arc A H B at B, it would meet the axis in a point which, like all points on the axis, '■ corresjionds to itself" From that jioint the projection of that tangent on the lower plane would be parallel to iS' 77, since they are to meet at infinity. Or, ii SJ is parallel to the tangent at B, then ./ will lie the projection of J' at infinity, where *S'/ meets the tangent; J will l)e therefore one point of the projection of said tangent on the lower plane; while another point would lie, as previously stated, that in which the tangent at B meets the axis. 152. Analogously in Fig. 94, the tangents at M and N meet the axis, as at F and E; but the projectors OM and ON go to points of tangency at infinity; M and N are on a "vanishing line"; hence OM is parallel to tlie tangent at infinity, that is, to the asymptote (see Art. 134) through F; while the other asymptote is a parallel through A' to N. 153. As in Fig. 93 the projectors from ;S' to all points "f the arc above the level of \ is parallel t(.) it. is I^ig- SS- 50 THEORETICAL AND PRACTICAL GRAPHICS. the centre of homology or perspective- centre. All points in the plane ir are their own perspectives, or, in other words, correspond to themselves. Therefore B" is one point of the projection or perspective of the line A B, heing the intersection oi AB with tt. The line v, parallel to A B, would meet the :F'ig'. ss. latter at infinity; therefore v, in the vanishing plane <^', would be the projection upon it of the point at infinity. Joining v with B", and cutting v B" by rays OA and OB, gives A' B' as the relief- perspective of A B. The plane through and A B cuts ^ in B" n, which is an axis of homology for AB and A' B', exactly as mn in Fig. 92 is for A^B^ and A^B.^. X-ig-. ©7". As DC in Fig. 96 is parallel io A B, a parallel to it through is again the line Ov. LINK-MOTION CURVES. 51 The trace of Z)C on tt is C". Joining v with C" and cutting v C" by rays D, DC, obtains D' C" in the same manner as A' B' was derived. The originals of A' B' and CD' are parallel lines; but we see that their relief - perspectives meet at v. The vanishing plane is therefore the locus" of the vanishing points of lines that are parallel on the original object, while the plane of homology is the locus of the axes of homology of corresponding lines; or, differently stated, any line and its relief -perspective will, if produced, meet on the plane of homology. 156. Fig. 97 is inserted here for the sake of completeness, although its study may be reserved, if necessary, until the chapter on projections has been read. In it a solid object is represented at the left, in the usual views, plan and elevation; GL being the ground line or axis of intersection of the planes on which the views are made. The planes v and ^' are interchanged, as compared with their positions in Pig. 96, and they are seen as lines, being assumed as perpendicular to the paper. The relief- perspective appears between them, in plan and elevation. The lettering of A B and D C, and the lines employed in getting their relief- perspectives, being identical with the same constructions in Fig. 96, ought to make the matter clear at a glance to all who have mastered what has preceded. Burmester's Grundzuge der Relief -Perspective and Wiener's Darstellende Geometric are valuable reference works on this topic for those wishing to pursue its study further; but for special work in the line of homological plane figures the student is recommended to read Cremona's Projective Geometry and Graham's Geometry of Position, the latter of which is especially valuable to the engineer or architect, since it illustrates more fully the practical application of central projection to Graphical Statics. LINK - MOTION CURVES 157. Kinematics is the science which treats of pure motion, regardless of the cause or the results of the motion. It is a purely kinematic problem if we lay out on the drawing-board the path of a point on the connecting-rod of a locomotive, or of a point on the piston of an oscillating cylinder, or of any point on one of the moving pieces of a mechanism. Such problems often arise in machine design, especially in the invention or modification of valve -motions. Some of the motion - curves or point -paths that are discovered by a study of relative motion are without special name. Others, whose mathematical properties had already been investigated and the curves dignified with names, it was later found could be mechanically traced. Among these the most familiar examples are the Ellipse and the Lemniscate, the latter of which is employed here to illustrate the general problem. The moving pieces in a mechanism are rigid and inextensible, and are always under certain conditions of restraint. "Conditions of restraint" may be illustrated by the familiar case of the con- necting-rod of the locomotive, one end of which is always attached to the driving-wheel at the crank-pin and is therefore constrained to describe a circle about the axle of that wheel, while the other end of the rod must move in a straight line, being fastened by the "wrist-pin" to the "cross- head," which slides between straight "guides." The first step in tracing a point -path of any mechanism is therefore the determination of the fixed points, and a general analysis of the motion. * Locus Is the liatin for place; and in rather untechnlcal language, although in the exact sense in which it is used mathe- matically, we may say that the locus of points or lines is the place where you may expect to find them under their conditions of restriction. For example, the surface of a sphere is the locus of all points equidistant from a fixed point (its centre). The locus of a point moving In a plane so as to remain at a constant distance from a given fixed point, is a circle having the latter point as its centre. 52 THEORETICAL AND PRACTICAL GRAPHICS. 158. We have given, in Fig. 98, two links or bars, MN and SP, fastened at N and P by pivots to a third link, NP, while their other . extremities are pivoted on stationary axes at M and S The only movement possible to the point N is therefore in a circle about M; while P is equally limited to circular motion about S. The points on the link NP, with the exception of its 2 MN 2 MS 3 THE LEMNISCATE AS A LINK-MOTION CURVE extremities, have a compound motion, in curves whose form it is not easy to predict and which differ most curiously from each other. The figure-of-eight curve shown, otherwise the "Lemniscate of Bernoulli," is the point -path of Z, the link NP being supposed prolonged by an amount, P Z, equal to NP. Since NP is constant in length, if N were moved along to F, the point P would have to be at a distance NP from i^, and also on the circle to which it is confined; therefore its new position /, is at the intersection of the circle Psr by an arc of radius P N, centre F. Then Ff, prolonged by an amount equal to itself, gives /, , another point of the Lemniscate, and to which Z has then moved. All other positions are similarly found. If the motion of N is toward D it will soon reach a limit. A, to its further movement in that direction, arriving there at the instant that P reaches a, when NP and PS will be in one straight line, SA. In this position any movement of P either side of a will drag N back over its former path; and unless P moves to the left, past a, it would also retrace its path. P reaches a similar ''dead point" at v. To obtain a Lemniscate the links NP and PS had to be equal, as also the distance .1/8 to MN. Bv varying the proportions of the links, the point- paths would be correspondingly aft'ected. INSTANTANEOUS CENTRES.— CENTRO IDS. 53 By tracing the path of a point on PN produced, and as far from iV as ^ is from P, the student will obtain an interesting contrast to the Lemniscate. If M and S were joined by a link, and the latter held rigidly in position, it would have been called the fixed link; and although its use would not have altered the motions illustrated, and it is not essential that it should be drawn, yet in considering a mechanism as a whole, the line joining the fixed centres always exists, in the imagination, as a link of the complete system. INSTANTANEOUS CENTRES. — CENTROIDS. 159. Let us imagine a boy about to hurl a stone from a sling. Just before he releases it he runs forward a few steps, as if to add a little extra impetus to the stone. While taking those few steps a peculiar shadow is cast on the road by the end of the sling, if the day is bright. The boy moves with respect to the earth; his hand moves in relation to himself, and the end of the sling describes a circle about his hand. The last is the only definite element of the three, yet it is sufiicient to simplify otherwise difficult constructions relating to the complex curve which is described relatively to the earth. 54 THEORETICAL AND PRACTICAL GRAPHICS. A tangent and a normal to a circle are easily obtained, the former being, as need hardly be stated at this point, perpendicular to the radius at the point of tangency, while the normal simply coincides in direction with such radius. If the stone were released at any instant it would fly off in a straight line, tangent to the circle it was describing about the hand as a centre; but such line would, at the instant of release, be tangent also to the compound curve. If, then, we wish a tangent at a given point of any curve generated by a point in motion, we have but to reduce that motion to circular motion about some moving centre; then, joining the point of desired tangency with the — at that instant — position of the moving centre, we have the normal, a perpendicular to which gives the tangent desired. A centre which is thus used for an instant only is called an instantaneous centre. 160. In Fig. 99 a series of instantaneous centres are shown and an important as well as inter- esting fact illustrated, viz., that every moving piece in a mechanism might be rigidly attached to a certain curve, and by the roUing of the latter upon another curve the link might be brought into all the positions which its visible modes of restraint compel it to take. 161. In the "Fundamental" part of Fig. 99 ^ -B is assumed to be one position of a link. We next find it, let us suppose, at A' B', A having moved over A A', and B over BB'. Bisecting A A' and B B' by perpendiculars intersecting at 0, and drawing A, OA', OB and OB', we have A A' = d^^B OB', and evidently a point about which, as a centre, the turning oi AB through the angle 6^ would have brought it to A' B'. Similarly, if the next position in which we find AB is A" B", we may find a point s as the centre about which it might have turned to bring it there; the angle being 6^, probably different from 0^. N and m are analogous to and s. If Os' be drawn equal to Os and making with the latter an angle 6^, equal to the angle AOA', and if Os were rigidly attached to A B, the latter would be brought over to A' B' by bringing s' into coincidence with Os. In the same manner, if we bring s' n' upon sn through an angle 0^ about s, then the next position. A" B", would be reached by A B. 0' s' n' m' is then part of a polygon whose rolling upon Osnm would bring AB into all the positions shown, provided the, polygon and the line were so attached as to move as one piece. Polygons whose vertices are thus obtained are called central polygons. If consecutive centres were joined we would have curves, called centroids*, instead of poh-gons- the one corresponding to Osnm being called the fixed, the other the roUiiig centroid. The perpen- dicular from upon A A' is a normal to that path. But were A to move in a circle, the normal to its path at any instant would be simply the radius to the position of A at that instant. If, then, both A and B were moving in circular paths, we would find the instantaneous centre at the intersection of the normals (radii) at the points A and B. 162. In Fig. 98 the instantaneous centre about which the whole link XP is turning, is at the intersection of radii MN and SP (produced); and calling it X we would have XZ for the normal at Z to the Lemniscate. 163. The shaded portions of Fig. 99 illustrate some of the forms of centroids. The mechanism is of four links, opposite hnks equal. Unlike the usual quadrilateral fulfillino- this condition, the long sides cross, hence the name "anti-i)arallelogram." The "fixed link (a)" corresponds to 31 S of Fig. 98, and its extremities are the centres of rotation of the short links, whose ends, / and f„ describe the dotted circles. For the given position T is evidently the instantaneous centre. Were a bar pivoted at T and *Eeuleaux' nomenclature; also called cenirodes t)y a number of writers on Kinematics. TROCHOIDS. 55 fastened at right angles to "moving link (a)," an infinitesimal turning about T would move "link (a)" exactly as under the old conditions. By taking "link (a)" in all possible positions, and, for each, prolonging the radii through its extremities, the points of the fixed centroid are determined. Inverting the combination so that "moving link (a)" and its opposite are interchanged, and proceeding as before, gives the points of "rolling centroid (a)." These centroids are branches of hyperbolas having the extremities of the long links as foci. By holding a short link stationary, as "fixed link (b)," an elliptical fixed centroid results; "rolling centroid (b)" being obtained, as before, by inversion. The foci are again the extremities of the fixed and moving links. Obviously, the curved pieces represented as screwed to the links would not be employed in a practical construction, and they are only introduced to give a more realistic effect to the figure and possibly thereby conduce to a clearer understanding of the subject. 164. It is interesting to notice that the Lemniscate occurs here under new conditions, being traced by the middle point of "moving link (a)." The study of kinematics is both fascinating and profitable, and it is hoped that this brief glance at the subject may create a desire on the part of the student to pursue it further in such works as Reauleaux' Kinematics of Machinery and Burmester's Lehrbuch der Kinematik. 165. Before leaving this topic the important fact should be stated, which now needs no argument to establish, that the instantaneous centre, for any position of a moving piece, is the point of contact of the rolling and fixed centroids. We shall have occasion to use this principle in drawing tangents and normals to the TROCHOIDS which are the principal Roulettes, or roll -traced curves, and which may be defined as follows: — If, in the same plane, one of two circles roll upon the other without sliding, the path of any point on a radius of the rolling circle or on the radius produced is a trochoid. 166. The Cycloid. Since a straight line may be considered a circle of infinite radius, the above definition would include the curve traced by a point on the circumference of a locomotive wheel as it rolls along the rail, or of a carriage wheel on the road. This curve is known as a cycloid,* and is shown in Tnabc, Fig. 100. It is the proper outline for a portion of each tooth in a certain case of gearing, viz., where one wheel has an infinite radius, that is, becomes a "rack." Were Tg a ceiling -corner of a room, and T^^ the diagonally opposite floor -corner, a weight would slide from T^ to T^, more quickly on guides curved in cycloidal shape than if shaped to any other curve, or if straight. If started at s, or any other point of the curve, it would reach T^^ as soon as if started at T^. 167. In beginning the construction of the cycloid we notice, first, that as T VD rolls on the straight line AB, the arrow DRT will be reversed in position (as at D^T^ as soon as the semi- circumference T2>D has had rolling contact with AB. The tracing point will then be at T^, its maximum distance from A B. When the wheel has rolled itself out once upon the rail, the point T will again come in contact with the rail, as at r^. *"Altliougli the invention of the cycloia is attributed to Galileo, it is certain that the family of curves to which it belongs had been known and some of the properties of such curves investigated, nearly two thousand years before Galileo's time, if not earlier. For ancient astronomers explained the motion of the planets by supposing that each planet travels uniformly round a circle whose centre travels uniformly around another circle."— Proctor, Geometry of Q/cloids. 56 THEORETICAL AND PRACTICAL GRAPHICS. The distance TT^^ evidently equals 2irr, when r=TR. We also have TD^—D^T^^ = irr If the semi-circumference TZD (equal to ttt) be divided into any number of equal parts, and the path of centres RR^ (again =7rr) into the same number of equal parts, then as the points 1, 2, etc., come in contact with the rail, the centre R will take the positions R^,R^, etc., directly above the corresponding points of contact. A sufficient rolling of the wheel to bring point 2 upon A B would evidently raise T from its original position to the former level of 2. But as T must always be at a radius' distance from i2, and the latter would by that time be at R^, we would find T located at the intersection (n) of the dotted line of level through 2 by an arc of radius R T, centre R^. Similarly for other points. The construction, summarized, involves the drawing of lines of level through equidistant points of division on a semi-circumference of the rolling circle, and their intersection by arcs of constant radius (that of the rolling circle) from centres which are the successive positions taken by the centre of the rolling circle. It is worth while calling attention to a point occasionally overlooked by the novice, although almost self-evident, that, in the position illustrated in the figure, the point T drags behind the centre R until the latter reaches R^, when it passes and goes ahead of it. From R^ the line of level through 5 could be cut not alone at c by an arc of radius cR^ but also in a second point; evidently but one of these points belongs to the cycloid, and the choice depends upon the direction of turning, and upon the relative position of the rolling centre and the mo\'ing point. This matter requires more thought in drawing trochoidal curves in which both circles have finite radii, as will appear later. IFig. lOO. 168. Were points T^ and T^^ given, and the semi -cycloid T^T^^ desired, we can readily ascertain the "base," AB, and generating circle, as follows: Join T^ with T,^; at any point of such line, as X, erect a perpendicular, xy; from the similar triangles xyT,^ and T^D.T,,, having angle <^ common and angles equal, we see that xy.xT,, T D • D T :2r:irr::2:Tr::l:-_^- or, very nearly, as 14 : '22. If, then, , we lay off xT,, equal to twenty-two equal parts on any scale, and a perpendicular, xy, fourteen parts of the same scale, the line yT,, will be the base of the desired curve; while tlie diameter of the generating circle will be the perpendicular from T^ to y T^, prolonged. 169. To draw thfe tangent to a cycloid at any point is a simple matter, if we see the analogy between the point of contact of the wheel and rail at any instant, and the hand used in the former illustration (Art. 169). At any one moment each point on the entire wheel may be considered as describing an infinitesimal arc of a circle whose radius is the line joining the point M-ith the point of contact on the rail. The tangent at iV, for example, (Fig. 100), would be t N, perpendicular to the normal, No, joining N with o; the latter point being found by using N as a centre and THE CYCLOID.— COMPANION TO THE CYCLOID. 57 cutting ^5 by an arc of radius equal to ml, in which m is a point at the level of JV on any position of the rolling circle, while I is the corresponding point of contact. The point o might also have been located by the following method: Cut the line of centres by an arc, centre iV, radius TR; would obviously be vertically below the position of the roUing centre thus determined. 170. The Companion to the Cycloid. The kinematic method of drawing tangents, just applied, was devised by Roberval, as also the curve named by him the "Companion to the Cycloid," to which allusion has already been made (Art. 120) and which was invented by him in 1634 for the purpose of solving a problem upon which he had spent six years without success, and which had foiled Galileo, viz., the calculating of the area between a cycloid and its base. Galileo was reduced to the expedient of comparing the area of the cycloid with that of the rolling circle by weighing paper models of the two figures. He concluded that the area in question was nearly but not exactly three times that of the rolling circle. That the latter would have been the correct solution may be readily shown by means of the "Companion," as will be found demonstrated in Art. 172. 171. Suppose two points coincident at T (Fig. 101) and starting simultaneously to generate curves, the first of these points to trace the cycloid during the rolling of circle TVD, while the second is to move independently of the circle and so as to be always at the level of the point tracing the cycloid, yet at the same time vertically above the point of contact of the circle and base. This makes the second point always as far from the initial vertical diameter, or axis, of the cycloid, as the length of the arc from T to whatever level the tracing point of the latter has then reached; that is, MA equals arc T Hs; R equals quadrant Tsy. Adopting the method of Analytical Geometry, and using as the origin, we may reach any point. A, on the curve, by co-ordinates, as Ox, x A, of which the horizontal is called an abscissa, the vertical an ordinate. By the preceding construction Ox equals arc sfy, while xA equals sw — the sine of the same arc. The "Companion" is therefore a curve of sines or sinusoid, since, starting from 0, the abscissas are equal to or proportional to the arc of a circle, while the ordinates are the sines of those arcs. It is also the orthographic projection of a 45° -helix. This curve is particularly interesting as "expressing the law of the vibration of perfectly elastic solids; of the vibratory movement of a particle acted upon by a force which varies directly as the distance from the origin; approximately, the vibratory movement of a pendulum; and exactly the law of vibration of the so-called mathematical pendulum."* (See also Art. 356). 172. From the symmetry of the sinusoid with respect to RRg and to 0, we have area TA0R = EC0R,; adding area D EL R to both mem- bers we have the area between the sinusoid and TD and DE equal to the rectangle RE, or one-half the rect- angle D EKT; or to g 'r r x 2 r- = Trr^, the area of the rolling circle. As T ACE is but half of the entire sinusoid, it is evident that the total area below the curve is twice that of the generating circle. The area between the cycloid and its "companion" remains to be determined, but is readily ascertained by noting that as any point of the latter, as A, is on the vertical diameter of the circle * Wood, Elements of Co-ordinate Geometry, p. 209. T^g. lOl. 58 THEORETICAL AND PRACTIQAL GRAPHICS. passing through the then position of the tracing point, as a, the distance, A a, between the two curves at any level, is merely the semi-chord of the rolling circle at that level. But this, evidently, equals Ms, the semi-chord at the same level on the equal circle. The equality of Ms and A a makes the elementary rectangles Mss^m^ and AA^a^a equal; and considering all the possible similarly -constructed rectangles of infinitesimal altitude, the sum of those on semi-chords of the rolling circle would equal the area of the semi-circle TDy, which is therefore the extent of the area between the two curves under consideration. The figure showing but half of a cycloid, the total area between it and its "companion" must be that of the rolling circle. Adding this to the area between the "companion" and the base makes the total area between cycloid and base equal to three times that of the rolling circle. 173. The paths of points carried by and in the plane of the rolling circle, though not on its circumference, are obtained in a manner closely analogous to that employed for the cycloid. In Fig. 102 the looped curve, traced by the arrow-point while the circle CHM rolls on the base A B, is called the Curtate Trochoid. To obtain the various positions of the tracing point T describe a circle through it from centre R. On this circle lay off any even number of equal arcs, and draw radii from R to the points of division; also "lines of level" through the latter. The radii drawn intercept equal arcs on the rolling circle CHM, whose straight equivalents are next laid off on the path of centres, giving R^, R^, etc. While the first of these arcs rolls upon A B, the point T turns through the angle TR 1 about R, and reaches the line of level through point 1. But T is always at the distance R T (called the tracing radius) from R; and, as R has reached Ri in the roUing supposed, we will find Ti — the new position of T — by an arc from R^, radius TR, cutting said line of level. x^xir. loa. After what has preceded, the figure may be assumed to be self-interpreting, each position of T having been joined with the position of R which determined it. 174. Were a tangent wanted at any point, as T,, we have, as before, to determine the point of contact of rolling circle and line when T reached T,, and use it as an instantaneous centre. T was obtained from i?,; and the point of contact must have been vertically below the latter and on A B. Joining such point to T, gives the normal, from which the tangent follows in the usual way 175. The Prolate Trochoid. Had we taken a point inside of the circle CHM and constructed its path, the only difference between it and the curve illustrated would have been in the name and the HYPO-, EPI- AND PERI-TROGHOIDS. 69 shape of the curve. An undulating, wavy path would have resulted,, called the prolate trochoid; but, as before, we would have described a circle through the tracing point; divided it into equal parts; drawn lines of level, and cut them by arcs of constant radius, using as centres the successive positions of R. A bicycle pedal describes a prolate trochoid relatively to the earth. HYPO-, EPI- AND PERI -TROCHOIDS. 176. Circles of finite radius can evidently be tangent in but two ways — either externally, or internally; if the latter, the larger may roll on the one within it, or the smaller may roll inside the larger. When a small circle rolls within a larger, the radius of the latter may be greater than the diameter of the rolling circle, or may equal it, or be smaller. On account of an interesting property of the curves traced by points in the planes of such rolling circles, viz., their capability of being generated, trochoidally, in two ways, a nomenclature was necessary which would indicate how each curve was obtained. This is included in the tabular arrangement of names below, and which was the outcome of an investigation* made by the writer in 1887 and presented before the American Association for the Advancement of Science. In accepting the new terms, advanced at that time. Prof. Francis Reuleaux suggested the names Ortho - cycloids and Cyclo - orthoids for the classes of curves of which the cycloid and involute are respectively representative; orthoids being the paths of points in a fixed position with respect to a straight line rolling upon any curve, and cyclo - orthoid therefore implying a circular director or base -curve. These appropriate terms have been incorporated in the table. For the last column a point is considered as within the rolling circle of infinite radius when on the normal to its initial position, and on the side toward the centre of the fixed circle. As will be seen by reference to the Appendix, the curves whose names are jjreceded by the same letter may be identical. Hence the terms curtate and prolate, while indicating whether the tracing point is beyond or within the circumference of the rolling circle, give no hint as to the actual form of the curves. In the table, R represents the radius of the rolling circle, F that of the fixed circle. NOMENCLATURE OF TROCHOIDS. Position of Tracing or Describing Point. Circle rolling upon Straight Line. F=co Circle rolling upon circle. Straight Line rolling upon Circle. R=co External contact. Internal contact. Larger Circle rolling. Smaller circle rolling. Ortho-cycloids. Epitrochoids. 2 R > F. 2 R< F. 2 R = F. Cyclo-orthoids. Peri trochoids. Major Hypotrochoids. Minor Hypotrochoids. Medial Hypotrochoids. On circumference i rvrlnirf of rolling circle. | ^V^^^'^- (a) Epicycloid. (a) Pericycloid. (d) Major Hypocycloid. (d) Minor Hypocycloid. Straight Hypocycloid. Involute. Within Prolate Circumference, Trochoid. (b) Prolate Epitrochoid. (c) Prolate Peritrochoid . (e) Major Prolate Hypotrochoid. (f) Minor prolate Hypotrochoid. (g) Prolate Elhptical Hypotrochoid. Prolate Cyclo-orthoid. Without Curtate Circumference. Trochoid. (c) Curtate Epitrochoid. (b) Curtate Peritrochoid \ (f) Major Curtate Hypotrochoid. (e) Minor Curtate Hypotrochoid. (g) Curtate Elliptical Hypotrochoid. Curtate Cyclo-orthoid. 177. From the above we see that the prefix epi {over or iipon^ denotes the curves resulting from external contact; hypo {wilder) those of internal contact with smaller circle rolling; while peri (about) indicates the third possibility as to rolling. * Re-printed in substance in the Appendi.^, 60 THEORETICAL AND PRACTICAL GRAPHICS. 178. The construction of these curves is in closest analogy to that of the cycloid. If, for example, ire desire a viajar hypocydoid, we first draw two circles, mVP, mxL, (Fig. 103), tangent internally, of which the rolling circle has its di- ameter greater than the radius of the fixed circle. Then, as for the cycloid, if the tracing -point is P, we divide the semi - circumference mVP into equal parts, and from the fixed centre, F, describe circles through the points of division, as those through 1, 2, 3, 4 and 5. These replace the "lines of level" of the cycloid, and may be called circles of distance, as they show the varying distances of the point P from F, for definite amounts of angular rotation of the former. For if the circle PVm were simply to rotate about R, the point P would reach m during a semi - rotation, and would then be at its maximum distance from F. After turning through the equal arcs P-1, 1-2, etc., its distances from F would be i^^a and Fb respectively. If, however, the turning of P about R is due to the roUing of circle PVm upon the arc mxz, then the actual position of P, for any amount of turning about R, is determined by noting the new position of R, due to such rolling, as R^, R^, etc., and from it as a centre cutting the proper circle of distance by an arc of radius R P. Since the radius of the smaller circle is in this case three - fourths that of the larger, the angle mFz (135°), at the centre of the latter, intercepts an arc, mxz, equal to the 180° -arc, viVP, on the smaller circle ; for equal arcs on tmequal circles are subtended by angles at the centre which are inversely proportional to the radii. As a proportion we would have Fm:Rm:: 180° : 135°. (In an inverse proportion between angles and radii, in two circles, the "means" must belong to one circle and the " extremes " to the other). While arc mVP rolls upon arc mxz, the centre R will evidently move over circular arc R---R^. Divide mxz into as many equal parts as m F P and draw radii from F to the points of division ; these cut the path of centres at the successive positions of R. When arc m 5-4, for example, has rolled upon its equal muv, then R will have reached R.^; P will have turned about R through angle PR2^mR4:, and will be at n, the intersection of bfg — the circle of distance through 2 — by an arc, centre R,, radius R P. Similarly for other points. 179. General solution for all trochoidal curves, illustrated by epi- and peri- trochoids. To trace the path of any point on the circumference of a circle so rolling as to give the epi- or j)er/- cycloid, requires a construction similar at every step to that of the last article. The same remark applies equally to the path of a point within or beyond the circumference of the rolling circle. This is shown in Fig. 104, before describing which in detail, however, we will summarize the steps for any and all trochoids. Letting P represent the tracing point, R the centre of the rolling circle and F that of the fixed circle, we draw (1) a circle through P, centre R; (2) a circle (path of centres) through R, centre F; (3) ascertain by a proportion (as described in the last article) how many decrees of arc on either circle are equal to the prescribed arc of contact on the other; (4) on the path of centres lay EPI- AND PERI TROCHOIDS. 61 off — from the initial position of R and in the direction of intended rolling — whatever number of degrees of contact has been assigned or ascertained for the fixed circle, and divide this arc by radii from F into any number of equal parts, to obtain the successive positions of jR, as R^, R^, etc.; (5) on the circle through P lay off^ — -from the initial position of P, and in the direction in which it will move when the assigned rolling occurs — the same number of degrees that have been assigned or calculated as the contact arc of the rolling circle, and divide such arc into the same number of equal parts that was adopted for the division of the path of centres; (6) through the points of division obtained in the last step draw " citcles of distance " with centre F, numbering them from ^^TQENERAToj^ P; (7) finally, to get the suc- cessive positions of P, use RP (the " tracing radius ") as a con- stant radius, and cut each circle of distance by an arc from the like -numbered position from R, selecting, of course, the right one of the two points in which said curves will always intersect when not tangent. In Fig. 104 the path of the point P is determined (a) as car- ried by the circle called "first generator," rolling on the exterior of the "first director"; (b) as carried by the "second generator" which rolls on the exterior of the "second director" — which it also encloses. In the first case the resulting curve is a prolate epi- trochoid; in the second a curtate peritrochoid ; but such values were taken for the diameters of the circles, that P traced the same curve under either condition of roUing.* These (before reduction with the camera) were 3" and 2" for first generator and first director, respectively. For the epitrochoid a semi -circle is drawn through P from rolling centre R; similarly with centre p for the peritrochoid. Dividing these semi -circles into the same number of equal parts, draw next the dotted "circles of distance" through these points, all from centre F. The figure illustrates the special case where the two sets of "circles of distance" coincide. The various positions of P, as Pi, Pa, etc., are then located by arcs of radii RP or p P, struck from the successive positions of i? or p and intersecting the proper "circle of distance." *Kegardlng their double generation refer to tlie Appendix. In iUustrating botli methods in one flgure It -wiU add greatly to the appearance and also the Intelligibility of the drawing if colors are used, red for one construction and blue for the other. 62 THEORETICAL AND PRACTICAL GRAPHICS. For example, the turning of P through the angle PRl about R would bring P somewhere upon the circle of distance through point 1; but that amount of turning would be due to the rolling of the first generator over the arc m Q, which would bring n upon Q and carry R to i2,; P would therefore be at ?„ at a distance RP from R^, and on the dotted arc through 1. Similarly in relation to p. ^^'hen s reached A-, in the rolling, we would find P at P^. Each position of P is joined with each of the centres from which it could be obtained. SPECIAL TKOCHOIDS. 180. The Ellipse and Straight Line. Two circles are called Cardanic* if tangent internally and the diameter of one is twice that of the other. If the smaller roll in the larger, all points in the plane of the generator will describe ellipses except points on the circumference, each of which will move in a straight line — a diameter of the director. Upon this latter property the mechanism known as "White's Parallel Motion" is based, in which a piston-rod is pivoted to a small gear-wheel whiqh rolls on the interior of a toothed annular wheel whose diameter is twice that of the pinion. 181. The Li'inagon and Cardioid. The Limagon is a curve whose points may be obtained by drawing random secants through a point on the circumference of ^'igr- los. a circle, and on each laying off a constant distance, on each side of the second point in which the secant cuts the circle. In Fig. 105 let v and d be random secants of the circle 0ns; then if nv, np, ca and c d are each equal to some con- stant, b, we shall have v, p, a and d as four points of a Limajon. Refer points on the same secant, as a and d, to and the diam- eter Os; we then have d=p=^0 c + cd = '2,r cos6 + h, while Oa = 2 r cos 6 — b ; hence the polar equation is p = 2r cos 6±b. When 6:= 2 7- the Limagon becomes a Cardioid.^ (See Fig. 106). 182. All Lima9ons, general and special, may be generated either as epi- or peri-trochoidal curves: as epi- trochoids the generator and director must have equal diameters, any point on the circumference of the generator then tracing a Cardioid, while any point on the radius (or radius produced) describes a Limayon; as |)«ri- trochoids the larger of a pair of Cardanic circles must roll on the smaller, the Cardioid and Limagon then resulting, as before, from the motion of points respec- tively on the circumference of the generator, or within or ivithout it. 183. In Fig. 106 the Cardioid is obtained as an epicycloid, being traced by point P during one revolution of the generator PHm about an equal directing circle msO. As a Lima5on we may get points of the Cardioid, as y and z, by drawing a secant through and laying off sy and sz each equal to 2r. 184. The Limagon as a Trisedrk. Three famous problems of the ancients were the squaring of the circle, the duplication of the cube and tlie trisection of an angle. Among the interesting cur\-es invented by early mathematicians for the purpose of solving one or the other of these problems, were the Quadratrix and Conchoid, whose construction is given later in this chapter- but it has been found that certain trochoids may as readily be employed for trisection, among them the Lima- gon of Fig. 106, frequently called tlie Epitrochoidal Trisectrix. When constructed as a Limagon we find points as G and A', on any secant RX of the circle called "path of centres," by making -S'A' and SG each equal to the radius of that circle. »Term due to Eeuleaux, and based upon the fact that Cardano (16tli century) was probablv the flr«,t tn i„„„ „ . ., paths described hy points during their rolling. tFrom Cardis. the Latin for A^t mveatigate the SPECIAL TROCHOIDS. 63 185. To trisect an angle, as MR F, by means of this epitrochoid, bisect one side of the angle, as FR, at m; use mR and mi^ as radii for generator and director respectively of an epitrochoid hav- ing a tracing radius, RF, equal to twice that of the generator. Make RN^RF and draw NF; this will cut the Limagon FT^RQ (traced by point F as carried by the given generator) in a point T^ . The angle T^RF will then be one-third of NRF, which may be proved as follows: F reaches T^ by the rolling of arc mn on arc mn^. These arcs are subtended by equal angles, <^, the circles being equal. During this rolling R reaches R^, bringing R F to R^T^. In the triangles T^R^F and RFR^ the side FR^ is common, angles 4> equal, and side R^T^ equal to side RF; the line RT^ is there- fore parallel to R^F, whence angle T^RF must also equal <^. In the triangle RFR^ we denote by the angles opposite the equal sides RF and RiF; then 2^-f <^ = 180°, or 0= g — ■ ^^ triangle NRF we have the angle at F equal to 6 — <^, and 2 (0 — tl)) + x+4> = 18O°, which gives x=24>, by substituting the value of from the previous equation. 186. The Involute. As the opposite extreme of a circle rolling on a straight line we may have the latter rolling on a circle. In this case the rolling circle has an infinite radius. A point on the straight line describes a curve called the involute. This would be the path of the end of a thread if the latter were in tension while being unwound from a spool. In Fig. 107 a rule is shown, tangent at m to a circle on which it is supposed to roll. Were a pencil -point inserted in the centre of the circle at j (which is. on the line ux produced) it would trace the involute. When j reaches a, the rule will have had rolling contact with the base circle over an arc uts---a whose length equals line uxj. Were a the initial point, we would obtain 6, c, 64 THEORETICAL AND PRACTICAL GRAPHICS. etc., by making tangent mb^^^arc ma; tangent nc^arc na. Each tangent thus equals the arc from the initial point to the point of tangency. 187. The circle from which the involute is derived or evolved is called the evolute. Were a hexagon or other figure to be taken as an evolute, a corresponding involute could be derived; but the name "involute," unqualified, is understood to be that obtained from a circle. From the law of formation of the involute, the rolling line is in all its positions a normal to the curve; the point of tangency on the evolute is an instantaneous centre, and a tangent at any point, as /, is a perpendicular to the tangent, fq, from / to the base circle. Like the cycloid, the involute is a correct working outline for the teeth of gear-wheels; and gears manufactured on the involute system are to a considerable degree supplanting other forms. A surface known as the developable helicoid (see Figs. 209 and 270) is formed by moving a line so as to be always tangent to a given helix. It is interesting in this connection to notice that any plane perpendicular to the axis of the helix would cut such a surface in a pair of involutes.* 188. The Spiral of Archimedes. This curve is generated by a point having a uniform motion around a fixed point— the poZe— combined with uniform motion toward or from it. In Fig. 107, with as the pole, if the angles 6 are equal, and OD, OE and Oy, are in arith- metical progression, then the points D, ^ and y, are points of an Archimedean Spiral.' This spiral can be trochoidally generated, simultaneously with the involute, by inserting a pencil point at y in a piece carried by-and at right angles with-the rule, the point y being at a distance, *Tlie day of writing the aljove article the foUowlng item appeared In the New York Evdina Po,f ■ " Vi=ft„.= *,,,„, Observatoi-y, Greenwich, will hereafter miss the great cylindrical strncture which has fo a qlner centuiv 1^ °^^ the largest telescope possessed by the Observatory. Notwithstanding its size the Astronomer Z™^ ZTlL 1 T^""^^ the Lords Commissioners a telescope more than twice as large as the old one The onti^r^L r « P™""^'^"^ ''^'■°"g^ new instrument will render it one of the three most powerful ^telescopes at present" In existence P'^™'?'!""^^ embodied In the featnreof the building which is to shelter the new telescope is tha't its dCe! of th™; feet d^amete^ w'n ''"''"'''*"r' tower having a diameter of only thirty-one feet. Technically the form adonted i c the curve has an undulation (ir wave -form towards the pole. «A series of curyes muoh more closely resembling those of a shell can be obtained by tracing the paths of points on the piston-rod of an oscillating cylinder. See Arts. 167 and 158 for the principles of their construction. THE CONCHOID.— THE QUADRATRIX. 67 Ov=^c+ Ora; On=^c — Om; we may therefore express the relation to of points on the curve by the equation p = c±0 m^c±asec- rigr. iio- 194. Mention has ab'eady been made (Art. 184) of the fact that this was one of the curves invented in part for the purpose of solving the problem of the trisection qf'an angle. Were mOx (or <^) the angle to be trisected we would first draw pqr, the superior branch of a conchoid having the constant, c, equal to twice Om. A parallel from m to the axis will intersect the curve at q; the angle pOq will then be one -third of <^: for since h q=2 0m we have m q = 2 m cos ft ; also mq: Om: -.sind: sin P; hence 20mcosfi:Om::sind:8inj3, whence sin ^^2 sin ft cos /3=: sin 2 ^ (from known trigonometric relations). The angle 6 is therefore equal to twice yS, which makes the latter one -third of angle 4>. 195. To draw a tangent and normal at any point v, we find the instantaneous centre o on the principle that it is at the intersection of normals to the paths of two moving points of a line, the distance between said points remaining constant. In tracing the curve, the motion of (on Ov) is — at the instant considered — in the direction Ov; Oo is therefore the normal. The point m oi Ov is at the same moment moving along MN, for which mo is the normal. Their intersection o is then the instantaneous centre, and o v the normal to the conchoid, with v z perpendicular to o ■« for the desired tangent. 196. This interesting curve may be obtained as a plane section of one of the higher mathemat- ical surfaces. If two non- intersecting lines — one vertical, the other horizontal — be taken as guiding lines or directrices of the motion of a third straight line whose inclination to a horizontal plane is to be constant, then horizontal planes will cut conchoids from the surface thus generated, while every plane parallel to the directrices will cut hyperbolas. From the nature of its plane sections this surface is called the Conchoidal Hyperholoid. (See Fig. 219). THE QUADRATRIX OP DINOSTRATUS. 197. In Fig. Ill let the radius T rotate uniformly about the centre; simultaneously with its movement let MN have a uniform motion parallel to itself, reaching AB &i the same time with radius T; the locus of the intersection oi MN with the radius will be the Quadratrix. Points 68 THEORETICAL AND PRACTICAL GRAPHICS. exterior to the circle may be found by prolonging the radii while moving M N away from A B. As the intersection of M N with OB is at infinity, the former becomes an asymptote to the curve as often as it moves from the centre an additional amount equal to the diameter of the circle; the number of branches of the Quadratrix may therefore be infinite. It may be proved analytically that the curve crosses ^ at a distance from equal to 2 r -^ ir. 198. To trisect an angle, as T a, by means of the Quadratrix, draw the ordinate ap, trisect p T hy s and x and draw sc and xm; radii Oc and Om will then divide the angle as desired: for by the conditions of generation of the curve the line MN takes three equi- distant i^arallel positions while the radius describes three equal angles. THE CISSOID OF DIOCLES. 199. This curve was devised for the purpose of obtaining two mean proportionals between two given quantities, by means of which the duplication of the cube might be effected. The name was suggested by the Greek word for ivy, since "the curve appears to mount along its asymptote in the same manner as that parasite plant climbs on the tall trunk of the pine."* This was one of the first curves invented after the discovery of the conic sections. Let C (Fig. 112) be the centre of a circle, ACE a, right angle, NS and MT any pair of ordinates paraMel to ^•ig-. 112. and equidistant from CE; then a secant from A through the extremity of either ordinate will meet the other ordinate in a point of the cissoid. A T and NS give P; A S and M T give Q. The tangent to the circle at B will be an asymptote to the curve. It is a somewhat interesting coincidence that the area between the cissoid and its asymptote is the same as that between a cycloid and its base, viz.,- three times that of the circle from which it is derived. 200. Sir Isaac Newton devised the following method of obtaining a cissoid by continuous motion- Make^lFWC; then move a right-angled triangle, of base = T^(7, so that the vertex F travels alon^ *Leslie. Oeometrical Analysis. 1821. THE QISSOID.—THE TRACTRIX. 69 the line DE while the edge JK always passes through V; then the middle point, L, of the base FJ, will trace a cissoid. This construction enables us readily to get the instantaneous centre and a tangent and normal; for Fn is normal to FC — the path of F, while nV is normal to the motion of / toward JV; the instantaneous centre n is therefore at the intersection of these normals. For any other point as P we apply the same principle thus: With radius AC and centre P obtain x; draw Px, then Vz parallel to it; a vertical from x will meet Vz at the instantaneous centre y, whence the normal and tangent result in the usual way. • The point y does not necessarily fall on nV. Since nV and FJ are perpendicular to / F they are parallel. So also must Vz be parallel to Px, regardless of where P is taken. 201. Two quantities m and n will be mean proportionals between two other quantities a and h if m^ = Mo and n''=mb; that is, if in'=a^b and if n' = a6l If 6^2 a we will find, from the relation m^^a'^b, that m will be the edge of a cube whose volume equals 2 a'. To get two mean proportionals between quantities, r and b, make the smaller, r, the radius of a circle from which derive a cissoid. Were APR the derived curve we would then make Ct equal to the second quantity, 6, and draw B t, cutting the cissoid at Q. A line A Q would cut off on Ct a distance Cv equal to m, one of the desired proportionals; for m' will then equal r''b, as may be thus shown by means of similar triangles: Cv:MQ::CA:MA whence Cv'= ''\^T ■ (1) r MO Ct:MQ::CB:BM " Ct=^^ (2) MQ-.MA-.-.SN-.AN:: \/AN.BN:AN, whence MQ= ^^^^^^^ (3) From (2) we have MQ = ^^^^^^^^ . • . •' (4) MA'' (AN B N) " (3) " " MQ^='^ ^ AN^ ^^) Replacing M Q^ in equation (1) by the product of the second members of equations (4)- and (5) gives Cv^ (i-e., m'}^r''h. By interchanging r and 6 we obtain n, the other mean proportional; or it might be obtained by constructing similar triangles having r, b and m for sides. THE TEACTRIX. 202. The Tractrix is the involute of the curve called the Catenary (Art. 214) yet its usual con- struction is based on the fact that if a series of tangents be drawn to the curve, the portions of such tangents between the • points of tangency and a given line will be of the same length ; or, in other words, -the intercept on the tangent, between the directrix and the curve, will be constant. A practical and very close approximation to the theoretical curve is obtained by taking a radius QR (Fig. 113) and with a centre a, a short distance from R on QR, obtaining b, which is then joined with a. On a 6 a centre c is similarly taken for another arc of the same radius, whence c d is obtained. A sufficient repetition of this process will indicate the curve by its enveloping tangents, or a curve may actually be drawn tangent to all these lines. Could we take a, b, c, etc., as mathematically consecutive points' the curve would be theoretically exact. The line QS ia an asymp- tote to the curve. 70 THEORETICAL AND PRACTICAL GRAPHICS. The area between the completed branch RPS and the lines QR and QS would be equal to a quadrant of the circle on radius QR. 5 =AeR 203. The surface generated by revolving the trac- trix about its asymptote has been employed for the foot of a vertical spindle or shaft, and is known as Schiele's Anti- Friction Pivot. The step for such a pivot is shown iii sectional view in the left half of the figure. Theoretically, the amount of work done in overcoming friction is the same on all equal areas of this surface. In the case of a bearing of the usual kind, for a cylindrical spindle, although the pressure on each square inch of surface would be constant, yet, as unit areas at different distances from the centre would pass over very different amounts of space in one revolution, the wear upon them would be necessarily unequal. The rationale of the tractrix form will become evident from the following x-ig.. lis. consideration : If about to split a log, and ha^dng a choice of wedges, any boy would choose a thin one rather than one with a large angle, although he might not be able to prove by graphical statics the exact amount of advantage the one would have over the other. The theory is very simple, how- ng.. n.^. ever, and the student may profitably be introduced to it. Suppose a ball, c, (Fig. 114) struck at the sa,me instant by two others, a and h, moving at rates of six and eight feet a second respectively. On a c and b c prolonged take ce and ch equal, respectively, to six and eight units of some scale; complete the parallogram having these lines as sides; then it is a well-known principle in mechanics* that cd — the diagonal of this parallel- ogram—will not only represent the direction in which the ball c will move, but also the distance — in feet, to the scale chosen — it will travel in one second. Evidently, then, to balance the effect of balls a and b upon c, a fourth would be necessary, moving from d toward c and traversing dc in the same second that a and b travel, so that impact of all would occur simultaneously. These forces would be represented in direction and magnitude (to some scale) by the shaded triangle c'd'e', which illustrates the very important theorem that if the three sides of a triangle — taken like c'e', e'd', d'c', in such order as to bring one back to the initial vertex mentioned — represent in magnitude and direction three forces acting on one jDoint, then these forces are balanced. Constructing now a triangle of forces for a broad and thin wedge, (Fig. 115) and denoting the force of the supposed equal blows ^ by i^ in each triangle, we see that the pressures are greater for the thin wedge than for the other; that is, the less the inclination to the vertical the greater the pressure. A pivot so shaped that as the pressure between it and its step increased the area to be traversed diminished would therefore, theoretically, be the ideal; and the rate of change of curvature of the tractrix, as its generating point approaches the axis, makes it, obviously, the correct form. X-ig-- lis. »For a demonstration the student may refer to Banklnes Applied Mechanics, Art. 51. THE TRACTRIX.— WITCH OF AGNES I.-CAR TES IAN OVALS. 71 204. Navigator's charts are usually made by Mercator's projection (so-called, not being a projection in the ordinary sense, but with the extended signification alluded to in the remark in Art. 2). Maps thus constructed have this advantageous feature, that rhumb lines or loxodromics — the curves on a sphere that cut all meridians at the same angle — are represented as straight lines, which can only be the case if the meridians are indicated by parallel lines. The law of convergence of meridians on a sphere is, that the length of a degree of longitude at any latitude equals that of a degree on the equator multiplied by the cosine (see foot-note, p. 31) of the latitude; when the meridians are made non- convergent it is, therefore, manifestly necessary that the distance apart of originally equi- distant parallels of latitude must increase at the same rate; or, otherwise stated, as on Mercaior's chart degrees of longitude are all made equal, regardless of the latitude, the constant length repre- sentative of such degree bears a varying ratio to the actual arc on the sphere, being greater with the increase in latitude; but the greater the latitude the less its cosine or the greater its secant; hence lengths representative of degrees of latitude will increase with the secant of the latitude. Tables have been constructed giving the increments of the secant for each minute of latitude; but it is an interesting fact that they may be derived from the Tractrix thus; Draw a circle with radius QR, centre Q (Fig. 113); estimate latitude on such circle from R upward; the intercept on QS between consecutive tangents to the Tractrix will be the increment for the arc of latitude included between parallels to Q S, drawn through the points of contact of said pair of tangents.* On map construction the student is referred to Chapter XII, or to Craig's Treatise on Projections. THE WITCH OF AGNESI. 205. If on any line S Q, perpendicular to the diameter of a circle, a point S be so located that S Q:AB::PQ: QB then S will be a point of the curve called the Witch of Agnesi. Such point is evidently on the ordinate P Q prolonged, and vertically below the intersection T of the tangent at A by the secant through P. X-lg-- US- H Tp-,. D 1 ^^^, ■n, ii\^ Aj\ !\ ^..f^v i V "^wjP^ W- '- \ V ^i><~~^~jft~-JF\ "\ \ asymptote The point E, at the same level as the centre 0, is a diameter's distance from the latter. The tangent at B is an asymptote to the curve. The area between the curve and its asymptote is four times that of the circle involved in its construction. The Witch, also called the Versier((, was devised by Donna Maria Gaetana Agnesi, a brilliant Italian lady who was appointed in 1750, by Pope Benedict XIV, to the professorship of mathematics and philosophy in the University of Bologna. THE CARTESIAN OVAL. 206. This curve, also called simply a Cartesian, after its investigator, Descartes, has its points connected with two foci, F' and F", by the relation m p' ±:n p" =k c, in which c is the distance between the foci, while m, n and k are constant factors. ♦ Leslie. Geometrical Analysis. Bdinburgh, 1821. 72 THEORETICAL AND PRACTICAL GRAPHICS. Salmon states that we owe to Chasles the proof that a third focus may be found, sustaining the ^igr- ii'7. same relation, and expressed by an equation of similar form. (See Art. 209). The Cartesian is symmetrical with respect to the axis — the line joining the foci. 207. To construct the curve from the first equation we may for convenience write mp'±np''^kc in the form p' n „ kj;^ m m' or by denoting — by 6 and — by d, it takes the yet more simple form p'±bp"^=d. Then p" will have two values, according as the positive or negative sign is taken, being respectively — r — and — r — ', the former is for points on the inner of the two ovals that constitute a complete Cartesian, while the latter gives points on the outer curve. To obtain p' d- take F' and F" (Fig. I^igr- IIB. 118) as foci; F'S^=d; S K a,t some random acute angle 6 with the axis, and make SH^-; that is, make F' S: S H::h -.1. Then from F' draw an arc tfP, of radius less than d, and cut it at P by an arc from centre F", radius ST, Tt being a parallel to F' H; then P is a point of the inner oval; for St=d — p', and ST=p"; there- fore p" -.d — p' -.-.rr-.d, whence p" = T^ . b 208. If an arc x%jK be drawn from i^", with radius, F' x, greater than d, we may find the second value of p", viz., ^-^— , by drawing xQ parallel to F' H to meet HS prolonged; for QS will equal -d . in which p' = F'x. Again using F" as a centre, and a radius QS = p", gives points R and M of the larger oval. The following are the values for the focal radii to the four points where the ovals cut the axes. (See Fig. 117). For A, p" = P^ =c + p' whence p' = F'A = '^ + ^ ' 1 — b „ d — p' a, p" = ~^ = c + p' ^, P =—r— = c-p' P' = F' a = d — bc 1 + 6 P' = F' B = d+bc b, p _d-p' ■■c — p' p' = F' b = 1 + 6 d — be 1-6 The construction -arcs for the outer oval must evidently have radii between the values of p' for A and B above; and for the inner oval between those of a and 6. CARTESIAN OVALS-CAUSTICS. 73 E'ig-. i.±s. The numerical values from which Fig. 118 was constructed were m = 3; n=2; c^l; k=8. 209. The Third Focus. Fig. 118 illustrates a special case, but, in general, the- method of finding a third focus F'" (not shown) would be to draw a random secant F^r through F', and note the points P and G in which it cuts the ovals — these to be taken on the same side of F', as two other points of intersection are possible; a circle through P, G and F" would cut the axis in the new focus sought. Then denoting by C the distance F' F'", we would find the factors of the original equation appearing in a new order; thus, k p' ±:n p'" = mC, which — for purposes of construction — ma}' be written p' ± 6'p"'= <^'- If obtained from the foci F" and F'" the relation would be m p'" — Ap"^=bmC", in which C" equals F" F"'. Writing this in the form p'" — Bp" = ±D we have the following interesting cases: (a) an ellipse for D positive and -B = — 1 ; (b) an hyperbola for D positive and B= -\- 1; (c) a limacon for D^C'B; (d) a cardioid for B^ + 1 and D:=C'. 210. The following method of drawing a Cartesian by continuous motion was devised by Prof Hammond: A string is wound, as shown, around two pulleys turning on a common axis; a pencil at P holds the string taut around smooth pegs placed at random at F^ and F^; if the wheels be turned with the same angular velocity, and the pencil does not slip on the string, it will trace a Cartesian having F^ and F.^ as foci.* If the pulleys are equal the Cartesian will become an ellipse; if both threads are wound the same way around either one of the wheels the resulting curve will be- an hyperbola. 211. It is a well-known fact in Optics that the incident and reflected ray make equal angles with the normal to a reflecting surface. If the latter is curved then each reflected ray cuts the one E-ig-. 120. next to it, their consecutive intersections giA'ing a curve called a caustic by reflection. Probably all have occasionally noticed such a curve on the surface of the milk in a glass, when the light was properly placed. If the reflecting curve is a circle the caustic is the evolute of a limaQon. In passing from one medium into another, as from air into water, the deflection which a ray of light undergoes is called refraction, and for the same media the ratio of the sines of the angles of incidence and refraction (^ and , Fig. 120) is constant. The consecutive intersections of refracted rays give also a caustic, which, for a circle, is the evolvte of a Cartesian Oval. The proof of this statement + involves the property upon which is based the most convenient method of drawing a tangent to the Cartesian, viz., that the normal at any point divides the angle between the focal radii into parts whose sines are proportional to the factors of those radii in the equation. If, then, we have obtained a point G on the outer oval from the relation mp' ±np" ^^hc, we may obtain the tangent at G by laying ofl on p' and p" distances proportional to m and n, as Gr and Gh, Fig. 118, then bisecting rh at j and drawing the normal Gj, to which the desired tangent is a perpendicular. At a point on the inner oval the distance would not be laid off on a focal radius produced, as in the case illustrated. * American Journal of Mathematics^ 1878. t Salmon. Higher Plane Curves. Art. 117. 74 THEORETICAL AND PRACTICAL GRAPHICS. CASSIAN OVALS. 212. In the Cassian Ovals or Ovah of Cassini the points are connected with two foci by the relation p'p"=k\ i.e., the product of the focal radii is equal to some perfect square. These curves have already been alluded to in Art. 114 as plane sections of the annular torus, taken parallel to its axis. i^igr- iss. In Art. 158 one form— the Lemniscate— receives special treatment. For it the constant k' must equal m^ "the square of half the distance between the foci. When k is less than m, the curve becomes two separate ovals. 213. The general construction depends on the fact that in any semicircle the square of an ordinate equals the product of the segments into which it divides the diameter. In Fig. 122 take F, and F, as the foci, erect a perpendicular F^^S to the axis F^F^, and on it lay off F^R equal to the constant, it. Bisect F^F^ at and draw a semicircle of radius OR. This cuts the axis at A and B, the extreme points of the curve; for k'=F,Ax F,B. Any other point T may be obtained by drawing from F^ a circular arc of radius F^t greater than F^A; draw t R, then Rx perpen- dicular to it; xF^ will then be the p", and F,t the p', for four points of the curve, which will be at the intersection of arcs struck from F^ and F^ as centres and with those radii. To get a normal at any point T draw T, then make angle F.,Ts=O^F,T 0; Ts will be the desired line. THE CATENAEY. 214. If a flexible chain, cable or string, of uniform weight per unit of length, be freely sus- pended by its extremities, the curve which it takes under the action of gravity is called a Catenary, from catena, a chain. A simple and practical method of obtaining a catenary on the drawing-board, would be to insert two pins in the board, in the desired relative i)Osition of the points of suspension, and then attach to them a string of the desired length. By holding the board vertically, the string would assume the catenary, whose points could then be located with the pencil and joined in the usual manner with the irregular curve. Otherwise, if its points are to be located by means of an equation, we take axes in the plane of the curve, the y-axis (Fig. 123) being a vertical line through the lowest point T of the catenary, while the x-axis is a horizontal line at a distance m below T. The quan- tity m is called the parameter of the curve, and is equal to the length of string which represents the tension at the lowest point. THE CATENARY.— THE LOGARITHMIC SPIRAL. 75 m The equation of the catenary ' is then 2/=-^(«"' + e logarithms'' and has the numerical value 2.7182818 +. By taking successive values of x equal to m, 2 m, 3 m, etc., we get the following values for y: 7?1 / 1 \ a;= m...y^-^{e-\ — 1 which for m = unity becomes 1 .64308 ^ in which e is the base of Napierian ^ ™ / 1 , 1 \ 11 3.76217 " 10.0676 27.308 To construct the curve we therefore draw an arc of radius B^=m, giving T on the axis of y as the lowest point of the curve. m For x=^OB = m we have y =£ P = 1.54308; for a; ^ a = -^ we have i/ ^ a n = 1.03142. The tension at any point P is equal to the weight of a piece of rope of length B P=^P C + m. At the lowest point the tangent is horizontal. The length of any arc TP is proportional to the angle between TO and the tangent P F at the upper extremity of the arc. 215. If a circle RLB be drawn, of radius equal to m, it may be shown analytically that tangents P S and Q R, to catenary and circle respectively, from points at the same level, will be parallel: also that PS equals the catenary -arc Pr T; S therefore traces the involute of the catenary, and SiS S B always equals RO and remains perpendicular to PS (angle ORQ being always 90°) we have the curve TSK fulfilling the conditions of a tractrix. (See Art. 202.) If a parabola, having a focal distance m, roll on a straight line, the focus will trace a catenary having m for its parameter. The catenary was mistaken by Galileo for a parabola. In 1669 Jungius proved it to be neither a parabola nor hyperbola, but it was not till 1691 that its exact mathematical nature was known, being then established by James Bernouilli. THE LOGARITHMIC OR EQUIANGULAR SPIRAL. 216. In Fig. 124 we have the curve called the Logarithmic Spiral. Its usual construction is based on the property that any radius vector, as p, which bisects the angle between two other radii, OM and ON, is a mean proportional between them; i.e., p^ ^0 S'^O M X N If M and G are points of the spiral we may find an intermediate point K by drawing the ordinate K to a, semicircle of diameter OM+OG; a perpendicular through G to GK will then give D, another point of the curve, and this construction may be repeated indefinitely. Radii making equal angles with each other are evidently in geometrical progression. This spiral is often called Equiaiigular from the fact that the angle is always the same between iRankire. Applied Mechanics. Art. 175, 2In the expression 102 = 100 the ctuantity "2" la called the logarithm of 100, It heing the exponent of the power to which 10 must be raised to give 100. Similarly 2 would be the logarithm of 64, were 8 the base or number to be raised to the power Indicated. 76 THEORETICAL AND PRACTICAL GRAPHICS. a radius vector and the tangent at its extremity. Upon this property is based its use as the out- line for spiral cams and for lobed wheels. The curve never reaches the pole. The name logarithmic spiral is based on the property that the angle of revolution is proportional to the logarithm of the radius vector. This is expressed by p = a*, in which 6 is the varying angle, and a is some arbitrary constant. To construct a tangent by calculation, divide the hyperbolic logarithm ' of the ratio M:OK (which are any two radii whose values are known) by the angle between these radii, expressed in circular measure;^ the quotient will be the tangent of the constant angle of obliquity of the spiral. 217. Among the more interesting properties of this curve are the following: Its involute is an equal logarithmic spiral. Were a light placed at the pole, the caustic — whether by reflection or refraction — would be a logarithmic spiral. The discovery of these properties of recurrence led James Bernouilli to direct that this spiral be engraved on his tomb, with the inscription — Eadeni Mutata Resurgo, which, freely trans- lated, is — I shall arise the same, though changed. Kepler discovered that the orbits of the planets and comets were conic sections having a focus at the centre of the sun. Newton proved that they would have described logarithmic spirals as they travelled out into space, had the attraction of gravitation been inversely as the cube instead of the square of the distance. THE HYPERBOLIC OR RECIPROCAL SPIRAL. 218. In this spiral the length of a radius vector is in inverse ratio to the angle through which T^ig-. 1S5. it turns. Like the logarithmic spiral, it has an infinite number of convolutions about the pole, which it never reaches. The invention of this curve is attributed to James Bernouilli, who showed that Newton's conclusions as to the logarithmic spiral (see Art. 217) would also hold for the hyperbolic spiral, the initial velocity of projection determining which trajectory was described. To obtain points of the curve divide a circle m 6 8 (Fig. 125) into any number of equal parts, and on some initial radius Om lay off some unit, as an inch; on the second radius 2 take On On -y; on the third -g-, etc. For one -half the angle 6 the radius vector would evidently be 2 On, giving a point s outside the circle. ~ = a0, in which r is the radius vector, a some numerical con- The equation to the curve is stant, and 9 is the angular rotation of r (in circular measure) estimated from some initial line. iTo get the hyperbolic logarithm of a numljer multiply its common logarithm by 2 3026 0.6236!'^ = a017«33"''^'"'' 360 ° = 2 . r, which, for r = 1. becomes 6.28318; 180° = 3.14169; 900 = 1.5708; 60<. = 1.0472; 45° =0.7854; THE HYPERBOLIC SPIRAL.— THE LITUUS. 77 The curve has an asymptote parallel to the initial line, and at a distance from it equal to — units. a r-igr- iss. To construct the spiral from its equation, take as the pole (Fig. 26); Q as the initial line; a, for convenience, some fraction, as — ; and as our unit some quantity, say half an inch, that will make — of convenient size. Then, taking Q as the initial line, make P=^ — =2", and draw PR parallel to Q for the asymptote. For ^ = 1, that is, for arc KH=--radim OH, we have r == — = ii", giving H for one point of the sniral. Writing the eauation in the form r^=— ^ . and — = 2", giving H for one point of the spiral. Writing the equation in the form r = — a d expressing various values of 6 in circular measure we get the following: 6 = 30° ==0.5236; r = M=B'.'8+ : e = 45° = 0.7854; r=ON=2'.'55; e = 90° = 1.5708; r = fi'=l'.'2+ : 6 =180° = 3.14159; r = r= .6366, etc. The tangent to the curve at any point makes with the radius vector an angle , which is found by analysis to sustain to the angle the following trigonometrical relation, tan ^ = 6; the circular measure of 6 may therefore be found in a table of natural tangents, and the corresponding value of obtained. THE LITUUS. — THE IONIC VOLUTE. 219. The Lituus is a spiral in which the radius vector is inversely proportional to the square root of the angle through which it has revolved. This relation is shown by the equation r= ■ _ also written a^ 6=-^ When 6^0 we find r = co , which makes the initial line an asymptote to the curve. In Fig. 127 take Q as the initial line, as the pole, a ^ 2, and as our unit 3" ; then 1 » — = 1J". a For 6 = 90° = IT (in circular measure 1.5708) we have r = 0M=l".2 +. For ^ = 1 we have the radius T making an angle of 57°. 29 + with the initial line, and in length equal to - units, 78 THEORETICAL AND PRACTICAL GRAPHICS. 1. e.. For 61=45°=^ (or 0.7854) r will be 0R=1".1+. Then 0H = OR for in rotating to H the radius vector passes over four 45° angles, and the radius must therefore be one- half what it was for the first 45° described. O M OV Similarl]^, 0K=^; M = -^, etc.; this rela- tion enabling the student to locate any number of points. To draw a tangent to the curve we employ the relation tan 4>^2 6, ^ being the angle made by the tangent line Avith the radius vector, while & is the angular rotation of the latter, in circular measure. Architectural Scrolls. — The Ionic Volute. The Lituus and other spirals are occasionally employed as volutes and other architectural ornaments. In the former application it is customary for the spiral to terminate on a circle called the eye, into which it blends tangentially. Usually, in practice, circular -arc approximations to true spiral forms are employed, the simplest of which, for the scroll on the capital of an Ionic column, is Fig. ist- (a,). probably the following: Taking A P, the total height of the volute, at sixteen of the eighteen "parts" into which the module (the unit of proportion ^ the semi- diameter of the column) is divided, draw the circular eye with radius equal to one such part, the centre dividing AP into segments of seven and nine parts respectively. Next inscribe in the eye a square with one diagonal vertical; parallel to its sides draw (see enlarged square mnop) 2—4 and 3 — 1, and divide each into six equal parts, which number up to twelve, as indicated. Then (returning to main figure) the arc AB has centre 1 and radius 1 — A. With 2 as a centre draw arc B C; then CD from centre 3, etc. In the complete drawing of an Ionic column the centre of the eye would be at the intersection of a vertical line fi-om the lower extremity of the cyma reversa with a hori- zontal through the lower line of the echinus. To complete the scroll a second spiral would be required, constructed according to the same law and beginning at Q, where A Q ia equal to one -half part of the module. THE GEOMETRIE DESCRIPTIVE OF GASPARD MONGE. 105 CHAPTER IX. OKTHOGEAPHIC PROJECTION UPON MUTUALLY PERPENDICULAE PLANES. 283. In this and the succeeding chapter, as also in nearly all of the latter part of this work, the principles of what has been generally known as Descriptive Geometry are either examined or applied. In Art. 19 — which, with Arts. 2, 3 and 14, should be reviewed at this point — reasons are given for calling this science Mange's Descriptive. Certain German writers call it Monge''s Orthogonal Projec- tion. The popular titles Mechanical Drawing, Practical Solid Geometry, Orthographic Projection, etc., are usually merely indicative of more or less restricted applications of Monge's Descriptive to some special industrial arts ; and working drawings of bridge and roof trusses, machinery, masonry and other con- structions, are simply accurately scaled and fully dimensioned projections made in accordance with its principles. Monge's service to mathematics and graphical science, which, according to Chasles", inaugurated the fifth epoch in geometrical history, consisted, not in inventing the method of representing objects by their projections — for with that the ancients were thoroughly familiar, but in perceiving and giv- ing scientific form to the principles and theorems which were fundamental to the special solutions of a great number of graphical problems handed down through many centuries, and many of which had been the monopoly of the Freemasons. Emphasizing in this chapter the abstract principles of the subject, treating it as a pure science, and giving a mathematical outline of the field of its appli- cation, I leave for the next chapter its more practical aspect, including the modifications in vogue in the draughting offices of leading mechanical engineers. Statements without proof are given whenever their truth is reasonably self-evident. FUNDAMENTAL PRINCIPLES. 284. The orthographic projection of a point on a plane is the foot of the perpendicular from ^ig-- ise. the point to the plane. In Fig. 156 the perpendiculars Pp' and Pp give the pro- jections, p' and p, of the point P. The planes of projection are shown in their space posi- tion ; one, H, horizontal, the other, V, vertical. The projection, p, on H, is called the plan or horizontal projection (h. p.) of P. The point p' is the elevation or ver- tical projection (v. p.) of P. Projections on the vertical plane are denoted by small letters with a single accent or " prime." Projections on H are small letters unaccented. A point may be named by its space-letter, the capital, or by its projections; thus, we may speak of the point P or of the point p p'. * Apergu Historique sur I'Orlgln et le Developpement des Methodes en GSomgtrie. 106 THEORETICAL AND PRACTICAL GRAPHICS. "We shall call Pp the lA-projector of P, since it gives the projection of P on H. Similarly, P^' is the Y-prqjedor of P. A projector-plane is then the plane containing both projectors of a point, and is evidently perpendicular to both V and H by virtue of containing a line perpendicular to each. 285. V and H intersect in a Une called the ground line, hereafter denoted by G. L. They make with each other four diedral angles. The observer is always supposed to be in the first angle, viz., that which is above H and in front of V. We shall call it Qj, or the first quadrant. Q,, is then the second quadrant, back of V but above H. Q3 is below Q,, while Q4 is immediately below the first angle. Points R and 8, in the second and third quadrants respectively, have their elevations, / and s', on opposite sides of G. L., while their plans, r and s, are on the back half of H. The point T, in Q,^, has its v. p. at t', below G. L., and its h. p. at t, in front of G. L. 286. In making a drawing in the ordinary way, and not pictorially, we suppose the planes V and H brought into coincidence by revolution about G. L., the upper part of V uniting with the back part of H, while lower V and front H merge in one. The arrow shows such direction of revolution, after which any vertical projection, as p', is found at p\, on a line pz perpendicular to G. L. and containing the plan p. This is inevitable, from the following consideration : Any point when revolved about an axis describes a circle whose centre is on the axis and whose plane is perpendicular to the axis ; but as the projector-plane of P contains p' — the point to be revolved, and is perpendicular to G. L. (the axis) because perpendicular to both H and V, it must be the plane of rotation of p', which can therefore only come into H somewhere on p z (produced). In Fig. 157 we find Fig. 156 represented in the ordinary way. Only the pro- ^ig. 157. jections of the points appear. V and H are, as usual, considei"ed indefinite in ex- tent, and their boundaries have disappeared. A projector-plane is shown only by the line, perpendicular to G. L., in which its intersections with V and H coincide after revolution. 287. A point (p p') in the first angle has its h. p. below G. L. and its v. p. above. For the third angle the reverse is the case, the plan, s, above, and the eleva- tion, s', helow. The second and fourth angles are also opposites, both projections, r /, being above G. L. for the former, and both, t if, below for the latter. 288. For any angle the actual distance of a point from H, as shown by any H-projector Pp (Fig. 166), is equal to p' z (either figure)— the distance of the v. p. of the point from G. L. Simi- larly, the V-projector of a point, as P / (Fig. 156), showing the actual distance of a point from N, is equal to the distance of the h. p. of the point from G. L. If one projection of a point is on G. L. the point is in a plane of projection. If in H, the elevation of the point will be on G. L. ; similarly the plan, if the point lie^ in V. rig-- a.5s. 289. The projection of a line is the line containing the projections of all its points. The projection of a straight line will be a straight '^^- =1-^®- line; for its extremity-projectors, as ^a and Bh (Fig. 158), would determine a plane perpendicular to H and containing AB; in such plane all other H-projectors must He, hence meet H in a6, J y which is straight because the intersection of two planes. In Fig. 159 we see the line .1 B of Fig. 158, orthographic- ally represented. 290. A plane containing the projectors of a straight hne is called a projecting plane of the line. (A B b a, Fig. 158). IP FUNDAMENTAL PRINCIPLES OF MONGE'S DESCRIPTIVE. 107 A Y-projecting plane is the plane through a hne and perpendicular to the vertical plane.. The 'H.-projeding plane of a line is the vertical plane containing it. The intersection of a surface by a given line or surface is called a trace. The trace of a line is E"3.g-. iso. a point; of a surface is a line. If on H -it is called a horizontal trace (A. t) ; on V, a vertical trace, abbreviated to v. t. 291. The projection of a curve is in general a curve, the trace — upon V or H — of the cylindrical surface* whose elements are the pro- jectors of the points of the curve. (See Figs. 160 and 161). The projection of a plane curve will be equal and parallel to the original curve when the latter is parallel to iFig:. isi. the plane on which it is projected. 292. All lines, straight or curved, Ijdng in a plane that is perpendicular to V, will be projected on V in the v. t. of the plane. A similar remark apphes qj to the plans of lines lying in a vertical plane. Fig. 162 illustrates these statements. The plane P'QP, being perpendicular to V— as shown ng. ISS- by the h. t. (P Q) being perpen- dicular to G. L — all points in the plane, as A, B, C, D, E, F, will have their projections on V in the trace P'Q. Plane M'N M is vertical, since M' N—i\& vertical trace — is |m perpendicular to G. L ; its h. t. {N M) therefore contains the h. p. of every point in the plane. Since the triangle ABC and the curve D E F are not parallel to H or V their exact size and shape would not be shown by their projections; these could, however, be readily obtained by rotating their plane into H, about the trace P Q, or into V about P' Q. Such rotation, called rabatmentf, is described in detail in Art 306. 293. From Fig. 163 it is evident that the h. t. of a line A B, must be at the intersection of the plan a 6 by a perpendicular to G. L. from s', where the elevation a' b' crosses G. L. Similarly, the v. t. of the line is on its V. p., immediately below r, the intersection of G. L. by the plan a b ■ hence the rule : To find the hor- - ^''^- ^^■^■ i^igr- iss. izontal trace of a line prolong the q. vertical projection until' it meets the ground line ; then draw a perpendicular to the plan of the line. An analogous construc- tion gives the vertical trace of the line. Fig. 164 shows, orthographically, the same projections and traces as Fig. 163. • See Remark, Art 8. t From the French rabattemmt. 108 THEORETICAL AND PRACTICAL GRAPHICS. 294. The inclination, 6, of the line to H, is that of the line to its plan a 6; the angle <^, made with V, is that of the line to its elevation, a! V. 295. Any horizontal line has a plan, a b (Fig. 165), equal and parallel to itself. Its elevation, a!h', is -parallel to G. L.,^at the same distance from it as the line from I^, -aiHtd 'makes the same angle with V as its plan does with G. L. Such line can evi- dently have no h. t. Its v. t. would he found by the rule in the preceding article. In the extreme case of perpendicularity to V the v. p. would reduce to a point. 296. A line parallel to V but oblique to H has its h. p. parallel to G. =^^e- =^®'^- i^igr- lee. l_^ makes with H the same angle that its v.- p. does with G. L., and has its h. t. found by the usual rule. (Pig. 167.) 297. A vertical line has no v. t. ; is projected on H in a point; has its v. p. parallel and equal to itself. (See CD, Fig. 168). A line parallel to both V and H is parallel to their intersection, has no traces, and each projection equals the line. (See M N, Fig. 168.) Fig. 169 shows, orthographically, the lines A B, CD and M N oi the two preceding figures. 298. A plane is represented by its traces. Like H and V, any plane is considered indefinite in extent when drawn in the usual way ; though our pictorial diagrams show them bounded, to add to the appearance. A horizontal plane has but one trace, that on V. A plane parallel to V, as JfiV^O, Fig. 170, has no vertical trace, and its horizontal trace MN is par- S'lg-- IT-O- p' n! r P' i ■■-«: ^^ G ^^=^ ■1 \ 1 ^^^ \ M ^ R'- i^iir- ISS. t c m' 1 t d cid in h t. a b n- allel to G. L. If parallel to G. L., while inclined to H and V, a plane has parallel traces. (R S, R' S', Fig. 170). Fig. 171 shows the planes of Fig. 170, as usually represented. The traces of a plane not parallel to G. L. must meet ° at the same point on G. L. 299. We have stated that if a plane is perpendicular iFigr- ivi. -Q' — L to a plane of projection, its trace on the other plane is perpendicular to G. L. This is the only case S^ig-. 1'7'S. m which the angles between the ground line and the traces of a plane equal the dihedral angles made with H and V by the plane. That such equality exists may be thus demonstrated: The dihedral angle between two planes equals the plane angle between two lines, cut from the planes by a third plane , ,, , ,, perpendicular to both; plane P' Q R (Fig. 172) is by hypothesis perpendicular to V; hence the ground Ime and P'Q are the hues cut by plane V from two other planes to which It IS perpendicular; and their angle 6 is, therefore, the measure of the dihedral angle between H and PQR. The same course of reasoning may be appHed to each trace of both planes FUNDAMENTAL PRINCIPLES OF MONGE'S DESCRIPTIVE. 109 300. In Fig. 173 we have a plane oblique to both H and V. Two important Hnes are drawn in it which we shall have frequent occasion to use. Any line parallel to H is necessarily a horizontal line; but when also contained by a plane it is called a "horizontal" of the plane. It evidently must be parallel to the h. t. (Q R) of the plane. Any line parallel to V can have that fact discovered by the paralleUsm of its plan to G. L. ; FLs- ITS. E'lg'. 17''&. fi'y' V. p. of horizontal ^igr- ITS. but when contained by a plane, we shall call it a V-parallel of the plane. Such hne will evidently be parallel to the v. t. of the plane in which it lies. The oblique lines CD and E F, with those just described, illustrate the additional fact that the traces of lines in a plane will be found on the traces of the plane. This furnishes the following — and usual— method of determining a plane, i. e., by means of two lines known to lie in it: Prolong the lines until they meet the plane of projection ; their H-traces joined give the h. t. of the plane. Similarly for the vertical trace of the plane. Two intersecting lines or two parallel lines determine a plane. Three points not in a straight line, or a point and straight line may be reduced to either of the foregoing cases. 301. In any plane, a line making with H or V the same angle as the plane, is called a line of declivity of the plane. Fig. 175 shows a line of declivity with respect to H. Both the line and its plan must be perpendicular to the h. t. of the plane; for the inclination of a line to its plan is that of the line to H; and if, at the same time, the measure of a x^ig-. IT'S. dihedral angle, such lines must, by elementary geometry, be perpendicular to the intersection of the planes. Fig. 176 shows Fig. 175 orthographically. Fig. 177 gives a line of declivity with respect to V. 302. Were the plane P'QR (Fig. 175) rotated on its x-ig-. itt. line of declivity (a b, a' V) it would make an increasing angle with H until it was perpendicular to it; that is, if a plane contains a line, the limits of its inclinations are 90° and that of the g — ^ line. On the other hand, the line of declivity might be turned in the plane until " ^f^ horizontal. The limits of the inclinations of lines in a plane are therefore 0° and that of the plane. A plane may make 90° with H and be parallel to V, or vice versa; or it may be perpendicu- lar to both H and V; the limits of the mm of its inclinations to H and V are thus 90° and 180°. If parallel to G. L., but inclined to H and V, the sum of the inchnations of the plane is the lower limit, 90°. no THEORETICAL AND PRACTICAL GRAPHICS. x'igr- IT'S. If equally inclined to both H and V, but cutting G. L., the traces of a plane will meet the latter at equal angles. 303. If a right line, as A B (Pig. 178), is perpendicular to a plane P' Q R, its projections will be ipig-- ±'7s. perpendicular to the traces of the plane: for the plan of the line lies in the h. t. of its H-projecting plane; the latter plane is — from its definition — perpendicular to H, and also to the given plane by virtue of containing the given line; hence is perpen- dicular to the h. t. of the given plane, since such h. t. is the line of intersection of the latter with H. The same principles apply to the relation of the elevation of the line to the v. t. of the plane. 304. Any plane perpendicular to both H and V is called a pi-ofile plane. Such plane (P, Fig. 179) is used when side or end views of an object are to be projected. To bring V, H and P into one plane we suppose the latter first rotated into V aljout their line of intersection, o L, then both V and P about G. L. into H. Projections on the profile plane are usually lettered with a double accent, the same as for any point revolved into or parallel to V. When, as in Fig. 179 all the projectors of a line A F lie in the same projector- plane, both projections of the line will be perpendicular to G. L. The most convenient method of dealing with such line is to project it upon a profile plane and revolve the latter into V; or the profile projector- plane through the line might be directly revolved into V, carrying the line with it. The former method is pictorially illustrated in Fig. 179. Orthographically, the operation is shown in Fig. 180. In this, as in many other constructions, we make use of the following principles : (a) all projections of one point on two or more vertical planes will be at the same height above H ; (b) if rotation occur about an axis that is perpendicular to H, each arc described about that axis by a point revolved, will be projected on H as an equal circular arc ; similarly as to V, if the axis is perpendicular to it. Since in Figs. 179 and 180 we rotate upon a vertical axis, a projector, as A a", will be seen in o s, drawn through the plan of A and perpendicular to the h. t. of P. From s a circular arc, s s^ (Fig. 180), from centre L, will be the plan of the arc described by a" of Fig. 179. From s^ a vertical to the level of a' gives a". Similarly the projection /" is obtained, which, joined with a", gives a"f", the profile view of the line A F. 305. As far as our view of what is in the first angle is concerned, the rotation just described amounts, practically, to the turning of H and V through an angle of 90°, so that instead of facing V we see it "edgewise," as a line. Mo; while H appears also as a line, G L s^. We thus get an "end view" into the angle. All figures lying in profile planes are then '^■- "X^ ti seen in their true form. \'' In Fig. 181 let us start with the entire system of angles thus turned. The ground line appears as a point, T; H and V as lines; and two lines, "si AB and CD-each of which lies in a profile plane-are shown in their true length and inclination. FUNDAMENTAL PRINCIPLES OF MONGERS DESCRIPTIVE. Ill Perpendiculars to H from A, B, C, and D, give their plans a^,b,,c^ and d^. V-projectors give d', c', etc., the heights of the elevations of the points. Revolving the whole system into its usual position, remembering, meanwhile, that the profile plane, P, turns on a vertical axis, (as in Fig. 182,) which divides P into parts which are on opposite sides of the axis both before and after revolution, we find c^ at c- di at d; Oi at x. Assuming S S' as the plane of the line CD, and that the plane oi AB is R P', at a given distance T Q from S S', we find a' and 6' on R P' at the same level as A and B; while a is derived from x, and b from bi as shown. The elevations cf and d' on S S' are on the level of G and D respectively. 306. The term rabatment, already employed to indicate the rotation of a plane about one of its traces until it comes into a plane of projection, is also used to denote the rotation of a point or line into H or ^" about an axis in such plane. Restoration to an original space-position, after revolving, will be called counter-rabatment or counter-revolution. E-ig-. 1B3. In Pig. 183 we have a^ as the rabatment of ^-1 into H, about an axis m n. B a^ is equal to B A — the actual distance of A from the axis, and which is evidently the hypothenuse of the triangle A B a, whose altitude is the H-projector of the point and whose base is the BX' ^ h. p. of the real distance. "Were the axis parallel to and not in H we would state the prin- ciple thus: In revolving a point about, and to the same level with, an horizontal axis it will be found on a perpendicular drawn through the h. p. of the point to the axis, and at a distance from the latter equal to the hypothenuse of a right-angled triangle whose altitude equals the difference of level of point and axis, and whose base is the h. p. of the real distance. Were the axis in or par- allel to V, the base of the triangle constructed would be the v. p. of the desired distance, and the altitude would be — in the first case — the V-projector of the point, and — in the second case — the dif- ference of distances of point and axis from V. In any case, the vertex of the right angle, in the triangle constructed, is the projection of the original point on the plane in which or parallel to which hes the axis. In usual position the foregoing construction would appear thus : with the point given by its projections, (a a', Fig. 184), let fall a perpendicular through a to ran; prolong this le'igr- is-4. indefinitely ; make a the vertex of the right angle in a right triangle of base B a, altitude a' s ; then the hypothenuse of such triangle, used as radius of arc a^ a^ Q- (centre B) gives a^ as the revolved position desired. 307. In applying the foregoing principles in the following problems we shall frequently find it convenient to employ the right cone as an auxiliary surface. All the elements of such a cone are equally inclined to its base, and a tangent plane to -the cone makes with the base the same angle as the elements. ment of tangency is a line of declivity of the plane with respect to the base of the cone. If the base of the cone is on H the h. t. of a tangent plane will be tan- gent to the base of the cone ; similarly for its v. t. were the base in V. 308. Prob. 1. From the projections of a line to determine (a) its traces; (b) its actual length ; (c) its inclinations, and 4>, to H and "V respectively. (a) The traces of the given line, when it is oblique to both H and V, as in Fig. 186, are found by the rule given in Art. 293. 112 THEORETICAL AND PRACTICAL GRAPHICS. v.t. 3. at"'-...^^ / 'S \o 1 ~~--~^, X ~'~^~--^\ct In /^^^^ ^-~-.. h.t. bi ,- / n a ....\w 1 a \ 1 aS;^ (b) The actual length of a line may be found either (1) by rabatment into H or V, or (2) by rotation until parallel to H or V. By the first method rabat the line on its plan a 6 as an axis. It will show the true length on H in tti 6i, the distance a a^ equalling a' o — the original height of the point A above H ; similarly, b bi equals b' n. Notice that a^ a and 6i b must be perpendicular to the axis, and that each is the projection of a circular arc, described by the point revolved. The h. t. being the point where the revolved line meets the axis of rotation, is common to both the rabatment, a-^ b^, and the space-position of the line. If we make a' a" and b' b" perpendicular to a' b' and equal to a and b n respectively, we have in a" b" the real length, shown on V. By rotation till parallel to a plane of projection, as H, either extremity of the line may be brought to the level of the other, when the new plan will show the actual length. Thus, (Fig. 187), using i^ig-. ±s7. the V-projector of b b' as an axis, d may be brought to a", at the level of 6', by an arc, centre 6', radius a' b' . The circular arc a' a" thus described has its h. p. in ttj a, the distance from V having been constant during the rotation, since the axis was perpendicular to V. In a^ b we then have the real length sought. If we rotate the line on a vertical axis through a, until b reaches b-^, we will find the v. p. of the revolved point at 6", on its former level. The new pro- jection, d b", is again the real length, now projected on V. (c) The inclinations, 6 and 4>, to H and V respectively. Either of the foregoing constructions for showing the real length of a line solves also the problem as to inclination. Thus, in Fig. 186, the rabatted lines make with their axes of rabatment the angles sought. In Fig. 187 we have d b" inclined at the angle 6 to H, while b a^ makes with a a, the angle <^. When the line lies in a profile plane, the traces, length and inclination are found by means of the operation described in Art. 305 and illustrated by Fig. 181. In that figure, were c d and (f d' given, we would carry c and d about T as a centre to Ci and d„ whence perpendiculars to their former levels would give C and D; joining the latter we would have CD, whose v.t. is at z; h. t. (not shown) at a distance Ty above T; while 6 and <^ are seen in actual size at y and z. 309. Prob. 2. To determine the projections of a line of given length, having given its angles, 6 and <^, with H and V respectively. If with the line we generate a vertical right cone, of base angle 6, four elements could be found on the cone, each of which would make the angle <^ with V, and therefore fulfill all the conditions. The sum of 6 and 4> can obviously not exceed 90°, and when equalling that limit there is but one solution and the line can then be contained by a profile plane. For data take length of line, 2"; 61= 44°; = 30°. From any point 6" on G. L., draw m V a hne h" d, of the given length and at the angle 6 = 44° with G. L. The plan of this line is a b", which use as the radius of the base of a semi-cone of vertical axis a d. Remembering that the inclination of a Hne determines the length of its projection, we next ascertain how long the projection of a two-inch Hne will be when inclined 30° to a plane. Drawing d T at X'ig- ISS. MONGE'S DESCRIPTIVE. — ELEMENTARY PROBLEMS. 113 In i^xir- iss. 30° to a' b", and projecting b" perpendicularly upon it at n, we find a' n as the invariable length sought. Arc n b' from centre a', gives a' h' as the v. p. of an element of the cone, whence a b fol- lows, as the plan of the desired line. As arc n b', continued, would cut G. L. at x, whence s and m could be derived, we wt)uld find s a and m a as the plans of two more elements fulfilling the conditions. Also, in line with b b', one more point (omitted to avoid confusing the solutions) could be found, on the rear of the cone. 310. Prob. o. To determine the plane containing (a) two intersecting lines; (b) two parallel lines. Fig. 189 let o' be the point of intersection of the lines A B and CD. Prolong the lines and obtain their traces, as in Art 293. R Q, the h. t. of the plane, is the line connecting m and n, the horizontal traces of the lines. Similarly, P' Q passes through the vertical traces e' and /'. (b). Parallel lines have parallel projections, and the traces of their plane connect like traces of the lines. 311. Prob. 4- To show the actual size of the angle between iivo lines. If the actual angle is 90° it will be projected as such when •one or both of its sides are parallel to the plane of projection: for, if both are parallel, the traces of the projecting planes — in which lie the projections of the lines — will evidently be perjDen- dicular to each other ; if either side of the angle in space be then rotated about the other side as an axis, it will turn in its previous projector- plane, and its projection will still fall upon the sam^ "trace as before. In general, to show the true size of any angle, rotate its plane either into or parallel to H or V. In Fig. 189 the angle whose vertex is o o' is shown by obtaining the plane R Q P' of its sides, then rabatting about R Q into H. The vertex reaches o^ after describing an arc whose plane is perpendicular to R Q and which is projected in o Oj. The actual space-distance of from r — the point on the axis, about which it turns — is the hypothenuse of a right triangle whose altitude o = o' s' and whose base is o r. Joining o, with m and n — the intersections of the axis by the lines revolved — gives moin, the angle sought. To obtain the angle without having the traces of its plane we may use as an axis the line connecting points — one on each line — and equidistant from a plane of projection. Thus, in Fig. 190, we find a' and ^' at the same level, and a g for the plan of the line connecting them ; then o rotates to Oj about a g, giving a Oj^g for the desired angle. ^^^- ^®^- 312. Prob. 5. To draw a horizon- tal and a V-parallel in a plane. (Fig. 191.) As shown by Figs. 173 and 174, any horizontal line has a v. p. par- allel to G. L., and — if contained by a plane— must be parallel to the h. t. of '■~-- --''' the plane, and meet V on its v.t.; there- fore any line b' d, parallel to G. L., will do for the v. p. of the desired line ; the intersection of P' Q FLs- ISO. 114 THEORETICAL AND PRACTICAL GRAPHICS. by h' c' will be the v. t. of the line, a vertical through which to G. L. gives c— one point of the plan c b, whose known direction enables it to be immediately drawn. A V-parallel is parallel to the v. t. of the plane in which it lies, and meets H on the h. t. of the plane; hence assume dl as the h. t. of the desired line, project to d' and draw d' a' parallel to' P'Q; then da, parallel to G. L., represents (with d' a') a V-parallel of the plane. Additional conditions might be assigned to either kind of line, as, for example, the distance from the plane to which the line is parallel; the quadrant; the length of the line, or that it should contain a certain point of the plane. 313. Prob. 6. Having one projection of a point in a plane, to locate the other prelection. If a point is on a line, its projections are vertically above each other on the projections of the line; if, there- fore, the plan, s, (Fig. 191) of the point is given, determine a horizontal line through the point and in the plane. This will be h s c, parallel to R Q. From c, where it meets G. L., a vertical to P' Q gives its v. t., through which draw c' b' parallel to G. L. The desired v. p. is then s', on c' b' and vertically above s. Were the elevation of the point given we would find its plan on the h. p. of a V-parallel drawn through the point and in the plane. 314. Prob. 7. To pass a plane through three points not in the same straight line. Any two of the three lines that would connect the points by pah's would determine the plane by the first case of Prob. 3 ; while the line joining any two of the points, together with a parallel to it through the third point, becomes the second case of the same problem. 315. Prob. 8. To pass a plane through one line and parallel to another draw through any point of the first hue a parallel to the second line; such parallel will, with the first line, determine the plane. 316. Prob. 9. To pass a plane through a given point and parallel to a given plane. Two fines through the given point and parallel to any pair of lines in the given plane will determine the required plane. In Fig. 192, with 0' as the given point, and M'iVJIf as the given plane, assume in the latter any two Imes, ^ £ and CD; parallel to these lines and through o o' draw ST and E F, whose traces will determine those of the plane sought. MONGERS DESCRIPTIVE. — ELEMENTARY PROBLEMS. 115 X'lg-. IS-i- Since parallel planes have parallel traces, one line throiigh o o' would suffice. For example, s t, s' if, parallel to a b, a' b', gives the traces t and s', through which draw R Q and Q P', parallel respectively to the like traces of the given plane. 317. Prob. 10. To pass a plane through a given point and perpendicular to a given line. Since, by Art. 303, the traces of the desired plane will be perpendicular to like projections of the line, draw through the given point, o o', Fig. 193, either a horizontal or a V-par- s-ig-. iss. allel of the desired plane ; the trace of either line, and the known directions of the required traces, suffice to solve the problem. The plan, n, of a horizontal, will be perpendicular to a b — the plan of the given line ; through s', the v. t. of the horizontal, we draw P' Q perpendicular to a' b', for the v. t. of the plane; then Q R perpendic- ular to a 6 for the desired h. t. A ^''-parallel through o o' is obtained by drawing o' y' perpendicular to a' b', and o y parallel to G. L. ; through y — the h. t. of the V-parallel — we then draw R Q in the known direction, then, from Q, the trace Q P' perpendicular to a! V . 318. Prob. 11. To determine (a) the angles 6 and , made with H and V respectively, by a given plane; (b) the angle between the traces of the plane. From the properties of the cone and its tangent plane, mentioned in Art. 307, we may solve the problem by generating a cone with a line of declivity of the plane and ascertaining the inclination of such line. In Fig. 194 let P' Q R be the plane. The projections a' b' and a b — the latter perpendicular to R Q— represent a line of declivity of the plane with respect to H. With it, and about a' a as an axis, generate a semi-cone. When the generatrix reaches V, either at b" a' or o' n, its inclination to G. L. shows the base angle 6 of the cone and therefore the inclination of the given plane. With respect to V the- construction is analo- gous. A line of declivity with respect to V has its V. p. perpendicular to P' Q, (Fig. 195). Using d on the h. t. of the plane, as the vertex of a semi-cone of horizontal axis d d', we find the base of the latter tan- gent to P' Q at c'. Carrying c' to the ground line, about d' as a centre, and joining it with d gives the angle cf sought. This problem might also be readily solved by rabatting the fine of decUvity into a plane of projec- jection. Thus, making d' d" perpendicular to c' d' and equal to d' d, we find the angle <^ between cf d' and c' d". For a plane parallel to G. L. use an auxiUary profile plane, rotating its line of intersection with the given plane as in Fig. 196. (h). The angle between the traces is obtained by rabatment of the given plane about either of its traces. In Fig. 195, using] trace i? Q as an axis and rotating QP' about it, any point, c', thus turned, will describe an arc projected in a perpendicular through c to Q R. Q being on the axis is constant during this rotation, and the {distance from it to (/ will be the same after as before revolution; therefore cut c c, by an arc, centre Q, radius Q , to H and V respectively. This is, obviously, the converse of Prob. 11 and is, practically, the same construction in reverse. The required plane will be tangent, simultaneously, to two semi-cones of base angles 6 and 4>, and having axes (a) in V and H respectively, and (b) in the same profile plane. Assume in V any vertical line a! a" as the axis of the ^-cone, and draw from any point of it, as a', a line a' b", at 6° to G. L; use a" b", the plan of this line, as radius of the base of the vertical semi-cone, to which the desired h. t. of the required plane will be tangent. The line a" s shows the perpendicular distance from a to the point of intersection of the two elements of tangency of the re- quired plane with the cones ; hence the generatrix of the \' the saine matliematieal nature. 349. Warjicd Suilnrrs nf Traiifpnslli,,,!. Tlie most important of these are the hi/perholic parahohnd ; tlic dliiitlcnl JniprrhnU/id or Jnipcrholnid nf one slwet : oninidal surfaces such as the rnnn-auiciis of Wallis and the aovdd nf Pincbr : the u'orpcd hdimid; the nmchoidal hyjierbniold of CaUdan ; the cyllmlroid oj Firzicr, and the wnrprd nrrli, also called tlie arnu:. dc vnchi'. lieferrinn- to iVrt. oi:-;, regarding the nurnher of conditions that may be imposed on a moving — i=-. 210. ^'^s- z±±. HYI'EKliOLle' I'ARAllOLelD. ELLinUAL IirrEKHciLOID. line, let us first consider all tlie possible' surlae-es having three stroiiiJif dircrtriees. Evidently such directrices, no twn of whicli lie in one plane, eitlrer are or are not parallel to some plane; for iiny two of them will invariably be parallel to some [)lane, and the third either is or is not parallel to that jjlane. Should tin.' former be the case the resulting surface would lie the In/perbolic parid>oloid, which may alsn be deKned as the warped surface having (((■') slrniijht ilirertriees and a piianc director; if however, tlie dire(.'triees haxe tlie se(.-()nd position sui)}ioseeen pre-empted by Frezier for the surface defined in Art. 3(J0. ■See Pliickej-'s Netie Oeometrie den Haumes, p. 97: al.so Maimheini'.:* Gco/nefrie De-'icriplicc, p. 43.?. V2C, 'I'lIEdRETICA L AM) I' L'A CTlcA L C UA T IFICH. V\'S. 21") reprosonts a hhmIi'I of tin/ coiinid uihIit consideration; Imt IVir the exact niatheniatical Hurfac(/ "the dianietev (if tli<; central (^vliudci- nnist l>e conrcived to l.e eN'anesccnt and tlie radiating wires must lie extemled to inlinity." (Bull). ^\'el•e A B and CJ> { Fiu'. 21.';) to lie the coinnion directrices of hyiierlxjlie paraholoids wliose plane directors were various positions of P R Q, rotated ahout the common perpendicular B C, their lines of strietion would he elements of a conoid of Pliieker. The same surfact: will result if ;i right line he moved so that, while perpendicular to and inter- wecting the axis of a circular eylindei-, it shall follow a douhle-eurved directrix uhtained hy wrapping -.u'ound tlie cylinder a, slnn^ind (Art. 171j or Jinrnionir cnrrc, U\o waves of wliich reach once around. 357. The irnrjiei'l Inlicoid has fir its dirci'trices a lielix and its axis, the generatrix making a ■constant angle witli the lattia-. The last condition is eipii\-alent to saying that the surface has a ■cone director. ^ OBLlciCE UELICdIDS. r.ieilT 11 K LI C( 1 1 li. KICHT IIKLICOIDS. If tlie angle made Jiy the elements with tlie axis is uruU'. the surface is called the obliipie hdicoid. It is the acting surface of ordinary trianuular-threaded screws, and of many screw propellers. Plane sections pierpendieular to tlie axis of this surface are j\rchiinede;m spirals. 358. Wlien the elements are perpendicular to the axis the cone director heeomes a plane direc- I^ir- SIS. tor ami the surface is called the rl,,/,l helicoid, laiiiilini' to all as the inidei- surface of a siiiral staircase. It is the acting surficc of a si|uaiv^tlircadc,l scrc« nnd. li'ei|Ucnllv, of s<-rew-propellers. WARPED SURFACES. — DOUBLE CURVED SURFACES. 127 The right heUcoid also belongs among conoidal surfaces, and its axis is — like that of the cono- cuneus — a line of striction. 359. The conchoidal hyperboloid, invented by Catalan, has two non-intersecting, rectilinear dire,ctrices — one horizontal, the other vertical — the generatrix making a constant angle with th.e latter. Planes parallel to both directrices will cut hyperbolas from the surface, while horizontal sections will be conchoids. (See Arts. 193 and 196). 360. The cylindroid of Frezier has a plane director and two curved directrices. In its usual form it may be imagined to be thus derived from a cylinder of revolution : Suppose a cylinder A BCD (Fig. 220) on H and parallel to V — the plane director; for curved directrices employ the ellipses cut from the cylinder by non-parallel, vertical, section-planes, ^ig-- 220. ab, c d, taken on opposite sides of some vertical right section ; if owe of these ellipses be shifted vertically, in its own plane, the lines joining the new positions of its points with their former points of connection on the other directrix will be elements of a cylindroid. This surface has been suggested for the soffit (under surface) of a descending arch. Any plane containing the line in which the planes of the curved direc- trices intersect will cut congruent' curves from the cylinder and cylindroid ; while planes parallel to such line will cut plane sections of the same area. 361. The warped arch or corne de vache has three linear directrices, one straight the others curved ; the latter are equal circles in parallel planes, while the straight directrix ^s.s- sai. is perpendicular to the planes of the circles and passes through the middle point of the line joining their centres. In oblique or skew arch construction one of the best known methods is that in which the soffit of the arch is a Corne de Vache.^ cornk dk vache. CYLINDROID OF FRBZIBK. DOUBLE CURVED SURFACES. 362. Double Curved Surfaces are surfaces that cannot be generated by a right line. 363. Double Curved Surfaces of Revolution. The sphere is the most familiar example under this head, the generatrix being a semi-circle and the axis its diameter. After it come the ellipsoids— the prolate spheroid and the oblate spheroid — generated by rotating an ellipse about its major or minor axis :^5.g. aas. , respectively ; the paraboloid of revolution, generatrix — a parabola, axis — that of the curve; the hyperboloid of revolution of separate nappes, formed by rotating the two branches of an hyperbola about their transverse axis; and the torus — annular or not — generated by revolving a circle about an axis in its plane but not a di- ameter. (See Fig. 222; also Arts. 112-114). 364. The revolution of other plane curves — as the involute, tractrix, cycloid, conchoid, etc., gives double curved surfaces of frequent use in architectural constructions and the arts.' 365. Double Curved Surfaces of Transposition. Of the innumerable surfaces possible under this head, we need only mention here the serpentine, generated by a sphere whose centre travels along a helix ;•, the ellipsoid of three unequal axes, which would result from turning an ellipse about one of its axes- in such manner that, while remaining an ellipse, its other axis should so vary in length that its extremities would trace a second ellipse; the elliptical paraboloid, whose plane sections perpendicular THE TOKIJS. 1 Congruent figures, if superposed, win coincide throughout. 2 For a comparison of the relative merits of these methods refer to Skew Arches, hy E. W. Hyde. (Van Nostrand's Science,- Series, No. 15.) ssee Note, p. 64; also Art. 203. 128 THEORETICAL AND PRACTICAL GRAI'HICS. to the axis are clli|is(.s, while sections containiiit: tlie axis are paratiolas ; the elh'ptical Injperboloid of onr rKtiijic, generatiMl hy tnniinu- a varialjle hj'perliola aliont its real axis so tliat its arc shall follow an elliptical direetrix ; tlie cUijilical Jti/jicrholoiil nf tivo nappes, analogous to the two-napped h3'perboloitl E^ig-. 2E3. E-ig-- SS-i- E'ig-- 23S. .SEtSJ'ENTINK. ELLII'MOID. ELLIPTICAL IIYl'EKLOLOID. of revolution, )jut having elliptical instead of circular sections perpendicular to its axis ; and the cydide*, whose lines of curvature (see Art. 375j are all circles and each of whose normals intersects two conies — an ellij^se and hyperliola — whose planes are mutuallj' perpendicular and having the foci of each at the extreniities of the transverse axis of the other. The cyclide is the envelope of all ELLIPTICAL HYIIII.iLell, oh TWO NAPPES. THE CYCLIBE «F BUPIX. si-heres (a; having tlieir centres on one of the conies and (1.) tangent to any si.here whose centre is on the other conic. The torus is a special case of the cyclide. TUISULAR SURFACES. — (lUADRIO SIIRFACES OR CONICOiriS. 306. Among the surfaces we have descriljed, the tonis and serpcnUnc belong to the tamily called inhiiliir, since each is tlie eiivclojie af a sphere of constant railius. 367. i^urfoces wliose ].lane sections are invariaf.ly conies are called eonlenllan draw concentric semi -circles acd and hsr, of diameters x and y respectively, join- int;- their extremities by straight lines ah, c d. At a distance apart of z inches draw the upper and lower limits of the elevations, and project to these levels from the points of the plan. In the side view the thickness of the shell of the cylinder is shown by the distance between e"f" and s"t" — the latter so drawn as to indicate an invisible limit or line of the object. The line shading would usually be omitted, the shade lines generally sufficing to convey a clear idea of the form. 391. Half of a hollow, hexagonal prism. In a semi -circle of diameter a d step off the radius three times as a chord, giving the vertices of the plan a bed of the outer surface. Parallel to b r, and at a distance from it equal to the assigned thickness of the prism, draw ef terminating it on lines (not shown) I^ig-- 23S- X^igr- SS-Sfc- apart, and a, b, s, x, etc., projected to them. The elevations of the opening are between levels m'm" and k'k", one inch apart and equi- distant from the upper and lower outhnes of the views. The dotted construction lines and the lettering will enable the student to recognize the three views of any point without difficulty. 393. In Fig. 237 we have the same object as that illustrated by Fig. 236, Init now represented as cut by a vertical plane whose horizontal trace is vy. The parts of the bLx^k that are actually cut by the plane are shown in section - lines in the elevations. This is done here and in some later examples merely to aid the beginner in understanding the views; but, in, engineering prar- tire, section -lining is ravel i/ do drawn through b and c at 60° to ad. From e and / draw ek &nd fg, parallel respectively to ab and rd. Drawing a'c" and m't" as upper and lower limits, project to them as in preceding problems for the front and side elevations. 392. Working drawimg of a hollow, prismatic block, standing obliquely to the vertical and profile planes. Let the block be 2"x3"xl" outside, with a square open- ing l"xl"xl" through it in the direction of its thickness. Assuming that it has been required that the two -inch edges should be vertical, we first draw, in Fig. 236, the plan asxb, 3"xl", on a scale of 1:2. The inch -wide opening through the centre is indicated by the short- dash lines. For the elevations the upper and lower limits are drawn 2" X^ig-- 23S. 3 ^— -. jHr^^ ^ \ i ^^\^ r-::v ~'"-- V y ' X ' _^^ .^, \ ! \ \ '^ ^Y'^ 'y^ _G — 1- 1 ] L i \ >.ft" ! ' c • |-.__ VI .i' ^ 1 1 vf ■ J j i- i ._ 1/ (', k 1 /lone on, mews not pcrpendicnlar to the section plane. PROJECTION OF SOLIDS.— WORKING DRAWINGS. 137 SECTIONAL VIEW 394. Suppression of the ground line. In machine drawing it is customary to omit the ground line, since the forms of the various views — which alone concern us— are independent of the distance of the object from an imaginary horizon- tal or vertical plane. We have only to remember that all elevations of a point are at the same level; and that if a ground line or trace of any vertical plane is wanted, it will be perpendicular to the line joining the plan of a point with its projection on such vertical plane. (Art. 286.) 395. Sections. Sectional views. Although earlier defined (Art. 70), a re-statement of the distinction between these terms may well precede problems in which they will be so frequently employed. When a plane cuts a solid, that portion of the latter which comes in actual contact with the cutting plane is called the section. A sectional view is a view perpendicular to the cutting plane, and showing not only the section but also the object itself as if seen through the plane. When the cutting plane is vertical such a view is called a sectional elevation; when horizontal, a sectional plan. 396. Working draiving of a regular, pentagonal pyramid, hollow, truncated by an oblique plane; also the development, or "pattern,''' of the outer surface below the cutting plane. For data take the altitude at 2"; inclination of faces, 0° (meaning any arbitrary angle); inclination of section plane, 30°; distance between inner and outer faces of pyramid, ^". (1) Locate v and v' (Fig. 238) for the plan and elevation of the vertex, taking them sufficiently apart to avoid the overlapping of one view upon the other. Through v draw the horizontal line ST, regarding it not only as a centre line for the plan but also as the h. t. of a central, vertical, refer- ence plane, parallel to the ordi- nary vertical plane of projection. PLAN 138 THEORETICAL AND PRACTICAL GRAPHICS. SECTIONAL VIEW PLAN (The student should note that for convenient reference Fig. 238 is repeated on this page.) On the vertical Hne vv' (at first indefinite in length) lay off v' s' equal to 2", for the altitude (and axis) of the pyramid, and through s' draw an indefinite horizontal line, which will contain the V. p. of the base, in both front and side views. Draw v'h' at 61° to the horizontal. It will represent the v. p. of an outer face of the pyramid, and h' will be the v. p. of the edge a 6 of the hose. The base ahcde is then a regular pentagon circumscribed about a circle of centre v and radius i;i = s' 6'. Since the angle avh is 72° (Art. 92) we get a starting corner, a or b, by drawing « a or -y 6 at 36° to /ST, to intercept the vertical through b'. The plans of the edges of the pyramid are then v a, vb, v c, vd and v e. Project d to d' and draw v' d' for the elevation of V d; similarly for v e and v c, which happen in this case to co- incide in vertical projection. For the inner surface of the pyramid, whose faces are at a perpendicular distance of ^" from the outer, begin by drawing g'V parallel to and \" from the face projected in b'v'; this will cut the axis at a point t' which will be the vertex of the inner sur- face, and g' t' will represent the elevation of the inner face that is parallel to the face avb — v' b' ; while gh, vertically above g' and included between va and vb, will be the plan of the lower edge of this face. Complete the pentagon gh — k for the plan of the inner base; project the corners to b' d' and join with t' to get the ele- vations of the interior edges. The section. In our figure let G' H' be the section plane, sit- uated perpendicular to the ver- tical plane and inclined 30° to the horizontal. It intersects v' d' the edges v' c' and v' e' at points projected in and q. A like construction gives m and boundary of the section. The inner edge g' t' is cut by the section plane at I', which projects to both vh and v g, giving the parallel to mn through I. The inner boundary of the section may then be completed either by determining all its vertices in the same way or on the principle that its sides will be parallel to those of the outer polygon, since any two planes are cut l)y a third in parallel lines. in p', which projects upon vd at p. Similarly, since G' H' cuts o', we project from the latter to v c and ve, obtaining The polygon mnopq is then the plan of the outer The line m'p' is the vertical projection of the entire section. PROJECTION OF SOLIDS.— WORKING DRAWINGS. 139 (2) The side elevation. This might be obtained exactly as in the five preceding figures, that is, by actually locating the side vertical, or profile, plane, projecting upon it and rotating through an arc of 90°. In engineering practice, however, the method now to he described is in far more general use. It does not do away with the profile plane, on the contrary presupposes its existence, but instead of actually locating it and drawing the arcs which so far have kept the relation of the views constantly before the eye, it reaches the same result in the following manner : A vertical line fS" T' is drawn at some convenient distance to the right of the front elevation ; the distance, from S T, of any point of the plan, is then laid off horizontally from S' T', at the same height as the front elevation of the point. For, as earlier stated, S T was to be regarded as the horizontal trace of a vertical plane. Such plane would evidently cut a profile plane in a vertical line, which we may call S'T', and let the S' T' of our figure represent it after a ninety -degree rotation has occurred. The distances of all points of the object, to either the front or rear of the vertical plane on S T would, obviously, be now seen as distances to the left or right, respectively, of the trace S' T', and would be directly transferred with the dividers to the lines indicating their level. Thus, e" is on the level of e', but is to the right of S'T' the same distance that e is above (or, in reality, behind^ the plane ST; that is, e" d" equals eu. Similarly d"h" equals ib; n"x" equals nx. It is usual, where the object is at all symmetrical, to locate these reference planes centrally, so that their traces, used as indicated, may bisect as many lines as possible, to make one setting of the dividers do double work. (3) True size of the section. Sectional view. If the section plane G' H' were rotated directly about its trace on the central, vertical plane S T, until parallel to the paper, it would show the section m'p' — mnopq in its true size; but such a construction would cause a confusion of lines, the new figure overlapping the front elevation. If, however, we transfer the plane G' H' — keeping it parallel to its first position during the motion — to some new position S"T", and then turn it 90° on that line, we get tn^n^o^p-^q^, the desired view of the section. The distances of the vertices of the section from S" T" are derived from reference to ST exactly as were those in the side elevation; that is, mia;i = ma; = m"a;". We thus see that one central, vertical, reference plane, ST, is auxiliary to the construction of two important views ; S' T' represents its intersection with the profile or side vertical plane, while S" T" is its (transferred) trace upon the section plane G'H'. For the remainder of the sectional view the points are obtained exactly as above described for the section; thus c'ci^i is perpendicular to S" T" ; e^u^ equals eu, and c^Ui equals cu. (4) To determine the actual length of the various edges. The only edge of the original, uncut pyramid, that would require no construction in order to show its true length, is the extreme right- hand one, which — being parallel to the vertical plane, as shown by its plan v d being horizontal — is seen in elevation in its true size, v'd'. Since, however, all the edges of the pyramid are equal, we may find on v'd' the true length of any portion of some other edge, as, for example o'c', by taking that part of v'd' which is intercepted between the same horizontals, viz.: o"'d'. Were we compelled to find the true length of o'c', oc, independently of any such convenient relation as that just indicated, we would apply one of the methods fully illustrated by Figs. 183, 184 and 187, or the following "shop" modification of one of them: Parallel to the plan oc draw a line yz, their distance apart to be equal to the difference of level of o' and c', which difference may be obtained from either of the elevations; from the plan o of the higher end of the line draw the common perpendicular of, and join / with c, obtaining the desired length fc'. (5) To show the exact form of any face of the pyramid. Taking, for example, the face ocdp, revolve o p about the horizontal edge c d until it reaches the level of the latter. The actual distance 140 THEORETICAL AND PRACTICAL GRAPHICS. of from c, and of p from d will be the same after as before this revolution, while the paths of o and 2^ during rotation will be projected in lines or and pw, each perpendicular to cd; therefore, with c as a centre, cut the perpendicular or by an arc of radius /c— just ascertained to be the real length of oc, and, similarly, cut piv by an arc of radius dw = p'd'; join r with c, w with d, draw w r and we have in cdivr the form desired. (6) The development of the outer surface of the truncated pyramid. With any point F as a centre (Fig. 239) and with radius equal to the actual length of an edge of the pyramid (that is, equal to v'd', Fig. 238) draw an indefinite arc, on which lay off the chords DC, C B, B A, A E, ED, equal respectively to the like -lettered edges of the base abcde; join the extremities of these chords with V: then on D F lay off DP=d'p'; make C 0= E Q= d'o'" = the real length of c'o'; also BN= AM=d'm"'=the actual length of a'm' and b'n'; join the points P, 0, etc., thus obtaining the development of the outer boundary of the section. The pattern A.B^CDE^ of the base is obtained from the plan in Fig. 238, while NMq^p^o^ is a duplicate of the shaded part of the sectional view in the same figure. (7) In making a model of the pyramid the student should use heavy Bristol board, and make allowance, wherever needed, of an extra width for overlap, slit as at x, y and z (Fig. 289). On this ^xg-- S3©. overlap put the mucilage which is to hold the model in shape. The faces will fold better if the Bristol board is cut half way through on the folding edge. 397. For convenient reference the characteristic features of the Third Angle Method, all of which have now been fully illustrated, may thus be briefly summarized : (a) The various views of the object are so grouped that the plan or top view comes above the front elevation; that of the bottom below it; and analogously for the projections of the right and left sides. (b) Central, reference planes are taken through the various views, and, in each view, the distance of any point from the trace of the central plane of that view is obtained by direct transfer, with the dividers, of the distance between the same point and reference plane, as seen in some other view, usually the plan. 398. To draw a truncated, pyramidal block, having a rectangular recess in its top; angle of sides, 60°; lower base a rectangle 3" x 2", having its longer sides at 30° to the horizontal; total height recess 1^" X ^", and i" deep. (Fig. 240.) The small oblique projection on the right of the plan shows, pictorially, the figure to be drawn. 6 ". 10 ) PROJECTION OF SOLIDS.— WORKING DRAWINGS. 141 The plan of the lower base will be the rectangle abde, 8" x 2", whose longer edges are inclined 30° to the horizontal. Take AB and mn as the H- traces of auxiliary, vertical planes, perpendicular to the side and end faces of the block. Then the sloping face whose lower edge is de, and which is inclined 60° to H, wiU have d^^y for its trace on plane mn. A parallel to mw and ^" from it will give s^, the auxiliary projection of the upper edge of the face aved, whence sv — at first indefinite in length — is derived, parallel to de. Similarly the end face bt-sd is obtained by projecting db upon AB at b^, drawing b^z at 60° to AB and terminating it at s, by CD, drawn at the same height (t^xt") as before. A parallel to b d through s^ intersects vs^ at s, giving one corner of the plan of the upper base, from which the rectangle stuv is completed, with sides parallel to those of the lower base. r" t" 1— A 1-V— I di \ As the recess has vertical sides we may draw its plan, o pqr, directly from the given dimen- sions, and show the depth by short -dash lines in each of the elevations. The ordinary elevations are derived from the plan as in preceding problems; that is, for the front elevation, a'u's'd', by verticals through the plans, terminating according to their height, either on a' d' or on u's', ■^" above it. For the side elevation, e"v"t"b", with the heights as in the front elevation, the distances to the right or left of s" equal those of the plans of the same points from 8 i, regarding the latter as the h. t. of a central, vertical plane, parallel to V. The plane ST oi right section, perpendicular to the axis KL, cuts the block in a section whose true size is shown in the line -tinted figure gih^k^li, and whose construction hardly needs detailed treatment after what has preceded. The shaded, longitudinal section, on central, vertical plane KL, also interprets itself by means of the lettering. 142 THEORETICAL AND PRACTICAL GRAPHICS. The true size of any face, as auve, may be shown by rabatment about a horizontal edge, as a e. As V is actually 3%" above the level of e, we see that v e (in space) is the hypothenuse of a triangle of base ve and altitude ^". Construct such a triangle, v v^e, and with its hypothenuse v^e as a radius, and e as a centre, obtain Vj on a perpendicular to ae through ■;; and representing the path of rotation. Finding u^ similarly we have au^v^e as the actual size of the face in question. If more views were needed than are shown the student ought to have no difficulty in their construction, as no new princijDles would be involved. 399. To draw a hollow, pentagonal prism, 2" long; edges to be horizontal and inclined 35° to V; base, a regular pentagon of 1" sides; one face of the prism to be inclined 60° to H; distance between inner and outer faces, ^". In Fig. 241 let HK he parallel to the plans of the axis and edges; it will make 35° with a horizontal line. Perpendicular to H K draw mw as the h. t. of an auxiliary, vertical plane, upon which we may suppose the base of the prism projected. In end view all the faces of the prism would be seen as lines, and all the edges as points. Draw a^b^, one inch long and at 60° to mn to represent the face whose inclination is assigned. Completing the inner and outer pentagons allowing Y' for the distance between faces, we have the end view complete. The plan is then PROJECTION OF SOLIDS. — WORKING DRAWINGS. 143 obtained by drawing parallels to HK through all the vertices of the end view, and terminating all by vertical planes, ad and gh, parallel to m7i and 2" apart. The elevations will be included between horizontal lines whose distance apart is the extreme height z of the end view; and all points of the front elevation are on verticals through their plans, and at heights derived from the end view. The most expeditious method of working is to draw a horizontal reference line, like that of Fig. 243, which shall contain the lowest edge of each elevation; measuring upward from this line lay off, on some random, vertical line, the distance of each point of the end view from a line (as the parallel to mn through b^ in Fig. 241, or xy in Fig. 243) which repre- sents the intersection of the plane of the end view by a horizontal plane containing the lowest point or edge of the object; horizontal lines, through the points of division thus obtained, will contain the projections of the corners of the front elevation, which may then be definitely located by vertical lines let fall from the plans of the same points. For example, e' and /', Fig. 241, are at a height, z, above the lowest line of the elevation, equal to the distance of e^ from the dotted line through b^; or, referring to Fig. 243, which, owing to its greater complexity, has its construction given more in detail, the distance upward from M to line G is equal to g^g.^ on the end view; from If to Q equals q-^q^, and similarly for the rest. Since the profile plane is omitted in Fig. 241 we take M' N' to represent the trace upon it of the auxiliary, central, vertical plane whose h. t. is MN; as already explained, all points of the side elevation are then at the same level as in the front elevation, and at distances to the right or left of M' N' equal to the perpendicular distances of their plans from M N. For example, e" s" equals e s. The shade lines are located on the end view on the assumption that the observer is looking toward it in the direction HK. 400. Projections of a hollow, ■pentagonal prism, cut by a vertical plane oblique to V. Letting the data for the prism be the same as in the last problem, we are to find what modification in the appear- ance of the elevations would result from cutting through the object by a vertical plane PQ (Fig. 242) and removing the part hxdi which lies in front of the plane of section. Each vertex of the section is on an edge of the elevation and is vertically below the point where P Q cuts the plan of the same edge ; the student can, therefore, readily convert the elevations of Fig. 241 into reproductions of those of Fig. 242 by drawing across the plan of Fig. 241 a trace P Q, similarly situated to the P Q of Fig. 242. Supposing that done, refer in what follows to both Figures 241 and 242. Since P Q contains h we find h' as one corner of the section. Both ends of the prism being vertical, they will be cut by the vertical plane P Q in vertical lines; therefore h'V is vertical until the top of the prism is reached, at l'. Join I' with x', the latter on the vertical through x — the intersection of PQ with the right-hand top edge ed, e'd'. From x the cut is vertical until the interior of the prism is reached, at o', on the line 6-4. We next reach w' on edge No. 4. The line o'w' has to be parallel to x'V (two parallel planes are cut by a third in parallel lines); but from lo' the interior edge of the section is not parallel to I'h', since PQ is not cutting a vertical end, but the inclined, interior surface. The other points hardly need detailed description, being similarly found. The side elevation is obtained in accordance with the principle fully described in Art. 396 and summarized in Art. 397 (b). M' N' represents the same plane as MN; e" s" equals es, and anal- ogously for other points. 401. In his elementary work in projections and sections of solids the student is recommended to lay an even tint of burnt sienna, medium tone, over the projections of the object, after which 144 THEORETICAL AND PRACTICAL GRAPHICS. any section may be line-tinted; and, if he desires to further improve the appearance of the yiews, distinctions may be made between the tones of the various surfaces by overlaying the burnt sienna with flat or graded washes of India ink. FRONT ELEVATION 402. Projections of an L-shaped bloch, after being cut by a plane oblique to both V and H; the block also to be inclined to V and H, and to have running through it two, non-communicating, rectangular openings, whose directions are mutually perpendicular. If the dotted lines are taken into account the front elevation in Fig. 243 gives a clear idea of the shape of the original solid. The end view and plan give the dimensions. Requiring the horizontal edges of the block to be inclined 30° to V, draw the first line xy ai 60° to the horizontal; the plans of aU the horizontal edges will be perpendicular to xy. Let the inclination of the bottom of the block to H be 20°. This is shown in the end view by drawing m^p^ at 20° to xy. All the edges of the end view of the object will then be parallel or perpendicular to m^p^ and should be next drawn to the given dimensions. WORKING DRAWINGS. — PLANE SECTIONS OF SOLIDS. 145 PLAN The central opening, 6jcZi«.,0i, through the larger part of the block, has its faces all ^" from the outer faces. In the plan this is shown by drawing the lines lettered of at a distance of ^" from the boundary lines, which last are indicated as 1-|" apart. The opening qiT^Sj^t^ has three of its faces i" from the outer surfaces of the block, while the fourth, giTi, is in the same plane as the outer face h^e^. The cutting plane X Y gives a section which is seen in end view in the lines c^g^, i^j-^ and Tc^l^; while in plan the section is pro- jected in the shaded portion, obtained, like all other parts of the plan, by perpendiculars to xy from all the points of the end view. For the front eleva- tion draw first the "reference line." To provide against overlapping of projections the reference line should be at a greater distance below the lowest point, I, of the plan, than the greatest height (a^a^ of the end view above xy. Then on MW lay off from M the heights of the vari- ous horizontal edges of the block, deriving them from the end view. Thus a^a^ is the height of A a' from M; from M to level B equals h^h^, etc. Next project to the level A from points ao of the plan, getting edge a' a' of the elevation, and similarly for all the other corners of the block. Notice that all lines that are 'parallel on the object will he parallel in each projec- tion (except when their projections coin- cide) ; also that in the case of sections, those outlines will be parallel which are the inter- section of parallel planes by a third plane. These principles may be advantageously employed as checks on the accuracy of the construction by points. The construction of the side elevation is left to the student. 146 THEORETICAL AND PRACTICAL GRAPHICS. FRONT ELEVATION. With section made by vertical plane P Q Reference line SIDE ELEVATION. With section by plane S T. Shade lines on this view are located for pictorial effect and not in accord- ance with shop rule. WORKING DRAWINGS. — PLANE SECTIONS OF SOLIDS. 147 403. Projections and sections of a block of irregular form, with two mutually perpendicular openings through it, and vnth equal, square frames projecting from each side. In Fig. 244 the side elevation shows clearly the object dealt with, while we look to the end view for most of the dimensions. The large central opening extends from w^w.^ to x^y^. The width of the main portion of the block is shown in plan as 2\", between the lines lettered Ae. The square frames project Y' ^om the sides, while the width of the central opening between the lines wa; is f". Two section planes are indicated, ST across the end view, and PQ — a vertical plane — across the plan ; the section made by plane S T is, however, shown only in fringed outline on the plan, though fully represented on the side elevation. The front elevation shows the section made by plane P Q, with the visible portion of that part of the object that is behind the cutting plane. Although detailed explanation of this problem is unnecessary after what has preceded, yet a brief recapitulation of the various steps in the construction of the views may be appreciated by some, before passing on to a more advanced topic. (a) EF, the first line to draw, is the trace of the vertical plane on which the end view is projected, and is at an angle of 60° to a horizontal line in order that the edges of the object (as a a, bb ee) may be inchned at 30 ° to the front vertical plane, which we may assume as one of the conditions of the problem. (b) A rotation of the object through an angle 6° about a horizontal axis that is perpendicular to EF, as, for example, the edge through /, is shown by the inclination of the end view to E F at an angle a^fiE^O°. (c) Drawing the end view at the required angle to EF we next derive the plan therefrom by perpendiculars to EF, terminating them on parallels io EF (as the lines ae, wx, nh, etc.,) whose distances apart conform to given data. (d) The elevations. For these a common reference line E' F' f" is taken, horizontal, and sufficiently below the plan to avoid an overlapping of views. For the front elevation any point as b', is found vertically below its plan b, and is as far from E' F as 6, is — perpendicularly — from E F. The height at which the section plane P Q cuts any line is similarly obtained. Thus at z it cuts the vertical end face of the block in a line which is carried over on the end view in the indefinite line Zz^; the portions of Zz.^ which lie on the end view of the frame g^h^i^ are the only real parts to transfer to the front elevation, and are seen on the latter, vertically below z and running from z' down; their distances from E' F' being simply those from Z, on the end view, transferred. The side elevation. Any point or edge is at the same level on the side elevation as on the front; hence the edge through b" is on b'b' produced. The distances to the right or left of M' N equal those of the corresponding points on the plan from MN; thus o"j' equals oj, etc. 404. Changed planes of projections. In the problems of Arts. 399-403 the employment of an "end view" — which was simply an auxiliary elevation— has prepared the student for the further use of planes other than the usual planes of projection; and if the auxiliary plan is now mastered he is prepared to deal with any case of rotation of object about vertical or horizontal axes, since new and properly located planes of projection are their practical equivalent. In Fig. 245 the object is represented in its initial position by the line-tinted figures marked "first plan" and "first elevation." The third and fourth elevations show somewhat more pictorially that it is a hollow, truncated, triangular prism, having through it a rectangular opening that is per- pendicular to the front and rear faces. 148 THEORETICAL AND PRACTICAL GRAPHICS. CHANGING PROJECTION-PLANE EQUIVALENT TO ROTATING OBJECT. 149 (a) Rotation about a vertical axis, or its equivalent, a change in the vertical plane of projection. Result: second elevation derived from first plan and elevation. Let the axis be one of the vertical edges of the object, as that at d in the first plan; also let the rotation be through an angle Ydo or 6°, (6 being taken, for convenience, equal to the angle YaF, which — with the line pq — will be employed in a later construction). If we were actually to rotate the object through an angle 6 the new plan would be the exact counterpart of the first, but its horizontal edges would make an angle with their former direction, and the new elevation would partly overlap the first one. To avoid the latter unnecessary complication, as also the duplication of the plan, we make the first plan do double duty, since we can accomplish the equivalent of rotation of the object by taking a new vertical plane that makes an angle with the plane on which the first elevation was made. This equivalence will be more evident if some small object, as a piece of India rubber, is placed on the " first plan " with its longer edges parallel to ah, and is then viewed in the direction of arrow No. 2 through a pane of glass standing vertically on XZ; after which turn both the object and the glass through the angle 9 until the glass stands vertically on e' j' and then "sdew in the direction of arrow No. 1. The second plane may be located anywhere, .as long as the angle 6 is preserved; XZ, making angle 6 at x^ with e'j', is, therefore, a random position of the new plane, and the projection upon it is our "second elevation." Since the heights of the various corners of an object remain unchanged during rotation about a vertical axis we will find all points of the second elevation at distances from the reference line XZ that are derived from the first elevation, and laid off on lines drawn perpendicular io X Z from the vertices of the plan: thus aO is perpendicular to XZ, and* Oo"' equals o' a' ; c"J' equals c'j', etc. (b) Rotation about a horizontal axis, or its equivalent, the adoption of a new horizontal plane. Result: second plan derived from first plan and second elevation. Having in the last case illustrated the method of complying with the condition that rotation should occur through a given angle (which is incidentally shown again, however, in the next con- struction) we now choose an axis pq so as to illustrate a different kind of requirement, viz. : that during rotation the heights of any two points of the object, which were at first at the same level as the axis, shall be in some predetermined ratio, regardless of the amount of rotation. In the figure it is assumed that e' {d) is to be at one -fifth the height of j'j, and that rotation shall occur about an axis passing through the lower end o' of the vertical edge at a. By drawing ad and aj, dividing the latter into five equal parts, and joining d with n — the first point of division from a — we obtain the direction dn, parallel to which the axis pq is, drawn through a. The distance dp is then one -fifth of jq, and they shorten in the same ratio, as rotation occurs. After locating the axis the next step is, invariably, the drawing of an elevation upon a plane perpendicular to the axis. This we happen, however, to have already in our "second elevation," having, in the interest of compactness, so taken 6 in the preceding case that the vertical plane XZ would be perpendicular to the axis we are now ready to use. Any rotation of the object about pq will, e-^ndently, not change the form of the "second elevation" but simply incline it to X Z. But, as before, instead of actually rotating the object, which would probably give projections overlapping those from which we are working, we adopt a new plane If JV as a horizontal plane of projection, so taken that it fulfills either of the following conditions : (a) that the object should be rotated about p q through an angle J' N =^ fi; (b) that the corner /' should be higher than by an amount x, MN being drawn tangent to an arc having J' for its centre, and /'/ (equal to x) for its radius. 150 THEORETICAL AND PRACTICAL GRAPHICS. Eeference to Fig. 246 may make it clearer to some that MN \s the trace of the new plane upon the vertical plane whose h. t. is XZ; that ON Hes vertically below the line XZ and is as truly perpendicular to the axis of rotation as is XZ; also that in Fig. 245 a view in the direction of arrow No. 3 (i. e., perpendicular to M N) is equivalent to a view ^^.g. a-as. perpendicular to the plane V in Fig. 246 after the whole assem- blage of planes and object has been rotated together about HOH until the "new plane" takes the position out which the first horizontal plane has just been rotated. (The remainder of the references are to Fig. 245.) The second plan is obtained by drawing Pj Q^, parallel to M N, to represent the transferred trace P Q of a vertical reference plane taken through some edge h and parallel to the plane Z N of the second elevation ; then any point d^ is as far from Pi Q , as the same point d on the first plan is from P Q; similarly, from point w^ to Pi Qi equals distance wh. p- (c) Further rotation about ver- tical axes. To show how the foregoing processes may be duplicated to any desired extent let us suppose that the object, as represented by the second plan and elevation, is to be rotated through an angle <^ about a vertical line through b^. If the rotation actually occurred, the plan biG^ would take the posi- tion b^G^, and the other lines of the plan would take corresponding positions in relation to a vertical plane on PiQi- A new vertical plane on b^Q^, at an angle <^ to b^G^, will, however, evidently hold the same relation to the plan as it stands, and transferring such new plane forward to O'iJ, we then obtain the points of the new (third) elevation by letting fall perpendiculars to O'Pi from the vertices of the second plan, and on them laying off heights above O'Pi equal to those of the same points above MN in the second elevation. Thus j' 9 equals J'f; W 6 in the fourth equals W"6 in the second. The fourth elevation is a view in the direction of arrow No. 5, giving the equivalent of a ninety - degree rotation of the object from its last position. To obtain it take a reference line r r through some point of the second plan, and parallel to 0' Ri] then R R' represents the vertical plane on r r, transferred. From R R' lay off — on the levels of the same points in the third elevation — distance IC" = c^Ci] AW" = w^w^, as in preceding analogous constructions. THE DEVELOPMENT OF SURFACES. 405. The development of surfaces is a topic not altogether new to the student who has read Chapter V and the earlier articles of this chapter;* so far, however, it has occurred only incidentally, but its importance necessitates the following more formal treatment, which naturally precedes problems on the interpenetration of surfaces, of which a " development " is usually the practical outcome. *The following articles shouia be carefully reviewed at this point: 120; 191; 344-6; 389, and Case 6 of Art. 396. THE DEVELOPMENT OF SURFACES. 151 n^_ S'i'T- A development of a surface, using the term in a practical sense, is a piece of cardboard or, more generally, of sheet -metal, of such shape that it can be either directly rolled up or folded into a model of the surface. Mathematically, it would be the contact -area, were the surface rolled out or unfolded upon a plane. The "shop" terms for a developed surface are "surface in the flat," "stretch-out," "roll-out"; also, among sheet -metal workers it is called a pattern; but as pattern -making is so generally under- stood to relate to the patterns for castings in a foundry, it is best to employ the qualifying words sheet-metal when desiring to avoid any possible ambiguity. 406. The mathematical nature of the surfaces that are capable of development has been already discussed in Arts. 344-346. Those most frequently occurring in engineering and architectural work are the right and oblique forms of the pyramid, prism, cone and cylinder. 407. In Art. 120 the development of a right cylinder is shown to be a rectangle of base equal to 2irr and altitude h, where h is the height of the cylinder and r is the radius of its base. 408. The development of a right cone is proved, by Art. 191, to be a circular sector, of radius equal to the slant height R of the cone, and whose angle 6 is found by means of the proportion R : r :: 360° : 6; r being the radius of the base of the cone. 409. The development of a right pyramid is illustrated in Art. 889, and in Case 6 of Art. 396. 410. We next take up right and oblique prisms, and the oblique pyramid, cone and cylinder; while for the sake of completeness, and departing in some degree from what was the plan of this work when Arts. 345 and 346 were written, the regular solids will receive further treatment, and also the developable helicoid. 411. The development of a right prism. Fig. '■• 247 represents a regular, hexagonal prism. The six faces being equal, and eb c f showing their actual size, we make the rectangles A B C D, Bi BEFC, etc., each equal to ebcf; then AA-^ r " equals the perimeter of the upper base, and we have the rectangle AA^B^D for the development sought. 412. The development of a right prism below a cutting plane. last article develop first as if there were no section to be taken into account. This gives, as he b a before, a rectangle of length AA^ and of altitude a d, divided into six equal parts. Then project, from each point where the plane cuts an edge, to the same edge as seen on the development. 413. Right section. Rectified mrve. Developed curve. A plane perpendicular to the axis of a surface cuts the latter in a right section. The bases of right cones, pyramids, cylinders and prisms fulfil this condition and require no special construction for their determination; but the development of an oblique form usually involves the construction of a right section and then the laying off on a straight line of a length equal to the perimeter of such section. Should the right section be a curve its equivalent length on a straight line is called its rectification, which should not be confounded with its development, the latter not being necessarily straight. 414. The development of an oblique prism, when the faces are equal in width. In Fig. 249 an oblique, hexagonal prism is shown, with x for the width of its faces. Since the perimeter of a right section would evidently equal 6 a; we may directly lay off x six times on some perpendicular fl ■" ■^^m^mb^ a A B E H « 1 1 1 d g^ 3 C F < I Taking the same prism as in the -Pig-- S-iB- B E ' ■i" 152 THEORETICAL AND PRACTICAL GRAPHICS. to the edges, as that through a. The seven parallels to a&, drawn at distances x apart, will contain the various edges of the prism as it is rolled out on the plane; and the positions of the extremities are found by perpendiculars from their original positions. The initial position Oi6, is parallel to but at any dis- tance from ab. The base edges are evidently unequal. 415. The development of an oblique prism lohose faces are unequal in width. In Fig. 250 c' d' h' g' is the elevation of the prism; np a. plane of right section. To get 1-2-3-4, the true shape of the right section, we require abhfe, the' plan of the prism.' Assuming that to have been given imagine next a vertical reference plane standing on ab. The right section plane np cuts the edge c' d' at n, which is at a distance x in front of the assumed reference plane. Make n2 = x. Similarly make Fig-, aso. o3 = 2/, and p4: = z; then 1-2-3-4 is the right section, seen in its true size after being revolved about the trace of the right -section plane upon the assumed reference plane. Prolong p n indefinitely, and on its extension make l'-2'=l-2; 2'-3' = 2-3, etc. Parallels to c' d' through the points of division thus obtained will contain the edges of the developed prism, and their lengths are definitely determined by perpendiculars, as ,.-' '■--, h'h", f'f", from the extremities of the orig- inal edges. 416. The development of an oblique cylinder, having a circular base and ellipticcd right section. Let am'n'Jc, Fig. 261, be an oblique cylinder with circular base. Take any plane of right section, as a'k'. Draw various elements, as those through b', c', etc., and from their lower extrem- ities erect perpendiculars to ai, as cc^, terminating them on the arc af^lc, which represents the half base of the cylinder. On cc' make c' c" = cc-^^; on ee' take e'e" = ee,, and similarly obtain other points on the elements, through which the curve a' c" e" g" h' can be drawn, this being one -half of the curve of right section, shown after revolution about its shorter diameter. Making KA equal to the rectified semi-ellipse just obtained, lay off .4 C= arc a' c" ; C E == arc c" e", etc., and through the points of division thus obtained on KA draw indefinite parallels to the axis of the cylinder. These will represent the elements on the development, and are limited by the dotted lines drawn per- pendicular to the original elements and through their extremities. The area a.^k.^NM is the development of one-half of the cylinder, the shaded area representing all between a'k' and the base ak. *In the interest of compactness the "First Angle" position of the views (Art. 385) is employed in Figs. 250, 253 and 255. THE DEVELOPMENT OF SURFACES. 153 V c to length 417. The development of an oblique pyramid. The development will evidently consist of a series of E-ig. asi. triangles having a common vertex. To ascertain the length of any edge we may carry it into or parallel to a plane of projection. Thus in Fig. 252 the edge vb is carried into the vertical plane at vb". Its true length is the hypothenuse of a right-angled triangle of base ob = ob", and altitude v o . In Fig. 253 a pyramid is shown in plan and elevation. Making oa" ^ V a we have ^^s- sss. v' a" for the actual length of edge v' a', a construction in strict >'^ analogy to that of Pig. 252. The plan V b being parallel to the base line shows that v' b' is the actual length of that edge. By carrying «Ci, where it becomes parallel to V, and then projecting c^ to c" we get v' c" for the true of edge v'c'. rig. sbs. To illustrate another method make vVi = v'o; then v^d by rabatment into H. i>, is the real length of v'd', shown 154 THEORETICAL AND PRACTICAL GRAPHICS. E-ig-. as^. For the development take some point v^ and from it as a centre draw arcs having for radii the ascertained lengths of the edges. Thus, letting v.^A represent the initial edge of the development, take ^ as a centre, ad as a radius, and cut the arc of radius v^d at D; then Av,^D is the develop- ment of the face avd, a'v'd'. With centre D and radius dc obtain C on the arc of radius v'c"; similarly for the remaining faces, completing the development v.^ — AD...A^. The shaded area Wj — TP...T is the development of that part of the pyramid above the oblique plane s'p', found by laying off, on the various edges as seen in the development, the distances along those edges from the vertex to the cutting plane; thus v^N = v' n', the real length of v'm'; v^P = «iPi; the length oi v' p' ; v^S—v's', the only elevation showing actual length. 418. The development of an oblique cone. The usual method of solving this problem gives a result which, although not mathematically exact, is a sufficiently close approximation for all practical purposes. In it the cone is treated as if it were a pyramid of many sides. The length of any element is then found as in the last problem. Thus in Fig. 254 an element v c is carried to v c" about the vertical axis v o. In Fig. 255 we have v'.a g for the elevation of the cone, and o — abc...g for the half plan. Make ob" = ob; then v' b" is the real length of the element whose plan is o b. Similarly, c, d, e and / are carried by arcs to ag and there joined with v'. For the development make v^A equal and parallel to v' a, and at any distance from it. With Vj as a centre draw arcs with radii equal to the true lengths of the elements; then, as in the pyramid, make A B= arc ab ; B C = arc be, etc. The greater the number of divisions on the semi -circle ab...g the more closely will the develop- ment approximate to theoretical exactness. 419. The five regular convex solids, with the forms of their developments, are illustrated in Figs. 256-265. They have already been defined in Art. 346, and that five is their limit as to number is thus shown: The faces are to be equal, regular polygons, and the sum of the plane angles forming a solid angle must be less than four right angles; now as the angles of equilateral triangles are 60° we may evidently have groups of three, four or five and not exceed the limit; with squares there can be groups of three only, each 90°; with regular pentagons, their interior angles being 108°, groups of three ; while hexagons are evidently impracticable, since three of their interior angles would exactly equal four right angles, adapt- ing them perfectly — and only — to plane surfaces. (See Fig. 131.) ^'i.g. 2SS- Fig-. EST-. X-lgr. 25S. TETRAHEDRON. Flgr- 2S9. DODECAHEDRON. ngr. seo. OCTAHEDRON. 1C05AHEDR0N. THE FIVE REGULAR CONVEX SOLIDS. 155; The dihedral angles between the adjacent faces of regular solids are as follows: 70° 31' 44" for Fig-.SSl the tetrahedron; 90° for the cube; 109° 28' 16" for the octahedron; 116° 35' 54" for the dodecahedron; and 138° 11' 23" for the icosahedron. Fig. ses. A sphere can he inscribed in each regular solid and can also as readily be circumscribed about it. The relation between c/, the diameter of a sphere, and e, the edge of an inscribed regular solid, is illustrated graphically by Fig. 266, hut may be otherwise expressed as follows : d : e :: Vs" : -v/ "2; for the cube die:: ^~^ : 1 d : e :: ' = r seo= 9 = the radius of the osculatory circle = the radius of curvature. For another proof, involving the radius of curvature of an ellipse, see Olivier, Cours de QiomHrie Descriptive Third Ed., p. 197. ' i^igr- THE INTERSECTION OF SURFACES. 157 the radius of the cyhnder on which the helix originally lay, and e is the angle at which the helix crosses the elements. To de- termine p draw on an elevation of the cyhnder, as in Fig. 274, a hne ab, tangent to the helix at its fore- most point, as in that position its inclination 6 is seen s't^. st"*. in actual size; then from 0, where a b crosses the extreme element, draw an in- definite line, OS, par- " allel to cd, and cut it at m by a line am that is perpendicular to a 6 at its intersection with the front element ef of the cylinder; then om = p = r sec' 6. For we have oa = on sec0 = r sec ; and on {= r) : oa :: o a : om; whence om = r sec' e = p. The circumference of circle p equals 2 -r r sec 6, the actual length of the heUx, as may be seen by developing the cylinder on which the latter Ues. The elements which were tangent to the hehx maintain the same relation to the developed helix, and appear in their true length on the development. The student can make a model of one nappe of this surface by wrapping a sheet of Bristol board, shaped like Fig. 273, upon a cylinder of radius r in the equation r sec' = p; or a two- napped helicoid by superposing two equal circular rings of paper, binding them on their inner edges with gummed paper, making one radial cut through both rings, and then twisting the inner edge into a helix. THE INTERSECTION OF SURFACES. 421. When plane -sided surfaces intersect, their outline of interpenetration is necessarily composed of straight Unes; but these not being, in general, in one plane, form what is called a twisted or warped polygon; also called a gauche polygon. 422. If either of two intersecting surfaces is curved their common line will also be curved, except under special conditions. 423. When one of the surfaces is of uniform cross section— as a cylinder or a prism— its end view will show whether the surfaces intersect in a continuous line or in two separate ones. In Cases a, b, c, d and g of Fig. 275, where the end view of one surface either cuts but one limiting line of the other surface or is tangent to one or both of the outlines, the intersection will be a continuous line. Two separate curves of intersection will occur in the other possible cases, illustrated by e and /, in which the end view of one surface either crosses both the outhnes of the other or else hes wholly between them. A cylinder will intersect a cone or another cyhnder in a plane curve if its end view is tangent to the outlines of the other surface, as in d and s-ig-. svs. g. Fig. 275. Two cones may also intersect in a plane curve, but as the conditions to be met are not as readily illustrated they will be treated in a special problem. (See Art. 489). ^^ 158 THEORETICAL AND PRACTICAL GRAPHICS. Fig', ays. Eef er ence Line 424. In general, the line of intersection of two surfaces is obtained, as stated in Art. 379, by passing one or more auxiliary surfaces, usually planes, in such manner as to cut some easily con- structed sections — as straight lines or circles — from each of the given surfaces; the meeting - points of the sections lying in any auxiliary surface will lie on the line sought. The application of the principle just stated is much simplified whenever any face of either of the surfaces is so situated that it is projected in a line. This case is amply illustrated in the problems most immediately following. The beginner will save much time if he will letter each projection of a point as soon as it is determined. 425. The intersection of a vertical triangular prism by a horizontal square prism; also the developments. The vertical prism to be 1^" high and to have one face parallel to V; bases equi- lateral triangles of 1" side. The horizontal prism to be 2" long, its basal edges f", and its faces inclined 45° to H ; its rear edges to be parallel to and ■|-" from the rear face of the horizontal prism. The elevations of the axes to bisect each other. Draw e i horizontal and 1" long for the plan of the rear face of the vertical prism. Complete the equilateral triangle egi and project to levels 1^" apart, obtaining e' f g'h', i' j', on the elevation. Construct an end view g" i" j" h", using t" j" to represent the reference line et, transferred. The end view of the horizontal iDrism is the square a" b" c" d", having its diagonal horizontal and upper and lower bases of the other prism, and with its corner from i"j". The plan and front elevation of the horizontal rived from the end view as in preceding constructions. Since the lines eg and gi are the iDlans of vertical faces their intersection by the edges a, h, r and d of the hori- n, m, I, p, q, r — and project to the elevations of the same edge a a meets the other prism at o and k, whicli project o' and k'. Similarly for the remaining points. The development of the vertical prism is shown in the shaded rectangle EJ', of length Sgi and altitude e' f. (See Art. 411). The openings o^p.q.r, and kj^m^n^ are thus found: For p^, which represents ]/, make GP=gp, the true distance ofp' from g'h'; then Pp^ = xp'. Similarly, 0G = og, and Oq, = yq'. midway between the h " one - eighth inch prism are next de- of one prism we note zontal prism — as at edges. Thus the to the level of a" at THE INTERSECTION OF PLANE-SIDED SURFACES. 159 E'ig. 377. ref. line The right half of the horizontal prism, o'a'c'q', is developed at r^h^h^r^ after the method of Art. 4"12. 426. The intersection of two prisms, one vertical, the other horizontal, each having an edge exterior to the other. The condition made will, as already stated (Art. 423), make the result a single warped polygon. Let abed, 1" x ^", be the plan of the vertical prism, which stands with its broader faces at some convenient angle cwg to V. From it construct the front and side elevations, taking a reference plane through d for the latter. Let the horizontal prism be triangular (isosceles section) one face inclined 45° to H; another 30° to H; the rear edge to be from that of the vertical prism. Begin by locating g" one -fifth inch from the right edge, draw f" g" at 45°, making it of sufficient length to have /" exterior to the other prism; then f'e" at 30° to H, terminated at e" by an arc of centre g" and radius g"f"; finally e" g". The edges e', /' and g' of the front elevation are then projected from e", f" and g". The rear edge g in the plan meets the face a d at s, which projects to s' on the elevation of the edge through g'. Moving forward from s, the next edge reached, of either solid, is a, of the vertical prism. To ascertain the height at which it meets the other prism we look to the end view, finding q'' for the entrance and t" for the exit. Being on the way up from g" to e" we use q", reserving t" until we deal with the face g"f". Projecting q" over to q' on edge a a' draw q' s', dotted, since it is on a rear face. Returning to a and moving toward b we next reach the edge e, whose intersection p with ab is then projected to edge e' at p' and joined with q'. For the next edge, b, we ng-. 27s. obtain o' from the side eleva- tion, projecting from the inter- section oi f" e" by the edge b". Moving from b toward w, projecting to the front eleva- tion from either the plan or the side elevation according as we are dealing with a hori- zontal edge or a vertical one, we complete the intersection. The development of the vertical prism is shown in Fig. 278. As already fully described, dd, = perimeter abed in Fig. 277; aQ=a'q'; bO=b'o'; ax=ax (of Fig. 277)- a- /S = vertical distance of s' from a'c', etc. 160 THEORETICAL AND PRACTICAL GRAPHICS. Although not required in shop work the draughtsman will find it an interesting and valuable E'ig-. aso. exercise to draw and shade either sohd after the removal of the other ; also to draw the common solid. The former is illustrated by Fig. 279; the latter by Fig. 280. 427. The intersection of two prisms, one vertical, the other oblique but with edges parallel to V. Let abcd....a'r' (Fig. 281) be the plan and elevation of the vertical prism. Let the oblique prism be (a) inclined 30° to H; (b) have its rear edge j%" back of the axis of the vertical prism; (c) have its faces incHned 60° and 30° respectively to V; (d) have a rectangular base 1^" X |". These conditions are fulfilled as follows: Through some point o' of the edge e' o' draw an indefinite line, o'f, at 30° to H, for the elevation of the rear edge, and //, also indefinite in length at first but ^" back of s, for the plan. E-ig-. sei- ,,-' ' Take a reference plane MN through s, and, as in Art. 397 (b), construct an auxiliary elevation on MX, transferring it so that it is seen as a perpendicular to o'f, thus obtaining the same view of the prisms as would be had if looking in the direction of the arrow. To construct this make o" f" equal to fV; c^™v f" I" at 60° to MX, and on it com- plete a rectangle of the given dimensions, after which lay off the points of the pentagonal prism at the same distances from 71/ iV in both figures. Project back, in the direction of the arrow, from /", g", h" and i'^ to the front elevation, and draw g' i' and the opposite base each perpendicular to o'f and at equal distances each side of o'. For the intersection we get any point n' on an oblique edge, as g', by noting and projecting from THE INTERSECTION OF PLANE-SIDED SURFACES. 161 S^igr- 2S2. X'igr- SS3. n where the plan , is the inclination of the shorter edges of the base to V. The intersection. Without going into a detailed construction for each point of the outline of interpenetration it may be stated that each method of the preceding article is illustrated in this ^-:-—'iy--^,p7 1 '■^ 1 ^2 162 THEORETICAL AND PRACTICAL GRAPHICS. problem, and that there is no special reason why either should have a preference in any case except where by properly choosing between them we may avoid the intersection of two lines at a very acute angle — a kind of intersection which is always undesirable. In the interest of clearness only the vmhle lines of the inter- section are indicated on the plan. (a) Auxiliary plane perpendicular to V. To find m, the intersection of edge s d with the face vhe, take a' d' as the trace of the auxiliary plane containing the edge in question; this cuts the limiting edges of the face at i' and n' which then project back to the plans of the edges at i and n. Drawing « i we note m, where it crosses « d, and project m to m' on s' d'. Had ni failed to meet s d within the limit of the face vhe we would conclude that our assumption that ad met that face was incorrect, and would then proceed to test it as to some other face, unless it was evident on inspection that the edge cleared the other solid entirely, ^>„ as is the case with sh, -"' / s'6', in the present in- stance. By using s'V as an auxiliary plane the student will obtain a graphic proof of fail- ure to intersect. (b) Auxiliary plane vertical. This case is illustrated by using v g as the trace of an auxiliary vertical plane containing the edge vg,v'g'. Thinking this edge may possibly meet the face sba we proceed to test it on that assumption. The plane vg crosses sa at I, and sb at p; these project to I' on s' n' and to p' on s'i'; then p'l' meets v' g' at q', which is a real instead of an imaginary intersection since it lies between the actual limits of the face considered. From q' a vertical to v g o-ives q. The order of obtniniag and roimecting the points. The start may be with an>/ edge, but once under way the progress should be uniform, and each point joined with the precedin- as soon as obtained. Two points are connected only when both lie on a single i\ice of each pyramid. THE INTERSECTION OF SINGLE CURVED SURFACES. 163 Supposing that q' was the point first found, a look at the plan would show that the edge sa of the oblique pyramid would be reached before vh on the other, and the next auxiUary plane would therefore be passed through sa to find uu'; then would come vh and sd. Running down from m on the face sdc we find the positions such that inspection will not avail, and the only thing to do is to try, at random, either a plane through vh or one through sc; and so on for the remaining points. The developments. No figure is furnished for these, as nearly all that the student requires for obtaining them has been set forth in Art. 396, Case 6. The only additional points to which attention need be called are the cases where the intersection falls on a face instead of an edge. For example, in developing the vertical pyramid we would find the development of j' by drawing v' j', prolonging it to o', and projecting the latter to o, when fxo would be the real distance to lay off from / on the development of the base; then laying off the real length of v' j' on v' o' as seen in the development we would have the point sought. Similarly, for tt', draw vx; make v^x^^vx, and tJ^'^i = a-ltitude v' z' ; then v^x^ is the true length oi vx (in space); also, making v^t^^vt and drawing t^t^, we find v^t^^ to lay off in its proper place on the development of the same face vfg. 430. An elbow or T-joint, the intersection of two equal cylinders whose axes meet. Taking up curved surfaces the simplest case of intersection that can occur is the one under consideration, and which is illustrated by Pig. 285. iFig-. sss. The conditions are those stated in Art. 423 for a plane intersection, which is seen in a' b' and is actually an ellipse. The vertical piece appears in plan as the circle m q. To lay off the equidistant elements on each cyhnder it is only necessary to divide the half plan of one into equal arcs and project the points of division to the elevation in order to get the fall elements, and where the latter meet a' b' to draw the dotted elements on the other. The development of the horizontal cylinder is shown in the line -tinted figure. The curved boundary, which represents the developed ellipse, is in reality a sinusoid. (Eefer to Art. 171). The relation of the developed ele- ments to their originals, fully de- scribed in Art. 120, is so evident as to require no further remark, except to call attention again to the fact that their distances apart, e^f^, f^g^, etc., equal the rectification of the small arcs of the plan. 431. To turn a right angle with a pipe by J p ^r a four-piece elbow. This problem would arise in carrying the blast pipe of a furnace around a bend. Except as to the number of pieces it differs but slightly from the last problem. Instead of one joint or curve of intersection there would be three, one less than the number of pieces in the pipe. (Fig. 286). 164 THEORETICAL AND PRACTICAL GRAPHICS. Let oqs show the size of the cyhnders employed, and be at the same time the plan of the vertical piece o's'n'a'. Until we know where a' n' will lie we have to draw o' a' and s' n' until they meet the elements from ;S" and T', and get the joint mM' as for a two-piece elbow. On mM' produced take some point ■;;', use it as a centre for an arc t'xyt" tangent to the extreme elements; divide this arc, between the tangent points, into as many equal parts as there are to be joints in the turn; then tangents at x and y— the intermediate points of division— will determine the outer limits of the joints at a', h' and /. Draw a'v', finding n' by its intersection with ss'; then n'V parallel to a'h', and similarly for the next piece. The deveh-pmenU of the smaller pieces would be equal, as also of the larger. One only is shown, laid out on the developed right section on v' x. The lettering makes the figure self - interpreting. 432. The intersection of two cylinders, when each is partially exterior to the other. The given con- dition makes it evident, by Art. 423, that a continuous non- plane curve will result. Let one cylinder be ver- Fig. 2S7. ^^. 2" in diameter and 2" M in sTt T — Iz TT ^ high. This is shown in half plan in hkl, and in front and side elevations between hori- zontals 2 " apart. Let the second cylinder be horizontal; located midway be- tween the upper and lower levels of the other cylinder; its diameter f". On the side elevation draw a circle a" b" c" d" of -f-" diameter, locating its centre midway between k" I" //and k-iN\ and in such posi- tion that a" shall be exterior to k " k^. The elevation of the horizontal cylinder is then pro- jected from its end ^-iew, and is shown in part without con- struction lines. The curve of intersection is obtained by selecting particular elements of either cylinder and noting where they meet the other surface. The foremost element of the vertical cylinder is k . . .k' n' m' Its side elevation, k" k^, meets the circle at n" and m", which give the levels of n' and m' respectively. On the horizontal cylinder the highest and lowest elements are central on the plan and meet the vertical cylinder at e, which projects down to the elements d' and h'. The front and rear elements, c and a, would be central on the elevation. The vertical line drawn from the intersection of element c with the arc hkl gives the right-hand point of the curve of intersection, at the level of a'. Any element w gx may be taken at random, and its elevation found in either of the following ways: (a) Transfer gz, the distance of the element from M N, to s" x on the side elevation, and draw xg" and g"y', to which last (prolonged) project g at g'; or (b) prolong gx io meet a THE INTERSECTION OF SINGLE CURVED SURFACES. 165 S^^ig-. 2SS. semi -circle on a c at g'" ; make a'y'^xg'" and draw y' g'. The same ordinate xg'", if laid off below a, would obviously give the other element which has the same plan gx, and to which g projects to give another point of the desired curve. / ■ / 433. The intersection of a vertical cone by a horizontal pmm. Let the cone have an altitude, tow', of 4"; diameter of base, 3". (As the cylinder is entirely in front of the axis of the cone only one- half of the latter is represented.) For the cylinder take a diameter of i"; length 8^"; axis parallel to V, |" above the base of the cone, and |-" from the foremost element. Draw ns parallel to p'r' and ■§■" from it; also g' m' horizontal and f" from the base; their intersection s is the centre of the circle a"d'c"m', of f" diameter, which bears to the element p' r' the relation assigned for the cylinder to the foremost element; said circle and p'ww' are thus, practically, a side elevation of cylinder and cone, superposed upon the ordinary view. The dimensions chosen were purposely such as to make one element of the cone tangent to the cylinder, that the curve of intersection might cross itself and give a mathematical "double point." The width d h, of the plan of the cylinder, equals m' d'. The plan of the axis (as also of the highest and lowest elements, a' and c') will be at a distance sg' from w. Any element as x' y' h' is shown in plan parallel to pq, and at a distance from it equal either to h' y' if on the rear or to h' x' if on the front. The element through v, on which /' falls, is not drawn separately from 6/ in plan, since v J' and m' g' are so nearly equal to each other; but / must not be considered as on the foremost element of the cylinder, although it is apparently so in the plan. For the intersection pass auxiliary horizontal planes through both surfaces; each will cut from the cone a circle whose intersection with cylinder-elements in the same plane will give points sought. A horizontal plane through the element a' would be represented by a' o', and would cut a circle of radius o' z' from the cone. In plan such circle would cut the element a at point 1, and also at a point (not numbered) symmetrical to it with respect to w Q. Similarly, the horizontal plane through the element x' h' cuts a circle of radius I' h' from the cone; in plan such circle would meet the elements x and y in two more points (5 aiid 3) of the curve. As the curve is symmetrical with respect to wQw' the construction lines are given for one -half only, leaving the other to illustrate shaded effects. The small shaded portion of the elevation of the cylinder is not limited by the curve along which it would meet the cone, but by a random curve which just clears it of the right-hand element of the cone. 434. To find the diameter and inclination of a cylindrical pipe that will make an elbow with a conical 166 THEORETICAL AND PRACTICAL GRAPHICS. ■S"i.s- SBS- fipe on a givm plane section of the latter. Let vab be a vertical cone, and cd the elliptical plane section on which the cylindrical piece is to fit. The diameter of the desired cyhnder will equal the shorter diameter of the ellipse c d. To find this bisect c d at e ; draw fh horizontally through e, and on it as a diameter draw the semi- circumference fgh; the ordinate eg is the half width of the cone, measured on a perpendicular to the paper at e, and is therefore the radius of the desired cylinder. In Pig. 290, the base N G equals twice ^^s- zso. g e oi Fig. 289. At first indefinite perpen- diculars are erected at N and G, on one of which a point C is taken as a centre for an arc of radius equal to cd in Fig 289. The angle <^ being thus determined is next laid off in Fig. 289 at c, and cdN"G" made the exact duplicate of CDNG, com- pleting " the solution. The developments are obtained as in Arts. 120 and 191. 435. To determine the conical piece which will properly connect two unequal cylinders of circular section, whose axes are parallel, meeting them either (a) in circles or (b) in ellipses; the planes of the joints being parallel. (a) When the joints are circles. To determine the conical frustum b ehc prolong the elements e b and he to v; develop the cone v . ..eh as in Art. 418, and on each element as seen in the develop- ment lay off the real distance from v to the upper base b c. Thus the element whose plan is v-^^k is of actual length vki and cuts the upper base at a distance vn from the vertex, which distance is therefore laid on vk^ wherever the latter ap- pears on the development. (b) When the joints are ellipses. Let the elliptical joints no and qr be the bases of the conical piece qnor. To get the development complete the cone by prolonging qn and or to m; prolong qr and drop a perpendicular to it from w; find the minor axis of the ellipse qr as in the first part of Art. 434 and having con- structed the ellipse proceed as in Art. 418, since in Fig. 255 the arc abc.g is merely a special case of an ellipse. 436. The projections and patterns of a bath-tub. Before taking up more difficult problems in the intersection of curved surfaces one of the most ordinary apphcations of Graphics is introduced, partly by way of illustrating the fact that the engineer and architect enjoy no monopoly of practical projections. In Fig. 292 the height of the main portion of the tub is shown at a' d'. Let it be required that the head end of the tub be a portion of a vertical right cone whose base angle c'b'a' equals the flare of the sides, such cone to terminate on a curve whose vertical projection is o'n'z'a'. Draw E-xg-. SSi. THE INTERSECTION OF SINGLE CURVED SURFACES. 167 IFlg-. ESS. two lines, b' I' and c' /', at first indefinite in length and at a distance a' d' apart. Take a' d' vertical, and regard it not only as the projection of the elements of tangency of the flat sides with the conical end, but also as the elevation of part of the axis, prolonging it to represent the latter. Use v, the plan of the axis, as the centre for a semicircle of radius v c, whose diameter ed is the width of the bottom of the tub. Project c to c'; make angle v'c'd' equal to the predetermined flare of the sides; prolong v' c' to b' and o' ; project b' to b on vc prolonged and draw arc abm with radius bv, obtaining am for the width of the plan of the top. The plan of one -half the curve o'n'z'a' is shown at onzvi and is thus found: Assume any element v'x'y'; prolong it to z'; obtain the plan vxy and project z' upon it at z. Similarly for n and as many inter- mediate points as it might seem desirable to obtain. Assuming that the foot of the tub is composed of an oblique cone whose section, his, with the bottom is equal to ecd, and whose base angle is h'i'k', we project i to i', draw i' k' a.t the given angle to the base, project k' to k, and through the latter draw the semicircle rkq with radius bv, obtaining the plan of the upper base. Joining the tangent points r and s, h and g, we have rs and Ag as the elements of tangency of sides with end. Their elevations coincide in h'V, which meets h' i' at v", whose plan is v-^ on hq. Fig', ass. The development. Fig. 293 is the development of one -half of the tub. EM equals b' c'; VO equals v' o'; VZ equals v'e", the true length of v'z', obtained, as in previous constructions, by car- rying z to Zj, thence to level of z'. Similarly at the other end. (Reference Articles 191, 408, 418.) 437. The intersection of a vertical cylinder and an oblique cone, their axes intersecting. Let MBd and M'R'P'N' be the projections of the cylinder; v'.a'h' and v.anbm those of the cone. The axes meet o' at an angle which is arbitrary. 168 THEORETICAL AND PRACTICAL GRAPHICS. ^ig:- 2S-4. The ellipse anbm is supposed to be constructed by one of the various methods employed when the axes are known; and in this case we get the length of 7nn from a' b' and its position from n', while a 6 is vertically above a'b'. (a) Solution by auxiliary vertical planes. Any vertical plane vis will cut elements from the cylinder at e and I; also, from the cone, elements which meet the base at s and t. Project s and ^ to s' and t', join the latter with the vertex v' and note I' and e' (just below d') where they cross the vertical projection of the elements from I and e; these will be points in the desired curve of intersection. By assuming a sufficient number of vertical planes through V the entire curve can be determined. (b) Solution by auxiliary spheres. If two surfaces of revo- lution have a common axis they will intersect each other in a circle whose plane is perpendicular to that axis.* This property can be advantageously applied in problems of inter- section. "With o' — the intersection of the axes — as a centre, we may draw circles with random radii o'/', o'i, and let these represent spheres. The sphere f'g'w intersects the cone in the circle f g'; the cylinder in the circle h'k'. These circles inter- sect each other at a; in a common chord whose extremities are points of the curves sought. They are both projected in the point X. A second pair of circular sections, lying on the same auxiliary sphere, are seen at pq and rw, their intersection z being another point in the solution. The point y results from taking the smaller sphere. 438. Intersection of a cylinder and cone, their axes not lying in the same plane. In Fig. 295 let the cylinder be vertical and the cone oblique, the axis of the latter being parallel to V and inclined 6° to H, and also lying at a distance x back of the axis of the cylinder. The auxiliary surfaces employed may preferably be vertical planes through the vertex of the cone, since each will then cut elements from both cylinder and cone. Thus, vfe is the h. t. of a vertical plane which cuts e V, e'v' from the cone, and the vertical element through / from the cylinder; these meet in vertical projection at /', one point of the desired curve. The plan of the intersection obviously coincides with that of the cylinder. E-lg-. 2SS. * By the aeflnition of a surface of revolution (Art. 340) any point on it can generate a circle about its axis. If, then, two surfaces have the same axis, any point common to both surfaces would generate one and the same circle which must also lie on both surfaces and therefore be their line of intersection. THE INTERSECTION OF SINGLE CURVED SURFACES. 169 '■•yQ This is one E'lgf. sse. 439. Conical elbow; right cones meeting at a given angle and having an elliptical joint. of the cases mentioned in Art. 423 as not admitting of illustration i^ in the same way as when dealing with surfaces of uniform cross ''/'i\ section, but a plane mtersection is nevertheless secured as with cylinders by making the extreme elements of the cones intersect. Let vx hi Fig. 296 be the axis of one of the cones. If xyz is the required angle between the axes bisect it by the line ym, and draw the joint cd parallel to such bisector. The right cone which is to meet abed on cd must be capable of being cut in a section equal to cd hj a, plane making an angle 6 with its axis, and must obviously have the same base angle as the original cone; since, however, the upper portion vdc of the given cone fulfills these conditions we may employ it instead of a new cone, rotating it about an axis p t which is per- pendicular to the plane of the ellipse dc and passes through its centre. The point o, in which the axis vx meets the plane dc, wiU then appear at s, by making op^ps; sv', drawn parallel to yz, will be the new direction of vo; and an arc from centre d with radius cv will give v', which is then joined with d and c to complete the construction. If the length of the major axis of the elliptical joint had been assigned, as e/ for example, that length would have first been laid off from some point e on the extreme element and parallel to ym, then from / a parallel to ve, giving g on vc; then gh parallel and equal to ef, gives the joint in its proper place. 440. Right cones intersecting in a non- plane curve; axes meeting at an oblique angle. Let one conej v'.a'b', (Fig. 297) be vertical; the other, oblique, its axis meeting v' o' at an angle 6. The plane a' b' of the base of the vertical cone cuts the other cone in an ellipse whose, longer axis is e'f. As in Art. 434 determine g' h', the semi-minor axis of this ellipse. Project e', g' and /' up to e, g and /; make 170 THEORETICAL AND PRACTICAL GRAPHICS. gh, and gh^ each equal to g'h'; then on ef and h^h^ aa axes construct the ellipse ehJK^ as in Art. 131. Tangents from v^ to the ellipse complete the plan of the oblique cone. (a) The curve of intersection, found hy auxiliary planes. In order that each auxiliary plane shall contain an element (or elements) of each cone, it must contain both vertices and therefore the hne v'v", which joins them; hence its trace on the plane e'a'b' must pass through the trace, t't, of such hne on that plane. Take tx as the horizontal trace of one of these auxiliary planes. It cuts elements starting at i and I on the base of the oblique cone. One of ihe elements cut from the other cone is v p, which in vertical projection (v' p') crosses the elevations of the other elements at q' and r', two points of the curves sought. Since the extreme elements of the cones are parallel to V we will have c' and d' — the intersections of their elevations — for two more points of the curve. Having found other points by repeating the same process the curve c'q'rd' is drawn through them, and the cones may then be developed as in Art. 191. (b) Method by auxiliary spheres. Since the axes intersect we may use auxiliary spheres as in Case (b) of Art. 437. Thus, with o' — the intersec- tion of the axes — as a centre, take any radius o' h and regard arc kyz a,a rep- resenting a portion of a sphere which cuts the cones in hs and y z. These meet at w, one point of the curve of intersection c' q' d'. 441. Intersecting cones, bases in the same plane but axes not. Let v.Jcbfg and e.sQhj be the plans of the cones; v.'p'd' and e.'Q'c' their elevations. As argued in Case (a) of the last problem, the auxiliary planes must con- tain the line joining the vertices; their H- traces would therefore, in the gen- eral case, pass through the trace of that line upon the plane of the bases; but, in the figure, both vertices ha^•- ing been taken at the same height above the bases, the line which joins them must be horizonicd, hence pandhi to the H- traces of the auxiliaries: that is, AT, ST, QR, etc., are parallel to V e. It happens that the trace MX of the foremost auxiliary plane is tangent to both bases, hence contains but one element of each cone and determines but one point of the desired curve. These elements, ae and b r, meet at n, while their elevations intersect at n'. THE INTERSECTION OF SINGLE CURVED SURFACES. 171 Each of the other planes, except X Y, being secant to both bases, will cut two elements from each cone, their mutual intersections giving four points of the curve of interpenetration. Thus, in plane P, the element e meets vk in q and v d in x, while element h e gives I and m on the same elements. The plane X Y being tangent to one base while secant to the other gives but two points on the curve sought. Order of connecting the points. Starting with any plane, as MN, we may trace around the bases either to the right or left. Choosing the former we find, in the next plane, the point h to the right of a on one base, and d similarly situated with respect to b on the other; therefore m, on he and dv, is the next point to connect with n. Elements oe and fv give the next point, then ue and gv locate s, after which those from j and w give the last before a return movement on the base of the f-cone. As nothing new would result from retracing the arc gfd we continue to the left from w, although compelled to retrace on the other base, since planes beyond j would not cut the v-cone. The element ii e is therefore taken again, and its intersection noted with an element whose projection happens to be so nearly coincident with vz that the latter is used. Continuing along arcs och and ikb we reach the plane MN again, the curves ilx and qnm crossing each other then at n — the point lying in that plane. Such point is called a double 'point, and occurs on non- plane curves of intersection at whatever point of two intersecting surfaces they are found to have a common tangent plane. Tracing to the left from a and to the right from b the elements e and d v are reached, in the plane OP. Their intersection x is joined with n on one side and with the intersection of Se and gv on the other. Soon the tangent plane X F is again reached and a return movement necessitated, during which the arc XSQOa is retraced, while on the other base the counter-clockwise motion is continued to the initial point b, completing the curve. Visibility. The visible part of the intersection in either view must obviously be the intersection of those portions of the surfaces which would be visible were they separate, but similarly situated with respect to H and V. In plan the point n lies on visible elements, and either arc passing through it is then visible till it passes (becomes tangent to, in projection) an element of extreme contour as at m or t, when it runs from the upper to the under side of the surface and is concealed from view. The point w would be visible on the 'u-cone but for the fact that it is on the under side of the e - cone. A similar method of inspection will determine the visible portions of the vertical projection of the curve, which will not be identical with those of the plan. In fact, a curve wholly visible in one view might be entirely concealed in the other. 442. The intersection of a vertical cylinder and an oblique cone, their axes in the same plane. If in Art. 440 the vertex v' were removed to infinity the ■y-cone would become a vertical cylinder; the line v' v" would become a vertical line through v" ; t would be vertically above v" ; but the method of solving would be unchanged. 443. In general, any method of solving a problem relating to a cone will apply with equal facility to a cylinder, since one is but a special case of the other. The line, so frequently used, that passes through the vertex of a cone in the one problem is, in the other, a parallel to the axis of the cylinder. Planes containing both vertices of cones become planes parallel to both axes of cylinders. In view of the interchangeability of these surfaces it is unnecessary to illustrate by a separate figure all the possible variations of problems relating to them. 172 THEORETICAL AND PRACTICAL GRAPHICS. 4AA. Intersection of two cones, two pyramids, or of a cone and a pyramid, when neither the hoses nor axes lie in one plane. One method of solving this problem has been illustrated in Art. 429, where the intersection was found by using auxiliary planes that were either vertical or perpendicular to V; we may as easily, however, employ the method of the last problem, viz., by taking auxiliary planes so as to contain both vertices. This will be illustrated for the problems announced, by taking a cone and pyramid; and, for convenience, we will locate the sur- faces so that one of them will be ver- tical, and the base of the other will be perpendicular to V, since the problem can always be reduced to this form. Let the cone v'.a'b', v.cdB, (Fig. 299) be vertical, and the pyramid o'.r' q' p', o.rqp, inclined. We will assume that the projec- tions of the pyramid have been found as in preceding problems, from assigned data, using oo^, o' p', (taken perpen- dicular to the base r' q') as the refer- ence line. Join the vertices by the line v' o', vo, and prolong it to get its traces, ss' and tt', upon the planes of the bases. All auxiliary planes containing the line v o, v' o', must intersect the planes of the two bases in lines pass- ing through such traces. Prolong r' q' to meet the plane a' b' at A'. Project up from X, get- ting yz for the plan of the intersection of the two bases. We may assume any number of auxiliary planes, some at random, but others more definitely, as those through edges of the pyramid or tangent to the cone. Taking first one through an edge, as or, we have trz for its trace on the pyramid's base, then zs for its trace on H. The elements cv and dv which lie in this plane meet the edge or at e and /, giving two points of the curve. These project to o' r' at e' and ./'. FIRST ANGLE METHOD. 173 The plane sy, tangent to the cone along the element uv, has the trace yt on the base of the pyramid, and cuts lines j o and k o from its faces. These meet vu at two more points of the curve, their elevations being found by projecting j to j' and k to k', drawing o' j' and o' k', and noting their intersections with v' u'. To check the accuracy of this construction for either point, as I, draw vv^ perpendicular to vu and equal to v'u', join v^ with u, and we have in v v^u the rabatment of a half section of the cone, taken through the element vu and the axis; then 11^, parallel to vv^, will be the height of I' above the base a' b'. With one exception, any auxiliary plane between s y and s z will give four points of the inter- section. The exception is the plane s F, containing the edge o g, and which, on account of hap- pening to be vertical, requires the following special construction if the solution is made wholly on the plan: Rabat the plane into H; the elements it contains will then appear at Av^ and -Bt^j, while the edge oq will be seen in Oj gj (by making 00^^=0' 0, and ggi = g'Q); elements and edge then meet at J, and N^ which counter - revolve to J and N. We might, however, get elevations first, as /', by the intersection of element A' v' with edge o'g'; then / from /'. In the interest of clearness several lines are omitted, as of certain auxiliary planes, hidden por- tions of the ellipses, and the curves in which s r g (the rear face) cuts the cone. The student should supply these when drawing to a larger scale. 8ee the latter fart of this chapter for further problems on the intersection of surfaces. MONDE'S DESCRIPTIVE GEOMETRY.-FIRST ANGLE METHOD. 445. In this method — the first, and so long the only one employed, and whose use would probably be still universal but for the reason given in Art. 383 — the object is located in front of the vertical plane and above the horizontal, as illustrated in Art. 385. While acquaintance with what has preceded in this chapter would be an advantageous prelim- inary to the study of the First Angle treatment of figures, yet it is not absolutely essential; but if for any reason, as, for example, with reference to its applications to perspective or stone cutting, the First Angle Method is taken up in advance of the other, it is assumed that the student will first thoroughly familiarize himself with Arts. 284-330, and 335-379. 446. To determine one projection of a point on a given surface, having given the other. This problem is of frequent recurrence and is illustrated in Figs. 300 and 301, in which the more familiar sur- faces are shown in their most elementary positions, i. e., with axes either perpendicular to or parallel to a plane of projection. The required projection is in each case enveloped in a small circle. Where two solutions are possible both are given. The general solution of this problem, for all surfaces, is as follows: Through the given projection draw a line on the surface, preferably a straight line, but otherwise the simplest curved section possible; obtain the other projection of this auxiliary line and project upon it from the given projection. (a) Right cone, axis vertical. In No. 1 of Fig. 300 the element v' x' is drawn through the given projection a'. Projecting x' down to x and y we draw the plans vx and vy and project a' upon each. (b) Right cone, axis vertical. Solution by auxiliary circle. In No. 2 draw through the given pro- jection a' the line m' n' parallel to the v. p. of the base. It represents a circle of diameter m'n', which is seen in full size in plan, and upon which a' projects in the two possible solutions. (c) Right cone, axis parallel to the ground line. As before, a' represents the given projection. The 174 THEORETICAL AND PRACTICAL GRAPHICS. element v' a' meets the base at, i', whose real distance in front of or to the rear of the vertical diameter of the base is seen at i'h', found by rotating the semi -base on b' c' as an axis until it is seen in full size at h' f c'. The counter-revolution is shown in plan, and the two solutions indicated. E'xg'. 300- 6' i ' o' I 1 / (d) Right cylinder, axis parallel to the ground line. In No. 4 the two rectangles represent the projections of the cylinder on H and V. To find the plans of the elements whose common eleva- tion passes through a' rotate the end of the cylinder into V, using as an axis the vertical line t c'. (The vertical -diameter method of the last case would answer equally well.) The arcs show the paths of the various points. Then m' n' iDrojects to both p" and q", M'hich are transferred to p and q by arcs "from r and s. s-ie-. 301. (_g) Sphere. In Fig. 301 a horizontal section through the given projection, a', cuts a circle seen in full size in plan, upon which a' projects in the two solutions. (f) Annidnr Uirus. This surface, also known as the anchor ring, is generated bj' revolving a circle about an axis in its plane but not a diameter. It has the same mathe- matical properties whether the axis is ex- terior to the circle or is a tangent or a chord; but obviously there would be no hole in the surface except in the former case. In Fig. 301 one -half of the ring is shown in plan and elevation, either shaded section showing the size of the generating circle. The axis is a vortical line through o. The axis being vertical, a horizontal plane through a' will cut two circles from the torus, of radii equal respectively to o' p' and o'q'. These are seen full size in plan, and upon them a' projects. 447. As the representation of any surface of revolution, when its axis is oblique to one or both planes of projection, necessitates the drawing of the oblique projection of a circle, the solution of the latter problem is a natural preliminary to constructions involving the former. 448. To obtain the projection of a circle when its plane is oblique to the plane of jirojcction. Proof that such projection is an ellipse. In Fig. 302 let abed, . . . a' c', be the projections of a circle lyino- in a horizontal plane. Using as an axis of rotation the diameter b d, which is perpendicular to V let CIRCLES AND CYLINDERS OBLIQUE TO PROJECTION-PLANE. 175 QR C will then represent C its ■QS suppose the plane of the circle to rotate through an angle 0; new position. Since the axis is perpendicular to V any points, as a and e, of the original circle, will describe arcs parallel to V and therefore seen in their true size in elevation, as at a' A' and e' E', while their plans, a A and eE, will be parallel to G.L; their new positions. A, E, are then, evidently, the intersections of verticals from A' and E' with horizontals through a and e. In Analytical Geometry the " greater auxiliary circle " of an ellipse has for its diameter the major axis of the latter curve; and, by analysis, the relation is established that, when measured on the same perpendicular to such major axis, an ordinate of the circle will be to the corresponding ordinate of the ellipse as the major axis of the ellipse to its minor axis. If, there- fore, we can establish this relation between the circle abed and the curve Ab C d, the latter must be an ellipse. In the elevation we have, from similar triangles, the pro- portion o' E' : o' m :: o' A' : o' s. But o' E' = o' e' = ex; o' m = Ex; o' A' = o' a' ^ a; and o' s = o A : the proportion may X-ig-. 303. therefore be written ex : Ex:: oa: A :: 2 a (=: b d) : 2 A (= A C). 449. Working drawing of a hor- izontal cylinder 4" long, 2" in diame- ter, axis 1" above H and inclined 30° to V. Draw first the plan cgsx (Fig. 303) which is simply a rect- angle 2" X 4", with longer sides at 30° to the ground line. The ends eg and ex are circles 2" in diame- ter and vertical. Rotate the base eg about the vertical tangent c.a'b' until it takes the position e g^, when its elevation g" n" c' d" will equal the circle of which it is the pro- jection. The centre of such circle will be at the height (1") assigned for the axis. Note various points, Aq, as g^g", f^f", d^d", and then, by / a construction in strict analogy to ■f^ that of the last problem, counter- revolve them into the original plane. Their paths of rotation will be hor- izontal arcs, seen in full size in plan, but as horizontal straight lines in elevation, as g" g', .f"S\ d" d'. 176 THEORETICAL AND PRACTICAL GRAPHICS. The new elevations are then the hitersections of verticals from g , f, d to tlie levels at which they rotated, giving points of the ellipse g'J'd' c'. The other end of the cylinder might have been obtained similarly, but the figure illustrates the use of the horizontal diameter sx, s'z', as an axis, when qq^ shows the distance to lay off above and below v' to get the levels of the elements whose common plan is pq; that is, for p' g' and for p"'g"' 450. To project a circle when its plane is oblique to both planes of projection; also to draw a tangent at a given point. To avoid multiplicity of lines we will assume that in Fig. 304 P Q P' — the plane of the circle — has been already determined from assigned inclinations, by means of Art. 319. Let it be required that the circle lying in that plane shall have a given radius (o d), and that its centre shall be at assigned distances, h ' h and o n, from H and V respectively. To fulfill the condition as to height of centre draw a' h' at assigned height h' h above G. L., to represent the v. p. of a hor- izontal of the plane. On the plan a 6 of such horizontal note the point o which fulfills the con- dition as to assigned distance {on) from V; this will be the h. p. of the centre and projects to o' . By Art. 306 rabat o into H, about P Q, as an axis. It takes the position o,, about which draw a circle with the prescribed radius o d. The diameter d^j^, which is parallel to the axis PQ, remains so during counter-revolution, and at d/ (passing through the original o) becomes the major axis of the ellipse, since it is the only diameter which is horizontal and therefore projected on H in its actual size. Project d and / to d' and /' on a' V , since its elevation must evidently be parallel to G. L. The minor axis of an ellipse being always perpendicular to the major' must in this case be the space -position oi c^e^. It will be part of the line of declivity (Art. 301) cut from plane PQP' by an auxiliary vertical plane RSP', and which appears at mJSi when the latter plane is carried into H about RS as an axis. iR,S = SP'). On such line we find o^ representing o, and make c^e^^ df for the auxiliary view of the minor axis sought, whence c and e are derived by counter-revolution. We find c' at height c' y ^ c^ c; similarly e' z = e^ e. But one diameter can be parallel to V, and in this case it must be a V- par- allel (Art. 300) of PQP; therefore through o' (and bisected by it) draw g' h,' parallel to P' Q and of length dj. Its plan gh IB parallel to G. L. To draw a tangent /' at any point, as t, find b;^ <, from which t was derived, and draw t^% tangent to the drcle. / rotation of the plane, and, when t^ returns . As X is on the axis PQ it remains constant during to t, the tangent becomes tx, whence t' %' as usual. OBLIQUE CONE.— PROBLEM IN BELTING. 177 451. To pryect a cone whose axis is inclined 6° to B. and 4>° to V; altitude 3"; diameter of base, 2". By Art. 309 we find va and v'a', (Fig. 305), the projections of the axis. Although all diameters of the base are perpendicular to the axis of the cone only one can be so projected in each view, but it will be that one which, being parallel to the plane of pro- jection, is seen in actual size (Art. 311)- therefore make be and d' e' each 2" long, and perpendicular respectively to V a and v'a'. Their other projections are parallel to G. L. The shorter axis of the base, in plan, lies in a vertical meridian plane (Art. 340) ; in the elevation it is in the meridian plane perpendicular to V. Using the former for illustration, rabat it into H, when v a will appear at Vja,, by making vv^^v'u, and aaj^a'a;'. Then m'n', 2" long and perpendicular to v^a^, is the auxiliary view of the diameter which, in counter- revolution, appears as the shorter axis mn of the ellipse. Make j/'m'^mmj, and z'n' = 7i7ij to get m' and n'. From f" g" we get f'g', the minor axis of the elevation of the base, by a construction in strict analogy with that described for mn. The determination of other points of each view is after the method of Art. 450. 452. To determine an intermediate or guide pulley to connect a pair of band -wheels running on two lines of shafting which make any angle with each other. This machine-shop problem is introduced here to show one of the practical applications of Art. 450. The principle of the construction is based both on theory and experience, and is, briefly stated, that the belt must be led on to a wheel in its plane. That is, the point where the belt leaves the mid-plane of one wheel must lie in the mid-plane of the next wheel. If rotation is to be reversible the mid -plane of the intermediate pulley will be determined by tangents to each of the main pulleys from some point of the line in which their mid -planes intersect. In Fig. 306 let A and B be the given pulleys, rotating on vertical and horizontal axes respect- ively, and in the directions indicated by the arrows. Their mid -planes intersect in the line mn, m'n', at any point of which, as a a', it is possible to draw tangents to both wheels; two such tan- gents, as ab, a'b', and ad, a'd', will therefore determine the mid-plane of an intermediate pulley which will direct the belt as desired. Determine P'QP, the plane of these tangents, and rabat it about QP into H; ab then becomes OiSi, and ad appears (partially) at aj^ (Art. 306). Bisect the angle b^aj^ by the line w^ z, and on it find a point Oj with which as a centre a circle can be drawn that will be tangent to both a^b^ and fti^i, its radius x to be that assigned for the pulley. After counter-revolution the circle appears as the ellipse efgh, found by the method of the last article, with, however, the following special features which somewhat simplify the solution: First, the bisector w^z becomes zio, z being on the axis; hence o^ projects directly to zw at o; second, as tangent a 6 is i)arallel to QP we have in og the semi -minor axis, and avoid a separate construction to obtain it. 178 THEORETICAL AND PRACTICAL GRAPHICS. For the elevation project z to z', draw z' a' and project o upon it. Next g to g' and draw upon which-prolonged-we project up from h. The other points may be found as m the last article. The axis of the shaft is shown only in the plan. It is, of course, perpen- dicular to the plane of the wheel and therefore to QP. ««> The pulley found takes the belt from wheel B and runs it upon A. Another auxiliary wheel would be needed to lead the belt off A and upon B and would be found by dealing with some point yy' in the same way as with a a'.^ ^, S-ig-. 30S. I'o', 453. To construct the projections signed position in any oblique plane, having said figure for its base. In the plane P' Q R has been already signed conditions, and proceed di- some figure — a rectangle for ex- data ; sides in ratio 2:3; longer horizontal; lowest corner at a of a plane figure having an as- and of a solid of given altitude Fig. 307 we will assume that determined to agree with as- rectly to the location in it of ample — with the following sides at a given angle to the given height 1 1' from H. Let M'NM be an auxiliary vertical plane, cutting P' Q R in a line of declivity which appears at nSj after rabatment on MAI In the plane P'QP assume the horizontal t'c', to, at the assigned height, and on it take some point cc' for the lowest corner. By prolonging tc we get c" as the auxiliary view of c; no" then equals the real distance of c from R Q, and, used as a radius for arc c"4, leads to t\ — the position of c after rabatment about R Q. Draw c^b^ at the prescribed angle to RQ and complete the rect- angle ai&iC,di with sides in the assigned ratio. Any corner, as d, of the plan is then found by projecting d^ upon n X, thence by an arc (centre n) to d", from which a parallel to R Q meets d,y (the path of rotation) at d. The elevations of the corners are at heights above G. L. equal to the distances of a", b", etc., from MN. This is indicated for a' by a"f, the arc fp x, and the line x a'. FIGURES IN INCLINED POSITIONS. — TANGENCIES. 179 If we assume that the rectangle we have been considering is the base of a pyramid -J-" high we have merely to erect at o" , (derived from o — the centre of the revolved rectangle) a perpendicular o" v" of the prescribed altitude, and join v" with the corners of the base. Also find v and v' by the process just out- / lined. / By turning the illustra- /. tion so that MN will be ,' horizontal, the student will I find that by looking in the \ direction Q R, the auxiliary \ view v."a"b"d"c" will ap- \ pear as the ordinary eleva- \ tion, and the rotation from H to the space -position will be somewhat more clearly seen. 454. A plane and a surface are tangent to each other if they have a common point through which, if an auxiliary plane be passed, the line cut by it from the plane will be tangent to the curve which it cuts from the surface. 455. Taiifjeiit planes to developable surfaces, a few problems in which are next given, are solved by means of the principles stated in Arts. 368-376, which should be reviewed at this point. 466. A plane, tangent to a cone at a given point, will be determined by (a) the element through the point, and (b) a tangent to the base at the extremity of the element. tt' the point at iFigr- SOS. 458. Let v.'a'h', v. ah, (Fig. 308) be an inverted cone; which the plane is to be tangent. Draw the element v c, v'c', containing the given point. Its h. t. is s,. through which draw P Q parallel to c m — the latter a tangent to the bas& at the extremity of the element. Join Q with m', the v. t. of the tangent, c m and we have in PQ P' the plane sought. 457. A plane, tangent to a cone and containing a given point in space, would be determined (a) by the line joining the given point with the vertex of the cone, and (b) by either tangent that could be drawn to the base from the trace of the first line upon its plane. In Fig. 309 we have an oblique projection of this case, the orthographic being left for the stu- dent. V. A B is the given cone ; the given point. A plane, tangent to a cone and parallel to a given line, would be determined (a) by a line drawn through the vertex of the cone and parallel to the given line, and 180 THEORETICAL AND PRACTICAL GRAPHICS. X-iir. 31i. (b) by either tangent (as in the last problem) drawn to the cone's base from the trace of the first line upon its plane. In Fig. 309, were vi n the Hne parallel to which a tangent plane was required, then V T, parallel to mn, would with either TM or TN determine a plane fulfilling the conditions. Were m n the direction of the sun's rays, either tangent plane would be a plane of rays, and its line of contact would be an element of shade. The area between the tangents TS and TX (Fig. 309) would be cut off from the light by the cone and would therefore be the shadow of the latter. This case is illustrated orthographically in Fig. 310. Through the vertex draw the ray vt, v't', parallel to the direction of light indicated by the arrows. Tangents ts, tx, from t — the h. t. of such ray — are the H- traces of the two possible tangent planes. To find the vertical trace of plane Mt we may draw through the ver- tex V a "horizontal" of the plane (300). Its projections are v c, v' c', and its V. t. is c', through which the vertical trace M' J' would be drawn to meet Mt on the ground line. Similarly for the v. t. of the plane Nxt. 459. A plane, tangent to a ajlinder at a given point. Art. 376 again indicates the method of solution. With o' (Fig. 311) as the given point, draw the element ab through it, and the tangent c d at its ex- tremity. The latter is the h. t. of the required plane, and d is one point of the v. t. Since the V. t. of the element ab, a' b\ is too remote we resort to a " horizontal " of the plane and through the point. This (o e, o'e') has its v. t. at e', which joins with d for the v. t. required. The oblique cylinder with circular base (right section elliptical) is used merely for convenience, to illustrate a general solution. The other possible tangent to the base, parallel to c d, would be the h. t. of the other solution. 460. A plane, tangent to a cylinder and containing a point exterior to it, would be determined (a) by a line through the point parallel to the axis of the cylinder, and (b) by either tangent to the base of the cylinder from the trace of the first line upon the plane of the base. This is shown pictorially in Fig. 312, where is the given point; ng. sis. O T the parallel to the axis ; T, the trace of T on the plane of the base; TM and TN the tangents, either of which- — with OT — deter- mines a plane fulfilling the conditions. 461. A plane, tangent to a cylinder and parallel to a given line, is determined most readily by making it parallel to a plane which can be passed, by Art. 315, through the given line and parallel to the axis of the cylinder. Since in Fig. 313 the cylinder is taken parallel to both H and V, a plane through the given line ab, a'b', and parallel to the axis must therefore have traces (M N, M' N') parallel to G. L. R S R' is any auxiliary, profile plane. It cuts a line from the MJV- plane which, revolved, is \ "\^ ^/ ^\d u y /e / r 1 / / t a' /' 466. Warped ■■ncrfm^es with thrw linedr (lirertr'u-c.-i. The element through an assumed point on one directrix may be found by making the point the vertex of a surface having either of the other lines as a directrix. Tliis auxiliary surface — which will be either pUum or conical — will be pierced by the third directrix in one or more points, through which the desired line(s) will pass. 467. ]Vnrpe(i .■mrfdccs having two flirectncrx and a cmir (or plane) ilircctor. To find an element through an assumed point of either directrix make sudi point the vertex of a cone similar to the cone director. The elcnnent sought will be the hne joining the vertex with the intersection of the aux- iliary CDue with the second directrix. W'ith a plane director an element would be foun<. Before dealing ^-itli imy particular ease of tangency to a warped surface attention is again called to the fact, stated in Art. 377, that a plane, tangent to a warped surface at a gi\'en point, must be, in general, a. secant plane elsewhere; for, by the law of generation of a warped surface, c(jnsecutive eleuKnits can not lie in the same plane, hence a tangent plane, of which an element is always ' plane iMintaining an element of a warped surface will (if not parallel to the elements, ^vhich it might be for any surface having a plane director) be tangent to the surface at some point of the element. F(jr the eletnent will be intersected somewhere by the curve cut from the surface by the plane. At such intersection a tangent line to the curve would, by Art. 454, lie in the tangent plane to the surface; and, by Art. 374, the element through that if a tangent plane. Hyperboloid point would be the other determining line WARPED Sir UFA GES M> THKIR TANGENT PLANES. 18?. X'ig-. 33.5. Pninf and not liiic contuct is thus seen to he the rule in the tangency of a jilane to a warped surface; although, with surfaces like the conoid of Fig. 314, a plane parallel to the plane director and tangent at either r' or s' would he tangent all along the element through the point. 470. The warped hyperboloid of recohifidn ; projertioiiK and tangent plane. Let the surface have a Tninimurn diameter of ^"; inclination of elements to H, 60°; height of surface, 1^". The i)rojectloiix. In Fig. 315 draw two limiting planes, M' N' iind 0' P', 1^" apart. Take o-r/'/' for the axis, and draw abrd, \" in diameter, for the plan of the gorge circle. Its elevation is u'c'. midway hetween the upper and lower bases. Througli h' a line m'n', at 60° to H, is the elevation of an element parallel to V. Its plan in n is next drawn, tangent (in projection, not in space) to the gorge circle. The circle through /// and // completes the plan of the surface. It will be noticed that the line .I'l/' has the same plan as ■vi' n' and fulfills the same conditions; that is, its points are all <3qually distant from the axis; its rotation about o' n" would there- fore result in the generation of the same surface. The elevation may be completed like Fig. 77 liy drawing more elements, or like Fig. 314, ])y assuming points of the sur- face, which — like t' and 7" — have a common jilan i, carrying them by arcs tf» the meridian plane Q R, and thence projecting to the lerel.^ of the elevations of the same points, getting t" and I"' on the hyperbolic contour, or meridian licction. The tangent -plane. The hyperboloid being one of the two possible doubly -ruled warped surfaces (Art. 350) its tangent plane at any point % is most readily found by drawing through it the elements €2 and /z, and finding their plane XYZ; although oue element, together with the tangent to the " parallel " or horizontal circle through z, would suffice to determine it. (Art. 340.) 471. The hyperbolic 'paraboloid; projections, also traces of a tangent plane. Fig. 213 reappears in Fig. 316 for convenient reference, although the stu- dent should review Arts. 349-353 at this point. (a) In accordance with the definition " having two straight directrices and a plane director" we see that the surface could be generated by moving R C ui:)on A B and D 0, keeping it always parallel to P R Q, as a plane director ; or by moving A B upon A D and B C, keeping it parallel to the hor- izontal plane as a plane director. (b) The elements of one set would be the sections of the surface by planes parallel to the plane director; hence they divide the elements of the other set proportionally. We might, therefore, obtain elements of an hyperbolic paraboloid by simply dividing two non- plane right lines into proportional parts and joining corresponding points. The plane director would be parallel to any pair of such elements, and would be found by Art. 315. 184 THEORETICAL AND PRACTICAL GRAPHICS. (c) CQ is the line of intersection of the plane directors; BC and NM are the two elements- one of each set-whose directions are perpendicular to CQ; their intersection M is called the vertex of the surface, and a parallel to GQ through M would be its axis. _ The surface is divided symmetrically by two mutually perpen.licular planes, called pnnapal diametric planes, which contain the axis and bisect the angles between the elements meetmg at the When the plane directors are mutually perpendicular, as in the figure, the surface is called right or isosceles; otherwise it would be oblique or scalene. (d) If the elements parallel to Pi? Q are projected upon it their projections will cross at the point M, which is the projection of NM, that element of the other set which is also the common perpendicular of the first set. Such point M is called a point of concourse. For the horizontal set of elements the point C is similarly a point of concourse. As consecutive elements are non-plane, while all of one set are parallel to a plane director, we- see that while no element can intersect another of the same set it must meet all of the other set. (e) The tangent plane at any point, as N (Fig. 316), would be determined by the elements AD and MN passing through the point. (Art. 374). (f) The surface in projection. In Fig. 317 let abed, a'b'c'd', be a warped quadrilateral analogous. ^ ^ to Fig. 210. AB and CD can then represent a pair of elements of one generation ; A D and B C of the other. Let p be the plan of a point on the surface. To find its eleva- tion draw a series of elements by the method indicated in the latter part of Case (b) of this article, and cut them by a secant plane containing p. Thus, divide A D and B C into eight equal parts, for example, and join the like- numbered points of division. Cut these elements by any auxiliary vertical plane MN, containing p. The curve obtained is shown in elevation at s't'x'y', upon which p projects at p' P' Q P is the plane director for the elements indicated, and is determined by c d and a parallel to a 6 through some point r oi c d. The other plane director (not shown) would be similarly found., (g) An element containing a known point, as pp', of the surface. Through the point pass a plane- (by Art. 316) parallel to the plane director of either set; it will cut either directrix of that set in a point which joins with the given point for the element sought. 472. In arches having warped soffits, those surfaces of voussoirs which have to be normal to the soifit are hyperbolic paraboloids, unless the "twist" of the theoretical normal surface from its TANGENT PLANES TO WARPED SURFA CES.—RA CCO B DMENT. 185 limiting plane is so slight that a plane bed may be substituted without imperilling stability. (See Art. 475.) Another and more readily observed application of this surface is in the " cow - catchers " of loco- motives, which are usually made of two hyperbolic paraboloids, symmetrically placed with reference to a vertical, longitudinally -central plane of the engine, the "elements" being bars, either parallel to such vertical plane as a plane director or else horizontal. 473. A conoidal surface. Conoidal surfaces are defined in Art. 354, and several of them illustrated in Figs. 214, 215 and 217. (a) The orthographic projections of the cono-cuiieus of Wallis are shown in Fig. 314, and sufficiently treated in Art. 468 (b). (b) To get a tangent plane to the cono - cunevs. To solve without resorting to an auxiliarj'' surface we need to know the nature of plane sections parallel to the base. That they are ellipses may be thus established: In Fig. 818 let H be the plane director; CDE the circular base; AB the right line directrix, and rxd a plane section parallel to the base. The semi -axes dz and DZ are equal; and from the similar triangles Bcz and B G Z, and the equality of xy to wz and of XY to W Z, we have xy : cz : : X Y : CZ, a characteristic of ellipses having equal major (or minor) axes. (c) A tangent plane at some point x would then be determined directly by the element GX and the tangent 1 1 to the elliptical arc dxc; or, indirectly, by obtaining It as an element of an auxiliary surface, as described in the next article. 474. Raccordment.* We have ^. „^^ seen (Art. 469) that a tangent plane to a warped surface has, usually, point and not line contact with it; it is, however, possible for two warped surfaces to be mutu- ally tangent at every point of a common element. The surfaces are then said to raccord along that element. Raccordment exists whenever two warped surfaces have a com- mon element and a common tan- gent plane at each of any three points that may be taken upon it; a secant plane at either of such three points will then cut from the surfaces lines which will be tangent to each other, and which could be used as directrices of their respective surfaces. Three points of common tangency must be established, since, by Art. 343, three conditions are imposed on the genera- trix of a warped surface, and each of them must be consistent with tangency to ensure raccordment. Although the conoid is employed in illustration of the foregoing the conclusions are perfectly general, and applicable to all warped surfaces. In Fig. 318 regard AB, dxc and DXC as sections of the conoid BA-CDE by three parallel planes. Draw tangents to these sections at the points where they are met by some element as G X. A B, being a straight section, may be regarded as its own tangent (Art. 370), while 1 1 * On account of the utility of an auxiUary raccordlng surface in passing tangent planes to warped surfaces, this topic is presented at this point in order to apply it to the warped surfaces yet to be treated. l.S(i THKOBKTICAL AND PRACTICAL (^RAl'HKS. and TL mx- obviously tangent in the ordinary sense; and us all three are parallel to the same }>lane they will, if used as directrices for a warped surface, give an hyperbolic paraboloid (Art. 349) which would evidently have a tangent plane in common with the conoid at each of the three points (I. -x; and A'; it would therefore raccord with the conoid along the element (I A'. Since TB \8 the horizontal trace of the paraboloid BKLT we may obtain It by joining • with /, which is the intersection of TB by the trace m-n of an auxiliary plane parallel to A'. Were the section d x >■ and its tangent not in a plane parallel to V we would obtain a I'according ■HKi-ped hyperboloid by using A B, LT and the new It as directrices. (Art. ;-!49.) Since an infinite number of sets of parallel planes or of non-parallel planes could lie passed through C, :<■ and A", in each of which a section of the conoid would — with its tangent — lie, it is possible to have an infinite number of hyperbolic })arab(iloids, or of warped hyperboloids, raccording with a given warped surface along the same element. In a t'ase like that of Fig. 318, if ^^•e regard the plane director, it is only necessary to sliow that there are two tangent planes and a plane director in common, in order to ha^'e the usual num- ber of conditions on the generatrix of each surface, and at the same time ensure raccordment. Let Pxq and S X M be any curves in the planes cutting dxc and DXC from the conoid, and having tangents in conmion with them, at x and A'; a surface generated with these new curNCs as directrices, and having the same plane director as the paraboloid BKLT and the conoid, would raccord with them both, along the element :c A', as would also lie the case if the third directrix, A B, were substituted for the plane director. 475. yoniiaJ hyperbolic pnraholoid. Were the surface BKLT (Fig. HIS) rotated 90° on ^' A' as an axis, the lines A B, It and L 7' would become normals to the sections to which they are now tangent, and tlie paraboloid would be iiormnl to the conoid along the entire element. A hyperbolic ])ara- R' x'iir. 313- _, boloid that shall lie normal to any A\'ar|ied surface along a given element may therefore he found as readily as a raccording surface. 476. The Ciinie dc Vache (Cow's Horn). To represent this surface in conformity Avith tlie definition in ,Vvt. ;->6], and as tlierc illustrated, draw (Fig. 819) the semi -circle m'n'ji', in p, parallel to ^^ and the njixd semi -circle, r'h't', in V. Join their centres by the line jj, and through its middle (tint, /, draw " /, o', })erpendicular to ^', for the straiglit directrix. Any plane t'on- taining a I, as n o' P\ for example, will be perpendicular to ^', and will cut the curved directrices in points, as hb\rc'. which, joined, will give an element of the surface. Similarly, the plane ii o' q' gives the element k'q',kcj. (See Art. 4(i(;. ) qiie element at ./■' is obviously horizontal. A taiif/i'iit jiliine, at any point, .<./ l-)raw b' c\ bra, the element containing the point. An auxihary hyperbolic i)araboloid, raccording with the given surtiu'e alons b' o\ bm, may THE CORNK DE VA CHE. — THE PLUCKER VOXOIV. 1S7 be generated by mo\4ng the latter upon //(/' — which is a tangent «/, \', and upon two other lineii that are tangent, at points of the assumed element, to sections made with the surface by planes Tparallel to ^' (Art. o49.) In (■':'', r t', we have one such tangent, and in h'P',hi, another, the plane no' P' being a tan- gent plane at -( because determmed by an element ha, b'o', and a straight directrix o' a, each of which possesses the distinctive property of a tangent. (Art. 374.) A second element, /)',(/', h y, of the same set as be, is found by Art. 466 thus: Pass the plane a:ri/' containing one directrix, hi, h' P', and some point, as i\ of another. The v. t. (xy') will be parallel to o' P', since one of the determining lines is a V-parallel. This plane cuts the third directrix at y', whence y' e' and ye follow, for the line sought. Ijustly, draw through s- a parallel to the ground line, for the plan of an element parallel to V and (jn the paraboloid. It meets ey at z, which projects to z', giving s' z' for the v. p. of the ele- ment. R S R' is then the desired tangent plane at .i a, and s'z', sz, each an element of one set in an auxiliary raccording surface. 477. The roiuiid of Plilcker. (Cayley's cylindroid.) This surface plays the same role in the deter- mination of the combined effect of two simultaneous r-ig-. sso. twists or wrenches on a solid body, as the parallel- ogram of forces in finding the resultant of tw(j forces acting simultaneously upon a point in their plane. (a) Referring to Figs. '2Vo and 215 and the first definition of Art. 856, we get elements of this sur- face as lines of striction of hyperbolic paraboloids thus : In Fig. 320, which is an orthographic pro- jection of the A BCD of Fig. 213, let GA (B' A') and CD, CD', be horizontal lines having C B' for their common perpendicular; then with ») a. as a plane director we have CE (perpendicular to )/) /i) for the line of striction of the set of ele- ments parallel to mrt, and E' F' for its elevation. With p q as the li. t. of a plane director, A H will be the plan of the element through A, and CJ (J' K') the corresponding line of striction, and therefore another element of the conoid under con- struction. Others may be similarly found. (b) The conoid of Pliicker may be obtained by taking H for its plane director; for its curved directrix an elliptical section of a vertical right cylinder of circular base; for its straight directrix the element of the cylinder at either extremity of the major axis of the elliptical directrix. Any circular cylinder having for its axis the straight directrix just indicated will cut the surface (assuming the elements to be indefinitely extended) in the double -curved directrix employed in Case (c). (c) To represent the surface in accordance with the second definition of Art. 366 draw first a rectangle A. B CD (Fig. 321) as the development of a cylinder whose length is twice the wave length of the sinusoid. The fact that a helix projects as a sinusoid may be availed of, and the latter curve now ob- 188 THEORETICAL AND PRACTICAL GRAPHICS. tained by the usual method for the former. Thus, extend the central line, m" z" of the development, to some point o, which use as the centre of a circle whose radius equals the amplitude (w w") of s^s- ssi- A C D E F G H A"_ D" O" each wave; then construct as a helix the curve in" t" w'' y" z'\ 120, making, however, two complete undulations. Draw next the plan and elevation of the cylinder on whose develop- ment we have been working, and locate upon it the various elements pre- viously represented. Thus, a,A' m' represents AB of the development, and upon it in" projects at m'. SD appears at s,D'D", upon which s' is found from s". By connecting other points similarly obtained we get m's't'w'y'x' for the curved directrix of the conoid. Horizontal lines through its points and the axis, as those through x, 1, 2, 3, etc., are then portions of elements. Their plans, extended to diameters, represent all of the surface included by the cylinder. (d) To draw a tangent, as s' Q', at any point .s' of the curved directrix, revert to the point s" from which s' was derived; draw s" p (from s,p^) as a tangent to a helix (by Cases (b) and (c) of Art. 420); prolong ps" to q; then on the plan make tangent sQ equal sub -tangent .S"^, and project Q to Q', when Q' s' will be the desired tangent. (e) The tangent plane at any point of the curved directrix would be determined by the element and the tangent at that point to the curved directrix. The tangent plane R.IR', at any random point of the surface, as k, is most readily found by means of an auxiliary hyperbolic paraboloid, racmrding with the conoid along the element through the point. Its determining lines are kl (plan parallel to s Q) and sc, .s-'/. (See Art. 473.) It is an interesting fact that all tangent planes to the Plucker conoid (and by Art. 469 every plane containing an element is a tangent plane) will cut it in ellipses.* 478. The right helicoid. This is a warped surface of the conoidal family, having a helix and its axis for directrices, and a plane director perpendicular to the axis. (Arts. H57 and 358) To project it orthographically draw a helix in the usual manner (Art. 120), and a series of horizontal lines through the axis and terminating on the curve. Thus, in Fig. 322 the helix .a'd'ha" is the curved directrix, and the horizontal Unes (radii, in plan) are the elements ^ A tangent plane at some point//' is determined by (a) the element f'o' containing the point, ♦For the more recent developments of the Theory of Sorevva and iiDDlications r.r ti,. , -, .... translation of Prof. Ball's work; Berlin, Ueorg Relmer, 1889. appricatlons of the conoid ol PKlcker see Gravellus' VAllPED HELICO IDS ISi) ■Fi-g. 333. and (1)) l)y tlie tanycnt to thr helix at /'. From C^ase (c) of Art. 420 we wee that f A, the plan s-ig-. 322. of sueh tangent, must equal the are fda, and that A is the trace of the tangent; M N, drawn parallel to the element /o, is then part of the h. t. of the tangent plane, since / o, / o" is a horizontal (Art. 300) of the plane. The V. t. of the plane would pass through that of the element. Wnx' the point of desired tangency not on the helical directrix a separate helix would have to be constructed, containing the point. jVlthough the j>itch (or yv> in one revolution) may be the same for two helices, the one nearer the axis woidd oliviously have the greater steep- ness or declivity. 47i). The oblvpir heltaiid. This warped surface (see Fig. 216) has a helix and its axis as directrices, the elements making a constant aoite angle with the latter; hence it comes in the class of surfaces having a cone director; a riiiht cone also, since the obliquity is constant. To project it orthographically construct a helix n'k'y' (Fig. 323), as in Art. 120; draw an element, (I ' 0, a g, parallel to V and at the assigned inclina- tion; then, since its extremities must ascend at the same rate, join /i ', in', /', etc., with p(.)ints s, t and q, whose vertical heights above o equal those of »', III' and '(' respectively aliove the level of a'. The visible contour of the elevation will not be straight, but a curve, tangent to the projection of the elements; their envelope, in other words. (Art. 335.) A taiif/ciit phiiw at any point would be found exactly as for a right helicoid, viz., with an eleinent, and a taiKjent to the helix through the point. Any section of tliis helicoid, by a plane perpendicular to the axis, will lie a uphill of Arrliimedex (Art. 188). This can be shown by taking a series of elements whose plans make equal angles with each other, and carrying them parallel to V. Being then both jjarallel and equidistant, in space, it will be found that they will, if produced, cut a given liorizontal plane at distances from the axis that are in arithmetical progression. 4S0. The (jniei-dl ri/linrlrirtil helicoid. The right and oblique helicoids just described, and the developable helicoid of Arts. 346 and 420, are but sjieeial cases of the general cylindrical helicoidal surface, having a cone director, and two helical directrices lying on con-axial cylinders. Fig. 324 illustrates such a helicoid, the helical directrices lying on cylinders of diameters ag and /J II resi)ectively, only the smaller being shown in elevation. The elements hi, ck, etc., are tangent both //( »p(ice and in jihm to the inner cylinder. The shortest method of construction is to draw the plan circles, and with the larger construct the outer helix a'd'g', of the given pitch and in the usual way (Art. 120); draw tangent bi, jiarallel to G. L., as the jjlan of the element parallel to V, and whose actual inclination to H may therefore be seen on the elevation; make such elevation {h' i') at the desired angle to G. L. As i' is then one point of the inner helical directrix we may draw the latter through it, making it of the same pitch as the outer helix; then k,m, etc., will project upon the inner helix at k'm,', to be joined with c', e', etc., for the elevations of elements. If hi ))(' prolonged to x the latter point will trace U))on the the larger cylinder a helix iden- /< ^ ST ^ V -^ >), s^^ i- X 190 THEORETICAL AM) PRACTICAL (IRAPHICS. tical in form with that traced by h, although not coinciding ^rith it, except undt-r the peculiar conditions described in the next article. The portion / x, i' x\ of the element hx, will generate a nappe is upward, but otherwise hke the other. X portion of such generation is suggested at ■FJ-S- -3-2.-. of the helicoid whose concavity .'/'"■'. Were the generatrix fir, V x\ to lengthen eciually on each side of i, the nappes MXO and m' x.' ir' would evidently a2Jproach each other until iinally they would intersect in a helix. Further elongation of h x would sinii)ly result in additional helical iii.tersectiiins; in other w(irds, a helicoidal surface, if indefinitely extended, will intersect itself in an infi- nite number of helices. To find one such helix jiass a vertical plane through hx, V f' \ determine the curve in which this plane cuts the nappe MX OP fextended) by joining the points in which the successive irig-. 32s. elements on that napi^e meet the jilane; then b' x' will mee't such curve in the point which generates the helical intersec- tion of the nappes. 481. In Fig. 325 an inter- esting case of helical intersec- tion is shown, the conditions being such that the outer helix is traced twice, it being the path of each extremity of the generatrix. To obtain the surface in this form the diameter of the iimer cylinder must reduce to zero, and the elements must interst'ct the axis at an angle o'n'q', or 6, whose tangent equals 0' q' -i- n' q' ; that is, //"(/' the pitch divided by 2 r. Positions of the generatrix that differ by a send -revolution, as c' li' and 0' 11', will intersect each other, as at »/, showing that the .helical directrix is also the first of the helices in which the svu'face intersects itself. The contour line ,'/'r'.s-' is called by Bardin a s[)ecies of hyi)er- bola, since it has »/ ■ is finite. (b) Ceiierid tmrjicd helicoid, (plane director), with r finite and 6=0. A J! FED HE LI CO IDS. ]91 (c) Derelopahh' helicoiii, whenever 6 = ft, and r is finite and either (d) Stirface of trinnguhir -threaded srreiv, with r = 0, and either greater or less than 90°. (e) Surface of square-threaded sa-ew, for r = and 6=90°. The student will obtain some interesting results Ijy constructing case (a) with various values of 0, contrasting in particular, the form in which the advancing half of the generatrix is lower than its point of tangency on the inner cylinder, with the opposite case of relative position. The meridian and right sections of the various helicoids also present some interesting features. 4S8. Helicoids of radially expanding pitch. These result from the combination of a unJfrrm motion of rotation of a generatrix, with a variable motion of its points, axially, in such manner that the pitches of the helices described by the points are proportional to the distances of the latter from the axis. Such a helicoidal surface has been employed for the screw propeller on the theory that it would tend to counteract the centrifugal action of the water which the screw had set in motion. Since each point of the generatrix traces a helix of uniform pitch, the intersection of the surface by any circular cylinder con- axial with it will be an ordinary helix. 484. (hnatrurtion. of a helicoid of radiedly expanding pitch. In the upper portion of Fig. 326 we have in a" I' h" and o'p'...z' two of the helices on a surface of this kind, each obtained in the usual way; that is, for the former, by dividing the half- pitch, h" G, into the same number of equal parts (six) as the half plan ulk..g, and projecting a to a", I to I', etc., upon the dotted horizontals through the points of division. Similarly, divide s'z", the pitch of the helix generated by o', into the same number of equal parts by horizontal lines, upon which project o, p, q, etc., from the plan of the same helix. The line h"z'j' is the position of the generatrix a" x" [a M) after the semi - rotation supposed, and x"j' therefore the half- pitch of the point that travels along the axis. (It is assumed that the generating line may be of indefinite length, and merelj^ has the two hehces as directrices). In a" o', I'p', ¥ q', etc., we have portions of the generating line (not equal portions) included between the helices. In the figure, j' x" is taken at one-half h" G, but any desired prf)portion may obviously be assumed. Probably the most interesting special form of this surface is that called Holm's conchoidal screw, from the fact that by employing only that part of the surface which was generated by a fixed por- tion of the initial line, he included the helicoid within a surface of revolution whose meridian section was the "superior" branch of a conchoid. In the figure the surface as thus limited beojns at FA less than or equal to J'. FLg 3SS. warped; HELIOOip WITH ; RADIALLY WARPED HELICOID WITH AXIALLY EXPANDING PITCH (elevation) and is thus constructed: — 10l> THEORETICAL ANT) PRACTICAL GRAPHICS. I^lg-. 3Sr7. Make j'P, P<1, eelerated move- ment of the receding mass. 4S7. To driiir (I helicoid if axiidhj expanding pitch. In Fig. ;'>2(i the axis )i i is divided into iiarts H Iff, vir, etc., that are in some ratio to eaidi other,; in arithmetical progression in this case. Horizontal hnes are then drawn tln-ougli the points of division, upon which — as for tlie ordinary hchx — the i)oints (', h, c, (/, etc., are projected from the plan to obtain the helix sliown, the cli'ments in plan making equal angles with eacli other. Since tlie ordinari/ helix develops into a straight line, it is obvious tliat a helix of the kind under consideration will develop into a curve. 488. The coiiclinldid hi/jicrholoid of (_'atalerpendiculai- INTERSECTION OF CONE BY PLANE. 197 PLg. 333- to V. It cuts from the cone an ellipse seen in e'f, which is also the v. p. of the major axis. Bisect e'f at o'. Through o' take a right section mn. Rotate half the latter till parallel to V, when it will appear as 'mo"n. Then o' o" is the semi -diameter of the elhpse sought, which can be constructed by any of the methods involving simply the knowledge of the axes; or, as we shall see later, by employing the plan eyfx, whose construction is next described. The horizontal 'projection of the curve of intersection. Since the plane P Q P' is at 90° to Y we can project directly from e\ t', etc., where the trace P'Q crosses the elevations of elements, to the plans of the latter. Thus e' and /' project at e and /, upon the plans of the extreme elements. On s k and s g, whose common elevation is s' g', we get t and j from t'. To get z, on the foremost element s b, s'b', project w' to z", thence to Zj and by arc (centre s) to z. Project o' upon ef at o, through which xy (equal to 2o'o") at 90° to ef will be the h. p. of the minor axis, seen in true size because horizontal. True size of the section, found ly revolu- tion into the horizontal plane. At fiy^e^x^ the true ellipse is shown by revolution of its plane into H, about PQ as an axis. The arcs described by the various points will be projected upon V in their true size, all with centre Q: thus, e' describes e'iL; similarly, o' (x, y) reaches 0; then L and project at e-^^x-^, and i/i, upon perpendiculars to PQ as ee^ and yy-^, which are the plans of the arcs of rotation. True size of the section, shown by revolution into the vertical plane. Through the v. p. of any point revolved, as e', draw a line at 90° to P'Q to represent the v. p. of the arc of rotation, and on it lay off e' e'' equal to the distance of the point in space from the axis P'Q, which distance, being horizontal in this case, is shown by the distance er of its plan from G. L. Similarly, o' a;" equals ux; o'y" equals uy. The tangent line at any point of the curve. Let a tangent be desired at some point q. MR is the h. t. of a plane that is tangent to the cone along the element s i on which q lies, and i2 is a point of the intersection of the tangent plane and section plane; Rq is, therefore the line of intersec- tion of those planes, hence the tangent line required. (Art. 505). When q reaches q^ the tangent is Rq^, R being constant during the rotation, being on the axis. Make QR" equal to QR; then R" q" is the tangent, rotated into V. The development of the cone. With radius SC (Fig. 334) equal to the slant height of the cone, draw an arc CA C, subtending an angle of 144°, determining the latter by calculation in the propor- tion R (a' s') : r (a s) : : S60 : ; since in unequal circles equal arcs are subtended by angles at the centre which are inversely proportional to the radii. Locating the elements S L, SD, etc., on the sector CSC, lay off on each the distance from S of 198 THEORETICAL AND PRACTICAL GRAPHICS. ■F^S- S3S. the point where that element ia cut by the plane m Fig. 333. Thus, SE=s'e'; SW=s'z", the true length oi s' w' i i j. 7 d • The tcunjent line m devdopnent appears at L R, drawn at 90° to SL and made equal to IR ip ^508.' The shortest distance hetioeen two points on the surface of a cone, i.e., their geodesic. (Art. 382). The shortest distance between two pomts^ would be a straight line on the develop- ment. Hence in Fig. 334 BK will be the geodesic between K and B. It crosses SA at 1 and SG at 2. Locate 1 and 2 on the same elements as seen on the cone in Fig. 333; then the curve g'lb' is the ele- vation of the geodesic. Co7iical helix. In Art. 191 the curve- traced by a point which combines a uni- form approach toward the vertex of a cone with a uniform rotation about its axis is called a conical helix, from the analogy of its generation to that of the cylindrical helix. But if we follow Javary and base the definition upon the geodesic property of the locus, the curve we have |ust ^o^istructed is also entitled to- the same name, although there is evidently nothing in its form to suggest what is usually understood by a helical curve. 509. Litcrscction of a ■pyramid by a plane; also sectional view. In Fig. 335 let v.ab..f r.'a'd' be a verti- cal pyramid, cut by a plane R' L R that is perpendicular to the vertical plane. Project s', where R' L crosses v'a' (the elevation of an edge), down to s upon v a, the h. p. of the same edge. Similarly, get m from m', n from n', and complete the shaded plan of the section. Another way of getting all points of the section but one, is to use the intersections of the trace R L with the H- traces of the various faces. Thus, by the first method, get from 0', with which to start; then, as ed is the h. t. of the face ved, we shall have j as one point of the intersection of that face with the given plane, and j for the- line itself, op being that portion of it which lies within the limits of the face considered. In like PLANE IiXTERSEOTIONS OF DEVELOPABLE SURFACES. 199 manner, pg, fe and RL would all meet in one point, and, correspondingly, all analogous sets of three. For the sectional view v".' BDE project all points upon R'LR by perpendiculars, as b'x; rotate R'LR upon RL till vertical, at R" L R; transfer it to Qh^ and finally rotate it into V on the left. 510. Plane sections of pyramids, cones, etc., are homologous with the bases of the surfaces cut; and the intersection of the cutting •plane with the plane of the base is an axis of homology. (Art.- 146). In Fig. 335, if vv' is a centre of projection, then e is the projection of p; d of o, etc. Also, po and ed meet at j on the trace RL; mn and b c meet at i; and similarlj- for the other lines and their projections. R L is, therefore, an axis of homology. 511. The intersection of an oblique cone by a plane oblique to both H and V; also, a tangent to the section. In Fig. 386 let v'.r'b', o.cbk, be the cone; PQP' the plane. Let fall a ver- tical line through the vertex of the pyramid. Its plan win be o, while v'y' will be part of its elevation, and y' is the elevation of its inter- section with the plane P Q P', found by Art. 322. In d, k, oj, etc., we have the traces of auxiliary rerticcd planes through the vertex. These planes must cut PQP' in lines passing through y'. Hence, for any one, as ob, note r— its intersection with the trace P Q, and project to •G. L. at r', which join with y' ; then 2/'r' is the v. p. of the intersection of plane PQP' with plane E-ig-. ss'z. ob, and at/' meets v'b', the highest element cut by f that auxiliary. The point on the element o i in the same plane is similarly found. The plans of the points /', m', etc., are then derived from the elevations. The tangent line, at any point nn' of the curve, is thus found : Draw the element o n k through the point. Make t k tangent to the base at k, for the h. t. of a tangent plane. It meets P Q at t, one point of the intersection of the section and tangent planes. Then tn is such intersection, and therefore the tangent sought. (Art. 505). The elevation of the tangent is then t'n'. 512. The intersection of an oblique cylinder by a plane oblique to both H and V; cdso, a tangent to the curve. In Fig. 337 let P' QP be the section plane. As aux- iliary j)lanes parallel to the axis of the cylinder will be the most convenient, take znn' as one such, ver- tical, and cutting P'QP in the line a'n'. Then all lalanes parallel to znn — a,a y j, cl — will intersect PQP' in lines e'g', c'q', etc., which will be parallel to a'n'. Any auxiliary plane cl cuts elements from the cylinder which will meet c'q' in points of the curve, as s', from which s is obtained. 200 THEORETICAL AND PRACTICAL GRAPHICS. The tangent line is drawn, as for the cone, by making Mio tangent to the base at i — the extremity of the element through /', the point of tangency — and joining w with u; also w' with w'. (Art. 505). 513. The intersection of nn hyjjerboloid of revolution by a phine that is oblique to both H and V. This problem may be taken as illustrating also the general problem of the intersection of a plane with a surface of revolution whose axis is perpendicular to a plane of projection. Let P'QP (Fig. 338) be the cutting plane, and let the hyperboloid be found as follows: Having given the altitude = 2 s's", the gorge diam- eter = A' F, and 45° for the inclination of the elements, draw circle y Cz for the plan of the gorge; p's'o' through s' and at 45° to Q L; jDroject p' and o' at p and a upon a line laarallel to Q L and tangent to y z. Then the circle of radius Sp is the base of the surface. (Art. 468). ^ Since the surface is doubly ruled draw f'k' through s' and at 45°, for the other element having op for its plan. Take any plan B on op), and project up to both o' p' and f'k'. Carry B by arc with centre S to D, in the meridian plane rt,S' parallel to V; then project D to the level of the original elevations, getting D' and a point on TF, not lettered but sym- metrical with D' as to the axis X F of the hyperbola. Similarly for other points in the hyperbolic contour or meridian section. The section. Auxiliary horizontal planes will cut circles from the hyperboloid, and straight lines from the jjlane P Q P'. WcV is one such horizontal plane. It cuts the circle cTq from the hyperboloid, and the line d b from P (I P'. These meet at c and b, two points of the section sought. These plans project up to the plane K'd', giving c' and b' An analogous process with each of the other auxiliaries, and with some which are omitted from the figure, gi-\-es the elliptical section shown. The highest and lowest 'points, h' and i', lie in that meridian plane a LP' which is perpendicular to P'QP. To find i' carry the line of declivity S a. (a'x') to .Saj, when — being parallel to V — its elevation a"x' will be parallel to P' Q, x' being the intersection of P Q P' with the axis. Then where a"x' crosses the h3'perbola J X R' will give the level of the desired point i' on a'x'. Simi- larly, the crossing of a" <;' with D' Y p" gives the level of h. 514. Plane section of an hyperboloid of revolution, found by weans of auxiliary cones. As illustrating the use of other auxiliaries than tlie customary planes, a few lines on Fig. 338 show how the same points could have been otlierwise obtained. Use x' — the intersection of plane PQP' with the axis s's" — as the vertex of a series of vertical cones. Each of these will, in general, cut the hyperboloid in two circles, and the plane P(lP' in two right lines. The intersection of these circles and lines will be points of the curve. Let n N in be the base of one auxiliarj^ cone. It cuts the plane P (I P' in the lines n s and A' N. One point of each circle in which the cone cuts the hyperboloid will be found on the clement INTERSECTING SURFACES. 201 ^ig-. 33S- of. To find either, pass a plane tiirough op, o'p' and the vertex x' (S). Its h. t. is mr, passing through p — the h. t. of the element, and through t, the h. t. of x't', St, which is a line drawn through the vertex and parallel to the element. Then Sm is the element cut from the cone by the plane containing op, and q therefore a point of the circular intersection of the cone and hyperboloid. The circle of radius Sq then gives b and c on the elements lying in both cone and plane. Joining iS with r — the second intersection of mpl with the base — would give a second element on the cone. It would meet op at a point which would be used like q to obtain two more points on the elements nS and i\^S'. 515. The intersection of surfaces with bases in one plane, found by means of one auxiliary plane. Taking two pyramids (Fig. 339) to illustrate the general problem, we note first the points A, B, j, k, etc., in which the bases of the two surfaces meet, they being, evi- dently, points of the lines of inter- section sought. Any auxiliary plane parallel to the bases would cut sections of the same form as the bases. Srclx is one such section, and o pm a another, determined as follows: Knowing that the altitudes of the pyramids are in this case as 2 to ^ any plane which bisected the edges and altitude of the smaller pyra- mid v.abc would be one- third of the way from base to vertex on the pyramid V.efgh; join, therefore, the middle points of the edges of v.ab cd, while on the other take fp one -third of fV, and similarly locate m, n and o, for the second section. jS" r, parallel to a b, meets po — the parallel to ef—a.tE; then B E is the intersection of the planes of faces Vef and vab, real, however, only to C, where the actual boundary of Vef is reached and the intersection runs on to the face Vh e. But, having A, we have only to draw CA to complete that part of the construction. Similarly, join the intersec- tion of any two edges of the bases with the intersection of the corresponding edges in the auxiliary sections, using the line thus obtained only up to the point where it reaches the actual limit of either of the faces on which it lies. 516. In orthographic projection a similar problem to the last is worked in Fig. 340, in plan only, the drawing of an elevation being left for the student. The altitudes of the pyramids, 5" and 4", are indicated at their vertices, v and w. The section r^s^t^ bisects the edges of the w- pyramid, being taken 2" above the base. The section aim^o^pi, at the same level as r^sj^, is therefore two -fifths of the way from the base mo pq to the vertex v. Starting at any point, as /, which is the intersection of the base lines of the faces oqm and lort, draw a line toward the intersection (j) of a■^m■^ and r-^t^, since these 202 THEORETICAL AND PRACTICAL GRAPHICS. are the parallels, at the upper level, to the base lines whose intersection with each other gave /. At e, where /j runs off the face v qm, we turn towards J, one point of the intersection of face vmo with wrt, but can use eJ no farther than d, where the limit of face wrt is reached, and the intersection becomes that of faces wts and vmo. Continuing this process (for which nearly all the construction lines are given) results in the outlines abcdef on the one side, and ghknil on the other. For the developments the student may refer, if necessary, to Art. 396 (Case 6), or to Art. 429. E'igr- 3-40- 517. The intersection of two cylinders, two prisms, or of a cylinder and prism, when their bases lie in the same plane. Each plane, in a system of auxiliary planes passed parallel to the axes of the two surfaces, would cut their surfaces — if at all — in lines parallel to the axes, which would meet at points of the desired outline of intersection. In Fig. 341, taking two prisms to illustrate the method just outlined, let azxm be the lower base of the one, and kwdnf that of the other. As one way of determining the directions of the edges, let it be required that the edge starting from m shall meet the one beginning at n at some point ss', whose position is assigned; then msp is the plan of one edge, and parallel to it will be the edges and axis of the prism to which it belongs. Similarly, nsl is a plan, and gives a direction for the other prism. Project m to m'; draw m's'p', and parallel thereto draw the other edges a'O', z' R' x'Q'. Then n's'l' and its parallels complete the elevation of the other prism. It having been predetermined that mp and nl should intersect, we have in mn the horizontal trace of the plane containing those two edges, and all other auxiliary planes parallel thereto will have H- traces parallel to mn. These are seen in xh, ah, etc. INTERSECTING PRISMS. 203 The plane mno cuts the face dwvg in a Ihie ot which meets the edge mp in the point t. Imagining the auxiliary plane advancing from 77ino— its rear position, and, at first, noting inter- sections on the right-hand side of each surface, we find cc/i as the next position in which it con- tains' an edge of either surface; then the edge x meets the line from h at h^. Further advance Fig. S-il. brings us to the w-edge, which meets the face xzQ at r^. The A -edge next meets the same face at k^, and then the /-edge meets a line from e. The edge x is then found to meet the ,fn face on a line running up from y. The next move 204 THEORETICAL AJ^D PRACTICAL GRAPHICS. ngf. S'iS- completes the circuit of the right-hand base, with the exception of the vertex d, a plane through which would be entirely exterior to mxza, showing that the edge d y clears the other prism. We next return from n to o, while moving from m to z by the way of o; and the first plane to contain an edge is fe, resulting in f^ on fi. Next ac gives c,; ks gives u^, while r falls on the auxiliary plane through w. Plane ab gives b^, which joins with t to complete the solution. The elevation of the intersection is most readily obtained by projecting up to the edges the points that have just been determined in plan. The visible portions of the surfaces are evidently not the same in the two views. 518. Intersection of any surface of revolution by a cylindrical surface, by means of auxiliary cylinders. In Fig. 342 Ave have the elhpsoid T' U' W Z' {TW) as the surface of revolution. Let it be inter- sected b)^ an oblique cylinder having an ellij)tical base, a g y zx. A series of random horizontal planes, M'N', P'Q', R'S', cut circles from the ellipsoid, as m'n', c'r', each of which can be made the directrix of a cyl- indrical surface, whose elements will be parallel to those of the given cylinder. These auxiliary cylin- ders will intersect, if at all, in a common element, which meets the original plane at a point of the curve sought. Taking, in particular, the plane P'Q' with which to illustrate, we have the circle C'r' projected at C p C (centre o), and also at IJy, drawn from centre s, where o s, i's' is parallel to the elements of the cylinder. Then Ip and yz are the elements cut from the original cylinder by the auxiliary cylinder having base IJy; and p and z, their intersections with C p C, are points in the desired curve, and project in elevation upon P'Q' at p' and z' Similarly, q' K' projects both to SRm and to KKx (centre t). Then x q meets S R m at q, which projects to q' on the plane R'S'. 519. The intersection of a surface of revolution icith a coniccd surface, by means of auxiliary cones. Substituting auxiliary cones for the auxiliary cylinders of the last problem, the solution would be in strictest analogy to the one tliere described. 520. The intersection of a -iphere by an oblique cone whose vertex is at the centre of the sphere. In Fig. 343 a quarter sphere is shown in s'M'R' and MsN. The cone is s'.a'b', s.apf It cuts from the cone a circle of diameter G_a g'z', centre P'Q' is an auxiliary horizontal plane x'; seen in plan in NKb, centre x. Q'g' is the radius of the circle cut from the sphere by P'Q', of which a quadrant is seen in plan] at mNk. N and h, the intersections of the auxihary circles in the plane P'Q' are then the I XTF.RS ECTI Xii SI' i; FACES. — [) K V K L () P M K X 'P OF CON F. 205 l" i "''■ \^ 1 ; I'l ; ' ' ' ^'-"-'^ ' ' I (fl ^iVr-'i'^l ''',-''// 3,^^-C"^'^V^i^'' y ''^ \ \ \ \'^ f,--yw ''' \ \ A \ \ /^-'''''^^^ \ ^^ \ "■ ^ '~'^ ""-J / 1 /U- \..-'^ ^^K^^'d^J/ ^\ \ \ '\ \. ~~ -\- '"/"- 1/.^ / '-X. / /^''~" '"^ >>v /' / jX^^ "~'^'5><^ ^^^■t' '^'i^ ]>l;ms (if t\vi> jioints nl' the elirve souuht. Tliesr aiv ]>ri))('<'t(.:(l uixm J'' (J' fur the rlev.'itiiniH. The luLi'licst ;uiil liiwi.'st jioiiits are thfisc Iviiii;' in tlic vertical meridian ]ilanc s k' ]>. Tn lind the former, carry .^' /) to -s- o, jiarallel to \'. Then o'.s' is its new v. p. This cuts the spliei'ical FLg. 3-i3. contour at ,/', the Icrcl of /•', wliich lies on the cleA'ation (not J..' drawn) of the element ■•< ji. A similar jirocedure \vith element •N' 7 ,i:i\'es /', the lowest point. • ilil. 77((' ihrrhiiniiciit (if |ier part of Fi.n'. 044, F 'P Y heing the rectifi- cation of the arc it: Fn in Fig. o4."). Then, with P'lv (Fig. 344) made e(iual to Fir' (Fig. o4o), 7"/ to Tl', etc., we lia\'e the cur\-e ivlii, Fig. o44, for the cylindrical de\-elo]imeut of the arc v>'Pii'. Draw next the arc ir"P'ii'\ Fig. o44, E-ig-. 3.4s. of radius e(iual to tliat of tile sphere, and of l(ii(/th ei|Ual to the ir hi ahove it; make the elements iS'.r,, Stj,. Sf,, equal to the true Icmjllix of the s[)aee elements as de- rived from their pr(jje<'tions in Fig. rU.'l. \\'e then lia\-e in the Hgui'e & .'■ , 7 , /, the development of a ]Mirtion of the gi\-en cone. ^)'1'1. Ih'iinli ShiuliiKj fif Ptii'rsirtiiii/ Siir- fiiccs. I'rohlems in intei'section are not onlv a. valuahle test of a student's aci|Uaiiitauce ■with the properties of surfaces and of his p()^^'er to apply genei'al principles to special cases, hut also afford an csjiecialh' a(h'an- tageous lielil in wdni'h to exhiliit skill -with the lii'ush, particidai'ly if to the ahility to represent foi'iu there he added a thorough aciniaintance «ilh the gi'ometrical construction of shadows. i\s a rule, ordy elevations \\-ould he sha.deil. In illustration of the fireg(jing Figs. -'Uo and o4(i arc inti'odnced. lleing reproductions of photo- graphs of pl.aster casts, they show iiiiliirn/ as distinguishe(l li-oni fdiinnhniiiil light and shade eheets. X^Lg. 3iS, 206 THEORETICAL AND PRACTICAL GRAPHICS. CHAPTER XI. TEIHEDEALS, OE THE SOLUTION OF SPHEEICAL TEIANGLES BY PEOJECTION. 523. The solid angle contained by three planes meeting at a given point is called a trihedral angle. If the vertex of a trihedral is at the centre of a sphere, as in Fig. 347, the planes of the sides will cut the surface of the sphere in three arcs of great circles, forming the sides of what is called a spherical triangle. The arcs (sides) are the measures of the plane angles whose common vertex is the centre of the sphere. The dihedral angle P Q between the planes of tw( > sides, as a and r, is measured by whatever arc, P Q, is included by the planes upon the great circle having for its pole the intersection (J?) of the sides. Since tangents at R to the arcs a and c would be parallel to OP and Q respectively, they would include the same angle as the planes in which they lie. The solution of a spherical triangle consists in the determination of the dihedral angles between the jjlanes of its sides, and the plane angles subtend- ing the latter. 524. In Fig. 347 the sides P R, P Q and QR oi the spherical triangle RPQ will lie referred tu as a, b and c respectively, the dihedral angle ojiposlte each side being denoted by the same letter capitalized. Thus, the dihedral angle on edge P is called the angle ('. If C", that pole of the arc QR which lies on the side of tlie plane op])osite to the trihedral, be joined by arcs of great circles to the analogous poles of the other ares, ) two sides and the angle ()])iioKite one of them; (4) the three anales two angles and the included side; (fi) two angles and a side opposite one of them. Although not often necessary, we may always reduce the last three eases to the form of the first three by means of the supplementary triangle. For exam])le. Problem 4 mav be worked liv and with anv (2) two sides (5) SOLUTION OF SPHERICAL TRIANGLES. 207 solving Prob. 1, using sides Avhich are the supplements of the angles given; then the su])plements of the angles thus determined will be the sides desired. The student will find it to his advantage to make cardboard models illustrating various cases. This can readily be done by cutting sectors of different sizes, and folding them on two of their radii. Thus, O.R(JPR' (Fig. 34S) is the sector which, folded on OP and (^, would illustrate the triangle of Fig. o47. •")'2G. The following properties of the spherical triangle must be kept in mind: (1) The greater side lies opposite the greater angle, and conversely; (2) the sum of any two sides nmst be greater than the third; (3) the sum of the sides must be less than four right angles; (4) the sum of the angles must be greater than two right angles and less than six; (5) each angle must be less than 180°. 527. To solre a xjiheriral trinii.gle having gioen the three m'des. (1) Take « = o2°; &=50°; c=42°. In Fig. 349 lay out these angles at 0, obtaining the development of the trihedral on the plane of the side b. If we now rotate the faces a and c upon P and Q respectiveljr, until R and R , coincide, the trihedral will take its space -form. Make D^ E; then, after the rotation supposed, the points D and E will coincide, each having turned in a vertical plane perpendicular to its axis of rotation. D s and Es are the traces of these planes of rotation, and s is the plan of the united D and E; s is therefore the plan of the united OR and OR^, that is, of the space- position of the third edge of the trihedral. At s draw perpendiculars to s D and s E. Cut the former at S^ by an arc of radius x D, centre X, and the latter at S, by an arc from centre y, radius E y. Then s S^ obviously equals s S^ , while «j-.S'j (C) and syS.^ (A) are two of the three angles sought. The plane of the third angle, B, being perpendicular to the edge s just determined, draw p q at 90° to Os to represent its trace on the plane of the side b. This plane will cut the faces a and r, when in their space -positions, in lines perpendicular to their common edge, and seen in development ed, pt and q r. Arcs t z and r z, from centres 'p and q, intersect at z, giving pzq as the angle B sought. (2) Salving upon both H and V we may lay out in the latter (Fig. 350) the three faces o, 6, c, with edge OQ perpendicular to (i. L. Make R= R^; join P with R; draw ^ig-. sso. arc PT from centre Q and cut it from centre R^ by an arc of radius TR^ = PR. Then Q T is obviousl.y the position taken by face b when the latter has carried with it the face P^ Let the angle A be the other given element of the jiroblem. Draw (j ;/, at !)()° to (j. as the h. p. of a line of declivity of the face r; carry y to o from centre 7, and make 7 /■=.!; ;• meet- ing the vertical gr at a point "f the v. t. of the face c. Draw Qr. Pxitate face a upon OP. The arc described by R cuts (Jr at \ ^ n- Y ,-*" \ s Q ^ J- vi Given 6,C,A \ 1 (2) Solved with the nfie of both H and Y we would draw (Fig. 36(1) the fac-c h in H, with one edge OQ periDendicular to G. L. Make angle ]1'(JP=A. Draw xy, a random line of declivity of the face a; cany x to x" (centre y), and draw from x" a line making angle C with G. L. This meets a vertical from 7/ at .(/', one point of the trace of a plane making angle C with H. Py' is- then the v. t. of face a. It crosses WQ at !<', which — with — gives the' common edge of faces a and c. 532. Given, two angles and the xide opjiosite one. In Fig. 357 draw first the given side c, and at some point Q draw Q E, at 90° to Qy to represent the plane of rotation of E. On this line lay off the given angle .1, and find i'~'' h\ arc of radius Q £; then derive s from S as here- tofore and join with for the plan of the space -edge of faces n and c. Through S draw a line Se at an angle C to Q G. Using Se as the generatrix of a vertical cone having Ss for its axis, e would trace the arc etfg; ^'ig-. 3S7. and P, tangent to the latter, would be a trace of face n. Then tl( = Se) gives /, through which draw OR for the outer edge of face n. 533. (-rireii, the three angles. In Fig. 358 take Rz for the ground line, and P at 90° to it' for the edge com- mon to II and b. Draw PS' at angle O to Pz. Then S' P 0, perpendicular to the vertical ])lane, is the face a in its space - ijosition. We have next to adajDt the i^roblem of Art. 319, which required a plane whose inclinations to two mutually per- pendicular planes were given. In the present case the planes a and b are not at 90° to each other; but tlie same principle applies, that liv resting a plane E-ig. ssa. against two cones having A and B for l)asal angles and whose axes intei-seet, each axis being perpendicular to one of the given planes, the conditions are met. Let P lie the point of intersec- tion of the axes of the auxiliary coui's, and JP the axis of a ver- tical cune whose base angk^ eijuals A. All planes tangent to this ,1-C()nu will be at the con- stant distance Pn from P, liut will contain the vertex ,/, and make the desired angle with tho face /). Since OPS' is perpen- dicular to V, an axis at 90° to it and containing the point P will lie, like the other, in \\ and is seen in the hue Pq. Tangent to a circle (or sphere) of radius Pn draw a line .r ry at angle B to PS'. It meets Pq at q, the vertex of the P-cone. Then, as the desired plane must contain both vertices, Jq is its v. t., and z 1 0, tangent to the liase of the .-1-cone, is its li.t Given c,A,C SPHERICAL PROJECTIONS.-CARTOGRAPHY. 211 CUAPTBR XII. ORTHOGRAPHIC— STEREOGRAPHIC.—GNOMONIC.—NICOLISl'iS GLOBULAR.— DE LA HIRE'S METHOD.— SIR HENRY JAMES' METHOD.— MERCATOR'S.— CONIC — BONNE'S METHOD.— RECTANGULAR POLYCONIC— EQUIDISTANT POLYCONIC— ORDINARY POLYCONIC — GLOBE COVERING. 534. In attempting to represent the whole or any portion of the earth's surface or the celestial sphere upon a sheet of paper the draughtsman is confronted with the impossibility of avoiding a distortion of some kind, owing to the fact that he is dealing with a double- curved surface, which can not be '' develoiDcd " or directly rolled out upon a plane. He is, therefore, compelled either to adopt some one of the many methods of applying the principles of perspective projection to the problem, or one of the equally large number of methods which, while not true projections in the ordinary sense, are now included under that head, owing to the extended mathematical significance of the term. In either case he will find that the system has been devised with a view to preserv- ing between the original surface and its representation some relation which may be regarded as essential for the purpose for which the map or chart is to be used, but which can usually be attained only at the sacrifice of some other relation which it would be desirable to maintain. Thus, if he preserves upon his drawing the equality of the angles between the planes of the various circles that are usualh' represented, he cannot at the same time have all areas reduced in a constant ratio; and other desirable conditions are often found to be as mutually exclusive. For an extended mathematical treatment of the various systems invented for the representation of the earth or the celestial sphere, as also for the tables essential to the construction of maps, the student is referred to Germain's Traite des Projections and Craig's Treatise on Projections, which, with De Morgan on Gnomonic Projection, have been the principal sources from which the writer has drawn. Adopting Craig's classification, we group all the methods under the following heads: (a) Orthomorfhic Projections, in which similarity of form is secured between areas on the sphere and on the map. (b) Equirnloii Pnjections, in which different areas are reduced in the same proportion. (c) Zenithal Projections, in which all points on the sphere that are equidistant from the assumed centre of projection are projected in a cii'cle whose centre is the projection of the assumed perspec- tive centre. (d) Projections by Ikrelojivient, in which the spherical surface is first represented upon a tangent or secant developable surface, and the latter then rolled out ujjon a plane. 535. 71ie great circles customarily represented are the equator, the ecliptic and the meridians. Meridians contain the poles of the sphere, and their planes are perpendicular to that of the e(iuator. The celiptir is the ai:)2?arent jjath of the sun, and its plane makes an angle of 23 ° 27 ' (usually called 23 J °) with the equator, cutting the latter at the equinoctial points. The EqiiiiKie.lifd C'olure is the meridian containing the equinoctial points. The Solstitial Culiire is a meridian whose plane is perpendicular to that of the equinoctial colure. It cut.i the eclifjtic at the solstitial points. 212 THEORETICAL AND PRACTICAL GRAPHICS. 636. The small circles usually projected lie in planes parallel to that of the equator, and are, in particular, the tropics and the polar circles. The Tropics of Cancer and Ca.jrricorn are the northern and southern limits, respectively, of the torrid zone, and being 282 ° from the equator they touch the ecliptic at the solstitial points. The Polar Circles; are the Arctic and the Antarctic, each 232-" from a pole of the earth. 537. The axis of any circle is a straight line through its centre and perpendicular to its plane. It meets the surface of the sphere in points called the poles of the circle. 538. The polar distance of a circle is its distance from either of its poles, measured on the arc of a great circle. The plane on which the projection is made is frequently called the priviitire. If it contains the centre of the sphere it cuts the surface in the primitive circle. 539. The line of measures of a circle is the intersection of the primitive plane by a plane con- taining the axes of the given and primiti^'e circles, and is jierpendicular to the intersection of those planes. 540. The point of sight (centre of projection) may be either at infinity — giving parallel, and, ordi- narily, orthographic projection — or at a finite distance, giving a perspective or central projection. ORTHOGRAPHIC PROJECTION. 541. Orthographic projection — a special case of zenithal — is believed to have been first applied to the sphere by Hipparchus. It is not used for terrestrial maps, but solely for celestial charts. In comparison with other systems orthographic projection has relatively greater distortions, both of form and area, for those portions of the sphere near the plane of pro- jection. The inconvenience of constructing elliptical projections is another objection to it. To show how the principal lines would appear by this method Fig. 359 is presented, the elevation having for the primitive the plane of the solstitial colure, that being the one usually selected for celestial charts constructed by this method. In this case all meridians appear as ellipses, while parallels are projected in straight lines. The derivation of meridians from the plan is obvious, each a])pearing on the latter as a diameter. The equinoctifd colure is projected in the straight line A".S", and the equinoctial points at C. The solstitial points are seen at a:' and y'. 542. Orthographic eijuiitori(d jinjcction, that is, with the equator as the primitive circle, is illustrated l)y the plan in Fig. 359, the meridianx being, as already stated, diameters; while the partdleU i)rnj(.ft in concentric circles. 543. The orthograi)liic projection of the sphere upon the plane of a meridian or other circle making any given angU' with tin- equator may )y the method of rotations (auxiliary planes) treated in Art. 4(j4. STEREOGR A PHIC I'HO.TECTIOX. SqaatoT ))(' most readilv )btained 1 544. This projection, called l)y the ab()\-e name only since 1613, was devised about 130 B. ('. Itv Hipparchus, who called it a planisjihere. It is an orthomorphic projection, and from tlie fact that it not only possesses the distinctive ])roperty of preserving similarity of form between infinitesimal SPHERICAL PROJECTIONS. 213 surfaces and their projections, but also that all circles on the sphere are projected as circles, it is the most convenient system so far devised, and at present the most employed for both geographical and astronomical purposes. The eye may be located at any point on the surface of the sphere, but is usually taken either at some point on the equator, giving a meridional projection, or else at one of the poles, when an equatorial stereographic projection results. The projection is always made upon the plane of that great circle whose pole is the assumed point of sight. Only the hemisphere opposite to the eye is projected. 545. The fact that every circle is projected, stereographically, into a circle, has been already established in Arts. 135-6, but for convenience the demonstration is repeated here. In Fig. 360, regarding at first only the triangle on the left, let o MN be a view of an oblique cone having a circular base M N. E'lg. sso. Take a section plane m n so as to make with o iV an angle mno equal to the angle o M N. Then will mn be circular. Such a section is called sub -contrary. Take any section p q, parallel to MN, which will obviously be circular. From the similar triangles pmx and nxq we have p x:nx: :mx:xq, whence px x xq^^mx x nx. But px x qx equals the square of the semichord common to the two sections at x; hence mn is circular. Turning now to the sphere ORMT, let RT be the primitive circle (plane of projection), and (its pole) the centre of projection or position of the eye. Let NM be any circular section of the sphere. Then ONM is a visual cone, and nm, the projection of N M, will be a circle, being a sub -contrary section of the cone. 546. To establish the orthomorphic property of a stereographic projection we have to show that the angles between the projections of circles equal those between the original curves. In Fig. 361 let be the position of the eye, and PN & tangent at N to the circle RNT. Then np is the projection of N P. Let B N he the tangent at N to some other circle containing that point, and let n c be the trace of the plane ONE on the plane of projec- tion RcT; then will the angle pnb be equal to the angle PNB between the tangents, i. e., the angle between the planes of the circles to which they are tangent. For we may take some point S, on ON prolonged, and draw SP parallel to R T, and SB parallel to nc; then the plane oi SPB is parallel to the primitive, and the angle PSB obviously equal to pnc. The angle OnT, being between chords, is measured by i(OT+RN); but this equals i(OR + RN) which measures the angle between tangent PN and chord NO. Hence OnT= S P= PNS; whence PN=PS. Simi- larly, we may prove BN=BS. The triangles BNP and BSP have the sides of the one equal to those of the other, each to each, and being, therefore, equal in all their parts, we find angle B NP^^B S P=pnb. Since a curve and its tangent are always projected as a curve and a tangent, and since the angle between two circles is that between their tangents at a common point, the proposition is established. c 3S1- ) r/ \ \ p 214 THEORETICAL AND PRACTICAL GRAPHICS. FLg. ssa. A * /j / 1 /L- N /^'i~ ""n^. „/^ ' \ Nft fi "''''< i-~^ \ \ ! ^N « ■--, \ \ p ^, ~--, w aV- T v^ jy 547. Stereographic meridian projection. As usually constructed, the plane of projection is that of the meridian of Greenwich, the actual longitude of the eye being then either 90° or 270°. The meridian through the eye is, however, called the first meridian, and graduation made therefrom each way, from 0° to 90°. To project a parallel of latitude, as a 6 (Fig. 362), draw a tangent at either extremity, as a, to meet the radius o N prolonged. This tangent, t a, is the radius of the arc acb in which the parallel is projected. For imagine the sphere rotated 90° to the right, so that the eye moves from o to E, and NS becomes the projection of the primi- tive. The projector Ea cuts the primitive at c, which remains con- stant during counter -rotation. Then the centre t, of arc acb, must be at the intersection i of o iV" by a perpendicular to chord o c at its middle point. In the triangles a t x and txc we have the angles p and <^ equal, being opposite equal sides. But p is the angle between two chords, and is measured by i (^S E + aN) = i (NE+ a N)= i aN E, which, as the measure of <^, shows that if a £■ is a chord, a t must be a tangent. To project a meridian of longitude. Let the meridian to be projected make an angle 6 with the plane of the primitive meridian, WNES, upon which it is to be projected. Draw SP at 6° to SN; then P is the centre and PS the radius of the arc NTS in which the meridian projects. This is established as follows: Let Pig. 363 represent a top view of a sphere; the point of sight on the equator; MB the plan of a s-iE-. sss. meridian making 6° with the primitive, mWE; then m6 is the plan of the circle in which MB is stereographically projected; and P, bisecting m h, is the centre of the projection. Now as MOB is a right angle we have P = Pb; hence angle POB equals Pb 0, or ^. But y Arrowsmlth it has been erroneously accredited to him. 216 THEORETICAL AND PRACTICAL GRAPHICS. a horizontal diameter for the projection of the equator, and a vertical diameter for that of the central meridian. Then, for ten -degree intervals, divide the quadrants into nine equal parts, number- ing each way from the equator. Also divide the horizontal and vertical radii into nine equal parts, numbering each way from the centre. The parallels are then drawn as circular arcs through like- numbered divisions each side of the equator, while the meridians are circular arcs containing the poles and the divisions on the equator. 555. De La Hire's Perspective Projection. This projection, often erroneously termed globular, was devised by De La Hire in 1701. In it the eye is taken on the prolonged axis of the primitive, and at a distance from the surface of the sphere equal to the sine of 45°. Its classification is obviously under zenithal projection. With a radius of unity the sine of 45° = N/i Hence, in Fig. 368, if CT=nX=s/i, then Ox and xE, which are the projections of the 45°- arcs NX and X E, will be equal. Other equal arcs will have projections very nearly equal. This is its only prac- tical advantage, as it is neither orthomorphic nor equivalent, and involves elliptical projections for all circles not parallel to the primitive. 556. Sir Henry James' Perspective Prelection is an interesting case of zenithal, devised for the purpose of reducing the misrepresentation to a minimum. Like De La Hire's, the eye is taken exterior to the sphere, but in this case at a distance equal to one -half the radius. For a hemisphere this is regarded as the best possible system of projection. By taking a plane of projection parallel to the ecliptic and touching one of the tropics, or, in other words, by adding a 23J°-zone to the hemisphere. Colonel James obtained America, Europe, Asia and Africa in one projection, claiming it to include "two-thirds of the sphere." This has been shown by Captain A. R. Clarke to be an underestimate, the exact figures being seven -tenths; while the same writer shows that for minimum distortion with the new primitive the eye should be at a distance of |^r outside the surface, instead of ^. PROJECTION BY DEVELOPMENT. — CYLINDEIC. — CONIC. — POLYCONIC. 557. With the eye at the centre of the sphere we may project the various circles of the latter upon either a cylinder that is tangent or secant to the sphere, or upon a tangent or secant cone. By then developing the auxiliary surface we will have in the one case a cylindric and in the other a conic projection. 558. In square cylindric projection the auxiliary cjdinder is tangent along the equator. The meridians then appear as straight lines perpendicular to the rectified equator, while the parallels — which projected as circles — develop into straight lines at 90° to the meridians, the distance of each from the equator being the tangent of its latitude. This projection is only occasionally used, the exaggerations involved being too great to make it serviceable except for a short distance each side of the equator. 559. Mercator's projection (also called a reduced chart) differs from the last described only as to the spacing of parallels. This spacing is, however, so effected that on the resulting map the angles are preserved between any two curvilinear elements of the sphere; in other words, INIercator's is an orthomorphic projection. Since meridians actually converge on a sphere at such rate that the length of a degree of longi- tude at any latitude equals that of a degree on the equator multijjlied l)y the cosine of the latitude it is obvious that when they are represented as non- convergent the distance apart of originally SPHERICAL PROJECTIONS. 217 equidistant parallels of latitude should increase at the same rate; or, otherwise stated, as on Merca- tor's chart degrees of longitude are all made equal, regardless of the latitude, the constant length representative of such degree bears a varying ratio to the actual arc on the sphere, being greater with the increase in latitude; but the greater the latitude the less its cosine or the greater its secant; hence lengths representative of degrees of latitude will increase with the secant of the latitude. The increments of the secant for each minute of latitude can be ascertained from taWes. Navigators' charts are usually made by Mercator's projection, since upon them (as upon the square cylindric) rhumb lines or loxodromics — the curves on a sphere that cross all the meridians at the same angle — are represented as straight lines. A loxodromic not being also a geodesic, the mariner takes for his practical shortest course between two points the portions of those different loxodromics which most nearly coincide with the great -circle arc through the points. 560. In conic projection, if the auxiliary cone be tangent along a parallel of latitude, the meridians will project as elements of the cone; the parallels into circles. On the development the parallels become concentric arcs on the sector into which the cone develops, the radius of each being the slant -height distance from the parallel to the vertex. The meridians obviously develop into radii of the sector. 561. If tangent to the sphere near the equator the vertex of a cone is inconveniently remote. Even when tangent along a parallel of latitude more medially situated this method gives undue distortion, except for a narrow zone on which the parallel of contact is central. Many methods have been devised for the purpose of obviating these difficulties, a few of which are next briefly mentioned. 562. Mercator suggested the substitution of a secant for a tangent cone, choosing its position with reference to the balancing of certain errors. By this niethod a large map of Europe was made in 1554. Euler carried out the same idea with greater exactness, fulfilling his self-imposed conditions that the errors at the northern and southern limits should not only equal each other, but also the maximum error near the mean parallel. 563. Bonne, in 1752, applied the following method (its invention is variously accredited) which was later adopted (1803) by the French War Department, and has been extensively used in European topographical work: Assuming a central meridian and a central parallel, a cone is made tangent to the sphere on the parallel. The central meridian is then rectified on the element tangent to it, and using the cone's vertex as a centre circular arcs are drawn through (theoretically) consecu- tive points of the developed meridian. The zones between the consecutive parallels on the sphere then develop in their true areas upon a plane. Each meridian is drawn upon the map so as to cut each developed parallel at the same point as on the sphere. The parallel of tangency cuts each meridian at a right angle. Bonne's method evidently comes under the head of equivalent projections, as it preserves the area though not the form of all elementary quadrilaterals. When extended to include the whole earth in one view the map has a peculiar shape, some- what like a crescent with full, rounded ends, and quite broad at the centre. By taking the equator for the "central parallel" a projection results, due to Sanson, called sinusoidal by d'Avezac, a,nd often credited to Flamstead. When applied to the entire sphere it resembles two equal and opposite parabolas with their extremities joined. 564. Polyconic Projection. A method largely used in England and employed by the United States Government on its Coast and Geodetic Survey, is based mpon the use of a separate tangent cone for each parallel to be developed. 218 THEORETICAL AND PRACTICAL GRAPHICS. 565. In Rectangular Polyconic Projection the rectangularity of the quadrilaterals between meridians and parallels is preserved. It is thus constructed: In the elevation, Fig. 369, let b"b', d"d', f"f' FLg. 3SS. be parallels of latitude ; their plans will be the concentric circles shown in the upper figure. Nf, V'd', v'b', are the elements of cones, tangent to the sphere on the indicated parallels. Taking the meridian Edh N as the central meridian, rectify it at E" D B in the lower figure, getting the lengths from E'd'b'. Make E"F=E'f, etc. Through F an arc of radius Nf is the development of the parallel /"/'. The arc through B has radius v'b'. On the plan draw meridians Nl, No, etc. Then lay off on each developed parallel the distances included on it between the meridians just drawn. Thus, DK and KM equal the rectified arcs dk and km. B TZ equals the true length of b tz. When the parallel of tangency is so near the equator as to make the vertex of the auxiliary cone inconveniently remote, tables are employed giving the rectilinear coordinates of points on the developed \f, parallels. 566. Equidistant Polyconic Projection is a modification of the method just described, resulting in a representation in which two parallels will include equal arcs on all meridians. This method is used in Government work, for small areas. To draw it a central meridian E"FDB (Fig. 369) and a central parallel DKMW are drawn as in the rectangular polyconic system, and the meridians also found in the same manner, or by the use of tables. From the points D, K, M, W, where the central parallel intersects the meridians, the equal lengths FD, D B, are laid off on the meridians, giving points through which the other parallels may be drawn. 567. Ordinary Polyconic Projection. This method sacrifices the rec- tangular intersection of meridians with parallels (except on the central meridian) in order to preserve the lengths of the degrees on the parallels. Drawing the usual central meridian in its true length, the parallels are developed as for the rectangular polyconic; but on each parallel the degrees of longitude are laid off in their actual lengths, and points thus obtained through which to draw the meridians. This method is in general use by the U. S. Government for the maps of its Coast Survey. 568. The foregoing is as extended an excursion into this attractive field as the limits of this treatise will permit, but it should be understood to be but a glance, and that a large number of interesting methods must go unnoticed, the student being referred to the authorities earlier mentioned, in case he wishes to pursue the subject further. RECTANGULAR POLYCONIC SHADES AND SHADOWS. 219 CHAPTEB XIII, SHADES AND SHADOWS OF MISCELLANEOUS SUKFACES. 5(i0. The shadows cast hj an object which is illumined 1)_7 either the sun or some other source of light are, in tlie matliematical sense, projections, and the rays of light become the projectors. 670. The shmle of an object is that pai't of its in\n surface which receives no direct rays from the si.iurce of light, while the shinloiv is the ilarkened i)ortion of s(jme (jther surface from which the original object excludes the light. The rays through all points of a given line will determine either a pilaae of rays or a cylinder of rays, according as the line is straight nr curved. 571. The tine of shade on an object is tlie l)oundary between the illumined and the unillumined portions, and its shadow tVirms the boundary of the shadow cast 1iy the oliject. If the ol)ject is curved, the line of shade is the line of contact of a tangent cylinder of ra3's, each element of which would be tangent to the oljject at a pcunt at which the cylinder and the ■^^s- 3'70- object would have a comnKjn tangent plane of rays. For convex plane -sided surfaces the line of shade is the warped polygon f)rmed by the edges contained by non- secant planes of rays. 572. It is the province of Descriptive Geometrv, in its apjilication to tliis tojiic, merel}' to determine the rigid outlines of shadows and shades. The delicate effects of cross and reflected lights, which alwaj^s exist in nature in greater or less degree, can only lie theorized aliout in a general way and can be most suc- cessfully imitated in di'aughting l)y Working from a model or Ijy the aid of phot(jgraplis. Figs. 3711 and 371, which are half-tone reproductioiis (E R F) ^= 1 -^ \/2, which, in a table of natural tangents, corresponds to the value indicated. Shadows thus cast are seen to be true oblique or clinographic projections (Art. 14), although orthographic projections are usually employed as auxiliaries in their determination. 576. In pictorial illustration of a few general principles Fig. 372 is drawn in oblique projection. In so elementary a figure as the cube the edges to be drawn as shade lines are evident from the outset, since, for the given direction of rays, the back, right and lower faces are obviously in the shade. We proceed then directly to find the shadow of the warped poh'gon ABCDEHA establishing at the same time some of the principles that are of most frequent use. 577. The shadmv of a point upon a surface, being the intersection of the surface by a ray through the point, is found where the ray meets its projection on the surface. In Fig. 372 the point C is projected on the plane MP at c. The ray Cc^ meets its projection cci at Ci, which is, therefore, the shadow sought. 678. Any line, straight or curved, is equal and parallel to its shadow, when the plane receiving the shadow is parallel to the line casting it. Having c, we therefore draw c^b^ and b^a^ parallel to C B and A B respectively. 579. A line that is perpendicular to a plane will cast a shadow upon it whose direction is that of the projection of rays upon the plane. The trace, cc,, of the vertical plane of rays through CD, contains the shadow d,c,, and the projections of both rays Cc^ and Dd^. SHADES AND SHADOWS. 221 580. Parallel lines cast parallel shadows on a plane, since they are the intersection of parallel planes of rays by a third plane. 581. In orthographic projection the construction of Fig. 372 is shown in Fig. 373. The rays are ^"Mr- s^s. shown on V in e'x, f'y and c'z, and each projects to the proper plan to give the trace of a ray on H. 582. The shade and shadow of a vertical pyramid. As the point where a line meets a surface is the beginning of the shadow of the line on the surface, we have merely, in Fig. 874, to join t, the trace of the ray through the vertex, with b and d, where ^^^- ^'^'^• the edges of shade (s 6 and s d) meet H, to include the shadow sought. Planes of rays through the other edges would in each case be secant to the pyramid, and therefore useless. 583. Shadows on vertical, horizontal and oblique planes are illustrated by Fig. 375, in which the pier flanking four steps receives on its inclined face the shadow of a vertical post, and in turn casts a shadow on the steps. The shadow of the post. To find the point y", y^, where the ray through the vertex y meets the face abed, regard y B aa the trace of a vertical plane through the ray; note A and B, where it cuts bounding lines of the inclined sur- face; project these at A' and B', upon the elevations of the same edges, and draw A' B' for the V. p. of the intersection of these planes. This receives the ray from y' at y", which projects down to 2/i. A similar construction gives u", which joins with y" for the shadow of the edge y'u". For the shadow of edge ry, r'y', we may regard the face a'b'c'd' as extended upward and to the left, sufficiently for a re - application of the method described, D' giving a direction for the line y"D', which is a real shadow only to the edge a'd'. The vertical edges of the post, whose plans are u and r, cast shadows on H in the direction of the projection of rays. The shadow of the pier on the steps. The vertical edge at b casts a shadow beginning at the foot of the line, and running — in the direction of plans of rays — to b^, the h. t. of the ray b' N. At &i the shadow of the line b' c' begins, and its direction — upon horizontal planes — is found by treating cc' as if it actually cast a shadow on the ground, t then being the h. t. of the ray from c', and tbi the direction sought. At k, however, the shadow begins to fall on the front of the lowest step, and we project up to k' for its v. p. To get the direction of shadows on the fronts of the steps, assume on b' c' some point i, i'; imagine the front of step No. 1 extended to catch the ray through i' at V, then toward I' the shadow M runs from k'. This direction once established, we have only to get one point of each shadow P and Q in order to draw them in; while R, S and T may be drawn parallel to bit, when one point of each is known. The shadow M runs off the front of step 1 at 5, which projects down to 6 for the beginning of shadow R. The latter is parallel to 6i«, and at 7 meets the lower edge of the front of the second step. It projects to 8, through which the shadow P is drawn parallel to k'l'. To determine where this process will terminate we may definitely locate the shadow of cc', either as a preliminary or at any stage of the work, thus: The ray c't' meets the level of the top step at s'; this projects upon ct at s, which is outside of the actual limits of step 4 and therefore 222 THEORETICAL AND PRACTICAL GRAPHIC S. Fig. 375. unreal. The plan ct oi the same ray meets the front of 4 at x, which projects at x', and this, being between the limits of the front of the step, is therefore a real shadow. At x' the shadow has an angle, corresponding to that between b' c' and c'f. Its direction above x' is most easily determined thus: Assume some point, as z', whose shadow is likely to fall on the top step; find its shadow Zj, and through it draw the line o q par- allel to the line casting the shadow; then project o from the front edge of the upper step up to m'n', the v. p. of the same edge, and there join with x'. 584. The shadow of a cylindri- cal abacus upon a similarly shaped column. Let MBN (Fig. 376) be the plan of the cylindrical column, and ab G that of the abacus. The vertical plane of rays yz gives, by its tangency at t, the element of shade t'r' on the abacus. A simi- lar tangent plane of rays b x gives the element of shade px' on the column. It contains bb', that point of the abacus which casts the last shadow, x', on the column. Any other vertical plane of rays, as d, will cut a point dd' from the abacus, and an element from the column, the latter catching the ray from the former at q. The point s^ig-. st-©. last found is *' necessarily the highest in the shadow, as it lies in that plane of rays which contains the axis; in other words, the meridian plane of rays. 585. The shadow of a rectangular block resting on a vertical semi -cylinder. Let mnop (Fig. 377) be the plan of a block whose front coincides with the section -lined surfaces of the cylinder. The edge a't' will cast an elliptical .shadow a"e"t', which is the intersec- tion of the inner cylindrical surface by the plane of rays through a't'. To find it regard ac, be, xy as traces of vertical planes of rays, as in the last problem; draw rays SHADES AND SHADOWS. 223 a ray back from a, Fie-. 377. The shadow on H. Fig. 37-3. t' a' b' a'a", b'e", etc., through the points casting shadows, and note where each ray meets the element lying in the same plane with it. The shadow a"ai is cast by an equal length of a'k', found by drawing The remainder of a'k' would cast the shadow ac upon any horizontal plane on which the object might be regarded as resting. 586. The shade and shadow of a vertical, inverted, hollow cone. In Pig. 378 let a's'f, abf, represent the cone. The lines casting shadows will be the elements of shade and a portion of the base. The elements of shade will be the lines of contact of tangent planes of rays. Each of these planes will contain the ray st, s't', through the vertex, and will cut the plane of the base in tangents to the latter; hence from tt — the trace of the ray — ^draw ta and th tangent to the base; then as and hs are the plans of the elements of shade. Of these the latter only is visible in elevation; and the surface b's'f on its right is the visible portion of the shade. The ray t's' has its horizontal trace at s^, one point of the shadow on H. As the arc aedb must cast on H a shadow that is equal and parallel to itself (Art. 578) find Si, the centre of the latter, by the ray from b'{s); then the arc a-^hb,^, limited by tangents from s^, completes the shadow on H. The shadow on the interior. Any secant plane of rays through the vertex, as tsd, will cut a point c from the base on the side toward the light, and an element s d on the opposite side of the cone. The ray through c will then intersect the element at a point m which is on the limiting line aib of the interior shadow. Other points are analogously found, as i from n. The shadow obviously terminates at the tangent points, a and b. 587. Brilliant points. All are familiar with the marked contrast as to brilliance between the various portions of a highly polished surface. The point which seems brightest to the observer is, however, not actually the one receiving the light most directly, but is that from which the incident (direct) ray is reflected directly to the eye, in conformity with the well-known optical law that an incident ray and the same ray as reflected make equal angles with the normal to the reflecting surface. Since in orthographic projection the reflected ray is perpendicular to the paper, we proceed as follows to find a brilliant point: Obtain the bisector of the angle between a ray of light and a perpendicular to the paper; then pass a plane perpendicular to such bisector and tangent to the surface. Its point of contact will be the brilliant point sought. 588. To find the brilliant point on a surface of revolution. In illustration of the principles stated in the last article Fig. 379 is presented, in which one -quarter of a surface of revolution appears, Ms"g' being its meridian curve. The direct ray through G" is a'C",aC. The reflected ray from the same point is CD, C". To find the bisector of the angle whose plan is a CD, carry the ray a' C" into H, when it appears at ftjC, b"C". Ce^ bisects angle BCD, and in space becomes e'C", 224 THEORETICAL AND PRACTICAL GRAPHICS. Flgr, 373. --, /' eC. The vertical meridian plane through Ce may be rotated till parallel to V, when the bisector just mentioned will appear at fC", and the meridian curve will project in Ms"g'. Tangent to the latter and at 90° to f'C" draw op, which represents an edge view of a tangent plane. Its point of contact, s", counter - revolves to s'(s), the brilliant point desired. 589. The brilliant point and the curve of shade on a sphere. In Fig. 380 the sphere is represented by one view only, its elevation, acjb; the vertical plane of projection being understood to contain the centre, B. The brilliant point. A B, parallel to the arrow, is the projection of the ray through the centre, the ray itself making an angle of 35° 16' with the plane of projection (Art. 575). The reflected ray from B is projected in that point. If the plane of the incident and reflected ray be rotated into V about Aj as an axis, the former will appear at R B and the latter at b B. L B ia the bisector of the angle RBb, and D therefore the rabatted brilliant point, E being its true position. Were JK a tangent mirror and R a luminous point, the incident ray RD would be reflected to a point on Bh, at a distance from B ^^s- sso. equal to R B. The curve of shade. This will be the circle of contact of a tangent cylinder of rays. It may be found by points, thus: Take a series of auxiliary planes of rays perpendicular to the paper, as M N, PQ, etc. Each will cut the sphere in a circle which may be seen as such by a 90 "-rotation of the plane; a revolved ray can then be drawn tangent to the circle, and its true position found by counter revolution. In the auxiliary plane MN, for example, we have ikn for half the circle cut from the sphere, and a tangent thereto at y, by a ray parallel to RB, gives a point which, after counter-revolution, appears at d on the curve of shade c e bf The point e is similarly derived from s, and g from x. Parallels to A B and tangent to the spherical contour at \ and b give the extremities of the major axis of the ellipse in which the curve of shade projects. 590. The curve of shade upon a torus. In illustration the architectural torus (Fig. 381) is employed, although the same methods are applicable to the annular torus. The points on the "equator" of the surface, viz., m' and n', are the points of contact of two verti- cal planes of rays, each indicated by a;?/ on the plan. Points on the apparent contour of the elevation are determined by the tangency of planes of rays perpendicular to V, as ts' and that through Q, each at 45° to the horizontal. The highest and lowest points, oo' and uu', which mmt lie in the meiidian plane of rays, N K, are found by revolving the meridian section in that plane about the vertical axis through C, until ' par- allel to V, when it will be projected in the given elevation. The ray AC, A'C, being revolved at the same time, becomes A,C, A"C', the latter then making 35° 16' with H. Parallel to A" C draw tangents to the elevation (only one, P, drawn) and counter -revolve the contact points into the first position of the plane NN. They appear at o and u, from which the elevations are derived. SHADES AND SHADOWS. 225 Points in any assumed meridian plane. Let D Ce he any meridian plane. Project the ray A C, A'C upon it at CB; rotate the plane to A^F, when E-igr. sei. A goes from B to b and thence projects to the level of A', gi^ang A'" C for the revolved trace upon D C B of a plane of rays perpendicular thereto. Tangents, as the one through R, parallel to A"'C', give the level of the points of shade in the meridian plane selected, and, after projection upon A^F, counter -revolve to D C B as at e, whence e'. Points in the meridian profile plane L X, as s" and Z, are at the same level as those on the apparent contour, owing to the equality of the angles A^CA and A C X. 591. The shadow on the interior of a niche, cast by its ovm outlines. The figure (382) shows the plan ABC, and elevation, A'R'C, of the surface, which may be defined as a vertical semi - cylinder, capped by a quarter sphere. The direction of the rays being given by sa, s'a',, a series of vertical planes of rays are passed. One of these is ee^, containing point e', which casts a shadow, and cutting from the cylinder an element which catches the ray from e' at e". The plane of rays mn contains a point oo', whose shadow is received by the spherical interior, but which involves no difficulty in its determination, as the ray through o and the circle catching the ray may be shown in their true relation thus: Project upon mn at q; then qn is the radius of the small circle cut from the top, and n o^m is the revolved position of the circle. Make nr^ equal to n'r', and draw o^r^ for the revolved ray through o. It cuts the circle n OiM at a point which counter -revolves to b, and thence projects upon o'n' at b'. The tangency of a plane of rays (MM) perpendicular to V gives the point x at which the shadow begins. The point p", at which the curve leaves the spherical part, can be exactly determined by a special construction, but is located with sufficient accuracy if enough points have been obtained on either side, by the previous method, for the drawing of a fair curve. 592. The curve of shade on a warped surface. This is . most readily determined by connecting the points of contact of tangent planes of rays, applying the principle that any plane containing an ""- --''' element of a warped surface is at some point a tangent plane to the surface and if also a plane of rays its traces will contain those of any ray intersecting the element. w\--,^ 226 THEORETICAL AND PRACTICAL GRAPHICS. E'igr- ses. 593. The shadow of any line upon a warped surface. Various methods may be employed, among them the following: (a) Pass a plane of rays so as to cut the line casting the shadow in a point; through the point draw a ray to meet the curve cut from the surface by the plane. (b) A plane of rays, containing an element on which the shadow falls, will cut the line casting the shadow at the point whose shadow falls on the element selected. (c) If two lines cast intersecting shadows upon a plane, a ray drawn back from such intersection will meet the line that is nearest the plane in the shadow cast upon it by the point where the same ray meets the more distant line. 594. The shades and shadows of warped helicoidal surfaces. Illustrating with the triangular -threaded screw, whose surfaces are helicoidal, let a'Q and a'e (Fig. 383) be the generatrices of the upper and lower surfaces, respectively, the point a' generating the outer helix a'is(P, while e and Q generate inner d helices of the same pitch as the outer. (Art. 478). 695. The shadow of the outer helix, P D K', on the surface of the thread below. Assuming Rw, R'T', for the direction of light, a vertical plane of rays Rjw will cut the helix in a point R, R', and the heli- coid in a curve m'Z8, the latter catching the ray from R' in the shadow, T', cast by it. The curve u'ZS is found by projecting u, j, w and 3, which are the plans of the intersections of the plane with various elements, up to the elevations of the same elements. For the elevation of j, which falls on element cd, the points d and j are carried to h and j,; h then projects to k', whence D k' for the revolved elevation of element c d. Upon it j^ appears at j', which in counter-revolution returns to Z. Vertical planes of rays parallel to R w are shown in ah, b g, If, and with each the process just described is repeated. 596. Points of a curve of shade, by means of a declivity cone. The direction of light with which we have been dealing so far in the case of this screw puts the entire under -surface of the thread in the shade; but were any portion illuminated, as would occur with light as indicated by tc t's' a curve of shade would have to be determined, point by point, one method for which is as foUows: Obtain a tangent plane to the helicoid at some point of any helix. This will be determined by an element and a tangent to the helix. (Art. 478). For the outer helix a'r' the tangent plane at a' a would be xea', as e is the h. t. of the element a's', while x is the h.t. of the tangent at a' (found by making a x equal to the rectified arc a d h). This plane cuts the axis at s', but is evidently not a plane of rays, since the H- trace, t, of the ray through s', does not fall on that of the plane. Since, however, all planes that are tangent to a helicoid at points on the same helix are equally inclined to H, we may find the point of contact of a tangent plane of rays by generating a cone with a line of declivity of the plane just found, passing a plane of rays tangent to said cone, and then finding the parallel plane of rays that is tangent to the helicoidal surface. SHADES AND SHADOWS. 227 The base of the " declivity cone "is on q, of radius o c, the plan of that line of declivity which lies in a plane with the axis. In aco we see the plan of the constant angle between elements and lines of declivity in the series of planes tangent along the particular helix in question. The ray through the vertex s' of this cone has t for its trace; tn is therefore the trace of a plane of rays tangent to the cone, and en the plan of its line of declivity. A parallel plane of rays, tangent to the outer helix, would then contain an element of the helicoid which would be projected as much to the left of en as ac is to the left of co; that is, angle mem is made equal to oca, and in(m') would be that point of the outer helix which belonged to the curve of shade. A similar process for the inner and any intermediate helices would give points of a curve of shade whose shadow could be found by either of the methods given in Art. 693. 597. When any two surfaces intersect, the shadows cast by either on the other may be found by applying the general principles of Art. ,593, care being taken to so avail one's self of known properties of the surfaces as to simplify the construction as much as possible. 228 THEORETICAL AND PRACTICAL GRAPHICS. CHAPTER XIV. DEFINITIONS AND PRINCIPLES. —AKCHITECTUEAL PBKSPECTIVB FOE EXTERIORS. — PERSPECTIVE OF SHADOWS. — PERSPECTIVE OF INTERIORS BY THE METHOD OF SCALES. 598. A drawing is said to be in perspective when its lines correctly represent those of a given object as it would appear from a point of view located at a given finite distance from both it and the plane upon which the drawing is made. If the representation is not only correct geometrically but is also shaded and colored, it is said to be in aerial perspective; otherwise it is simply a linear perspective. The construction of the latter is obviously a preliminary to all artistic work in oils or water colors. Perspective plane. The plane on which the drawing is made is called the picture plane or per- spective plane, and is always understood to be vertical; it will therefore be frequently denoted by the same letter (V) heretofore employed for the vertical plane of projection. It is usually taken between the eye and the object, in order that the perspective may be smaller than the object itself. 599. The general principles and definitions may be illustrated by Fig. 384, which is a pictorial representation of the various elements involved. The picture plane is the vertical surface BZRK, later transferred to X Z"R"Y. Point of sight. — Visual ray. — Visual plane. S is the supposed position of the eye, and is variously termed point of sight, perspective centre and centre of the pic- ture. Any line through S is called a visual ray, and any plane containing it a visual plane. 600. ABCDEG (Fig. 384) is a rectangular block whose perspective is to be constructed. It is so placed that one of its faces — A BCD — is in the perspective plane, making that face its own perspective. Visual rays, S F, S E, S H, inter- sect the plane V at points /, e, h, which are the perspectives of the original points. Joined with A, D and they give— with A D CB— the perspective of that part of the block which is visible from S. To find the trace, /, of ray S F, a. vertical visual plane may be taken through the ray. Gs is the horizontal trace of such a plane, and om its vertical trace. The ray meets the latter at /. Similarly for other rays. Other methods are given in later articles. The figure illustrates the fact that the perspective of a vertical line is always vertical; for the verti- cal visual plane through EH must cut a vertical plane V in a vertical line, part of which is the perspective eh of the original. LINEAR PERSPECTIVE. — GENERAL DEFINITIONS. 229 601. Horizon. The point of sight, S, is projected upon the picture plane at s'. A horizontal visual plane will cut the perspective plane in a horizontal line through s', called the horizon. 602. Vanishing points. The convergence, in a drawing or photograph, of lines representing others known to be parallel on the original object, is a familiar phenomenon. To determine the point of convergence or vanishing point of any set of parallels, we have only to obtain the trace on V of a visual ray drawn parallel to the system of lines; for such trace on V may evidently be regarded as exactly covering the point at infinity at which we may conceive the set of parallels as meeting. The vanishing point of a line is one point of its perspective. The horizon is the locus of the vanishing points of all horizontal lines. 603. Vanishing point of perpendiculars. A perpendicular is a line at 90° to V. A visual ray parallel to a perpendicular must obviously be the projecting ray through the eye; and s', therefore, the vanishing point of perpendiculars. 604. Diagonals and their vanishing points. Horizontal lines making with V an angle of 46° are called diagonals. Sw' and Sw", the diagonals through the eye, meet the horizon at the vanishing points of diagonals, w' and w", also known as points of distance, since they are as far from the pro- jection of the eye as the space -position of the latter is from V. 605. Lines parallel to the perspective plane have their vanishing points at infinity; or, in other words, the lines and their perspective representations are parallel. This is illustrated hj EH and eh in Fig. 384. 606. Perspective by trace and vanishing point. Since any point in the perspective plane is its own perspective, we may obtain the indefinite perspective of any line, as FA, by joining A — its trace on V — with its vanishing point, the latter being s' in this case, as the line mentioned is a perpendicu- lar. The visual ray SF then intersects the indefinite perspective A s' at /, when Af is readily seen to be the definite perspective of A F. (For application see Art. 612). 607. Perspective by .diagonals and perpendiculars. The student has already discovered, without formal statement of the principle, that a point is determined in perspective by its being the inter- section of the perspective of two lines passing through the original point. "We saw in the last article that the perspective of F was obtained as the intersection of a ray SF with a line As\ which might be regarded either as the trace of a visual plane or as found by joining trace of line with vanishing point. Obviously any pair of lines may be drawn through a point, and the inter- section of their perspectives noted; but the auxiliaries which, on account of their convenience, are most frequently used in perspectives of interiors, are the diagonals and perpendiculars already defined. In the figure, FD is the diagonal of the square top of the block; Sw' is parallel to F D, and w' therefore the vanishing point of diagonals, to use in getting the perspective {Dw') of said diago- nal. This intersects A s' (perspective of perpendicular A F) at /. 608. Ha-^dng illustrated pictorially the principles most employed in linear perspective, we have next to show how they are applied to the orthographic projections which are usually all that the draughtsman has, with which to start his constructions. Obviously, the perspective plane cannot be rotated backward into coincidence with the paper on which the object is represented in plan, without the latter drawing being in most cases overlapped by the perspective representation. The usual — probably because the most natural — way to avoid this difficulty, is to imagine the plane V trans- ferred forward to some position Z"XYR", where, if rotated into the paper about its trace XF, the perspective will clear the auxiliary views. 609. In Fig. 385 the method just described is illustrated as applied to the block of Fig. 884, and both figures may be referred to in the following description: 230 THEORETICAL AND PRACTICAL GRAPHICS. E'lg-. see. Fig-. 3SS. Horizon '--/,-—- 's K B K is the trace (and orthographic representation) of the entire perspective plane BZRK, and XY its transferred position, upon which the elevation of the object is drawn {A'B'C'D'). The plan ADEF is drawn back of PK, and in the same relation to it as the object to V. Ts is the same in each figure. The horizon is located in relation to the ground line XF in Pig. 385, at a distance from it equal to Ts' (or Ss) in Fig. 384. The vanishing point of diagonals, w', is at a dis- tance from s' equal to s T. We have now merely to apply- either of the methods previously de- scribed, thus: (a) Es (D's') is the visual ray through E (the latter being projected on V at D'). Erh is the vertical visual plane through this ray; rA is its vertical trace, and e is the trace of the visual ray and therefore the perspective sought. (b) D' is the trace of ED, and D' s' is its indefinite perspective, drawn to the vanishing point of perpendiculars. Ray sE meets the trace Bk at r, whence e for the perspective desired. (c) Wanting the perspective of F, we may draw through it the perpendicular FA and the diagonal FD. These, being on the top of the block, meet V at the level A'D'. A's' is therefore the perspective of the perpendicular, D'w' that of the diagonal, and their intersection / the point desired. 610. The method by inverted plan. In Fig. 386 the same perspective as in the preceding figures is obtained by assuming that the object has been rotated 180° about B K, so that it appears, inverted, in front of the perspective plane. Diagonals and perpendiculars then give the same result as before. Any diagonal, as G C, being inverted, is drawn in perspective in its true direction, D'lo'. , ^' ^^"^-^ This method is quite convenient when w ,.--" dealing with plane figures; but for large and complicated objects the conception of inversion is confusing, and renders it far inferior to the other. To show, however, its serviceability in the field indicated, as also the device of circumscribing polygons — usually resorted to in case of curves — Fig. 387 is given. 611. The perspective of a eirele. In Fig. 3S7 the circle is circumscribed by an octagon. Through the various points of circle and octagon diagonals and perpendiculars are drawn. These vanish, in perspective, at the points w', tr" and s", and by their intersections either give points of the circle directly, or lines to which the perspective of the circle can be sketched in tangentiallv. ^^ .-;-'■"'''/ ARCHITECTURAL- PERSPECTIVE METHODS. 231 The perspective of a circle will be a circle only when it is parallel to V, or when the visual cone to its points is cut by V in a sub -contrary section. In all other cases it is an ellipse, when the circle is on the opposite side of V from the eye. 612. Perspective by Trace and Vanishing Point, with special reference to its application to architectural constructions {exteriors). In further illustration of the method of Case (b) of Art. 609, which is generally used in drawing the perspective of the exteriors of residences and other architectural constructions, Fig. 388 is pre- sented. The object dealt with has not only horizontal and vertical lines but also edges inclined at Fig-. 3SS. various angles to a horizontal plane, so as to illustrate the method of dealing with any direction of line that can occur in a house perspective. The only reason for not presenting the plans and elevations of some actual building, is the impossibility of reducing such a series of views to the neces- sary limits of our illustration without sacrificing clearness as to the constructions made therewith; but if for the given elevations and plan there were substituted the elevations of a house, together with a general roof and wall plan of that part of the house which is visible from the point of view selected, the procedure from that point would be identical with that shown above. 232 THEORETICAL AND PRACTICAL GRAPHICS. The central view gives the best idea of what the object is like, a block devised simply with reference to compact illustration of the various principles involved. Lines of height. Vertical planes through the edges of the plan, as Diy, K^Z, X Y, are first drawn, and their traces shown on the (transferred) perspective plane in the verticals yx, 9-10, Zz, etc. These, in architect's parlance, are lines of height, since upon each is laid off the true height of lines in the plane it represents. These heights are most conveniently located by projecting over directly from one or the other of the elevations. Thus, the height of K^ C^ is seen at on the front elevation, and at C" on the other. Projecting from the latter, we have z as the height at which K^L^ would meet the perspective plane; its trace, therefore, to use as in Art. 606. Similarly, 6 5i produced gives OU for its line of height, and BA cuts it at the trace U. PiQi gives the trace 9-10, upon which PQ projects, giving 10 for the height of both P and Q. 613. The vanishing points. With s^ as the plan of the eye in its relation to the original position of V, draw the horizontal line s^I parallel to the longer lines of the plan, and project I to T" on the horizon, for the vanishing point of that set. Similarly draw Sit^, parallel to those hori- zontal lines of the object that are perpendicular to the first set, getting T' for the other vanishing point of horizontal lines. Were there horizontal lines in other directions on the object, their vanish- ing points would have to be determined by an analogous process. The vanishing point of lines making angle 6 (see side elevation) with the horizontal is at i', found thus: Draw s^I parallel to the plans of the lines whose inclination is 0. At I draw a perpendicu- lar to s^I and prolong it to meet, at i, a line Sji making 6 (at s^') with s^I. Make T"i' equal to li, when i' is recognized as the trace of a visual ray parallel to the lines whose inclination was given; for when the triangle s^il is rotated upon its base Sj/ until vertical, and then placed with said base at the level of the eye, we would evidently find i at i'. The vanishing point of Iwies inclined 0° to H is found by duplicating the last procedure in every detail, s^t^t^ being then the triangle whose altitude t.^t^ is laid off vertically from T', and toward which 8' 2' and 5' 6' converge. The remaining construction is as follows: Vertical visual planes are drawn through s^ and all points of the object. To avoid complicating the lines only a few of these are shown, s^G^, s^P^, s^F^, s,J. Taking K^P^s^ as illustrative of all, we draw its vertical trace kkp. The line of heights for the point P being 9-10, project P upon the latter at 10; then 10-T' is the perspective of Q-P^, and its intersection p with the vertical kk is the perspective sought. K^, in the same visual plane, has its height projected from K to z, upon the line of heights through Z; then z T' gives k. The perspectives of all the other points might be similarly found; but with two or three points thus obtained we may find the various edges by means of the vanishing points, thus: Starting with e, for example, prolong i'e to meet at / the trace of visual plane s,F,; then fT', stopping at g on trace s^G,. As gk and fe have the same vanishing i)oint, we find k as the intersection of ^t' and z T'. Then T'k prolonged gives I and c on traces of visual planes (not drawn) through i, and C, . The point d beino; found independently, we join it with c for the edge c d, for which we might also find a vanishing point thus: Obtain the base angle of a right triangle of base C,D„ and altitude equal to height of C above R; then use this angle (which we may call jS) and the direc- tion DjCi exactly as 6 and F^J were used in the construction giving vanishing ])oint i'. 614. Perspective of Shadows. These might be obtained from their orthographic projections in e^-ery case, but usually a shorter method is employed. Both ways are illustrated in Fig. 3S9. The object whose perspective and shadows are to be constructed is a hollow rectangular block, PERSPECTIVE OF SHADOWS. 233 X'igr. 3SS- _,,,y,-^ R whose plan is fb c d, and whose height is seen at A B. The corner h being in the perspective plane, we have va A B the perspective of the front edge. The vanishing points R and L having been found from s, as in the last problem, draw AR and B R, and terminate them on the trace n g of the visual plane s m. Similarly, terminate A L and B L upon the trace d h of the vertical visual plane s/. Then FR and CL give the rear corner E, etc. The shadow. Let c'm' be the orthographic elevation of the edge whose plan is c. Then if a ray of light through c (c') has the projections ca;,, c' X, we shall have x^ for the shadow of cc'. In the same way the shadow might be completed in orthographic pro- jection, a portion only, being, however, actually indicated. Then, treating x^ like any other point whose perspective is desired, we would find r — the vanishing point of hori- zontal lines parallel to mx^, and draw Dr for the perspec- tive of the plan of a ray; then X, the intersection of Dr with the trace ox of the vertical visual plane sx^, is the perspective of the shadow of C OE being horizontal, its shadow on H is in reality parallel to it, and, perspectively, has the same vanishing point; hence draw from x toward L to complete the visible portion of the shadow. CD being a vertical line, has its shadow Dx in the direction of the projection of rays on H. 615. The perspectives of shadows, without preliminary construction of their orthographic projections, are thus obtained: In Fig. 389, with c' X and cx^ as the orthographic projections of a ray, draw s'r', s t, for the parallel visual ray, when r' is seen to be the vanishing point of rays. Then r is obviously the vanishing point of horizontal projections of rays; and for shadows on horizontal planes the two points thus found are sufficient. For the shadow of we have merely to take the direct ray Or', and the plan Dr of the same ray, and note their intersection, x. For shadows on a set of parallel planes that are not horizontal, we would replace r by the vanishing point of projections of rays on the planes in question. 616. Perspective by the method of scales, (a) In Fig. 890 let s be the point of sight, mn the horizon, and m and n vanishing points of diagonals. Attention is called again, by way of review, to the fact that the real position of the eye in front of the perspective plane is shown by either ms or ns, and that all horizontal lines inclined 45° to V will converge to m or n. 234 THEORETICAL AND PRACTICAL GRAPHICS. We now apply the properties of the 45 "-triangle thus: To cut ofi' any distance, perspectively, upon a perpendicular to V, lay off the same distance parallel to V and draw a diagonal. If the room is to be twenty -two feet deep, make Ch equal to that number of units and draw the diagonal h n, cutting off the perpendicular from C at i, making Ci the perspective of the given depth. The rectangle A BCD having been laid off in the perspective plane, from given dimensions and to the same scale, draw from A, D and B toward s, terminating these perpendiculars on a rectangle obtained by drawing id and ij, then j/ and df parallel to the corresponding sides of the larger rectangle. (b) Reduced vanishing points. In case the point of sight has been taken at such a distance from the perspective plane as to throw m and n beyond convenient working limits, we may get the same r''_ -;-:-4=""*^^fNS^^— 1 \V>13 "'■■■-\ii3 c e g result by bisecting or trisecting sn and taking the same proportion of the distance to be laid off. Thus the point i might be obtained by bisecting Ch and drawing a line to the middle of s n. VCe might equally well lay off from C toward h any other fraction of Ch, and draw thence to a point on the horizon whose distance from s was the same fraction of sn. (c) The perspective of the steps. Let it be required to draw a flight of three steps leading to a doorway in the left wall. If the lowest step is to be six feet from the front of the room, make D a equal six feet and draw a m, cutting £> d at a corner of the step in question. If the steps are to be three feet wide, make D h equal to three units, and draw b s for the trace of the vertical plane of the sides of the steps. Making ac three feet, draw cm, getting point 9, which should be even with the first corner found. PERSPECTIVE BY THE METHOD OF SCALES. 235 The widths of the steps being laid off from c at e and g, and their heights at 1, 2, 3 on D A^ their perspective is completed by a process which should need no further description. (d) The doorway in the left wall. Assuming this to be the same width as the landing, which — as seen at gr — \B e-v-idently four feet; and also that the walls of the hallway are in the planes of the front and back of the landing, draw vertical Hues from the left-hand corners of the latter, terminating them by a perpendicular Ps drawn from a point P whose height (ten units) is that of the top of the doorway. 3-4 shows the height of step from landing to hallway. The pm-spective of the door is obtained in this case on the supposition that it is open at an angle of 54°, for which a vanishing point (not shown) lies on sm prolonged, and from which a hne J'v gives the direction vJ, and similarly H' H for the top of the door. To find / we may draw D^ at 54° to a vertical Hne (the 45°- angle indicated is an error, should be 36°) so as to represent a four- foot door swung through the proper arc, when by project- ing up DK to 4-i, which is the level of the bottom of the door, a perpendicular Ls wiU cut J'v at J. Then a vertical line from / will cut the line H' at H. ■\Yere the door actually open 45°, the edge Jv would pass through m. The hallway on the right has its corner 7 at a distance of thirteen feet from C, and is seven feet high. The width of the passage may be ascertained by the student. The method of getting the perspective of a door by means of an auxiliary circle is shown in Fig. 391. (e) The location of the light, I. To locate the light five feet from the right wall, move five units from B, to E, when Es will be the trace, on the ceiling, of the vertical plane containing the light. If the light is to be five feet below the ceiling, mark off five units down from E, when Gs will be a horizontal line giving the level of I. Finally, to have the light a definite distance back, say eighteen feet, make Bo eighteen units on the front edge of the ceiling; draw on and get t, when txl^y will be a plane at the required depth, and its intersection I with Gs will be the position of the light. (f) The shadows. As in any other shadow construction, we have to note, in any case, where a direct ray through a point meets the projection of the same ray. All horizontal projections of rays will pass through Zj, which is the projection of the light on the floor. For the triangular block FM, we take a direct ray Z8, through any point of the edge casting the shadow, and Z,8 for the projection of the same ray; then 8 is the shadow of the point selected. At F, where the edge meets the floor, the shadow begins, hence P-8 is the direction and FN the extent of the shadow cast on the floor, and, obviously, NM that received by the side wall. (g) The shadow of the door. JHll^ is a vertical plane of rays containing JH. It cuts the left wall in a line Wz, found by continuing the trace from l^ to meet D d, erecting therefrom a vertical line and cutting it at TT by a ray from I through H. The shadow of Jv on the landing has the same vanishing point as Jv. When it meets the side wall it joins with v, since there the line casting the shadow meets the surface receiving it. The shadow of J is at the intersection of ray I J with the- trace (not drawn) of the plane of rays J HI upon the top of the block. This would be found thus: Where the h. t. of said plane cuts the edge 9-10 draw a vertical line, and from the intersection of the latter with the top edge of the landing draw a line to the point below I on the line 14 -s. This will give the direction of the shadow of HJ on the landing, since the line 14 -s is at the level of the top of the landing. (h) The shadows of the steps. The plane yEGl has the trace l^p on the floor; and if on the vertical pq we lay off distances equal to the heights of the steps and draw vanishing hues to s, these will cut 11^ at points which may be regarded as the projections of the light upon the planes 236 THEORETICAL AND PRACTICAL GRAPHICS. of the tops of the steps, and should be used in getting the directions of the shadows of vertical lines upon said tops. The direction of the shadow of the vertical edge at 9 is given by Zi9, which is made definite as to length by a ray from I through its upper extremity. The shadow on the floor is then parallel to the front edge till it meets the side wall, where it joins with the end of the line casting the shadow. The vertical edges of the second and third steps would cast shadows whose directions would be found by means of the points above l^ on 12-s and 13-s, and which would run obliquely across the tops, instead of covering them entirely, as shown, incorrectly, by the engraver. 617. The perspective of a right lunette, the intersection of two semi- cylindrical arches of unequal heights. Let AMB be the front of one of the arches and DNC the opposite end, at a distance back which may be found by drawing from the vanishing point of diagonals, T, a line TD to meet A B pro- duced, giving a point whose distance from A is that sought. Let the smaller passage be at a distance A X back of A, and equal to X F in width. Continue the vertical plane on AD to the level eS oi the highest element of the smaller arch, and in that plane construct Pmno — the perspective of half of a square whose sides equal X Y. In this draw the perspective of a semicircle Phgo. At ee', dd', etc., we see the amounts by which the elements of the side cylinder extend past the plane A enF to their intersection with the main arch, and these in perspective are ordinates of the curve o'e"g'. For any one, as bb', draw bS cutting the semicircle Pko at o and g. Horizontals through these points will be those elements of ^'ig-. 3si. the smaller cylinder that lie in the hori- zontal plane bb' S; and the perpendicular b' S cuts them at the points a' and g' of the intersection. 618. The perspective of a door, found by means of an auxiliary circle in perspective. Let QP, Pig. 391, be, perspectively, the width of the given door. Construct PKO, the perspective of the circle that P would describe as the door opened. If Q P were to swing to Q8, the prolongation of the latter would give 1 on the horizon for its vanishing point, which joins with / for the direction of the top edge, the latter being then limited at 2 by a vertical through 3. Similarly, KQ prolonged to the horizon, gives a vanishing point from which a line through / gives the top edge //. 619. The perspective of a groined arch. If the axes of two equal cylinders intersect, the cylinders themselves will intersect in plane curves, eUipses. In the case of arches intersecting under these con- ditions the curves are called groins, and the arches groined arches, when that part of each cylinder is constructed which is exterior to the other, as in Pigs. 392 and 393. Were G joined with M in Pig. 392 and the line then moved up on the groin curves Go and Mg to h, it would generate between those curves one -quarter of the surface of a cloistered arch, but the curves would stiU be called groins. Pig. 392 represents in oblique projection that portion of the structure which is seen in perspec- tive above the piUars in Pig. 393. THE PERSPECTIVE OF A GROINED ARCH. 237 The pillars are supposed to be square, and to stand at the corners of a square floor set with square tiles. Each pillar rests on a square pedestal and is capped by an abacus of the same size except as to thickness. Taking the perspective plane coincident with the faces of the pedestals and abaci, make ah It, Fig. 393, of any assumed size; prolong bl until Iv s-ig-. sss. equals the height assigned to the pillar; then complete the front of the abacus on qv as an edge, to given data. Locate on tl a, point whose distance from I equals that of the front face of pillar from the perspective plane, and draw therefrom the perpendicular hGs; also draw the diagonal I h d, giving h for a starting corner on the base of pillar. A parallel to 1 1 through h is cut by the diagonal tEd at E. The same diagonal gives G and two points on the diagonally -opposite pillar, corresponding to E and G. The vertical line through h is cut by a diagonal from the t;- comer of the abacus at the point where that edge meets the abacus, and the completion of the perspective of the top is identical with that just described for its base. The prolongation of Aw meets a diagonal from q at the point where the front semicircle begins on the top of the abacus. Joining it with the corresponding point on the other abacus and bisecting such hne gives the centre of the front curve, which may be drawn with the compasses, as the circle is parallel to the perspective plane. Similarly for the back semicircle. The perspectives of the groins and side semicircles. As the cylinders on which these curves lie are either parallel to or perpendicular to the paper we may refer to them as the parallel and perpen- dicular cylinders, respectively. The elements of the latter will converge in perspective to the point of sight, as TO c and y Le, Fig. 393. On the other cylinder they will be parallel in perspective. A horizontal plane of section, as that through a b (Fig. 392) or B T (Fig. 393) will cut a square o6c)i from the outside of the structure, and two elements from each cylinder. Either diagonal of this square, as a c (Fig. 392) will cut the elements in points of the groin. In Fig. 393 the diago- nal Bd cuts the elements ms and y s &i c and e, two points of the groin. These points also belong to elements of the parallel cylinder, and the latter, if drawn, will meet the side walls in points of the side semicircles.- This is shown in Fig. 392 by drawing the element fi to meet the trace be at i. In perspective this is seen in the horizontal element through e (Fig. 393) which meets the perspective perpendicular i?s at point / of the side curve. A number of planes should be treated like B T to give enough points for the accurate drawing of the curves. The tiled floor is made of squares whose diagonal is seen in true size at gj. By laying off the latter on a and drawing diagonals to the vanishing points d, the floor is rapidly laid out. The shadow of the left-hand front pillar and of its abacus. Assume r^ and r for the vanishing point of rays and of their horizontal prelections, respectively, remembering that these points must be on the same vertical hne, since a ray and its plan determine a vertical plane, which can intersect another vertical plane only in a vertical line. The ray from I to r^ meets its projection 6 r at the shadow of I on the floor, whence a perpen- dicular to s would give the direction of the shadow of Is. Join h with r; it runs off the pedestal at s, whence a ray to r^ will give the shadow s" on the floor, from which s"r is one boundary of 238 THEORETICAL AND PRACTICAL GRAPHICS. the shadow of the pillar. This meets the further pedestal, upon whose front the shadow runs up vertically, being the shadow of a vertical line. On the top of the pedestal the shadow continues ^Lg*- 3S3. toward r till it meets the face of the pillar, where it again runs vertically until merged in the shadow of the aljacus q r. The shallow of the abacus qv on the diagonally -opposite pillar consists of a horizontal shadow cast by a small portion of the lower front edge; a vertical portion cast by ? < \ "-v n" t: \ 71 ' o' 1 /i\ f k i' / r r A / h 1 / s 3W, c f4 ^11 / : Jc I 4 b \ 246 THEORETICAL AND PRACTICAL GRAPHICS. HORIZONTAL PROJECTION OR ONE -PLANE DESCRIPTIVE GEOMETRY. 637. One- Plane Descriptive Geometry or Horizontal Projection is a method of using orthographic projections with but one plane, the fundamental principle being that the space -position of a point is known if we have its projection on a plane and also know its distance from the plane. Thus, in Fig. 402, a with the subscript 7 shows that there is a point A, vertically above a and Figr- -ioa. at seven units distance from it. The significance of 63 is then evident, and to show the line in its true length and inclination we have merely to erect perpen- diculars a A and Bh, of seven and three units respectively, join their extremities, and see the line A B in true length and inclination. In this system the horizontal plane alone is used; One- plane Descriptive is «^ ^''» therefore applied only to constructions in which the lines are mainly or entirely horizontal, as in the mapping of small topographical or hydrographical surveys, in which the curva- ture of the earth is neglected; also in drawing fortifications, canals, etc. The plane of projection, usually called the datum, or reference plane, is taken, ordinarily, below all the points that are to be projected, although when mapping the bed of a stream or other body of water it is generally taken at the water line, in which case the numbers, called indices or refer- ences, show depths. 638. A horizontal line evidently needs but one index. This is illustrated in mapping contour lines, which represent sections of the earth's surface by a series of equidistant horizontal planes. !Figr- -5:03- +. HILL CONTOUR ^ In Fig. 408 the curves indicate such a series of sections made by planes one yard, metre or other unit apart, the larger curve being assumed to lie in the reference or datum plane, and there- fore having the index zero. The profile of a section made by any vertical plane MN would be found by laying off — to any assumed scale for vertical distances — ordinates from the points where the plane cuts ^^-e- ^o-i. the contours, giving each ordinate the same number of units as are in the index of the curve from which it starts. Such a section is shown in the shaded portion on the left, on a ground line PQ, which represents MN transferred. 639. The steepness of a plane or surface is called its slope or declivity. A line of slope is the steepest that can be drawn on the surface. A scale of slope is obtained by graduating the plan of a line of slope so that each unit on the scale is the projection of the unit's length on the original line. Thus, in Fig. 404, if m w and B are horizontal lines in a plane, one hav- ing the index 4 and the other 9, the point B is evidently five units above A, and the five equal divisions between it and A are the projections of those units. HORIZONTAL PROJECTION OR ONE-PLANE DESCRIPTIVE. 247 The scale of slope is often used as a ground line upon which to get an edge view of the plane. Thus, if B B' is at 90° to B A, and its length five units, then B' A is the plane, and <^ is its inclination. The scale of slope is always made with a double line, the heavier of the two being on the left, ascending the plane. As no exhaustive treatment of this topic is proposed here, or, in fact, necessary, in view of the simplicity of most of the practical applications and the self-evident character of the solutions, only two or three typical problems are presented. 640. To find the intersection of a line and -plane. Let a 15 & 30 be the line, and XY the plane. Draw horizontal lines in the plane at the levels of the indexed points. These, through 15 and 30 on X Y, meet horizontal lines through a and 6 at e and d; ed is then the line of intersection of X F and a plane containing ab; hence c is the intersection of the latter with X Y. The same point c would have resulted if the lines a e and b d had been drawn in any other direction while still remaining parallel. E-lir. -405. Fiir- -3=os. 641. To obtain the line of inter- section of two planes, draw two hori- zontals in each, at the same level, and join their points of intersection. In Fig. 406 we have m n and qn as horizontals at level 15, one in each plane. Similarly, xy and y s are horizontals at level 30. The planes intersect in y n. Were the scales of slope parallel, the planes would intersect in a horizontal line, one point of which could be found by introducing a third plane, oblique to the given planes, and getting its intersection with each, then noting where these lines of intersection met. 642. To find the section of a hill by a plane of given slope. Draw, as in the problem of Art. 640, horizontal lines in the plane, and find their intersections with contours at the same level. Thus, in Fig. 403, the plane X Y cuts the hill in the shaded section nearest it, whose outlines pass through the points of intersection of horizontals 10, 20, 30 of the plane, with the like -numbered contour lines. 248 THEORETICAL AND PRACTICAL GRAPHICS. CHAPTEB, XVI. OBLIQUE OR CLINOGRAPHIC PROJECTION. — CAVALIER PERSPECTIVE. — CABINET PROJECTION. MILITARY PERSPECTIVE. E'ig-. "iOT. 643. If a figure be projected upon a plane by a system of parallel lines that are oblique to the plane, the resulting figure is called an oblique or clinographic projection, the latter term being more frequently employed in the applications of this method to crystallography. Shadows of objects in the sunlight are, practically, oblique projections. In Fig. 407, ABnm is a rectangle and mxyn its oblique projection, the parallel projectors Ax and By being inclined to the plane of projection. 644. When the projectors make 45° with the plane this system is known either as Cavalier Perspective, Cabinet Projection or Military Perspective, the plane of projection being vertical in the case of the first two, and horizontal in the last. 645. Cavalier Perspective. — Cabinet Prcjection. — Military Perspective. As just stated, the projectors being inclined at 45° for the system known by the three names above, we note that in this case a line perpendicular to the plane of pro- jection, as ^m or Bn (Fig. 407), wiU have a projection equal to itself. It is, therefore, unnecessary to draw the rays for lines so situated, as the known original lengths can be directly laid out on lines drawn in the assumed direction of projections. Let abcd.n be a cube with one face coinciding with the vertical plane. If the arrow m indi- cates one direction of rays making 45° with V, then the ray hn, parallel to m, will give h as the projection of n, and from what has preceded we should have ch equal to en, and analogously for the remaining edges, giving abcd.i for the cavalier perspective of the cube. Similarly, E K H is a correct projection of the same cube for another direction of projectors, and we may evidently draw the oblique edges in any other direction and still have a cavalier perspec- tive, by making the projected line equal to the original, when the latter is perpendicular to the jilane of projection. 646. Oblique projection of circles. Were a circle inscribed in the back face of the cube DKG (Fig. 407) the projectors through the points of the circle would give an oblique cylinder of rays, whose intersection with the vertical plane DX would be a circle, since parallel planes cut a cylinder in similar sections. We see, therefore, that the oblique projection of a circle is itself circular when the plane of projection is parallel to that of the circle. In any other case the oblique projection of a circle may be found like an isometric projection (see Art. 631), viz., by drawing chords of the circle, and tangents, then representing such auxiliary lines in obhque view and sketching the curve (now an ellipse) through the proper points. Fig. 408 illustrates this in full. OBLIQUE PROJECTION.— CAVALIER PERSPECTIVE. 249 647. Oblique ■projection is even better adapted than isometric to the representation of timber fram- ings, machine and bridge details, and other objects in which ^'"^^ '*°®- straight Hnes — usually in mutually perpendicular directions — predominate, since all angles, curves, etc., lying in planes par- allel to 'the paper, appear of the same form in projection, while the relations of lines perpendicular to the paper are preserved by a simple ratio, ordinarily one of equality. 648. When the rays make with the plane of projection an angle greater than 45°, oblique projections give effects more closely analogous to a true perspective, since the fore- shortening is a closer approximation to that ordinarily exist- ing from a finite point of view. This is illustrated by Fig. 409, in which an object ABDE, known to be 1" thick, has its depth represented as only \" in the second view, instead of full size, as in a cavalier perspective, the front faces being the same size in each. Provided that the scale of reduction were known, abcdhf would answer as well for a working drawing as a 45 "-projection. 649. By way of contrast with an isometric view the timber framing represented by Pig. 398 is Fig-- -ail. ^■ig-. -aio. drawn in cavalier perspective in Fig. 410. Eeference may advantageously be made, at this point, to Figs. 44, 45 and 46, which are oblique views of one form. The keystone of the arch in Fig. 400, whose isometric view is shown in Fig. 401, appears in oblique projection in Fig. 411; the direction of lines not parallel to the axes of the circumscribing prism being found by "offsets" that must be taken in E'lir. -iis. axial directions. 650. Shadows, in oblique projection. As in other pro- jections, the conventional direction for the light is that A|< of the body - diagonal of the oblique cube. The edges to draw in shade lines are obvious on inspection. (Fig. 412). 651. An interesting application of oblique projection, earlier mentioned, is in the drawing of crystals. Fig. 414 illustrates this, in the representation of a form common in fluorite and called the tetrahexahe- dron, bounded by twenty -four planes, each of which fulfills the condition expressed in the formula 250 THEORETICAL AND PRACTICAL GRAPHICS. 00 : n : 1 ; that is, each face is parallel to one axis, cuts another at a unit's distance, and the third at some multiple of the unit. The three axes in this system are equal, and mutually per- pendicular; but their projected lengths are a a', bb', cc'. The direction of projectors which was assumed to give the lengths shown, was that of i?iV in Fig. 413, derived by turning the perpendicular CN through a horizontal angle CNM= 18° 26', and then elevating it through a vertical angle MNS = 9°28'; values that are given by Dana as well adapted to the exhibition of the forms occurring in this system. The axes once established, if we wish to construct on them the form co : 2 : 1, we lay off on A ^^ ^^ N ° R^^ 0^ ^ EDEON" each (extended) one-half its own (projected) length; thus cc" and c' c'" each equal o c' ; bb" equals ob, etc. Then draw in light lines the traces of the various faces on the planes of the axes. Thus a'b" and a"b each represent the trace of a plane cutting the c-axis at infinity, and the other axes at either one or two units distance; the former intercepting the two units on the 6 -axis and the one on the a -axis, while for a"b it is exactly the reverse. Through the intersection of a'b" and a"b' a line is drawn parallel to the c-axis, indefinite in length at first, but determinate later by the intersection with it of other edges similarly found. The student may develop in the same manner the forms oo:3:l; (x>:2: 3- oo:3:4- oo:4:5. THE NOMENCLATURE AND DOUBLE GENERATION OF TROCHOIDS. THE NOMENCLATURE AND DOUBLE GENERATION OF TROCHOIDS. [The anomalies and inadequateness of the pre-existing nomenclature of troohoidal curves led to an attempt on the part of the writer to simplify the matter, and the following paper is, in substance, that presented upon the subject before the American Association for the Advancement of Science, in 1887. Two brief quotations from some of the communications to which it led will indicate the result. From Prof. Francis Eeuleaux, Director of the Royal Polytechnic Institution, Berlin: "/ agree with pleaswe to your discrimination of major, minor and medial hypotrochoids and will in future apply these novel designations," Prom Prof. Eichard A. Proctor, B.A., author of Geometry of Cycloids, etc.: ^'Your system seems complete and satisfactory. I was conscious that my own suggestions were but partially corrective of the manifest anomalies in former nomenclatures." The final outcome of the investigation, as far as technical terms are concerned, appears on page 59, in a tabular arrange- ment suggested by that of Kennedy, and which is both a modification and an extension of his ingenious scheme. The property of double generation of trochoids, when the tracing -point is not on the circumference of the rolling circle, is even at present writing not treated by some authors of advanced text-books who nevertheless emphasize it for the epi-, hypo- and peri - cycloid. This fact, and the importance of the property both in itself and as leading to the solution of a, vexed question, are my main reasons for introducing the paper here in nearly its original length ; although to the student of mathematical tastes the original demonstration presented may prove to be not the least interesting feature of the investigation. The demonstrations alone might have appeared in Chapter V — their rightful setting had this been merely a treatise on plane curves, but they would there have unduly lengthened an already large division of the work, while at that point their especial significance could not, for the same reason, have been sufficiently shown.] That would he an ideal nomenclature in wHcli, from the etymology of the terms chosen, so clear an idea could be obtained of that which is named as to largely anticipate definition, if not, indeed, actually to render it superfluous. This ideal, it need hardly be said, is seldom realized. As a rule we meet with but few self-explanatory terms, and the greater their lack of suggestiveness the greater the need of clear definition. Instances are not wanting of ill -chosen terms and even actual misnomers having become so generally adopted, in spite of an occasional protest, that we can scarcely expect to see them replaced by others more appropriate. Whether this be the case or not, we have a right to expect, especially in the exact sciences, and preeminently in Mathematics, such clearness and comprehensiveness of definition as to make ambiguity impossible. But in this we are frequently disappointed, and notably so in the class of curves we are to con- sider. Toward the close of the seventeenth century the mechanician De la Hire gave the name of Roulette — or roll -traced curve — to the path of a point in the plane of a curve rolling upon any other curve as a base. This suggestive term has been generally adopted, and we may expect its complementary, and equally self - interpreting term, Glissette, to keep it company for all time. By far the most interesting and important roulettes are those traced by points in the plane of a circle rolling upon another circle in the same plane, such curves having valuable practical applications in mechanism, while their geometrical properties have for centuries furnished an attractive field for investigation to mathematicians. The terms Cycloids and Trochoids have been somewhat indiscriminately used as general names for this class of curves. As far as derivation is concerned they are equally appropriate, the former being from /cfeXos, circle, and cfSos, form ; and the latter from rpbxos, wheel, and efSos. Preference has, however, been given to the term Trochoids by several recent writei-s on mathematics or mechanism, among them Prof. K. H. Thurston and Prof. De Volson Wood ; also Prof. A. B. W. Kennedy of England, the translator of Eeuleaux' Theoretische Kinematik, in which these curves figure so largely as cen- troids. Adopting it for the sake of aiding in establishing uniformity in nomenclature I give the following definition: If two circles are tangent, either externally or internally, and while one of them remains fixed the other rolls upon it without sliding, the curve described hy any point on u. radius of the rolling circle, or on a radius produced, will be a Trochoid. Of these curves the most interesting, both historically and for its mathematical properties, is the cycloid, with which all are familiar as the path of a point on the circumference of a circle which rolls upon a straight line, i. e. the circle of infinite radius. The term "cycloid" alone, tor the locus described, is almost universally employed, although it is occasionally qualified by the adjectives right or common. Of almost equally general acceptation, although frequently inappropriate, are the adjectives curtate and prolate, to indicate trochoidal curves traced by points respectively without and within the circumference of the rolling circle (or generator as it will hereafter be termed) whether it roll upon a circle of finite or infinite radius. As distinguished from curtate and prolate forms all the other trochoids are frequently called common. Should the fixed circle (called either the base or director) have an infinite radius, or, in other words, be a straight line, the curtate curve is called by some the curtate cycloid; by others the curtate trochoid; and similarly for the prolate forms. Since uniformity is desirable I have adopted the terms which seem to have in their favor the greater number of the authorities consulted, viz., curtate and prolate trochoid. It should also be further stated here, with reference to this word "trochoid," that it is usually the termination of the name of every curtate and prolate form of trochoidal curve, the termination cycloid indicating that the tracing point is on the circumference of the generator. With the base a straight line the curtate form consists of a series of loops, while the prolate forms are sinuous, like a wave line ; and the same is frequently true when the base is a circle of finite radius ; hence the suggestion of Prof. Clifford that the terms looped and wavy be employed instead of curtate and prolate. But we shall see, as we proceed, that they would not be of universal applicability, and that, except with a straight line director, both curtate and prolate curves may be, in form, looped, wavy, or neither. And we would all agree with Prof. Kennedy that as substitutes for these terms "Prof. Cay ley's kru- nodal and ac- nodal hardly seem adapted for popular use." It is therefore futile to attempt to secure a nomenclature which shall, throughout, suggest both the form of the locus and the mode of its construction, and we must rest content if we completely attain the latter desideratum. We have next to consider the trochoids traced during the rolling of a circle upon another circle of finite radius. At this point we find inadequacy in nomenclature, and definitions involving singular anomalies. The earlier definitions have been summarized as follows by Prof. E. A. Proctor, in his valuable Geometry of Cycloids: — {epicycloid ) i. is the curve traced out by a point in the circumference of a circle which rolls without sliding hypocycloidj J f & (■ external ) on a fixed circle in the same plane, the two circles being in -^ \ contact." (^ internal J As a specific example of this class of definition I quote the following from a more recent writer; — "If the gen- erating circle rolls on the circumference of a fixed circle, instead of on a fixed line, the curve generated is called an epicycloid if the rolling circle and the fixed circle are tangent externally, a hypocycloid if they are tangent internally." (Byerly, Differential Calculus, 1880.) In accordance with the foregoing definitions every epicycloid is also a hypocycloid, while only some hypocyoloids are epicycloids. Salmon {Higher Plane Curves, 1879) makes the following explicit statement on this point: — "The hypo- cycloid, when the radius of the moving circle is greater than that of the fixed circle, may also be generated as an epicycloid." To avoid any anomaly Prof. Proctor has presented the following unambiguous definition : — f epicycloid ) " An \ f. is the curve traced out by a, point on the circumference of a circle which rolls without sliding ( hypocycloid J ° {outside *) y of the fixed circle." inside j This certainly does away with all confusion between the epi- and Aypo-curves, but we shall find it inadequate to enable us, clearly, to make certain desirable distinctions. By some writers the term external epicycloid is used when the generator and director are tangent externally, and similarly, internal epicycloid when the contact is internal and the larger circle is rolling. Instead of internal epicycloid we often find external hypocycloid used. It will be sufScient, with regard to it, to quote the following from Prof. Proctor : " It has hitherto been usual to define it (the hypocycloid) as the curve obtained when either the convexity of the rolling circle touches the concavity of the fixed circle, or the concavity of the rolling circle touches the convexity of the fixed circle. There is a manifest want of symmetry in the resulting classification, seeing that while every epicycloid is thus regarded as an external hypocycloid, no hypocycloid can be regarded as an internal epicycloid. Moreover an external hypocycloid is in reality an anomaly, for the prefix 'hypo,' used in relation to a closed figure like the fixed circle implies interiorness." To avoid the confusion which it is evident from the foregoing has existed, and at the same time to conform to that principle which is always a safe one and never more important than in nomenclature, viz., not to use two words where one will suffice, I prefer reserving the term " epicycloid " for the case of external tangency, and substituting the more recently suggested name pericycloid for hoth "internal epicycloid" and "external hypocycloid. " The curtate and prolate forms would then be called periiroehoids. By the use of these names and those to be later presented we can easily make distinctions which, without them, would involve undue verbiage in some cases, and, in others, the use of the ambiguous or inappropriate terms to which exception is taken. And the necessity for such distinctions frequently arises, especially in the study of kinematics and machine design. Take, for example, problems like many in the work of Keuleaux already mentioned, relating to the relative motion of higher kinematic pairs of elements, the centroids being circular arcs and the point -paths trochoids. In such cases we are quite as much concerned with the relative position of the rolling and fixed circles as with the form of a point-path. In solving problems in gearing the same need has been felt of simple terms for the trochoidal profiles of the teeth, which should imply the method of their generation. Although they have not, as yet, come into general use, the names pericycloid and peritrochoid appear in the more recent editions of Weisbach and Keuleaux, and will undoubtedly eventually meet with universal acceptance. Yet strong objection has been made to the term "pericycloid" by no less an authority than the late eminent mathematician, Prof. W. K. CliflFord, who nevertheless adopted the "peritrochoid." I quote the following from his Elements of Dynamic : — "Two circles may touch each other so that each is outside the other, or so that one includes the other. In the former case, if one circle rolls upon the other, the curves traced are called epicycloids and epitrochoids. In the latter case, if the inner circle roll on the outer, the curves are hypocycloids and hypotrochoids, but if the outer circle roll on the inner, the curves are epicycloids and peritroohoids. We do not want the name pericycloids, because, as will be seen, every pericycloid is also an epicycloid; but there are three distinct kinds of trochoidal curves." As it will later be shown that every j)evi- trochoid can also be generated as an epi- trochoid we can scarcely escape the conclusion that the name peritrochoid would also have been rejected by Prof. Clifford, had he been familiar with this property of double generation as belonging to the curtate and prolate forms as well. But it is this very property, possessed also by the Ai/po- trochoids, which 'necessitates a more extended nomenclature than that heretofore existing, and I am not aware that there has been any attempt to provide the nine terms essential to its completeness. These it is my principal object to present, and that they have not before been suggested I attribute to the fact that the double, generation of curtate and prolate trochoidal curves does not seem to have been generally known, being entirely ignored in many treatises which make quite prominent the fact that it is a property of the epi- and hypo -cycloids, while, as far as I have seen, the only writer who mentions it proves it indirectly, by showing the identity of trochoids with epicyclics and establishing it for the latter. As it is upon this peculiar and interesting feature that the nomenclature, as now extended, depends, the demonstra- tions necessary to establish it are next in order. For the epi- and hypo -cycloid probably the simplest method of proof is that based upon the instantaneous centre, and which we may call a kinematic, as distinguished from a strictly geometrical, demonstration. It is as follows : x^igr- ±. E'ig'. a. Let P (Pigs. 1 and 2) be the centre of the fixed circle, and r that of a rolling circle, the tracing point, P, being ore the circum- ference of the latter. The point of contact, q, is — at the moment that the circles are In the relative position indicated — an instanta- neous centre of rotation for every point in the plane of the rolling circle; the line Fg, joining such point of contact with the tracing point, is therefore a normal to the trochoid that the point P is tracing. But if the ''-■^- ^R^cTog ^ normal Py be produced to intersect the fixed circle in u, second point, Q, it is evident that the same infinitesimal arc of the trochoid would be described with Q serving as instantaneous centre as when g fulfilled that office. The point P will, therefore, evidently trace the same curve, whether it be considered as in the circumference of the circle r or in that of a second and larger circle, E, tangent to the fixed circle at Q. It is ■worth while, in this connection, to note what erroneous ideas with regard to these same loci were held by some writers as late as the middle of this century, — ideas whose falsity it would seem as if the most elementary geometrical X'lg'. 3- construction would have exposed. Eeuleaux instances the following statement made by Weissenborn in his Cyclischen Kurven (1856) : " If the circle described about to„ roll upon that described about M, and if the describing point, Bj, describe the curve BjPjPj as the inner circle rolls upon the arc B 5 6 , then, evidently, if the smaller circle be fixed and the larger one rolled upon it in a direction opposite to that of the former rotation, the point of the great circle which at the beginning of the operation coincided with Bj describes the same line BoPjP,." The fallacy of this statement is to us, perhaps, in the light of what has preceded, a little more evident than Weissenborn 's deduction ; although, as Beuleaux says, " his ' evidently' expresses the usual notion, and the one which is suggested by a, hasty pre-judgment of the case. In point of fact B„ describes the pericycloid BjB'B'''', which certainly differs sufficiently from the hypocycloid BgPiPj." We have next to consider the curtate and prolate epi-, hypo- and peri - trochoids. As previously stated, I have seen no direct proof that they also possess the same property of double generation, hut find that the kinematic method lends itself with equal readiness to its demonstration. Por the hypotroehoids, let E, Pig. 4, be the centre of the first rolling circle or generator, F that of the first director, and P the initial position of the tracing point. The Initial point of tangency of generator and director is m. Let the generator roll over any arc of the director, as mQ. The centre R will then be found at E,, and the tracing point P at Pj. The point of contact, Q, will then he the instantaneous centre of rotation for Pj, and Pj Q will, therefore, be a normal to the trochoid for that particular position of the tracing point. The motion of P is evidently circular about E, while that of E is in a circle about P. The curve P Pj Pj P^ is that portion of the hypotrochoid which is described while P describes an arc of 180° about E, the latter meanwhile moving through an arc of 108 ° about P, the ratio of the radii being 3 : 5. Now while tracing the curve indicated the point P can be considered as rigidly connected with a second point, p, about which it also describes a circle, p meanwhile (like E) describing a circle about P. Such a point may be found as follows: — Take any position of P, as P^, and join it with the corresponding position of E, as R^; also join E, to P. Let us then suppose PjE and EjP to be adjacent links of a four-link mechanism. Let the remaining links, Ppj and PjPj, be parallel and equal to PjEj and E^P respectively. Taking P as the fixed point of the mechanism let us suppose P, moved toward it over the path P2P3....Pg. Both E^ and pj will evidently describe circular arcs about P; while the motion of Pj with respect to pj will be in a circular arc of radius PjPj. We may, therefore, with equal correctness, consider p^ as the centre of a generator carrying the point P,, and PaP a new line of centres, intersected by the normal Pj Q in a second instantaneous centre, q, which, in strictest analogy with Q, divides the line of centres on which it lies into segments, p^q and P y, which are the radii of the second generator and director respectively; q being, like Q, the point of contact of the rolling and fixed circles for the instant that the tracing point is at Pj. The second generator and director, having p^q and g'P respectively for their radii, are represented in their initial positions, p being the centre of the former, and p. the initial point of contact. The second generator rolls in the opposite direction to the first. It is important to notice that whereas the tracing point is in the first case within the generator and therefore traces the curve as a prolate hypotrochoid, it is without the second generator and describes the same curve as a curtate hypo- trochoid. If we now let E and P denote no longer the centres, but the radii, of the rolling and fixed circles, respec- tively, we have for the first generator and director 2 E > P, and for the second 2 E < P. It occurred to me that a distinction could very easily be made between trochoids generated under these two opposite relations of radii, by using the simple and suggestive term major hypotrochoid when 2 E is greater than F, and minor hypotrochoid when the opposite relation prevails. We would then say that the preceding demonstration had estab- lished the identity of a major prolate with a minor curtate hypotrochoid. Similarly the identity of major curtate and minor prolate forms could be shown. If the tracing point were on the circumference of the generator the trochoids traced would be, by the new nomen- clature, major and minor hypo -cycloids. It is worth noticing that for both hypo - cycloids and hypo - trochoids the centre ¥ is the same for both generations, and that the radius ¥ is also constant for both generations of a hypo - cycloid, but variable for those of a hypo - trochoid. DOUBLE GBNBRATIOlf OF HYPOTROCHOIDS. Having given the radii of generator and director for the construction of a hypo - trochoid, the method just illustrated will always give the lengths of the radii of the second rolling and fixed circles. The accuracy of the values thus obtained may be checked by simple formulae derived from the same figure, as follows : — Eadii being given for generation as a major hypotrochoid, to find corresponding values for the identical minor hypo- trochoid. Let F, denote the radius P Q [ = P m ] of the first director. " Fj " " '' Pj [ = P/i ] " " second " " r " " " EjQ [ = Em] " " first generator. " p " " " P2? [= PM ] " " second " " tr " " tracing radius of the first generation, i. e., the distance E^Pj (or E P) of tracing point from centre of first generator. Let tp equal the second tracing radius = pjP, = /> P. Prom the similar triangles Q P y and Q K , Pj we have P, : Pj : ;