THEORETICAL AND PRACTICAL GRAPHICS 
 
 AN EDUCATIONAL COURSE 
 
 THEORY AND PRACTICAL APPLICATIONS OF MECHANICAL DRAWING 
 
 FREDERICK NEWTON \yiLLSON, C. E., 
 
 Pfo/i'ssnr It/ (Tfnphirs iu the John C. drfcn School of Scie7}ce. 
 
 DEC • ■? «| 
 
 "7- 
 
 ADVANCE SHEETS ISSUED TO THE CLASS OF 'gS. 
 
 1*^ 
 
COPYRIGHT BY F. N. WILLSON. 
 
 ON ILLUSTRATIONS, 1890 : WITH TEXT, 1892. 
 
 ALL RIGHTS RESERVED. 
 
 THE PRINCETON PRESS. 
 
 G. S. ROBINSON 8i CO., UNIVERSITY PRINTER 
 
 PRINCETON, N. J, 
 
 X 
 
THEORETICAL AND PRACTICAL GRAPHICS. 
 
 CH APT JEM I. 
 
 FIRST PRINCIPLES, WITH GENERAL SURVEY OF THE FIELD OF GRAPHIC SCIENCE. 
 
 x-igr- 1. 
 
 1. Geometrically considered, any euniliination of points, lines and .surfac&s is called a figure. 
 A figure lying wlioUy in one plane is called a plane fir/are; 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 valual;)le 
 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 ce^iire of projection. 
 
 In Fig. I let S be an assumed centre of projection and A any j)oint in 
 space. The straight line SA, joining S with A, is called a projecting line or 
 ray, or simply a projector, and its intersection, a, with any line CD, its projection 
 upon that line. It is otherwise expressed by saying tliat A is in-ojected upon 
 CD at a. 
 
 In the same way the iioint 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 signitlcancc, being employed to denote the representation 
 of any curve or surface upon any other. 
 
 3. A figure, as ABC (Fig. 2), is i)rojected 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 which the 
 projectors intersect the plane, is then the projection of the first, or original figure. 
 
 Thr jilane ui)on which the projection is made is called the flaiui of projection. 
 
 4. Were abe (Fig. 3) the original figure and MN the plane of projection, then 
 would ABC be the projection desired. Each figure may thus be considered a 
 projection of the other for a given position of the centre of projection, and when so 
 related figures are said to correspond to each other. Points on the same projector, 
 as a and A, are called corresponding points. 
 
 ' Were S the muzzle of a gun, and B a bullet speeding from It toward the plane. It would be projected against or through 
 the plane at h. The appropriateness of the term "projection" Is obvious. 
 
 - In Fig. 3 the projectors meet the plane between the centre S and the given figure. 
 
2 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 5. Having indicated what projections are and how obtained, it will he 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 
 descriiDtive class, however, we find involved only those properties dependent ujjon relative position.^ 
 Descriptive properties are unaltered by projection while l3ut few metrical properties are jjrojective. The 
 jjrovince of projection is obvious. 
 
 6. Descriptive Geometry is that branch of mathematics in which figures are . represented and their 
 descriptive pro23erties investigated and demonstrated by means of projection. 
 
 ^toS„ 
 
 DIVISIONS OF PROJECTIONS. 
 
 7. All projections may be divided into two general classes. Central and Parallel. 
 If the centre of i^rojection 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 
 j)rojection is thus seen to be merely a sjjecial case of central jjrojection, 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. 
 
 Fig-- s. Eemakk. — A straight line is said to generate a oonical surface (see Fig. 5) when it constantly 
 
 passes through a fixed point (the vertex), and is guided in its motion by 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. 
 
 ^s'S-s- s. 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. 6) 
 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 terras conical and cylindrical as applied to projection is obvious. 
 
 We have now to mention the more important sub-divisions of jjrojections, 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 Conical) projection we have : — • 
 
 9. Projective Geometry {Geometry of Position). While in its most general sense this science includes 
 all central projections (and therefore those of Articles. 10-13) yet in its ordinary acceptation it may 
 be defined as that branch of mathematics in which — with the centre of projection considered as a mathe- 
 matical point at a finite distance from the line or plane of projection — the descriptive properties of geomet- 
 rical figures are investigated and established. 
 
 1 The following will illustrate under metrical: 
 
 (a). The lateral area of a cylinder is equal to the product of the perimeter of its I'ight section by an element of the surface, 
 (b). Two tetrahedrons which have a trihedral angle of the one equal to a trihedral angle of the other, are to each other 
 as the products of the three edges of the equal trihedral angles. 
 
 2 Examples under the descriptive class: 
 
 (a). If a line is perpendicular to a plane, any plane containing the line will also he perpendicular to the plane, 
 (b). To locate the centre of a sphere whose surface shall contain four definitely located points in space, 
 (c). To determine the form of the curve of intersection of two given arches. 
 
DEFINITIONS. 3 
 
 Its chief practical ajiplication is in Graphical Statics in which the stresses in bridge and roof trusses or other engineering 
 •constructions are determined grapliioallv h_v means of diagrams. 
 
 10. Perspective. — If the centre of projection is the eye of the observer, the projection is called a per- 
 spective or scenographic projection, or — more commonly — simply a perspective. The plane of projection is 
 then called the perspective plane or pictxire plane and is always vertical. The position of the eye is 
 calletl the point of .iiijht or .<<tntion point, and the projectors are called visual rays. 
 
 Applied in the graphical construction usually preliminary to art work in water colors or oil ; also in architectural 
 perspectives and in scientific illustrations of machinery, etc. 
 
 It may be remarked that any projection, central or parallel, presents to the eye the saine appearance as 
 the jigure projected would if viewed from the centre of projection. 
 
 11. Relief-perspective. — This differs from the perspective just defined in requiring, in addition to the 
 usual perspective plane, a second plane parallel to it called a vanishing plane, the required represen- 
 tation appearing in relief between the two planes— a solid perspective, so to speak. 
 
 Employed chiefly in the construction of bas-reliefs and theatre decorations. 
 
 12. Shadows (artificial light). — If the centre of projection is an artificial light, as the electric or tliat 
 of a candle — either of which may, without apprecialile error, be treated in graphical constructions as 
 a mere point — the projectors will be rays of light and the projection will Ije the shadow of the figure 
 projected. 
 
 Employed in obtaining shadow eti'ects in interiors. 
 
 13. Photogrammetry or Photometrography, the application of photography to surveying, the centre of 
 projection becoming the ojitical centre of the lens. 
 
 14. Under Parallel (or CylindricaV) projection we have : — 
 (a). Oblique or Clinographic, and 
 
 (b). Perpendicular or Orthographic, also called Orthogonal or Rectangular. 
 These divisions are based upon the direction of the projectors with respect to the plane of pro- 
 jection, they being — as the names imply — inclined to it in oblique projection and perpendicular to it 
 in orthographic. 
 
 OBLIQUE PROJECTION. 
 
 15. The shadow of an object in the sunlight would be its oblique projection, the sun's rays being 
 practically parallel. 
 
 16. Oblique projection is usually called Clinographic when employed in Crystallography. 
 
 17. In its other applications, when not simply called oblique, this projection is variously termed 
 Cavalier Perspective, Cabinet Projection and Military Perspective — the plane of projection being vertical in 
 the first and second while in the last it is horizontal. 
 
 Oblique projection gives a pictorial eft'ect closely analogous to a true perspective, yet is far more simple in its construc- 
 tion, and is much used for showing the form or method of assemblage of parts, or details, of machinery and architectural 
 work. 
 
 ORTHOGRAPHIC PROJECTION, UPON A SINGLE I'LAXE. 
 
 IS. When l)ut one plane of projection is employed the only iiiipcirtant ajiplications of ortho- 
 graphic projection having si)ecial names are — 
 
 (a). One-Plane Descriptive, otherwise called HorizoaUd Projection. 
 Employed chiefly in fortification and general topographical work, in which the lines and surfaces represented are mainly 
 horizontal. 
 
 (b). Axonomelric (including Isometric) Projection. 
 
 Has the same range of application as oblique projection, viz., to objects whose lines lie mainly in directions mutually 
 perpendicular to each other. 
 
4 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 OETHOGRAPHIC PROJECTION UPON MUTUALLY PERPENDICULAR PLANES. 
 
 19. When uiDon two (or more) mutually perpendicular planes orthographic projection becomes 
 the Geometrie Descriptive of Gaspard Monge, who reduced its principles to scientific form in the latter 
 part of the eighteenth century. 
 
 The tendency — a logical one — toward the general adoption of the title "Descriptive Geometry" in 
 the broad sense of Art. 6 would make it seem advisable to appropriate the name Mongers Descriptive 
 to this — the most important division of graphic science, that we may find in it a hint as to its source 
 and at the same time pay to its inventor the honor of perpetual association of his name with his 
 creation. As originally defined by Monge it is the application of orthographic jarojection, 
 
 (a) to the exact representation upon a plane surface, as that of a drawing-board, of all objects 
 capable of rigorous definition, and 
 
 (b) to the solution of problems relating to these objects in space and involving only their properties 
 of form and position. 
 
 It might with propriety be divided into pm-e and ajjpKgcZ, the former being the abstract science, 
 in which the mathematical relations existing between figures and their projections are examined and 
 applied in the solution of certain fundamental problems of the point, line and plane; while the latter 
 division would naturally include the application of these principles and methods to the solution of 
 problems relating to the various elementary and higher mathematical surfaces and to machine drawing 
 and design, shades, shadows, jjerspective, stone-cutting, spherical projections, crystallography, pattern- 
 making, carpentry, etc. 
 
 ADDITIONAL REMARKS ON NOMENCLATURE. 
 
 The student may find the following serviceable by way of enabling him to get clear ideas of the distinctions between various 
 systems. 
 
 The term Geometry, unqualifled, is usually understood to refer to the synthetic method of investigation of the form, position, 
 ratio and naeasurement of geometrical figures, the reasoning being from particular to general truths by the aid of diagrams. In 
 centra-distinction to other geometries it is frequently called Euclidean, after the celebrated Greek geometer, Euclid, (about 330-275 
 B. C.) who organized its theorems and problems into a science. 
 
 In Coordinate (or Analytical) Geometry the figure considered is referred to a system of coordinates and the relation existing 
 between the coordinates of every point of the figure is expressed by means of an equation in which the coordinates are repre- 
 sented by algebraic symbols. The operations performed are algebraic and the method of reasoning from general to particular 
 truths. 
 
 Although the invention of Analytical Geometry has been attributed to Descartes it is now recognized that he neither originated 
 the use of coordinates nor the representation of curves and surfaces by means of equations. As the first to give complete scientific 
 form to the analytic method his name has, however, justly been given to the most important division of coordinate geometry, 
 Cartesian. But the winters of the present day under that head do not by any means confine themselves to the system of coordinates 
 employed by him, wliich consisted of intersecting straight lines usually perpendicular to each other. 
 
 In selecting the title "Projective Geometry" for the science defined in Art. 9, the eminent Cremona says, "I prefer not 
 to adopt that of Higher Geometry, {Geometric siiperieiire, hohere Geometric) because that to which the title 'higher' at one time seemed 
 appropriate, may to-day have become very elementary ; nor that of Modern Geometry {neuere Geometrie) which in like manner ex- 
 presses a merely relative idea, and is moreover open to the objection that although the methods may be i-egarded as modern, yet 
 the matter is to a great extent old. Nor does the title Geometry of Position {Geometric der Lage) as used by Staudt seem to me a suit- 
 able one, since it excludes the consideration of the metrical properties of figures. I have chosen the name of Projective Geometry as 
 expressing the true nature of the methods, w'hich are based essentially upon central projection or perspective. And one reason 
 which has determined this choice is that the great Poncelet, the chief creator of the modern methods, gave to his immortal book 
 the title of " Traite des proprietSs vrojectives des figures. {133-3).^^ 
 
 Cremona further states that "there is one important class of metrical properties (anharmonic properties) which are projective, 
 and the discussion of which therefore finds a place in the Projective Geometry." But the positional definition since given by 
 Staudt for the anharmonic ratio of four points, which removes these properties from the class metrical to the class descriptive 
 (which last are always jTrojective), to that extent justifies the title employed by him, w^hile making Cremona's choice none the 
 less a fortunate one. 
 
 There are other geometries, belonging to what may be called speculative mathematics, based upon "quasi-geometrical notions, 
 those of more-than-three-dimenslonal space, and of non-Euclidean two-and-three-dimenslonal space, and also of the generalized 
 notion of distance."* 
 
 The following will illustrate a method of arriving at a conception of non-Euclidean two-dimensional geometry. "Imagine 
 the earth a perfectly smooth sphere. Understand by a plane the surface of the earth, and by a line the apparently straight line 
 (in fact an arc of a great circle) drawn on the surface. What experience would in the first instance teach w'ould be Euclidean 
 two-dimensional geometry ; there would be intersecting lines, which, produced a few miles or so, would seem to go on diverging, 
 and apparently parallel lines which would exhibit no tendency to approach each other ; and the inhabitants might very well 
 conceive that they had by experience established the axiom that two straight lines cannot enclose a space, and also the axiom as 
 to parallel lines. A n^iore extended experience and more accurate measurements would teach them that the axioms were each 
 of them false; and that any two lines, if produced far enough each way, would meet in two points; they would, in fact, arrive at 
 a spherical geonietry accurately representing the properties of the two-dimensional space of their experience. But their original 
 Euclidean geometry would not the less be a true system; only it would apply to -an ideal space, not the space of their experience."* 
 
 * Cayley. 
 
FREE-HAND DRA WIXG. 
 
 CHAPTBR II. 
 
 ARTISTIC AND TECHNICAL FREE-HAND DRAWING.— SKETCHING FROM MEASUREMENT.— FREE- 
 HAND LETTERING.— CONVENTIONAL REPRESENTATIONS. 
 
 20. DraAviugs, if classified as to the method of their production, are either free-hand or mechanical ; 
 while as to purpose they may be working drawings, so fully dimensioned that tliey can be worked 
 from and what they represent may be manufactured ; or finished drawings, illustrative or artistic in 
 character and therefore shaded either with pen or brush and having no hidden parts indicated by 
 dotted lines as in the preceding division. Finished drawings also lack figured dimensions. 
 
 Working drawings of ])ai-ts or " details " only of a structure are called detail drawings ; and 
 the representation of a structure as a whole with all its details in their i>roper relative position, 
 hidden parts indicated by dotted lines, etc., is termed a general or assembly drawing. 
 
 21. While mechanical drawing is involved in making the various essential views — plans, eleva- 
 tions and sections — of all engineering and architectural constructions and in solving the problems of 
 form and relative position arising in their design, yet to the engineer the ability to sketch effectively 
 and rapidly, free-hand, is of scarcely less importance than to handle the drawing instruments skil- 
 fully ; while the success of an architect depends in still greater measure upon it. 
 
 We must distinguish, however, between artistic and technical free-hand work. The arcliitect must 
 be master of both ; the engineer necessarily only of the latter. 
 
 To secure the adoption of his designs the architect relies largely upon the eS'ective way in which 
 he can finish, either with pen and ink or in water-colors, the perspectives of exterior and interior 
 views ; and such drawings are judged mainly from the artistic standpoint, ^^'hile it is not the prov- 
 ince of this treatise to instruct in such work a word of suggestion may properly be introduced for 
 the student looking forward to architecture as a profession. He should procure Linfoot's Picture Mak- 
 ing in Pen and Ink, Miller's Essentials of Perspective and Delamotte's Art of Sketching from Nature; and 
 with an experienced arcliitect or artist, if possible, Ijut otherwise by himself, master tlie ]irinoiples 
 and act on the instructions of these writers. 
 
 22. Since the camera makes it, fortunately, no longer essential that a civil engineer should l)e a 
 landscape artist as well, his free-hand work has become more restricted in its scope and more rigid 
 in its character, and like that of the machine designer it may properly be called technical, from its 
 object. Yet to attain a sufficient degree of skill in it for all practical and commercial purposes is 
 possible to all and among them many who could never hope to produce artistic results. It is con- 
 fined mainly to the making of vxrrking sketches, conventional rep^-esentalions and free-hand lettering, and 
 the equipment therefor consists of a pencil of medium grade as to hardness; lettering pens— Falcon 
 or CJillott's 803, with Miller Bros. "Carbon" pen. No. 4; either a note-book or a sketch-ldock or 
 pad, and — fur sketching from measurement — a two-foot pocket-rule; calipers, both external and inter- 
 nal, for taking outside and inside diameters; a pair of pencil compasses for making an occasional 
 circle too large to be drawn absolutely free-hand ; and a steel tape-measure for large work if one 
 can have assistance in taking notes, but otherwise a long rod graduated to eighths. 
 
6 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 23. In the evolution of a machine or other engineering project the designer places his ideas on 
 paper in the form of rough and mainly free-hand sketches, beginning with a general outline, or 
 " skeleton " drawing of the whole, on as large a scale as possible, then filling in the details, separate 
 — and larger — drawings of which are later made to exact scale. While such preliminary sketches are 
 not drawn literally " to scale " it is obviously desirable that something like the relative proportions 
 should be preserved and that the closer the approximation thereto the clearer the idea that they 
 will give to the draughtsman or workman who has to work from them. A habit of close observa- 
 tion must therefore be cultivated, of analysis of form and of relative direction and proportion, by all 
 who would succeed in draughting, whether as designers or merely as copyists of existing construc- 
 tions. While the beginner belongs necessarily in the latter category he must not forget that his aim 
 should be to place himself in the ranks of the former both by a thorough mastery of the funda- 
 mental theory that lies back of all correct design and by such training of the hand as shall facilitate 
 the graphic expression of his ideas. To that end he should improve every opportunity to put in 
 practice the following instructions as to 
 
 SKETCHING FEOM MEASUREMENT, 
 
 as each structure sketched and measured will not only give exercise to the hand but also jarove a 
 valuable object lesson in the proportioning of parts and the modes of their assemblage. 
 
 A free-hand sketch may be as good a working drawing as the exactly scaled — and usually 
 inked — drawing that is generally made from it to be sent to the shop. 
 
 While several views are usually required yet for objects of not too complicated form, and whose 
 lines lie mainly in mutually perpendicular • directions, the method of representation illustrated by Fig. 
 7, is admirably adapted and obviates all necessity for additional sketches. 
 
 It is an oblique projection (Art. 17) the theory of whose construction will be found in a sub- 
 sequent chapter, but with regard to which it is sufficient at this point to say that the right angles 
 of the front face are seen in their true form while the other right angles are shown either of 30°, 
 60°, or 120° ; although almost any oblique angle will give the same general effect and may be 
 adopted. Lines parallel to each other on the object are also parallel in the drawing. 
 
 Draw first the front face, whose angles are seen in their true form; then run the oblique lines 
 off in the direction which will give the best view. 
 
 24. While Fig. 7 gives almost the pictorial effect of a true perspective and the object requires 
 no other description, yet for complicated and irregular forms it gives i^lace to the plan-and-eleva- 
 tion mode of representation, the plan being a top and the elevation a front view of the object. And 
 if two views are not enough for clearness as many more should be added as seem necessary, includ- 
 ing what are called sections, which represent the object as if cut apart by a plane, separated and a 
 
SKETCHING FROM MEASUREMENT. 7 
 
 view obtained perpendicular to the cutting plane, showing the internal arrangement and shape of parts. 
 
 In Fig. 8 we have the same object as in Fig. 7, but repre^iented by the method just mentioned. 
 The front view (elevation) is evidently the same in both figures, except that Fig. 8 has dotted lines 
 to indicate the recess which is in full sight in Fig. 7. 
 
 The view of the top is placed at the top in conformity to the now quite general practice as to 
 location, viz., grouping the various sketches about the elevation, so that the view of the left end is 
 at the left, of the right at the right, etc. 
 
 i^gr- s. 
 
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 3 
 
 -^ 
 
 
 y 
 
 ,c 
 
 
 1. 
 
 h^ 1 
 
 /e'Li ; 
 
 
 H 
 
 
 5' ^^^ 
 
 fc — -Zio ^ 
 
 > 
 
 L 
 
 
 -£L£VAT/ON- 
 
 
 
 In these views, which fall under Art. lU as to theoretical construction, entire surfaces are fro- 
 jected as straight lines, as G B C H in the straight Une H' C". Were this a metallic surface and 
 "finished " or " machined " to smoothness, as distinguished from the surface of a rough casting, that 
 fact would be denoted by an "/" on the line H' C which represents the entire surface, the cross- 
 line of the "/" cutting the line of the surface obliquely, as shown. 
 
 25. In sketching, centre-lines and all imj)ortant centres should be located first and measure- 
 menta taken from them or from finished surfaces. 
 
 Feet and inches are abbreviated to "Ft.," and " L\.," as 4 Ft. Gi Ix: also written 4' Ql" and 
 occasionally 4 Ft. 6J." A dimension should not be written as an improper fraction, '/" t^"" '^^" 
 ample, but as a mixed number, 11". Fractions should have horizontcd dividing lines. 
 
 Not only should dimensions of successive parts be given but an "over-all" dimension which, it 
 need hardly be said, should sustain the axiom regarding the whole and the sum of its parts. 
 
 Dimensions should read in line with the line they arc on and either from the bottom or the 
 right hand. 
 
 The arrow tips should touch the lines between which a distance is given. 
 
 An opening should be left in the dimension line for the figures. 
 
 Extension lines should be drawn and the dimension given outside the drawing whenever such 
 course will add to the clearness. 
 
THEORETICAL AND PRACTICAL GRAPHICS. 
 
 In case of very small dimensions the arrow tips may be located outside the lines, as in Fig. 9, 
 and the dimension indicated by an arrow as at A or inserted as at B if there is room. 
 
 Should a piece of" uniform cross-section be too long to be represented in its proper relative length 
 on the sketch, it may be broken as in Fig. 9 and the form of the section (which in the case sup- 
 posed will be the same as an end view) may be inserted with its dimensions, as in the shaded figure. 
 
 E'iir 
 
 ■X^fi 
 
 4-" C/ut77,neT, 
 
 ■ G 8 
 
 aJibrevixitecf^ ^io° i-j" i^Ti, 
 
 tn 
 
 «* 
 
 an- 
 
 The same dimension should not appear on each view, but each dimension must be given at least 
 once on some view. 
 
 In riveted work the "pitch" of the rivets, i. e., their distance from centre to centre ("c. to c") 
 should be noted, as also that between centre lines or rows, and of the latter from main centre lines. 
 Similarly for bolts or holes. If the latter are located in a circle note the diameter of the circle con- 
 taining their centres. Note that the hole for a rivet is usually about one-half the diameter of the 
 forged head. In measuring nuts take the width between parallel sides (" width across the flats ") 
 and abbreviate for the shape, as "sq.," "hex.," " oct." 
 
 For a piece of cylindrical shape a frequently used symbol is the circle, as 4" for 4" diam- 
 eter; but it is even clearer to use the abbreviation of the latter word, viz., "diam." 
 
 In taking irotes on bolts and screws the outside diameter is sufficient if they are "standard," 
 that is, proportioned after either the Sellers (U. S. Standard) or Whitworth (English Standard) sys- 
 tems, as the proportions of heads and nuts, number of threads to the inch, etc., can be obtained 
 from the tables in the Appendix. If not standard note the number of threads to the inch. Record 
 whether a screw is right- or left-handed. If right-handed it will advance if turned clock-wise. The 
 shape of thread, whether triangular or square, would also be noted. 
 
 On cog, or "gear," wheels obtain the distance between centres and the number of teeth on each 
 wheel. The remaining data are then obtained by calculation. 
 
 In taking bridge notes there would be required general sketches of front and end view ; and of 
 the flooring system, showing arrangement of tracks, ties, guard-beam and sidewalk ; a cross-section ; 
 also detail drawings of the top and foot of each post-connection in one longitudinal line from one 
 end to the middle of the structure. In case of a double-track bridge the outside rows of posts are 
 alike but differ from those of the middle truss. 
 
 The foregoing hints might be considerably extended to embrace other and special cases, but ex- 
 perience will prove a sufficient teacher if the student will act on the suggestions given and will 
 remember that to get an excess of data is to err on the side of safety. He should take his notes 
 on as large a scale as possible and so index them that drawings of parts may readily be understood 
 in their relation to the whole. 
 
 It need hardly be added that the hints given are intended to be merely a partial summary of 
 the instructions which would be given in the more or less brief practice in technical sketching which, 
 presumably, constitutes a part of every course in Graphics ; and that unless the draughtsman can be 
 under the direction of a teacher he will be able to sketch much more intelligently after studying 
 
LETTERING. -CONVENTIONAL BEPRESENTA TIONS. 9 
 
 more of the theory involved in Meclianical Drawing and given in the later pages of this work. 
 
 LETTERING. 
 
 26. In Figs. 7 and 8 the dimensions are given in numerals belonging to the Extended Gothic 
 alphabet below, now growing in favor for this purpose. B3' giving feet to the 1 and 4, as shown 
 
 i^ig-. 10. 
 
 ABCDEFGHIJKLMNOPQRSTUVWXYZ& 
 
 1234567890 
 
 in those figures, they are no longer strictly Gothic in form yet their appearance is enhanced in the 
 oj^inion of many. 
 
 The Italic Gothic and Italic Roman (Fig. 11) are also favorite types witli the engineering profession 
 
 for general lettering as well as for dimensioning, and the draughtsman should thoroughly master 
 
 ■ ITALIC 
 
 GOTH/C ■:• 
 
 — 6Ta&o ■ 
 
 ABODE rGH/JKLMNOPClFt S TUVWXYZ 
 
 ■:■ ITALIC •:■ ROMAN •.'• 
 
 jLS C D:EFaiIIJ':KLMNOI'Q US 
 
 123 45 TZTV W^r 1^ Z 67 8 9 O 
 
 cvh c defahJilklrrhTLop qrs h.LV wjcy z 
 
 them as among the most serviceable he can employ. They should, like the Gothic, be invariably 
 drawn free-hand when less than a quarter of an inch in height. Use Miller Bros. " Carbon " pen. 
 No. 4, or any equally flexible fine stub for Gothic forms, and either the " Falcon " or Gillott's No. 
 303 for Roman and Italic Roman letters. 
 
 CONVENTIONAL REPRESENTATIONS. 
 
 27. Conventional rei)resentations of the natural features of the country or of the materials of 
 construction are so called on the assumption, none too well founded, that the engineering profession 
 
 E^igr- 12. 
 
 rig. 13. 
 
 has agreed in convention that they shall indicate that which they also more or less resemble. While 
 there is no universal agreement in this matter there is usually but little ambiguity in their use, 
 especially in those that are drawn free-hand, since in them there can be a nearer a2)proach to the 
 natural appearance. This is well illustrated by the sections of river-beds presented in Figs. 12 and 
 
10 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 13, the former showing, m addition to the rock cutting, the method of indicating a mud or sand 
 bed with small random boulders. Water either in section or as a receding surface may be shown by 
 jDarallel lines, the spaces between them increasing gradually. 
 
 The conventional representations of wood, masonry and the metals will be found in a succeeding 
 chapter, after hints on coloring have been given, the foregoing figures appearing at this point merely 
 to illustrate the third division of technical free-hand work. Those, however, who have already had 
 some practice in drawing may undertake Figs. 12 and 13 with Inirnt umber undertone for the earthy 
 bed, paZe blue or india ink tint for the rock-section and prussian blue for the water-lines. 
 
THE DRAUGHTSMAN'S EQUIPMENT. 
 
 11 
 
 CHAPTER III. 
 
 DRAWING INSTRUMENTS AND MATERIALS.— INSTRUCTIONS AS TO USE.— GENERAL PRELIMINARIES 
 
 AND TECHNICALITIES. 
 
 28. The draughtsman's equipment for graphical work should be the hest con- '^'^s- i-^- :F3.g-- is. 
 sistent with his means. It is mistaken economy to buy inferior instruments. 
 The best obtainable will l^e found in the end to have been the cheapest. 
 
 The set of instruments illustrated in the following figures contains onlj^ those 
 which may be considered ab.solutely essential for the beginner. 
 
 THE DRAWINC; PEN. 
 
 The right line pen (Fig. 14) is ordinarily used for drawing straight lines, 
 with either a rule or triangle to guide it ; but it is also employed for the draw- 
 ing of curves when directed in its motion by curves of wood or hard rubber. 
 For average work a pen about five inches long is best. 
 
 The figure illustrates the most approved tj^pe, i. e., made from a single piece 
 of steel. The distance between its points, or " nibs," is adjustable by means of 
 the screw H. An older form of pen has the outer blade connected with the in- 
 ner by a hinge. The convenience with which such a pen may be cleaned is 
 more than offset by the certainty that it will not do satisfactory work after the 
 joint has become in the slightest degree loose and inaccurate through wear. 
 
 29. If the points wear unecjually or become blunt the draughtsman may 
 sharpen them readil}' himself upon a fine oil-stone. The process is as follows : 
 
 Screw up the blades till they nearly touch. Incline the pen at a small angle 
 to the surface of the stone and draw it lightly from left to right (supposing 
 ^-ig-. IS- ^ the initial position as in Fig. 16.) Before reaching the right 
 end of the stone begin turning the pen in a plane perpendic- 
 ular to the surface and draw in the opposite direction at the 
 same angle. After frequent examination and trial to see that 
 the blades have become ecjual in length and similarly rounded the process is com- 
 pleted by lightly dressing the outside of each blade separately upon the stone. 
 No grinding should be done on the inside of the blade. Any " burr " or rough 
 edge resulting from the ojaeration may be removed with fine emery paper. For 
 the best results, obtained in the shortest possible time, a magnifying glass should be used. The stu- 
 dent should take particular notice of the shape of the pen points when new, as a standard to be 
 aimed at when compelled to act on the above suggestions. 
 
 30. The pen may be supplied with ink by means of an ordinary writing pen dipped in the ink 
 and then passed between the blades ; or liy using in the same manner a strip of Bristol board 
 about a quarter of an inch in width. Should any fresh ink get on the outside of the pen it must 
 
 ri 
 
12 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 E'ig-- iT-. 
 
 be removed ; otherwise it will be transferred to the edge of the rule and thence to the paper, caus- 
 ing a blot. 
 
 31. As with the pencil, so with the pen, horizontal lines are to be drawn from left to right, 
 while vertical or inclined lines are drawn either from or towards the worker, according to the posi- 
 tion of the guiding edge with respect to the line to be drawn. If the 
 line were m n, Fig. 17, the motion would be away from the draughts- 
 man, i. e., fi"om ri toward m ; while o p would be drawn toward the 
 worker, being on the right of the triangle. 
 
 32. To make a sharply defined, clean-cut line — the only kind al- 
 lowable — the pen should be held lightly but firmly with one blade 
 resting against the guiding edge, and with Ijoth points resting equally 
 upon the paper so that they may wear at the same rate. 
 
 33. The inclination of the pen to the paper may best be about 70°. When properly held the 
 f)en will make a line about a fortieth of an inch from the edge of the rule or triangle, leaving 
 visible a white line of the jsaper of that width. If, then, we wish to connect two points by an 
 inked straight line the rule must be so placed that its edge will be from them the distance indi- 
 cated. 
 
 It need hardly be said that a drawing-\)en should not be pushed. 
 
 The more frequently the draughtsman will take the trouble to clean out the point of the pen 
 and supply fresh ink the more satisfactory results will he obtain. When through with the pen clean 
 it carefully and lay it away with the points not in contact. Equal care should be taken of all the 
 instruments and for cleaning them nothing is superior to chamois skin. 
 
 DIVIDEES. 
 
 34. The hair-spring dividers (Fig. 15) are employed in dividing lines and spacing off distances, 
 and are capable of the most delicate adjustment -by means of the screw G and spring in one of the 
 legs. When but one pair of dividers is purchased the kind illustrated should have the preference 
 over plain dividers, which lack the spring. It will, however, be frequently found convenient to have 
 at hand a pair of each. Should the joint at F become loose through wear it can be 'tightened by 
 means of a key having two j^rojections which fit into the holes shown in the joint. 
 
 35. In spacing off distances the pressure exerted should be the slightest consistent with the loca- 
 tion of a point, the puncture to be merely in the surface of the paper and the points determined 
 
 by lightly pencilled circles about them, thus O ' In laying off several equal distances 
 
 along a line all the arcs described by one le'ig-- is. leg of the dividers should be on the 
 
 same side of the line. Thus, in Fig. 19, with b the first centre of turning, the leg x describes the 
 
 _/, JT 
 
 arc R, then rests and pivots on c while the leg y describes the arc S; x then traces arc T, &c. 
 
THE COMPASSES.— THE BOW-PENCIL AND PEN. 
 
 13 
 
 COMPASS SET. 
 
 ?)6. The compasses (Fig. 20) resemble the dividers in form and may be used to perform the same 
 office, hut are usuallj'' employed for the drawing of circles. Unlike the dividers one or both of the 
 
 legs of compasses are detachable. Those illustrated have one perma- 
 nent leg, with pivot or " needle-point " adjustable by means of screw R. 
 The other leg is detachable by turning the screw 0, when the pen leg 
 L M (Fig. 21) may be inserted for ink work ; or the lengthening bar on 
 the right (Fig. 22) may be first attached at and the pencil or pen 
 leg then inserted at I, where large work is involved. The metallic point 
 held by screw <S' is usuallj' replaced by a hard lead, sharpened as in- 
 dicated in Art. 54. 
 
 37. When in use the legs should be bent at the joints P and L 
 so that thej' will stand perpendicular to the paper if the coinpasses are 
 held in a vertical plane. The turning luay be in either direction but 
 is usually "clock-wise;" and the compasses maj' be slightly inclined 
 toward the direction of turning. AMren so used, and if no undue pres- 
 sure be exerted on the pivot leg, there should be but the slightest punct- 
 ure at the centre, while the pen points having rested equally upon 
 the paper have sustained equal wear and the resulting line has been 
 sharply defined on both sides. Obviously the legs must be re-adjusted 
 as to angle for any material change in the size of the circles wanted. 
 
 The compasses should be held and turned liy the milled head, 
 which projects above the joint N. 
 
 Dividers and compasses should open and shut with an absolutely 
 uniform motion and somewhat stiffiy. 
 
 «i 
 
 BOW-PENCIL AND PEN. 
 
 38. For extremely accurate work, 
 in diameters fi-om one-sixteenth of an inch to about 
 two inches, the bow-pencil (Fig. 23) and how-pen (Fig. 
 24) are especially adapted. The pencil-bow has a 
 needle-point, adjustable by means of screw E, which 
 gives it a great advantage over the fixed j^ivot-point 
 of the bow-pen, not alone in that it permits of 
 more delicate adjustment for unusually small work 
 but also because it can be easily replaced by a new 
 one m case of damage ; whereas an injury to the 
 other renders the whole instrument useless. For very 
 small circles the needle point should project very sligMly beyond the 
 pen-point ; theoretically by only the extremely small distance the 
 needle-point is expected to sink into the paper. 
 
 The spring of either bow should be strong; otherwise an at- 
 tempt at a circle will result in a Sf)iral. 
 
 It will save wear upon the threads of the milled heads A and 
 C if the draughtsman will press the legs of the bow together with 
 his left hand and run the head up loosely on the screw with his 
 right. 
 
 E'ig-. 23. 
 
 S'ig-- 2^. 
 
14 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 39. To the above described — which we may call the minimum set of instruments — might be ad- 
 vantageously added a pair of bow-spacers (small dividers shaped like Fig. 24) ; beam-compasses, for 
 extra large circles; parallel-rule; proportional dividers and an extra — and larger — right-line pen. 
 
 40. The remainder of the necessary equipment consists of paper ; a drawing-board ; T-rule ; trian- 
 gles or " set squares ; " scales ; pencils ; India ink ; water colors ; saucers for mixing ink or colors ; 
 brushes; water-glass and sponge; irregular (or "French") curves; India rubber; erasing knife; pro- 
 tractor; file for sharpening pencils, or a pad of fine emery or sand paper; thumb-tacks (or "draw- 
 ing pins"); horn centre, for making a large number of concentric circles. 
 
 PAPER AND TRACING CLOTH. 
 
 41. Drawing paper may be purchased by the sheet or roll and either unmounted or mounted, 
 i. e., " backed " by muslin or heavy card-board. Smooth or " hot-pressed " jDaper is best for drawings 
 in line-work only ; but the rougher surfaced, or " cold-pressed, should always be emjjloyed when 
 brush-work in ink or colors is involved : in the latter case also either mounted f)aper should be used 
 or the sheets " stretched " by the process described in Art. 44. 
 
 42. The names and sizes of sheets are: — 
 Cap 13 X 17 Elephant 23 x 28 
 
 Demy 15 x 20 Atlas 26 x 34 
 
 Medium 17 X 22 Cokmibia 23 x 35 
 
 Royal 19 X 24 Double Elephant 27 X 40 
 
 Super Royal 19 x 27 Antiquarian 31 x 53 
 
 Imperial 22 x 30 
 
 43. There are many makes of first-class i^apers, but the best known and still probably the most 
 used is Whatman's. The draughtsman's choice of jjaper must, however, be determined largely bj' the 
 value of the drawing to be made upon it and by the probable usage to which it will be subjected. 
 
 Where several copies of one drawing were desired it has been a general practice to make the 
 original, or " construction " drawing, with the pencil, oia paper of a medium grade, then to lay over 
 it a sheet of iracing-cloth and copy upon it, in ink, the lines underneath. Upon placing the tracing 
 cloth over a sheet of sensitized paper, exposing both to the light and then immersing the sensitive 
 paper in water a copj' or print of the drawing was found upon the sheet, in white lines on a blue 
 ground — the well-known bhie-print. The time of the draughtsman may, however, be economized as 
 also his purse, by making the original drawing in ink upon Crane's Bond paj^er, which combines 
 in a remarkable degree the qualities of transparency and toughness. About as clear blue-prints can 
 be made with it as with tracing-cloth yet it will stand severe usage in the shop or the drafting- 
 room. 
 
 Better papers may yet be manufactured for such purposes and the progressive draughtsman will 
 be on the alert to avail himself of these as of all genuine improvements upon the materials and 
 instruments heretofore employed. 
 
 44. To stretch paper tightly upon the board lay the sheet right side up*, place the long rule 
 with its edge about one-half inch back from each edge of the paper in turn and fold up against 
 it a margin of that width. Then thoroughly damjjen the back of the paper with a full sponge, except 
 on the folded margins. Turning the paper again face up, gum the mai'gins with strong mucilage or 
 glue and quickly but firmly press opposite edges down simultaneously, long sides first, exerting at the 
 same time a slight outward pressure with the hands to bring the paper down somewhat closer to 
 
 *The "right sifle " of a sheet is, presamably, that towards one when— on holding it up to the light— the manufaoluier's 
 name, in water-mark, reads correctly. 
 
TRACIXG-C'LOTH. — DRA WIXG B OARD.— T-RULE.— TRIAXGLES. 
 
 15 
 
 the board. Until the gum " sets," so that the paper adheres perfectly where it should, the latter 
 should not shrink ; hence the necessity for so completely soaking it at first. The sponge may he ap- 
 plied to the face of the paper provided it is not rubbed over the surface, so as to damage it. When 
 drying, the stretch should be horizontal and no excess of water should be left standing on the surface ; 
 otherwise a water-mark will form at the edge of each pool. 
 
 45. When iva.cmg-cloth is used it must be fastened smoothly, with thumb-tacks, over the di-awing 
 to be copied and the ink lining done upon the glazed side, any brush work that may be required 
 — either in ink or colors — being always done upon the dull side of the cloth after the outlining has 
 been completed. 
 
 Tracing-cloth, like drawing paper, is most convenient to work upon if perfectly flat. When either 
 has been purchased b}' the roll it should therefore be cut in sheets and laid away for some time 
 in drawers to become flat before needed for use. 
 
 nii.VWIXG BOARD. 
 
 46. The drawing board should be slightly larger than the paper for which it is designed and 
 of the most thoroughly seasoned material, preferably some soft wood, as f)ine, to facilitate the use of 
 the drawing-pins or thumb-tacks. To i^revent warping it should have battens of hard wood dove- 
 tailed mto it across the back, transversely to its length. The back of the board should be grooved 
 longitudinally to a depth equal to half the thickness of the wood, which weakens the board trans- 
 versely and to that degree facilitates the stifiening action of the battens. 
 
 It will be found convenient for each student in a technical school to possess two boards, one 
 20" X 28" for paper of Super Royal size, which is suitable for much of a beginner's work, and another 
 28" X 41" for Double Elephant Sheets (about twice Super Royal size) which are well adapted to large 
 drawings of machinery, bridges, &c. A large board may of course be used for small sheets and the 
 expense of getting a second board avoided ; but it is often a great convenience to have a medium- 
 sized board, especially in case the student desires to do some work outside the draugh ting-room. 
 
 THE T-RULE. 
 
 47. The T-rule should be slightly shorter than the drawing board. Its head and blade must 
 have absolutely straight edges and be so rigidly combined as to admit of no lateral play of the lat- 
 ter in the former. The head should also be so fastened to the blade as to be le^'el with the surface 
 of the board. This permits the triangles to slide freely over the head, a great convenience when 
 the lines of the drawing run close to the edge of the paper. (See Fig. 32.) 
 
 TRIAKGLES. 
 
 48. Triangles, or " set-squares " as they are also called, can be obtained in various materials, as 
 hard rulilier, celluloid, pear-wood, mahogany and steel; and either solid (Fig. 25) or open (Fig. 26). The 
 open triangles are preferable and two are required, oire with acute angles of 30° and 60°, the other 
 with 45° angles. Hard rubber has an advantage over metal or wood, the latter being likely to warp 
 and the former to rust, unless nickel-plated. 
 
 The most frequently recurring problems involving the use of the triangles are the following : — 
 E-ig. 2s. _j^g jiy j^i-fn^ parallel lines place either of the edges 
 
 against another triangle or the T-rule. If moved along 
 in either direction each of the other edges will take a 
 series of parallel positions. 
 
 50. To draw a line perpendicular to a given line 
 place the hypothenuse of the triangle, o a, (Fig. 26), 
 so as to coincide with or be parallel to the given line; then a rule or another triangle against the 
 
 E'ig-- 26. 
 
16 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 base. By then turning the triangle so that the other side, o c, of its right angle shall be against 
 the rule, as at Oj Cj, the hypothenuse will be found perpendicular to its first position and therefore 
 to the given line. 
 
 51. To construct regulai- hexagons place the shortest side of the 60° triangle against the rule (Fig. 
 ■Fis- a'7- 27) if two sides are to be horizontal, as /e and he of hexagon H. For 
 
 vertical sides, as in H', the position of the triangle is evident. By making 
 a h indefinite at first and knowing h c, the length of a side, we may obtain 
 a by an arc, centre b, radius b c. 
 
 If the inscribed circles were given the hexagons might also be obtained 
 hy drawing a series of tangents to the circles with the rule and triangles 
 in the jsositions indicated. 
 
 THE SCALE. 
 
 52. But rarely can a drawing be made of the same size as the object, or " full-size," as it is 
 called. The lines of the drawing therefore usually bear a certain ratio to those of the object. This 
 ratio is called the scale and should invariably be indicated. 
 
 If six inches on the drawing represent one foot on the object the scale is one-half and might be 
 variously indicated, thus: SCALE J; SCALE 1:2; SCALE 6 Ix=l Ft. Su.iLE 6"=1'. 
 
 At one foot to the inch anj' line of the drawing would be one-twelfth the actual size and the 
 fact indicated SCALE 1 lN.=l FT. 
 
 Although it is a simiDle matter for the draughtsman to make a scale for himself for any partic- 
 ular case yet scales can be f)urchased in great variety, the most serviceable of which for the usual 
 range of work is of box-wood, 12" long, (or 18", if for large work) of the form illustrated hj Fig. 
 :Fig.. as. 28 and graduated -5%: ^^■. \: \: f: \: f: 1: li; 3 inches to the foot. This 
 
 is known as the architect's scale in contradisthiction to the engineer\s, which is 
 decimallj' graduated. It Avill, however, be frequently convenient to have at 
 \\\\W\\\\\WV4 hand the latter as well as the former. 
 When in use it should be laid along the line to be spaced and a light dot made upon the 
 paper with the pencil, opposite the proper division on the graduated edge. A distance should rarely 
 be transferred from the scale to the drawing hy the dividers, as such procedure damages the scale if 
 not the paper. 
 
 a Jl 10 .9 8 7 6 5 4 3 3 1 
 
 53. For special cases diagonal scales can readily be constructed. If, for example, a scale of 3 inches 
 to the foot is needed and measuring to fortieths of inches, draw eleven equidistant, parallel lines, enclos- 
 ing ten equal spaces, as in Fig. 29, and from the end A Inj off A B, B C, &c., each 3 inches and 
 representing a foot. Then twelve parallel diagonal lines in the first space interceiat quarter-inch spaces 
 on A B or a b, each representative of an inch. There being ten equal spaces between B and b, the 
 distance s x, of the diagonal b m from the vertical b B, taken on any horizontal line .s x, is as many 
 
SCALES. — PENCILS. — INK. 17 
 
 tenths of the space m £ as there are spaces between sx and 6; six, in this case. The princii^le of 
 construction may be generalized as follows : — 
 
 The distance apart of the vertical lines represents the units of the scale, whether inches, feet, 
 rods or miles. Except for decimal graduation, divide the left-hand space at top and bottom into as 
 many spaces as there are units of the next lower denomination in one of the original units (feet, 
 for yards as units; inches in ease of feet, &c.)- Join the points of division by diagonal lines. And 
 if 2- is the smallest fraction that the scale is de.signed to give rule x+l equidistant horizontal lines, 
 giving X equal horizontal spaces. The scale will then read to ^.th of the intermediate denomination 
 of the scale. 
 
 When a scale is properly used the spaces on it which represent feet and inches are treated as if 
 they were such in fact. On a scale of one-eighth actual size the edge graduated H inches to the foot 
 would be employed, each 1] inch space on the scale would be read as if it were a foot; and ten 
 inches, for example, would be ten of the eighth-inch spaces, each of which is to represent an inch 
 of the original line being scaled. The usual error of beginners would be to divide each original 
 dimension by eight and lay off' the result, actual size. The former method is by far the more ex- 
 peditious. 
 
 THE PENCILS. 
 
 •54. For construction lines afterward to be inked the pencils should be of hard lead, grade 6H if 
 Faber's or VVH if Dixon's. The pencilling should be light. It is easy to make a groove in the 
 paper by exerting too great pressure when using a hard lead. The hexagonal form of pencil is 
 usually indicative of the finest quality and has an advantage over the cylindrical in not rolling off 
 when on a board that is slightly inclined. 
 
 Somewhat softer pencils should be used for drawings afterwards to be traced and for the prelim- 
 inary free-hand sketches from which exact drawings are to be made ; also in free-hand lettering. 
 
 Sharpen to a chisel edge for work along the edges of the T-rule or triangles, but use another 
 pencil with coned point for marking off distances with a scale, locating centres and other isolated 
 points and for free-hand lettering ; also sharpen the compass leads to a point. Use the knife for cut- 
 ting the wood of the pencil, beginning at least an inch from the end. Leave the lead exposed for 
 a quarter of an inch and shape it as desired either with a knife or on a fine file or a pad of fine 
 emerj' paper. 
 
 THE INK. 
 
 55. Although for many purposes some of the liquid drawing-inks now in the market, partic- 
 ularly Higgins', answer admirably, yet for the best results, either with pen or brush, the draughtsman 
 should mix the ink himself with a stick of India — or, more correctly, China ink, selecting one of the 
 higher-priced cakes of rectangular cross-section. The best will show a lustrous, almost iridescent fi-act- 
 ure and will have a smooth, as contrasted with a gritty feel when tested by rubbing the moistened 
 finger on the end of the cake. 
 
 Sets of saucers, called " nests," designed for the mixing of ink and colors, form an essential part 
 of an equipment. There are usually six in a set and so made that each answers as a cover for the 
 one below it. Placing from fifteen to twenty drojDS of water in one of these the stick of ink should 
 be rubbed on the saucer with moderate pressure. 
 
 To properly mix ink requires great patience, as with too great f)ressure a mixture results having 
 flakes and sand-like particles of ink in it, whereas an absolutely smooth and rather thick, slow-flow- 
 ing liquid is wanted, whose surface will reflect the face like a mirror. The final test as to suificiency 
 of grinding is to draw a broad line and let it dry. It should then be a rich jet black with a slight 
 
18 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 lustre. The end of the cake must be carefully dried on removing it from the saucer, to prevent its 
 flaking, as it will otherwise invariably do. 
 
 One may say, almost without qualification, and particularly when for use on tracing-cloth, the 
 thicker the ink the better; but if it should require thinning, on saving it from one day to another — 
 which is possible with the close-fitting saucers described — add a few drops of water, or of ox-gall if 
 for use on a glazed surface. 
 
 When the ink has once dried on the saucer no attempt should be made to work it up again 
 into solution. Clean the saucer and start anew. 
 
 AVATER COLORS. 
 
 56. The ordinary colored writing inks should never be used by the draughtsman. They lack the 
 requisite " body " and are corrosive to the pen. Winsor and Newton's water colors, in the form 
 called " moist," and in " half-pans " are the best and most convenient for color work either with pen 
 or brush. Those most frequently employed in engineering and architectural drawing are Prussian 
 Blue, Carmine, Light Red, Burnt Sienna, Burnt Umber, Vermilion, Gamboge, Yellow Ochre, Chrome 
 Yellow, Payne's Gray and Sepia. 
 
 Although hardly properly called a color, Chinese White may be mentioned at this point as a 
 requisite, and obtainable of the same form and make as the colors above. 
 
 DRAWING-PINS. 
 
 57. Drawing-pins or thumb-tacks, for fastening paper upon the board, are of various grades, the 
 best and at present the cheapest being made from a single disc of metal one-half inch in diameter, 
 from which a section is partially cut, then bent at right angles to the surface, forming the point of 
 the pin. 
 
 IRREGULAR CURVES. 
 
 58. Irregular or French curves, also called siveeps, for drawing non-circular arcs, are of great vari- 
 ety and the draughtsman can hardly have too many of them. They may be either of pear wood or 
 hard rubber. A thoroughly equipped draughting office will have a large stock of these curves, which 
 may be obtained in sets, and are known as railroad curves, ship curves, spirals, ellipses, hyperbolas, 
 parabolas and combination curves. 
 
 If but two are obtained (which would be a minimum stock for a beginner) the forms shown in 
 Fig. 30 will probably jDrove as serviceable as an}'. When employing them for inked work the pen 
 F ig- 30 - should be so turned, as it advances, that its blades will maintain the same 
 
 relation (parallelism) to the edge of the guiding curve -as they ordinarily 
 do to the edge of the rule. And the student must content himself with 
 drawing slightly less of the curve than might apparently be made with one 
 setting of the sweep, such course being safer in order to avoid too close an 
 approximation to angles in what should be a smooth curve. For the same 
 reason, when placed in a new position, a portion of the irregular curve must coincide with a part 
 of that last inked. 
 
 The pencilled curve is usually drawn free-hand after a number of the points through which it 
 should pass have been definitely located. In sketching a curve free-hand it is much more naturally 
 and smoothly done if the curve is concave toward the hand. 
 
 INDIA RUBBER. 
 
 59. For erasing pencil lines and cleaning the j^aper India rubber is required, that known as 
 "velvet" being recommended for the former purpose, and either "natural" or "sponge" rubber for 
 the latter. Stale bread crumbs are equally good for cleaning the surface of the paper after the lines 
 have been inked, but will damage pencilling to some extent. 
 
ERASERS. — PRO TRACTORS. — BRUSHES. 
 
 19 
 
 One end of the velvet rubber may well be wedge-shaped for erasing lines without damaging 
 others near them. 
 
 INK ERASER. 
 
 60. The double-edged erasing knife gives the quickest and best results when an inked line is to 
 be removed. The point should rarely be employed. The use of the knife will damage the paper 
 more or less, to partially obviate which rub the surface with the thumb nail or an ivory knife 
 handle. 
 
 PROTRACTOR. 
 
 61. For laying out angles a graduated arc called a "protractor" is employed. 
 
 E'ig-. 31. 
 
 These are made of various materials, as metal, horn, celluloid, Bristol board and tracing paper. 
 The two last are quite accurate enough for ordinary purposes, though where the utmost precision 
 is required one of German silver should be obtained, having a movable arm and vernier attachment. 
 
 The graduation may advantageously be to half degrees for average work. 
 
 To lay out an angle (say 40°) with the protractor, the radius, C H, would be made to coin- 
 cide with one side of the desired angle; the centre C, with the desired vertex; and a dot made with 
 the pencil opposite division numbered 40 on the graduated edge. The line, M C, through this point 
 and C, would comj^lete the construction. 
 
 BRUSHES. 
 
 62. Sable-hair brushes are the best for laying flat or graduated tints, with ink or colors, upon 
 small surfaces ; while those of camel's hair, large, with a brush at each end of the handle, are as 
 well adapted for tinting large surfaces. Reject any brush that does not come to a perfect point on 
 being moistened. Five or six brushes of different sizes are needed. 
 
 PRELIMINARIES TO PRACTICAL WORK. 
 
 63. The first work of a draughtsman, like most of his later productions, consists of line as dis- 
 tinguished from brush work, and for it the paper may be fastened upon the board with thumb-tacks 
 only. 
 
 There is no universal standard as to size of sheets for drawings. As a rule each draughting 
 office has its own set of standard sizes, and system of preserving and indexing. The columns of 
 the various engineering papers present frequent notes on these points, and apparently the best system 
 of preserving and recording drawings, tracings and corrections is in process of evolution. For the 
 student the best plan is to have all drawings of the same size bound in neat but permanent form 
 at the end of the course. The title-pages, which presumably have also been drawn, will sufficiently 
 distinguish the different sets. 
 
 In his elementary work the student may to advantage adopt two sizes of sheets which are con- 
 siderably employed, 9" X 13" and its double 13" x 18" ; sizes into which a " Super Royal " sheet 
 naturally divides, leaving ample margins for the mucilage in case a " stretch " is to be made. 
 
20 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 A " Double Elephant " sheet being twice the size of a " Super Royal " divides equally well into 
 plates of the above size, but is preferable on account of its better quality. 
 
 To lay out the rectangles upon the paper, first locate the centre (see Fig. 32) by intersecting 
 diagonals, as at 0. These should not be drawn entirely across the sheet, but one of them will 
 necessarily pass a short distance each side of the point where the centre lies — judging by the eye 
 alone ; the second definitely determines the point. If the T-rule will not reach diagonally from 
 corner to corner of the paper (and it usually will not) the edge may be practically extended by plac- 
 ing a triangle against but projecting beyond it, as in the upper left-hand portion of the figure. 
 
 64. The T-rule being placed as shown, with its head at the left end of the board — the correct and 
 usual position — draw a horizontal line X F, through the centre just located. The vertical centre line 
 is then to be drawn, with one of the triangles placed as shown in the figure, i. e., so that a side, 
 as m n or t r, is perpendicular to the edge. 
 
 It is true that as long as the edges of the board are exactly at right angles with each other 
 we might use the T-rule altogether for drawing mutually perpendicular lines. This condition being, 
 however, rarely realized for any length of time, it has become the custom, a safe one — so long as 
 rule and triangle remain "true," to use them as stated. 
 
 The outer rectangles for the drawings (or " plates," in the language of the technical school) are 
 completed by drawing parallels, as /JV and Y N, to the centre lines, at distances from them of 9" 
 and 13" resj)ectively, laid off from the centre, 0. 
 
 An inner rectangle, as abed, should be laid out on each plate, with proper margins ; usually at 
 least an inch on the top, right and bottom, and an extra half inch on the left as an allowance for 
 binding. These margins are indicated by x and s in the figure, as variables to which any conven- 
 ient values may be assigned. The broad margin x in the upper rectangle will be at the draughts- 
 man's left hand if he turns the board entirely around — as would be natural and convenient — when 
 ready to draw on the rectangle Q Y. 
 
EXERCISES FOR PEN AND COMPASS. 21 
 
 CMAPTBR IV. 
 
 GRADES OF LINES.— LINE TINTING.— LINE SHADING.— CONVENTIONAL SECTION-LINING.— 
 FREQUENTLY RECURRING PLANE PROBLEMS.— MISCELLANEOUS PEN AND 
 
 COMPASS EXERCISES. 
 
 65. The various kinds of lines emploj-ed in mechanical drawing are indicated in the figure below, 
 and while getting his elementary practice with the ruling-pen the student may group them as shown 
 or in 
 
 any other symmetrical arrangement, either original with himself or suggested by other designs. 
 
 ^i-S- 33- 
 
 
 
 ^■^^ CENTRE LINE, if red. 
 
 "\ 
 
 ^^^ CENTRELINE, if black. 
 
 "\^ 
 
 ,..-'' FOR ORDINKRY OUTLINES. 
 
 ^-^. 
 
 HIDDEjl LINE 
 
 \^ 
 
 1 
 MEDIUM, Continuous. 
 
 "dotted LI N E_ usually employed as a construction line, \^ 
 
 SH(>DE 
 
 LI^E 
 
 -f- 
 
 "DOTTED LINE'1 line of motion in Kinematic Geometry. 
 
 DIMENSION LINE, if red. 
 
 DIMENSION LINE, black. 
 
 ^-.i 
 
 When drawing on tracing cloth or tracing paper, for the purpose of making blue-prints, all the 
 lines will preferably be black, and the centre and dimension lines distinguished from others by the 
 alternation indicated, as also by being somewhat finer than those employed for the light outlines of 
 the object drawn. Heavy, opaque, red lines may, however, be used, and will blue-print as fanit 
 white lines. There is at present no universal agreement among the members of the engineering pro- 
 fession as to standard dimension and centre lines. Not wishmg to add another to the systems 
 already at variance, but preferring to facilitate the securing of the uniformity so desirable, I have 
 presented those for some time employed by the Pennsylvania Railroad and now taught at Cornell. 
 The so-called " dotted " line is actually composed of short dashes. Its use, as a " line of motion," 
 was suggested at Cornell. When colors are used without intent to blue-print they may be drawn 
 
22 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 as light, continuous lines. Colors will further add to the intelligibility of a drawing if employed 
 for construction lines. Even if red dimension lines are used the arrow heads should invariably be black. 
 They should be drawn free-hand, with a writing pen, and their points touch the lines between which 
 they give the distance. 
 
 66. The utmost accuracy is requisite in pencilling, as the draughtsman should be merely a copy- 
 ist when using the pen. On a complicated drawing even the kind of line should be indicated at 
 the outset, so that no time will be wasted, when inking, in the making of distinctions to which 
 thought has already been given during the process of construction. No unnecessary lines should be 
 drawn ; or exceeding of the intended limit of a line if it can be avoided. 
 
 If the work is symmetrical, in whole or in part, draw centre lines first, then main outlines; and 
 continue the work from large parts to small. 
 
 The visible lines of an object are to be drawn first; afterward those indicated as concealed. 
 
 All lines of the same quality may to advantage be drawn with one setting of the pen, to ensure 
 uniformity ; and the light outlines before the shade lines. 
 
 In drawing arcs and their tangents ink the former first, invariably. 
 
 All the iiiking may best be done at once, although for the sake of clearness, in making a large 
 and comf)licated drawing, a portion — usually the nearest and visible parts — may be inked, the draw- 
 ing cleaned and the pencilling of the construction lines of the remainder continued from that point. 
 
 The inking of the centre, dimension and construction lines naturally follows the completion of 
 the main design. 
 
 67. In Fig. 34 we have a straight-line design usually called the " Greek Fret," and giving the 
 student his first illustration of the use of the " shade line " to bring a drawing out " in relief" 
 The law of the construction will be evident on ^ig-- s-a. 
 
 a b 
 
 examination of the numbered squares. 
 
 Without entering into the theory of shadows 
 at this point we may state briefly the " shop 
 rule" for draAving shade lines, viz., right-hand and 
 lower. That is, of any jjair of lines making the 
 same turns together or representing the limit of 
 the same flat surface, the right-hand line is the 
 heavier if the pair is vertical, but the loiver if they 
 run horizontally; always subject, however, to the 
 proviso that the line of intersection of two illuminated planes is never a shade line. 
 
 68. Another interesting exercise in ruled lines is the curve called the parabola, represented by 
 
 
 
 
 
 1 1 1 
 1 1 2 1 3 1 4 
 
 5 1 6 
 
 7 
 
 8 
 
 9 
 
 r^ 
 
 
 
 
 
 
 1 2 13 
 
 4 1 5 
 
 6 
 
 7 
 
 
 
 
 
 
 
 
 
 1 2 
 
 3- 1 4 
 
 5 
 
 
 
 
 
 
 
 
 
 
 1 
 
 2 i 3 
 
 
 
 
 
 
 
 
 
 
 
 
 1 1 2 
 
 3 
 
 4 
 
 5 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ] 
 
 
 
 
 
 1 2 13 
 
 4 
 
 M 
 
 its tangents (Fig. 35). The angle CAE may be assumed at pleasure, and on the finished drawing 
 
SECTION- LINING. — LINE-SHADING. 
 
 the numbers may be omitted, being given merely to show the law of construction. Like numbers 
 are joined and all spaces equal. 
 
 Some interesting mathematical properties of the curve will be found in a subsequent chapter. 
 
 69. A pleasing design that will test the beginner's skill is that of Fig. 36. It is suggestive of 
 a cobweb; and a skillful free-hand draughtsman could make it more realistic by adding the spider. 
 
 The 60° triangle may be used for the heavy diagonals and parallels to them, the T-role giving the 
 horizontals. 
 
 70. The even or flat effect of equidistant parallel lines is called line-tinting; or, if representing 
 an object that has been cut by a plane, as in Fig. 37, it is called section-lining. 
 
 The section, strictly speaking, is the part actually in contact with the 
 ^D cutting plane ; while the drawing as a whole is a sectional view, as it also 
 shows what is back of the plane of section — the latter being, presumably, 
 transparent. 
 
 Adjacent pieces have the lines drawn in different directions in order to 
 distinguish sufficiently between them. 
 
 The curved effect on the semi-cylinder is evidently obtained by prop- 
 erly varjing both the strength of the lines and the spacing. 
 
 71. The difference between the shading on the exterior and interior 
 of a cylinder is sharply contrasted in Fig. 38. The two right-hand curves might preferably be more 
 elliptical, and tangent to the straight lines on which they terminate. 
 
 ^^g-. 3S. 
 
 The spacing of the lines, in section-lining, depends upon the scale of the drawing. It may run 
 down to a thirtieth of an inch or as high as one-eighth ; but from a twentieth to a twelfth of an 
 
24 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 inch would be best adapted to the ordinary range of work. Equal spacing and not fine spacing 
 should be the object, and neither scale nor patent section-liner should be employed, but distances 
 gauged by the eye alone. 
 
 72. A refinement in execution which adds considerably to the effect is to leave a white line 
 between the top and left-hand outlines of each piece and the section lines. When purposing to pro- 
 duce this effect rule light pencil lines as limits for the line-tints. 
 
 73. If the various pieces shown in a section are of different materials there are four ways of 
 denoting the difference between them: — 
 
 (a.) By the use of the brush and certain water-colors, a method considerably employed in Europe, 
 but not used to any great extent in this country, probably owing to the fact that it is not appli- 
 cable where blue-prints of the original are desired. 
 
 The use of colors may, however, be advantageous!}' adopted when making a highly finished, 
 shaded drawing ; the shading being done first, in india ink or sepia, and then overlaid with a flat 
 tint of the conventional color. The colors ordinarily used for the metals are 
 Payne's gray or India ink for Cast Iron. 
 
 • Gamboge " Brass (outside view). 
 
 Carmine " Brass (in section). 
 
 Prussian Blue " Wrought Iron. 
 
 Prussian Blue with a tinge of Carmine " Steel. 
 
 IF'ig-. 3S_ 
 
 Last Iran. 
 
 BtEEl. 
 
 Wr't. Iran. 
 
 PElTlTiL. If. 5. 
 
 Standard Sections 
 
 
 Brass. 
 
 StnnE. 
 
 Wnnd. 
 
 Capper. 
 
 
 
 Brick. 
 
C X VEXTIO NA L SECTION-L IX IX G. 
 
 25 
 
 More natural effects can also be given b}' the use of colors, in representing the othiBr materials 
 of construction ; and the more of an artist the draughtsman proves to be the closer can he approx- 
 imate to nature. 
 
 Pale blue may be used for water lines; Burnt Sienna, whether grained or not, suggests wood; 
 Burnt Umber is ordinarily employed for earth; either Light Red or "\''enetian Red are well adapted 
 for brick, and a wash of India ink having a tinge of blue gives a fair suggestion of masonry; al- 
 though the actual tint and surface of any rock can be exactly represented after a little practice with 
 the brush and colors. These points will be enlarged upon later. 
 
 (6.) By section-lining with the drawing pen in the conventional colors ju.st mentioned, a iirocess 
 giving very handsome and thoroughly intelligible results on the original drawing but, as before, un- 
 adajjted to blue-printing and therefore not as often used as either of the followinu' methods. 
 
 (c.) B}'- section-lining uniformly in ink throughout and printing the name of the material upon 
 each piece. 
 
 , (d) By alternating light and heavy, continuous and broken 
 lines according to some law. Said " law " is, unfortunately, by 
 no means universal, despite the attempt made at a recent con- 
 vention of the American Society of Mechanical Engineers to secure 
 uniformity. Each draughting office seems at present to be a 
 law unto itself in this matter. 
 
 74. As ailbrding valuable examp)les for further exercise with 
 the ruling pen the system of section-lines adopted by the Penn- ^^^^ 
 sylvania Railroad is presented on the opposite page. The ivovd //^ 
 section is an exception to the rule, being drawn fi-ee-hand with 
 a Falcon pen. 
 
 By way of contrasting free-hand with me- 
 
 ^^i^- ^O. 
 
 chanical work Fig. 40 is introduced, in which rubble masonry 
 the rinss showing annual growth are drawn 
 
 as concentric circles with the compass. 
 
 In Fig. 41 a few other sections appear, 
 ^^ selected from the designs of M. N. Forney 
 and F. \m\ Vleck, and which are fortunate arrangements. 
 
 75. Figs. 42 and 43 are profiles or outlines of mouldings, 
 such as are of frequent occurrence in architectural work. It is 
 good practice to convert such views into oblique projections, giving the effect of solidity ; and to 
 further bring out their form by line shading. Figs. 44-46 are such representations, the front of each 
 "being of the same form as Fig. 42. The obhque lines are all parallel to each other and — where 
 
 COURSED RUBBLE MASONRY 
 
 u.jht i,„i:.: :„k. 
 
 VULCANITE 
 India Ink. 
 
 BRICK 
 Venetian Red. 
 
 CONCRETE 
 Yellow Ochre. 
 
 ^'ig-- -is 
 
 X'ig- 
 
 J >) 
 
 
 
 -^ \^ 
 
 <_ 
 
 1^ 
 
 =.i 
 
 1 
 
 IFig-- 
 
 visible throughout— of the same length. Their direction should be chosen with reference to best ex- 
 hibiting the peculiar features of the object. Obviously the view in Fig. 44 is the least adajated to 
 the conveying of a clear idea of the moulding, while that of Fig. 46 is evidently the best. 
 
26 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 76. The student may, to advantage, design jDrofiles for mouldings and line-shade them, after 
 converting them into oblique views. As hints for such work two figures are given (47-48), taken 
 
 X^ig-. -5:5- ^ig-- -is. 
 
 from actual constructions in wood. By setting a moulding vertically, as in Fig. 49, and projecting 
 horizontally from its points, a front view is obtained, as in Fig. 50. 
 
 ^^ig- -iS- 
 
 E^ig-. SO. 
 
 IPig-. -5;S. 
 
 77. The reverse curves on the mouldings may be drawn with the irregular curve, (see Art. 58); or, 
 if composed of circular arcs to be tangent to vertical lines, by the followiag ^^er- si. 
 construction : — 
 
 Let M and N be the points of tangency on the verticals Mm and Nn; 
 and let the arcs be tangent to each other at the middle point of the Ime M N. 
 Draw Mn and N m perpendicular to the vertical lines. The centres, c and Cj, 
 of the desired arcs are at the intersection of M n and Nm by perpendiculars 
 to If iV from X and y, the middle points of the segments of M N. 
 
 78. The light is to be assumed as coming in the usual direction, i. e., descending from left to right 
 at such an angle that any ray would be projected on the paper at an angle of 45° to the horizontal. 
 
 In Fig. 43 several rays are shown. At x, where the light strikes the cylindrical portion most 
 directly — technically is normal to the surface — is actually the brightest part. A tangent ray s t gives 
 t the darkest part of the cyliirder. The concave portion beginning at o would be darkest at o and 
 get lighter as it approaches y. 
 
 Flat parts are to be left either white, if in the light, or have equidistant lines if in the shade; 
 unless the most elegant finish is desired, in which case both change of space and gradation of line 
 
 must be resorted to as in Fig. 52, which represents a front view of a 
 hexagonal nut. The front face being ijarallel to the paper receives an 
 even tint. An inclined face in the light, as a b hf, is lightest toward 
 the observer, while the unillumined face th d g is exactly the reverse. 
 Notice that to give a fled, effect on the inclined faces the spacing- 
 out as also the change in the size of lines must be more gradual than 
 when indicating curvature. (Compare with Figs. 46 and 50.) 
 
 ■Fi.g. S2- 
 
REiMARKS ON SHAD ING.-P LANE PROBLEMS. 27 
 
 If two or more illuminated flat surfaces are parallel to the pajjer (as t g b h, Fig. 52) but at 
 different distances from the eye, the nearest is to be the lightest; if unilluminated, the reverse would 
 be the case. 
 
 79. In treating of the theorj' of shadows distinctions have to be made, not necessary here, between 
 real and apparent brilliant points and lines. We may also remark at this point that to an experi- 
 enced draughtsman some license is always accorded, and that he can not be expected to adhere 
 rigidly to theory when it invoh'es a sacrifice of effect. For examijle, in Fig. 46 we are unable to 
 see to the left of the, theoretically, lightest part of the cylinder, and find it, therefore, advisable 
 to move the darkest part p^st the point where, according to Fig. 43, we know it in reality 
 to be. The professional draughtsmen who draw for the best scientific papers and to illustrate the 
 circulars of the leading machine designers allow themselves the latitude mentioned, with most pleasing 
 results. Yet until one may be justly called an expert he can depart but little from the narrow 
 confines of theory without being in danger of producing decidedly peculiar effects. 
 
 80. As irom this point the student will make considerable use of the compasses, a few of the more 
 important and frequently recurring plane problems, nearly all of which involve their use, may well 
 be introduced. The proofs of the geometrical constructions are in several cases omitted, but if desired 
 the student can readily obtain them by reference to any synthetic geometry or work on plane problems. 
 
 All the problems given (except No. 20) have proved of value in shop practice and architectural work. 
 
 The student should again read Arts. 48-51 regarding special uses of the 30° and 45° triangles, 
 which, with the T-rule, enable him to employ so many " draughtsman's " as distinguished from 
 " geometrician's " methods ; also Arts. 36 and 37. 
 
 81. Prob. 1. To draw a perpendicular to a given line at a given pointy as A, (Fig. 53), use the tri- 
 angles, or triangle and rule as previously described ; or lay off equal distances A a, A b, and with 
 a and b as centres draw arcs o s t, m s n, with common radius greater than one-half a b. The required 
 perpendicular is the line joining A with the intersection of s'j.g. 53; 
 
 JVI X « K ^ 
 
 these arcs. «- A i 
 
 82. Prob. 2. To bisect a line, as M N, use its extremities 
 
 exactly as a and b were employed in the preceding construe- -^ 
 tion, getting also a second pair of arcs (same radius for all o. / 
 the arcs) intersecting above the line at a point we may call x. ^^^^ 
 The line from s to x will l^e a bisecting perpendicular. nv. -"'^^ f 
 
 83. Prob. 3. To bisect an angle, as AVE, (Fig. 54) take on its sides any equal distances Va, V 
 
 b. Use a and b as centres for intersecting arcs having a com- ^^s- s-a. 
 mon radius. Join V with x, the intersection of these arcs, for 
 the bisector required. 
 
 84. Prob. 4. To bisect an arc of a circle, as am b (Fig. 54) 
 bisect the chord a c 6 by Prob. 2 ; or, by Prob. 3, bisect the 
 angle a V b which subtends the arc. ^^ ■. \I2l. 
 
 85. Prob. 5. To construct an angle equal to a given angle, let B be the given angle, (Fig. 55) draw 
 any arc a b with centre 0, then, with same radius, an indefi- i^ig-- ss. 
 
 nite arc 7)i B, centre V. Use chord of a 6 as a radius and 
 from centre B cut the arc m B at x. Join V and x. 
 
 86. Prob. 6. To pass a circle through three points a, b and 
 
 c, join them by lines a b, b c, bisect these lines by perpen- 
 diculars and the intersection of the latter will be the centre T^ 
 of the desired circle. 
 
28 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 ^igr- se. gy_ Prob. 7. To divide a line into any number of equal parts draw 
 
 from one extremity as A (Fig. 56) a line A C making any random 
 angle with the given line A B. With a scale point off on AC as 
 many equal parts (size immaterial) as are required on A B ; four, for 
 example. Join the last point of division (4) with B and parallels 
 to such line from the other points will divide A B similarly. 
 88. A secant to a curve is a line cutting it in two points. If the secant A B be turned to the 
 s'ig-. ST-. jgf^ about ^-i the point B will aj^proach A and the line at one time coincide 
 
 ^^^;::rr::^=^r'' ' with .1 C. If turned so that B reaches and coincides with A the line will 
 
 :Fig-. se- 
 
 .B\ then touch the curve in but one point and is then called a. tangent. 
 The normal to a curve is perpendicular to the tangent, at the point of tangency. In a circle it 
 coincides in direction with the radius to the point of tangency. 
 
 89. Prob. 8. To draw a tangent to a circle at a given point draw a radius to the point. The per- 
 pendicular to this radius at its extremity will be the required tangent. Solve with triangles. 
 
 90. Prob. 9. To draw a tangent to a circle from a point withoid, join 
 the centre C (Fig. 58) with the given point A ; describe a semi-circle 
 on yl C as a diameter and join A with D, the intersection of the arcs. 
 A D C equals 90°, being inscribed in a semi-circle ; A D is therefore 
 perpendicular to radius CD at its extremity, hence tangent to given 
 circle. 
 
 91. Prob. 10. To draio a tangent at a given point of a circular arc 
 whose centre is unknown or inaccessible, locate on the arc two points equidistant from the given point 
 and on opposite sides of it ; join them by a chord and draw through the given point a parallel to 
 the latter. 
 
 92. A regular polygon has all its sides equal, as also its angles. If of three sides it is called the 
 equilateral triangle; four sides, the square; five, pentagon; six, hexagon; seven, heptagon ; eight, octagon; 
 nine, nonagon or enneagon; ten, decagon; eleven, undecagon; twelve, dodecagon. The angles of the 
 more important regular jjolygons are 
 
 Triangle 
 
 Square 
 
 Pentagon 
 
 Hexagon 
 
 Octagon 
 
 Decagon 
 
 Dodecagon 
 
 120° at the centre 
 
 90° 
 
 
 
 72° 
 
 
 
 60° 
 
 
 
 45° 
 
 
 
 36° 
 
 
 
 30° 
 
 
 
 60° at the circumference. 
 90° " " 
 
 I'lg-. ss. 
 
 108° " " 
 
 120° " " " 
 
 135° " " " 
 
 144° " " '■ 
 
 150° " " 
 93. For the polygons most frequentlj' occurring there are many special 
 methods of construction. All but the pentagon and decagon can be readily 
 inscril:)ed or circumscribed about a circle by the use of the T-rule and triangles. 
 For example, draw a b (Fig. 59) with the T-rule and c d perpendicular to it 
 ^ with a triangle. The 45° triangle will then give the square a c b d. The same 
 triangle in two positions would give ef and g h, whence a g, g c, &c., would be 
 sides of a regular octagon. 
 
 94. The 60° triangle used as in Art. 51 would give the hexagon; and alter- 
 nate vertices of the latter, joined, would give the equilateral triangle. Or the radius of the circle 
 stepped off six times on the circumference and alternate points connected would resvilt similarly. 
 
PLANE PROBLEMS. 
 
 29 
 
 95. Prob. 11. An additional method for inscribing an equilateral triangle in a cir- 
 cle, when one vertex of the triangle is given, as A, is to draw the diameter, A B, through 
 A, and use the triangle to obtain the sides A and A D, making angles of 30° 
 with A B. D and C will then he the extremities of the third side of the triangle 
 sought. 
 
 96. Prob. 12. To inscribe a circle in an equilateral triangle draw a perpendicular " 
 from any vertex to the opposite side. The centre of the circle will lie on such 
 line, two-thirds of the distance from vertex to base, while the radius desired will 
 be the remaining third. 
 
 97. Prob. 13. To inscribe a circle in any triangle bisect any two of the interior 
 angles. The intersection of these bisectors will be the centre and its perpendicu- 
 lar distance from any side the radius of the circle sought. 
 
 98. Prob. 14. To inscribe a pentagon in a circle draw mutually perpendicular 
 diameters (Fig. 62); bisect a radius as at s; draw arc ax of radius s a and cen- 
 tre s; then chord a x = af, the side of the jjentagon to be constructed. 
 
 99. Prob. 15. To construct a regular polygon of any number of sides, length of 
 side given. 
 
 Let A B (Fig. 63) be the length assigned to a side, and a regular polygon of 
 X sides desired. Take x equal to nine for illustration, draw a semi-circle with A B 
 jy trial into x (or 9) equal parts. Join B with x — 2 points of division, 
 or seven, beginning at A, and prolong all l:iut the last. With 7 as a 
 centre, radius A B, cut line B 6 at m by an arc and join m with 7, 
 giving another side of the required polygon. Using m in turn as a 
 centre, same radius as before, cut B 5 (produced) and so obtain a 
 third vertex. 
 
 This solution is based on the familiar principles (a) that if a reg- 
 
 AA ular polygon has x sides each interior angle equals ~- — —, and (b) 
 
 that the diagonals "drawn from any vertex of the polygon make the same angles with each other as 
 
 = -Jths of the two 
 
 
 as radius and divide it 
 
 Fig-- S3- 
 
 with the sides meeting at that vertex. For a; = 9 we have .-1 B s = 
 right angles about B. 
 
 100. Prob. 16. Another solution of Prob. 15. Erect a 
 perpendicular H R (Fig. 64) at the middle point of the 
 given side. With If as a centre, radius M S, describe 
 arc S A and divide it by trial into six equal parts. 
 Arcs through these points of division, using ^ as a centre, 
 and numbered up fi-om six, give the centres on tire ver- 
 tical line for circles passing through M and S and in 
 which MS would be a chord as many times as the num- 
 ber of the centre. 
 
 101. For any unusual number of sides the method 
 by." trial and error" is often resorted to, and even for 
 ordinary cases it is by no means to be desiDised. By 
 it the dividers are set " by guess " to the probable chord 
 of the desired arc and, supposing a heptagon wanted, 
 the chord is stejDped off seven times around the cir- 
 
 180° (9—2) 
 9 
 
 ^ig-_ S-^. 
 
30 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 PLATE L 
 
PLANE PROBLEMS. 
 
 31 
 
 cumference; care being taken to have the points of the dividers come exactly on the arc and also to 
 avoid damaging the pajjer. If the seventh step goes past the starting point the dividers require 
 closing ; if it falls short, the original estimate was evidently too small. Obviously the change in setting 
 the dividers ought in this case to be, as nearly as possible, one-seventh of the error; and after a 
 few trials one should " come out even " on the last step. 
 
 102 Prob. 17. To lay off on a gwen circle an arc of the same length as a given straight line} Let t 
 (Plate I, Fig. 1) be one extremity of the desired arc ; t s the given straight line and tangent to the 
 circle ; t m equal one-fourth of t s, and s x drawn with centre m, radius m s. Then the length of the 
 arc t X is a close approximation to that of the line t s. 
 
 103. Prob. 18. To lay off on a straight line the length of a given circular arc ' or, technically, to 
 rectify the arc, let af (Plate I, Fig. 3) be the given arc; ai the chord prolonged till fi equals one- 
 half the chord af; and ae an arc drawn with radius a i, centre i. Then fe approximates closely 
 to the length of the arc af 
 
 104. Prob. 19. To obtain a straight line equal in length to a given semi-circle,'' draw a diameter 
 h of the given semi-circle (Plate I, Fig. 2) and a radius inclined at an angle of 30° to the radius 
 c h. Prolong the radius to meet the line b h I, drawn tangent to the circle at h. From k lay off 
 the radius three times, reaching n. The line n o equals the semi-circumference to four places of 
 decimals. 
 
 105. Prob. 20. To clraic a circle tangent to two straight lines and a given circle. (Four solutions.) 
 This problem is given more on account of the valuable exercise it will prove to the student in ab- 
 solute precision of construction than for its probable practical applications. Fig. 4 (Plate I) illus- 
 trates the geometrical principles involved and in it a circle is required to contain the points s and 
 a and be tangent to the line m v^. Draw first any circle containing s and a, as the one called " aux. 
 
 1 These methods of approximation were devised hy Prof. Eanklne. They are sufflolently accurate for arcs not exceeding 
 60°. The error varies as the fourth power of the angle. The complete demonstration of Prob. 17 can be found in the Philo- 
 sophical Magazine of October, 1867, and of Prob. 18 in the November issue of the same year. 
 
 - In his Graphical Statics Cremona states this to be the simplest method known for rectifying a semi-circumference. Accord- 
 ing to Bottcher it is due to a Polish .Jesuit, Kochansky, and was published in the Acta Eruditorum Lipsiae, 1685. 
 
 Demonstration. Calling the radius unit;/, the diameter would have the numerical value J. 
 
 Then on = x/oh-: + hn^ = ^/oh^ -1- (kn — kh)-' = \/4 -f- (3 — tan 30°)= = 3.14153 + 
 
 The tangent of an angle (abbreviated to "tan.") is a trigonometric function whose numerical value can be obtained from 
 a table. A draughtsman has such frequent occasion to use these functions that they are given here for reference, both as lines 
 and as ratios. 
 
 Trigonometric Functions as Ratios, 
 e = the given angle = CA B 
 h = hypothenuse of triangle CAJ3 
 a = A B = side of triangle adjacent to vertex of 9 
 = B C= side of triangle opposite to d 
 
 Then sin 9 = ? ; cos 9 = ?. ; 
 
 Trigonometric Functions as Lines. 
 
 Cu-tangent of Q 
 
 Ian 9 = — = 
 
 a 
 
 sec e = 'i = 
 a 
 
 eosec e = — 
 
 
 
 cotan 9 = - ; 
 
 h' 
 
 sin 9 , 
 cos e ' 
 
 reciprocal of cosine. 
 " " sine 
 
 cos < 
 sin { 
 
 reciprocal of tan 9. 
 
 As lines, the functions 
 
 The prefix " co " suggests " complement," the co-sine of 9 is the sine of the complement of 9, 
 may be defined as follows : 
 
 The sine of an arc (e. g., that subtended by angle 9) is the perpendicular {C B) let fall from one extremity of the arc 
 upon the diameter passing through the other extremity. If the radius A C, through one extremity of the arc, be prolonged 
 to cut a line tangent at the other extremity, the intercepted portion of the tangent is called the tangent of the arc, and the 
 distance, on such produced radius, from the centre of the circle to the tangent, is called the secant of the are. 
 
 The co-sine, co-secant and co-tangent of the arc are respectively the sine, secant and tangent of the complement of the 
 given arc. 
 
32 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 circle." Join s to a and prolong to meet m v^ at I: From k draw a tangent, k g, to the auxiliary 
 circle. With radius k g obtain m and i on the line m v. A circle through s, a and m or through 
 s, a and i will fulfill the conditions. For k g' = k s x k a, as kg is a tangent and ^' s a secant. 
 But k i ■== k g therefore k i' = k s x ka which would make ki a, tangent to a circle through s, a and i. 
 
 In Fig. 5 (Plate I) the construction is closely analogous to the above and the lettering identi- 
 cal for the first half of the work. The " given circle " is so called in the figure ; the given lines 
 are P v and R v. Having drawn the bisector, v e, of the angle P v R, locate s as much below v e as 
 a (the centre of the gi\'en circle) is above it, the line a s being perpendicular to v e. Draw v^ m k i 
 parallel to v p and at a distance from it equal to the radius of the given circle. Then s, o, k and 
 m v^ of Fig. 5 are treated exactly as the analogous points of Fig. 4, and a circle obtained (centre d) 
 containing a, s and i. The required circle will have the same centre d but radius d w, shorter than 
 the first by the distance iv i. Treat s, a, and m, (Fig. 5), similarly, getting the smallest of the four 
 possible circles. 
 
 The remaining solutions are obtained by using the points a and s again but in connection with 
 a line y z parallel to v R and inside the angle, again at a perpendicular distance from one of the 
 given lines equal to the radius of the given circle.^ 
 
 This problem makes a handsome plate if the given and required lines are drawn in black; the 
 lines giving the first two solutions in red; and the remaining construction lines in blue. 
 
 106. Prob. 21. 
 
 To draw a tangent to two given circles (a problem, that may occur in connecting 
 band-wheels by belts) join their centres, c and o (Fig. 67) and 
 at s lay off s in and s n each equal to the radius of the smaller 
 circle. Describe a semi-circle o h k c on o c as a diameter. Carry 
 m and n to k and h, about o as a centre. Angles c k o aiid 
 c h are each 90°, being inscribed in a semi-circle ; and c k is 
 parallel to a b, which last is one of the required tangents ; while 
 c h is jjarallel to t x, a second tangent. Two more can be sim- 
 ilarly found. 
 To unite tivo inclined straight lines by an arc tangent to both, radius given. Prolong 
 the given lines to meet at a (Fig. 68). With a as a centre and 
 the given radius describe the arc m n. Parallels to the given lines and 
 tangent to arc inn meet at d, from which perpendiculars to the given 
 lines give the points of tangency of the :e'5-s- es. 
 
 required arc which is then drawn with A -^^'n 
 
 the given radius. ' * ^ ! :^ 
 
 108. Prob. 23. To draw through a q^ 
 
 -M, 
 
 given jwint a line which loill — if produced— pass through the inaccessible 
 
 Prob. 22. 
 
 E'ig'. es. 
 
 1 This solution is taken from Benjamin Alvord's Tangencies of Circles and of Spheres, published hy the Smithsonian Institution. 
 That valuable pamphlet presents geometrical solutions ot the ten problems of Apollonius on the tangencies of circles and also 
 of the fifteen problems on the tangencies of spheres. These solutions are all based on the principle illustrated by Fig. 57, and 
 applied in Prob. 20, that the tangent line or tangent curve is the limit of all secant lines or curves. The ten problems on the 
 tangencies of circles, as proposed by the Greek geometer were the following :— 
 
 Prob. 1. To draw a circle through three points. One solution. 
 
 Prob. 2. Circle through two points and tangent to a given straight line. Two solutions. 
 
 Prob. 3. Circle through a given point and tangent to two straight lines. Two solutions. 
 
 Prob. 4. Circle through two points and tangent to a given circle. Two solutions. 
 
 Prob. 5. Circle through a given point, tangent to a given straight line and a given circle. Four solutions. 
 
 Prob. 6. Circle through a given point and tangent to two given circles. Four solutions. 
 
 Prob. 7. Circle tangent to three straight lines, two only of which may be parallel. Four solutions. 
 
 Prob. S. Circle tangent to two straight lines and a given circle. Four solutions. 
 
 Prob. 9. Circle tangent to two given circles and a given straight line. Eight solutions. 
 
 Prob. 10. Circle tangent to three given circles. Eight solutions. 
 
PLANE PROBLEMS.— TAPERING CIRCULAR ARCS. 
 
 33 
 
 E'ig-- 70- 
 
 intersection of two lines. Join the given point e with any point /' on A B and also witli some point 
 gr on CD. From any point h on A B draw hi p)arallel to / <;, then i k parallel to g e and h k 
 parallel to ,/' e. The line k e will fulfill the conditions. 
 
 109. Prob. ^4. To draiv an oval upon a given line. Describe a circle on the given line, m n, (Fig 
 
 70) as a diameter. With in and n as centres describe arcs, m x, nx 
 radius hi n. Draw m v and n t through v and t, the middle points 
 of the quadrants y m, y n. Then in s and n r are the portions of 
 in X and n x forming part of the oval. Bisect n c at f] and draw q x. 
 Also bisect c q at z and join the latter with x. Bisect y b in d and 
 draw / d from /, the intersection of n s and q x. Use / as a centre 
 and fs as radius for an arc sk terminating on f d. The intersec- 
 tion, h, of kf with a; z is then the next centre and hk the radius 
 of the arc k I which terminates on h y produced. The oval is then 
 
 n 
 
 completed with y as & centre and radius y I. The lower jjortion is symmetrical with the upper and 
 therefore similarly constructed. 
 
 110. Where exact tangency is the requirement novices occasionally endeavor to conceal a failure 
 to secure the desired object by thickening the curve. Such a course usually defeats itself and makes 
 more evident the error they thus hofie to conceal. With such instruments of precision as the draughts- 
 man employs there is but little, if any, excuse for overlaf)ping or falling short. 
 
 i^igr "71. ^ common error in drawing tangents where the lines are of appreciable 
 
 thickness is to make the outsides of the lines touch ; whereas they should 
 have their thickness in common at the point of tangency, as at T (Fig. 
 70), where, evidently, the centre-lines a and b of the arcs would be exactly 
 tangent, while the outer arc of M would come tangent to the inner arc of 
 N. 
 
 111. When either a tube or a solid cylindrical i^iece is seen in the direction of its axis the 
 ^Lg. Ts.. outUne is, evidently, simply a circle; and often the only way to determine 
 
 which of the two the circle rejDresented would be to notice which part of said 
 ^(tt' OIOQ -G^ end view was represented as casting a shadow. In Fig. 72, if the shaded 
 arcs can cast shadows, the si^ace imide the circles must be open, and the fig- 
 ure would rej^resent a portion of the end view of a boiler with its tubular 
 openings. 
 
 By exactly reversing the shading, the effect of which can be seen by turn- 
 ing the figure upside down, it is converted into a drawing of a number of 
 solid, cylindrical jjins projecting from a plate. 
 
 The tapering begins at the points where a diameter at 45° to the horizontal would cut the cir- 
 cumference. 
 
 To get a perfect taper on small circles use the bow-pen and after making one complete circle 
 add the extra thickness by a second turn which is to begin with the jjen-point in, the air, the pen 
 being brought down gradually upon the paper and then, while turning, raised from it again. 
 
 On medium and large circles the requisite taper can be obtained by a different process, viz., by 
 using the same radius again but by taking a second centre, distant from the first by an amount equal 
 to the proposed width of the broadest part of the shaded arc • the line through the two centres to 
 be perpendicular to that diameter which i^asses through the extremities of the taper. 
 
 -e 
 
 O 
 
 -0 
 
 
 ^0 
 
 ^Q\eB 
 
 eepo^e 
 
 G @0^ 
 
 WiWt^ 
 
34 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 112. As an]][exercise in concentric circles Fig. 73 Tvill prove a good test of skill. It is a fair 
 representation of a gymnasium ring, the " annular torus " of mathematical works, and possessing 
 
 ng- TS- 
 
 some remarkable properties, chief anions which is. the fact that it is the only surface of revolution 
 known fi-om which circles can be cut by^three different svstems of nlanes. 
 
 \ »fr [M 
 
 E'ig-- '7S,. 
 
 113. In Fig. 74 the same surface is shown, in the centre, in outline only. The axis of the 
 surface would be a perpendicular to the paj^er at A. If M N represents a plane perpendicular to 
 the paper and containing the axis, then Fig. X will show the shape of the cut or sectimi. As M N 
 was but one of the positions of a plane containing the axis and as the surface might be generated 
 by rotating MN with the circle ab about the axis, it is evident that one of the three systems of planes 
 must contain the axis. 
 
 When a surface can be generated bj^ revolution about an axis one of its characteristics is that 
 any plane perpendicular to the axis ivill cut it in a circle. The circles of Fig. 73 may then be, for the 
 moment, considered as parallel cuts by a series of jjlanes perjjendicular to the axis, a few of which 
 may be shown in mn, op, &c. (Fig. X). Each of these planes cuts two circles from the surface; 
 the plane o p, for example, giving circles of diameters c d and v lo respectively. 
 
 * OUvier, Memoires de Geometrie Descriptive. Paris, 1851. 
 
ANNULAR TORUS. — WARPED HYPERBOLOID. 35 
 
 A plane perpendicular to the iiaper, on P Q, vrould be a bi-tangent plane, because tangent to 
 the surface at two jioints, P and Q; and such plane would cut tuv over-lapping circles fi-om the 
 torus, each of them running partly on the inner and partly on the outer jjortion of the surface. 
 For the proof that such sections are circles the student is probabh' not prepared at this point, but 
 is referred to OUvier's Seventh Memoir. 
 
 114. Another interesting fact with regard to the torus is that a series of j)lanes parallel to, but 
 not containing the axis, cut it in a set of curves called the Cassian ovals, of which the Lemniscate 
 of Art. 158 is a special case, and which would result frona using a plane parallel to the axis and 
 tangent to the surface at a point on the smallest circle, as at a, (Fig. 74.)* 
 
 115. Fig. Y is given to illustrate the fact that from mere untapered outlines, such as compose 
 the central figure, we cannot determine the form of the object. By shading ehf and DNr, the form 
 shown in Fig. Y would lie instantly recognized without the drawing of the latter. An angular object 
 must therefore have shade lines, as also the end view of a round object ; but a side view of a cyl- 
 indrical jiiece must either have uniform outlines or be shaded with several lines. 
 
 Thus, in Fig. 76, A would represent an angular piece while B would indicate a circular cylin- 
 der; if elliptical its section would be drawn at one side as shown. 
 
 3 
 
 116. Before presenting the crucial test for the learner — the railroad rail— two additional jDractice 
 exercises, mainly in ruling, are given in Figs. 77 and 78. The former, shows that, like the parabola, 
 the circle and hyperbola can be represented by their enveloping tangents. The upper and lower 
 figures are merel_y two views of the surface called the warped hyperboloid, from the hyperbolas which 
 constitute the curved outhnes seen in the upper figure. The student can make this surface in a 
 few moments by stringing threads through equidistant holes arranged in a circle on two circular 
 discs, of the same or different sizes, but having the same number of holes in each disc. By attaching 
 weights to the threads to keep them in tension at all times, and giving the upper disc a twist, the 
 surface will change from C3dindrical or conical to the hyperboloidal form shown. 
 
 Gear wheels are occasionally constructed, having their teeth upon such a surface and in the 
 direction of the lines or elements forming it; but the hyperljoloid is of more interest mathematically 
 than mechanically. 
 
 Begin the drawing by pencilling the three concentric circles of the lower figure. When inking, 
 omit the smaller circle. Draw a series of tangents to the inner -circle, each one beginning on the mid- 
 dle circle and terminating on the outer. Assume any vertical height, t s', for the upper figure, and 
 draw H' 31' and P' R' as its upper and lower limits. H' M' is the vertical projection, or elevation, 
 of the circle H K M N, and all points on the latter as 1,2,3,4, are projected, by perpendiculars to 
 H' M', at /, 2', 3', If, etc. All points on the larger circle P Q,R are similarly projected to P' R'. The 
 extremities of the same tangent are then joined in the upper view, as 1' with 1 (a). 
 
 Part of each line is dotted to represent its disappearing upon an invisible portion of the sur- 
 face. The law of such change on the lower figure is evident from inspection ; wdiile on the eleva- 
 
 • These curves can also be obtained by assuming two foci, as if for an ellipse, but taking the product of the focal radii as 
 a constant quantity, some perfect square. If pp' = 36" then a point on the curve would be found at the intersection of arcs 
 having the foci as centres, and for radii 2" and 18", or 4" and 9", etc. When the constant assumed is the square of half the 
 distance between the foci, the Lemniscate results. 
 
36 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 E-lg-. 77. 
 
 2' 3' 4' 
 
 3((I)2W1 !(,>) 
 
 32 31 
 
 13 14 
 
TAPERING LINES. — RAIL SECTIONS. 
 
 37 
 
 E'ig'. TS- 
 
 tion the point of division on each line is exactly above the point where the other view of the same 
 line runs through H M in the lower figure. 
 
 117. To reproduce Fig. 78 draw first the circle afbn, then two circular arcs which would con- 
 tain a and b if extended, and whose greatest distance from the original circle is x, (arbitrar_y). Six- 
 teen equidistant radii as at a, c, d, etc., are next in order, of which the rule and 45° triangle give 
 those through a, d,f and h. At their extremities, as m and n. lay off the desired width, y, and draw 
 toward the points thus determined lines radiating from the centre. Terminate these last upon the inner 
 arcs. Ink by drawing from the centre, not through or toward it. 
 
 All construction lines should be CTased before the tapering lines are filled in. The "filling in" 
 may be done ver_y rapidly by ruling the edges of the line in fine at first, then opening the pen 
 slightly and beginning again where the opening between the lines is apparent and ruling fi-om there, 
 adding thickness to each edge on its inner side. It will then be but a moment's work to fill in, free- 
 hand, with the Falcon pen or a fine-pointed sable-brush, between the now heavy edge-lines of the 
 
 taper. To have the pen make a coarse line 
 when starting from the centre would destroy 
 the effect desired. 
 
 118. The draughtsman's ability can scarcely 
 be put to a severer test on mere outline work 
 than in the drawing of a railroad rail, so many 
 are the changes of radii involved. 
 
 As previously stated, where tangencies to 
 straight lines are required, the arcs are to be 
 f drawn first, then the taiigents. 
 
 Figs. 79 and 80 are photo-engravings of 
 rail sections, showing two kinds of "' finish." 
 Fig. 80 is a "working drawing" of a Penn- 
 sylvania Railroad rail, full-size. If finished with 
 shade lines, as in Fig. 79, section-lined with 
 Prussian blue, and the dimension lines drawn 
 in carmine this makes one of the handsomest 
 plates that can be undertaken. 
 
 A still higher effect is shown in the wood- 
 cut of the title-jDage, the rail being represented in oblique projection and shaded. 
 
 Begin Fig. 80 by drawing the vertical centre-line, it being an axis of symmetry. Upon it lay 
 off 5" for the total height, and locate two points between the top and base at distances from them 
 of 1\" and I" respectively; these to be the points of convergence of the lower lines of the head and 
 sloping sides of the base. From these points draw lines, at first indefinite in length, and inclined 
 13° to the horizontal. The top of the head is an arc of 10" radius, subtended by an angle of 9°. 
 This changes into an arc of y^" radius on the upper corner, with its centre on the side of said 9° 
 angle. The sides of the head are straight lines, drawn at 4° to the vertical and tangent to the corner arcs. 
 The thin vertical portion of the rail is called the web and is ^" wide at its centre. The outlines of the 
 web are arcs of 8" radius, subtended by angles of 15°, centres on line marked "centre line of bolt holes." 
 The weight per yard of the rail shown is given as eighty-five pounds, from which we know the 
 area of the cross-section to be eight and one-half square inches, since a bar of iron a yard long 
 and one square inch in cross section weighs, approximately, ten pounds. (10.2 lbs., average.) 
 
38 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 The proportions given are slightly different from those recommended in the report* of the com- 
 mittee appointed by the American Society of Civil Engineers to examine into the jjroper relations to 
 
 each other of the sections of railway wheels and rails. There was quite general agreement as to 
 
 the following recommendations: a top radins of twelve inches; a quarter-inch corner radius; vertical sides 
 
 ^'ig-- eo. 
 
 ^ "Tf: 
 
 to the web; a lower-corner radius of one-sixteenth inch, and, lastly, a broad head relatively to the depth. 
 
 •Transactions A. S. C. E. January, 1891. 
 
EXERCISES FOR THE IRREGULAR CURVE. 
 
 39 
 
 CHAPTER V. 
 
 THE HELIX.— COyiC SECTIONS.— HOMOLOGICAL PLAXE CUEYES AND SPACE-FIGURES —LINK-MOTION 
 CUEVES.-CENTROIDS.-THE CTCLOID.— COMPANION TO THE CYCLOID.- THE CURTATE TRO- 
 CHOID.-THE PROLATE TEOCHOID.-HYPO-, EPI-, AND PEKI-TEOCHOIDS.-SPECIAL TROCHOIDS- 
 ELLIPSE, STRAIGHT LINE, LIMAgON, CARDIOID, TRISECTRIX, INYOLUTE, SPIRAL OF ARCHI- 
 MEDES.— PARALLEL CUEYES.— CONCHOID.— QUADEATEIX.— CISSOID.— TRACTRIX.— WITCH OF 
 AGNISL-CARTESIAN OYALS.-CASSIAN OYALS.-CATENAEY.-LOGAEITHMIC SPIEAL.— HYPER- 
 BOLIC SPIRAL.-THE LITUUS. 
 
 119. There are many curves which the draughtsman has frequent occasion to make whose con- 
 struction involves the use of the irregular curve. The more important of these are the Helix ; Conic 
 Sections — Ellipse, Parabola and Hyperbola ; Link-motion curves or jJoint-paths ; Centroids ; Trochoids ; the 
 Involute and the Sj^iral of Archimedes. Of less practical importance, though equally interesting 
 geometrically, are the other curves mentioned in the heading. 
 
 The student should become thoroughlj^ acquainted with the more important geometrical proper- 
 ties of these curves, both to facilitate their construction under the varying conditions that may arise 
 and also as a matter of education. Considerable space is therefore allotted to them here. 
 
 At this point Art. 58 should be reviewed, and in addition to its suggestions the student is fur- 
 ther advised to work, at first, on as large a scale as possible, not undertaking small curves of sharp 
 curvature until after acquiring some faciht}^ in the use of the curved ruler. 
 
 THE HELIX. 
 
 120. The ordinar}' helix is a curve which cuts all the elements of a cyhnder at the same angle. 
 Or we may define it as the curve which would be generated by a jjoint having a uniform motion 
 
 around a straight Hue combined with a uniform motion parallel to the line. 
 
 i^igr- SI. 
 
 ffluuuuuuuuuuuuumuuijuuuuuuuuuuuu 
 
 Din 
 
 UDDI 
 
 [ffl 
 
 lEa 
 
 The student can readily make a model of the cylinder and helix by 
 drawing on thick paper or Bristol-board a rectangle A" B" C" D" (Fig. 81) 
 and its diagonal, D" B" ; also equidistant elements, as m" b," n" c," etc. Allow 
 at the right and bottom about a quarter of an inch extra for overlapjDing, as 
 
 shown by the Hues x y and s z. Cut out the rectangle z x ; also cut a series of vertical 
 slits between Z>" C" and zs and turn the divisions up at right angles to the paper; put mucilage 
 between B" C" and xy; then roll the paper up into cylindrical form, bringing A" D" f h" in front 
 
40 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 of and upon the gummed portion, so that A" U' will coincide with B" C". The diagonal U' B" will 
 then be a helix on the outside of the cylinder, hut half of which can be seen in front view, as 1/ 
 7', (see right-hand figure) ; the other half, 7' A', being indicated as unseen. 
 
 To give the cylinder more permanent form it can be pasted to a cardboard base by mucilage 
 on the under side of the divisions turned up along its lower edge. 
 
 The rectangle A" B" C" D" is called the development of the cylinder ; and any surface like a cyl- 
 inder or cone, which can be rolled out on a plane surface and its equivalent area obtained by 
 bringing consecutive elements into the same plane, is called a developable surface. " The elements m" b", 
 n" c", etc., of the development stand vertically sX b, c, d . . . . g of the half plan, and are seen in the 
 elevation at m' 6', n' d, o' d', etc. The point 5', where any element, as c', cuts the helix, is evidently as 
 high as S" where the same point appears on the development. We may therefore get the curve U 7' A' 
 by erecting verticals from b, c, d, . . . g, to meet horizontals from the points where the diagonal D" 
 B" crosses those elements on the development. 
 
 The length D" G" obviously equals 2 tt r, in which r = D. 
 
 The height A' Z>', that the curve attains in winding once around the cylinder, is called the pitch 
 of the helix. 
 
 The construction of the helix is involved in the designing of screws and screw-propellers, and in 
 the building of winding stairs and skew-arches. 
 
 Mathematically the helix and its orthographic projection are well worth study, particularly the 
 latter when the helix crosses the elements at 45° ; it becoming then identical with Roberval's curve 
 of sines, otherwise known as the companion to the cycloid. 
 
 For a helix on a cone refer to the article on the Spiral of Archimedes. 
 
 THE CONIC SECTIONS. 
 
 121. The ellipse, parabola and hyperbola are called conic sections or conies because they may be 
 obtained by cutting a cone by a plane. We will, however, first obtain them by other methods. 
 
 According to the definition given by Boscovich, the eUipse, parabola and hyperbola are curves in 
 which there is a constant ratio between the distances of points on the curve from a certain fixed 
 point (the focus) and their distances from a fixed straight line (the directrix). 
 
 Referring to the parabola. Fig. 82, if S and B are points of the curve, F the focus and X Y the' 
 directrix, then, if S F: S T: : B F: B X, we conclude that B and S are points of a conic section. 
 
 122. The actual value of the ratio (or eccentricity) may be 1 or either greater or less than unity. 
 When SF equals ST the ratio equals 1, and the relation is that of equality, or parity, which sug- 
 gests the parabola. 
 
 123. If it is farther fi'om a point of the conic to the focus than to the directrix the ratio is 
 greater than 1 and the hyperbola is indicated. 
 
 124. The ellipse, of course, comes in for the third possibility as to ratio, viz., less than 1. Its 
 construction by this principle is not shown in Fig. 82 but later, the method of generation here 
 given illustrating the practical way in which, in landscape gardening, an elliptical plat would be laid 
 out; it is therefore called the construction as the "gardener's ellipse." 
 
 Taking A G and D E a,s representing the extreme length and width, the points F and -F, (foci) 
 would be found by cutting ^ C by an arc of radius equal to one-half A C, centre D. Pegs or pins 
 at F and F^, and a string, of length A B, with ends fastened at the foci, complete the preliminaries. 
 The curve is then traced on the ground by sliding a pointed stake against the string, as at P, so 
 that at all times the parts F^ P, F P, are kept straight. 
 
CONIC SECTIONS. 
 
 41 
 
 125. According to the foregoing construction the ellipse may be defined as a curve in which the 
 sum of the distances from any point of the curve to two fixed points is constant. That constant is evidently 
 the longer or transverse (major) axis, A C. The shorter or conjugate (or minor) axis, D E, is perpen- 
 dicular to the other. 
 
 "With the compasses we can determine P and other points of the ellipse by using F and F^ as 
 centres, and for radii any two segments of A C. Q, for example, gives A Q and C Q as segments. 
 Then arcs from F and F^, with radius equal to Q C, would intersect arcs from the same centres, 
 radius Q A, in four points of the ellipse, one of which is P. 
 
 126. By the Boscovich definition, we are enaliled to construct the f)arabola and hj'perbola also 
 by continuous motion along a string. 
 
 For the parabola place a triangle as in Fig. 82, with its altitude G X toward the focus. If a 
 string of length G X be fastened at G, stretched tight from G to miy point B, by ijutting a pencil 
 at B, then the remainder B X swung around and the end fastened at F, it is then, evidently, as 
 far from P to P as it is from B to the directrix; and that relation will remain constant as the tri- 
 angle is slid along the directrix, if the pencil point remains against the edge of the triangle so that 
 the portion of the string fi-om G to the pencil is kept straight. 
 
 s'ig-- ea. 
 
 127. For the hyperbola, (Fig. 82), the construction is identical with the preceding, except that 
 the string fastened at / runs down the hypothenuse and equals it in length. 
 
 128. Referring back to Fig. 35, it will be noticed that the focus and directrix of the parabola 
 axe there omitted ; but the former would be the point of intersection of a perpendicular from A 
 upon the line joining C with E. A line through .1, parallel to C E, would be the directrix. 
 
 129. Like the ellipse, the hyperbola can be constructed by using two foci, but whereas in the 
 ellipse (Fig. 82) it was the sum of two focal radii that was constant, i. e., FP+F^P^FD + 
 
42 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 F^D= AC (the transverse axis), it is the difference of the radii ^^er- ss. 
 
 that is constant for the hyperbola. 
 
 In Fig. 83, if A B is to be the transverse axis of the 
 two arcs, or "branches," which malve the complete hyperbola, 
 then using p and p to represent any two focal radii, as F Q 
 and F^Q, or F R and F, R, we will have p — p = A B (the 
 constant quantity). 
 
 To get a point of the curve in accordance with this prin- 
 ciple we may lay off from either focus, as F, any distance ■ 
 greater than F B, as F J, and with it as a radius, and F as 
 a centre, describe the indefinite arc JR. Subtracting the con- 
 stant, A B. from F J, by making J E = A B, we use the 
 remainder, F F, as a radius, and F^ as a centre to cut the 
 first arc at R. The same radii will evidently determine three 
 other points fulfilling the conditions. 
 
 130. The taiigent to the hyperbola at anj^ point, as Q, bisects the angle F Q F-^, between the 
 focal radii. 
 
 In the ellij^se, (Fig. 82), it is the external angle between the radii that is bisected bj^ the tan- 
 gent. 
 
 In the parabola, (Fig. 82), the same principle ajjijlies, but as one focus is supposed to be at 
 infinity, the focal radius, B G, toward the latter, from any point, as B, would be parallel to the axis. 
 The tangent at B would therefore bisect the angle F B X. 
 
 131. The ellipse as a circle viewed obliquely. If A R M B F (Fig. 84) were a circular disc and we 
 
 were to rotate it on the diameter A B, it would become 
 narrower in the direction FE until, if sufficiently turned, 
 only an edge view of the disc would be obtained. The 
 axis of rotation, A B, would, however, still ajDpear of its 
 original length. In the rotation sujaposed, all points 
 not on the axis would describe circles about it with 
 their planes jjerpendicular to it. M, for example,' 
 
 .would move in the arc of a circle part of which is 
 shown in M Af, , which is straight, as the plane of the 
 arc is seen " edge-wise." If instead of a circular disc 
 we were to take one of elliptical form, as A C B D, and 
 turn it uijon its shorter axis CD, it is obvious that B 
 would apparently approach on one side while A ad- 
 vanced on the other; and when B was directly in firont 
 of, and projected at k, we would have the ellipse projected in the small circle r t k. Having given 
 then the two axes of a desired ellipse, &s A B and C D, use them as diameters of concentric circles 
 and from their common centre, 0, draw random radii, as M, T, K. AVhere any radius, M, 
 meets the outer circle, drojD a perjDendicular M M^ to the longer axis, to meet a line m M^, parallel 
 to the same longer axis and passing through in, where M cuts the smaller circle. 
 
 132. If T S is a tangent at T to the large circle, then when T appears at T^ we shall have 2\ 
 S a,s a, tangent to the ellipse at the point derived from T; S having remained constant, being on 
 the axis of rotation. 
 
CONIC SECTIONS. 
 
 43 
 
 rE-ig-. ss. 
 
 Similarly, if a tangent at R^ were wanted we may first find r, corresponding to R^; draw the 
 tangent r J to the small circle ; then join J, (constant point), with R^. 
 
 133. Occasionally we have given the length and inclination of a pair of diameters of the ellipse, 
 
 making oblique angles with each other. Such diameters 
 are called conjugate and the curve may be constructed 
 upon them thus: Draw the axes T D and H K at the 
 assigned angle DOH; construct the parallelogram MN 
 X Y ; divide D M and D into the same number of 
 equal parts ; then from K draw lines through the points 
 of division on D 0, to meet similar lines through H and 
 the divisions on D 31. The intersection of like numbered 
 lines will give points of the ellijDse. 
 
 134. It is the law of expansion of a perfect gas that the volume is inversely as the jDressure. 
 That is, if the volume be doubled the pressure drops one-half; if trebled the laressure becomes one- 
 third, etc. Steam not being a jierfect gas departs somewhat from the above law, but the curve in- 
 dicating the fall in pressure due to its expansion is coiiipared with that for a j^erfect gas. 
 
 To construct the curve for the latter let us suppose C L K G (Fig. 86) to be a cylinder with a 
 volume of gas C G b c behind the piston. Let c b indicate the ^^s- se. 
 
 pressure before expansion begins. If the piston be forced ahead 
 by the expanding gas until the volume is doubled, the pressure 
 will drop, bj' Boyle's law, to one-half and will be indicated by 
 t d. For three volumes the pressure becomes vf, etc. The curve 
 CSX is an hyperbola, the special case called equilateral. 
 
 Suppose the cylinder were infinite in length. Since we cannot 
 conceive a volume so great that it could not he doubled, or a pressure so small that it could not 
 be halved, it is evident that theoretically the curve c x and the line G K will forever approach each 
 other yet never meet ; that is, they will be tangent at infinity. In such a case the straight line is 
 called an asymptote to the curve. 
 
 135. Although the right cone (i. e., one having its axis perpendicular to the plane of its base) 
 is usually employed in obtaining the ellipse, hyiierbola and parabola, yet the same kind of 
 
 y sections can be cut from an oblique or scalene cone of circular base, 
 as V. A B, Fig. 87. Two sets of circular sections can also be cut 
 fi'om such a cone, one set, obviously, by planes parallel to the base, 
 the other by planes like C D, making the same angle with the lowest 
 element, V B, that the highest element, V A, makes with the base. 
 The latter sections are called sub-contrary. 
 
 To prove that the section x y is a, circle we note that both it 
 and the section m n — the latter known to be a circle because i^arallel 
 to the base — intersect in a line perpendicular to the paper at o. 
 This line pierces the front surface of the cone at a point we may 
 can r. It would be seen as the ordinate o r (Fig. 88), were the fi-ont half of circle 
 m n rotated until parallel to the paper. Then or' = om X on. But fi-om the ^^ 
 similar triangles in Fig. 87 we have o m : o y :: o x : o n whence o y X o x = o m X o 7i 
 ^ or', thus showing that the section x y is circular. 
 
 Were the vertex of a scalene cone removed to infinity the cone would become an oblique cylinder 
 with circular base; but the latter would possess the property just established for the former. 
 
 
 
 \ 
 
 
 
 ) 
 
 2 
 
 d 
 
 \e if \f, K 
 
 
 G b 
 
 x^igr- ss. 
 2- 
 
44 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 136. The most interesting practical application of the sub-contrary section is in Stereographic 
 Projection, one of the methods of representing the earth's surface on a . ^^s-- ss. 
 map. The especial convenience of this projection is clue to the fact 
 that in it every circle is projected as a circle. This results from the 
 relative position of the eye (or centre of projection) and the plane of ^^ 
 projection; the latter is that of some great circle of the earth and the 
 eye is located at the pole of such circle. 
 
 137. In Fig. 89 let the circle ABE represent the equator; MN 
 the plane of a meridian, also taken as the j^lane of projection; A B 
 
 any circle of the sphere ; E the position of the eye : then a b, the projection of ^ 5 on plane M N, 
 
CONIC SECTIONS. 
 
 45 
 
 (see Art. 4), is a circle, being a sub-conti-ary section of the visual cone E. A B, as the student 
 can easily prove. 
 
 138. We now take uj:) the conic sections as derived from a right cone. 
 
 A complete coiie (Fig. 90) lies as much above as below the vertex. To use the term adopted 
 from the French, it has two nappes. 
 
 Aside from the extreme cases of perpendicularity to or containing the axis the inclination of a plane 
 cutting the cone may he 
 
 (a) Equal to that of the elements,* therefore i^arallel to oiw element, giving the parabola, as M 
 Q My^ (Fig. 90) ; the plane h qM being parallel to the element V U and therefore making with the 
 base the same angle, 9, as the latter. 
 
 (b) Greater than that of the elements, causing the plane to cut both najDpes and giving a two- 
 branch curve, the hyperbola, as DJE and fhg (Fig. 90). 
 
 (c) Less than that of the elements, the plane therefore cutting all the elements on one side of the 
 vertex, giving a closed curve, the ellipse; as KsH, Fig. 91. ^^s- ©i- 
 
 139. Figures 90 and 91, with No. 4 of Plate 2 are 
 not only stimulating examples for the draughtsman but 
 they illustrate probably the most interesting fact met 
 with in the geometrical treatment of conic sections, viz., 
 that the spheres which are tangent simultaneously to the 
 cone and the cutting planes, touch the latter in the foci 
 of the conies ; while in each case the directrix of the 
 curve is the line of intersection of the cutting plane 
 and the plane of the circle of tangency of cone and 
 sphere. 
 
 To establish this we need only employ the well known 
 principles that (a) all tangents from a point to a 
 sphere are equal in length, and (b) all tangents are 
 equal that are drawn to a sphere fi-om ]ooints equidis- 
 tant from its ceiatre. In both figures all jjoints of the 
 cone's bases are evidently equidistant fi-om the centres 
 of the tangent spheres. 
 
 140. On the upper nappe, (Fig. 90), let S H be the circle of contact of a sphere which is tan- 
 gent at Fi to the cutting plane P L K. The ^\a.ne P H^, of the circle cuts the plane of section in 
 Pm. If D is any point of the curve D J E, J another jjoint, and we can prove the ratio constant 
 (and greater than unity) between the distances of D and J fi-om F^ and their distances to P m, 
 then the curve DJE must be an hyperbola, by the Boscovich definition ; F^ must be the focus and 
 P m the directrix. 
 
 D F^ is a tangent whose real length is seen at X S. J F^ and J S are equal, being tangents to 
 the sphere fi-om the same point. We have then the proportion XS:GR::JS:JR or DF^:DZ:: 
 J F^: J R. Since JS and its equal J F^ are greater than J R, and the ratio J F^ to J R is, constant, the 
 proposition is established. 
 
 141 . For the parabola on the lower napjae, since the plane M e k is inclined at the angle 9 
 made by the elements, we have Q A= Q B (ojjposite equal angles) and Q B equal Q F (equal tan- 
 gents). M F=B W=Mo, therefore M F: M o:: Q F: Q o, and it is as far fi-om M to the focus F as 
 to the directrix s x, fulfilling the condition essential for the parabola. 
 
 *See Eemark, Art. 4. 
 
46 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 142. For the ellipse K s H, (Fig. 91), we have PX and NT as the lines to be proven direc- 
 trices, and F and F^ the points of tangency of two spheres. Let s be any point of the curve under 
 consideration and VL the element containing s. This element cuts the contact circles of the spheres 
 in a and A. A plane through the cone's axis and parallel to the paper would contain o t, o v and 
 V n. Prolong v n to meet a line V B that is parallel to K H. Join B with a, producing it to meet 
 PX at r. In the triangles asr and aVR we have sa:sr:: V a: V R. But sa=sF (equal tangents) 
 and similarly Va=Vn; therefore sF:sr:: Vn: V R, which ratio is less than unity; therefore a is a 
 point on an elliiase.* 
 
 The plane of the intersecting lines V a and R r cuts the plane ilf i\^ in AT which is therefore 
 parallel to ar; therefore sA:s T:: Vn: V R. But s A = s F^ therefore s F^: s Tr. Vn: V R, the same 
 ratio as before. 
 
 143. If the plane of section P N were to approach parallelism to F C the point R would ad- 
 vance toward n, and when VR became Vn the plane would have reached the position to give the 
 parabola. 
 
 * Schlouiilcli, Geometrie des Maasses, 1S74. 
 
CONIC SECTIONS AS HOMOLOGOUS FIGURES. 
 
 47 
 
 144. The proof that A' s H (Fig. 91) is an ellipse when the curve is referred to two foci is as 
 follows: KF= Km; KI\ = Kt; therefore K F + K F^ = Km + Kt = t m = x n = 2 K F + F F^ = 
 2HF+FF,; i. e., K F = H F,. 
 
 Since s F = s a and .s F^ = s A we have sF+sFi = sa + sA = Aa=tm=xn=IIK. The 
 sum of the distances fi-om any point s to the two fixed points F and F^ is therefore constant and 
 equal to the longer axis H K. 
 
 HOMOLOGOUS PLANE AND SPACE FIGURES. 
 
 145. Before leaving the conic sections their construction will be given bj' the methods of Pro- 
 jective Geonretry. (See Art. 9.) 
 
 In Fig. 92, if S^ is a centre of projection, then, by Art. 4, the figure A^ B^ C is the central 
 projection of A^ B^ Cy The points A'- and A^ are corresponding points. Similarly B^ and B^, C" and Oj. 
 
 For S^ as the centre of projection the figure A^ B^ C^ corresponds to A^ B^ C^. 
 
 If we join S.^ with Si and prolong to 0, to meet the jDlane of the figures A^ B^ 0^ and A.^ B^ C[, 
 we find that the point sustains the same relation to these two figures that S^ and S, do to the 
 figures just taken in connection with them. Using the technical term trace for the intersection of a 
 line with a i^lane or of one plane with another, we see that A^ c^ is the trace, on the vertical j^lane, 
 of any plane containing .4^ B^. This plane cuts the " axis of homology," t^ m, in c^. As A^ B^ lies 
 in the plane of S^ and A^ B^, and in the horizontal plane as well, it can only meet the vertical plane 
 in Co, the point of intersection of all these planes. Similarly we find that A^ C" and A^ C^, if 
 prolonged, meet the axis at the same point; corres]3ondingly B^ C and B^ C^ meet at a^. But A}- 5' 
 and A^ B^, being corresponding lines, lie in the plane with S^, though belonging to figures in two 
 other planes ; they must, therefore, meet also at the same point, c^ ; and similarly for the other 
 lines in the figures used with S,. ^tg. ss. 
 
 Finally, the line A^ A^ being the hor- 
 izontal trace of the plane determined by 
 the lines joining A^ with S^ and S^ must 
 contain the horizontal trace, 0, of the line 
 joining S^ with S^. But this puts A^ and 
 A-i into the same relation with that A^ 
 and A' sustain to S^; or that of A^ and 
 A^ to Si. Hence we rightly conclude that 
 on one plane we can take a point as a 
 centre of projection, and a figure A^ B, G,, 
 and from them derive a second figure, A^ 
 jBj Ci, which corresponds to the assumed 
 figure in the same way as if they lay in 
 different planes. Figures so related in one 
 plane are called homological figures and the 
 centre, 0, a centi-e of homology. 
 
 146. Had A^ £, C, been a circle, and all 
 its points joined with S^, then from what has 
 preceded we know that A^ jB' C" would have 
 been an ellipise; as also would have been 
 the case were A^ B^ C^ a circle used in con- 
 nection with (§2- I^ut our conclusions should enable us to substitute a circle for A^ B.^ C^ and using 
 
48 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 in tJbe same plane with it get an ellipse in place of the triangle A^ B^ C^. Before illustrating this 
 it is necessary to show the relation of the axis to the other elements of the problem and to supply 
 a test as to the nature of the conic. 
 
 147. First as to the axis, and employing again for a time a space figure (Fig. 93), it is evident 
 that raising or lowering the horizontal plane c X Y parallel to itself, and with it, necessarily, the 
 axis, would not alter the kind of curve that it would cut from the cone S. H A B, were the elements 
 of the latter prolonged. But raising or lowering the centre, S, would decidedly affect the curve. 
 Where it is, there are two elements of the cone, S A and S B, which would never meet the plane 
 c X Y. The shaded plane containing those elements meets the vertical plane in " vanisliing line (a)," 
 parallel to the axis. This contains the projections, A and B, of the points at infinity where the 
 lower plane may be considered as cutting the elements SA and SB. Were Sand the shaded plane 
 raised to the level of H, so that " vanishing line (a) " should become tangent to the base, there would 
 
 ■ be one element, S H, of the cone, parallel to the lower plane, and the section of the cone by the latter 
 would be the parabola; while the figure as it stands would indicate the hyperbola. The former 
 would have but one point at infinity; the latter, two. 
 
 148. Raise the centre S so that the vanishing line does not cut the base and, evidently, no line 
 from S to the base would be parallel to the lower i^lane; but the latter would cut all the elements 
 on one side of the vertex, giving the ellijDse. 
 
 149. Bearing in mind that the projection of the circle AHBt is on the lower plane jDroduced, 
 if we wish to bring both these figures and the centre S into one plane, without destroying the relation 
 between them, we may imagine the end plane Q L X removed and rotation of the remaining system 
 
 VANISHING LINE f a} 
 
 occurring about cr ^ in a manner exactly similar to that which would occur were i oj c a system of 
 four pivoted links and the point o pressed toward c. The motion of S would be parallel and equal 
 
CONIC SECTIONS AS HOMOLOGOUS FIGURES. 
 
 49 
 
 to that of 0, and, like the latter, would evidently maintain its distance from the vanishing line and 
 describe a circular arc about it. The vanishing line would remain parallel to the axis. 
 
 150. From the foregoing we see that to obtain the hyperbola by projection of a circle from a 
 point in the plane of the latter we would require simply a secant vanishing line, M N (Fig. 94), 
 and an axis of homology parallel to it. Take any point P on the vanishing line and join it with 
 any point K of the circle. P K meets the axis at y ; therefore the line h y that corresponds to P K 
 must also meet the axis at ?/. OP is analogous to S A of Fig. 93 in that it meets its corresponding 
 line at infinitj', i. e., is parallel to it. Therefore y I, parallel to P meets the ray K at /r, cor- 
 responding to A'. Then AT joined with any other point R gives K z. -Join z with the point k 
 just obtained and jsrolong R to intersect them, obtaining r, another point of the hyperbola. 
 
 151. In Fig. 93, were a tangent at B drawn to arc A H B, it would meet the axis in a point 
 which, like all points on the axis, corresponds to itself From that point the projection of that 
 tangent on the lower plane would be parallel to S B, since they are to meet at infinity. ( )r if S J 
 is parallel to the tangent at B then ./ will be the jjrojection of .7' at infinity, where S J meets 
 the tangent; / will be therefore one point of the projection of said tangent on the lower plane; 
 while another point would be, 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 If is parallel to the tangent at infinity, that is to the asymptote (see Art. 134) through F; 
 while the other asymptote is a parallel through E to N. 
 
 153. As in Fig. 93 the projectors from S to all points of the arc above the level of S could 
 cut the lower plane only by being produced to the right, giving the right-hand branch ol the hyper- 
 bola; so, in Fig. 94, the arc M H N, above the vanishing line, gives the lower branch of the hyperbola. 
 To get a point of the latter, as h, and having already obtained any point x of the other branch, 
 join H with A' (the original of x) and get its intersection, g, with the axis. Then x g h corresponds 
 to g X H, and the ray OH meets it at A, the projection of H. 
 
 The cases should be worked out in which the vanishing line is tangent to the circle or exterior to it. 
 
 154. The homological figures with which we have been dealing were plane figures. But it is 
 possible to have space figures homological with each other. 
 
 In homological space figures corresponding lines meet 
 at the same point m a plane, instead of the same point 
 on a line. A vanishing plane takes the jjlace of a van- 
 ishing line. The figure that is in homology with tlie 
 original figure is called the relief-perspective of the latter. 
 (See Art. 11.) 
 
 Kemarkably beautiful effects can be ol^tained by the 
 construction of homological space figures, as a glance at 
 Fig. 95 will show. The figure represents a triple row 
 of groined arches and is from a photograjjh of a model 
 designed by Prof. L. Burmester. 
 
 Although not always requiring the use of the irregular curve and therefore not strictly the material 
 for a toi^ic in this chapter, its close analogy to the foregoing matter may justify a lew words at this 
 point on the construction of a relief-i^erspective. 
 
 155. In Fig. 96 the plane P Q is called the plane of homology or picture-plane, and — adopting 
 Cremona's notation — we will denote it by it. The vanishing plane, M N, or 4>, is parallel to it. is 
 
 ^'ig-- S5- 
 
50 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 the centre of homology or perspective-centre. All points in the plane v 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, being the intersection oi A B with tt. The line v, parallel to A B, would meet the 
 
 latter at infinity; therefore v, in the ' vanishing plane, 4>', 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 -n- in B" n, which is an axk of homology for 
 AB and A' B', exactly as mn in Fig. 92 is for A^ B^ and A^ B.,. 
 
 As Z) C is parallel io A B (Fig. 96) a parallel to it through is again the line v. 
 
LINK-MOTION CURVES. 51 
 
 The trace of D C on tt is C". Joining v with C" and cutting v C" by rays D, C, obtains 
 U C in the same manner as A' B' was derived. The originals of A' B' and C D 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 jDlane 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 ; G L being the ground line or axis of intersection 
 of the planes on which the views are made. The planes tt and 4>' are interchanged, as comioared 
 with their positions in Fig. 96, and they are seen as lines, being assumed as peri^endicular 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 Grundzilgc der Relief- Perspective and Wiener's Darstellende Geometrie 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 i^ractical application of central projection to Grapliical 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 
 Ijoint on one of the moving pieces of a mechanism. Such problems often arise in machine design, 
 especiallj^ 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 si3ecial 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 j)roblem. 
 
 The moving pieces in a mechanism are rigid and ine.xtensible 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 j)oint-path of any 
 mechanism is therefore the determination of the fixed points and a general analysis of the motion. 
 
 * LofUi is tliu Latin for place; and in rather untechnical language, althougli in the exact sense in which it is used mathe- 
 matically, we may say that the /ocks 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 locns of all iioints 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 S P, fastened at N and P by 
 pivots to a third Hnk, N P, 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 JV P, with the exception of its 
 
 ^ig-- se_ 
 
 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 N P. Since NP is constant in length, if N were moved along to E the point P would 
 have to be at a distance N P from E and also on the circle to which it is confined ; therefore its 
 new position, /, is at the intersection of the circle P s r by an arc of radius P N, centre F. Then 
 Ff prolonged by an amount equal to itself gives J\ , another point of the Lemniscate, and to which 
 Z has then moved. All other j^ositions are similarly fou.nd. 
 
 If the motion of N' is toward D it will soon reach a limit. A, to its fui'ther movement in that 
 direction, arriving there at the instant that P reaches a, when NP and PS will be in one straight 
 line, S A. 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 jDath. P reaches a similar 
 " dead point " at v. 
 
 To get the. curve illustrated, the links N P and P S had to be equal, as also the distance M S 
 to M N. By varying the jDroportions of the links, the point-paths would be correspondingly affected. 
 
INSTANTANEOUS CEN TRES. — CENTR IDS. 
 
 53 
 
 By tracing the path of a point on P N' procluced, and as far from iV as Z is from P, the 
 student will obtain an interesting contrast to tlie 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. 
 
 I.NSTA.NTANEOUS CENTRES. CENTEOIDS. 
 
 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 
 
 S'ig-. S3. 
 
 toy 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 sufficient to simplify otherwise difficult constructions relating to the complex curve which is 
 described relatively to the earth. 
 
64 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 tlie point of desired tangency with the — 
 at that instant — position of the moving centre, we have the nnnnal, 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 rolling of the latter upon another curve, the link might be brought into 
 all the jjositions which its visible modes of restraint compel it to take. 
 
 161. In the "Fundamental" part of the figure AB 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 perjaendiculars intersecting at 0, and drawing A, A', OB, and OB', we 
 have A A' ^ 6^^= B B', and evidently a point about which, as a centre, the turning of AB 
 through the angle ^i would have brought it to A' B'. Similarly, if the next position in which we 
 find A B 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 0,, probably different from 6,. 
 
 The points n and m are analogous to and s. 
 
 If s' be drawn equal to s and making witli the latter an angle 0^, equal to the angle 
 A A', and if Os were rigidly attached to AB the latter would be brought over to A' B' by 
 bringing s' into coincidence with s. 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 s n m would bring A B 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 jDolygons ; 
 the one corresponding to s n m being called the fixed, the other the rolling centroid. The perpen- 
 dicular from uj)on A A' is a normal to that path. But if A were to move in a circular arc 
 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) to the points A and B. 
 
 162. In Fig. 98 the instantaneous centre about which the whole link N P is turning is at the 
 intersection of radii MN and SP (produced); and calling it A' we would have XZ 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 links equal. Unlike the usual quadrilateral fulfilling 
 this condition, the long sides cross, hence the name "anti-parallelogram." 
 
 The "fixed link (a)" corresponds to MS 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 
 
TROCHOIDS. 55 
 
 fastened at right angles to "moving link (a)," an infiniteishaal turning al)out T would move "link 
 (a) " exactl_y 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 i)roceedhig as before, gives the points of 
 "rolling centroid (a)." 
 
 These centroids are branches of hyperbolas having the extremities of the long links as foci. 
 
 Bj' 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 
 possilily 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 jDursue it further in such works 
 as Reuleaux' Kinematics of Machinery and Burmester's Lehrbwh der Kinematik. 
 
 Before leaving this toiiic the important fact should be stated, which now needs no argument to 
 ■establish, that the instantaneous centre for any position of the moving piece, is the point of contact 
 of the rolling and fixed centroids. 
 
 16-5. We shall have occasion to use this principle in drawing tangents and normals to the 
 
 TROCHOIDS 
 
 "which are the principal RoulcUes, or roll-traced curves, and which may be defined as follows: — 
 
 If in the same plane one of two circles rolls upon the other without sliding, the path of any 
 point on the 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 bj' 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 T^ 
 a, ceiling-corner of a room, and Tj, the diagonally opposite floor-corner, a weight would slide from 
 
 T^ to 7",, more quickly on guides curved in cycloidal shape than if shaped to any other curve or 
 if straight. If started at c or any other point of the curve it would reach Ty, as soon as if started 
 at T,. 
 
 167. In beginning the construction of the cycloid we notice first that as TVD rolls on the 
 straight line A B the arrow D R T will be reversed in position (as at D-^ T^ as soon as the semi- 
 circumference T3D has had rolling contact with A B. The tracing j^oint 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 T,. 
 
 » "Although the invention of the cycloid 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 
 Tound a circle whose centre travels uniformly around another circle." — Froctor, Geometry of Cycloids. 
 
56 
 
 THEORETICAL AND PRACTICAL GRAPHICS . 
 
 The distance T Tj, evidently equals 3 -k r, when r = T R. 
 
 If the semi-circumference T3D (equal to it r) be divided into any number of equal parts; and 
 also the path of centres R Rg (again = -n-r) into the same number of equal parts, then as the jDoints 
 1, ^, etc., come in contact with the rail, the centre R will take the positions i?,, R^, etc., directly 
 above the corresponding points of contact. A sufficient rolling of the wheel to bring point ^ upon 
 AB would evidently raise T from its original position to the former level of 3. But as T must always 
 be at a radius' distance from R, 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 3 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-circuinference 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 i^^, when it passes and goes ahead of it. Prom R. the line of level 
 through 5 could be cut not alone at c by an arc of radius c J?, 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 the relative position of the rolling centre and the moving point. This matter requires 
 more thought in drawing trochoidal curves in which both circles have finite radii, as will appear 
 later. 
 
 lOO- 
 THE CYCLOID. 
 
 168. Were jDoints T^ and T^, given, and the semi-cycloid T^ Ty^ desired, we can readily ascertain 
 the " base," A B, and generating circle, as follows : — .Join T|. with T^^ ; at any point of such line, as 
 X, erect a perpendicular, x y ; from the similar triangles x y T^^ and T^ D^ T^^ having angle <^ common 
 and angles & equal we see that 
 
 xy:xT,,::T,D,:D,T,^::3r:7rr::2:-7T::l:'^ or, very nearly, as U : 22. 
 
 If, then, we lay ofi' x T,, equal to twenty-two equal parts on any scale, and a periDcndicular, x y, 
 fourteen parts of the same scale, the line y T^^ will l^e the base of the desired curve; while the di- 
 ameter of the generating circle will be the perpendicular from Ts to y T^, prolonged. 
 
 169. To draw a 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. 159). 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 with the point 
 of contact on the rail. The tangent at N, for exaniijle, (Fig. 100), would be t A\ perpendicular 
 to the normal, No, joining N with o; the latter point being found by using A^ as a centre and 
 
THE CYCLOID. — COMPANION TO THE CYCLOID. 
 
 57 
 
 cutting A B by an arc of radius equal to m I, in which m is a jDoint at the level of N 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 N, radius 
 TR; would obviously be verticallj' below the f>osition of the rolling 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 
 GalUeo, 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 jaaper 
 models of the two figures. He concluded that the area in question was nearl}' 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 simultaiieously 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 fioint tracing the cj'cloid, 
 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 THs; R equals quadrant T s y. 
 
 Adopting the method of Analytical Geometry, by using as a reference point, or origin, 
 we may reach any point, ^4, on the curve, by co-ordinates, as x, x A, of which the horizontal is 
 called an abscissa, the vertical an ordinate. By the preceding construction Ox equals arc sfy, while 
 X A equals s w — the sine of the same arc. The " Companion to the Cycloid " is therefore a curve 
 of sines or simisoid, since starting from the abscissas are equal to or proportional to the arc of a 
 circle while the ordinates are the sines of those arcs. 
 
 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 i^endulum ; and exactly the 
 law of vibration of the so-called mathematical pendulum."* 
 
 172. From the symmetry of the 
 sinusoid with respect to R R^ and to 
 we have area TAO R= E C R,; 
 adding area D E L R to both mem- 
 bers we have the area between the 
 sinusoid and T D and D E equal to 
 the rectangle R E, or one-half the rect- 
 angle D E K T; or to 
 
 TT r X » r = 
 
 TT r ", the area of the rolling circle. 
 
 As T A C E 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 
 
 ^ig-. lOl. 
 
 *Wooa. Elements of Co-ordinate Geometry, p. 209. 
 
58 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 passing through the then position of the tracing point, as a, the distance, A a, between the two 
 curves at any leA'el, is merely the semi-chord of the rolhng circle at that level. But this, evidently, 
 equals Ms, the semi-chord at the same level on the equal circle. The equality of M s and A a 
 makes the elementary rectangles M s s^ m, and A A^ a, a equal ; and considering all the possible 
 similarljr constructed rectangles of infinitesimal altitude, the sum of those on semi-chords of the 
 rolUng 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 C H M. While the first of these arcs rolls upon 
 A B the point T turns through the angle T R 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) fi-om R; and, as R has 
 reached i^i in the rolling supposed, we find T^ — the new position of T — by an arc from R^, radius 
 TR, cutting said line of level. 
 
 After what has preceded, the figure may be assumed to be self-interpreting, each positioii of T 
 being 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 R,; 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-TROCHOIDS. 
 
 59 
 
 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. 
 
 HYPO-, EPI- AND PERI-TEOCHOIDS. 
 
 176. Circles of finite radius can evidentlj^ 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. AYhen 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 waj's, a nomenclature was necessary which should 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 Oyclo-orthoids for the classes of curves 
 of which tlie cycloid and involute are respecti\'ely 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 loithin 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 preceded 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 
 
 Descrtbing 
 
 Point. 
 
 Circle rolling 
 
 upon 
 Straight Line. 
 
 Circle rolling upon circle. 
 
 Straight Line 
 
 rolling upon 
 
 Circle. R = 00 
 
 External 
 contact. 
 
 Internal contact. 
 
 Ortho-cycloids. 
 
 Epitrochoids. 
 
 Larger Circle 
 rolling. 
 
 Smaller circle rolling. 
 
 Cyclo- 
 orthoids. 
 
 2 K > E. 
 
 2 It < F. 
 
 2R = P. 
 
 Peritrocholds. 
 
 Major 
 Hypotroehoids. 
 
 Minor 
 Hypotroehoids. 
 
 Medial 
 Hypotroehoids. 
 
 On circumference 
 of rolling circle. 
 
 Cycloid. 
 
 (a) Epicycloid. 
 
 (a) Pericycloid 
 
 (d) Major 
 Hypocyclolds. 
 
 (d) Minor 
 Hypocycloid. 
 
 Straight 
 Hypocycloid. 
 
 Involute. 
 
 WitUn 
 Circumference. 
 
 Prolate 
 Trochoid. 
 
 (b) Prolale 
 Epitrochoid. 
 
 (CI Prolate 
 Peritrochoid. 
 
 (e) Major Prolate 
 Hypotrochoid. 
 
 (f) Minor Prolate 
 Hypotrochoid. 
 
 (g) Prolate 
 Elliptical 
 Hypotrochoid. 
 
 Prolate 
 Cyclo-orthoid. 
 
 Without 
 Circumference. 
 
 Curtate 
 Trochoid. 
 
 (c) Curtate 
 Epitrochoid, 
 
 (b) Curtate 
 Peritrochoid. 
 
 (f) Major 
 Curtate 
 Hypotrochoid. 
 
 (e) Minor 
 Curtate 
 Hypotrochoid. 
 
 (g> Curtate 
 Elliptical 
 Hypotrochoid. 
 
 Cui'tate 
 Cyclo-orthoid. 
 
 177. From the above we see that the prefix epi {over or ivpon) denotes the curves resulting 
 from external contact; hypo {under) those of internal contact with smaller circle rolling; while 
 peri {about) indicates the third possibility as to rolling. 
 
 * Re-printed in substance in the Appendix. 
 
60 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 :S'LS. 103. 
 
 178. The construction of these curves is in closest analogy to that of the cycloid. If, for 
 example, we desire a major hypocycloid we first draw two circles, m V P, m x L, (Fig. 103), tangent 
 
 internally, of which the rolling circle has its' 
 diameter greater than the radius of the fixed 
 circle. Then, as for the cycloid, if the tracing- 
 point is P, we divide the semi-circumference m V P 
 into equal parts and from the fixed centre, F, de- 
 scribe circles through the points of division, as 
 those through 1, 3, 3, 4 and -5. These replace the 
 "lines of level" of the cycloid, and may be called 
 circles of distance, as they show the distance from 
 F of the jooint P, for definite amounts of angular 
 rotation of the latter. For, if the circle P Vm 
 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, 1-3, etc., its distances 
 from F would be i^o and Fh resisectively. 
 
 If, however, the turning of P about R is due 
 to the rolling of circle P V m upon the arc m x z 
 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 intercei^ts an are, mxz, equal to the 180° arc, m V P, on the 
 smaller circle. Equcd arcs on unequal circles are subtended by angles at the centre ivhich are inversely 
 proportioned to the radii. 
 
 While arc mVP rolls ui^on arc mxz the centre R will evidently move over circular arc 
 R — iig. Divide mxz into as many equal jjarts as mVP and draw radii from F to the points of 
 division ; these cut the j^ath of centres at the successive positions of R. When arc m 5-4, for 
 example, has rolled upon its equal m u v then R will have reached R., ; P will have turned about 
 R through angle PR2 = mR4 and will be at n, the intersection of h f g — the circle of distance through 
 3 — by an arc, centre R^, radius RP. Similarly for other points. 
 
 179. To trace the path of anj^ point on the circumference of a circle so rolling as to give the 
 epi- or pm-cycloid requires a construction similar at every step to that just illustrated. The same 
 remark applies equally to the determination of the paths of points within or beyond the circum- 
 ference of the rolling circle, as will be seen by reference to Fig. 104, in which the path of the 
 point P is determined (a) as carried 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. Proceeding in the usual way, a semi-circle is drawn 
 through P from each rolling centre, R and p. Dividing these semi-circles into the same number of 
 equal parts draw next the clotted " circles of distance " through these points, all from centre F. 
 The figure illustrates the special case where the larger set of "circles of distance" divides both semi- 
 circumferences into equal parts. The successive positions of P, as Pj, P.^, etc., are then located by 
 
EPI- AND PERI-TROCHOIDS. 
 
 61 
 
 arcs of radii R P or p P, struck from the successive jjositions of R or p and intersecting the proper 
 "circle of distance." For example, the turning of P through the angle PRl ahout R would bring 
 P somewhere upon the circle of distance through jDoint 1; but that amount of turning would be 
 due to the rolling of the first generator over the arc m Q, which would carry R io R^; P would 
 
 
 therefore be at P^ , at a distance R P from R^ and on the dotted arc through 1. Similarly in 
 relation to p. Each position of P is joined with each of the centres from which it could be 
 obtained. 
 
62 
 
 THEORETICAL AND PRACTICAL GRAPHICS: 
 
 SPECIAL TROCHOIDS. 
 
 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 jjroperty the mechanism known 
 as " White's Parallel Motion " is based, in which a piston-rod or other piece intended to have 
 reciprocating rectilinear motion is pivoted to a small gear-wheel or pinion which rolls on the interior 
 of a toothed annular wheel of twice the diameter of the i^inion. 
 
 181. The Limagon and Cardioid. The Limajon is a curve whose points may be obtained by 
 i^ig-- 105. drawing random secants through a point on the circumference of 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 
 ns; then if ?) v, n p, c a 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. 
 Referring any point as d to and the diameter s, we have 
 
 Od = p = c -f cd = 
 
 + b, while a ^ i2 r cos i 
 
 whence the general polar equation for the Limagon, p ^ 2 r cos 
 
 0--^ e + b. 
 
 When b = 2 r the Limagon becomes the heart-shaped curve called the Cardioid.^ 
 
 182. All Limagons, general and special, may be generated either as epi- or peri-trochoidal 
 curves: as ei^i-trochoids the generator and director must have equcd diameters, any point on the 
 circumference of the generator then tracing a Cardioid, while any point on the radius (or radius 
 produced) describes a Limagon; as peri-trochoids the larger of a jjair of Cardanic circles must roll 
 on the smaller, the Cardioid and Limagon then resulting, as before, from the motion of points 
 respectively on the circumference of the generator or loithin or without it. 
 
 183. In Fig. 106 the Cardioid is obtained as an epicycloid, being traced by point P during one 
 revolution of the generator P H m about an equal directing circle m s 0. 
 
 As a Limagon we may get points of the Cardioid, as y and z, by drawing a secant through 
 and laying off -s y and s z each equal to 2 r. 
 
 184. The Limacon as a Trisectrix. Three famous problems of the ancients were the squaring of 
 the circle, the duplication of the cube and the trisection of an angle. Among the interesting curves 
 invented by early mathematicians for the purpose of solving the latter problem were the Quadratrix 
 and Conchoid, whose construction is given later in this chapter ; but it has been found that certain 
 trochoids also f)0ssess this interesting property, among them the Limagon of Fig. 106, frequently 
 called the Epitrochoidal Trisectrix. Its construction as an epitrochoid need not be described in detail 
 after what has preceded. 
 
 As a Limagon we would find points as G and X by drawing from R a secant RX to the circle 
 •called " path of centres " and making S X and S G each equal to the radius of that circle. 
 
 185. To trisect an angle, as M R F, bisect one side of the angle as RF at m; use m R and 
 m F as radii for generator and director respectively of an epitrochoid having a tracing radius, R F, 
 equal to twice that of the generator. Make R N= R F and draw N F; this will cut the Limagon 
 
 * Term due to Keuleaux and baaed upon the fact that Cardano <16th century) was probably the first to investigate the 
 paths described by points during their rolling. 
 f From Cardis, the Latin for heart. 
 
SPECIAL TROCHOIDS. 
 
 63 
 
 FT^RQ, (traced by point F as carried by the given generator) in a point T^. The angle T^ R F 
 will then be one-third of N R F, which may be proved as follows : F reaches T^ by the rolling of 
 arc m n on arc m7i, . These arcs are subtended by equal angles, <^, the circles being equal. During 
 this rolling R reaches Ri , bringing R F to R^ T^. In the triangles 7\ R^ F and R F R^ the side 
 F Ri is common, angles <f> equal and side Ri T^ equal to side R F; the line R T^ is therefore 
 parallel to i?i F, whence angle T^ R F must also equal 4>- In the triangle R F R^ we denote by 6 
 
 iso° — <t> 
 
 the angles opposite the equal sides RF and R^F; then 26 + (l>=^lS0° or 6 = 
 
 In triangle 
 
 N R F we have the angle at F equal to 6 — (^, and 2 (0 — <^)-|-x+</)= 180° which gives x = 2 <j> 
 by substituting value of from previous equation. 
 
 ■Fi-S- lOS- 
 
 
 186. The Involute. As the opposite extreme of a circle rolling on a straight line we may have 
 the latter rolUng 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 u 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 line u x produced) it would trace 
 the involute. When j reaches a the rule will have bad rolUng contact with the base circle over an 
 arc ut s — a whose length equals line u x j. Were a the initial point we may obtain b, c, etc., by 
 making tangent m b = arc m a ; tangent n c ^ arc n a, etc. Each tangent thus equals the arc from 
 initial point to point of tangency. 
 
64 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 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 tangenc}^ on the evolute is an instantaneous centre, and a tangent at any 
 point as / is a perjjendicular to the tangent, f q, fi'om /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 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 jDlane jaerpendicular 
 to the axis of the helix would cut such a surface in a pair of involutes.' 
 
 188. The Siyiral of Archimedes. This is the curve that would be generated by a point having a 
 uniform motion around a fixed point — the pole — combined with uniform motion toward or fi'om the 
 pole. 
 
 In Fig. 107, with as the pole, if the angles 6 are equal and D, E and 1/3 are in arith- 
 metical progression then the points D, E and 1/3 are points of an Archimedean Spiral. 
 
 This spiral can be trochoidally generated, simultaneously with the involute, by inserting a pencil 
 point at 2/ in a piece carried by — and at right angles with — the rule, the point y being at a distance, 
 X y, from the contact-edge of the rule, equal to the radius s of the base circle of the involute ; for 
 
 * The day of writing the ahove article the foUowing item appeared in the New York Evening Post: "Visitors to the Royal 
 Observatory, Greenwich, will hereafter miss the great cylindrical structure which has for a quarter century and more covered 
 the largest telescope possessed by the Observatory. Notwithstanding its size the Astronomer Royal has now procured through 
 the Lords Commissioners a telescope more than twice as large as the old one. . . . The optical peculiarities embodied in the 
 new instrument will render it one of the three most powerful telescopes at present in existence. . . . The peculiar architectural 
 feature of the building which is to shelter the new telescope is that its dome, of thirty-six feet diameter, will surmount a 
 tower having a diameter of only thirty-one feet. Technically the form adopted is the surface generated by the revolution of 
 , an involute of a circle.^'* 
 
SPECIAL TROCHOIDS. 
 
 65 
 
 i^ig-- ice. 
 
 after the rolling of u x over an arc u t we shall have t x, as the portion of the rolling line between 
 X and the point of tangency, and x y will have reached x^ ?/,. If the rolling be continued y will 
 evident])'- reach 0. We see that y = u x and y^ = t x^ ; but the lengths m x and t x, are propor- 
 tional to the angular movement of the rolling line about 0, and as the spiral may be defined as 
 that curve in which the length of a radius vector is directly propoi-tional to the angle through which 
 it has turned about the pole, the various positions of y are evidently points of such a curve. 
 
 189. Were the pole, 0. given and a portion only of the spiral, we could draw a tangent at 
 any point, //, , liy determining the circle on which the spiral could be trochoidally generated, then 
 the instantaneous centre for the given position of the tracing-point, whence the normal and tangent 
 would be derived in the usual way. The radius Ot of the base circle would equal ivy, — the 
 difference between two radii vectores Oy and Oz which include an angle of 57? 29 -f, (the angle 
 which at the centre of a circle subtends an arc equal to the radius). The instantaneous centre, t, 
 would be the extremity of that radius which was perpendicular to .y, . The normal would be 
 1 2/j and the tangent T T, perpendicular to it. 
 
 190. This spiral is the proper outline for a cam to convert uniform rotary into uniform recti- 
 linear motion, and when comljined with an equal ami opposite spiral gives tlie well known form 
 called the heart-cam. As usually constructed the acting curve is not the true spiral but a curve 
 whose points are at a constant distance from 
 the theoretical outline equal to the radius of 
 the Mction-roller which is on the end of the 
 piece to be raised. A small portion of such ; 
 
 a "parallel curve" is indicated in the upper ;k; 
 
 part of Fig. 107. 
 
 191. If a point travel on the surface of 
 a cone so as to combine a uniform motion 
 around the axis with a uniform motion toward 
 the vertex it will trace a conical helix whose 
 orthographic projection on the plane of the 
 base will be a Spiral of Archimedes. 
 
 In Fig. 108 a top and front view are 
 given of a cone and heUx. The shaded por- 
 tion is the development of the cone, that is, 
 the area equal to the convex surface and 
 which — if rolled up — would form the cone. 
 To obtain the development draw an arc 
 A' G" A" of radius equal to an element. The 
 convex surface of the cone will then be rej^re- 
 sented by the sector A' 0' A" whose angle 6 
 may be found bj^ the proportion A : 0' A' :: 
 6 : 360°, since the arc A' Ct" A" must equal 
 the entire circumference of the cone's base. 
 
 The student can make a paper model of 
 the cone and lielix Ijy cutting out a sector of 
 a circle, making allowance for an overlap on which to put tlie mucilage, as shown by the dotted 
 lines 0' y and y v z in the figure. 
 
 
66 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 The development of the conical helix is evidently the same kind of spiral as its orthographic 
 projection. 
 
 PARALLEL CURVES. 
 
 192. A parallel curve is one whose points are at a constant normal distance from some other 
 curve. Parallel curves have not the same mathematical pfoperties as those from which they are 
 derived, except iia the case of a circle; this can readily be seen from the cam figure under the last 
 heading, in which a point, as Si, of the true spiral, is located on a line from which is by no 
 means in the direction of the normal to the curve at S,, upon which lies the point S^ of the 
 parallel curve. 
 
 Instead of actually determining the normals to a curve and on each laying off a constant 
 distance we may draw many arcs of constant radius, having their centres on the original curve; the 
 desired parallel will be tangent to all these arcs. 
 
 In strictly mathematical language the parallel curve is the envelope of a circle of constant radius 
 whose centre is on the original curve. We may also define it as the locus of consecutive inter- 
 sections of a sj'stem of equal circles having their centres on the original curve. 
 
 If on the convex side of the original the parallel will resemble it in form, but if ivithin the 
 two may be totally dissimilar. This is well illustrated by Fig. 109 in which the parallel to a 
 Lemniscate is shown. 
 
 The student will obtain some interesting results by constructing the parallels to ellipses, parabolas 
 and other plane curves. 
 
 THE CONCHOID OF NICOMEDES. 
 
 193. The Conchoid, named after the Greek word for shell* may be obtained by laying off a constant 
 length on each side of a given line MN (the directrix) upon radials through a fixed point or 
 pole, (Fig. 110). If »7i u = m 11 = s a; then v, n and x are points of the curve. Denote by 
 a the distance of from MN and use c for the constant length to be laid off; then if a <i c 
 there will be a loop in that branch of the curve which is nearest the pole, — the inferior branch. 
 If a = c the curve has a point or cusp at the pole. When « > c the curve has an undulation or 
 wave- form towards the pole. 
 
 * A series of curves much more closely resembling those of a shell can he ohtained hy tracing the paths of points on the 
 piston-rod of an oscillating cylinder. See Arts. 157 and 158. 
 
THE CONCHOID. — THE QUADRATRIX. 
 
 67 
 
 Ov=c+ Ora; n ^= c — Oin; we may therefore express the relation to of points on the 
 curve by the equation p=f± n = c ± a sec <^. 
 
 194. Mention has ah-eady been made of the fact that this was one of the curves invented for 
 the purpose of solving the problem of the trisection of an angle. Were the angle m x (or <j>) we 
 would first draw p q r, the sujierior branch of a conchoid having the constant, c, equal to twice 
 m. A parallel from m to the axis will intersect the curve at q; the angle ]) q will then be 
 one-third of ^ : for since b q = 2 m we have m q = :3 vi cos /3; also m q : m : : sin 6 : sin /3 (the 
 sides of a triangle being proportional to the sines of the opposite angles) ; therefore 2 0m cos /3 : 
 m : : sin 6 : sin 13, whence sin 6* = 5 sin (i cos /3 = sin 2 (i (fi-ora known trigonometric relations). The 
 angle <^ is therefore equal to twice /3, making the latter one-third of angle <f>. 
 
 195. To draw a tangent and normal at any point v we find the instantaneous centre o on the 
 princii^le that it is at the intersection of normals to the paths of two moving j)oints of a line, the 
 distance between said points remaining constant. The motion of v in tracing the curve is — at the 
 instant considered — in the direction Ov; Oo is therefore the normal. The point in of v is at 
 the same moment moving along M N, for which m o is the normal. Their intersection o is then 
 the instantaneous centre and o r the normal to the conchoid, with v z perjjcndicular to o v for the 
 desired tangent. 
 
 196. This interesting curve may be obtained as a plane section of one of the higher mathematical 
 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 every horizontal plane 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 Hyperboloid. 
 
 THK QUAIIKATRIX 01- r)IX0STR.\TUS. 
 
 197. In Fig. Ill let the radius T rotate uniformly about the centre; simultaneously with its 
 movement let M N have a uniform motion parallel to itself, reaching A B &t the same time with 
 radius T: the locus of the intersection of M N 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 oi M N with OB is at infinity the former becomes an asymjrtote 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 / i \ ^^s- m. 
 
 be infinite. It may be proved analytically that the curve 
 crosses A at a distance fi'om equal to 3 r -^ ir. 
 
 198. To trisect an angle, as T a, by means of the 
 Quadratrix draw the ordinate a-p, trisect p T hj s and % 
 and draw sc and x in ; 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 parallel 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, loy means of which one of the great problems of the Greek geometers — the dupli- 
 cation 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, A C E a, right angle, N S and M T any pair of ordinates parallel to 
 
 E'ig'- 11.2. 
 
 and equidistant from CE; then either secant from A through the extremity of one ordinate will 
 meet the other ordinate in a point of the cissoid; P and Q, for example, will be points of the 
 curve. 
 
 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 fi:om which 
 it is derived. 
 
 * Leslie. Geometrical Analysis. 1821. 
 
THE CISSOID. — THE TRACTRIX. 69 
 
 200. Sir Isaac Newton devised the following method of obtaining a cissoid by continuous motion : 
 Make AV= AC; then move a right-angled triangle, of base = V C, so that the vertex F travels along 
 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 i^n, is normal to FC — the path of F, while nz is normal to the motion of / toward 
 / V ; 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 equal to A C and from centre P obtain 
 x; draw P x, then Vz parallel to it ; a vertical from x will meet Vz at the instantaneous centre y, 
 from which the normal and tangent result in the usual way. 
 
 201. Two quantities m and n will be mean proportionals between two other quantities a and b 
 if m' = a a and n- = m b ; that is, if m" = a'^ b and if n^ = a b". 
 
 If 6 = -^ a we will have m for the edge of a cube whose volume will lie twice that of a^, when 
 m' = a' b. 
 
 To get two mean projjortionals 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 C t 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 nv' will then equal r'' b, as maj^ 
 be thus shown by means of similar triangles : — 
 
 Cv: MQ:: CA:MA whence ^' »' = ~jif^ (1) 
 
 r M Q 
 Ct: MQ:: CB:BM " ^ ^ = Jm ^^^ 
 
 MQ:iMA::SN:AN:: VAN.BN: A N, whence M Q = ^^^^^-^^^ .... (3) 
 
 1 ^^r^ B M (_C t = b) ,,. 
 
 From (2) we have M Q = ^ — ■ (4) 
 
 r 
 
 " (3) " " MQ^-=^^^^f§^^^ (5) 
 
 Replacing M Q' in equation (1) by the product of the second members of equations (4) and (5) 
 gives Cv'^ (i. e., m') = /■- b. 
 
 By interchanging r and b we obtain n, the other mean proportional ; or it might be obtained 
 by constructing similar triangles having a, b and m for sides. 
 
 THE TEACTEIX. 
 
 202. The Tractrix is the involute of the curve called the Catenary (later described) yet its usual 
 'construction 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 Q R 
 (Fig. 113) and with a centre a, a short distance from R on Q R, obtaining b which is then joined 
 with a. On aba. centre, c is similarly taken, 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 Q .S' is an asymptote to the curve. 
 
70 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 The area between the completed branch R P S and the hnes Q R and Q S would be equal to 
 a quadrant of the circle on radius Q R. 
 
 M _„_-_ . 9 =S^R 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 in 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 
 ^'ig-. lis- consideration: — If about to split a log, and having 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 simjDle, how- ^^^^ il-±^. 
 
 ever, and the student may profitably be introduced to it. 
 Suppose a ball, c, (Fig. 114) struck at the same instant 
 by two others, a and 6, moving at rates of six and eight 
 feet a second respectively. On a c and b c jjrolonged take 
 c e and c h equal, respectively, to si.r and eight units of 
 some scale ; complete the parallelogram having these lines 
 as sides; then it is a Avell 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. Obviously, then, to balance the effect of 
 balls a and b upon c, a fourth would be necessary, moving from d toward c and traversing d c 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 point, 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 bloivs 
 by F in each triangle, we see that the jiressures 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 
 ''^''''''Slro~~^ change of curvature of the tractrix as its generating point approaches 
 
 the axis makes it, obviously, the correct form. 
 
 E'lg-. IIS- 
 
 * FoL' a denionslrixtion tlie student may refer to Rankine's Applied Mechanics, Art. 51. 
 
THE TRACTRIX.— WITCH F A G XISL — CARTESIAN 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 rhurah 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 bj^ the cosine (see footnote, 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 Mercator'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 Q R, centre Q (Fig. 113); estimate latitude on such circle from R upward; the intercept on 
 Q S 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 the subject of map construction the student is referred to Craig's Treatise on Projections. 
 
 THE WITCH OF ,\CtXI.SI. 
 
 205. If on any line S Q, perpendicular to the diameter of a circle, a point .S' be so located 
 that S Q : A B :: P Q : <i B then S will be a point of the curve called the Witch of Agnki. 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. 
 
 The point E, at the same level as the centre 0, is a diameter's distance from the flatter. 
 
 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 Versiera, was devised bj' Donna Maria Gaetana Agnisi, a l^rilliant 
 Italian lady, who was apj^ointed QToO) liy Pope Benedict XI^^ to the j^rofessorship of mathematics 
 and philosophy in the University of Bologna. 
 
 THE l-ARTESI.\N OVAL. 
 
 206. This curve, also called simply a Cartes-Ian, after its investigator, Descartes, has its points 
 connected with two foci, F' and F", by the relation m p' dz n p" = k c, in which c is the distance 
 between the foci while m, a and /.■ are constant factors. 
 
 * Leslie. Geometrical Anatiisii. Edinbnrgli, 182], 
 
72 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 ^ig-. Xi.'T. 
 
 Salmon states that we owe to Chasles the proof that a third focus may be found, sustaining the 
 
 same relation, 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 m p' ± n p" ^ k e in the form p' 
 
 11 ,, he 
 — p = — ; or 
 m m 
 
 71 Jc C 
 
 by denoting — by 6 and — by d it takes the yet more simple form 
 
 p' =L b p" = d. Then p" will have two values according as the positive 
 
 d 
 
 _L_- n n f\ 1 
 
 b 
 
 inner of the two ovals that constitute a complete Cartesian, while the latter gives points on the 
 outer curve. 
 
 d — p' 
 
 or negative sign is taken, being respectively — ; and 
 
 -; the former is for points on the 
 
 To obtain p" = 
 
 take F' and F" (Fig. 
 
 E-lir- liS- 
 
 118) as foci; F' S=d; SK at some random acute 
 anale B with the axis, and make SH^ ,/ that 
 
 
 
 is, make F' S : S H :: b : 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 S T = p" ; there- 
 
 j. It 1 / '^ J 1 „ d — p' 
 
 lore p : a — p ■.:.-■. d whence p = — -, — . 
 
 208. If an arc -x y k be drawn from F' 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 — -J — , in which p' = F' x. Again using F" as a centre, and a radius Q S = 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" = ^^^ = c + p' whence p' = F' A = ^ '^ ^ '^ 
 
 „ d — p' , 
 
 a, p = — T" = c + p 
 
 ±>, p = 
 
 
 b, p = 
 
 d- 
 
 p' = F' a = 
 p' = P' £ = 
 p' =. F'b = 
 
 1 — b 
 
 d — bc 
 
 1 + b 
 
 d + b c 
 
 1 + b 
 
 d — be 
 
 The construction-arcs for the outer oval must evidently have radii betioeen the values of p' for ^4 
 and B above; and for the inner oval betioeen those of a and b. 
 
CA R TESIA X VA L S. — CA US TICS. 
 
 73 
 
 X^ig-- US. 
 
 The numerical values from which Fig. 118 was constructed were m = 3; n= 2; c= 1; i = 3. 
 
 209. The Third Focus. Figure 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' dr n p'" = m C, which — for purposes of construction — 
 may be written p' ± // p'" = rf'. 
 
 If obtained from the foci F" and F"' the relation would be in p'" — k p" ^ ±71 C", in which C 
 equals F" F'". ^^'riting this in the form p'" — B p" = ± D we have the following interesting cases: 
 (a) an ellipse for D positive and 5 = — 1 ; (Ij) an hyperbola for D positive and B = + I; (c) a 
 limagon for D = C B ; (d) a cardioid for 5 = + 1 and D = C. 
 
 210. The following method of drawing a Cartesian by continuous motion was 
 devised bj^ Prof Hannnond : — A string is wound, as shown, around two jjulleys 
 turning on a common axis ; a pencil at P holds the string taut around smooth 
 pegs placed at random at F^ and i^, ; if the wheels be turned with the same 
 angular velocity and the pencil does not slij> 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 loay around cither one of the wheels the resulting curve will be 
 an hyperbola. 
 
 211. It is a well-known fact 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 next to 
 
 it, their consecutive intersections giving a curve called a caustic 
 by rejiection. Probably all have occasional]}' 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 lima(;on. 
 
 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 {6 and <^, Fig. 120,) is constant. 
 The consecutive intersections of refracted rays give also a caustic, 
 which, for a circle, is the evolute 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 anj^ jDoint 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 m p' ± n p" = k c we may obtain the tangent at G by laying 
 off on p' and p" distances proportional to 7); and n, as G r and G h, 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 l^e laid off on a focal radius produced, as 
 in the case illustrated. 
 
 rig. 120- 
 
 1 American Journal of Mathematics, 1878, 
 
 - Salmon, nir/her Plane Curves. Art. 117. 
 
74 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 CASSIAN OVALS. 
 
 212. In the Cassian Ovals or Ovals 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. 
 
 E'ig-- laa. 
 
 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 le.ss 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 Fi R equal to the constant k. 
 Bisect Fi i^2 at and draw a semicircle of radius R. 
 This cuts the axis at ^4 and B, the extreme jDoints of 
 the curve; for A' = F^ A X F^ B. Any other point T 
 may be obtained bj' drawing from F^ a circular arc of 
 radius F-^t greater than F^A; draw t R, then Rx perpen- 
 dicular to it; X Fj will then be the p" and Fit 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, T s = 6 = F^ T ; Ts will be 
 the desired line. 
 
 THE CATENARY. 
 
 214. If a flexible chain, cable or string, of uniform weight per unit of length, be freely 
 suspended 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 ijins in the board, in the desired relative position of the points of susjaension, 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 he located by means of an equation, we 
 take axes in the plane of the curve, the y-a.xis (Fig. 123) being a vertical line through the lowest 
 point T of the catenary, while the a;- axis is a horizontal line at the distance m below T. The 
 quantity 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. 
 
 ni I i- ^ . 
 
 The equation of the catenary' is then y=-^[ 
 
 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: — 
 
 a; = TO -.I/ = — f f + - ] which for »! = iiidty becomes 1.54308 
 
 7)1 / ., 1 \ 
 
 x=^2m...y = —\^e'+ -^ j 
 
 ■'} 
 
 in which e is the base of Napierian 
 
 ng-. 3.S3. 
 
 x=3m..,y = —{^e'+ -j 
 
 TO / , 1 \ 
 x = im...y= Y(e'+ -j) 
 
 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. 
 
 For a; = B ^ m vn 
 
 have y = B P = 1.54308; for :c = a= — we have y = a n = 1.03142. 
 
 The tension at any point P is equal to the weight of a piece of rope of length BP^PC+m. 
 
 At the lowest point the tangent is horizontal. The length of any arc TP is proportional to the 
 angle 6 between T C and the tangent P F at the upper extremity of the arc. 
 
 215. If a circle R L B be drawn, of radius equal to to, it maj^ lie 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 as SB always equals RO and remains perpendicular to P S (angle R (J being alwa3's 90°) 
 we have the curve TSK fulfilling the conditions of a tractrix. (See Art. 202.) 
 
 If a parabola, having a focal distance to, 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 LOGAHITH.MIC 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, M 
 and iV, is a mean proportional between them ; i. e., p' = 5' = i/ X N. 
 
 If M and G are points of the spiral we maj' find an intermediate point K by drawing the 
 ordinate if to a semicircle of diameter M + G. A perpendicular through G to G K 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. 
 
 The curve never reaches the pole. 
 
 1 llankine. Applied Mechanics, Art. 175. 
 
 2 In the expression 102 = 100 the quantity "2" is caUed the togartthni of 103, it being the exponent of the power to which 
 10 must be raised to give 100. Similarly 2 wonld be the logarithm of 64 were 8 the base or number to be raised to the power 
 indicated. 
 
76 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 This spiral is often called Equiangular from the fact that the angle is always the same between 
 a radius vector and the tangent at its extremity. Ui^on this property is based its use as the out- 
 line for spiral cams and for lobed wheels. 
 
 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 is the 
 varying angle and a is some arbitrary constant. 
 
 To construct a tangent by calculation divide the h3'perbolic 
 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 tlie 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 jDroperties of recurrence led James 
 Bernouilli to direct that this spiral be engraved on his tomb, 
 with the inscription — Eadem Mutata Resurgo, which, freely trans- 
 lated, is — / 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 RECIPBOCAL SPIRAL. 
 
 218. In this spiral the length of a radius vector is in inverse ratio to the angle through which 
 it turns. 
 
 Like the logarithmic spiral it has an infinite number of convolutions aliout the pole, which it 
 never reaches. 
 
 The invention of this curve is attributed to James Bernouilli, who showed that Newton's con- 
 clusions as to the logarithmic spiral (see Art. 217) would also hold for the hyperbolic spiral, the 
 initial velocitj^ of projection determining which -trajectory was described. rpig-. 125. 
 
 To obtain points of the curve divide a circle m 5 S (Fig. 125) 
 into any number of equal parts, and on some initial radius m 
 lay off some unit, as an inch; on the second radius 3 take 
 
 On On 
 
 — ^; on the third -^-, etc. For one-half the angle the radius 
 
 vector would evidently be 2 n, giving a point 
 circle. 
 
 1 
 
 outside the 
 
 The equation to the curve is ^ = a 0, in which r is 
 
 the 
 
 1 To get the hyiierbolic logarithm of a number multiply its common logarithm by 2,302G. 
 = In eircnlar measure 360° = 27r7- which for r = l becomes 6.2831S ; 180° = 3.11159; 90^ = 1.3708: 60° 
 0.5236 ; 1 ° = 0.0174533. 
 
 = 1.0473 ; 45 ° = 0.7854 ; 
 
THE HYPERBOLIC SPIRAL.— THE LITUUS. 
 
 77 
 
 radius \-ector, a some numerical constant, and 6 is the angular rotation of /• (in circular measure) 
 estimated from some initial line. 
 
 The curve has an as^Tuptote parallel to the initial line and at a distance from it equal to 
 1 
 
 - units. 
 a 
 
 X'igr- 12S. 
 
 To construct the spiral fr-om its equation take as the pole (Fig. 126): OQ 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 draAv PR 
 a a 
 
 parallel to OQ for the asymptote. For 6=1, that is, for arc KH = radius OH we have 
 
 r =-= 2". orivinsi H. — one point of the spiral. Writing the equation in the form r = -.-r and 
 a oi ^^ 
 
 expressing various values of in circular measure we get the following: — 
 
 6 = 30° = 0.5236; /• = M= 3'.'8 + : ^ = 45° = 0.7854; r=0 X= ■r'.oo; 
 = 90° = 1.5708; r = .S' = 1'.'2 + : = 180° = 3.14159; r = T= .6366. etc. 
 The tangent to the curve at anj^ point makes with the radius vector an angle <^ which is found 
 
 by analysis to sustain to the angle the following trigonometrical relation, tan 0=6*; the circular 
 
 measure of 6 may therefore be found in a table of natural tangents and the corresponding value of 
 
 (t> obtained. 
 
 THE LITCrS. 
 
 219. The spiral in which the radius vector is inversely proportional to the square root of the 
 angle through which it has revolved is called the Lituus. This relation is shown by the equation 
 
 r = — ,vs) also written a" = - . 
 ay 6 r 
 
 When 6 = we find r = oo , making the initial line an asymptote to the curve. 
 
 In Fig. 127 take Q as the initial Une, as the pole, a = 2, and our unit a three-inch line; 
 
 then - = 11". 
 a 
 
78 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 For 6^=90° = 77 (in circular measure 1.5708) we have r = M=\".2 +. For e = 1 we have 
 the radius T making an angle of 57 ? 29 + with the initial line, and in length equal to - units, 
 
 H". 
 
 For 6 = Ab°= -r (or 0.7854) r 
 4 
 
 will be i? = 1'.'7 +. Then H = 
 
 R 
 
 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. Similarly' K 
 enabling the student to locate &ny number of points. 
 
 DM 
 
 DM 
 
 OV 
 
 , etc.; this relation 
 
 To draw a tangent to the curve we employ the relation tan (f> = 2 6. <^ being the angle made 
 by the tangent line with the radius vector, while 6 is the angular rotation of the latter, in circular 
 measure. • 
 
BRUSH TINTING AND SHADING. 79 
 
 CSAPTER VI. 
 
 TINTING — FLAT AND GRADUATED. — MASONRY, TILING, WOOD GRAINING, RIVER-BEDS AND 
 OTHER SECTIONS, WITH BRUSH ALONE OR IN COMBINED BRUSH AND LINE WORK. 
 
 220. Brush-work, with ink or colons, is either Jial or graduated; the former gives the effect of 
 a flat surface parallel to the paper on which the drawing is made, wliile g raded tints either show 
 curvature or — if indicating fiat surfaces — represent them as inclined to the paper, i. e., to the plane 
 of projection ; for either, the paper should be, as previously stated (Arts. 41 and 44) cold-pressed and 
 stretched. 
 
 The surface to be tinted should not be abraded by sponge, knife or rubber. 
 
 221. The liquid employed for tinting must be free from sediment; or, at least, the latter, if 
 present, must be allowed to settle and the brush dipjDed only in the clear portion at the top. Tints 
 may, therefore, best be mixed in an artist's water-glass rather than in anything shallower. In case 
 of several colors mixed together, however, it would be necessary to thoroughly stir up the tint each 
 tune Ijefore taking a brushful. 
 
 A tint prepared from a cake of high-grade India ink is far superior to any that can be made 
 bj' using the readj^-made liijuid drawing inks. 
 
 222. The size of Inrush should bear some relation to that of the surface to be tinted ; large 
 brushes for large surfaces and vice versa. The customary error of beginners is to use too small and 
 too dry a brush for tinting, and the reverse for shading. 
 
 223. Harsh outlines are to be avoided in brush work, especially in handsomely shaded drawings 
 in which, if sharply defined, they would detract fi-om the general effect. This will become evident 
 on comparing the spheres in Figs. 1 and 4 of Plate II. 
 
 If tinting and shading can be successfullj^ done with only pencilled limits there is then no excuse 
 for inking the boundaries ; but if, for the sake of definiteness, the outlines are inked at all it should 
 be before the tinting and in the finest of lines, preferably of "water-proof" ink; although any ink 
 will do ijrovided a soft sponge and plenty of clean water be applied to remove any excess that will 
 "run." The sponge is also to be the main reliance of the draughtsman for the correction of errors 
 in brush work; the water, however, and not the friction to be the active agent. An entire tint may 
 be removed in this way in case it seems desirable. 
 
 224. When beginning work incline the board at a small angle so that the tint will flow down 
 after the brush. For a flat, that is, a uniform tint, start at the upper outline of the surface to be 
 covered and with the brush full, yet not surcharged — which would prevent its coming to a good point — 
 pass lightly along from left to right and on the return carry the tint down a little farther, making 
 short, ([uick strokes, with the brush held almost perpendicular to the pajicr. Advance the tint as 
 evenly as possible along a horizontal line; work quicklj' between outlines, but more slowly along 
 outlmes, as one should never overrun the latter and then resort to " trimming " to conceal lack of 
 skill. It is possible for any one with care and practice to tint to yet not over boundaries. 
 
 The advancing edge of the tint must not dry until the lower boundary is reached. 
 
80 
 
 THEORETICAL AXD PRACTICAL GRAPHICS. 
 
 No portion of the paper, however small, should be missed as the tint advances, as the work is 
 likely to be spoiled bj^ retouching. 
 
 Should any excess of tint be found along the lower edge of the figure it should be absorbed 
 by the brush, after first removing the latter's surplus b^^ means of blotting paper. 
 
 To get a dark effect several medium tints laid on in succession, each one drying before the 
 next is apj)lied, give better results than one dark one. 
 
 The heightened effect described in Art. 72, viz., a line of light on the upper and left-hand edges, 
 may be obtained either (a) by ruling a broad line of tint with the drawing-peri at the desired 
 distance from the outline and instantlj', before it dries, tinting from it with the brush ; or (b) by 
 ruling the line with the pen and thick Chinese White. 
 
 225. A tint will spread much more evenl}' on a large surface if the paper be first slightly 
 dampened with clean water. As the tint will follow the water, the latter should be limited exactly 
 to the intended outlines of the final tint. 
 
 E'ig-- 123. 
 
 226. Of the colors frequently used bj' engineers and architects those which work best for flat 
 effects are carmine, Prussian blue, burnt sienna and Payne's gray. Sepia and Gamboge are, fortunately, 
 rarely required for even tints ; but the former works ideally for shading by the " dry " process 
 described in the next article ; and its rich Ijrown gives effects unapproachable with anything else. 
 It has, however, this peculiarity, that repeated touches upon a spot to make it darker produce the 
 opposite effect unless enough time elapses between strokes to allow each addition to dry thoroughly. 
 
 227. For elementary practice with the brush the student should lay flat washes, in India tints, 
 on from six to ten rectangles, of sizes between 2" x 6" and 6" x 10". If successful with these 
 his next work may be the reproduction of Fig. 128, in which H, V, P and S denote horizontal, 
 vertical, profile and section planes respectively. The figure should be considerably enlarged. 
 
 The plane V may have two washes of India ink; H one of Prussian blue; P one of burnt 
 sienna and S one of carmine. 
 
 The edges of H, V and P are either vertical or inclined 30° to the horizontal. 
 
BRUSH TINTING AND SHADING. 
 
 81 
 
 For the section-plane assume ') and m at pleasure, giving direction n m, to which JR and TX 
 are parallel. A horizontal, mz, through in gives z. From /*, a horizontal, n y, gives y on a b. 
 Joining y with z gives the "trace" of S on T. 
 
 228. Figures 129 and I'iO illustrate the use of the brush in the representation of masonry. 
 The former may be altogether in ink tints, or in medium burnt umber for the front rectangle of 
 
 ^-ig-- 12S. 
 
 p 
 
 1 
 
 pT^ 
 
 
 pBI. 
 
 
 W 
 
 
 each stone, and dark tint of the same, directly from the cake, for the bevel. Lightly pencilled 
 limits of bevel and rectangle will be needed ; no inked outlines required or desirable. 
 
 The last remark applies also to Fig. 130, in which " quarry- faced " ashlar masonry is represented. 
 If properly done, in either burnt umber or sepia, this gives a result of great beauty, especiallj' 
 effective on the piers of a large bridge drawing. 
 
 The darker portions are tinted directly from the cake, and are purposely made irregular and 
 "jagged" to reproduce as closely as possible the fractured appearance of the stone. 
 
 E'lg-- 130. 
 
 M'hen an '■ over-hang " or jutting jDortion is to be represented two brushes are required, one with 
 a medium tint, the other with the thick color, as before; an irregular line being made with the 
 latter the tint is softened out on the loiver side with the point of the brush having the lighter 
 tint. A light wash of the intended tone of the whole mass is quickly laid over each stone, either 
 before or qftei' the irregularities are represented, according as an exceedingly angular or a somewhat 
 softened and rounded effect is desired. 
 
82 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 229. Designs in tiling are excellent exercises not only for brush work in flat tints but also — in 
 their preliminary construction' — in precision of line work. The superbly illustrated catalogues of the 
 Minton Tile Works are, unfortunately, not accessible by all students, illustrating as they do, the finest 
 and most varied work in this line, both of designer and chromo-lithographer; but it is quite within 
 the bounds of possibility for the careful draughtsman to closely approach if not equal the standard 
 and general appearance of their work, and as suggestions therefor Figs. 131 and 132 are presented. 
 
 230. In Fig. 131 the upper boundary, a dh k, of a rectangle is divided at a, b, c, etc., into 
 equal spaces, and through each point of division two lines are drawn with the 30° triangle, as 6 x 
 and b r through b. The oblique lines terminate on the sides and lower line of the rectangle. If 
 the work is accurate — and it is worthless if not — any vertical line as mn, drawn through the inter- 
 section, m, of a pair of oblique lines, will pass through the intersection of a series of such pairs. 
 
 E^ig-- 131.. 
 
 The figure shows three of the possible designs whose construction is based on the dotted lines 
 of the figure. For that at the top and right, in which horizontal rows of rhombi are left white, we 
 draw vertical lines as s g and vi n from the lower vertex of each intended white rhombus, continuing 
 it over two rhombi, when another white one will be reached. The dark faces of the design are to 
 be finally in solid black, previous to which the lighter faces should be tinted with some drab or 
 brown tint. The pencilled construction lines would necessarily be erased before the tint was laid on. 
 
 The most opaque effect in colors is obtained by mixing a large proportion of Chinese white 
 with the water color, making what is called by artists a " body color." Such a mixture gives a 
 result in marked contrast with the transjaarent effect of the usual wash; but the amount of white 
 used should be sufficient to make the tint in reality a paste, and no more should be taken on the 
 brush at one time than is needed to cover one figure. 
 
 Sepia and Chinese white, mixed in the pro^Der proportions, give a tint which contrasts most 
 agreeably with the black and white of the remainder of the figure. The star design and the hexagons 
 in the lower right - hand corner result from extensions or modifications of the construction just 
 described, which will become evident on careful insi^ection. 
 
TILING. — BRUSH SHADING. 
 
 83 
 
 231. Fig. 132 is a Minton design with which many are familiar and which affords opportunity 
 for considerable variety in finish. Its construction is almost self-evident. The equal spaces ab, c d, 
 m n — which may be any width, x, — alternate with other equal spaces be which may preferably be 
 about 3 a- in width. Lines at 45°, as indicated, complete the preliminaries to tinting. 
 
 E-ig-- 3.32. 
 
 The octagons may he in Prussian blue, the hexagons in carmine and the remainder in white 
 and black, as shown ; or browns and drabs may be employed for more subdued effects. 
 
 SH.\DING. 
 
 232. For shading, by graduated tints, provide a glass of clear water in addition to the tint; 
 also an ample supply of blotting pajjer. 
 
 The water-color or ink tint may be considerably darker than for flat tinting; in fact, the darker 
 it is, provided it is clear, the more rapidlj' can the desired effect be obtained. 
 
 The brush must contain much less liquid than for flat work. 
 
 Lay a narrow band of tint quickly along the part that is to be the darkest, then dip the brush 
 into clear water and immediately apply it to the blotter, both to bring it to a good point and to 
 remove the surplus tint. With the now once-diluted tint carry the advancing edge of the band 
 slightly farther. Repeat the operation until the tint is no longer discernible as such. 
 
 The process may be repeated from the same starting point as many times as necessary to 
 produce the desired effect; but the work should be allowed to dry each time before laying on a 
 new tint. 
 
 Any irregularities or streaks can easily be removed, after the work dries, by retouching or 
 "stippling" with the point of a fine brush that contains but little tint — scarcely more than enough 
 to enable the brush to retain its jjoint. For small work, as the shading of rivets, rods, etc., the 
 process just mentioned, which is also called "dry shading," is especially adapted, and although 
 somewhat tedious gives the handsomest effects possible to the draughtsman. 
 
 233. Where a good, general effect is wanted, to l^e obtained in less time than would be required 
 for the preceding processes, the method by over-lapping flat tints may be adopted. A narrow band 
 of dark tint is first laid over the part to be the darkest. When dry this is overlaid by a broader 
 band of Ughter tint. A yet lighter wash follows, beginning on the dark portion and extending still 
 farther than its predecessor. The process is repeated with further diluted tints until the desired 
 ■effect is obtained. 
 
 Faintly-pencilled lines may be drawn at the outset as limits for the edges of the tints. 
 
84 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 This method is better adapted' for large work, that is not to be closely scrutinized, than for 
 drawings that deserve a high degree of finish. 
 
 234. As to the relative position and gradation of the lights and shades on a figure the student 
 is referred to Arts. 78 and 79 and the chapter on shadows ; also to the figures of Plate II, which 
 may serve as examples to be imitated while the learner is acquiring facility in the use of the brush 
 and before entering upon constructive work in shades and shadows. Fig. 3 of Plate II may be 
 undertaken first, and the contrast made yet greater between the upper and lower boundaries. Fig. 1 
 (Plate II) requires no explanation. In Fig. 133, we have a wood-cut of a sphere with the theoretical 
 dark or "shade" line more sharply defined than in the spheres on the plate. 
 
 ngr. J.33- ^ig'- 134:- 
 
 A drawing of the end of a highly-polished revolving shaft or even of an ordinary metallic disc 
 would be shaded as in Fig. 134. 
 
 Fig. 2 (Plate II) represents the triangular-threaded screw, its oblique surfaces being, in mathe- 
 matical language, warped helicoids, generated by a moving straight line, one end of which travels along 
 the axis of a cylinder while the other end traces or follows a helix on the cylinder. 
 
 The construction of the helix having already been given (Art. 120) the outlines can readily be 
 drawn. The method of exactly locating the shadow and shade lines will be found in the chapter 
 on shadows. 
 
 Fig. 4 (Plate II) when compared with Fig. 91 illustrates the possibilities as to the 
 representation of interesting mathenaatical relations. The fact may again l^e mentioned, on the 
 principle of "line upon line," as also for the benefit of any who may not have read all that has 
 preceded, that the spheres in the cone are tangent to the oblique plane at the foci of the elliiDtical 
 section. The peculiar dotted effect in this figure is clue to the fact that the original drawing, of 
 which this is a photographic reproduction by the gelatine process, was made with a lithographic 
 crayon upon a special pebbled paper inuch used by lithographers. The original of Fig. 1, on the 
 other hand, was a brush-shaded sphere on Whatman's paper. 
 
 235. Fig. 5 (Plate II) shows a " Phoenix column," the strongest form of iron for a given weight, 
 
 for sustaining comjDression. The student is familiar with it as an 
 element of outdc or construction in bridges, elevated railroads, etc. ; 
 and it is being increasingly employed in indoor work, notable examples 
 as to size being those of the New York World building. 
 
 By drawing first an end view of a Phtpnix column, similar to 
 that of Fig. 135, we can readily derive an oblique view like that of 
 the plate, by including it between parallels from all jjoints of the 
 former. The proportions of the columns are obtainable from the 
 tables of the company. 
 
 Fig. 135 is a cross-section of the 8-segment column, the shaded 
 portion showing the minimum and the other lines the maximum 
 size for the same inside diameter. 
 
 E'ig-. 13S- 
 
MA TERIA LS OF CONSTRUC TI N. 85 
 
 In a later chapter the proportions of other forms of structural iron will be found. Short 
 lengths of any of these, if shown in oblique view, are good subjects for the E"igr- iss. 
 
 brush, especially for " dry " shading, the effect to be aimed at being that 
 of the rail section of Fig. 136. 
 
 236. When some particular material is to be indicated, a Hat tint of 
 the proper technical color (see Art. 73) should be laid on with the brush, 
 either before or after shading. ^\'hen the latter is done with sepia it is 
 probably safer to lay on tlie Hat tint first. 
 
 A darker tint of the technical color should always be given to a cross- 
 section. For blue -printing, a cross -section may be indicated in solid black. 
 
 WOOD. RIVER - BEDS. MASOXR V, ETC. 
 
 237. While the engineering draughtsman is ordinarily so pressed for time as not to be able to 
 give his work the highest finish, j^et he ought to be able, when the occasion demands, to olitain 
 both natural and artistic efl'ects ; and to conduce to that end the writer has taken pains to illustrate 
 a number of ways of representing the materials of construction. Although nearly all of them may 
 be — and in the cuts are — rej^resented in black and white (with the exception of the wood-graining 
 on Plate II), yet colors, in combined brush and line work, are preferable. The student will, however, 
 need considerable practice with pen and ink before it will be worth while to work on a tinted figure. 
 
 238. Ordinarily, in representing wood, the mere fact that it is wood is all that is intended to 
 be indicated. This ma}' be done most simply by a series of irregular, approximately parallel lines, 
 as on the rule in Fig. 17, page 12. Make no attempt, however, to have the grain very irregular. 
 The natural unsteadiness of the hand in drawing a long line toward one continuously, will cause 
 almost all the irregularity desired. 
 
 If a better effect is wanted, yet without color, the lines may be as in Fig. 107, which represents 
 hard wood. 
 
 In graining, the draughtsman should make his lines totvard himself, standing, so to speak, at the 
 end of the plank ujDon which he is working. 
 
 The splintered end of a plank should be sharply toothed, in contradistinction to a metal or 
 stone fracture, which is what might be called smoothly irregular. 
 
 239. An examination of any jjiece of wood on which the grain is at all marked will show 
 that it is darker at the inner vertex of any marking than at the outer point. Although this 
 difference is more readily produced with the brush, yet it may be shown in a satisfactory degree 
 with the pen, by a series of after-touches. 
 
 240. If we fill the pen with a rather dark tint of the conventional color, draw the grain as in 
 the figures just referred to and then overlay all with a medium flat wash of some properly chosen 
 color, we get effects similar to those of Plate II. 
 
 On large timber -work the jjreliminary graining, as also the final wash, may be done altogether 
 with the brush ; as was the original of Fig. 9, Plate II. 
 
 End views of timbers and planks are represented by a series of concentric free-hand rings, the 
 spacing of which increases with the distance from the heart; these are overlaid with a few radial 
 strokes of darker tint. In ink alone the appearance is shown in Figs. 39 and 115. 
 
 241. The mixtures recommended for wood graining are something short of infinite in number; 
 Imt with the addition of one or two colors to those listed in the draughtsman's outfit (Art. 56) one 
 should be able to imitate the tint of any natural wood he might have in hand. 
 
86 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 COURSED RUBBLE MASONRY 
 Light India Inli. 
 
 Delicate rather than glaring tints should be employed. 
 
 No hard-and-fast rule as to the proportions of the colors can be given. In this connection one 
 recalls Sir .Joshua Reynolds' reply — "With brains" — to the one who inquired how he mixed his 
 paints. 
 
 Merely to indicate wood with a color and no graining use burnt sienna, the tint of Figs. 7, 8 
 and 10 of Plate II. 
 
 Drawing from the writer's experience and from the suggestions of various experimenters in this 
 line the following hints are presented: — 
 
 In every case grain first, then overlay with the ground tint, which should always be much 
 lighter than the color used for the grain. If possible have at hand a good specimen of the wood 
 to be imitated. 
 
 Hard Pine: Grain — burnt umber with either carmine or crimson lake; for overlay add a little 
 gamboge to the grain-tint diluted. 
 
 Soft Pine: Gamboge or yellow ochre with a small amount of l^urnt sienna. 
 
 Black Walnut: Grain — burnt umber and a very little . dragon's blood; final overlay of modified 
 tint of the same or with the addition of Payne's gray. ^'ig-- isv. 
 
 Oak: Grain — burnt sienna; for overlay, the same, with 
 yellow ochre. 
 
 Chestnut: Grain — burnt umber and dragon's blood; over- 
 lay of the same, diluted, and with a large proportion of gam- 
 boge or light yellow added. 
 
 Spruce: Grain — burnt umber, medium; add j^ellow ochre 
 for the overlay. 
 
 Mahogany: Grain — burnt sienna or umber with a small 
 amount of dragon's blood ; dilute and add light yellow for 
 the overlay. 
 
 Rosewood: Grain — replace the dragon's blood of mahogany- 
 grain by carmine, and for overlay dilute and add a little 
 Prussian blue. 
 
 242. River-beds in black and white or in colors have 
 been already treated in Art. 27, to which it is only necessary 
 to add that such sections are usually made quite narrow and, 
 preferably, — if in color — shaded out quite abruptly on the side 
 opposite the water. 
 
 243. The sections of masonry, concrete, brick, glass and vul- 
 canite, given on page 25 as pen and ink exercises, are again 
 
 j)resented in Fig. 137 for reproduction in combined brush and line 
 work. The appropriate color is indicated under each section. 
 
 , 244. Masonry constructions are roughly divided into rubble and 
 ashlar. 
 
 In ashlar masonry the bed -surfaces and the joints (edges) are 
 shaped and dressed with great care so that the stones may not 
 only be placed in regular layers or courses, but often fill exactly 
 some predetermined place, as in arch-construction, in which case the determination of their forms 
 and the derivation of the patterns for the stone-cutter involves the application of Mongers Descriptive. 
 
 ■ \ 
 
 1 
 
 " 1 
 
 X 
 
 ^ 
 
 ■ — - 
 
 m 
 
 V^ 
 
 -'T 
 
 / 
 
 ' 
 
 /' 
 
 RUGBLE MASONRY 
 
 Light India Ink. 
 
 WA/ '/)//''///// 
 
 wmmM. 
 
 BRICK 
 '/".nefian Red. 
 
 CONCRETE 
 Yellow Ochre. 
 
 1 l-w 
 
 
 ng-. 13S- 
 
 ■4r, 
 
 i 
 
 /I 
 
REPRESEXTATION OF MASONRY. 
 
 E-ig-. 1-iO- 
 
 Ricbble work, however, consists of constructions involving stones mainly "in the rough," liut mar 
 be either coursed or uncoursed. Fig. 138 is a neat example of uncoursed though partially dressed 
 or " hammered " rubble. In section, as shown in Fig. 137, it is merely necessary to rule section- 
 lines over the boundaries of the stones — a remark applying equally to ashlar masonry. 
 s-igr- iss- The other examples in this chapter 
 
 are of ashlar, mainly " quarry -faced," 
 
 tliat is, with ' the front nearly as rough 
 
 as when quarried. A beveled or 
 
 "chamfered" ashlar is shown in Figs. 
 
 129 and 140, the latter shaded in what 
 
 is probably the most effective way for 
 
 small work, viz., with dots, the effect 
 
 depending upon the number, not the 
 
 size of the latter. 
 
 Only a careful examination of the 
 kind and position of the linesj in' the other figures will disclose the secret of the variety in the 
 effects produced. For the handsomest results with any of these figures the pen-work — whether dotting 
 
 I^ig-. 1-41- 
 
 ^igf. 1^2- 
 
 or "cross-hatching"— should be preceded by an undertone of either India ink, umljer, Payne's gray, 
 coljalt or Prussian blue, according to the kind of stone to be represented. For slate use a pale blue; 
 
 S'igr- i-as- 
 
 E'iir- ±-- 
 
 for brown lree-.stone either an umber or sepia; while for stone in general, kind immaterial, use India 
 ink. 
 
88 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 CHAPTEB, VII. 
 
 FREE-HAND AND MECHANICAL LETTERING. — PROPORTIONING OP TITLES. 
 
 245. Practice in lettering forms an essential part of the elementary work of a draughtsman. 
 Every drawing has to have its title, and the general effect of the result as a whole depends largely 
 upon the quality of the lettering. 
 
 Other things being equal, the expert and rapid draughtsman in this line has a great advantage 
 over one who can do it but slowlj'. For this reason free-hand lettering is at a high premium,' and 
 the beginner should, therefore, aim not only to have his letters correctly formed and properly spaced., 
 but, as far as possible, to do without mechanical aids in their construction. When under great 
 pressure as to time it is, however, perfectly legitimate to emj^loy some of the mechanical expedients 
 used in large establishments as "short cuts" and labor-savers. Among these the principal are 
 "tracing" and the use of rubber types. 
 
 246. To trace a title one must have at hand complete printed alphabets of the size of type required. 
 Placing a fiiece of tracing-paper over the letter wanted, it is traced, with a hard pencil, the paper 
 then slipped along to the next letter needed and the process repeated until the words desired have 
 l)een outlined. The title is then transferred to the drawing by first running over the lines on the 
 back of the tracing- jjaper with a soft pencil, after which it is only necessary to re-trace the letters 
 with a hard pencil, on the face of the transfer -paper, to find their outlines faintly yet sufficiently 
 indicated on the paper underneath. 
 
 Carbon pajDer maj' also be used for transferring. 
 
 247. The process just described would be of little service to a ready free-hand draughtsman, 
 but with tlie use of rubber types, for the words most frequently recurring in the titles, a merely 
 average worker ma}^ easily get results which — in point of time — cannot be exceeded by any other 
 method. When emplo^dng such types either of the following ways may be adopted: (a) a light 
 impression may be made with the aniline ink ordinarily used on the pads, and- the outlines then 
 followed and the "filling in" done either with a writing- pen or fine- pointed sable-hair brush; or 
 (b) the impression may be made after moistening the types on a pad that has been thoroughly wet 
 with a light tint of India ink. The drawing-ink must then be immediately applied, free-hand, 
 with a Falcon j)en or sable brush, before the type -impression can drj'. The pen need only be f)assed 
 down the middle of a line, as on the dampened surface of the pajDer the ink will sjjread instantly 
 to the outlines. 
 
 248. The educated draughtsman should, however, be able not only to draw a legible title of the 
 simple character required for shop -work and in which the foregoing expedients would be mainly 
 serviceable, but be prejjared also for work out of the ordinary line and, if need be, quite elaborate, 
 as on a comf)etitive drawing. Such knowledge can only be gained by careful observation of the 
 forms of letters and considerable j^i'actice in their construction. 
 
 No rigid rules can be laid down as to choice of alphabets for the various possible cases. 
 Common-sense and a natural regard for the ''fitness of things" must be the determining factors. 
 
DESIGNING OF TITLES. 89 
 
 Obviously rustic letters would he out of place on a geometrical drawing, and other incongruities 
 will naturally suggest themselves. In addition to the hints in Art. 26 a few general principles and 
 instructions may, however, be stated to the advantage of the beginner. 
 
 249. In the first place, a title should be symmetrical with respect to a vertical centre-line, a 
 rule which may be violated but rarely and then, usually, when the title is to he somewhat fancy 
 in design, as for a magazine cover. 
 
 ElementaF|j Plates 
 
 M E P ^ A i I C A L D i AW I N G 
 
 drawn by QJi^rf kubf IBau Qlnrlmr ^+ +hE 
 
 LEaniNG TECHNICAL SCHDDL 
 
 Jan. —June, 3001. 
 
 250. If it he a complete as distinguished from a partial or szt6- title it will answer the following 
 questions which would naturally arise in the mind of the examiner : — 
 
 What is it? — Where done? — By whom? — When? — On what scale? 
 In answering these questions the relative valuation and importance of the lines are expressed by 
 the sizes and kinds of type chosen. This is a point requiring most careful consideration, as the final 
 effect depends largely upon a proper balancing of values. 
 
 — >-^- OF "«— f^ 
 
 PERFECTION SUSPENSION BRIDGE 
 
 —<i^i<^ design Ed tiy 4<«— 
 G-DDdwin^ Mackenzie V Carivrright 
 
 —-^^^m MINNEAPOLIS. MINN. #-^-^<— 
 
 SOALE 4 Ft. = I In. JU.ne 16, 2900- ■Jo.se Martinez, del. 
 
 2.51. The "By whom?" may cover two possibilities. In the case of a set of drawings made in a 
 scientific school it would refer to the draughtsman and his name might properly have considerably 
 greater prominence than in any other case. The upper title is illustrative of this point and at 
 the same time suggestive of a good general arrangement for such a case. 
 
 Ordinarily the "By whom?" will refer to the de.signer, and the draughtsman's name ought 
 to be comijaratively inconspicuous, while the name of the designer should be given a fair degree 
 of prominence. This, and other important points to be mentioned, are illustrated in the preceding 
 
90 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 arrangement, printed, like the upper title, fi-om types of which complete alphabets will be found in 
 the Appendix. 
 
 252. The abbreviation Del., following the draughtsman's name, is for Delineavit — He drew it — and 
 does not indicate what the visitor at the exhibition supposed, that all good draughtsmen hail from 
 Delaware. 
 
 253. The best designed titles are either in the form of two truncated pyramids having the 
 most important line as their common base, or else elliptical in shape. 
 
 254. The use of capitals throughout a line depends u^jon the style of type. It gives a most 
 unsatisfactory result if they are of irregular outline, as is amply evidenced by the words 
 
 each letter of which is exquisite in form but the comljination almost illegible. Contrast them with 
 the same style, but in capitals and small letters : — 
 
 M:erl;antral ^ra)^rt^g, 
 
 255. As to spacing, the visible white spaces between the letters should be as nearly the same 
 as possible. In this feature, as in others, the draughtsman can get much more pleasing results than 
 the printer, since the latter has each letter on a separate piece of metal, and can not adjust his 
 space to any particular combination of letters, such as FA, L V, W A or A V, where a better effect 
 would be obtained by placing the lower part of one letter under the s^ig-- i-^s- 
 
 upper part of the next. This is illustrated in Fig. 146, which may be ~ r' V /"* . \ ~ V*" H ~ p" J 
 contrasted with the printer's best spacing of the A and W in the word / / \ ' / r* i 
 " Drawings " of the last title. ^ -I ' — ' ^ ' 
 
 256. The amount of space between letters will depend upon the length of line that the word or 
 words must make. If an important word has few letters they should be "spaced out" and the 
 letters themselves of the " extended " kind, i. e., broader than their height. The following word will 
 illustrate. The characteristic feature of this type, viz., heavy horizontals and light verticals, is com- 
 mon to all the variations of a fundamental form frequently called Italian Print. 
 
 IB ^^ I ^ & E . 
 
 When, on the other hand, many letters must be crowded into a small space, a " condensed " 
 style of letter must be adojDted, of which the following is an example: — 
 
 Pennsylvania Railroail. 
 
 257. While the varieties of letters are very numerous, yet they are all but changes rung on a 
 few fundamental or basal forms, the most elementary of which is the 
 
 GOTHIC, ALSO CALLED HALF- BLOCK. 
 
 Letters like B, 0, etc., which have, usually, either few straight parts or none at all, may, for 
 the sake of variety as also for convenience of construction, be made partially or wholly angular; in 
 the latter case the form is called Geometric Gothic by some tyjae manufacturers. It is especially 
 appropriate for work exclusively mechanical. 
 
LETTERING. 
 
 91 
 
 The following complete Gothic alphabet is so constructed that whether designed in its "condensed" 
 or " extended " form the proper proportions may be easily j^reserved. 
 
 t u* 
 
 ? V < 
 
 x/ 
 
 tV '• 'f u ■ 
 
 Taking all the solid parts of the letters at the same width as the I, we will find any letter of 
 average uidih, as U, to be twice that unit, jjlus the oi^ening between the uprights, which last — being 
 indeterminate — we may call x, making it small for a "condensed" letter and broad as need be for 
 an " extended " form. 
 
 The word march would foot up 5 U + 3, disregarding — as we would invariably — the amount the 
 foot of the E projects beyond the main right-hand outline of the letter. In terms of x this makes 
 5 a; +13, as U = .c + 2. Allowing spaces of 11 unit width between letters adds 5 to the above, making 
 5 .r + 18 for the total length in terms of the I. Assuming x ecj^ual to twice the unit we would 
 have the whole word equal to twenty- eight units; and if it were to extend seven inches the width 
 of the solid parts would therefore be one- quarter of an inch. 
 
 A^'here the width of a letter is not indicated it is assumed to be that of the U. The ^V is 
 equal to 2 U — 1. This relation, however, does not hold good in all alphabets. 
 
 The angular corners are drawn usually with the 45° triangle. 
 
 The guide-hnes show what points of the various letters are to be found on the same level, and 
 should be but faintly jjencilled. 
 
 As remarked in Art. 26, the extended form of Gothic is one of the best for dimensioning and 
 lettering xoorhing drawings and is rapidlj^ coming into use by the profession. 
 
 258. The Full -Block letter next illustrated is easier to work with than the Gothic in the matter of 
 preliminary estimate, as the width of each letter — in terms of unit squares — is evident at a glance. 
 
 The same word march would foot up twentj^- seven squares without allowing for spaces between 
 letters. Calling the latter each hoo we would have thirty- five sciuares for the same length as iDefore 
 (seven inches) making one- fifth of an inch for the width of the solid jjarts. For convenience the 
 widths of the various letters are summarized : 
 1 = 3; C, G, 0, Q, S, Z = 4; A, B, D, E, F, J, P, R, T. ct = 5 ; H, K, N, U, V, X, Y = 6; M = 7 ; W = 8. 
 
 259. In case the preliminary figuring were only approximate and there were but two words in 
 the line, as, for example. Mechanical Draioing, a safe method of working would be to make a fair 
 allowance for the space between the words, begin the first word at the calculated distance to the 
 left of the vertical centre-line, complete it, then work the second word backward, beginning with the 
 
92 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 G as far to the right of the reference liae as the M was to the left. On completing the second 
 word any difference between the actual and the estimated length of the words, due to ov.er- or 
 under-width of such letters as M, W and I, will be merged into the space between the words. 
 
 N-;.jr 
 
 Z-J.'C 
 
 ■l^z^ 
 
 '6h6±2±\±h±\t6}-. 
 
 Tr3 
 
 
 E'ig-- ISO. 
 
 With three words in a line the same method might be adopted, the middle word being easily 
 placed half way between the others which, by the construction, would begin and terminate where 
 they should. 
 
 260. Note particularly that the top of a B is always slightly smaller than the bottom ; ^^s- ^'S.s. 
 similarly with the S. This is made necessary by the fact that the eye seems to exaggerate C'i C^ 
 the upper half of a letter. To get an idea of the amount of difference allowable compare L^ |^ 
 the following equal letters printed from Roman type, condensed. Although not so important 
 
 in the E still some difference between top and bottom may to advantage be made. Another refine- 
 ment is the location of the horizontal cross-bar of an A slightly below the middle of the latter. 
 
 261. While vertical letters are most frequently used yet no handsomer effect can be obtained 
 than by a well- executed inclined letter. The angle of inclination should be about 70°. 
 
 Beginners usually fail sadly in their first attempt 
 
 with the A and V, one of whose sides they give the 
 
 same slant as the upright of the other letters. In jDoint 
 
 of fact, however, it is the imaginary (though, in the 
 
 construction, pencilled) centre-line which should have that 
 
 inclination. See Fig. 150. 
 
 In these forms — the Roman and Italic Roman — the union of the light horizontals or "seriffs" 
 
 with the other parts is in general effected by means of fine arcs, called "fillets," drawn free-hand. 
 
 On many letters of this alphabet some lines will, however, meet at an angle, and only a careful 
 
 examination of good models will enable one to construct correct forms. Upon the - size of the fillets 
 
 the appearance of the letter mainly depends, as will be seen by a glance at Fig. 151, which repro- 
 duces, exactly, the N of each of two leading alphabet books. If the fillets 
 round out to the end of the spur of the letter a coarse and bulky appear- 
 ance is evidently the result; while a fine curve, leaving the straight 
 horizontals projecting beyond them, gives the finish desired. This is further 
 
 illustrated by alphabet No. 23 of the Appendix, a type which for clearness and elegance is a 
 
 triumph of the founder's art. If preferred, the D and R may be finished at the top like the P. 
 
 E'ig'. ISl. 
 
 JVJV 
 
n 
 □ 
 
94 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 262. The Roman alphabet and the Italic Roman are much used m topographical work, and 
 while testing the draughtsman's skill to the utmost amply reward him for his labor. 
 
 A text- book devoted entirely to the Roman alphabet is in the market, and in some works on 
 topographical drawing very elaborate tables of proportions for the letters are presented ; these answer 
 admirably for the construction of a standard alphabet, but in practice the proportions of the model 
 would be preserved by the draughtsman no more closely than his eye could secure. Except when 
 more than one -fifth of an inch in height these letters should be drawn entirely free-hand. 
 
 263. When a line of a title is curved no change is made in the forms of the letters; but if of 
 a vertical, as distinguished from a slanting or italic type, the centre-line of each letter should, if 
 produced, pass through the centre of the curve. 
 
 Italic letters, when arranged on a curve, should have their centre-lines inclined at the same angle 
 to the normal (or radius) of the curve as they ordinarily make with- the vertical. 
 
 264. An aljihabet which gives a most satisfactory appearance, yet can be constructed with great 
 rapidity, is what we may call the " Railroad " type, since the public has become familiar with it 
 mainly from its frequent use in railroad advertisements. 
 
 The fundamental forms of the small letters, with the essential construction lines, are given in 
 rectangular outline in the complete alphabet on the preceding page, with various modifications thereof 
 in the words below them, showing a large number of possible effects. 
 
 At least one plain and fancy capital of each letter is also to be found on the same page, with in 
 some instances a still larger range of choice. 
 
 No handsomer effects are obtainable than with this alphabet when brush tints are employed for 
 the undertone and shadows. 
 
 265. For rapid lettering on tracing -cloth, Bristol ,1^^ " ^^ 
 board or any smooth-surfaced paper a style long used /^ / "^ 
 
 c) icand SruiUiq 
 
 abroad and increasing in favor in this country is that 
 
 known as Round Writing, illustrated by Fig. 152, and 
 
 for which a special text-book and pens have been prepared by F. Soennecken. The pens are 
 
 E'ig-. 1E3. stubs of various widths, cut off obliquely, and when in 
 
 E„_-„— -Q-y a'eC "*® should not, as ordinarily, be dipped into the ink, but 
 
 11/ -^ the latter should' be inserted by means of another pen — 
 
 ' between the top of the Soennecken pen and the brass 
 
 "feeder" that is usually slipped over it to regulate the flow. 
 
 h, I -^ I The Soennecken Round Writing Pens are also by far 
 
 aniCal UraWinq the best for lettering in Old English, German Text and 
 
 ' *-' kindred types. 
 
 The improvement due to the addition of a few straight lines to an ordinary title will become 
 evident by comparing Figs. ^" i ■ E'^i-e-- is-4. 
 
 153 and 154. The judicious 
 use of "word ornaments" 
 
 such as those of alphabets _^^=.^ i p. 
 33, 42, 49 and of other ] 
 
 tj'pes in the Appendix, will wy i i i pv 
 
 p lemenhary PLfihe.^- 
 
 tjqjes m the Appendix, will w y i i i pv i 
 
 greatly enhance the appear- p^ eehanieQl U T Q W I P I d 
 
 ance of a title, without ma- 
 
 teriallj^ increasing the time expended on it. This is illustrated in the lower title on page 89. 
 
DESIGNS FOR BORDERS. 
 
 95 
 
96 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 266. Another effective adjunct to a map or other drawing is a neat border. It should be strictly 
 in keeping with the drawing-, both as to character and simplicit}'. 
 
 On page 95 a large number of corner designs and borders are presented. The principle 
 of their construction is illustrated by Fig. 155, in which the larger design shows the necessary 
 preliminary lines and the smaller the completed corner. It is evident in this, as in all cases of 
 interlaced designs, that we must lay off each way from the corner as many squares as there are 
 bands and spaces, and make a network of squares — or of rhombi, if the angles are acute — by 
 construction - lines through the points of division. 
 
 267. The usual rule as to shade lines applies equally to these designs, thus : Following any 
 band or pair of lines making the turns as one piece, if it runs horizontally 
 the loiver line is the heavier, while in a vertical pair the right-hand line 
 is the shade line. This is on the assumption that the light is coming 
 in the direction usually assumed for mechanical drawings, i. e., descending 
 diagonally from left to right. 
 
 In case a pair of lines runs obliquely the shade lines may be 
 determined by a study of their location on the designs of the plate of 
 borders. 
 
 It need hardl}^ be said that on any drawing and its title the light 
 should be supposed to come from hut one direction throughout and not be 
 shifted ; and the shade lines should be located according■l3^ This rule is always imperative. 
 
 In drawing for scientific illustration or in art work it is allowable to depart from the usual 
 strictlj' conventional direction of light if a better effect can therebj^ be secured. 
 
 268. A striking letter can be made by drawing the shade Hne only, as in Fig. 146, page 90, which 
 we may call "Full-Block Shade-Line," being based upon the aliAabet of Fig. 148, page 92, as to 
 construction. Owing to its having more i^rojecting parts it gives a much handsomer effect than the 
 
 
 
 
 
 t 
 
 r. 
 
 6 
 
 5 
 
 413 
 
 rr 
 
 "^ 
 
 
 -- 
 
 i3 
 
 
 
 
 
 " 
 
 |4 
 
 r' 
 
 
 ; \ 
 
 15 
 
 
 
 
 
 16, 
 
 V- 
 
 
 ? 
 
 \ 
 
 7 
 
 l 
 
 
 i, 
 
 1 
 
 I 1 
 
 I 1 ! 
 
 \ 
 
 alA 
 
 -H 
 
 ^Ml 
 
 
 L |1 
 
 \ 
 
 \\kuv--^Vi^(:Vv ^v\iiv>\^-ViVVi\E,c 
 
 The student will notice that the light comes from different directions in the two examples. 
 
 These forms are to the ordinarj' fully- outlined letters what art work of the "impressionist" 
 school is to the extremely detailed and painstaking work of many; what is actually seen suggests 
 an equal amount not on the paper or canvas. 
 
 269. While a teacher of draughting may - well have on hand, as reference works for his class, 
 such elaborate books on lettering as Prang's and Becker's, yet they will be found of only occasional 
 service, their designs being as a rule more highly ornate than any but the specialist would dare 
 undertake, and mainly of a character unsuitable for the usual work of the engineering or architec- 
 tural draughtsman, whose needs were especiallj' in mind when selectixig types for this work. 
 
 The Appendix affords a large range of choice among the handsomest forms recently designed by 
 the leading type manufacturers, also containing the best among earlier types; and with the "Rail- 
 road," Full- Block and Half- Block alphabets of this chapter, proportioned and drawn by the writer, 
 supplies the student with a practical "stock-in-trade" that will probably require but little, if any, 
 supplementing. 
 
COPYING PROCESSES.— DRAWING FOR ILLUSTRATION. 97 
 
 CMAFTBB nil. 
 
 BLTJE-PEINT AND OTHER COPYING PKOCESSES. — iMETHODS OF ILLUSTRATION. 
 
 270. While in a draughting ofi&ce the jjrocess described below is, at present, the only method 
 of copying drawings with which it is absolutely essential that the draughtsman should be thoroughly 
 acquainted, he may, nevertheless, find it to his advantage to know how to prepare drawings for 
 reproduction by some of the other methods in most general use. He ought also to be able to 
 recognize, usualty, by a glance at an illustration, the method by which it was obtained. Some brief 
 hints on these points are therefore introduced. 
 
 This is, obviously, however, not the place to give full particulars as to all these processes, even 
 were the methods of manipulation not, in some cases, still " trade secrets " ; but all important details 
 concerning them, that have become common jjroperty, may be obtained fi-om the following valuable 
 works: Modern Heliographic Processes,* by Ernst Lietze; Photo- Engraving, Etching and Lithography,^ by 
 "W. T. Wilkinson ; and Modern Reproductive Graphic Processes, * by Jas. S. Pettit. 
 
 THE BLUE -PRINT PROCESS. 
 
 271. By means of this process, invented by Sir John Herschel, any number of copies of a draw- 
 ing can be made, in white lines on a blue ground. In Arts. 43 and 45 some hints will be found 
 as to the relative merits of tracing-cloth and "Bond" jjaper, for the original drawing. 
 
 A sheet of paper may be sensitized to the action of light by coating its surface with a solution 
 of red prussiate of potash (ferrocyanide of potassium) and a ferric salt. The chemical action of 
 light upon this is the production of a ferrous salt from the ferric compound ; this combines with 
 the ferrocyanide to produce the final blue undertone of the sheet; while the portions of the i^aper, 
 fi:om which the light was intercepted by the lines of the drawing, become white after immersion in 
 water. 
 
 The proportions in which the chemicals are to be mixed, are, apparently, a matter of indiffer- 
 ence, so great is the disj^arity between the recipes of different writers ; indeed, one successful 
 draughtsman saj^s: "Almost any jDroportion of chemicals will make blue-jDrints." Whichever recipe 
 is adoj)ted — and a considerable range of choice will be found in this chapter — the hints immediately 
 following are of general application. 
 
 272. Any white paper will do for sensitizing that has a hard finish, like that of ledger paper, 
 so as not to absorb the chemical solution. 
 
 To sensitize the pajser dissolve the ferric salt and the ferrocyanide in water, separately, as they 
 are then not sensitive to the action of light. The solutions should be mixed and applied to the 
 paper only in a dark room. 
 
 Although there is the highest authority for "floating the paper to be sensitized for two minutes 
 on the surface of the liquid " yet the best American practice is to apply the solution with a soft 
 flat brush about four inches wide. The main object is to obtain an even coat, which may usually 
 
 *Pul>lisliea by the D. Van Nostrand Company, New York, t American Edition revised and published by Edward L. 
 Wilson, New York. 
 
98 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 be secured by a primary coat of horizontal strokes followed by an overlay of vertical strokes ; the 
 second coat applied before the first dries. If necessary, another coat of diagonal strokes may be 
 given to secure evenness. The thicker the coating given the longer the time required in printing. 
 A bowl or flat dish or plate will be found convenient for holding the small portion of the solution 
 required for use at any one time. The chemicals should not get on the back of the sheet. 
 
 Each sheet, as coated, should be set in a dark place to dry, either "tacked to a board by two 
 adjacent corners", or "hung on a rack or over a rod", or "placed in a drawer — one sheet in a 
 drawer", — varying instructions, illustrating the quite general truth that there are usually several 
 almost equally good ways of doing a thing. 
 
 273. To copy a drawing place the prepared paper, sensitized side up, on a drawing-board or 
 printing -frame on which there has been fastened, smoothly, either a felt pad or canton flannel cloth. 
 The drawing is then immediately placed over the first sheet, inked side up, and contact secured 
 between the two by a large sheet of plate glass, placed over all. 
 
 Exposure in the direct rays of the sun for four or five minutes is usually sufficient. The 
 progress of the chemical action can be observed by allowing a corner of the paper to project beyond 
 the glass. It has a grayish hue when sufficiently exjjosed. 
 
 If the sun's rays are not direct, or if the day is cloudy, a proportionately longer time is required,, 
 ruirning up in the latter case, from minutes into hours. Only experiment will show whether one's 
 solution is " quick " or " slow ; " or the time required by the degree of cloudiness. 
 
 A solution will print more quickly if the amount of water in it be increased or if more iron 
 is used ; but in the former case the print will not be as dark, while in the latter the results, as to- 
 whiteness of lines, are not so apt to be satisfactory. 
 
 Although fair results can be obtained with paper a month or more after it has been sensitized,, 
 yet they are far more satisfactory if the paper is prepared each time (and dried) just before using.. 
 
 On taking the print out of the frame it should be immediately immersed and thoroughly 
 washed in cold water for fi-om three to ten minutes, after which it may be dried in either of the 
 ways previously suggested. 
 
 If many prints are being made, the water should be frequently changed so as not to become 
 charged with the solution. 
 
 274. The entire jarocess, while exceedingly simple in theory, varies, as to its results, with the- 
 experience and judgment of the manipulator. To his choice the decision is left between the following 
 standard recipes for preparing the sensitizing solution. The " parts " given are all by weight. In 
 every case the potash should be pulverized, to facilitate its dissolving. 
 
 No. 1. (From Le G^nie Civil). 
 
 f Red Prussiate of Potash 8 parts. 
 
 Solution No. 1. 
 
 { 
 
 Water 70 parts. 
 
 f Citrate of Iron and Ammonia 10 parts. 
 
 iion No. 2. \ 
 
 { Water 70 parts. 
 
 Filter the solutions separately, mix equal quantities and then filter again. 
 
 Solution No. 
 Solution No. 2. 
 
 ■ '■ [ W 
 
 { 
 
 No. 2. (From U. S. Laboratory at Willett's Point). 
 
 Double Citrate of Iron and Ammonia 1 ounce. 
 
 ater 4 ounces. 
 
 Red Prussiate of Potassium 1 ounce. 
 
 Water 4 ounces. 
 
Stock Solution. 
 
 BLUE-PRINT PROCESS. 99 
 
 No. 3. (Lietze's Method). 
 5 ounces, avoirdupois, Red Prussiate of Potash. 
 32 fluid ounces Water. 
 
 "After the red prussiate of potash has been dissolved — which requires from one to two days — ■ 
 the liquid is filtered. This solution remains in good condition for a long time. Whenever it is 
 required to sensitize paper, dissolve, for every two hundred and forty square feet of paper 
 
 1 ounce, avoirdupois. Citrate of Iron and Ammonia, 
 ii fluid ounces Water, 
 
 and mis this with an equal volume of the stock solution. 
 
 The reason for making a stock solution of the red prussiate of potash is, that it takes a con- 
 siderable time to dissolve and because it must be filtered. There are many impurities in this 
 chemical which can be removed by filtering. Without filtering, the solution will not look clear. 
 The reason for making no stock solution of the ferric citrate of ammonia is that such solution soon 
 becomes moldy and unfit for use. This ferric salt is brought into the market in a very j^ure state 
 and does not need to be filtered after being dissolved. It dissolves very raj^idly. In the solid 
 form it may be preserved for an unlimited time, if kept in a well -stoppered bottle and protected 
 against the moisture of the atmosijhere. A solution of this salt, or a mixture of it with the solution 
 of red prussiate of potash, will remain in a serviceable condition for a number of days, but it will 
 spoil, sooner or later, according to atmospheric conditions. . . . Four ounces of sensitizing solution, 
 for blue prints, are amply sufiicient for coating one hundred square feet of paper, and cost about 
 six cents." 
 
 For copying tracings in blue lines or black, on a white ground, one may either employ the 
 recipes given in Lietze's and Pettit's works or obtain paper, already sensitized, from the leading dealers 
 in draughtsmen's supplies. The latter course has become quite as economical, also, for the ordinary 
 blue -print, as the preparing of one's own supjaly. 
 
 For copying a drawing in any desired color the following method, known as TilheCs, is 
 said to give good results : " The paper on which the copy is to appear is first dipped in a 
 bath consisting of 30 parts of white soap, 30 parts of alum, 40 jparts of English glue, 10 parts 
 of albumen, 2 jjarts of glacial acetic acid, 10 parts of alcohol of 60°, and 500 parts of water. It 
 is afterward jDut into a second bath, which contains 50 parts of burnt umber ground in alcohol, 
 20 parts of lampblack, 10 parts of English glue, and 10 parts of bichromate of potash in 500 j^arts 
 of water. They are now sensitive to light, and must, therefore, be preserved in the dark. In 
 preparing paper to make the positive jsrint another bath is made just like the first one, except that 
 lampblack is substituted for the burnt umber. To obtain colored positives the black is replaced by 
 some red, blue or other jDigment. 
 
 In making the copy the drawing to be copied is put in a photographic printing frame, and the 
 negative paper laid on it, and then exposed in the usual manner. In clear weather an illumination 
 of two minutes will suffice. After the exposure the negative is put in water to develop it, and the 
 drawing will ap^jear in white on a dark ground; in other words, it is a negative or reversed picture. 
 The paper is then dried and a positive made from it by placing it on the glass of a i^rinting- 
 frame and laying the positive jjajjer ujDon it and exposing as before. After placing the fi-ame in 
 the sun for two minutes the positive is taken out and put in water. The black dissolves off without 
 the necessity of moving back and forth." 
 
100 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 PHOTO -AND OTHER PROCESSES. 
 
 275. If a drawing is to be reproduced on a different scale from that of the original, some one 
 of the processes which admits of the use of the camera is usually employed. Those of most 
 importance to the draughtsman are (1) ivood engraving; (2) the "wax process" or cerography ; (3) 
 lithography, and (4) the various methods in which the photographic negative is made on a film of 
 gelatine which is then used directly — to print from, or indirectly — in obtaining a metal plate from, 
 which the impressions are taken. 
 
 In the first three named above the use of the camera is not invariably an element of the 
 process. 
 
 All under the fourth head are essentially photo -processes and their already large number is 
 constantly increasing. Among them may be mentioned photogravure, collotype, phototype, autotype, photo- 
 glyph, alberiype, heliotype, and heliogravure. 
 
 WOOD ENGRAVING. 
 
 276. There is probably no process that surpasses the best work of skilled engravers on wood. 
 This statement will be sustained by a glance at Figs. 14, 1-5, 20-24, 134, 136, and those illustrating 
 mathematical surfaces, in the next chaj)ter. Its expensiveness and the time required to make an 
 illustration by this method are its only disadvantages. 
 
 Although the camera is often emploj^ed to transfer the drawing to the boxwood block in which 
 the lines are to be cut, yet the original drawing is quite as frequently made in reverse, directly on 
 the block, by a professional draughtsman who is sujjposed to have at his disposal either the objeot 
 to be drawn or a photograph or drawing thereof The outlines are pencilled on the block and the 
 shades and shadows given in brush tints of India ink, re- enforced, in some cases, by the pencil, for 
 the deepest shadows. 
 
 The " high hghts " are brought out by Chinese white. A medium wash of the latter is also 
 usually spread upon the block as a general preliminary to outlining and shading. 
 
 The task of the engraver is to reproduce faithfully the most delicate as well as the strongest 
 effects obtained on the block with pencil and brush, cutting away all that is not to appear in black, 
 in the print. The finished block may then be used to print from directly, or an electrotype block 
 can be obtained from it which will stand a large number of impressions much better than the wood. 
 
 CEROGRAPHY. 
 
 277. For map -making, illustrations of machinery, geometrical diagrams and all work mainly in 
 straight lines or simple curves, and not involving too delicate gradations, the cerographic or " wax 
 process " is much employed. For clearness it is scarcely surjoassed by steel engraving. Figures 36, 
 90 and 107 are good sijecimens of the effects obtainable by this method. The successive steps in the 
 process are (a) the laying of a thin, even coat of wax over a copper i^late; (b) the transfer of 
 the drawing to the surface of the wax, either by tracing or — more generally — by photography; (c) 
 the re -drawing or rather the cutting of these lines in the wax, the stjdus removing the latter to the 
 surface of the cojDper; (d) the taking of an electrotype from the plate and wax, the deposit of 
 copper filling in the hnes from which the wax was removed. 
 
 Although in the preparation of the original drawing the lines may preferably be inked yet this 
 is not absolutely necessary, provided a pencil of medium grade be employed. 
 
LITHOGRAPHY.— PHOTO-ENGRAVING. 101 
 
 Any letters desired on the final plate may be also pencilled in their proper places, as the 
 engraver makes them on the wax with type. 
 
 A surface on which section -lining or cross-hatching is desired may have that fact indicated upon 
 it in writing, the direction and number of lines to the inch being given. Such work is then done 
 with a ruling machine. 
 
 Errors may readilj^ be corrected, as the surface of the wax may be made smooth, for recutting, 
 by passing a hot iron over it. 
 
 LITHOGEAPHY. — PHOTO - LITHOGEAPHY. — CHEOMO - LITHOGRAPHY. 
 
 278. For the lithographic process a fine-grained, imported limestone is used. The drawing is 
 made with a greasy ink — known as "lithographic" — upon a specially jjrepared paper, from which 
 it is transferred, under pressure, to the surface of the stone. The un- inked parts of the stone are 
 kept thoroughly moistened with water, which prevents the printer's ink (owing to the grease which 
 the latter contains) from adhering to any portion except that fr'om which the impressions are desired. 
 
 Photo- lithography is simply lithography, with the camera as an adjunct. The jDOsitive might be 
 made directly upon the surface of the stone bj' coating the latter with a sensitizing solution; but, 
 in general, for convenience, a sensitized gelatine film is exposed under the negative, and by subsecpient 
 treatment gives an image in relief which, after inking, can be transferred to the surface of the stone 
 as in the ordinary process. 
 
 Chroma -lithography, or lithography in colors, has been a very expensive jDrocess owing to its 
 requiring a separate stone for each color. Recent inventions render it probable that it will be much 
 simplified and the expense correspondingly reduced. The details of manipulation are closely analogous 
 to those for ink prints. 
 
 When colored plates are wanted, in which delicate gradations shall be indicated, chromo-Hthograpliy 
 may preferably be adopted ; although " half tones," with colored inks, give a scarcely less pleasing 
 effect, as illustrated hj Figs. 7-10, Plate II. But for simple line-work, in two or more colors, one 
 may jjreferably employ either cerography or i^hoto - engraving, each of which has not onl}^ an 
 advantage, as to expense, over any lithographic process, but also this in addition — that the blocks 
 can be used b}^ any printer; whereas lithograi^hing establishments necessarily not only prepare the 
 stone but also do the printing. 
 
 PHOTO - ENGEAVIXG. — PHOTO - ZINCOGRAPHY. 
 
 279. In this popular and rapid process a sensitized solution is spread upon a smooth sheet of 
 zinc and over this the photographic negative is placed. Where not acted on by the light the 
 coating remains soluble and is washed away, exposing the metal, which is then further acted on by 
 acids to give more relief to the remaining portions. 
 
 Except as described in Art. 281 this process is only adapted to inked work in lines or dots, 
 which it reproduces faithfully, to the smallest detail. Among the best photo - engravings in this book 
 are Figs. 12, 13, 50, 79 and 80. 
 
 280. The following instructions for the preparation of drawings, for reproduction by this process, 
 are those of the American Society of Mechanical Engineers as to the illustration of papers by its 
 members, and are, in general, such as all the engraving companies furnish on application. 
 
 "All lines, letters and figures must be perfectly black on a white ground. Blue prints are not 
 available, and red figures and fines will not appear. The smoother the paper, and the blacker the 
 ink, the better are the results. Tracing-cloth or paper answers very well, but rough paper — even 
 
102 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 Whatman's — gives bad lines. India ink, ground or in solution, should be used; and the best lines 
 are made on Bristol board, or its equivalent with an enameled surface. Brush work, in tint or 
 grading, unfits a drawing for immediate use, since only line work can be photographed. Hatching 
 for sections need not be completed in the originals, as it can be done easily by machine on the 
 block. If draughtsmen will iiadicate their sections unmistakably, they will be properly lined, and 
 tints and shadows will be similarly treated. 
 
 The best results may be expected by using an original twice the height aiad width of the 
 proposed block. The reduction can be greater, provided care has been taken to have the lines far 
 enough apart, so as not to mass them together. Lines in the plate may run from 70 to 100 to 
 the inch, and there should be but half as many in a drawing which is to be reduced one half; 
 other reductions will be in like proportion. 
 
 Draughtsmen may use photographic prints from the objects if they will go over with a carbon 
 ink all the lines which they wish reproduced. The photographic color can be bleached away by 
 flowing a solution of bi- chloride of mercury in alcohol over the print, leaving the pen lines only. 
 Use half an ounce of the salt to a pint of alcohol. 
 
 Finally, lettering and figures are most satisfactorily printed from tyjie. Draughtsmen's best, efforts 
 are usually thus excelled. Such letters and figures had therefore best be left in pencil on the 
 drawings, so they will not photograph but may serve to show what type should be inserted." 
 
 To the above hints should be added a caution as to the use of the rubber. It is likely to 
 diminish the intensity of lines already made and to affect their sharpness ; also to make it more 
 difficult to draw clear-cut lines wherever it has been used. 
 
 It may be remarked with regard to the foregoing instructions that they aim at securing that 
 uniformity, as to general appearance, which is usually quite an object in illustration. But where the 
 preservation of the individuality and ' general characteristics of one's work is of any importance what- 
 ever, the draughtsman is advised to letter his own drawings and in fact finish them- entirely, himself, 
 with, perhaps, the single exception of section -lining, which may be quickly done by means of Day^s 
 Rapid Shading Mediums or by other technical processes. 
 
 281. Half Tones. Photo -zincography may be employed for reproducing delicate gradations of light 
 and shade, by breaking up the latter when making the photographic negative. The result is called 
 a half tone and it is one of the favorite processes for high-grade illustration. Figs. 95 and 130 
 illustrate the effects it gives. On close inspection a series of fine dots in regular order will be 
 noticed, so that no tone exists unbroken, but all have more or less white in them. 
 
 The methods of breaking up a tone are very numerous. The first patent dates back to 18.52. 
 The iDrinciple is practically the same in all, viz., between the object to be photographed and the 
 plate on which the negative is to made there is interposed a " screen " or sheet of thin glass on 
 which the desired mesh has been previously photographed. 
 
 In the making of the "screen" lies the main difference between the variously -named methods. 
 In Meissenbach's method, by which Figs. 95 and 130 were made, a photograph is first taken, on the 
 "screen," of a pane of clear glass in which a system of parallel lines — one hundred and fifty to the 
 inch — has been cut with a diamond. The ruled glass is then turned at right angles to its first 
 position and its lines photographed on the screen over the first set, the times of exposure differing 
 slightly in the two cases, being generally about as 2 to 3. 
 
 This process is well adapted to the reproduction of "wash" or brush -tinted drawings, photographs, 
 etc. The object to be represented, if small, may preferably be furnished to the engraving company 
 and they will photograph it direct. 
 
PHOTOGRAPHIC ILLUSTRATIVE PROCESSES. 103 
 
 GELATINE FILM PHOTO - PROCESSES. 
 
 282. As stated in Art. 275, in wliicli a few of the above processes are named, a gelatine film 
 may be emjjloyed, either as an adjunct in a method resulting in a metal block, or to print from 
 directly ; in the latter case the prints must be made, on si^ecial paper, by the company preparing 
 the film. In the composition and manipulation of the film lies the main difference between otherwise 
 closely analogous processes. For any of them the company should be sujiplied with either the original 
 object or a good drawing or photographic negative thereof 
 
 Not to unduly prolong this chapter — which any intelligible distinction between the various 
 methods would involve, yet to give an idea of the general principles of a gelatine process I 
 conclude with the details of the preparation of a heliotype plate, given in the language of one 
 compan_y's circular. Figs. 1 — 5 of Plate II illustrate the effect obtained by it. 
 
 "Ordinary cooking gelatine forms the basis of the i^ositive j^late, the other ingredients being bichromate 
 of potash and chrome alum. It is a peculiarity of gelatine, in its normal condition, that it will absorb 
 cold water, and swell or expand under its influence, but that it will dissolve in hot water. In the i3repara- 
 tion of the plate, therefore, the three ingredients just named, being combined in suitable proportions, 
 are dissolved in hot water, and the solution is poured upon a level plate of glass or metal, and left 
 there to dry. When dry it is about as thick as an ordinary sheet of jjarchment, and is stripped 
 from the drying-plate, and jjlaced in contact with the previously-prepared negative, and the two 
 together are exposed to the light. The presence of the bichromate of potash renders the gelatine 
 sheet sensitive to the action of light; and wherever light reaches it, the plate, which was at first 
 gelatinous or absorbent of water, becomes leathery or waterproof In other words, wherever light 
 reaches the plate, it produces in it a change similar to that which tanning ^jroduces upon hides in 
 converting them into leather. Now it must be understood that the negative is made up of trans- 
 parent parts and opaque parts ; the transjjarent jjarts admitting the passage of light through them, 
 and the opaque parts excluding it. When the gelatine jjlate and the negative are placed in contact, 
 they are exposed to light with the negative uppermost, so that the light acts through the translucent 
 portions, and waterj^roofs the gelatine underneath them; while the opaque portions of the negative 
 shield the gelatine underneath them from the light, and consequently those jjarts of the plate remain 
 unaltered in character. The result is a thin, flexible sheet of gelatine, of which a portion is water- 
 proofed, and the other portion is absorbent of water, the waterproofed portion being the image which 
 we wish to rej^roduce. Now we all know the repulsion which exists between water and any form 
 of grease. Printer's ink is merely grease united with a coloring-matter. It follows, that our gelatine 
 sheet, having water applied to it, will absorb the water in its unchanged jjarts; and, if ink is then 
 roUed over it, the ink will adhere only to the waterj)roofed or altered j^arts. This flexible sheet of 
 gelatine, then, i^repared as we have seen, and having had the image imijressed upon it, becomes the 
 heliotype plate, capable of being attached to the bed of an ordinary printing-press, and printed in the 
 ordinary manner. Of course, such a sheet must have a solid base given to it, which will hold it 
 firmly on the bed of the press while printing. This is accomplished by uniting it, under water, with 
 a metallic plate, exhausting the air between the two surfaces, and attaching them by atmospheric 
 pressure. The plate, with the printing surface of gelatine attached, is then placed on an ordinary 
 platen printing-jDress, and inked up with ordinary ink. A mask of paper is used to secure white 
 margins for the prints; and the impression is then made, and is ready for issue." 
 
" The study of Descrvptive 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 coinbinations in space, 
 and to follow loiih precision their continual correspondence with the figures which are actually 
 traced — of thus exercising to the utmost, in the most certain and precise inanner, that im- 
 portant faculty of the human mind luhich 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 moderns 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 geometry; for the methods used are always conceived as 
 applicable to any figures whatever, the peculiarity of each having only a purely secondary 
 influence. ' ' Auguste Comte : Cours de PMlosopMe Positive. 
 
 ">yl mathematical problem may usually be attacked by what is termed in military par- 
 lance the method of ' systonatic 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 p^'oblem must be seen through and through before it can he attempted. 
 The entire scope of its conditions as well as each step toward its solution must be grasped 
 by the iinagination. It must be 'taken by assault.'" 
 
 George Sydenham Clarke, Captain, Eoyal Engineers. 
 
THE GEOMETRIE DESCRIPTIVE OF GASPARD MONGE. 
 
 105 
 
 CHAPTER IX. 
 
 ORTHOGRAPHIC PROJECTION UPON MUTUALLY PERPENDICULAR 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 Mongers Descriptive. Certain German writers call it Mongers Orthogonal Projec- 
 tion. The fiopular titles Mechanical Draining, Practical Solid Geometry, Orthographic Projection, etc., are 
 usual!}' merely indicative of more or less restricted applications of Monge's Descriptive to some sjjecial 
 industrial arts ; and working clraivings of bridge and roof trusses, machinery, masonry and other con- 
 structions, are simplj^ accurately scaled and fully dimensioned projections made in accordance with 
 its princiiDles. 
 
 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 jwoblems handed down through many centuries, and many of which 
 had been the monopoly of the Freemasons. Emphasizing in this chapter the abstract princiijles 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. 
 
 ^ig-. 15 e. 
 
 FUNDAMENTAL PRINCIPLES. 
 
 284. The orthographic projection of a point on a plane is the foot of the perpendicular fi-oni 
 the f)oint to the plane. 
 
 In Fig. 156 the perpendiculars P p' and Pp give the pro- 
 jections, p' and p, of the point P. 
 
 The i^lanes of j^rojection 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. jj.) of P. 
 
 Projections on the vertical plane are denoted by small 
 letters with a single accent or " i^rime." 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 jDoint P or of the point p p'. 
 
 * Aperyu Hlstoiique sur POrigin et le Developpement des Methodes en G6om6trie. 
 
106 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 ^ig-- 157". 
 
 We shall call Pp the H-projector of P, since it gives the projection of P on H. Similarly, P p' 
 is the Y-projector 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 line called the ground line, hereafter denoted by G. L. They make 
 with each other four diedral angles. 
 
 The observer is alwaj^s supposed to be in the first angle, viz., that which is above H and in 
 front of V. We shall call it Qi, or the first quadrant. Q^ is then the second quadrant, back of V 
 but above H. Q^ is below Q^, while Q^ is immediately below the first angle." 
 
 Points R and S, in the second and third quadrants respectively, have their elevations, r' 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 fi-ont of G. L. 
 
 286. In making a drawing in the ordinary way, and not pictorially, we sujDpose the planes V 
 and H brought into coincidence by revolution about G. L., the upjDer 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 p) z 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 onlj^ come into H somewhere on p z (produced). 
 
 In Fig. 157 we find Fig. 156 represented in the ordinary way. Only the pro- 
 jections of the points appear. V and H are, as usual, considered indefinite in ex- 
 tent, and their boundaries have disappeared. A projector- plane is shown only by 
 the line, peri^endicular 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', below. The second and fourth angles are also opposites, both projections, 
 r r', being above G. L. for the former, and both, 1 1', below for the latter. 
 
 288. For any angle the actual distance of a point from H, as shown by any H- projector Pp 
 (Fig. 156), 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 R r' (Fig. 156), showing the actual distance of a point from V, 
 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 lies in V. 
 
 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 ^^^- ^^®- 
 line; for its extremity-projectors, as ^a and Bb (Fig. 158), would 
 determine a plane j)erpendicular to H and containing AB; in 
 such plane all other H-projectors must lie, hence meet H in ab, 
 which is straight because the intersection of two planes. 
 
 In Fig. 159 we see the line A B oi Fig. 158, orthographic- 
 ally represented. 
 290. A jjlane containing the projectors of a straight line is called a projecting 
 plane of the line. (A B b a, Fig. 158). 
 
 ti ip 
 
 G — !- 
 
FUNDAMENTAL PRINCIPLES OF MONGERS DESCRIPTIVE. 
 
 107 
 
 ^'ig-. ieo. 
 
 A Y-projeding plane is the plane through a line and perpendicular to the vertical plane. 
 The 'S-projecting plane of a line is the vertical jjlane containing it. 
 
 The intersection of a surface by a given line or surface is called a trace. The trace of a line is 
 
 a point; of a surface is a line. If on H it is called a horizontal trace 
 (h. 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 l3e equal and parallel to the 
 original curve when the latter is parallel to :^i.s- isi. 
 
 the plane on which it is projected. 
 
 292. All lines, straight or curved, lying 
 in a plane that is perpendicular to V, will 
 be projected on "\^ in the v. t. of the plane. A similar remark applies q. 
 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 
 s-i-. 1S2. l^y 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'NM is vertical, since M' N—iis, 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 shccpe 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 rabatment t, 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 
 
 l^lan ab hy 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- - ^^s- is-a. 
 
 izonial trace of a 
 line prolong the g_ 
 vertical projection 
 
 X'ig-. 1S3. 
 
 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 Eemark, Art 8. 
 
 t From the French rabattement. 
 
108 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 ^igr, iss_ 
 
 294. The inclination, 6, of the line to H, is that of the line to its plan ah; the angle <^, made 
 with V, is that of the line to its elevation, a! V . 
 
 I^ig-- ISS. 
 
 295. Any horizontal line has a plan, a h (Fig. 
 165), equal and parallel to itself Its elevation, a!b', 
 is 'parallel to G. L., at the same distance from it as 
 the line from H, and makes the same angle with q- 
 V as its plan does with G. L. Such line can evi- 
 dently have no h. t. Its v. t. would be found by 
 the rule in the preceding article. In the extreme \ib 
 
 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. 
 
 ngr- isT. 
 
 S'ig-- iss. 
 
 
 
 
 
 1 
 d 
 
 m' 
 
 n' 
 
 f 
 
 
 
 
 b 
 
 c 
 
 d 
 
 r.i 
 
 
 h.t. 
 
 a 
 
 
 [n 
 
 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. (Fig. 167.) 
 
 297. A vertical line has no v. t. ; is projected 
 on H in a point; has its v. p. parallel and eqvial 
 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 Hues A B, CD and M N oi the two preceding figures. 
 
 298. A plane is represented by its traces. Like H and ^^s- iss- 
 V, any plane is considered indefinite in extent when drawn 
 
 ^^s- ^To. JQ i\^Q 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 M N K 0, Fig. 170, has no vertical trace, and its horizontal trace If iV is par-, 
 allel to G. L. 
 
 If parallel to G. L., while inclined to H and V, a ^^sr- i'7-i. 
 
 plane has parallel traces. (R S, R' S', Fig. 170). Fig. 171 , , ^' s' 
 
 shows the planes of Fig. 170, as usually represented. 
 
 The traces of a plane not parallel to G. L. must meet ' " ■ l 
 
 at the same point on G. L. 
 
 R g 
 
 299. We have stated that if a plane is i^ei'pendicular ^ n 
 
 to a plane of projection, its trace on the other plane is perpendicular to G. L. This is the only case 
 E'ig-. 172. jj-^ -nrhich 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 jalane 
 perpendicular to both ; plane P' Q R (Fig. 172) is by hypothesis perpendicular 
 to V; hence the ground hne and P'Q are the lines cut by jjlane 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 
 P'QR. The same course of reasoning may be applied to each trace of both planes. 
 
FUNDAMENTAL PRINCIPLES OF MONGE'S DESCRIPTIVE. 
 
 109 
 
 300. In Fig. 173 we have a plane obliqiie to both H and V. Two important hnes are drawn 
 in it which we shall have frequent occasion to vise. 
 
 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 parallelism of its plan to G. L. ; 
 
 E'ig-- ITS- ^'ig-- IT-'S;. 
 
 IFig-. IT-S- 
 
 but when contained by a plane, we shall call it a V-jxirallel of the plane. Such line will evidently 
 be parallel to the v. t. of the plane in which it lies. 
 
 The obHque 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 hnes until they meet the plane of projection ; their H-traces joined give the h. t. of the jjlane. 
 Similarlj' 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 dechvity with respect to H. Both the hne 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 
 
 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 hne of declivity with respect to V. 
 
 302. Were the plane P'QR (Fig. 175) rotated on its i-xgr. i7-r. 
 
 line of decKvity (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 — g^ 
 line. 
 
 On the other hand, the line of declivity might be turned in the plane until " ^r 
 
 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 sum 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°. 
 
 ^ig-. ir/s. 
 
110 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 ^ig-- ITS. 
 
 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 ^ -B (Fig. 178), is perpendicular to a plane P' Q. R, its projections will be 
 
 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 jalane, 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 profile 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 about 
 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 princii^les : 
 
 (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 perjjendicular 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 a s, drawn through the plan of A and perjDcndicular 
 to the h. t. of P. From s a circular arc, s Sj (Fig. 180), from centre L, will 
 be the i^lan of the arc described by a" of Fig. ,179. From Sj a vertical to the level of a' gives a". 
 Similarly the iDrojection /" 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 api^ears also as a line, G L Sj. We thus get an 
 " end view " into the angle. All figures lying in profile planes are then "'■ ^^'' 
 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, 
 A B and CD — each of which lies in a profile plane — are shown in their true length and inclination. 
 
 ^ig-- ISO. 
 
FUNDAMENTAL PRINCIPLES OF MONGE'S DESCRIPTIVE. 
 
 Ill 
 
 and fZ;. V-projectors give d', 
 that the profile 
 
 Perpendiculars to H from ^-1, B, C, and D, give their plans (\ , b^ 
 c', etc., the heights of the elevations of the points. 
 
 Revolving the whole system into its usual position, remembering, meanwhile, 
 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 Cj at C; 
 dj at d; «i at x. Assuming S S' as the plane of the line CD, and that the plane 
 ot A B is R P', at a given distance T Q from S S', we find a' and V on R P' at 
 the same level as A and B ; while a is derived from x, and b fi-om b^ as shown. 
 The elevations c' and d' on iS S' are on the level of C and D respectively. 
 
 306. The term rabatment, already employed to indicate the rotation of a plane about one of its 
 traces until it conies into a plane of projection, is also used to denote the rotation of a point or line 
 into H or V about an axis in such plane. Restoration to an original space-position, after revolving, 
 will be called counter-rabatment or counter-revolution. 
 
 E'ig-- las. In Fig. 183 we have a^ as the rabatment of A 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 
 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 j^oint 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 hj^pothenuse of a right-angled triangle whose altitude equals the difference of 
 level of point and axis, and whose base is the h. jj. 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 jjoint on the plane in which or parallel to lohich 
 lies the axis. 
 
 In usual position the foregoing construction would appear thus : with the point given by its 
 projections, {a a'. Fig. 18-1), let fall a perpendicular through a to mn; prolong this ^^^- ^®;*- 
 indefinitely; make a the vertex of the right angle in a right triangle of base B a, 
 altitude a' s ; then the hj'pothenuse of such triangle, used as radius of arc a^a^'^- 
 (centre B) gives a^ as the revolved position desired. 
 
 307. In applying the foregoing principles in the following problems we shall 
 fi'equently 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 f)lane 
 
 to the cone makes with the base the same angle as the elements. The ele- 
 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. Proh. 1. From the projections of a line to determine (a) its traces; (b) 
 its actual length ; (c) its inclinations, 6 and <j>, 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. 298. 
 
112 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 
 ct r---......^^^ 
 
 
 
 
 ! 
 
 "^^^^i/ 
 
 's 
 
 !o 
 
 '• "~~~^. X 
 
 '"" 
 
 "■----^la 
 
 i?i 
 
 
 
 / 
 
 ^^r--- 
 
 h.t 
 
 ta 
 
 (b) The actual length of a line may be found either (1) by rabatmeni into H or V, or (2) by 
 
 rotation until parallel to H or V. 
 
 By the first method rabat the Hne 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 ^tg. les. 
 
 the jjoint A above H ; similarly, b bi equals b' n. Notice that a^ a '"■*-r~-"9_' "'«" ;&" 
 
 aiid 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' U 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 
 the V-projector of b b' as an axis, a! may be brought to a", at the level of U, 
 by an arc, centre b', 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 6;, we will 
 find the v. p. of the revolved point at b", on its former level. The new pro- 
 jection, a' b", is again the real length, now projected on V. 
 
 (c) The inclinations, and <j>, 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 a' b" inclined at the angle 6 to H, while b a^ makes with a a^ the 
 
 angle <i>. When the line lies in a profile plane, the traces, length and inclination are found by means 
 
 of the ojDeration described in Art. 305 and illustrated by Fig. 181. In that figure, were c d and 
 
 (/ 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 C D, whose v. t. is at z ; 
 
 h. t. (not shown) at a distance Ty above T; while & and 4> are seen in actual size at y and s. 
 
 309. Prob. 2. To determine the projections of a line of given length, having given its angles, 6 and </>, 
 
 vnth H and V respectively. If with the line we generate a iFig- .ise. 
 
 vertical right cone, of base angle 0, 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 and <^ 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" ; 6 = 44 ° ; <^ = 30 °. From any point 
 
 h" on G. L., draw in V a line h" o', 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 a'. 
 
 Remembering that the inclination of a line determines the length of its projection, we next ascertain 
 
 how long the projection of a two-inch line will be when inclined 30° to a plane. Drawing a' T at 
 
MONGERS DESCRIPTIVE. — ELEMENTARY PROBLEMS. 
 
 113 
 
 In 
 
 i^iir. iss. 
 
 30° to a' b", and projecting b" iserpendiculavly upon it at n, we find a' n as the invariable length 
 sought. Arc n b' from centre a, gives a' V 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 U, continued, would cut G. L. at .c, whence s and m could be derived, we would 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. 3. To determine the plane containing (a) two intersecting lines; (b) tivo 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 two 
 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 perpen- 
 dicular to each other; if either side of the angle in sjjace 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 same 
 "trace as before. 
 
 In general, to shoio the true size of any angle, rotate its plane either into or ptaraUel 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 j^rojected 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 mo^n, 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 g" at the same level, and a g for the plan of the line 
 connecting them ; then o rotates to 0, about a g, giving a O^g for the 
 desired angle. ^'ig-- isi. 
 
 312. Prob. 5. To draxo 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 i:)arallel to the h. t. of 
 '■- -"'' the plane, and meet V on its v.t. ; there- 
 
 fore any line b' c', parallel to G. L., will do for the v. p. of the desired line ; the intersection of P' Q 
 
 ^S-S- ISO- 
 
 o 0.7=0 X 
 
114 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 by ¥ c' will be the v. t. of the Ime, 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 ]3lane; 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 jDlane 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 irrojection of a point in a plane, to locate the other projection. 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 b s c, parallel to R Q. From c, where it meets G. L., a vertical to 
 P' Q gives its v. t., through which draAV c' U 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 pairs 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 aiiother draw through any i^oint of 
 the first line a parallel to the second line; such parallel will, with the first line, determine the 
 plane. 
 
 316. Prob. 9. To pass a pAane through a given point and 'parallel to a given plane. Two lines 
 through the given point and parallel to any pair of lines in the given plane will determine the 
 required plane. 
 
 ^ig-- isa. 
 
 In Fig. 192, with o o' as the given point, and M' N M as the given plane, assume in the latter 
 any two lines, AB and CD; parallel to these lines and through o o' draw ST and E F, whose 
 traces will determine those of the plane sought. 
 
MONGE'S DESCRIPTIVE. — ELEMENTARY PROBLEMS. 
 
 115- 
 
 S'ig-. iS3. 
 
 :F'ig-. lS-3;- 
 
 Since parallel planes have parallel traces, one line through o o' would suffice. For example, s t, 
 5' f, parallel to « b, a' 6', 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 pciss a plane through a given jwint 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, 0', Fig. 193, either a horizontal or a Y-par- 
 allel of the desired plane ; the trace of either line, and the known 
 directions of the required traces, suffice to solve the j^roblem. The 
 plan, 11, of a horizontal, will be perpendicular to a h — 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 V-parallel through 0' is obtained 
 by drawing 0' y' perpendicular to a' b', and y i^arallel 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! U. 
 
 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 a' n, its inclination to G. L. shows the base angle 6 of the cone 
 and therefore the inclination of the given plane. 
 
 A^'ith 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 <^ sought.' This problem might also 
 
 be readily solved by rabatting the line of declivity into a plane of projec- 
 
 jection. Thus, making d' d" perpendicular to c d! and ecjual to d! d, we find 
 
 the angle (f> between c' d' and c' d". 
 
 For a jDlane parallel to G. L. use an auxiliary profile plane, rotating its 
 
 line of intersection with the given plane as in Fig. 196. 
 
 fb). The angle between the traces is obtained by rabatment of the given 
 plane about either of its traces. In Fig. 195, using trace R 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 
 
 cf will be the same after as before revolution; therefore cut c Ci by an arc, centre Q, radius Q <f ; 
 
 the desired angle is ji or c^Q R. 
 
 ■S'i.s- i-SS- 
 
 
 ^^s- 
 
 ise. 
 
 m' 
 
 t 
 N 
 
 ^,-^ .. 
 
 
 
 
 1 r 
 
 M 
 
 N 
 
 y 
 
 
 
 
116 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 ^ig-, IST"- 
 
 rig. ise. 
 
 319. Proh. 12. To obtain the traces of a plane, having given its inclinations, 6 and ^, to H and Y 
 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 & 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 
 (9 -cone, and draw from any point of it, as a!, a line a' h", 
 at 6° to G. L ; use a" h", 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" & 
 shows the perpendicular distance from a to the point of 
 intersection of the two elements of tangency of the re- 
 quired plane Avith the cones ; hence the generatrix of the 
 ^-cone must, when in H, be at an angle <f> with G. L., and 
 tangent to arc s hi of radius a" s, centre a" , and is there- 
 fore x^c". Draw a\ j/ cc for the half-base of the c^-cone. 
 The h. t. of the plane sought is then R Q, drawn through 
 c" and tangent to the base of the ^-cone; while the v. t. 
 is Q P', tangent to base .I'l y x. For the limits of ^ -f <^ 
 
 see Art 301. 
 
 320. Prob. IS. To draw two parcdlel planes at a given distance apart. To draw (Fig. 198) a plane 
 at a distance of \" from plane P' QR, rotate a hue of 
 declivity of P'QR into V at a! b". Draw c' d", parallel to 
 and i" from a' b", to represent (in V) the line of declivity 
 of the plane sought. It meets a a' (prolonged) at c', 
 through which draw c' N parallel to P' Q, and from N 
 the trace MN parallel to R Q. M' N M fulfills the con- 
 ditions. 
 
 321. Proh. 111.. To obtain the line of intersection of two 
 planes. As two points determine a line, we have merely 
 to find two points, each of which lies in both planes, and 
 join them. In Fig. 199 the Hne in .ipace which would 
 join a', the intersection of the vertical traces of the planes, with b, the corresponding point on the 
 
 horizontal traces, would be the required line. Its projections are a b, a' V . 
 Were the H-traces, of the planes parallel, the line of intersection would 
 be horizontal. Were the V-traces parallel, '^^^- ^°°- 
 
 their line of intersection would be a 
 V-parallel of each plane. 
 n^ ^- If both planes were parallel to G. L., 
 
 a profile plane might advantageously be employed in the solution. 
 The line sought would be parallel to G. L. 
 
 322. Prob. IB. To find the point of intersection of a line and 
 plane find the line of intersection of the given plane by any aux- 
 iliary plane containing the line; the given Hne will meet such line 
 of intersection in the desired point. In Fig. 200 m n v' is an aux- 
 ihary vertical plane through the given line a b, a' b'. The given plane, P' Q R, intersects the plane 
 
MONGE'S DESCRIPTIVE. — ELEMENTARY PROBLEMS. 
 
 117 
 
 I^ig. 201.- 
 
 m n v' in line m n, m' n'. The elevations a' b' and m' n' meet at o' — the v. p. of the point sought, 
 whose h. p. must then be on a vertical through o' and on both the other projections of the hues, 
 hence at o. 
 
 323. Proh. 16. To shoio the actual angle between two planes. 
 
 A plane perpendicular to the line of intersection of the planes will cut from them lines whose 
 
 plane angle equals the dihedral angle sought. 
 
 Let a b', a b be the line of intersection of the planes P' Q R and M' N M. Any line in n, drawn 
 
 perpendicular to a b, may be taken as the h. t. of an auxiliary plane perpendicular to the line of 
 
 intersection. The part s n, included be- 
 tween the H-traces of the given planes, 
 will — with the lines cut from the lat- 
 ter — form a triangle whose vertical 
 angle is that desired. The altitude of 
 this triangle is a line projected on c b, 
 and — in space — is a perjaendicular 
 from c to A B. To ascertain its length 
 rabat the H-projecting plane of A B 
 into V, about a' a as an axis. The 
 point c appears at c", and the line 
 A B at a b". The altitude sought ap- 
 pears in actual size at c" d", perpen- 
 dicular to a' b". From c lay off on 
 c b the length c d^ equal to c" d" ; 
 s di n, or /3, is the required angle. 
 
 A vertical from d" to G. L., thence 
 an arc to c 6 (from centre a), gives d, 
 
 the plan of the vertex of the angle in space. 
 
 When both planes are parallel to G. L. use an auxihary profile plane in solving. 
 
 324. Prob. 17. To determine the angle between a line and plane let fall a perpendicular from any 
 pomt of the line to the plane; the angle between the perpendicular and the given line will be the 
 complement of the desired angle. The principles involved are those of Arts. 303 and 311. 
 
 325. Proh. 18. To show in its true length the distance from a point to a plane. 
 
 If the plane is perpendicular to R or Y the distance is evident without any construction. Thus, 
 in Fig. 202, a perpendicular through the point to the plane is seen :^ts. sos. 
 
 in its true length, a' s', because in space it is parallel to V. f 
 
 Similarly i r is the actual length of the perpendicular fi-om t f 
 to the vertical plane v' f h. 
 
 MTien the plane is oblique to both H and V let fall a perpen- g 
 dicular from the point to the plane; find by Prob. 15 the intersec- 
 tion of such perpendicular with the plane, and show, by Prob. 1 (b), 
 the real length of the distance between the original point and the one thus obtained. 
 
 326. Prob. 19. To draiv a line of given inclination in a plane. We have seen, in Art. 301, that 
 we can assign to the hne no greater inclination than that of the plane. Starting with the hne mi 
 V (Fig. 203), make its intersection, a', with P' Q, the vertex of a cone whose base angle is the 
 inclination assigned to the line. With a b" as a radius describe the base of this cone. This cuts 
 
118 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 R Q — iliQ h. t. of the plane — in h and \, each of which, joined in space to a', would give a line 
 fulfilling the conditions. 
 
 The projections of the two solutions are a! b', a b ; a' b\ , a \. 
 
 E-ig-- 303. 
 
 ix" 
 
 ■^,/^ 
 
 ^^^ \ 
 
 ft' 
 
 9"^', 
 
 bi 
 
 '\ 
 
 
 / 
 
 / 
 
 If the line is to pass through a given jjoint of the plane 
 make that point the vertex of the cone employed in the con- 
 struction, taking the initial position of the generatrix parallel to Y. 
 327. Prob. SO. To obtain the traces of a plane containing a 
 given line and making a given angle with H or V. 
 
 Make any point 
 on the line the vertex 
 of a cone whose base 
 angle is the assigned 
 inclination ; the re- 
 quired plane will be 
 tangent to such cone, 
 and its traces will pass through the like traces of the 
 given line. 
 
 In Fig. 204 let a b, a' b' be the given line. On it 
 take any point a a' as the vertex of a cone generated 
 by a line a' s', a s, inclined 6° to H and, at first, parallel 
 to V. Since the H-traces of the two possible planes ful- 
 filling the conditions would be tangent to the like trace 
 of the the cone, draw from b — the h. t. of the given 
 line — the tangents b R and b M to the circle m s n. These cut CI. L. at Q and N, each of which 
 is then joined with x — the v. t. of the given line — to obtain the v. t. of one of the desired planes. 
 328. Prob. 21. To determine the projections of two intersecting lines, having given their inclinations to 
 H or V, and their angle with each other. 
 
 X'ig-- SOS. 
 
 It will be seen from Fig. 205 that, in space, either hue A B, A C, will be the hypothenuse of 
 a right-angled triangle whose plane is perpendicular to that plane of projection relative to which the 
 
MONGERS DESCRIPTIVE. — ELEMENTARY PROBLEMS. 119 
 
 inclinations are given ; that the altitudes (perpendiculars) of these triangles will not only be equal 
 but coincide (as in A a), and that the angles 6 and 6^, at their bases, will be the given inclina- 
 tions to the plane of projection. 
 
 Let the two lines, A C and A B, be inclined 6° and 6° respectively to H, and also make with 
 each other some angle /3; to find their projections. From any point on H, as A^ (Fig. 206), draw 
 two hues, A^B and A^K, making the angle P with each other. At any point on A^B, as B, draw 
 a line B S, making with ^-Ij B an angle equal to the inclination to H of the line A B in space. 
 From Ai draw A^ a-i perpendicular to B S. With yl, a^ as a radius, and A^ as a centre, describe a 
 circle. Tangent to this circle draw a line meeting ^4; K at C, at an angle equal to the other given 
 inclination, 6°. B and C are then the H-traces of the lines A B and A C ; and the line B C will 
 form part of the h. t. of their plane. For convenience take the vertical plane of projection perjjen- 
 dicular to the jjlane of the lines, so that the triangle ABC will be projected upon it in a straight 
 line. This condition will be met by taking G. L. perpendicular to B C. On V draw s' f parallel 
 to G. L., and at a distance from it equal to A^ a^ or A^ a.^, either of which represents the actual 
 height of the point A in space, when the lines A B and A C make the given angles with H. With 
 Q as a centre and radius Q m (in is the v. p. of A^) describe an arc, m a', intersecting s' t' at a'. 
 This arc is the v. p. of the arc that A^ describes about R Q in reaching its space-position. A^ o is 
 the h. t. of the plane in which A^ rotates. The h. p. of A is then at a — the intersection of o A^ 
 by a! n a, the projector-plane of A. The desired j)lans of the given lines are a B and a C, while 
 a' Q is their common elevation. 
 
 Limits of the conditions that may be assigned. The sum of the given inclinations and the angle 
 between the lines will reach its maximum, 180°, when the lines lie in a profile plane. 
 
 The length of but one of the lines may be iDredetermined. If assigned, such], length must be 
 made the hypothenuse of the first triangle constructed. The longer line will, obviously, make the 
 smaller angle. 
 
 329. Proh. 22. To determine the traces of tivo mutually perpendicular planes, having given their inclina- 
 tions to H or V. Two planes will be mutually perpendicular if either contains a line that is per- 
 pendicular to the other. To solve this problem we therefore determine, by the first step of Prob. 
 12, the traces of a plane at one of the given inclinations, 6, to H ; by Art. 303 erect a perpendic- 
 ular to this plane; by Art. 327, obtain the traces of a plane containing such perpendicular and 
 tangent to a cone (a) whose vertex is on the perpendicular and (b) whose base angle is the other 
 assigned inclination. 
 
 The p)lanes will intersect in a horizontal line when the sum of their dihedral angle and their 
 inclinations to H has its minimum value, 180°. 
 
 330. Proh. 23. To determine the common perpendicular to two lines not lying in the same plane. Only 
 a pictorial representation of the various steps is shown (Fig. 207) their orthographic counterparts 
 having been already fully described in detail. Let A B and CD be 
 the given lines. Pass a plane through either line parallel to the other. 
 The parallelogram represents such a plane, determined by CD and 
 C E, the latter a parallel to A B and drawn through any point of 
 D C. By Art. 303, droj) a perpendicular K T S, from any point of A B 
 to the auxiliary plane, and obtain its trace T on the latter by Art. 
 322. T M, drawn parallel to A B, will be the projection of the latter s' 
 on the plane. M N, a. parallel to TK through the intersection ot T M and D C^ will be the desired, 
 perpendicular, whose true length may be ascertained as in Art. 308. 
 
120 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 DEFINITIONS AND VARIOUS CLASSIFICATIONS OF THE MORE IMPORTANT LINES AND SURFACES. 
 
 In our grajAical constructions the draughtsman is concerned — not with the contents of solids — 
 but with the form and relative position of the surfaces that bound them, the lines in which such sur- 
 faces meet, and the points which are tlie intersections or extremities of such lines. After the acquaint- 
 ance with the relations between figures and their projections which may now be assumed if the pre- 
 ceding articles of this chapter have been mastered, a sort of bird's-eye glimpse into the province of 
 their application is in order; the remainder of this chapter is therefore devoted to definition, classi- 
 fication, illustration, and some of the important properties of the principal lines and surfaces with 
 which the mathematician and draughtsman have to deal, one object being to enlarge the student's 
 horizon as to the extent of the mathematico-graphical field and to give hmi some idea of its at- 
 tractiveness. 
 
 ALGEBRAIC AND TRANSCENDENTAL LINES AND SURFACES. — DEGREE. — ORDER. — CLASS. 
 
 331. Algebraic. — Transcendental. Taking up first the distinctions of Analytical Geometrj' (for defi- 
 nition see page 4) we find that a line or surface is algebraic if the relations of the co-ordinates of 
 its points can be expressed by an equation which may be reduced to a finite number of terms, in- 
 volving positive, integral powers of the variables. If, however, its equation involves other functions 
 — as trigonometrical, circular, exponential, logarithmic — the curve or surface is called transcendental. 
 
 382. Degree. — Order. The degree of an algebraic cur-^-e or surface is that of the equation express- 
 ing it. The term order is used synonymously with degree. 
 
 The degree, or order, of an algebraic curve is the same as the maximum number of points in 
 which it can be cut by a plane, and for a plane curve is the number of times it can be met by 
 a right line in its plane. 
 
 The straight line is the only line of the first order. 
 
 The conic sections are the only mathematical curves of the second order. 
 
 Of the other curves treated in Chapter V the Witch of Agnisi and the Cissoid are of the third 
 order ; the Limajon, Cardioid, Trisectrix, Cartesian ovals and Cassian ovals are of the fourth; and 
 the remainder — Helix, Sinusoid, the Trochoids in general. Catenary, Tractrix, Involute and the 
 Sf)irals — are transcendental. 
 
 Since transcendental equations expand into an infinite series their degree is assumed infinite, 
 from which we conclude that a straight line may have an infinite numlser of intersections with a 
 transcendental plane curve; analogously for a plane and a transcendental space curve. The former 
 may be illustrated by the cycloid, (Fig. 100), which is not comiDleted by rolling the generating cir- 
 •cle out once on a straight line, but consists, theoreticall}^, of an infinite series of similar arcs. 
 
 333. When all the plane sections of a surface are lines of a certain order the surface is of the 
 same order. 
 
 A jjlane is the only surface of the first order. 
 
 Conicoids or quadrics (see Art. 367) are all and the only surfaces of the second order. 
 
 Of the other surfaces defined in the following articles the Conoid of Pliicker (Cylindroid of 
 Cayley) is of the third order; the Cyclide of Dupin, Cono-cuneus of Wallis, the Cylindroid of 
 Frezier, Corne de Vache, Conchoidal Hyperboloid of Catalan and the Torus are of the fourth; and 
 the others are transcendental. 
 
 334. Class. The class of a plane cfiirve is the number of tangents that can be drawn to it from 
 a point in its jDlane. 
 
GENERAL DEFINITIONS OF LINES AND SURFACES. 121 
 
 The class of a non-plane curve is the number of osculating planes* that can be passed to it 
 through any point in space. 
 
 The class of an algebraic surface equals the number of possible tangent planes to it through any 
 line in space. 
 
 LINES AND SURFACES AS ENVELOPES. — GENERATION BY CONTINUOUS MOTION. — REVOLUTION. — TRANSPOSITION. 
 
 335. Envelopes. If the j^oints of a line are obtained as the consecutive intersections of the curves 
 of a series, the line would be tangent to all the curves of the system, and would be called their 
 envelope. On page 22 we find the parabola as the envelope of a series in which the right line 
 appears as the variable curve. Similarly we find the parallel curve to the lemniscate (page 66) as 
 the envelope of all circles of the same radius, whose centres are on that curve. 
 
 Analogously, a surface may be defined as the envelope of the various positions taken by another 
 surface. Thus the right cone is the envelope of all the possible planes containing a given point on 
 a line and making a constant angle with the line. 
 
 336. Lines and surfaces regarded as the resxdt of continuous motion. By the conception with which 
 the draughtsman is mainly concerned, a line is considered as generated by the motion of a point; 
 a surface by the motion of a line. He regards a straight line as generated by a jjoint moving 
 always in the same directiont; a curve as the path of a point whose direction of motion changes 
 continually; a plane as the surface generated by a moving straight line which glides along a fixed 
 straight line while jaassing always through a fixed point. 
 
 337. A curve is plane if any four consecutive positions of its generating point lie in one plane. 
 
 338. A space curve or curve of double curvature or non-plane curve is any curve not plane ; that is, 
 a curve generated by a moving point, no four consecutive laositions of which lie in the same plane. 
 
 Chief among space curves is the ordinary helix,, whose construction is given in Art. 120. After it 
 may be mentioned the conical helix (see Art. 191) and the spherical epicycloid, the latter being the 
 theoretical outline of the teeth of l)evel gears, and generated by the motion of a point on an ele- 
 ment of a cone which rolls on a cone having the same slant height, their vertices being common. 
 
 The spherical epicj'cloid evidently lies on the surface of a sphere whose radius equals an element 
 of either cone. 
 
 339. Whether it be straight or curved, the moving line that generates a surface is called the 
 generatrix, and in any of its positions is an element of the surface. Its motion may be either of rev- 
 olution or transposition. 
 
 340. Revolution imi^lies a straight line called an axis, which may or may not be in the plane 
 of the revolving line. Each point of the moving line describes a circle whose jDlane is perpendicular 
 to the axis and whose centre is on the axis. 
 
 Any plane containing the axis of a surface of revolution is called a meridian plane and cuts the 
 surface in a meridian curve. 
 
 All planes perpendicxdar to the axis of a surface of revolution intersect it in circles called parallels, 
 the smallest of which, so long as of finite radius, is called the circle of the gorge. 
 
 Any surface of revolution may also be generated as a surface of transposition. 
 
 * An osculating plane to a non-plane curve contains three consecutive points of the curve. 
 
 t Frequently also deflned as the shortest distance between two points. If we take into account the generalized notions of modern 
 geometry as to direction and distance we might define a straight line as the line which is completehj determined by two points; a curve as 
 any line not straight; a. plane as the surface containing all straight lines connecting its points by pairs; parallel straight lines as non-intersecting 
 straight lines in a plane. 
 
122 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 341. Transposition may be defined as any motion other than revolution, and involves certain 
 guiding lines called directrices, along which the generatrix glides ; and, frequently, certain surfaces 
 called directors, with respect to whose elements the generatrix of the new surface takes definite positions. ■ 
 
 The surface- director most frequently occurring is the cone, including its special case, the plane. 
 Every element of a cone -director will have a parallel element on the other surface. 
 
 RULED SURFACES. 
 
 342. Whether the motion be of revolution or transposition, any surface that can have a right 
 line generatrix is called a ruled surface. In contradistinction, all others are called double curved surfaces, 
 since any plane section thereof must be a curve. 
 
 As to curvature, ruled surfaces other than the plane are called single curved.' 
 
 343. The motion of a right line involves the consideration of " conditions of restraint " and 
 " degrees of freedom." If it is to move so as always to intersect another line, the generatrix is 
 under one condition of restraint, but, as to motion, has still two degrees of freedom. Require it in 
 all its positions to intersect tioo lines and it has still one degree of freedom. Imisose a third con- 
 dition, either to move so as always to intersect each of three lines — either straight or curved — 
 or, while intersecting two lines, to be parallel to some element of a surface -director, or to be tangent 
 to a given surface, and we have then determined the line in position thus far, that we have located 
 it upon a certain surface. Impose one more — a fourth condition — and the limit of motion is reached, 
 the line becoming a particular line on that particular surface. 
 
 344. Developable Surfaces. A surface which can be directly rolled out or unfolded upon a plane, 
 without undergoing distortion, is called a developable surface. 
 
 345. Plane- sided figures evidently belong to this division and, after the pyramid and prism, the 
 more impoi'tant are the regular polyhedrons, or solids bounded hj equcd, regular polygons whose planes 
 are all inclined to each other at the same angle. Of these, five are convex to outer space, i. e., no 
 face produced will cut into the solid. They are the tetrahedron, octahedron and icosahedron, bounded, 
 respectively, b\' four, eight and twenty equal, equilateral triangles; the cube or hexahedron, whose six 
 faces are equal squares; and the dodecahedron, whose surface consists of tivelve equal, regular, pentagons. 
 
 By allowing re-entrant angles the number of regular solids has been increased by four, named 
 after their investigator, Poinsot, and frequently also called star polyhedra, from their form.* 
 
 346. A single curved surface is called a developable or torse if consecutive positions of its rectilinear 
 generatrix lie mi the same plane. 
 
 If not only consecutive elements but any pKiir lie in the same plane we have either a cone or a 
 cylinder. 
 
 When only consecutive elements lie in the same plane we have developable surfaces of great 
 variety, but whose common characteristic is, that they are generated by a 
 straight line moving so that in every position it is a tangent to a curve of 
 double curvature. 
 
 That such a surface is developable may be thus demonstrated without refer- 
 ence to any figure. Suppose a, b, c and d to be four consecutive points of a 
 curve of double curvature. The line joining a and b is — by the ordinary defi- elliptical cone. 
 nition — a tangent to the curve. Similarly b c and c d are tangent and consecutive. Tangent a b 
 meets tangent b c ai b and their jDlane is tangent at b to the generated surface, since it contains 
 
 * For their construction, as also of the convex regular solids, see Rouchg et De Comberousse, Traite de G^om6trie, Fart II, 
 Book VII. 
 
DEVELOPABLE SURFACES.— WARPED SURFACES. 123 
 
 two lines, each of which is tangent to the surface at the same point. Also b c intersects c d at c. 
 But, since four consecutive iDoints of a curve of double curvature cannot lie in the same plane, the 
 tangent a b does not meet tangent c d. The plane ab c may, however, be rotated on b c into coin- 
 cidence with plane bed, and this process repeated indefinitely, bringing the entire surface into one 
 plane without disturbing the relative position of consecutive elements. 
 
 In Fig. 209 we have the developable helicoid, a representative surface of the family just described. 
 Its elements (the white lines) are all tangent to a helix. In Arts. 186 and 187 its upper and 
 lower outlines are described. „. „„„ 
 
 X^ig-- SOS. 
 
 Like the cone, complete surfaces of this family consist of two 
 nappes or sheets. In the cone both nappes meet at the vertex. On 
 other developables they meet on the curve whose tangents constitute 
 the surface. 
 
 Could the nappes of such a surface be as readily seijarated as 
 those of a cone either of them could as directly be develojsed upon a plane by rolling contact. 
 
 The moving line generates both nappes simultaneously, its point of tangeney separating it into 
 the portions on upper and lower najjpes respectively. 
 
 Whatever the mathematical name for the curve of double curvature employed it has the follow- 
 ing names as a line of the developable surface : (a) cuspidal edge, since any plane section of the 
 surface will have a cusp or point, on that curve; (b) edge of regression, since along it the surface is 
 most contracted. 
 
 347. The family of surfaces just described, as, in fact, all developable surfaces but two, must be 
 surfaces of transposition; for a straight line can have but three positions with respect to an axis about 
 which it is to revolve : — 
 
 (a) it may be parallel to the axis, when a cylinder of revolution will result, developable, since con- 
 secutive elements lie in the same plane, because parallel; 
 
 (b) it may intersect the axis, giving the cone of revolution, developable, because consecutive elements 
 lie in the same jjlane by virtue of intersecting; 
 
 (c) it may make an angle with the axis without intersecting it, giving consecutive positions not 
 lying in the same plane. 
 
 348. Warped Surfaces m- Scrolls. A warped surface or scroll is a ruled surface in which consecutive 
 elements do not lie in the -iame plane. 
 
 All warped surfaces but one must be surfaces of transposition : for we have seen, from the last 
 article, that any straight line that generates a surface of revolution either is or is not in the same 
 plane with the axis ; and that if the former is the case a developable surface is generated : if, then, 
 a loarped surface of revolution be possible, it must be due to the revolution of a right line about an 
 axis not in its plane — evidently the only other possible alternative. The proof that such motion 
 fulfills the essential condition of a warjjed surface is as follows: — 
 
 (a) Consecutive positions of the revolving line do not intersect, as — having no jsoint on the axis — 
 each point must describe a circle about the axis and cannot, therefore, coincide with its preceding 
 position ; 
 
 (b) Consecutive positions are not parcdlel, for if parallel in sjDace they would be in projection ; 
 while reference to Fig. 77, which is a plan and elevation of the surface in ciuestion, shows that they 
 are jDrojected as consecutive tangents to the projection of the circle of the gorge {A BCD), rendering 
 parallelism impossible. 
 
 Neither intersecting nor being parallel, consective elements do not, therefore, lie in the same jDlane. 
 
124 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 The warped surface of revolution is also called the hyperholoid of revolution since it may be gen- 
 erated by the revolution of its meridian curve — an hyperbola — about its conjugate (or imaginary) 
 axis. If the asymptote to the hyperbola be simultaneously revolved with it, a surface called an 
 asymptotic cone will be generated. The curves cut by a plane from the hyperboloid and its asymp- 
 totic cone will be of the same mathematical nature. 
 
 349. Warped Surfaces of Transposition.^ The most important of these are the hyperbolic paraboloid; 
 the elliptical hyperboloid or hyperboloid of one sheet; conoidal surfaces such as the cono-cuneus of Wallis 
 and the conoid of Piiicker ; the loarped. helicoid; the conchoidal hyperboloid of Catalan; the cylindroid of 
 Frkier, and the warped arch, also called the come de vache. 
 
 Referring to Art. 343, regarding the number of conditions that may be imi^osed on a moving 
 
 Ti.s- 21.0. 
 
 ^ig-. 211- 
 
 HyPERBOLIC PAKABOLOID. 
 
 ^"ig-- 212. 
 
 'M^ \ 
 
 HYPERBOLIC PARABOLOID. 
 
 ELLIPTICAL 
 
 line, let us first consider all the possible surfaces having three straight directrices. Evidently such 
 directrices, no two of which lie in one plane, either are or are not parallel to some plane ; for any 
 two of them will invariably be parallel to some plane, and the third either is or is not parallel to 
 that jjlane. Should the former be the case the resulting surface would be the hyperbolic paraboloid, 
 which may also be defined as the warped surface having two straight directrices and a plane director; 
 if, however, the directrices have the second position supposed, the surface generated is called the 
 hyperboloid of one sheet or the elliptical hyperboloid : these two (and their special forms) evidently ex- 
 haust the possibilities as to surfaces with only straight directrices. 
 
 350. The hyperboloid and hyperbolic paraboloid are further interesting as being the only sur- 
 faces which may be doubly-ruled, that is, may have two different sets of rectilinear elements ; for in 
 such a surface it must be possible for either directrix of one set of elements to become the generatrix 
 of a second set; hence, to become in its turn an element, each generatrix must be straight, thus re- 
 stricting the possibilities — as to number of surfaces — to the two named. 
 
 351. In a doubly-ruled warped surface each element of one generation, while intersecting no 
 other elements of its own set, meets all of the other set;, and any three of either set might be taken 
 as directrices for the other. 
 
WARPED SURFACES. 
 
 125 
 
 352. Each set of elements of the hyperbolic paraboloid is — by the second definition of the sur- 
 face — parallel to its own plane director. Section-planes parallel to the line of intersection of the 
 plane directors will cut the surface in parabolas; while other plane sections are hyperbolas. 
 
 The curvature of the hyperbolic paraboloid is called anticlastic. Its typical, saddle-like form is 
 best illustrated by Fig. 211. 
 
 353. A line of striction is the line — straight or curved — containing the shortest distance between 
 consecutive elements of a warped surface. For the hyperbolic paraboloid it is that element of one 
 set which is perpendicular to the plane director of the other set of elements. (See Fig. 213.) 
 
 354. A conoidal surface is generated by a line moving parallel to a plane director and so as 
 always to intersect a fixed right line or axis, fulfilling, at the same time, an additional condition, 
 either of meeting some fixed curve or of being tangent to a fixed surface. 
 
 E-ig-. SI'S: 
 
 EIGHT CONOID 
 
 THE COXO-CUNEtS 
 OF WALLIS. 
 
 HYPERBOLIC PAKABOLOID 
 WITH A 
 LINE OF STRICTION. 
 
 CONOID OF PLUCICER. 
 
 When the axis or straight directrix is perpendicular to the plane director we have a right conoid, 
 and the axis becomes the line of .striction. 
 
 355. With a circle as the curved directrix; an axis equal to, parcdlel to and directly opposite a 
 diameter of the circle; and with a plane director perpendicular to the axis, we would obtain the 
 right conoid known as the Cono-cuneus of Wallis. This surface (Fig. 214) has been employed with 
 pleasing effect in architectural constructions, in towers, arches and for the faces of wing-walls. 
 
 Plane sections parallel to the curved directrix are ellipses whose longer axes are equal to the 
 straight directrix. 
 
 356. Another interesting surface of the same family is the Conoid of Plilcker*, who.se mechanical 
 and kinematic properties are treated by Prof. R. S. Ball in his Theory of Screws; also by Clifford in 
 his Elements of Dynamic. The name cylindroid, under which it appears in those works, was suggested 
 by Cayley, but had much earlier been pre-empted by Frezier for the surface defined in Art. 360. 
 
 * See Piiicker's Neue Geomeirie des Raumes, p. 97; also Mannlieim*d G^omttrie Descriptive, p. -135. 
 
126 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 Fig. 215 represents a model of the conoid under consideration ; but for the exact mathematical 
 surface "the diameter of the central cylinder must be conceived to be evanescent and the radiating 
 wires must be extended to infinity." (Ball). 
 
 Were A B and CD (Fig. 21-3) to be the common directrices of hyperbolic paraboloids whose 
 plane directors were various positions of P R Q, rotated about the common perpendicular B C, their 
 lines of striction would be elements of a conoid of Pliicker. 
 
 The same surface will result if a right line be moved so that, while perpendicular to and inter- 
 secting the axis of a circular cylinder, it shall follow a double-curved directrix obtained by wrapping 
 around the cylinder a sinusoid (Art. 171) or harmonic curve, two waves of which reach once around. 
 
 357. The warped helicoid has for its directrices a helix and its axis, the generatrix making a 
 constant angle with the latter. The last condition is equivalent to saying that the surface has a 
 cone director. 
 
 E'ig'. SIS. E'ig'- SIT-. E-ig. SIS- 
 
 OBLIQUE HELICOIDS. KIQHT HELICOID. RIGHT HELICOIDS. 
 
 If the angle made by the elements with the axis is acute, the surface is called the oblique helicoid. 
 It is the acting surface of ordinary triangular-threaded screws, and of many screw propellers. 
 Plane sections perpendicular to the axis of this surface are Archimedean spirals. 
 358. When the elements are perpendicular to the axis the cone director becomes a plane direc- 
 
 ZFig. 21S. 
 
 CONCHOIDAL HTPEBBOLOID OF CATALAN. 
 
 "tor and the surface is called the right helicoid, familiar to all as the under surface of a spiral 
 ■staircase. It is the acting surface of a. square-threaded screw and, frequently, of screw-propellers. 
 
WARPED SURFACES. — DOUBLE CURVED SURFACES. 
 
 127 
 
 The right hehcoid also belongs among conoiclal surfaces, and its axis is — like that of the cono- 
 euneus — a line of striction. 
 
 359. The conchoidal hyperboloid. invented by Catalan, has two non-intersecting, rectilinear directrices 
 — one horizontal, the other vertical — the generatrix making a constant angle with the 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, s'lg-- aso. 
 
 a b, c d, taken on opposite sides of some vertical right section ; if one of 
 these ellipses be shifted vertically, in its oion j^lane, 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 sofBt (under surface) of a descending arch. 
 
 Anj' plane containing the line in which the planes of the cur^-ed 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. 
 
 861. The warped arch or come de vache has three linear directrices, one straight the others 
 curved; the latter are equal circles in parallel jDlanes, while the straight directrix s-ig-. 221. 
 
 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 sof&t of the arch is a Corne de Vache.' cokne de vache. 
 
 DOUBLE CURVED SURFACES. 
 
 CYLINDROID OF FREZIER. 
 
 362. Double Curved Surfaces are surfaces that cannot be generated l)y 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 
 
 ^ig-- 222. respectivelj' ; the prxraboloid of revolution, generatrix — a jjarabola, 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 j^ossible under this head, 
 we need only mention here the serpentine, generated hy 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 ellijjse, its other axis should so vary in length that its 
 extremities would trace a second elliiDse ; the elliptical paraboloid, whose plane sections perpendicular 
 
 THE TORrS. 
 
 1 Congment figures, if superposed, wiU coincide throughout. 
 
 2 For a corapavison of the relative merits of these methods refer to Skew Arches, by E. W. Hyde. (Van Nostrand's Science. 
 Series, No. 15.) 
 
 •'See Note, p. 64: also Art, 203. 
 
128 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 to the axis are ellipses, while sections containing the axis are parabolas; the elliptical hyperboloid of 
 one nappe, generated by turning a variable hyperbola about its real axis so that its arc shall follow 
 an elliptical directrix : the elliptical hyperboloid of two nappes, analogous to the two-napped hyperboloid ' 
 
 X"ig-- 223. 
 
 ng-. 22'3;. 
 
 T-i-S- 225. 
 
 SERPENTINK. 
 
 ELLIPSOID. 
 
 ELLIPTICAL HYPEKBOLOID. 
 
 of revolution, but having elliptical instead of circular sections perpendicular to its axis ; and the 
 cyclide*, whose lines of curvature (see Art. 37S.) are all circles and each of whose normals intersects 
 two conies — an ellipse and hyperbola — whose planes are mutually perpendicular and having the foci 
 of each at the extremities of the transverse axis of the other. The cyclide is the enveloj^e of all 
 
 ELLIPTICAL HYPEKBOLOID OF TWO NAPPES. THE CYCLIDE OF DUPIN". 
 
 spheres (a) having their centres on one of the conies and (b) tangent to any sphere whose centre 
 is on the other conic. 
 
 The torus is a special case of the cj^clide. 
 
 TUBULAR SURFACES. — QUADRIC SURFACES OR CONICOIDS. 
 
 366. Among the surfaces we have described, the torus and serpentine belong to the family called 
 iuhular, since «ach is the envelope of a sphere of constant radius. 
 
 367. Surfaces whose plane sections are invariably conies are called conicoids or quadrics. These 
 are the cone, cjdinder and sjDhere; ellipsoids; hyperboloids of one and two naiJjDes; ellii^tic and 
 hyperbolic paraboloids. In theory, all conicoids are ruled surfaces; but on some the right lines are 
 imaginary, while they are real on the cone, cylinder, hyperboloid of one sheet and hyperbolic para- 
 boloid. 
 
 TANGENTS AND NORMALS TO CURVES. 
 
 368. A tangent to a curve is a right line joining two consecutive points of the curve. 
 
 A normal to a curve is the right line perpendicular to the tangent at the point of tangency. 
 
 Both tangent and normal to a plane curve lie in its plane. 
 
 * For mathematical, oiJtioal and other properties of the eycliile see Salmon, Geometry of Three Dimensions, and the writings of 
 -J. Clerk Maxwell. 
 
GENERAL DEFINITIONS. 129 
 
 If the curve is the intersection of a surface by a plane, the tangent to it at any point will be 
 the line of intersection of the plane of the curve by a plane tangent to the surface at the given 
 point. 
 
 369. The tangent at any point of a non-plane curve, when the latter is the intersection of two 
 surfaces, is the line of intersection of two planes, each of which is tangent — at the given point — to 
 one of the surfaces. 
 
 LINES AND PLANES, TANGENT AND NORMAL TO SURFACES. 
 
 370. A straight hue joining two consecutive points of a surface is a tangent to it. 
 
 371. A tangent plane to a surface at any point is the locus of all the right lines tangent to 
 the surface at that point. ' 
 
 372. A right line perpendicular to a tangent plane at the point of tangency is a normal to the 
 surface. 
 
 378. Any plane containing the normal cuts from the surface a normal section. 
 
 TANGENT PLANES TO RULED SURFACES. 
 
 374. Since any element of a ruled surface fulfills the condition (Art. 310) necessary to make it 
 a tangent to the surface, it may be taken as one of the two right lines necessary to determine a 
 plane, tangent to the surface at any point of the element. 
 
 In case the surface is doubly ruled the two elements through the point would determine the 
 tangent plane. 
 
 375. In general, the line which — with an element — would determine a tangent plane to a ruled 
 surface at a given point, would be a tangent, at that point, to any curve passing through the latter 
 and lying on that surface. 
 
 376. A plane, tangent to a developable surface at any point, would, therefore, be determined by the 
 element containing the point and, jireferably, by a tangent to the base at the extremity of the ele- 
 ment, since to such a surface a tangent jjlane has line and not merely point contact. 
 
 There is, however, this difference to be noted between the two kinds of developable surfaces ; a 
 tangent plane to a cylinder or cone is tangent the entire length of an element, while, in the case 
 of a surface of which the developable helicoid is representative, a plane tangent to the lower nappe 
 along an element becomes, in general, a secant plane to the upper najjpe, and vice versa. 
 
 377. The tangent pilane at any point of a warped surface is found by Arts. 374 and 375, and is in 
 general a secant jjlane as well, tangency being along an entire element only in special cases. 
 
 TANGENT PLANES TO DOUBLE CURVED SURFACES. 
 
 378. A tangent plane to a double curved surface has, usually*, but one point of contact with it. 
 If a normal to the surface can readily be drawn then the tangent plane may be determined 
 
 most simply on the principle that it will be perpendicular to the normal, at the given point. This 
 method is especially applicable in problems of tangency to a sphere, since the radius to the point of 
 tangency is the normal to the surface; also to any tubular surface, since such may be regarded as 
 generated by the motion of a sphere, and at any point of the circle of contact of sijhere and 
 tubular surface they would have a common tangent plane. 
 
 •For an eiception refer to Fig. 74, page 34, where a bi-tangent plane to an annular torus is seen In profile view as the line 
 
130 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 In general, by taking the the tivo simplest curves that could be drawn through the point of 
 desired tangency and upon the surface, the tangent plane would be determined by the tangents to 
 these curves at that point. In Chajater V methods are given for drawing tangents to the more 
 important mathematical curves, among which we would find nearly all of the plane sections of the 
 surfaces defined in this chapter. 
 
 INTERSECTION OF SURFACES. 
 
 379. The line of interpenetration of two intersecting surfaces may be found by cutting them by 
 a series of auxiliary surfaces; the two sections of the former, cut by any one of the latter, will 
 meet (if at all) in jjoints of the desired line of intersection. 
 
 The surfaces should be so located with respect to H and V as to facilitate the drawing of the 
 auxiliary sections ; and the latter should, if possible, be straight lines or circles, in preference to 
 other forms less easy to represent. 
 
 RADII AND LINES OF CURVATURE. — OSCULATOEY CIRCLE AND PLANE. — GEODESICS. 
 
 380. Radius of Curvature. A circle is osculatory to a non-circular curve when it has three con- 
 secutive points in common with it. Its radius is called the radius of curvature of the curve, for the 
 middle one of the three points. 
 
 381. Line of Curvature. It is ascertained by analj'sis that among all the possible normal sections 
 at any point of a surface two may be found, mutually iserpendicular, whose radii of curvature are 
 respecti^-el}^ the maximum and minimum for that point ; such sections are called principal sections, 
 and their radii the principal radii for that point. 
 
 If ui)on any surface a line be so drawn that the tangent to it at any point lies in the direc- 
 tion of one of the principal sections at that point, the line is called a line of curvature of the 
 surface. 
 
 Smce at every point of a surface two principal sections are possible, there may also be drawn 
 through each point two lines of curvature, intersecting each other at right angles. Such curves are 
 shown in white lines on nearly all the preceding figures illustrating mathematical surfaces. 
 
 382. Geodesies. When a circle is osculatory to a non-plane curve its plane is called an osculating ■ 
 plane. At every point of a geodesic line on a surface the osculating plane is normal to the surface. 
 
 Either the greatest or shortest distance between two points of a surface would be measured on 
 the geodesic passing through them. Since the maximum or minimum distance between two points 
 on a sphere would be measured on the great circle containing them, such circle would be the geo- 
 desic on that surface. 
 
 A helix is the geodesic on a cylinder. 
 
PROJECTION OF SOLIDS. — WORKING DRAWINGS. 131 
 
 CHAPTER X. 
 
 CONVENTIONAL METHODS OF GROUPING PROJECTIONS. —WORKING DRAWINGS. 
 
 383. In the iweceding chapter the principal surfaces which occur in engineering and architectural 
 construction, and which are capable of mathematical definition, have been illustrated and described, 
 and a scientific foundation laid for the graphical solution of proljlems relating thereto. 
 
 In dealing with some of these surfaces, by way of illustration of general princii^les, the figures 
 will, for the most part, be found to be 'fully dimensioned, placing them under the head of what have 
 already been described, viz., vjorkincj draioing-s. While it is important and customary that working 
 drawings should be strictly "to scale" throughout, yet the workman is, ordinarily, held responsible 
 only for following the dunensions given in figures by the draughtsman. 
 
 Although in sheet -metal work, f)articularly in blast-furnace construction, very difficult graphical 
 problems often occm-, there are probably none that surj)ass in complication those occasionally arising 
 in designing masonry arches. To solve such the draughtsman should not only be familiar with all 
 that the preceding chapter contains, but have a thorough course in stereotomy, especially as applied 
 to stone -cutting. His practical prol)lem in any case would be (a) to draw the projections of the 
 arch or other design and derive from them (b) templets showing the exact form of the edges of the 
 stone; (c) heveU showing the dihedral angles between faces; (d) patterns, in sheet -metal, showing the 
 exact form of the various faces of the stone. The work of the mason is greatly facilitated if, in 
 addition to the foregoing, he is furnished with a pictorial representation of the stone upon which he 
 is to work, drawn either in isometric or oblique i^rojection. While the stone-cutter is not expected 
 to understand the theory of projections the machinist must be familiar with it to a certain extent; 
 but, in order to simplify for him the " reading " or comprehension of working drawings, a self- 
 interpreting method of grouping the views has been adopted by a large number of leading American 
 mechanical engineers, although the old method has still ardent advocates among prominent designers 
 and will probably remain the favorite for stereotomy and for purely mathematical work. We shall 
 call the new method that of the Third Angle, the old — or First Angle — method being that with which 
 the student has become familiar in the last chapter, and with which this chapter concludes. 
 
 384. TJdrd Angle Method. This is illustrated by the planes and the hollow, prismatic block 
 A B C D-F, in Fig. 228. Projectors fi-om its points to the three jilanes of projection give d'e'f'c' 
 for the elevation, ased for the plan and e"f"x"s" for the profile view or side elevation. Rotation of 
 the profile jilane about Q R as an axis, and in the dnection indicated by the arrow, carries the 
 side elevation into V at e"'f"'x"'s"'. Rotation of the first angle part of H doivnward, and the rear 
 of H npivard, on axis G. L., would carry the plan, a d e s, into V above d'e'f'c'. 
 
 The lettering of each projection makes it represent that side of the object which is nearest to 
 the plane on which the projection is made. This is one of the distinctive features of the method 
 under consideration, the other being that the rotations supposed bring the view of the top of the 
 object above the fi:ont elevation; the view of the right-hand side of the object (the observer is sup- 
 posed to be looking in the direction X TJ') appears at the right of said elevation. 
 
132 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 Were it desirable to project tlie face A B C D this method would involve a vertical plane on 
 W X, upon which the view would appear; then revolution anti - clockwise would bring the projection 
 into V at the left of d'e'f'c'. 
 
 rig". £2S. 
 
 We might obtain similar relative positions of the projections, were the object in front of V 
 and below the plane Q T XN (Fig. 229), by elevating 'that plane sufficiently to rotate it downward 
 about Q r so as to bring the shaded plan a d e s into V above the elevation. 
 
 r^ig-. ass. 
 
 385. First Angle Method. Disregarding Q T X N (Fig. 229) we have the object and planes 
 illustrating the first angle method, the lettering of each projection showing that it represents the 
 side of the object farthest from the plane. 
 
ORTHOGRAPHIC PROJECTION OF SOLIDS. 
 
 133 
 
 In the ordinary representation the same object would be shown by its three views as in Fig. 
 230. In the front and side elevations the short- dash lines indicate the invisible edges of the opening 
 in the block. 
 
 386. For the sake of more readily contrasting the two methods a group of views is shown in 
 Fig. 231, all above G. L. illustrating an object by the First Angle sj^stem, while all below H K 
 represents the same object by the Third Angle method. 
 
 X'ig-. 231. 
 
 r y ' 
 
 When looking at Figures 1, 2, 3 and 4 the observer queries: What is the object, in space, whose 
 front view is like Fig. 1, top view is Fig. 2, left side is like Fig. 3 and right side like Fig. 4? 
 For the view of the left side he might imagine himself as having been at first between G and H, 
 looking in the direction of arrow N, after which both himself and the object were turned, together, 
 to the right through a ninety- degree are, when the same side would be presented to his view 
 in Fig. 3. Similarly, looking in the direction of the arrow M, an equal rotation to the left, as 
 indicated by the arcs 1-2, 3-4, 5-6, etc., would give in Fig. 4 the view obtained from direction 
 
134 THEORETICAL AND PRACTICAL-GRAPHICS. 
 
 M. His mental queries would then be answered about as follows : Evidently a cubical block with 
 a rectangular recess — r' v' cV c' — in front; on the rear a prismatic projection, of thickness pA and 
 whose height equals that of the cube; a short cylindrical rhig projecting from the right face of the 
 cube; an angular projecting piece on the left face. 
 
 In Fig. 2 the- line rv is in short dashes as, in that view, the back plane of the recess r'v'.d'c' 
 would be invisible. In Fig. 4 the back plane of the same recess is given the letters, v"d", of the 
 edge nearest the observer from direction M. 
 
 Third Angle Method. Ignoring all above line H K we have in Fig. 5 the ■ same front elevation 
 as before, but above it the view of the top ; helow it the view of the bottom, exactly as it would 
 appear were the object held before one as in Fig. 5, then given a ninety-degree turn, around a' 6', 
 until the under side became the front elevation. 
 
 Fig. 7 may as readily be imagined to be olstained by a shifting of the object as by the rotation 
 of a plane of projection ; for by translating the object to the right, from its jjosition in Fig. 6, then 
 rotating it to the left 90° about h'n', its right side would appear as shown. The grouj^ 2-5-6-7-8 
 might, therefore, be conceived as derived by entirely surrounding the object by a system of mutually 
 perpendicular planes, parallel to its main lines; a projection made — on each plane — of the side nearest 
 it on the object; and the f)lanes then rotated into V in such manner, as to right- or left-handed 
 rotation, that the view presented to us is that which we would have of the object itself, were we 
 looking squarely at the face projected. 
 
 387. We shall for the present employ the Third Angle Method, preceding its application by a 
 general resume of terms, abl^reviations and hints as to notation and execution of problems. 
 
 H, V, P the horizontal, vertical and profile planes of projection respectively. 
 
 H- projector that projecting line which gives the horizontal projection of a point. 
 
 V- projector the projecting line giving the projection of a point oti V. 
 
 Projector -plane the profile plane containing the projectors of a point. 
 
 h. p the horizontal jirojection or plan of a point or figure. 
 
 V. p the vertical projection or elevation of a point or figure. 
 
 h. t horizontal trace, the intei-section of a line or surface with H. 
 
 V. t vertical trace, the intersection of a line or surface with V. 
 
 H- traces, V- traces plural of horizontal and vertical traces respectively. 
 
 G. L groiind line, the line of intersection of V and H. 
 
 V - parallel a line parallel to V and lying in a given plane. 
 
 A horizontal any horizontal line lying in a given plane. 
 
 Line of declivity the steepest line, with respect to one plane, that can be drawn in another plane. 
 
 Kabatment revolution into H or V about an axis in such plane. 
 
 Counter - rabatment or revolution . . restoration to original position. 
 
 388. For problems relating solely to the jioint, line and pilane, and not involving solids, the ffiveti lines should be 
 fine, continuous, black ; required lines heavy, continuous, black or red ; construction lines fine, continuous red or short-dash 
 black; traces of an auxiliary plane, or invisible traces of any plane, in dash - and - three - dot lines, red or black. 
 
 For Problems relating to Solid Objects. 
 
 (1) Pencilling. Exact; generally completed for the whole drawing before any inking is done*; the work usually 
 from centre lines, and from the larger — and nearer — parts of the object to the smaller or more remote. 
 
 (2) Inking of the Object. Curves to be drawn before their tangents; fine lines uniform and drawn before the shade 
 lines; shade lines next and with one setting of the pen, to ensure uniformity. 
 
 Begarding iapei-ing shade lines see Art. 111. The tapering should immediately follow the drawing of the curve, 
 
 (3) Location of Shade Lines. In architectural work these would be drawn in accordance with a given direction of 
 light. (See chapter on Shadows.) 
 
 *See Art. 66 for exception; also Arts. 25, 65, 73 and 115 for more aetailed information on many of the other points. 
 
THE CONSTRUCTION AND FINISH OF WORKING DRAWINGS. 
 
 135 
 
 In American machine-shop praciice the right-hand and lower edges of a plane surface are shade lines if they separate 
 it from invisible surfaces. Indicate curraiure by line- shading if not otherwise sufficiently evident. (See Art. 115.) 
 
 (4) Invisible lines of the object, black, invariably, in dashes about one- tenth of an inch in length. 
 
 (5) Inking of lines other than of the object. When no colors are to bo employed the following distinctions as to 
 kind of line are those most frequently made. The lines ma^y preferabl^v be drawn in the order mentioned. 
 
 Centre lines, an alternation of dash and two dots • 
 
 Dimension lines, a dash and dot alternatel_v, with opening left for the dimension 
 
 Ground line, (when it cannot be advantageously omitted) a continuous heavy line 
 
 Construction and other explanatorj' lines, in short dashes 
 
 (6) When colors are employed the centre and dimension lines may be fine, continuous, red; or the former may be 
 blue, if preferred. Construction lines may also be red, in short dashes or in fine continuous lines. Instead of using 
 bottled inks the carmine and blue may preferably be taken directly from Winsor and Newton cakes, "moist colors." 
 
 Drawings of developable and warped surfaces are much more effective if their elements are drawn in some color. 
 
 (7) Dimensions and Arrow -Tips. The dimensions should invariably be in black, printed free-hand with a writing- 
 pen, and should read in line with the dimension line they are on. On the drawing as a whole the dimensions should 
 read either from the bottom or right - hand side. Fractions should have a horizontal dividing line ; although there is 
 high sanction for the omission of the dividing line, particularly in a mixed niimber. Extended Gothic, Roman and Italic 
 Koman are the popular types for dimensioning. 
 
 The a7-row-tips are to be always drawn free-hand, in black; to touch the lines between which they give a distance; 
 and to make an acute instead of a right angle at their point. 
 
 389. Working drawing of a right pyramid; base, an equilateral triangle 0.9" on a side; altitude, x. 
 
 Draw fir.st the equilateral triangle a b c for the plan of the base, making its sides of the pre- 
 scribed length. For lack of any other conditions we may 
 locate the jjlan so that the edge a b shall be perpendicular 
 to the profile plane. 1 ; then the face v ab will appear 
 as a straight line, in profile view. Being a right pyramid 
 the vertex will be vertically above the centre of the base ; 
 hence v is equally distant from a, b and c; and v a, v h, 
 V c are the plans of the edges. 
 
 Parallel to G. L. and at a distance apart equal to the 
 assigned height, x, draw m v" and n c" as upper and lower 
 limits of the front and side elevations; then, as the h. p. 
 and V. p. of a jjoint are always in the same perpendicu- 
 lar to G. L., we project v, a, b and c to their resi^ective 
 levels by the construction lines shown, obtaining v'-a' b' c' 
 for the front elevation. 
 
 Projectors to the profile plane from the points of the plan give 1, 2, 3, which are then carried, 
 in arcs about 0, to L, 5, 4, and projected to their proper levels, giving the side elevation, v" b" c". 
 
 As the actual length of an edge is not shown in either of the three views, we employ the fol- 
 
 X-ig-. 23S. 
 
 lowing construction to ascertain it : Draw v i\ perpendicular to 
 V b and make it equal to x ; %\ b is then the real length of 
 the edge, shown by rabatmeut about v b. 
 
 The devdopriient of the pyramid may be obtained by draw- 
 ing an arc A B C, of radius = r, b, and on it laying off the 
 chords A B, B C, CA^ equal to ab, b c, c a of the plan ; then 
 V-A B C Ai is the plane area which, folded on T' C and V B, 
 ■would give a model of the pjT:amid represented. 
 
 ngr- 233. 
 
136 
 
 THEORETICAL AND PRACTICAL GRAPHICS. 
 
 390. Worhing drawing of a vertical, semi -cylindrical pipe: outer diameter, x; inner diameter, y ; 
 height, 2. :E,3.g. sa^. 
 
 For the pjlan draw concentric semi-circles a e d and h s c, of 
 diameters a; and y respectively, joining their extremities by 
 straight lines a b, c d. At a distance apart of s inches draw 
 the ufiper and lower limits of the elevations and project to 
 these levels from the points of the j)lan. 
 
 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. 
 
 Ordinarily the line shading would be omitted, the shade 
 lines on the various views sufficing to convey a clear idea of 
 the form. 
 
 391. Hcdf of a hollow, hexagonal prism. In a semi -circle of 
 diameter a d step off the radius three times as a chord, giving 
 
 ^3.gr- 23S. the vertices of the plan abed of the outer surface. Parallel 
 
 to b c, and at a distance from it equal to the assigned thickness 
 of the prism, draw ef, terminating it on lines (not shown) drawn 
 through b and c at 60° to a d. From e and / draw e h and / g, 
 parallel respectively to a 6 and c d. Drawing a' c" and m' f, as 
 upper and lower limits, project to them as in preceding problems 
 for the fi-ont and side elevations. 
 
 392. Working drawing of a hollow, prismatic block, standing 
 obliquely to the vertical and profile planes. 
 
 Let the block be 2" x 3" x 1" outside, with a square opening 
 1" X 1" X 1" 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 a s xb, 3" x 1", 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" 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 outlines of the 
 views. The dotted construction lines and the let- 
 tering will enable the student to recognize the 
 three views of any point without difficulty. 
 
 393. In Fig. 237 we have the same object as 
 in Fig. 236, but represented as cut by a vertical 
 plane whose h. t. is v y. The parts of the block 
 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 
 student in understanding the views ; but in engin- 
 eering practice section- lining is rarely done on views not perpendicular to the section plane. 
 
 ^^.^ 
 
 M 
 
 
 
 ,\ 
 
 ^^ J ^ 1 j,X^ 
 
 yi L_J_ 
 
 
 V" t--\- ' -' 
 1^1 """" 
 
 
 G 
 
 a 1.._4 ; ^ 
 
 : i i i L ~ 
 
 -N^ 
 
 '^ \ 
 
 
 
 6 Vi' 1 1 : 1 Ix' 
 
 s' 
 
 a 
 
 6| 
 
 ] s" : L 
 
 
 c'> 
 
 
 
 m 1 ;/ : 
 
 
 
 
 m" ; 
 
 j''J\. 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 d \f 
 
 fl_. 
 
 
 k I i 
 d" Ic" 
 
 d' 
 
PROJECTION OF SOLIDS. — WORKING DRAWINGS. 
 
 137 
 
 394. Svp-pression of the Ground Line. In machine drawings it is customary to omit the 
 ground line, since the forms of the various views — ^ig- ^av. 
 
 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 anj' 
 vertical plane is wanted, it will be perpendicular 
 to the line joining the jjlan of a point with its 
 projection on such vertical plane. 
 
 395. To draw a holloiu, pentagonal prism, 2" 
 long ; edc/es to be horizontal and inclined 35° to 
 V ; base, a regular j^entagon of 1" sides ; one face 
 of the prism to be inclined 60° to H. 
 
 In Fig. 238 let H K be parallel to the plans 
 of the axis and edges ; it will make 35° Avith a 
 horizontal line. Perpendicular to H K draw mn as the h. t. of an auxiliary vertical plane upon 
 
138 
 
 THEORETICAL AND PRACTICALGRAPHICS. 
 
 which we suppose the base of the prism projected. In end view all the faces of the prism would 
 be seen as lines, and we. may draw a 16,, one inch long and at 60° to m n, to represent the face 
 whose inclination is assigned. Completing the inner and outer pentagons, allowing, for example, I" 
 for the distance ' between faces, we have the end view complete. The plan is then obtained by 
 drawing parallels to HK through all, the vertices of the end view and terminating all by vertical 
 planes, a d and g h, parallel to m n a;nd 2" apart. 
 
 The elevations will be included between horizontal lines whose distance apart is the extreme 
 height a 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. These heights are estimated from the level of b^. 
 
 Since the profile plane is omitted, we take M' N' to represent the trace upon it of the auxiliary, 
 central, vertical plane whose h. t. is M N. 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' A'' equal to the perpen- 
 dicular 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 
 towards it in the direction of H K. 
 
 396. To draw a truncated pyramidal block having a rectangidar 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 tV 
 
 -"-" and Y' deep. The small oblique projection on the right of Fig. 239 shows, 
 
 pictorially, the figure to be drawn. 
 
 11 1 r 
 
 e ■ \> 
 
 The plan of the lower base will be the rectangle ab d e, 3" X 2", whose longer .edges are inclined 
 30° to the horizontal. " ■' 
 
ORTHOGRAPHIC PROJECTION. — THIRD ANGLE METHOD. 139 
 
 Take AB and m n 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 d e, and which is inclined 60 ° 
 to H, will have d, y for its trace on plane m n. A parallel to m n and ^" from it will give Sj , 
 the auxiliary projection of the upper edge of the face s v e d, whence s v — at first indefinite in length 
 — is derived, parallel to d e. Similarly the end face btsd is obtained by projecting db upon ^ J5 at 
 61, drawing b^ z at 60° to ^ i? and terminating it at s.^ by CD, drawn at the same height (^is") as 
 before. A parallel to 6 d through s., intersects v s, at s, giving one corner of the plan of the upper 
 base, from which the rectangle s t u v is completed with sides parallel to those of the lower base. 
 
 As the recess has vertical sides, we may draw its plan opqr 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 the preceding problems ; that is, for 
 the front elevation, o' u' s' d', by verticals through the plans ; for the side elevation, e" v" t" b", the dis- 
 tances to the right or left of s" equal those of the plans of the same points from s i, regarding the 
 latter as the h. t. of a central plane parallel to V. 
 
 The plane ST of right section, ■perpendicular to the axis K L, cuts the block in a section whose 
 true size is shown in the tinted figure gi \ \ /!„ and whose construction hardly needs detailed treat- 
 ment after what has preceded. The shaded longitudinal section, on central vertical plane K L, also 
 interprets itself by means of the lettering. 
 
 The true size of any face, as auv e, may he shown by rabatment about a horizontal edge, as a e. 
 As V is actually ■^" above the level of e, we see that v e (in space) is the hypothenuse of a tri- 
 angle of base v e and altitude yV'- Construct such a triangle, v v.^ e, and with its hypothenuse v.^ e 
 as a radius, and e as a centre, obtain v, on a line through v and perpendicular to a e and repre- 
 senting the path of rotation. Finding Ui similarly, we have aUiV^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 principles would be involved. 
 
 397. The last two problems illustrate all of the features of the Third Angle method, as practically 
 applied in machine drawing and design. They luay be thus summarized : 
 
 (a) The various views of the object are grouped around the front elevation, that of the top 
 being above it, that of the bottom below it, and analogously for the projections of the right and left 
 sides. 
 
 (b) The ground line is omitted, although its direction may be readily determined by insi^ection. 
 
 (c) Central reference- planes are taken through the various views, and, in each view, the distance 
 of any point from the trace of the central jjlane 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 — usual)}' the plan. 
 
C^HE remaining pages contain some illustrations for which the text is not complete, but regarding 
 which oral instruction will be given; also a large number of alphabets which will be serviceable 
 in the drawing of titles, and which are specimens of the type - founder's art, procured for, though not 
 designed with especial reference to, this work. 
 
Railroad Bridge 
 Post CannEGtinn 
 
 Upper Clinrd, 
 
 (In drawing the above the student will make horizontal, not slanting dividing linos in all fractions. Cenfe and dimension lines in continxious red 
 lines, unless for blue printing, in which, case all lines will be black.) 
 
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