.^'^ o -^^■^ - *^- I - aV „ - '(A x^^^. ^^j >>> ^^. ■%^ . ^ 3o .^-^ '''c^. c^'^ ..^ V ' kO^^ ^^' -^p. ■^ ,.<^^ PRACTICAL ENGINEERING DRAWING AND THIRD ANGLE PROJECTION F. N. \VlLLSON " IQ p£Q8il898 PRACTICAL ENGINEERING DRAWING AND THIRD ANGLE PROJECTION FOR STUDENTS IN SCIENTIFIC, TECHNICAL, AND MANUAL TRAINING SCHOOLS AND FOR ENGINEERING AND ARCHITECTURAL DRAUGHTSMEN, SHEET METAL WORKERS, Etc. c \'< A ^ % ^^f Fredefiek fleoaton Willson, C.E., fl.JVI., Professor of Descriptive GeoJ'tetry^ Stercoiomy and TeclLniccil Drawing ill the John C, Green School of Seience^ Princeton University. PUBLISHED BY THE AUTHOR PRINCETON, N. J. 1897 L. 2181 ALTHOUGH the chapters here presented are taken from Theoretical and Practical Giriphics — the author's more extended treatise on the theory and applications of descrij^tive geometry and mechanical drawing, they were iirejjared with a view to their separate issue in this form, and are independent of the other matter with whicli they are paged. COPvniGHTED, Ifl'je, BY FRED'K n. WILLSON. TABLE OF CONTENTS NOTE-TAKING, DIMENSIONING, ETC. Technical B'ree-Hand Sketching and Lettering. — Note -Taking from Measurement. —Dimension- ing. — Conventional Representations. Pages 5 - lo. THE draughtsman's EQUIPMENT. The Choice and Use of Drawing Instruments and the Various Elements of the Draughtsman's Equipment. — General remarks preliminary to instrumental work. Pages II -20. EXERCISES FOR PEN AND COMPASS. Kinds and Signification of Lines. — Designs for Elementary Practice with the Right Line Pen. — Standard Methods of Representing Materials. — Line Shading. — Plane Problems of the Right Line and "Circle, including Rankine's and Kochansky's approximations. — Exercises for the Compass and Bow -pen, including uniform and tapered cur\'es. — The Anchor Ring. — The Hy- perboloid. — A Standard Rail Section. Pages 21-38. ON HIGHER PLANE CURVES AND THE HELIX. Regarding the Irregular Curve. — The Helix. — The Ellipse, Hyperbola and Parabola, by various methods of construction. — Homological Plane Cur\"es, — Relief - Perspective. — Link - Motion Curves. — Centroids. — The Cycloid. — The Companion to the Cycloid. — The Curtate and Prolate Trochoids, — Hypo - , Epi - , and Peri- Trochoids. — Special Trochoids, as the Ellipse, Straight Line, Limaijon. Cardioid, Trisectrix. Involute and Spiral of Archimedes. — Parallel Curves, — Conchoid. — Quadratrix. — Cissoid. — Tractrix. — Witch of Agnesi. — Cartesian Ovals. — Cassian Ovals. — Catenary. — Logarith- mic Spiral. — Hyperbolic Spiral. — Lituus. — Ionic Volute. Pages 39-78. TINTING AND SHADING. Brush Tinting, Flat and Graduated. — Masonry, Tiling, Wood Graining, River- Beds, etc., with brush alone, or in combined brush and line work. Pages 79-87. THE LETTERING OF DRAWINGS. Free- Hand Lettering. — Mechanical Expedients. — Proportioning of Titles. — Discussion of Forms. — Half- Block, Full Block and Railroad Types. — Borders and how to draw them. (Alphabets in Appendix). Pages 88-96. BLUE-PRINTING AND OTHER PROCESSES. The Blue -print Process. — Photo-, and other Re- productive Graphic Processes, including Wood Engraving, Cerography, Lithography. Photo- lithography, Chromo-lithography, Photo-engrav- ing, "Half-Tones," Photo-gravure and allied processes.— How to Prepare Drawings for Illustra- tion. Pages 97- 103. THIRD ANGLE PROJECTION. — WORKING DRAWINGS. Projections and intersections by the Third Angle Method. — The Development of Surfaces, for Sheet Metal or Arch Constructions. — Working Drawings of Bridge Post Connection, — Struc- tural Iron. — Spur Gearing (Approximate Invo- lute Outlines). — Helical Springs, Rectangular and Circular Section. — Screws and Bolts (U. S. Standard), and Table of Proportions. Pages 131- 180. AXONOMETRIC (INCLUDING 'SOMETRIC) PRO- JECTION. —ONE - PLANE DESCRIPTIVE GEOMETRY. Orthographic Projection upon a Single Plane. — Axonometric Projection. — General Fundamental Problem, inclinations known for two of the three axes.' Isometric Projection vs. Isometric Draw- ing. — Shadows on Isometric Drawings. — Tim- ber Framings and Arch Voussoirs in Isometric View. — One-Plane Descriptive Geometry. Pages 241 - 247. OBLIQUE PROJECTION, Oblique or Clinographic Projection, Cavalier Per- spective, Cabinet Projection, Military Perspec- tive. — Applications to Timber Framings, Arch Voussoirs and Drawing of Crystals. Pages 248- 250. APPENDIX. Table of the Proportions of Washers. — Working Drawings of Standard 100 - lb. Rail, and of Allen - Richardson Slide Valve. — Designs for Variation of Problems in Chapters X, XV and XVI. —Alphabets. Pages 251 -268. FREE-HAND DRAWING. CHAPTER II. ARTISTIC AND TECHNICAL FEEE-HAND DRAWING.— SKETCHING FROM MEASUREMENT. — FREE- HAND LETTERING.— CONVENTIONAL REPRESENTATIONS. 20. Drawings, 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 they can be worked from and what they rejaresent may be manufactured; or finished drawings, illustrative or artistic in character and therefore shaded either with jjeu or brush, and ha^'ing no hidden jjarts indicated by dotted lines as in the jDreceding division. Finished drawings also lack figured dimensions. Working drawings of parts or " details " of a structure are called detail drawings; while the representation of a structure as a whole, with all its details in their prof)er 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 %dews — jilans, eleva- tions and sections — of all engineering and architectural constructions, and in solving the f)roblems 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 im23ortance than to handle the drawing instruments skill- 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 architect 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 effective 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 standjDoint. While it is not the province 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 jDrofession. He should j^rocure Linfoot's Picture Making in Pen and Ink, Miller's Essenticds of Perspective and Delamotte's Art of Sketching from Nature; and with an experienced architect or artist, if jjossible, but otherwise by himself, master the prin- ciples and act on the instructions of these writers. 22. Since the camera makes it, fortunately, no longer essential that a civil engineer should be 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 j^urposes 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 working sketches, conventional representations and free-hand lettering, and the equipment therefor consists of a pencil of medium grade as to hardness; lettering pens — Falcon or Gillott's 303, with Miller Bros. "Carbon" pen No. 4; either a note-book or a sketch-block or pad; also the following for sketching from measurement: a two -foot pocket -rule; calipers, both external and internal, 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 otlierwise a long rod graduated to eighths. 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 jDOSsible, 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 ob^'iousl}^ desirable that something like the relative proportions should be preserved and that the closer the approximation thereto the clearer the idea they will give to the draughtsman or workman who has to work from them. A habit of close observation must therefore be cultivated, of analysis of form and of relative direction and proportion, bj^ 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 categorj' 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 theorj^ that lies back of all correct design and hj such training of the hand as shall facilitate the graphic expression of his ideas. To that end he should improve every opjjortunity to put in practice the following instructions as to SKETCHING FROM MEASUREMENT, as each structure sketched and measured will not only give exercise to the hand but also prove a valuable object lesson in the proportioning of parts and the modes of their assemblage. A free-hand sketch maj' be as good a working drawing as the exactly scaled — and usuallj' inked — drawing that is generallj' made from it to be sent to the shop. "While several views are usually required, yet for objects of not too comijlicated form, and whose lines lie mainly in mutually perpendicular directions, the method of representation illustrated by Fig. 7, is admirablj' adapted,* and ob^dates 'all necessity for additional sketches. It is an oblique -projection FREE-HAND SKETCH OF TIMBER FRAMING. (Art. 17) the theory of ^\-hose construction will be found in a subsec^uent 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 i^arallel 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. (Refer to Figs. 42, 44, 45 and 46.) 24. While Fig. 7 gives almost the jjictorial effect of a true perspective and the object requires no other clescriiDtion, yet for comjjlicatecl and irregular forms it gives place to the plan - and - eleva - tion mode of representation, the plan being a top and the elevation a jront ^dew of the object. And * The figures iu this chapter are photo - reproductions of free-hand work and are intended not only to illustrate the text but also to set a reasonable standard for sketch -notes. SKETCHING FROM MEASUREMENT. 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 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 represented by the method just mentioned. The front %'iew (elevation) is evidently the same in hoth Figs. 7 and 8, except that in the latter we indicate by dotted lines the hidden 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, ^dz., 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. JO ^^ N 5 i —PLAN- M ^ K f V - -?" — V 1 1 1 1 f ■ jT -■ ■• D' ^ H c' %' '" fr— -Zio > - £L£yAT/0/\f- FREE- HAN O SKETCH OF TrMBER FRAMING, fN PLAN AND ELEVATION . In these views, which fall under Art. 19 as to theoretical construction, entire surfaces are fro- kcted as straight lines, as G B C H in the straight line 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 obliquely, as shown. CENTEE - LINES. — DIMENSIONING. 25. Dimensioning. In sketching, centre-lines and all important centres should he located first, and measurements taken from them or from finished surfaces. Feet and inches are abbreviated to "Ft.," and "In.," as 4 Ft. 6| In.: also written 4' 6|", and occasionally 4 Ft. 6f". A dimension sh(juld not be written as an improper fraction, i^" for ex- ample, but as a mixed number, 1| Fractions should have horizontal 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 sliould read in line with the line they are on, and either from the bottom or the right hand. The arrow tijis should toucli the lines between which a distance is given. THEORETICAL AND PRACTICAL GRAPHICS. Extension lines should be drawn and the dimension given outside the drawing whenever such- course will add to the clearness. (See D' F', Fig. 8.) An opening should always be left in the dimension line for the figures. 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 (as, for example, a rail, angle -iron, channel bar, Phoenix column or other form of structural iron) 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. If the kind of bar and the number of f)0unds per j^ard are known the dimensions can be obtained by reference to the handbook issued by the manufacturers. ^igr- s. 2.00 Z-ss. p/t. yo. f9'^- \i^? T^ i FREE-HAND . SKETCH OF A CHANNEL BAR. The same dimension should not appear on each view, but each dimension must be given at least once on some view. Notes on Riveted Work, Pins, Bolts, Screws and Nuts. In riveted work the " pitch " of the rivets, i. e., their distance from centre to centre (" c. to c") should lie noted, as also that between centre lines or rows, and of the latter from main centre lines. Similarly for bolts and holes. If the latter are located in a circle note the diameter of the circle containing their centres. Note that a 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" O (read "four inches, round," not around,) for 4" diameter; but it is even clearer to use the abbreviation of the latter word, viz., " diam." In taking notes 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 prop)ortions 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. Notes on Gearing. 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. Bridge Notes. In taking bridge notes there would be required general sketches of front and end view; of the flooring system, showing arrangement of tracks, ties, guard -beam and side -walk; 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. CUXVK.XTIoyAL REJ> UESEXTA T 1 X S. — F R E E- H A M) LETTERING. 9 All notes should Ije taken (in as lar}>-e a seale as possil-ile, and so indexed that drawings of parts ma)^ readily be understood in their relation to the whole. The foregoing hints might be considerably extended to embrace other and special cases, but experience 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. It need hardly be added that what lias preceded is intended to be merely a partial sunnnary of the instructions which would be o-iven 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 alile to sketch much more intelligently after studying more (jf the theory involved in Mechanical Drawing and given in the later pages of this work. COXVENTIOX.VL REPRESENTATIONS. 26. Tonventional representations 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 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 ft-ee-hand, since in them there can be a nearer approach to the natural appearance. This is well illustrated by Figs. 10 and 11. ki. In addition to a rock section Fig. 11 ("a) shows the method of indicating a mud or sand bed witli small random lioulders. Water either in section or as a receding surface may be shcjwn l\v parallel lines, the spaces between them increasing gradually. Conventional representations of wood, masonry and the metals will be found in Chapter VI, after hints on coloring have lieen given, the foregoing figures ajipearing at this point merely to illustrate, in black ami wliite, one of the important divisions of technical free-hand work. Those, however, who have already had some practice in drawing may undertake them either with pen and ink or in colors, in the latter case observing the in.structions of Arts. 237-241 for wood, while for the river 10 THE B E riCA L A ND PEA C TIC A L G B A PHIC S. sections they may employ burnt loiiber undertone for the earthy bed, pale blue or india ink tint for the rock, and pni>i>iiaH blue for the water lines. FREE-HAKI) LETTERIXG. 27. Although later on in this work an entire chapter is dev(^ted to the subject of lettering, yet at this point a word should be said regarding those forms of letters whieli ought to be mastered, early in a draughting course, as the most serviceable to the practical worker. . :Fig-. 3.2. ABCDEFGHIJKLMNOPQRSTUVWXYZ& 1234567890 ABCDEFGHIJKLMNOPQRSTUVWXYZdL I234567890 The first, known as the Gothic, is the simplest form of letter, and is illustrated in both its vertical and inclined (or Italic) forms in Fig. 12. It is much used in dimensioning, as well as for 12 (a.) titles. The lettering and numerals are Gothic in Figs. 7 and 8, with the exception ^. of the 1 and 4, which, by the addition of feet, are no longer a pure form although O^V^^M^ T' enhanced in appearance. In Fig. 12 (a) some modifications of the forms of certain numerals are shown; also the omission of the dividing line in a mixed number, as is customary in some offices. For Gothic forms a coarse j^en is necessary, and shading is to be avoided, the object being to get all lines of uniform weight. Miller Bros'. "Carbon" pen. No. 4, is well adapted to them for w^ork on a fairly smooth surface and with a free -flowing ink. Fig. 13 illustrates the Italic (or inclined) form of a letter which Avhen vertical is known as the Roman. The Roman and Italic Boman are much used on Government and other map work, and in ^ig". 13- ^ B C D I] F G H I J K L M W P Q E S 12 3 4 5 TUT W X T Z 67890 a~b c d e f CI Iviyl k I m n ojj q^r s tTi v w x ij z the draughting offices of many j)rominent mechanical engineers. Regarding them the student may profitably read Arts. 260-262. Make the spaces between letters as nearly uniform as possible, and the small letters usually about three - fifths the height of the capitals in the same line. For Roman and other forms of letter requiring shading use a fine pen; Gillott's No. 303 for small work, and a " Falcon " pen for larger. A form of letter much used in Europe and growing in favor here is the Soennecien Roimd Writing, referred to more particularly in Art. 265 and illustrated by a complete alphabet in the Appendix. The text -book and special pens required for it can be ordered through any dealer in draughtsmen's supplies. THE DRAUGHTS MAX'S EQUIPMENT. 11 CHAPTER III. DRAWING INSTRUMENTS AND MATERIALS. — INSTRUCTIONS AS TO USE. AND TECHNICALITIES. -GENERAL PRELIMINARIES Fig-- IS. S 4 28. The drauji'ht.snuiu s equipment for graphical work .should Ije the l.iest con- '^^s- 1-*- :Fig-. is. sistent with his means. It is mistaken economy to buy inferior instruments. The best obtainalile -will be found in the end to have Ijeen the cheapest. The set of instruments illustrated in the following figures contains only those which may lie considered a))solutely essential for the beginner. THE DU.VWINC PEN. The right line pen (Fig. 14) is onlinarilj- used for drawing .straight lines, with either a rule or triangle to guide it ; liut it is also employed for the draw- ing of curves when directed in its motion l>y curves of wood or liard rubber. For average work a pen alxiut five inches long is best. The figure illustrates the most ajiproved type, i. e., made from a single piece of steel. The distance between its ])oints, or "nibs," is adjustable l>y means of the screw H. An older form of pen has the outer blade connected with the inner by a liinge. The convenience ^\\ih. which such a pen may be cleaned is more than offset liy 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 unerj^ually or become blunt the draughtsman may sharpen them readily himself ujion a fine oil-stone. Tlic ])roce,ss 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 fi-om left to right (sujiposing the initial position as in Fig. Ifi). Before reaching the right enrl of the stone liegin turning the pen in a plane perpendic- ular to the surface, and draw in the opposite direction at the same angle. Alter frequent examination and trial, to see that the blades liave l)ecomc equal in length and similarly rounded, the jDrocess is completed by lightly dressing the outside of eacli blade separately upon the stone. Xo griniling shoulil l)e done on the inside of the blade. Any "burr" or rough cilge resulting from tlie operation may lie removed mth fine emery paper. For tlie liest results, obtained in the shortest pos.sible time, a magnifying glass should be- hould take particular notice c^f the shape of the pen when new, as a standard Tile student aimed at when compelled to act on the above suggestions. used, to be '■V). The ijcn may be supplied with ink by means of an ordinary writing pen dipped in the ink and then passed between the blades; or by u.sing in the same manner a strip of Bristol board about a quarti-r of an inch in width. Should any fresh ink get on the outside of the pen it must 12 THEORETICAL AND PR A C T T CA L G R A PHI C S. be removed ; otherwise it will be transferred to the edge of the rule mikI thence to the jjuper, caus- ing a blot. 31. As with the jjencil, so with the pen, horizontal lines are to be dra\\'n from Itfi to right, while vertical or inclined lines are drawn either from or toward the worker, according to tlie position 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 draughtsman, i. e., from n toward m ; while op would be drawn foivnrd the worker, being on the right of the triangle. 32. To make a sharply defined, clean-cut line — the only kind allowable — the pen should be held lightly but firmly with one blade resting against tire guiding edge, and with both jjoints resting equally upon the paper so that they may wear at the same rate. 83. The inclination of the pen to the paper may best be about 70°. When properly held the pen 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 paper 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 Ije from them the distance indicated. It need hardly be said that a drawing -ijen should not be pushed. The niore frequently the draughtsman will take the trouble to clean out the point t)f the pen and supjDly 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 slmuld lie taken of all the instruments, and for cleaning them nothing is superior to chamois skin. DIVIDERS. 34. The hair-sjiring di\dders (Fig. 15) are employed in dividing lines and spacing off distances, and are capiable of the most delicate adjustment by means of the screw G and si:)ring in one of the legs. When but one j)air of dividers is purchased the kind illustrated should have the preference over i^lain dividers, which lack the spring. It will, however, be frequently found convenient to have at hand a jDair of each. Should the joint at F become loose through wear it can be tightened by means of a key having two projections which fit into the holes shown in the joint. 35. In spacing oif distances the jDressure exerted should be the slightest consistent with the loca- tion of a point, the jDuncture to be merely in the surface of the paper and the f)oints determined by lightly pencilled circles about them, thus q q . In laying off several ecjual distances along a line all the arcs described by one ^igr- is- jgg 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 X'ig'- IS. M a _/., w arc R, then rests and pivots on c while the leg y describes the arc S; x then traces arc T, etc. THE COMPASSES.— HOW-ri'JXCIL .1X1) PEN. 13 COMPASS SET. ^ig-- SO- S'igr- 21- E'ig-- 22. .»J 36. The compasses (Fig. 20) rcseiuMe the dividers in tnrm and nuiij be used to jjerform the same office, but are usually employed for tlic 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, Avhen the pen leg LM (Fig. 21) may be inserted for ink work; or, where large work is involved, the lengthening bar on the right (Fig. 22) may be first attached at and the pencil or pen leg tlien inserted at /. The metallic point held by screw N is usually rejilaceil liy a liard lead, sharpened as indicated in Art 54. 37. When in use the legs should be bent at the joints I' and L, so that they will lie perjiendicular to the paper when the compasses are held in a vertical plane. The turning may be in either direction, but is usually " clock- wise ; " and the cunipasses may be slightly inclined toward the direction of turning. A\'hen so used, and if no undue pressure be exerted on the pivot leg, there should be but the slightest puncture at the centre, while the pen points having rested equally upon the paper have sustained ecpial 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 shuuld be held and turned l)y the milled head which 131'ojects above the joint i\'. Dividers and compasses should o]ien ane fastened smoothly, with thumb-tacks, over the drawing to be copied, and the ink lining done u])on the glazed side, any l)rush work that may be recjuired — either in ink or colors — lieing always done u]ion the dull side of the cloth after the outlining has been completed. If the glazed surface be first dusted with powdered j)i]je-clay applied with chamois skin it will take the ink much more readily. AMien erasure is necessary use the rublier. after which the surface may be restored for further pen -work \>y rubbing it with soapstone. Tracing -cloth, like drawing paper, is most convenient to work u])on if perfectly flat. When either has been puix-hased by the roll it should therefore be cut in sheets and laid away for some time in drawers to become flat before needed for use. I)HAWI.\(; BOARD. 46. The drawing board should l>e slightly larger than the pajjer for which it is designed and of the most thoroughly seasoned material, jjreferably some ^(ift wood, as pine, to facilitate the use of the drawing-pins or thumli -tacks. To ])revent warping it should have 1)attens of hard wood dove- tailed into 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, Avhich weakens the board trans- versely and to that degree facilitates the stifl'ening action of the battens. For work of moderate size, on stretched j>a23er, yet without the use of mucilage, the " 2:)anel " board is recommended, provided that both frame and jjanel are made of the best seasoned hard wood. 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 (abfiut twice Super Rf)yal size) which are well adapted to large drawings of machinery, bridges, etc. 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 draughting-room. THE T-KULE. 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 latter in the former. The head should also be so fastened to the blade as to be level with the surface of the board. This i^ermits the triangles to slide freely over the head, a great convenience when the Hnes of the drawing run close to the edge of the j^aper. (See Fig. 32.) The head of the T-rule should always be used along the left-hand edge of the drawing board. TRIANCJLES. 48. Triangles, or " set - squares " as they are also called, can be .obtained in various materials, as hard rubber, celluhjid, pear -wood, mahogany and steel; and either solid (Fig. 25) or oi^en (Fig. 26). The open triangles are preferable, and two are required, one 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 fomier to rust, unless plated. Celluloid is transparent and the most cleanly of all. 16 THEORETICAL AND PRACTICAL GRAPHICS. The most frequently .recurring problems involving the use of the triangles are the following: — ng-- as- 49. To draw parnlld lines place either of the edges E'igr- ss. against another triangle or the T-rule. If then 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 base. By then turning the triangle so that the other side, o c, of its right angle shall be against the rule, as at o^c,, the hypothenuse will be found perpendicular to its first position and therefore to the given line. iFigr- 27. 51_ 3}) construct regular hexagons place the shortest side of the 60° triangle against the rule (Fig. 27) if two sides are to be horizontal, as /e and b c of hexagon H For vertical sides, as in H ', the position of the triangle is evident. By making « b indefinite at first, and knowing be — 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 by drawing a series of tangents to the circles, with the rule and triangles in the positions indicated. THE SCALE. 52. Bu-t 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 i; SCALE 1:2; SCALE 6 In. = 1 FT. SCALE 6" = 1' . At one foot to the inch any line of the drawing would be one -twelfth the actual size, and the fact indicated in either of the ways just illustrated. Although it is a simple matter for the draughtsman to make a scale for himself for any par- ticular case yet scales can be purchased 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 by Fig. ^ig"- 3S- 28, and graduated _3_ ■ i 16- 8 li: inches to the foot. This is known as the arcliitecih scale in contradistinction to the engineer's, which is decimally graduated. It will, however, be frequently convenient to have at \\\\W\A\\v\\\4 hand the latter as well as the former. When in use it should he laid along the line to be spaced, and a light dot made upon the X 11 10 9 8 76 5432 1 b D I 1 1 1 1 1 1 1 1 1 1 j 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 / 1 1 ■ / 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I si X /// 1 1 1 1 t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Ic Id k If Ig Ih li h Ik ll I'll Sill 1ft B a paper with the pencil, opposite the proper division on the graduated edge. A distance should rarely SCALES. — PENCILS. — INKS. 17 be transferred from the scale to the drawing by the dividers, as such procedure damages the scale if not the paper. 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, enclosing ten equal spaces, as in Fig. 29, and from the end A lay off A B, B C, etc., each 3 inches and representing a foot. Tlien twelve parallel diagonal lines in the first space intercept quarter -inch spaces on AB or ab, each representative of an inch. There being ten equal siDaces between B and b, the distance s x, of the diagonal 6 m from the vertical b B, taken on any horizontal line s X, is as many tenths of the space in B as there are sjDaces between s x and b ; six, in this case. The principle of construction may be generalized as follows : — The distance apart of the vertical lines rejaresents 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 j'ards as units ; inches in case of feet, etc.). Join the points of division by diagonal lines ; and, if — is the smallest fraction that the scale is designed to give, rule x + 1 equidistant horizontal lines, giving X equal horizontal sfjaces. The scale will then read to -th of the intermediate denomination of the scale. When a scale is properly used, the sjDaces 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 1^ inches to the foot would be em2Dloyed; each 1^ inch sjsace 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. Tlie former method is the more expeditious. THE PENCILS. 54. For construction lines afterward to be inked the pencils should be of hard lead, grade 6H if Fabers 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 afterward 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 comjDass leads to a point. Use the knife for cutting 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 knive or on a fine file, or a laad of emery 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 Avith 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 fracture, and will have a smooth, as contrasted with a gritty feel when tested by rubbing the moist- ened finger on the end of the cake. 18 THEORETICAL AND PRACTICAL GRAPHICS. 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 fi-om fifteen to twenty drops 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 pressure a mixture results having flakes and sand -like particles of ink in it, whereas an absolutely smooth and rather thick, slow- flowing liquid is wanted, whose surface will reflect the face like a mirror. The fiiral test as to sufficiency of grinding is to draw a broad line and let it dry. It should then be a rich jet black, with a slight lustre. The end of the cake must be carefullj^ dried on removing it from the saucer to jjrevent its flaking, which 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. WATER 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. Very good colored drawing inks are now manu- factured for line work, but Winsor and Newton's water colors, in the form called " moist," and in "half-i^ans" are the best if not the 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. For some of their special uses see Art. 73. 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 f)oint of the pin. IRREGULAR CURVES. 58. Irregular or French curves, also called s^veeps, for drawing non- circular arcs, are of great variety, and the draughtsman can hardly have too many of them. They may be either of pear Fig-- 30. wood or hard rubber. A thoroughly equipjDed 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. Some very serviceable flexible curves are also in the market. If but two are obtained (which would be a minimum stock for a beginner) the forms shown in Fig. 30 will probably prove as serviceable as any. When employing them for inked work the pen 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 CURVES. — RUBBER. — ERASERS. — PRO TRACTORS. — BRUSHES. 19 the rale. And the student must content himself with drawing slightlj^ 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 hand is always kept on the concave side of the curve. INDIA EUBBER. 59. For erasing pencil -lines and cleaning the paper 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. One end of the velvet rubber may well be wedge-shaped in order to erase lines without damag- ing 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 jjoint should rarelj' 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 laving out angles a graduated arc called a "protractor" is used. Various materials are employed in the manufacture of protractors, as metal, horn, celluloid, Bristol board and tracing paper. The two last are quite accu- rate enough for ordinary j)urf)0ses, although where the utmost precision is required, one of German silver should be obtained, with a moveable arm and vernier attachment. The graduation may advantageously be to half degrees for average work. To lay out an angle (say 40°) T\ath a protractor, the radius CH (Fig. 31) should be made to coincide with one side of the desu-ed angle; the centre, C, with the desired vertex; and a dot made with the jDencil opposite division numbered 40 on the graduated edge. The line MC, through this point and C, completes the construction. BRUSHES. 62. Sable -hah- brushes are the best for laj'ing 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 better adapted for tinting large surfaces. Reject any brush that does not come to a jjerfect point on being moistened. Five or six brushes of different sizes are needed. PRELIinXAEIES TO PRACTICAL WORE. 63. The first work of a draughtsman, like most of his later j)roductions, consists of line as distin- guished from brush work, and for it the jjajjer may be fastened ujjon the board with thumb-tacks only. 20 THEORETICAL AND PRACTICAL GRAPHICS. 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 jDoints, and the best system of pre- serving and recording drawings, tracings and corrections is apparently 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 suf- ficiently 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. A "Double Elephant" sheet being twice the size of a "Super Royal" divides equally well into plates of the above size, biit is preferable on account of its better quality. To lay out four rectangles -a])on the paper locate first 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 ■s'lg. ss. 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 placing a triangle against but pro- jecting beyond it, as in the upper left- hand portion of the figure. The T-rule being placed as shown, with its head at the left end of the board — the correct and usual jDosition ■ — draw a horizontal line X Y, 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 mn or tr, 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, as 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 JN and Y N, to the centre lines, at distances from them of 9" and 13" respectively, laid off from the centre, 0. An inner rectangle, as ab c d, should be laid out on each plate, with proper margins ; usually at least an inch at 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 z in the figure, as variables to which any con- venient 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 QY. mimimmmmiiimmmmfmmmmmm^^^ EXERCISES FOR PEN AND COMPASS. 21 CHAPTEIt IV. GEADES OF LINES. — LINE TINTING. — LINE SHADING. — CONVENTIONAL SECTION-LINING.— FREQUENTLY RECURRING PLANE PROBLEMS.— MISCELLANEOUS PEN AND COMPASS EXERCISES. 65. Several kinds of lines employed in mechanical drawdng are indicated in the figure below. 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. Lgr- 33. -t- CENTRE LINE. If red. CENTRE UlNE. If black. V. ! "^^ FOR ORDINhRY OUTLINES. V^ HIDDEl[l LINE I MEDIUM, Continuous. "DOTTED LI N E_ usually employed as a constructio n line. I *^ SHADE Jjt^ LIME "DOTTED LINE'I line of_motibn in Kinematic Geometry. When drawing on tracing cloth or tracing jjaper, for the purpose of making blue -prints, all the lines wiU preferably be black, and the centre and dimension lines distinguished from others as indi- cated above, as also by being somewhat finer than those employed for the light outlines of the object. Heavy, opaque, red lines may, however, be used, as they will blue -print, though faintlj-. There is at present no universal agreement among the members of the engineering profession as to standard dimension and centre Unes. Not wishing to add another to the systems already at variance, but preferring to facilitate the securing of the unifonuity so desirable, I have presented those for some time employed by the Pennsylvania Railroad and now taught at Cornell University. The lines of Fig. 33, as also of nearly all the other figures of this work, having been printed from blocks made by the cerographlc process (Art. 277), are for the most part too light to serve as examples tor machine-shop work. Fig. 80 is a sample of P. K. R. drawing, and is a fair model as to weight of line for working drawings. 22 THEORETICAL AND PRACTICAL GRAPHICS. A dash -and -three -dot Hue (not shown in the figure) is considerably used in Descriptive Geometry, either to represent an auxiliary plane or an invisible trace of any plane. (See Fig. 238). 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 l^e drawn 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 arroio 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 copyist when using the pen. On a comjjlicated 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 alreadj' been given during the process of construction. No unnecessary lines should l)e drawn, or any exceeding of the intended limit of a line if it can possibly 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 to be indicated as concealed. All lines of the same qualitj^ 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 inking may best be done at once, although for the sake of clearness, in making a large and complicated drawing, a portion — usuallj^ the nearest and visible parts — may be inked, the draw- ing cleaned, and the iDencilling of the construction lines of the remainder continued frem that j)oint. The inking of the centre, dimension and construction lines naturally follows the completion of the main design. a b 1 1 2 1 3 4 5 6 7 8 9 r^ 1 2 3 4 ^ 6 7 1 2 3 4 5 1 2 3 — ■ 1 2 3 4 5 i i i 1 1 1 2 3 4 ^t 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 examination of the numbered squares. Without eiatering into the theory of shadows at this point we may state briefly the " shoj) rule " for drawing shade lines, viz., right-hand and loiver. ] That is, of any i^air 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 lower if they run horizontally; always subject, however, to the proviso that the line of inter- section of two illumin- ated planes is never a shade line. 68. The conic section called the parabola fur- nishes another interesting exercise in i-uled lines, when it is represented by its tangeiits as in Fig. 35. The angle CA E may be assumed at pleasure, and on the finished drawing the numbers may SECTION-LINING. — LIJ^E-SH A DING. 23 ^'ig-. 3S- be omitted, being given here merely to show the law of construction. All the divisions are equal, and like numbers are joined. Some interesting mathematical properties of the curve will be found in Chapter V. 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. Use the 60° triangle for the heavy diagonals and parallels to them; the T-rule for the hor- izontals. Pencil the diagonals first but ink them last. 70. The even or flat effect of equidistant parallel lines is called line - tinting ; or, if repre- senting 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 cutting j^Iane; 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 always as- sumed to be transparent. Fig-. 37. Adjacent jDieces have the lines drawn in different directions in order to distinguish sufficiently between them. The curved effect on the semi - cylinder is evidently obtained Ijy prop- erly varying both the strength of the line and the spacing. 71. The difference between the shading on the exterior and interior H of a cylinder is sharply contrasted in Fig. 38. On the concavity the darkest line is at the top, while on the convex surface it is near the Isot- tom, and below it the spaces remain unchanged while the lines diminish. A better effect would have been obtained in the figure had the engraver begun to increase the lines with the first decrease in the space between them. IPig-- 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 inch would be best adapted to the ordinary range of work. Equal s^Dacing and not fine spacing 24 THEORETICAL AND PRACTICAL GRAPHICS. 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 produce 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 applic- able where blue -prints of the original are desired. The use of colors may, however, be advantageously 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. X-ig-. 3S. Last Iran. StEE Wr't. Iran. PZlTlTiL. 5. 5. Btandai^d ^ectiens, Brass. StnnE. Wand. CappEr. Brick. CONVENTIONAL SECTION-LINING. 25 rig-. -41. More natural effects can also be given by the use of colors, in representing the other 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 Venetian Red are well adapted for brick, and a wash of India ink having a tinge of blue gives a fair suggestion of masonry; although the actual tint and surface of any rock can be exactly represented after a little practice wdth the brush and colors. These jjoints will be enlarged upon later. (b) By section - lining with the drawing pen in the conventional colors just mentioned, a process giving very handsome and thoroughly intelligible results on the original drawing, but, as before, unadapted to blue -printing and therefore not as often used as either of the following methods. (c) By 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 Societj' of Mechanical Engineers to secure uniformit}^ Each draughting office seems at present to be a law unto itself in this matter. 74. As affording valuable examples for further exercise with the ruling pen the 'system of section - lines adopted by the Pennsylvania Railroad is presented on the opposite page. The wood section is an exception to the rule, being drawn free-hand, with a Falcon \>en. T'^s- -SiO. gy ^yr^y gf coutrasting free-hand with me- chanical work Fio;. 40 is introduced, in which COURSED RUBBLE MASONRY Light India Ink, RUBBLE MASONRY Light India Ink, the rings showing annual growth are drawn as concentric circles with the comi^ass. In Fig. 41 a few other sections appear, ■0^ selected from the designs of M. N. Forney and F. Van Vleck, and which are fortunate arrangements. 75. Figs. 42 and 43 are profiles or outlines of mouldings, such as are of frequent occurrence m 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 olalique lines are all jjarallel to each other, and ^ where ^±S- "is. ^ig-- -43. X'ig'. 4^- BRICK Venetian Red. CONCRETE Yellow Ochre. ^ '^ \ n ""' -H r 1" V '^ =-*' i 'visible throughout — of the same length. Their direction should h& chosen with reference to best ex- hibiting the peculiar features of the object. Ob^^.ously the view in Fig. 44 is the least adapted 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 i^rofiles for mouldings and line -shade them, after converting them into oblique views. As hints for such work two figures are given (47-48), taken from actual construction in wood. By setting a moulding vertically, as in Fig. 49, and projecting horizontally fi-om its points, a fi-ont view is obtained, as in Fig. 50. Fig-. '47'. ■FLs- -ae. 77. The reverse curves X'ig-. SI. M n /'o yy ^v^ / m „¥■ \ c, N on the mouldings may be drawn with the irregular curve, (see Art 68); or, if composed of circular arcs to be tangent to vertical lines, by the follow- ing construction : — Let 31 and iV be the points of tangency on the verticals Mm. and N n, and let the arcs be tangent to each other at the middle point of the line MK Draw 3In and Nm perpendicular to the vertical lines. The centres, c and c,, of the desired arcs, are at the intersection of 31 n and Nm by per- pendiculars to 31 N from x and y, the middle points of the segments of 31 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 st gives t, the darkest part of the cylinder. The concave portion beginning at o would be darkest at o and get lighter as it apjDroaches y. Flat parts are either to be left white, if in the light, or have equidistant Hues 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, ll^fc hexagonal nut. The front face, being parallel to the paper, receives an even tint. An inclined face in the light, as abhf, is lightest toward the observer, while an unillumined face tkdg is exactly the reverse. Notice that to give a flat effect on the inclined faces the spacing - out as also the change in the size of lines must be more gradual than when indicating curA-ature. (Compare with Figs. 46 and 50.) E-igr- S2, REMARKS ON SHAD IN G. — PLANE PROBLEMS. 27 If two or more illuminated flat surfaces are i:)arallel to the paper (as tgb 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 maj^ 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 theor^y when it involves a sacrifice of effect. For example, in Fig. 46 we are unable to see to the left of the (theoretically) lightest jjart of the cylinder, and find it, therefore, advisable to move the darkest part past 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 from this point the student will make considerable use of the compasses, a few of the more important and frequentty 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 s^'nthetic geometry or work on plane problems. All the problems given (except No. 20) have proved of value in shoji 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 jmint, as A (Fig. 53), use the tri- angles, or triangle and rule as previously described ; or lav off equal distances Aa^ Ab, and with a and b as centres draw arcs ost, msn, with common radius greater than one -half ab. The required perpen- dicular is the line joining A with the intersection of these arcs. ., E"igr- ss- 82. Prob. 2. To bisect a line, as M N, use its extremities exactly as a and b were employed in the preceding construc- tion, getting also a second pair of arcs (same radius for all the arcs) intersecting abwe the line at a point we may call x. The line from s to x wiU be a bisecting perpendicular. m- 83. Prob. 3. To bisect an angle, as AVE, (Fig. 54), lay off on its sides any equal distances V a, x^ig-- s^. T' b. Use a and b as centres for intersecting arcs having a common radius. Join T^ with x, the intersection of these arcs, for the bisector required. 84. Prob. If. To bisect an arc of a circle, as a m b (Fig. 54), bisect the chord acb by Prob. 2; or, bj^ Prob. 3, bisect the angle aVb which subtends the arc. 85. Prob. 6. To construct an angle equal to a given angle, as 6 (Fig. 55), draw any arc a b with centre 0, then, with same radius, an indefinite arc m B, ^^s- ss. centre V; use the chord of a 6 as a radius, and from centre B cut the arc m B at x. Join V and x. Then angle AVB equals 6. 86. Prob. 6. To pass a circle through three points, a, b and c, join them by lines ab, be, bisect these lines by '^'^ perpendiculars, and the intersection of the latter will be the centre of the desired circle. 28 THEORETICAL AND PRACTICAL GRAPHICS. 87. Prob. I . To divide a s'igr- se. line into any nimiber 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. "W^ith a scale point off on ^ C as many equal parts (size immaterial) as are required on AB; four, for example. Join the last point of division (4) with B; then 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 .4 -B be turned to the left about A the point B will approach A, and the line will pass- through A C and other secant positions. When B reaches and coincides ^^s- s'r. with A the line is said to be tangent to the curve. (See also Art. 368.) A tangent to a mathematical curve is determined by means of known properties of the curve. For a random or graphiccd curve the method illus- trated by Fig. 57 (a) is the most accurate and is as follows : Through T, the point of desired tan- gency, draw random secants to points on either side of it, as A, B, D, etc. and prolong them to meet a circle having centre T and any radius. On each secant lay off— from its intersection with the circle— the chord of that secant in the random curve. Thus, am=TA; hn=TB; pcl=TD. From s where the curve m n o p q cuts the circle, draw s T, which will be a tangent, since for it the chord has its minimum value. A normal to a curve is a line 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 perpendicular to this radius at its extremity will be the required tangent. Solve with triangles. 90. Prob. 9. To draio a tangent to a circle from a point without join the centre C (Fig. 58) with the given point A; describe a semi- circle on AC as a diameter and join A with D, the intersection of the arcs. ADC equals 90°, being inscribed in a semi -circle; AD is then the required tangent, being perpendicular to CD at its extremity. 91. Prob. 10. To draio a tangent at a given point of a circidar ^ ^ iFig-- ss. arc whose centre is unknoivn or inaccessible, locate on the arc two points equidistant from the given point and on opjDOsite sides of it; the chord of these ijoints will be parallel to the tangent sought. 92. A regular polygon has all its sides equal, as also its angles. If of three sides it is called the equilatercd 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 imjaortant regular j)olygons are as follows : triangle, 120°; square, 90°; pentagon, 72°; hexagon, 60°; octagon, 45°; decagon, 36°; dodecagon, 30°. The angle at the vertex of a regular polygon is the supplement of its central angle. 93. For the polygons most frequently occurring there are many special methods of construction. All but the pentagon and decagon can be readily inscribed or circumscribed about a circle by the use of the T-rule and triangle. 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 a square, acb d. The same triangle in two positions- would give ef and gh, whence ag, g c, etc., would be sides of a regular octagon. PLANE PROBLEMS. 29 94. The 60 ° triangle used as in Art. 51 would give -the hexagon ; and alternate vertices of the latter, joined, would give an equilgiefal triangle. Or the radius ?)f ,J;he circle stepped off six times on the circumference, and alternate'' iDoints connected, would result similarly. 95. Prob. 11. An additional method for inscribing an equilateral triangle in a circle, ivhen one vertex of the triangle is given, as A, Pig. ^60, is to draw the diameter, A B, through A, and use the triangle to obtain the sides A C and A D, making angles of 30 ° with A B. D and C ^\•ill then be the extremities of the third side of the triangle sought. 96. Prob. 12. To inscribe a circle in an .equilateral triangle draw a perpendicular fi-om any vertex to the oisposite side. The centre of the circle will be on such line, two -thirds of the distance fi-om vertex to base, while the radius desired will be the remaining third. (Fig. 61). 97. Prob. IS. To insaibe a circle in any triangle bisect any two of the interior angles. , The intersection of these bisectors will be the centre, and its perpen- dicular distance from any side will be the radius ~of the circle sought. 98. Prob. H. To inscribe a pentagon in a circle draw mutually perpendicular s'lg-- es. diameters (Fig. 62) ; bisect a radius as at s ; draw arc a x of radius s a and centre s; then chord ax=af, the side of the pentagon to be constructed. 99. Prob. 15. To construct a regular polygon of any number of sides, the length of the side being 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 as radius and divide by trial into x (or 9) equal parts. Join B with x — 2 points of division, or seven, beginning at A, and prolong all but 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 (pro- duced) and so obtain a third vertex. This solution is based on the familiar principles (a) that if a regular 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 with the sides meeting at that vertex. 100. Prob. 16. Another solution of Prob. 15. Erect a perijendicular HR (Fig. 64) at the middle point of the given side. With iJf as a centre, radius MS, describe arc SA and divide it by trial into six equal parts. Arcs through these points of di\'ision, using ^ as a centre, and numbered up from six, give the centres on the ver- tical line for circles passing through M and S and in which MS would be a chord as many times as the number of the centre. 101. For any unusual number of sides the method S'igr- S3- 30 THEORETICAL AND PRACTICAL GRAPHICS. PLANE PROBLEMS. 31 by "trial and error" is often resorted to, and even for ordinary cases it is by no means to be despised. By it the di\'iders are set " by guess " to the probable chord of the desired arc, and, sup- posing a heptagon wanted, the chord is stepped off seven times around the circumference; care being taken to have the points of the di^dders come exactly on the arc, and also to avoid damaging the paper. If the seventh step goes past the starting point the di^dders require closing; if it falls short, the original estimate was evidently too small. Obviously the change in setting the di^dders ought in this case to be, as nearly as j^ossible, one -seventh of the error; and after a few trials one should " come out even " on the last step. 102. Proh. 17. To lay off on a given 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; ts the given straight line and tangent to the circle; tm equal one -fourth of ts, and sx drawn with centre m, radius m-s. Then the length of the arc tx is a close approximation to that of the line ts. 103. Prob. IS. 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 ai, centre i. Then fe approximates closely to the length of the arc af. 104. Proh. 19. To obtain a straight line equal in length to any given semi- circle,'' draw a diameter oh of the given semi -circle (Plate I, Fig. 2) and a radius inclined at an angle of 80° to the radius ch. Prolong the radius to meet the line b h k, drawn tangent to the circle at h. From k lay off the radius three times, reaching n. The line no equals the semi -circumference to four places of decimals. 105. Prob. SO. To drcvw 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 appHcations. Fig. 4 (Plate I) illus- trates the geometrical principles involved, and in it a circle is required to contain the points s and 1 These -metliods of approximation were devised by Prof, iiaukine. They are sufficiently accurate for arcs not exceeding 60°. The error varies as the fourth power of the angle. The complete demonstr.ition of Prob. 17 can be found in the Philo- sophical Magazine for October, 1867, and of Prob. 18 in the November issue of the same year. 2 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 Llpsiae, 16So. The demon- stration is as follows ; Calling the radius unity, the diameter would have the numerical value '2. Then in Fig. 2, Plate I, we have on = y/oh- -f lin- = \/ok- + {kn — k/i)- = \/4 -)- (3 — tan 30°)2 = 3.14159 -f 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 freguent occasion to use these functions that they are given here for reference, both as lines, and as ratios. Trigonometric Functions as Eatios. Trigonometric Functions as Lines. e = the given angle = CA B h = hypothenuse of triangle CAB a = A B = side of triangle adjacent to vertex of 9 o = £C=side of triangle opposite to e Then sin . 9 = j- ; cos i — tt ; Cu-tangent of Q tan e = cotan Q - sm I ' cos I : reciprocal of cosine. cos I ' bin ( : reciprocal of tan 9. A'«^ B The prefix "co" suggests "complement;" the co-sine of B is the sine of the complement of 5, &c. As lines the functions- may be defined as follows : The sine of an arc (e. g., that subtended by angle 6 in the figure) 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 extended radius, from the centre of the circle to the tangent, is called the secant of the arc. 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. a and be tangent to the line mv■^. Draw first any circle containing s and «, as the one called "aux. circle." Join « to a and prolong to meet mv^ at k. Prom 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 k s a, secant. But k i = k g, therefore k i^ ^=k s X k a, which makes k i 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 given 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, a, k and VI V I 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 lo, shorter than the first by the distance w 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 ijerpendicular 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; the remaining construction lines m blue. 106. Prob. 21. To draw a, tangent to two given circles (a problem that may occur in connecting ^'igr- ev. band-wheels by belts) join their centres, c and o, (Fig. 67) and at s lay off s m and s n each equal to the radius of the smallei'- 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 and c h o are each 90 °, being inscribed in a semi-circle ; and c ^ is parallel to a b, which last is one of the required tangents ; while c h is parallel to t x, a second tangent. Two more can be similarlj^ found. To unite two 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 m n meet at d, from which perpendiculars to the given lines give the points of tangency of the required arc, which is now drawn with the given radius. g n 108. Prob. 23. To drcm through a •given point a line lohich icill — if produced — pass through the inaccesssible f ll *Tliis solution is taken from Benjamin Alvord's Tangencies of Circles and of Spheres^ published by the Smithsonian Institute. Tliat valuable pamphlet presents geometrical solutions of tbe ten problems of Apollonius on tbe tangencies of circles, and also ■of tbe fifteen problems on the tangencies of spheres, all of which are valuable to the draughtsman, both geometrically and as ■exercises in precision. The solutions are based on the principle, illustrated by Fig. 57, that the tangent line or tangent curve is the limit of all secant lines or curves. The problems on the tangencies of circles are as follows, the number of solutions in each case being given : (1) To draw a circle through three points. One solution. (2) Circle through two points and tan- igent to a given straight line. Two solutions. (3) Circle through a given point and tangent to two straight lines. Two solu- tions. (4) Circle through two points and tangent to a given circle. Two solutions. (5) Circle through a given point, tangent to a given straight line and a given circle. Four solutions. (6) Circle through a given point and tangent to two given ■circles. Four solutions. (7) Circle tangent to three straight lines, two only of which may be parallel. Four solutions. (8) Circle tangent to two straight lines and a given circle. Four solutions. (Art. 105, above). (9) Circle . tangent to two given circles and & given straight line. Eight solutions. (10) Circle tangent to three given circles. Eight solutions, PLANE PROBLEMS.— TAPERING CIRCULAR ARCS. 33 intersection of two lines. Join the given point e with any point ,/' on A B and also with some point g on CD. From anj' point h on A B draw h i parallel to / g, then i k parallel to g e and h k parallel to f e. The line k e will fulfill the conditions. ~4. To drmo an oval upon a given line. Describe a circle on the gi\-en line, m n, (Fig 70) as a diameter. 'With m and n as centres describe arcs, m .r, n x radius »i a. Draw m v and n t through v and t, the middle points of the quadrants y in, y n. Then m s and n r are the portions of m X and n x forming part of the oval. Bisect n c at q and draw q x. Also bisect c q at z and join the latter with x. Bisect y h in d and draw f d fi-om /', the intersection of n s and q x. Use / as a centre and f s as radius for an arc sk terminating on f d. The intersec- tion, h, of If with .1- : is then the next centre and /; k the radius of the are /.■ I which terminates on h y produced. The oval is then completed with ;/ as a centre and radius y I. The lower portion is symmetrical with the upper and therefore similarly constructed. 110. Where exact tangencj' 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 thev' thus hope to conceal. With such instruments of precision as the draughts- man employs there is but little, if anj', excuse for overlapping or falUng short. ^^s- ''3.. i^ 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, evidentlj', 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. ^Mlen either a tube or a solid cylindrical piece is seen in the direction of its axis the outline is, evidently, simply a circle; and often the only way to determine which of the two the circle rei^resented would be to notice which part of said end view was represented as casting a shadow. In Fig. 72, if the shaded arcs can cast shadows, the space inside the circles must be open, and the fig- ure would represent 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 soUd, cylindrical jiins 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-jjen and after making one complete circle add the extra thickness by a second turn which is to begin with the pen-point in the air, the jjen being brought down gradually upon the paper and then, while turning, raised fi-om 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 fi-om 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 passes through the extremities of the taper. 111. ^^©i^ei^a ^ 34 THEORETICAL AND PRACTICAL GRAPHICS. 112. As an exercise in concentric circles Fig. 73 will prove a good test of skill. It is a fair representation of a gymnasium ring, the " annular torus " of mathematical works, and possessing some remarkable jiroperties, chief among which is the fact that it is the only surface of revokition known from which circles can be cut by three different systems of planes.* ng-. 7-4. E^igr- vs. Fitr X, / TORUS \ / C/ la y \D ip If 113. In Fig. 74 the same surface is shown, in the centre, in outhne only. The axis of the surface would be a perpendicular to the paper at A. If MN represents a plane peirpendicular to the paper and containing the axis, then Fig. X will show the shape of the cut or section. 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 by revolution about an axis one of its characteristics is that any plane perpendicular to the axis will cut it in a circle. The circles of Fig. 73 may then be, for the moment, considered as parallel cuts by a series of planes perjsendicular 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 iv respectively. * Olivier, Memoires de Geometrie Descriptive. Paris, 1851, . ANNULAR TORUS.— WARPED HYPERBOLOID. 35 A plane perpendiciilar to the paper, on P Q, n'ould be a bi-iangent plane, because tangent to the surface at two points, P and Q; and such plane would cut two over-lapjDing circles from the torus, each of them running partly on the inner and partly on the outer portion of the surface. For the proof that such sections are circles the student is probaljly 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 planes parallel to, but not containing the axis, cut it in a set of curves called the Cassian ovals, of which the Lenniiscate of Art. 158 is a special case, and which would result from 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 e/i/and DNr, the form shown in Fig. Y would be 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 piece must either have uniform outlines or be shaded with several lines. Thus, in Fig. 76, A would represent an angular piece w^hile B would indicate a circular cylin- der ; if elliptical its section would be drawn at one side as shown. 116. Before jjresenting the crucial test for the learner — the railroad rail— two additional practice exercises, mainly in ruling, are given in Figs. 77 and 78. The former shows that, like the parabola, the circle and hyperbola can be rejDresented by their envelojjing tangents. The upjDcr and lower figures are merely two views of the surface called the warped hyperboloid, from the hyperbolas which constitute the curved outlines 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 cylindrical or conical to the hyijerboloidal 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 hyperboloid is of more interest mathematically than mechanically. Begin the drawing by iDcncilling 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 u^jper figure, and draw H' M' 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 1', 2', 3', 4', etc. All points on the larger circle P QR are similarly projected to P' R'. The extremities of the same tangent are then joined in the ujjper 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 e\-ident from inspection ; while 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 ciuantily, 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. a(«)2(n) Urt) 32 31 TAPERING LINES. — RAIL SECTIONS. 37 ■FLs- 'na- 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 afhn, then two circular arcs which would con- taiii a and b if extended, and whose greatest distance ft-om the original circle is x, (arbitrary). Six- teen equidistant radii as at a, t, d, etc., are next m 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 /Vo?7i the centre, not through or toward it. All construction lines should be erased before the tapering lines are filled in. The " filling in " may be done very rapidly by ruling the edges of the line in fine at first, then opening the pen slightly and beginning again whei-e 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 jjen or a fine-pointed sable-brush, between the now heavy edge-lines of the tapier. 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 tangents. 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 Unes 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-page, the rail being represented in oblique projection and shaded. Begin Fig. 80 by drawing the vertical centre-hne, 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 II" 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 -f^" 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 iveh and is y^" 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 Ije 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 proper relations to each other of the sections of railway wheels and rails. There was quite general agreement as to the following recommendations: a top radius of twelve inches; a quarter-inch corner radius; vertical sides s'igr- eo- I 2^ IN. 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.— CONIC SECTIONS.— HOMOLOGICAL PLANE CUEYES AND SPACE-FIGURES.— LINK-MOTION CURVES.-CENTROIDS.— THE CYCLOID.— COMPANION TO THE CYCLOID.- THE CURTATE TRO- CHOID.-THE PROLATE TROCHOID.— HYPO-, EPI-, AND PERI-TROCHOIDS.— SPECIAL TROCHOIDS— ELLIPSE, STRAIGHT LINE, LIMA^'ON, CARDIOID, TRISECTRIX, INVOLUTE, SPIRAL OF ARCHI- MEDES. -PARALLEL CURVES— CONCHOID.— QUADEATRIX.— CISSOID.— TEA CTRIX.— WITCH OP AGNISI.-CARTESIAN OVALS.— CASSIAN OVALS.- CATENARY.— LOGARITHMIC SPIRAL.— 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 Hjq^erbola ; Link-motion curves or point-paths ; Centroids ; Trochoids ; the Involute and the Spiral of Archimedes. Of less practical importance, though ecjually 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 sjDace 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 facility in the use of the curved ruler. THE HELIX. 120. The ordinarj' helix is a curve which cuts all the elements of a cyhnder at the same angle. Or we maj' define it as the curve which would be generated by a point having a uniform motion around a straight hue combined with a uniform motion parallel to the line. X'ig-. SI. ffluuuuuuuuumuumuui The student can readily make a model of the cylinder and helix by drawing on thick paper or Bristol-board a rectangle A" B" 0" D" (Fig. 81) and its diagonal, D" B" ; also equidistant elements, as m" 6," n" c" etc. Allow at the right and bottom about a quarter of an inch extra for overlapping, as shown by the lines x y and .s z. Cut out the rectangle z x ; also cut a series of vertical slits between D" C" and zs and turn the divisions up at right angles to the paper; put mucilage between B" C" and x ij ; then roll the paper up into cylindrical form, bringing A" D" t" ft" in front 40 THEORETICAL AND PRACTICAL GRAPHICS. of and upon the gummed i^ortion, so that A" D" will comcide with B" C". The diagonal D" B" will then be a helix on the outside of the cylinder, but half of which can be seen in front view, as D 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 ujd 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 wi" 6", n" c", etc., of the development stand vertically &i b, c, d . . . . g of the half plan, and are seen in the elevation at m' b\ n' d, d d', etc. The point 3', where any element, as c', cuts the helix, is evidently as high as o" where the same point appears on the development. We may therefore get the curve ff 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" C" obviously equals 2 tt r, in which r = D. The height A' ly, 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 ijrojection 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 hyjDerbola 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 ellipse, parabola and hyperbola are curves in which there is a constant ratio between the distances of laoints on the curve fi:om 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 A' 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 from 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 C and D E a,s representing the extreme length and width, the points F and Fj^ ifoci) 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 Pj, and a string, of length A B, with ends fastened at the foci, complete the preUminaries. 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 elUpse may be defined as a cvirve in which the sum of the distances from any point of the curve to tivo 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 perperL- dicular to the other. With the comijasses 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 ^1 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 enabled to construct the parabola and hyperbola 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 any point B, by putting a pencil at B, then the remainder B X swung around and the end fastened at F, it is then, evidently, as far from B to F 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. ^ig-- ©2. 127. For the hyperbola, (Fig. 82), the construction is identical with the i^receding, 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 jjarabola are 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 A, parallel to O E, would be the directrix. 129. Like the ellipse, the hyperljola 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+FiP=FD + 42 THEORETICAL AND PRACTICAL GRAPHICS. F^D = A C (the transverse axis), it is the difference of the radii ^'^s- sa. that is constant for the hyperbola. In Fig. 83, if J. -B is to be the transverse axis of the two arcs, or "branches," which make the comijlete hyperbola, then using p and p to represent any two focal radii, as F Q and Fi Q, or FR and F^ R, we will have p — p = A B (the constant quantitjO- To get a point of the curve in accordance with this jsrin- cii^le 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 E, as a radius, and F-^ as a centre to cut the first arc at R. The same radii will evidently determine three other jioints fulfilling the conditions. 130. The tangent to the hyperbola at any point, as Q, bisects the angle F Q F^, between the focal radii. In the ellipse, (Fig. 82), it is the external angle between the radii that is bisected by the tan- gent. In the jaarabola, (Fig. 82), the same principle applies, 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 vieiuecl obliquely. If ARMBF (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 appear of its original length. In the rotation supposed, all points not on the axis would describe circles about it with their planes jDerpendicular to it. M, for example, 1 would move in the arc of a circle part of which is shown in M M-^ , 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 upon 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 front 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, as ^ -B and C D, use them as diameters of concentric circles and from their common centre, 0, draw random radii, as M, T, OK. Where any radius, M, meets the outer circle, drojD a perjaendicular M M^ to the longer axis, to meet a line m M^ , parallel to the same longer axis and passing through m, where M cuts the smaller circle. 132. If rS is a tangent at T to the large circle, then when T appears at T^ we shall have T^ S as a, tangent to the ellipse at the point derived from T ; S having remained constant, being on the axis of rotation. CONIC SECTIONS. 43 Similarly, if a tangent at R^ were wanted we may first find r, corresponding to R^; draw the tangent- ?•/ 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 elhpse, ^ig-. as. making oblique angles with each other. Such diameters N are called conjugate and the curve may be constructed iTpon them thus: Draw the axes TD and H K ai the assigned angle D H; construct the parallelogram M N X Y ; divide D M and D into the same number of equal parts ; then irom K draw lines through the points of division on D 0, to meet similar lines through H and the divisions on D M. The intersection of like nimibered lines will give points of the elliiDse. 134. It is the law of expansion of a j^erfect gas that the volume is inversely as the pressure. That is, if the volume be doubled the pressure drops one-half; if trebled the pressure becomes one- third, etc. Steam not being a perfect gas departs somewhat from the above law, but the curve in- dicating the fall in pressure due to its expansion is compared with that for a perfect 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 h indicate the ^^er- ss. pressure before expansion begins. If the piston be forced ahead by the expanding gas until the volume is doubled, the pressure will drop, by Boyle's law, to one-half and will be indicated by t d. For three volumes the j)ressure 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 be doubled, or a pressure so smaU that it could not be halved, it is evident that theoretically the curve c x and the line G K will forever ajjproach 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, hyperbola and parabola, yet the same kind of -tr sections can be cut from an oblique or scalene cone of circular l^ase, as T'! A B, Fig. 87. Two sets of circular sections can also be cut from such a cone, one set, obviously, by planes parallel to the base, the other by planes like CD, 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 parallel to the base — intersect in a line perpendicular to the paper at o. This iine pierces the front surface of the cone at a point we may call 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 o r'' = o m X o n. But fi-om the '^^ similar triangles in Fig. 87 we have o m : o y :: o x : o n whence oy X o x = o mX on = or', thus showing that the section xy ia 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. iF-igr- es. o 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 due 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 j^ 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 plane of projection; A B j. 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-contrary section of the visual cone E. A B, as the student can easilj' prove. 138. We now take up the conic sections as derived from a right cone. A complete cone (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 be (a) Equal to that of the elements,* therefore jiarallel to one element, giving the parabola, as M Q l/j (Fig. 90) ; the plane k q M being parallel to the element V U and therefore maktag with the base the same angle, 0, as the latter. (b) Greater than that of the elements, causing the plane to cut both nappes 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- si- 139. Figures 90 and 91, with No. 4 of Plate 2 are not only stimulating examples for the draughtsman Ijut 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 spihere. To establish this we need only emjjloy the well known princijjles that (a) all tangents fr'om a jjoint to a sphere are equal iu length, and (b) all tangents are equal that are drawn to a sphere fr-om jjoints equidis- tant from its centre. In both figures all jjoints of the cone's bases are evidently ec[uidistant fr-om the centres of the tangent spheres. 140. On the ujai^er naj^ije, (Fig. 90), let S H he the circle of contact of a sphere which is tan- gent at F^ to the cuttmg i)lane P L K. The plane P H^, of the cfrcle cuts the plane of section in P m. If D is any point of the curve D J E, J another point, and we can jjrove the ratio constant (and greater than unity) between the distances of D and J fr-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 / S are equal, being tangents to the sphere from the same point. We have then the proportion XS:GR::.IS:JR or D F^: D Z:: J F^: J R. Since / S and its equal .7 F^ are greater than / R, and the ratio J F^ to J R is constant, the proposition is established. 141. For the jiarabola on the lower napj^e, since the plane Mek is inclined at the angle & made by the elements, we have Q A= Q B (opposite 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 fr-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 iVT as the lines to be i^roven 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 containiiag s. This element cuts the contact circles of the spheres in a and A. A plane through the cone's axis and jiarallel to the paper would contain o t, o v and V n. Prolong v oi to meet a line V B that is parallel to K H. Join E 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:87-:: Vn: V R, which ratio is less than unity; therefore a is a point on an ellipse.* The ijlane of the intersecting lines V a and R r cuts the plane M iV 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. S2. * Schlomilcli, Geometrie des Maasses, 1874. CONIC SECTIONS AS HOMOLOGOUS FIGURES. 47 144. The proof that KsH (Fig. 91) is an elliiDse when the curve is referred to two foci is as follows : KF= K m ; KF, = Kt; therefore K F + K F, = K m + Kt= t m = x n = S K F + F F, = '2HF+FF,; i. e., K F = H F,. Since s F= s a and s F^ — s A we have s F + s F^ = s a + s A = A a = im = x n = H K. The sum of the distances from 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 by the methods of Pro- jective Geometry. (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 C^. For ^3 as the centre of projection the figure A^ B^ C[ corresponds to A^ B^ C^. If we join S, with S^ and prolong to 0, to meet the plane of the figures A^ B^ C, and A.^ B^ C„ we find that the jjoint sustains the same relation to these two figures that S^ and S„ do to the figures just taken in connection with theni. Using the technical term trace for the intersection of a line with a jjlane or of one plane with another, we see that A^ c^ is the trace, on the vertical j^lane, of any plane containing A^ B^. This plane cuts the " axis of homology," t^ m, in c^. As A^ B^ lies in the jjlane of S^ and A^ 5', and in the horizontal j^lane as well, it can only meet the vertical plane in c„, the point of intersection of all these jjlanes. Similarly we find that A^ C" and A^ C^, if prolonged, meet the axis at the same point; correspondingly B'^ C- and B^ C^ meet at a„. But A^ B'^ 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.^- ^i-s- ©3. Finally, the line A^ A^ being the hor- izontal trace of the plane determined by the hnes 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^ into the same relation with that A^ and A' sustain to S^; or that of A^ and A^ to »S'i. 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, ^Ij JBi C, , 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 centj-e of homology. 146. Had Ai B^ C, been a circle, and all its points joined with S^, then from what has preceded we know that A^ 5' C would have been an ellipse; as also would have been the case were A.^ B^ C„ a circle used in con- nection with (Sj. But our conclusions should enable us to substitute a circle for A^ B^ C„ and using 48 THEORETICAL AND PRACTICAL GRAPHICS. in the same plane with it get an ellipse in place of the triangle A^Bi Cj. 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 jDarallel 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 " vanishing line (a)," parallel to the axis. This contains the projections, A and B, of the f)oints at infinity where the lower i^lane may be considered as cutting the elements SA and SB. Were S and the shaded f)lane 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 jjoint at infinity; the latter, two. 148. Raise the centre S so that the vanishing line does not cut the base and, evidently, no Hne from S to the base would be parallel to the lower plane; but the latter would cut all the elements on one side of the vertex, giving the ellipse. 149. Bearing in mind that the projection of the circle A H B t is on the lower plane produced, 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 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. 4& 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 oljtain the hyperbola by projection of a circle from a point in the plane of the latter we would require simply a secant vanishing line, MX (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 ky that corresponds to P K must also meet the axis at i/. P is analogous to S ^ of Fig. 93 in that it meets its corresponding line at infinity, i. e., is parallel to it. Therefore y k, parallel to P meets the ray K at k. cor- responding to K. Then A' joined with an}' other point R gi\-es K z. .Join z with the point k just obtained aiid jKolong R io 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 jjoint which, like all points on the axis, corresjjonds to itself. From that point the projection of that tangent on the lower i^lane would be parallel to S B, since they are to meet at infinit\'. Or if S J is parallel to the tangent at B then ./ will be the projection of J' at infinity, where SJ meets the tangent; / will be therefore one jDoint of the projection of said tangent on the lower plane;, while another point would be, as previousl}' 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 M is parallel to the tangent at infinity, that is to the nsymptoie (see Art. 134) through F; while the other asymptote is a parallel through E to X. 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 MUX, 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 Ijranch,- join H with X (the original of x) and get its intersection, r/, with the axis. Then x g h corresponds to g X H, and the ray OH meets it at h, the pirojection 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 correspioncling lines meet at the same point in a plane, instead of the same point on a line. A vanishing pAanc takes the jilace of a \-an- ishing line. The figure that is in homology with tlie original figure is called the relief-per-fpective of the latter. (See Art. 11.) Remarkafily beautiful effects can be obtained 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 ii-om a photograph of a model designed Ijy Prof. L. Burmester. Although not always requiring the use of the irregular curve and tlierefore not strictly the material for a to})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-jiersijective. 155. In Fig. 96 the plane P Q is called the plane of homology or pidure-plane, and — adopting Cremona's notation — we will denote it by tt. The vanishing plane, M N, or ^', is parallel to it. is ■Fi-s- ©5- 50 THEORETICAL AND PRACTICAL GRAPHICS. the centre of homology or perspective-centre. All points in the plane n are their own perspectives or, in other words, corresimid to themselves. Therefore B" is one point of the projection or perspective of the line A B, being the intersection of 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 tt in B" n, which is an axis of homology for AB and A' B', exactly as mm in Fig. 92 is for A, B, and A, B,. Fig-- S'7- A C->i° \-- As D C is parallel to A B (Fig. 96) a parallel to it through is again the line v. LINK-MOTION CURVES. 51 The trace of D C on -n- is C". Joining v with ('" and cutting v C" by rays D, C, obtains IXC in the same manner as A' B' was derived. The originals of A' B' and C Z)' are parallel lines; but we see that their reliet-perspectives meet at v. The vanishing plane is therefore the locus* of the vanishing points of lines that are parallel on the original object; while the plane of homology is the locus of the axes of homology of corresponding lines ; or, differently stated, any line and its relief-persiJective will, if produced, meet on the plane of homolog}'. 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 tlie planes on which the views are made. The planes tt and <^' are interchanged, as compared with their positions in Fig. 96, and they are seen as lines, lieing assumed as perpendicular to the paper. The relief-perspective appears between them, in plan and elevation. The lettering of A B and D C, and the lines employed in getting their relief-perspectives, being- identical with the same constructions in Fig. 96 ought to make the matter clear at a glance to all who have mastered what has preceded. Burmester's Grundzi'igc der Relief- Perspective and \\'iener's Darstellende Geomctrie are valuable reference works on this topic for those wishing to pursue its studj^ further; ).)ut 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 \'aluable to the engineer or architect since it illustrates more fully the practical application of central projection to Graphical Statics. LINK-MOTIOX CURVES. 167. Kineiiiatics is the science which treats of j^ure motion, regardless of the cause or the results of the motion. It is a i^urely kinematic problem if we laj' out on the drawing-board the path of a point on the connecting-rod of a locomotive, or of a point on the jDiston of an oscillating cylinder, or of any point 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 siaecial name. Others, whose mathematical properties had already been investigated and the curves dignified with names, it was later found could be mechanically traced. Among these the most familiar examples are the Ellipse and the Lemniscate, the latter of which is employed here to illustrate the general problem. The moving pieces in a mechanism are rigid and inextensible and are always under certain conditions of restraint. " Conditions of restraint " may be illustrated by the familiar case of the con- necting-rod of the locomotive, one end of which is alwa3^s attached to the driving-wheel at the crank-pin and is therefore constrained to describe a circle about the axle of that wheel, while the other end of the rod must move in a straight line, being fastened by the "wrist-pin" to the "cross- head," which slides between straight " guides." The first step in tracing a point-path of anj^ mechanism is therefore the determination of the fixed points and a general analysis of the motion. ==■ Locus is the Latin for place; and in father untechnical language, although in the exact sense in which it is used mathe- matically, we may say that the Locus of points or lines is the place where you \n-Ay expect to find them under their conditions of restriction. For example, the surface of a sphere is the locus of all points equidistant from a fixed point (its centre). The locus of a point moving in a plane so as to remain at a constant distance from a given lixed 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 link, 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 N^ P, with the exception of its 2 MN _2 MS 3 THE LEMNISCATE AS A LINK-MOTION CURVE ■extremities, have a compound motion, in curves whose form it is not easy to jaredict 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 iVP is constant in length, if N were moved along to E the point P would have to be at a distance iV P irom 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 /; , another point of the Lemniscate, and to which Z has then moved. All other positions are similarly found. If the motion of N is toward D it will soon reach a limit, A, to its fm-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 ^1. In this position any movement of P either side of a will drag N back over its former path ; and unless P moves to the left, past a, it would also retrace its path. P reaches a similar ■" dead point " at v. To 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 proportions of the links, the point-paths would be correspondingly affected. INSTANTANEOUS CENTR ES. — CENTRO IDS. 53 By tracing the path of a point on P N' produced, and as far from N as Z is from P, the student will ol>tain an interesting contrast to the Lemniscate. If M and S were joined liy 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. INSTAXTAXEOUS CEXTEES. — CEXTROIDS. 159. Let us imagine a l.ioy al>out 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 l)y the end of the sling, if the day is bright. The Dy boy moves with respect to the earth ; his hand moves in relation to himself, and the end of the sling describes a circle about his hand. The last is the only definite element of the three, j'et it is sufficient to simplify otherwise difficult constructions relating to the complex curve which is described relatively to the earth. 54 T.HEORETICAL AND PRACTICAL GRAPHICS. A tangent and a normal to a circle are easily obtained, the former being, as need hardly be stated at this point, perpendicular to the radius at the point of tangency, while the normal simply coincides in direction with such radius. If the stone were released at any instant it would fly off in a straight line tangent to the circle it was describing about the hand as a centre; but such line would, at the instant of release, be tangent also to the compound curve. If then we wish a tangent at a given point of any curve generated by a point in motion, we have but to reduce that motion to circular motion about some moving centre; then joining the point of desired tangency with the — at that instant — position of the moving centre, we have the normal, a perpendicular to which gives the tangent desired. A centre which is thus used for an instant only is called an instantaneous centre- ISO. 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 positions 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' £', A having moved over A A' and B over B B'. Bisecting A A' and B B' by perpendiculars intersecting at 0, and drawing A, A', OB, and OB', we have A A ' = 0^ = B B', and evidently a point about which, as a centre, the turning of A B through the angle ^i would have brought it to A' B'. Similarly, if 'the next position in which we find AB is A" B", we may find a point s as the centre about which it might have turned to bring it there; the angle being 0.,, probably different from 0^. The points n and m are analogous to and s. If s' be drawn equal to s and making with the latter an angle 0^ , equal to the angle A A', and if s were rigidly attached to A B 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 polygons ; the one corresponding to s n m being called the fixed, the other the rolling centroid. The perjDen- dicular from upon 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 /j, 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 infinitesimal turning aljout T would move '' link (a) " exacth^ as under the old conditions. B}' taking "link (a)" in all possible positions and, for each, prolonging the radii through its extremities, the points of the fixed centroid are determined. Inverting the combination so that "moving link (a)" and its opposite are interchanged, and proceeding as before, gives the points of ■"rolling centroid (a)." These centroids are branches of hyperbolas having the extremities of the long links as foci. By holding a short link stationary, as "fixed link (b)," an elliptical fixed centroid results; "rolling centroid (b) " being obtained, as before, by inversion. The foci are again the extremities of Ihe fixed and moving links. Obviously the curved pieces represented as screwed to the links would not be emploj^ed in a practical construction, and they are only introduced to give a more realistic effect to the figure and jjossibly thereby conduce to a clearer understanding of the subject. 164. It is interesting to notice that the Lemniscate occurs here under new conditions, being "traced by the middle point of "moving link (a)." The study of kinematics is both fascinating and profitable, and it is hoped that this brief glance -at the subject may create a desire on the part of the student to pursue it further in such works as Reuleaux' Kinematics of Machinery and Burmester's Lehrbuch der Kinematik. Before leaving this topic the important fact should be stated, which now needs no argument to ■establish, that the instantaneous centre for any position of the moving piece, is the point of contact of the rolling and fixed centroids. 165. "We shall have occasion to use this principle in drawing tangents and normals to the TROCHOIDS which are the principal Roulettes, or roll-traced curves, and wliich may be defined as follows: — If in the same plane one of two circles rolls upon the other without sliding, the path of anj' 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 by a i^oint 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 T n a b c, 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^ 3, ceiUng-corner of a room, and T,, the diagonally opposite floor-corner, a weight would sHde from Tg to Tj2 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 T,., as soon as if started at r, 167. In beginning the construction of the cycloid we notice first that as T V D rolls on the straight fine A B the arrow D R T will be reversed in position (as at D-^ T^ as soon as the semi- circumference TSD has had rolling contact with A B. The tracing point will then be at T^, its maximum distance from A B. When the wheel has roUed itself out once ujion the rail the point T will again come in contact T\'ith the rail, as at T,,. * "Although the invention of the cycloid is iittributefl to Galileo, it is certain that the family of curves to which it belongs haa heen known and some of the properties of such curves investigated, nearly two thousand yeare 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."— Proctor, Geometry of Cycloids. 56 THEORETICAL AND PRACTICAL GRAPHICS. The distance T T^^ evidently equals 2 tt r, when r = T R. If the semi-circumference To D (equal to ir r) be divided into any number of equal parts; and also the path of centres R R^ (again = 7rr) into the same number of equal parts, then as the points 1, 2, etc., come in contact with the rail, the centre R will take the positions R^, R,, etc., directly above the corresponding points of contact. A sufficient rolling of the wheel to bring point 3 upon A B would evidently raise T from its original position to the former level of ~. 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 8 by an arc of radius R T, centre R^. Similarly for other points. The construction, summarized, iiivolves the drawing of lines of level through equidistant points of division on a semi-circumference of the rolling circle, and their intersection by arcs of constant radius (that of the rolling circle) from centres which are the successive positions taken by the centre of the rolling circle. It is worth while calling attention to a point occasionally overlooked by the novice, although almost self-evident, that in the position illustrated in the figure the point T drags behind the centre R until the latter reaches ii^, when it passes and goes ahead of it. From .R, the line of level through 5 could be cut not alone at c by an arc of radius c R, 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. E'igr- loo- , THE CYCLOID. 168. Were ^joints T^ and T^, given, and the semi-cycloid T^ T^.., desired, we can readily ascertain the " base," ^4 B, and generating circle, as follows : — -Join T|, with T,, ; at any point of such line, as X, erect a j^erpendicular, xyj. from the similar triangles x y T^, and T,. D.^ T^^ having angle common and angles 6 equal we see that xy:xT,,::T,D,:D,T,,::2r:7rr::2:7r::l:~ov, very nearly, as U : 22. If, then, we lay off x Tj^ equal to twenty-tioo equal parts on any scale, and a perpendicular, x y, fourteen parts of the same scale, the line y T-^.^ will be the base of the desired curve; while the di- ameter of the generating circle will be the perpendicular from Tg 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 examjDle, (Fig. 100), would be t N, i^erpendicular to the normal. No, joining N with o; the latter point being found by using iV 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 point 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 hne of centres by an arc, centre N, radius T R; would obviously be vertically below the position of the rolling centre thus determined. 170. Till' Companion to the Cycloid. The kinematic method of drawing tangents, just applied, was devised by Roberval, as also the curve named by him the " Companion to the Cycloid," to which allusion has already been made (Art. 120) and which was invented by him in 1634 for the purpose of solving a problem upon which he had spent six years without success and which had foiled Galileo, viz., the calculating of the area between a cycloid and its base. Galileo was reduced to the expedient of comjoaring the area of the cj'cloid with that of the rolhng circle by weighing paper models of the two figures. He concluded that the area in cj[uestion was nearly but not exactly three times that of the rolling circle. That the latter would have been the correct solution may be readily shown by means of the " Comi^anion," as will be found demonstrated in Art. 172. 171. Suppose two points coincident at T, (Fig. 101), and starting simultaneously to generate curvesi the first of these points to trace the cycloid during the rolling of circle T V D while the second is to move independently of the circle and so as to be always at the level of the point tracing the cycloid, yet at the same time vertically above the point of contact of the circle and base. This makes the second point always as far from the initial vertical diameter, or axis, of the cycloid as the length of the arc from T to whatever level the tracing point of the latter has then reached; that is, MA equals arc THs; RO equals quadrant Tsy. Adopting the method of Analytical Geometrj^, by using as a reference point, or origin, we may reach any point, ^-l, on the cur^'e, 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 tv — the sine of the same arc. The " Companion to the Cycloid " is therefore a curve of sines or sinusoid, since starting from the abscissas are ecjual to or proportional to the arc of a circle while the ordinates are the sines of those arcs. This curve is particularlj^ interesting as " expressing the law of the vibration of perfectly ■ elastic sohds ; of the vibratory movement of a particle acted upon by a force which varies directly as the distance from the origin ; apiDroximately, the vibratory movement of a pendulum ; 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 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 -, ir r x 3 r = IT 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 jjoint of the latter, as A, is on the vertical diameter of the circle ^-igr- loi. * Wood. 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 level, 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 Ms and A a makes the elementary • rectangles MsSim^ and A A^ a^ a equal; and considering all the possible similarly constructed rectangles of infinitesimal altitude, the sum of those on semi-chords of the rolling circle would equal the area of the semi-circle TDy, which is therefore the extent of the area between the two curves under consideration. The figure showing but half of a cycloid, the total area between it and its "companion" must be that of the rolling circle. Adding this to the area between the " companion " and the base makes the total area between cycloid and base equal to three times that of the rolling circle. 173. The paths of points carried by and in the plane of the rolUng circle, though not on its circumference, are obtained in a manner closel)^ analogous to that employed for the cycloid. In Fig. 102 the looped curve, traced by the arrow-point while the circle GHM 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 AB the point T turns through the angle TRl about R and reaches the line of level through point 1. But T is always at the distance R T (called the tracing raclm) from R; and, as R has reached R^ in the rolling supposed, we find T^ — the new position of T — by an arc from R-^, radius T R, cutting said line of level. rE-ig-- loa. 2 1^ ' ^ After what has preceded, the figure may be assumed to be self-interpreting, each position 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. Tj was obtained from i?, ; and the point of contact must have been vertically below the latter and on A B. Joining such jjoint to T, gives the normal, from which the tangent follows in the usual way. 17.5. The Prolate Trochoid. Had we taken a point inside^ of the circle C H M 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-TROCHOIDS. 176. Circles of finite radius can evidently be tangent in but two ways — either externally, or internally; if the latter, the larger may roll on the one within it, or the smaller may roll inside the larger. When a small circle rolls within a larger the radius of the latter may be greater than the diameter of the rolling circle, or may equal it, or be smaller. On account of an interesting property of the curves traced by points in the jilanes of such rolling circles, viz., their capability of being generated, trochoidally, in two ways, a nomenclature was necessarj' 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 acceptmg 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 the cycloid and involute are respectively representative; orthoids being the paths of points in a fixed position with resi^ect to a straight line rolling ujion 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 ivithin the rolling circle of infinite radius when on the normal to its mitial position and on the side toward the centre of the fixed circle. As will be seen bj^ 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 aa to the actual form of the curves. In the table R represents the radius of the rolling circle, F that of the fixed circle. NOMENCLATUKE OF TROCHOIDS. Position of Tracing Circle rolling upon Straight Line. Circle rolling upon circle. Straight Line rolling upon Circle. li = a> External contacL- Internal contact. or Epitrochoids. Larger Circle rolling. Smaller circle rolling. : Cyclo- orthoids. Describing j „ . Oi-tho-cycloias. Point. 1 2 R > F. 1 2 K < F. 1 2 R = P. Perltrochoids. Major Hypotrochoids. Minor Hypotrochoids. Medial Hypotrochoids. 1 On circumference i rvelold of rolling circle. v-jv, uiu. (a) Epicycloid. (a) Pericyclold (d) Jlajor 1 Hypocyclolds. (d) Minor Hypocycloid. Straight j Hypocycloid. Involute. Within Prolate Circnmference. Troehoid. (b) Prolate Epitrochoid. (c) Prolate Perltrochold. (e) Jlajor Prolate (f) Jlinor Prolate '^' Eu'im^cal ' Hypotrochoid. i Hypotrochoid. Hypotrochoid. Prolate Cyclo-orthoid. Without Circumference. Curtate 1(c) Curtate (b) Curtate 1 ^^ curSe '" Cumt'e *°^ E^ilpttcll Trochoid. ;j Epitrochoid. ' Perltrochold. |j HypotiSJhold. Hypotrochoid. Hypot^ochiid. Curtate Cyclo-orthoid. 177. From the above we see that the prefix qn (over or upon) denotes the curves resulting from external contact; %j50 (under) those of internal contact with smaller circle rolUng; while peri (about) indicates the third possibility as to rolling. * Ke-prlnted in substance in the Appendix. 60 THEORETICAL AND PRACTICAL GRAPHICS. :F'ig-- 1.03. 178. The construction of these curves is in closest analogy to that of the cycloid. If, for example, we desire a rtiajor hypocydoid we first draw two circles, mVP. 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 jjoints of division, as those through 1, 2, 5, ^ 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 point 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, 1-2, etc., its distances from F would be i^a_and Fh respectively. If, however, the turning of P about R is due to the rolling of circle P Vm 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 niFz (135°) at the centre of the latter intercepts an arc, m x z, equal to the 180° arc, m V P, on the smaller circle. Equal arcs on imequal circles are subtended by angles at the centre which are inversely proportioned to the radii. While arc mVP rolls upon arc mxz the centre R will evidently move over circular arc R — iig . Divide mxz into as many equal parts as m V P and draw radii from F to the points of division ; these cut the jDath of centres at the successive positions of R. When arc m 5-Jf, 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 hfg — the circle of .distance through 2 — by an arc, centre R,, radius R P. Similarly for other points. 179. To trace the path of any point on the circumference of a circle so rolling as to give the epi- or pen-cycloid requires a construction similar at every step to that just illustrated. The same remark ajsplies 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 pjeritrochoid. 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 jjarts draw next the dotted " circles of distance " through these points, all from centre F. The figure illustrates the sjDecial case where the larger set of " circles of distance " divides both serai- circumferences into equal parts. The successive positions of P, as P^, 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 P R 1 about R would bring P somewhere upon the circle of distance through point 1; but that amount of turning would be due to the rolling of the first generator over the arc m Q, which would carry R to R^; P would therefore be at Pj , 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 property 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 i^inion which rolls on the interior of a toothed annular wheel of twice the diameter of the pinion. 181. The Limacon and Cardioid. The Limafon is a curve whose points may be obtained by ^ig-- i-os. 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 0ns; then if n 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 Limagon. Referring any point as d to and the diameter s, we have d = p = c + c d = 3 r cos + h, while a = 2 r cos 6 — 6, whence the general polar equation for the Limagon, p = 2 r cos e + h. When h ■= 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 epi-trochoids the generator and director must have equal diameters, any point mi the circumference of the generator then tracing a Cardioid, while anj^ point on the radius (or radius produced) describes a Limagon; as peri-trochoids the larger of a pair of Cardanic circles must roll ■on the , smaller, the Cardioid and Limagon then resulting, as before, from the motion of jDoints respectively on the circumference of the generator or within or ivithout it. 183. In Fig. 106 the Cardioid is obtained as an epicycloid, being traced by point P during one revolution of the generator 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 purjaose 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 possess 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 io 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 resijectively of an epitrochoid having a tracing radius, R F, ■equal to twice that of the generator. Make R N = R F and draw NF; this will cut the Limagon *Tenn due to Keuleaux and based upon tlie fact that Cardano (16th century) was probably the fir^t to investigate the paths described by points during their rolling, t From CardU, the Latin for heart. SPECIAL TROCHOIDS. 63 F T^R Q (traced by point F as carried bj' 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 m 1\ . These arcs are subtended bj^ equal angles, 4> , the circles being equal. During this rolling R reaches R^ , bringing i? i^ to R^ T^. In the triangles Ti R^ F and R F R^ the side F Ri is common, angles <^ equal and side R^ T^ equal to side R F; the line R T^ is therefore parallel to Ri F, whence angle T^ R F must also equal <^. In the triangle R F R^ we denote by 6 180° — ^ 1S0° or 6 = 2 In triangle N R F we have the angle at F equal to 6 — <^, and J {6 — <^) + .r + <^ = 180° which gives x ^= 2 by substituting value of 6 from previous equation. x'lg-. loe. FOT_Cardioid 186. The Involute. As the opposite extreme of a circle rolling on a straight line we may have the latter rolHng 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 irom 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 m x produced) it would trace the involute. When j reaches a the rule will have had rolling contact with the base circle over an arc uts — a whose length equals Une u x j. Were a the initial point we may obtain b, c, etc., bj' making tangent m h = arc in a ; tangent n c ^ arc n a, etc. Each tangent thus equals the arc from initial point to i^oint of tangency. 64 THEORETICAL AND PRACTICAL GRAPHICS. 187. The circle from which the invokite 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 tangencj' on the evolute is an instantaneous centre, and a tangent at any point as / is a jDerpendicular 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 plane perpendicular to the axis of the helix would cut such a surface in a pair of involutes.* 188. The Spiral of Archimedes. This is the curve that would be generated by a point having a uniform motion around a fixed jDoint — the pole — combined with uniform motion toward or fr'om the pole. In Fig. 107,. with as the pole, if the angles are ec[ual and D, E and y^ are in arith- metical progression then the points D, E and y^ 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, xy, from the contact-edge of the rule, equal to the radius Os of the base circle of the involute; for * The day of writing tlie above article the following item aijpeared in the New York Eoening 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 Boj'al 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 after the rolling of u x over an are ii 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, y^. If tlie rolling be continued y will evidently reach 0. We see that y = u x and y^ = t x^; but the lengths u 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 proi^ortional 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, y^, by 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 lo y, — the difference between two radii vectores Oy and Oz which include an angle of -57? 29 +, (the angle which at the centre of a circle subtends an arc equal to the radius). The instantaneous centre, t, would be the extrenaity of that radius which was perpendicular to y^. The normal would be t i/i and the tangent T 1\ 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 and opposite spiral gives the well known form called the heart-cam. As usually constructed the acting curve is not the true spiral but a curve whose jioints are at a constant distance from 'J---- 'e'^s- a-cs. the theoretical outline equal to the radius of the friction-roller which is on the end of the piece to lie raised. A small portion of such a " parallel curve " is indicated in the upper 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 orthograjDhic projection on the plane of the base will be a Spiral of Archimedes. In Fig. 108 a to]:) and fi'ont view are given of a cone and helix. 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 repre- sented by the sector A' 0' A" whose angle 6 may be found b}' the proportion A : 0' A' : : 6:360°, since the arc A' G" A" must equal the entire circumference of the cone's base. The student can make a paper model of the cone and helix by cutting out a sector of a circle, making allowance for an overlap on which to put the mucilage, as shown by the dotted lines 0' y and yvz 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 properties as those from which they are derived, except in the case of a circle ; this can readily be seen from the cam fig-ure under the last heading, in which a point, as S-^, 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 jjoint S^ of the parallel curve. E-ig-. lOS. 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 system 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 loithin 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 M N (the directrix) upon radials through a fixed point or pole, (Fig. 110). If m v = m n = s x then v, n and x are points of the curve. Denote by a the distance of fi-om M A^ and use c for the constant length to be laid off; then if a < 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 a> c the curve has an undulation or wave -form towards the jDole. * A series of curves much more closely resembling those of a shell can be obtained by tracing the paths of points on the piston-rod of an oscillating cylinder. See Arts. 157 and 158. THE CONCHOID.— THE QUADRATRIX. 67 V = c + Om; On= c — Oin; we may therefore express the relation to of points on the curve by the equation p = c ± i7i = c ± « sec . X'ig-- llO- 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. ^^'ere the angle m x (or ^) we would first draw p qr, the superior branch of a conchoid having the constant, c, equal to twice Dm. A parallel from m to the axis will intersect the curve at q; the angle p Oq will then be one-third of <^: for since 6 f/ = -.' m we have m q ^ 2 m cos /3 ; also m q : m : : sin : sin fi (the sides of a triangle being proportional to the sines of the ojiposite angles) ; therefore 8 m cos (3 : m : : sin 6 : sin ji, whence sin d ^ " sin ft cos /3 = sin 2 ft (fi-om known trigonometric relations). The angle ^ is therefore equal to twice y3, making the latter one-third of angle 4>- 195. To draw a tangent and normal at any point v we find the instantaneous centre o on the principle that it is at the intersection of normals to the paths of two moving points of a line, the distance between said points remaining constant. The motion of c in tracing the curve is — at the instant considered — in the direction Ov; Oo is therefore the normal. The point m of Ov is at the same moment moving along M N, for which mo is the normal. Their intersection o is then the instantaneous centre and o v the normal to the conchoid, with v z perpendicular to o 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 j^'ane sections this surface is called the Conchoidal Hyperboloid. THE QUADR.XTRIX OF DINOSTEATl'S. 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 .1 B at the same time with radius OT; the locus of the intersection of .1/ .V witli the radius will he the Quadratriz. Points 68 THEORETICAL AND PRACTICAL GRAPHICS. ^ E'igr- 111- ,/z exterior to the circle may be found by prolonging the radii while moving If iV away from A B. As the intersection of MN with OB is at infinit}^ the former becomes an asymptote to the curve as often as it moves from the centre an additional amount equal to the diameter of the circle; the number of branches of the Quadratrix may therefore be infinite. It may be proved analytically that the curve crosses A at a distance from equal to 3 r -r- ir. 198. To trisect an angle, as T a, by means of the Quadratrix draw the ordinate ap, trisect p T hj s and x and draw sc and xm; radii Oc and Om will then divide the angle as desired: for by the conditions of generation of the curve the line MN takes three equi- distant 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, by 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, NS and 31 T any pair of ordinates parallel to i^ig-. iia. asynljitote 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 from which it is derived. * Leslie. Geometrical Analysis. 1821. THE GISSOID. — 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 hne DE while the edge JK always passes through V; then the middle point, L, of the base FJ, will trace a cissoid. This construction enables us readily to get the instantaneous centre and a tangent and normal; for Fn is normal to FC — the path of F, while nz is normal to the motion of J 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 io A C and from centre P obtain x; draw P x, then Vz p)arallel to it; a vertical from x will meet Vz at the instantaneous centre j/,. trom which the normal and tangent result in the usual way. 201. Two quantities in and n will be mean proportionals between two other quantities a and b if m^ ^ n a and n"' = m b ; that is, if m' ^ a'' b and if n" = a b''. If 6 = i? a we will have m for the edge of a cube whose volume will be twice that of a^, when m' = a'"' b. To get two mean proportionals between quantities r and h make the smaller, r, the radius of a circle from which derive a cissoid. Were APR the derived curve we would then make Ct equal to the second quantity, b, and draw B t, cutting the cissoid at Q. A line A Q would cut off on C t a distance C v equal to m, one of the desired proportionals ; for vi' will then equal r- b, as maj^ be thus shown by means of similar triangles : — Cv: MQ:: CA:MA whence C v' = - '^^^ (1) Of. MQ::CB:BM " '^^=''^ (2) M Q ■ 31 A -.-.SNiAN:: VAN. BN: A N, whence MQ= ^^^^^^^-^N . _ (3) From (2) we have M Q = ^^(^^=^) (4) " (3) " " MQ^-=^^^^^§^^^ (5) Keplacing M Q^ in equation (1) by the product of the second members of equations (4) and (5) gives Cv'' (i. e., m^) = r- b. By interchanging ;■ and b we obtain n, the other mean projDortional ; 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 6 which is then joined with a. On a b a 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 comecutire points the curve would be theoretically exact. The line Q S is an asymptote to the curve. 70 THEORETICAL AND PRACTICAL GRAPHICS. ^ig-. ll-i. 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 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-- 113- 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 simple, how- ever, and the student may profitably be introduced to it. SujDpose a ball, c, (Fig. 114) struck at the same instant by two others, a and b, moving at rates of six and eight feet a second respectively. On a c and b c prolonged take c e and c h equal, respectively, to six and eight units of some scale; complete the parallelogram having these lines as sides; then it is a well known principle in mechanics* that cd — the diagonal of this parallel- ogram — will not only represent the direction in which the ball c will move but also the distance — in feet, to the scale chosen — it will travel in one second. Obviously, then, to balance the effect of balls a and 6 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' c', 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 blows by F in each triangle, we see that the pressures are greater for the thin wedge than for the other; that is, the less the inclination to the vertical the greater the pressure. A pivot so shaped that as the pressure between it and its step increased the area to be traversed diminished would therefore, theoretically, be the ideal; and the rate of 'tS~o-~~J change of curvature of the tractrix as its generating jDoint approaches IFig-. lis. the axis makes it, obviousl}', the correct form. *Foi- a demonstration llie student may refer to Rankine's Applied Mechanics, Art. 51. THE TRAGTRIX.— WITCH OF A G NISI. — CARTESIAN OVALS. 71 204. Navigator's charts are usually made by Mercator^s projection (so-called, not being a projection in the ordinary sense, liut with the extended signification alluded to in the remark in Art. 2). Maps thus constructed have this advantageous feature, that rhumb lines or loxodromics — the curves on a sphere that cut all meridians at the same angle — are represented as straight lines, which can only be the case if the meridians are indicated by parallel lines. The law of convergence of meridians on a sphere is, that the length of a degree of longitude at any latitude equals that of a degree on the equator multiplied by the cosine (see footnote, p. 31,) of the latitude; when the meridians are made non-convergent it is, therefore, manifestlj^ 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 AC4NISI. 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 : Q B then S will be a point of the curve called the Witch of Agnisi. 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^ latter. The tangent at B is an asymptote to the curve. The area between the curve and its asymptote is four times that of the circle involved in its construction. The Witch, also called the Versiera, was devised by Donna Maria Gaetana Agnisi, a brilliant Italian lady, who was appointed (1750) by Pope Benedict XIV. to the professorship of mathematics and philosophy in the University of Bologna. THE CARTESIAN OVAL. 206. This curve, also called simply a Cartesicvn, after its investigator, Descartes, has its points connected with two foci, F' and F", by the relation m p' ± n p" = k c, in which c is the distance between the foci while m, n and k are constant factors. * Leslie. Geometrical Anal/isis. Edinburgh, 1831. 72 THEORETICAL AND PRACTICAL GRAPHICS. :rLg. ±±T. Salmon states that we owe to Chasles the proof that a thkd focus may be found, sustaming 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 w p' ± a p' h c in tlie form o' ± — o' m k c -; or n a' c by denoting — by b and — by d it takes the yet more simple form p' d= 6 p" = d. Then p" will have two values according as the positive d- p' or negative sign is taken, being respectively — r and .' - d ; the former is for points on the inner of the two ovals that constitute a complete Cartesian, while the latter gives points on the outer curve. d—p' Figr. lis. To obtain p" = take F' and F" (Fig. 118) as foci; F' S=d; SK at some random acute d a,ngle with the axis, and make S H = -■ that is, make F' S : S H :: b : 1. Then from F' draw an arc ■ = 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, A-p'±np"'= m C, which — for purposes of construction — may be written p' ± h' p'" — d' . If obtained from the foci /"'and F'" the relation would be m p"' — A'p"=±/iC', in which C equals F" F'". Writing this in the form p'" — B p" ^ ± D we have the following interesting cases: (a) an ellipse for D positive and B ^ — 1 ; (b) an hyperbola for D positive and 5 = + 1 ; (c) a lirnacon for D = C B ; (d) a cardioid for B = + 1 and D = C. 210. The following method of drawing a Cartesian by continuous motion was ^'-s- ns- devised by Prof Hammond: — A string is wound, as shown, around two pulleys turning on a common axis ; a pencil at P holds the string taut around smooth pegs placed at random at F, and F„ : if the wheels be turned with the same angular velocity and the jDencil does not slip on the string it will trace a Cartesian having F^ and F., as foci.' If the pulleys are equal the Cartesian will become an ellipse ; if both threads are wound the same icay around either 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 reflectiiig 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 reflection. Probably all have occasionally noticed such a curve on the surface of the milk in a glass, when the light was properly placed. If the reflecting curve is a circle the caustic is the evolutc of a limaron. 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 refi-action {0 and 4>, 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 any point divides the angle between the focal radii into parts whose sines are proportional to the factors of those radii in the equation. If, then, we have obtained a point G on the outer oval from the relation m p' ± n p" = he we may obtain the tangent at G bj' la3'ing off on p' and p" distances proportional to m and n, as Gr find G h, Fig. 118, then bisecting rh at / and drawing the normal G j, to which the desired tangent is a jDerpendicular. At a point on the inner oval the distance would not be laid off on a focal radius produced, as in the case illustrated. ig-. 12C- 1 American Journal of Mathematics, 1S78. - Salmon. Hiqhcr Plane Curves. Art. 117. 74 THEORETICAL AND PRACTICAL GRAPHICS. CASSIAN OVALS. 212. In the Gassian Ovals or Ovals of Cassini the points are connected with two foci by the relation p' p" = I-", 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. ^ig-- las. In Art. 158 one form — the Lemniscate — receives special treatment. For it the constant k'' must equal m^, the square of half the distance between the foci. When k is less than m the curve becomes two separate ovals. 213. The general construction depends on the fact that in an}' 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 peri^endicular F^ S to the axis Fi F^ and on it lay off F^ R equal to the constant Jc. Bisect F^ F, at and draw a semicircle of radius R. This cuts the axis at A and B, the extreme points of the curve; for k'' = F^ A X F-^ B. Any other point T m.ay be obtained by drawing from F^ a circular arc of radius F^ t greater than F^ A ; draw t R, then R x perpen- dicular to it; X F^ will then be the p" and F^t the p' for four jooints of the curve, which wdll 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^ TO; 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, fi-om catena, a chain. A simple and practical method of obtaining a catenary on the drawing-board would be to insert two pins in the board, in the desired relative position of the points of suspension, and then attach to them a string . of the desired length. By holding the board vertically the string would assume the catenary, whose points could then be located with the pencil and joined in the usual manner with the irregular curve. Otherwise, if its points are to be located by means of an equation, we take axes in the plane of the curve, the i/-axis (Fig. 123) being a vertical line through the lowest point T of the catenary, while the x-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. 75 The equation of the catenary' is then (/=-—((•"'+ e " logarithms" and has the numerical vakie 2.71S"2S18 +. By taking successive values of .v equal to m, 2 m, 3 m, etc.. we get the following values for y: — — (/ .-■ '»■/■. 1 \ x=om...y = ^\^e'+ -j . "I / X 1 \ :c = -Lm...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 :r = B = m have y = B P = 1.54308: for x = a = — we have */ = 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 T P is proportional to the angle between T C and the tangent P V at the upper extremity of the arc. 215. If a circle R L B be drawn, of radius equal to m, it may be shown anah^ically that tangents P S and Q R, to catenary and circle respectively, from points at the same level, will be parallel : also that P S equals the catenary-arc P r T ; S therefore traces the involute of the catenarj^, and as SB always equals RO and remains perpendicular to P S (angle R Q being always 90°) we have the curve TSK fulfilling the conditions of a tracirix. (See Art. 202.) If a parabola, having a focal distance m, roll on a straight line, the focus will trace a catenary having m for its parameter. The catenary was mistaken Ij}- Galileo for a parabola. In 1669 Jungius jjroved it to be neither a parabola nor hj'perbola, but it was not till l(i91 that its exact mathematical nature was known, being then established by .James Bernouilli. THE LOGARITHMIC OR EQUIAXGUL.iR SPIRAL. 216. In Fig. 124 we have the curve called the Logarithmic Spiral. Its usual construction is based on the jjroperty that any radius vector, as p. which bisects the angle between two other radii, M and N, is a mean proportional between them ; i. e., p- = S'' = M X X. If 31 and G are points of the spiral we may find an intermediate point K by drawing the ordinate K 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 Runkine. Applied Mechanics, Art. 175. -In ttie expression 10- = 100 the quantity *'*2" is called the togartthm of 109, it being the exponent of the power to which 10 must he raised to give 100. Similarlj- 2 would be the logarithm of 61 were 8 the 6ttse 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. Upon 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 firoportional to the logarithm of the radius vector. This is expressed by p = a^, in which 6 is the varying angle and a is some arbitrary constant. To construct a tangent by calculation divide the hyperbolic logarithm ' of the ratio M : K (which are any two radii whose values are known) by the angle between these radii, expressed in circular measure'; the quotient will be the tangent of the constant angle of obliquity of the spiral. 217. Among the more interesting properties of this curve are the following: — Its involute is an equal logarithmic spiral. Were a light placed at the pole, the caustic — whether by reflection or refraction — would , be a logarithmic spiral. The discovery of these properties of recurrence led James Bernouilli to direct that this spiral be engraved on his tomb, with the inscription — 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 HYPEBBOLIC OE RECIPROCAL 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 aljout 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 velocity of projection determining which trajectory was -described. s'i.s- iss. To oljtain points of the curve divide a circle m 5 8 (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 2 take 7h 01/ — ^; on the third — ^, etc. For one-half the angle the radius vector would evidently be 2 n, giving a point s outside the circle. 1 The equation to the curve is in which is the 1 To get the hypertolic logarithm of a number multiply Its common logarithm by 2.3026. = In circular measure 3G0° = 2 IT r which for r = 1 becomes 6,28318; 180° = 3.14159; 90o = 1.5708; 60 = 0..'J236 ; 1 ° = 0.0174533. . 1.0472 ; 45 o = 0.7854 ; 30 ° : TEE HYPERBOLIC SPIRAL.— THE L ITU US. 77 radius vector, a some nmiierical constant, and is the angular rotation of /• (in circular measure) estimated from some initial line. The curve has an asymptote parallel to the initial line and at a distance from it equal to - units. To construct the spiral fi-om its equation take as the pole (Fig. 126): OQ as the initial line: a for convenience, some fi-action, as -: and as our unit some quantitv. say half an inch, that will 4' make - of convenient size. Then taking Q as the initial line make P = - = 2" and draw PR a " a parallel to OQ for the asymptote. For 6=1. that is. for arc KH = radius OH we have ?-=-=2". givintj H. — one point of the spiral. Writing the equation in the form r=-.-r and expressing various values of 9 in circular measure we get the following: — e = .30° = 0.5236; ;■ = J/ = 3'.'8 + : 0= 45° = 0.7854; r=0 -Y = 2 '.'55 ; e = m° = 1.5708; r = S = i:-l + : 6 = 180° = 3.14159; r = T= .6366. etc. The tangent to the curve at any point makes with the radius vector an angle ^ which is found by analysis to sustain to the angle 6 the following trigonometrical relation, tan ^ 6; the circular measure of xn&y therefore be found in a table of natural tangents and the corresponding value of (^ obtained. THE Lixrrs. 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 Lituu-s. This relation is shown bv the equation r = 7^1 also written a.- 6 = - . ay 6 r When = we find r = oo . making the initial line an asymptote to the curve. In Fig. 127 take Q as the initial line. as the pole, a = 2, and our unit a three-inch line : then - = 1;". a THEORETICAL AND PRACTICAL GRAPHICS. For e = 90° = TT (in circular measiu-e 1.5708) we have r= M=l'!-2 +. - For 6=1 we have the radius T making an angle of 57 ? 29 + with the initial line, and in length equal to - units, i. e., U". For 6=45°= -f (or 0.7854) r will be OR=V:i +. Then 0H = 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. Similarlj^ K = —^; OM — —^, etc.; this relation enabling the student to locate an_y number of points. To draw a tangent to the curve Ave emjoloy the relation tan ^=26, <^ being the angle made bj' the tangent line with the radius vector, while is the angular rotation of the latter, in circular measure. BRUSH TINTING AND SHADING. 79 CHAPTER VI. TINTIJJG — FLAT AND GRADUATED. — MASONEY, TILING, "WOOD GRAINING, RIVER-BEDS AND OTHER SECTIONS, WITH BRUSH ALONE OR IN COMBINED BRUSH AND LINE WORK. 220. Brush-work, with ink or colors, is either flat or graduated. The former gives the effect of a flat surface parallel to the paper on which the drawing is made, while graded tints either show cun-ature, or — if indicating flat surfaces — represent them as inclined to the paper, i.e., to the plane of projection. For either, the paper should be, as pre^dously 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 emploj-ed for tinting must be tree from sediment; or at least the latter, if present, must be allowed to settle, and the brush dii3ped 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 ujj the tint each time before taking a brushful. A tint 23repared trom a cake of high-grade India ink is far superior to any that can be made by using the ready-made liquid drawing inks. 222. The size of brush 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 dr}' 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 from the general effect. This will become evident on comparing the spheres in Figs. 1 and 4 of Plate II. Since tinting and shading can be successfully done, after a little practice, with onlj' pencilled limits, there is but little 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 provided a soft sjDonge 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. AMien 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 unijorra tint, start at the upper outline of the surface to be covered, and with a brush fuU, 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, quick strokes, with the brush held almost perpendicular to the paper. Advance the tint as evenl}' as possible along a horizontal line; work quickly betireen outlines, but more slowly along outlines, 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 be allowed to dry until the lower boundary is reached. 80 THEORETICAL AND PRACTICAL GRAPHICS. No portion of the paper, however small, should he missed as the tint advances, as the work is likely to be spoiled by 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 by means of blotting paper. To get a dark effect several medium tints laid on in succession, each one drying before the next is applied, give better results than one dark one. The heightened effect described in Art. 72, viz., a line of light on the ui:)per and left-hand edges, may be obtained either (a) by ruhng a broad line of tint with the drawing-pen at the desired distance from the outline, and instantly, 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 evenly 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. 226. Of the colors frequently used by engineers and architects those which work best for flat efiects are carmine, Prussian blue, burnt sienna and Payne's gray. Sepia and Gamboge, are, fortunately, rarely required for uniform tints; but the former works ideally for shading bj^ the "dry" process described in the next article; and its rich brown gives effects unapproachable with anything else. It has, however, this peculiarity, that repeated touches ujDon a spot to make it darker produce the opposite effect, unless enough time elapses between the strokes to allow each addition to dry thoroughly. 227. For elementarj' 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 the planes H, V and P are either vertical or inclined 30° to the horizontal. BRUSH TINTING AND SHADING. 81 For the section -plane assume a and 7)!, at pleasure, gi^'ing direction nm, to which JR and TX are parallel. A horizontal, mz, through m gives z. From n a horizontal, ny, gives y on ab. Joining y with 2 gives the "trace" of S on V. 228. Figures 129 and 130 illustrate the use of the brush in the representation of masonry. The former ma}' be altogether in ink tints, or in medium burnt umber for the front rectangle of -^ '■ "~ ■■■■ " '-■■■■■-" - - ■ - — — — ^T — T^ — " — ""^^ "' '~""'?r* ■ :^' "" j '. ' ' 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 niasomy is represented. If properly done, in either burnt umber or sepia, this gives a result of great beauty, especially effective on the piers of a large drawing of a bridge. 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. Two brushes are required when an ''over -hang" or jutting portion is to be represented, one \vith a medium tint, the other with the thick color, as before. An irregular line being made with the latter, the tint is then softened out on the lower 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 after 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 suj^erbly 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 aiaproach 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, adhh, of a rectangle is divided at o, 6, c, etc., into equal spaces, and through each point of division two lines are drawn with the 30° triangle, as 6 a; and hr through h. 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 tlirough the inter- section, m, of a pair of oblique lines, will pass through the intersection of a series of such pairs. ^'i-S- 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 ^^'hich horizontal rows of rhombi are left white, we draw -^-ertical lines as s q and m 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 Avas laid on. The most opaque effect in colors is obtained by mixing a large portion 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 transparent 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 projDcr 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 inspection. TINTING. — BRUSH SHADING. 83 231. Fig. 132 is a Minton design with which many are famihar, and which affords opportunity for considerable variety in finish. Its construction is almost self-evident. The equal spaces, ab, cd, mn — which may be any width, x, — alternate with other equal spaces be, which may preferably be about 3x in width. Lines at 45°, as indicated, complete the preliminaries to tinting. E'igr- 132. The octagons may be 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. SHADING. 232. For shading, by graduated tints, provide a glass of clear water in addition to the tint; also an ample supjily of blotting jiaper. 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 rapidly 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 immediate^ aj^ply it to the blottei', both to bring it to a good point and to remove the surjjlus tint. With the now once -diluted tint carrj' the advancing edge of the band slightly farther. Repeat the oijeration until the tint is no longer discernible as such. The process may be ■ repeated from the same starting point as man}' times as necessary to produce the desired effect; but the work should be allowed to drj'' each time before laying on a new tint. Any irregularities or streaks can easih' 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 i^oint. 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 i^ossible to the draughtsman. 233. Where a good, general effect is wanted, to be obtained in less time than would be required for the preceding processes, the method of over-lapping flat tints may be adopted. A narrower band of dark tint is first laid over the part to be the darkest. A\'hen dry this is overlaid by a liroader band of lighter tint. A yet lighter wash follows, Ijeginning on the dark portion and extending still farther than its predecessor. The process is reijeated with further diluted tints until the desired effect is obtained. Faintly -pencilled lines may l)e 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 ujjon 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 theo- retical dark or "shade'' line more sharply defined than in the spheres on the plate. ^ig-- 133- ^ig-. a.3-4. J 'A drawing of the end of a highly -ijolished revolving shaft, or even of an ordinary metallic disc, would be shaded as in Fig. 184. 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 mathematical relations. The fact may again be mentioned, on the princii:)le 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 elliptical section. The peculiar dotted effect in this figure is due to the fact that the original drawing, of which this is a photographic reproduction by the gelatine process, was made with a lithographic craj'on upon a special pebbled paper much used by lithographers. The original of Fig. 1, on the other hand, was a brush -.shaded sphere on Whatman's jDaper. 235. Fig. 5 (Plate II) shows a "Phoenix column," the strongest form of iron for a given weight, for sustaining compression. The student is familiar with it as an element of outdoor construction in bridges, elevated railroads, etc.; also in indoor work in many of the higher office buildings of our great cities. By drawing first an end view of a Phoenix column, similar to that of Fig. 135, we can readily derive an oblique view like that of the jjlate, by including it between parallels from all points of the former. The ijroportions 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. ^ig-. 13E. MATERIALS OF CONSTRUCTION. 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 brush, especially for "dry" shading, the effect to be aimed at being that of the rail section of Fig. 136. 236. When some f)articular material is to be indicated, a flat tint of the proper technical color (see Art. 73) should be laid on with the brush, either before or after shading. When the latter is clone with sepia it is jjrobably safer to lay on the flat 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. S^S. i3S. WOOD. — RIVER - BEDS. — M.iSOXRY, ETC. 237. While the engineering draughtsman is ordinarih' so pressed for time as not to be able to give his work the highest finish, yet he ought to be able, when occasion demands, to obtain both natural and artistic effects; and to conduce to that end the writer has taken pains to illustrate a number of ways of representing the materials of construction. .\1 though nearly all of them may be — and in the cuts are — represented in black and white (with the exception of the wood - graining on Plate II), j-et 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 may be done most simply by a series of irregular, approximately - i^arallel lines, as in Fig. 10 or 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 continu- ously, will cause almost all the irregularity desired. If a better effect is wanted, yet without color, the lines vaay be as in Fig. 107, which represents hard wood. In graining, the draughtsman should make his lines toward himself, standing, so to S23eak, at the end of the plank upon which he is working. The splintered end of a plank should be sharjjly toothed, in contradistinction to a metal or stone fracture, which is what might be called smoothly irregular. 239. An examination of anj' piece 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 readih' produced with the brush, yet it may be shown in a satisfactorj' degree with the pen, by a series of after -touches. 240. If we fiU 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 preliminary graining, as also the final wash, may be done altogether with the brush; as was the original of Fig. 9, Plate II. End -^-iews of timbers and planks are conventionally represented by a series of concentric free- hand rings in which the spacing 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 color -mixtures recommended b\- different writers on wood graining are something short of infinite in number; but with the addition of one or two colors to those listed in the draughts- man's outfit (Art. 56 j one should be able to imitate nature's tints very closely. 86 THEORETICAL AND P RA C T I CA L G RAP HIC S. COURSED RUBBLE MASONRY Light India inli. No hard-and-fast rule as to the proportions of the colors can be given. In this connection we may quote Sir Joshua Re3'nolds' reply to the one who inquired how he mixed his paints. "With brains," said he. One general rule, however; always employ delicate rather than glaring tints. 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 burnt 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. ^^s- -ysv. 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 yellow 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. 26, to which it is only neces- sarj' to add that such sections are usually made quite narrow, and, preferably — if in color — shaded quite abruptly on the side ojjposite the water. 243. The sections of masonry, concrete, brick, glass and vul- canite, given on page 25 as pen and ink exercises, are again presented in Fig. 137, for reproduction in combined brush and line work. The appropriate color is indicated under each section. 244. Masonry constructions may be broadly 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 ma)' 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 the Descriptive Geometry of Monge. (Art. 283). RUBBLE MASONRY Liglit India Ink. VULCANITE India Ink, BRICK Venetian Red. CONCRETE Yellow Ochre. FLs- J.3S- iiMii'b,Mj ^-^ the upper half of a letter. To get an idea of the amount of difference allowable compare ^ ^ the following equal letters printed from Roman tj'pe, condensed. Although not so important in the E, some difference between top and bottom may still to advantage be made. Another refine- ment is the location of the horizontal cross-bar of an A slightly below the middle of the letter. 261. While vertical letters are most frequently iised, yet no handsomer effect can be ol;)tained 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 point 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 alphal^et 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- ■s^i-s- 151. duces, exactly, the N of each of two leading alphabet books. If the fillets ^ y "m Y' 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 No. 23 of the alphabets appended, a type which for clearness and elegance is a triumph of the founder's art. As usually constructed, however, the D and R are finished at the top like the P. 94 THEORETICAL AND PRACTICAL GRAPHICS. ■E-i-S- iSS. 262. The Roman alphabet and its incHned or italic form are much used in topographical work. 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. Usually the small letters should be about three- fifths the height of the capitals. Except when more than one- third of an inch in height, these letters should be 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 alphabet 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-clotli, Bristol board or any smooth - surfaced paper a style long used 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 stubs of various widths, cut off obliquely, and when in use should not, as ordinarily, be dipped into the ink, but 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. The Soennecken Round Writing Pens are also by far 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. 153 and 154. The judicioxis ~ use of "word ornaments," such as those of alphabets 33, 42, 49, and of several of the other forms illustrated, will greatly enhance the appearance of a title with- out materially increasing the time expended on it. This is illustrated in the lower title on page 89. QJloiiMd Wu^inq E-ig-. 153- plernen'ary pare^ ^ eeh ^ anieal D ra^vind E'igr- 3.5-4. D rawl n ^- V ee_ 1 ■ an lea DESIGNS FOR BORDERS. 95 96 THEORETICAL AND PRACTICAL GRAPHICS. V ~:l 51 4i3 ffi / u 266. Bordeis. 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 simplicity. On page 95 a large number of corner designs and borders is presented, one -third of them orig- inal designs, by the writer, for this work. The principle of their construction is illustrated by Fig. 155, in which the larger design shows the necessary preliminary lines, and the smaller the complete corner. It is evident in this, as in all cases of interlaced designs, that we must first lay off each way from the corner as many equal distances as there are bands and spaces, and lightly make a network of squares — or of rhombi, if the angles are acute — by pencilled construction -lines through the points of division. 267. Shade lines on borders. 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 shaded line. This is on the assumj^tion 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 shaded lines may be determined b}' a study of their location on the designs of the plate of borders. It need hardly be said that on any drawing and its title the light should be supposed to come from but one direction throughout, and not be shifted; and the shaded lines should be located accordingly. This rule is always imperative. In drawing for scientific illustration or in art work it is allowable to depart from the usual strictty conventional direction of light, if a better effect can thereby be secured. 268. A striking letter can be made by drawing the shade line only, as in Fig. 146, page 90, which we may call "Full -Block Shade -Line," being based upon the alphabet of Fig. 148, page 92, as to construction. Owing to its having more projecting parts it gives a much handsomer effect than the The student will notice that the light comes from different directions in the two examples. These forms are to the ordinary 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 books on lettering as Prang's, Becker's and others equally elaborate, yet they will be found of only occasional service, their designs being as a rule more highly ornate than any but the sijecialist would dare undertake, and mainly of a character unsuitable for the usual work of the engineering or architectural draughtsman, whose needs were especially in mind when selecting tjqjes for this work. The aljjhabets appended afford a large range of choice among the handsomest forms recently designed by the leading type manufacturers, also containing the best among former types; and with the "Railroad," Full-Block and Half- Block alphabets of this chapter, j^roportioned and drawn by the writer, supply the student with a practical "stock in trade" that it is believed will require but little, if any, supplementing. COPYING PROCESSES. — DRAWING FOR ILLUSTRATION. 97 CMAPTJEB nil. BLUE-PEINT AND OTHER COPYING PROCESSES. — METHODS OF ILLUSTRATION. 270. While in a draughting office the process 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, usuallj', 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 property, may be obtained from the following valuable works: Modern Hdiographic Processes,* by Ernst Lietze; Photo- Engraving, Etching and Lithography, 'i 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" paper, for the original drawing. A sheet of pajDer 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 i^roduce the final blue undertone of the sheet; while the portions of the paper, from 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, apparentlj^, a matter of indiffer- ence, so great is the disparity between the recipes of different writers ; indeed, one successful draughtsman says: "Almost any proportion of chemicals will make blue-prints." Whichever recipe is adopted — 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 paper 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 * Published 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 apjDlied 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, jjlaced 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 exposed. If the sun's rays are not direct, or if the day is cloudy, a proportionately longer time is required,, running 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 ta 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 j^rint out of the frame it should be immediately immersed and thoroughly washed in cold water for from 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 process, 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 Genie Civil). f Red Prussiate of Potash '. 8 parts. Solution Nc '' No. 1. I Water 70 parts. f Citrate of Iron and Ammonia 10 parts. Solution No. 2. \ ,,, ^ _. ( Water 70 parts. ■ Filter the solutions separately, mix equal quantities and then filter again. No. 2. (From U. S. Laboratory at Willett's Point). Solution No. 1. Solution No. 2. Double Citrate of Iron and Ammonia 1 ounce. Water 4 ounces.. Red Prussiate of Potassium ;..... 1 ounce. Water 4 ounces. Stock Soluti 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, 4} fluid ounces Water, and mix 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 pure state and does not need to be filtered after being dissolved. It dissolves very rapidly. 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 atmosphere. 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 i:)rints, are amply sufficient for coating one hundred square feet of iDajser, 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 supply. For copying a drawing in any desired color the following method, known as Tilhei''s, is said to give good results : " The iiiajjer on which the copy is to ajjpear is first dipjDed in a bath consisting of 30 parts of white soap, 30 parts of alum, 40 parts of English glue, 10 j^arts of albumen, 2 p)arts of glacial acetic acid, 10 parts of alcohol of 60°, and 500 parts of water. It la afterward put into a second bath, which contains' 50 jaarts of burnt umber ground in alcohol, 20 parts of lampblack, 10 parts of English glue, and 10 parts of bichromate of potash in 500 jjavta of water. They are now sensitive to light, and must, therefore, be preserved in the dark. In preparing paper to make the positive print another bath is made just like the first one, excei^t that lampblack is substituted for the burnt umber. To obtain colored positiA'es the black is replaced by some red, blue or other pigment. In making the copy the drawing to be copied is put in a p>hotographic 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 appear in white on a dark ground; in other words, it is a negative or reversed jDicture. The paper is then dried and a positive made from it by placing it on the glass of a printing- frame and laying the positive paper upon it and exi)osing 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) wood 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. W'OOD 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, 15, 20-24, 134, 136, and those illustrating mathematical surfaces, in the next chapter. Its expensiveness and the time required to make an illustration by this method are its only disadvantages. Although the camera is often employed 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 supposed to have at his disposal either the object to be drawn or a j)hotograph 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 lights" 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 surpassed by steel engraving. Figures 36, 90 and 107 are good specimens 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 plate; (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 copper; (d) the taking of an electrotj^pe from the plate and wax, the deposit of copper filling in the lines 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 jDroper 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. LITHOGRAPHY. — PHOTO - LITHOGEAPHY. — CHROJIO - LITHOGRAPHY. 278. For the lithographic process a fine-grained, imported limestone is used. The drawing is made with a greasy ink — known as " lithograiahic " — upon a specially prepared 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 j^revents the printer's ink (owing to the grease which the la/tter contains) from adhering to any jjortion except that from which the impressions are desired. Photo -lithography is simply lithography, with the camera as an adjunct. The positive might be made directly upon the surface of the stone by coating the latter with a sensitizing solution ; but, in general, for convenience, a sensitized gelatine film is exjiosed under the negative, and by subsec[uent treatment gives an image in relief which, after inking, can be transferred to the surface of the stone as in the ordinary process. Chromo- lithography, or lithography in colors, has been a very expensive process 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 -lithography may preferably be adopted ; although " half tones," with colored inks, give a scarcely less pleasing eifect, as illustrated by Figs. 7-10, Plate II. But for simple line -work, in two or more colors, one may preferably employ either cerography or jDhoto - engraving, each of which has not only an advantage, as to expense, over any lithograj)hic process, but also this in addition — that the blocks can be used b}' anj' jjrinter; whereas lithographing establishments necessarily not only prepare the stone but also do the printing. PHOTO - ENGRAVING. — PHOTO - ZINCOGRAPHY. 279. In this popular and rapid process a sensitized solution is spread ui^on 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 hues 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 indicate their sections unmistakably, the}' 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 and width of the proposed block. The reduction can be greater, provided care has been taken to have the lines far enough aj)art, 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 tyjje. 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 ijhotographic negative. The result is called a half tone and it is one of the favorite jarocesses 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 pateiat dates back to 1852. The i^rinciple 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 interjjosed a " screen " or sheet of thin glass on which the desired mesh has been previously photograjDhecl. 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 which a few of tlie above processes are named, a gelatine film may be employed, 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 special paper, by the company prei^aring the film. In the composition and manipulation of the film lies the main difference between otherwise closely analogous processes. For any of them the comiDany should be supplied 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 company's circular. Figs. 1 — 5 of Plate "II illustrate the effect obtained by it. " Ordinary cooking gelatine forms the basis of the jDositive plate, 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 exjiand under its influence, but that it will dissolve in hot water. In the prej)ara- tion of the plate, therefore, the three ingredients just named, being combined in suitable iiroportions, are dissolved in hot water, and the solution is poured upon a level jDlate of glass or metal, and left there to dry. When dry it is about as thick as an ordinary sheet of parchment, and is strij^ped from the drying-plate, and placed in contact with the previously-prepared negative, and the two together are exi^osed 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 produces 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 transparent parts admitting the passage of light through them, and the opaque parts excluding it. When the gelatine f)late 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 waterproofs the gelatine underneath them ; while the opaque portions of the negative shield the gelatine underneath them from the light, and consequently those parts of the plate remain unaltered in character. The result is a thin, flexible sheet of gelatine, of which a jDortion is water- proofed, and the other portion is absorbent of water, the waterproofed jjortion being the image which we wish to rejDroduce. 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 parts; and, if ink is then rolled over it, the ink will adhere only to the waterproofed or altered parts. This flexible sheet of gelatine, then, prepared as we have seen, and having had the image impressed upon it, becomes the heliotype plate, capable of being attached to the bed of an ordinary i^rinting - 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 accomjDlished 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-press, 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 Descriptive Geometry possesses an important philosophical peculiarity, quite independent of its high industrial utility. This is the advantage which it so pre-eminently offers in habituating the mind to consider very complicated geometrical combinations in space, and to follow with precision their continual correspondence with the figures which are actually traced — of thus exercising to the utmost, in the most certain and precise manner, that im- portant faculty of the human mind which is properly called ' imagination,' and tohich consists, in its elementary and positive acceptation, in rejiresenting 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 geoinetry of the ancients by the character of its solutions, on the other hand it appiroaches 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 tnodern geometry; for the methods used are always conceived as apjjlicable to any figures whatever, the peculiarity of each having only a purely seco?ida?-y influence." Acguste Comte : Cours de PMlosophie Positive. " A mathematical problein may usually be attacked by lohat is terined in military par- lance the method of ' systematic approach,;' that is to say, its solution may be gradually felt for, even though the successive steps leading to that solution cannot be clearly foreseen. But a Descriptive Geometry problem must be seen through and through before it can be attempted. The entire scope of its conditions as well as each step toioard its solution must be grasped by the imagination. It must be ' taken by assault. ' ' ' Geoege Sydenham Clarke, CaiJtain, Boyal Engineers. MECHANICAL DRAWINGS.— WORKING DRAWINGS. 131 CHAPTEB X. PROJECTIONS AND INTERSECTIONS BY THE THIRD- ANGLE . METHOD.— THE DEVELOPMENT OF SURFACES FOR SHEET METAL PATTERN MAKING. — PROJECTIONS, INTERSECTIONS AND TAN6ENCIES OF DEVELOPABLE, WARPED AND DOUBLE CURVED SURFACES, BY THE FIRST -ANGLE METHOD. 383. The me(*hanical drawings ]ireliminary \o the construction of machinery, blast furnaces, stone arclics. buildincrs, and, in fact, all architectural and engineering projects, are made in accordance with the ])rindplcs of Descripti^-e Geometry, ^^'llen fully dimensioned the}' are called working drawings. The oliject to be reiiresented is supposed to be jalaced in either the first or the third of the four angles formed by the intersection of a horizontal plane, H, with a vertical plane, Y. (Fig. '228). The representations of the object upon the planes are, in mathematical language, ■projections,* and are iilitained liy drawing perpendiculars to the planes H and V from the various points of the object, the jioint of intersection of each such projecting line with a plane gi^'ing a projection of the original point. Such drawings are, obviously, not ''views" in the ordinary sense, as the)' lack the perspective effect which is involved in having the j^oint of sight at a finite distance; yet in ordinary parlance the terms top vieii:, horizontal projection and plan are used synonymously; as are front view and front elevation with vertical projection, and side elevation with profile view, tlie latter on a j)lane perpendicular to both H and V and called the profile plane. ^ Until the last decade of the first century of Descriptive Geometry (1795-1895) problems were solved as far as j^ossible in the first angle. As the location of the oliject in tlie third angle — that is, below the horizontal plane and behind the vertical — results in a grouping of the views which is in a measure self-interpreting, the Third Angle Method is, however, to a considerable degree sujiplant- ing the other for machine-shop work. The advantageous grouping of the j^i'^^jections which constitutes the only — though a cpiite suf- ficient — justification for giving it special treatment, is this: The front view being alwaj's the central one of the grouji, the top view is found at the top; the view of the right side of the object appears on the right; of the left-hand side on tlie left. etc. 'Thus, in Fig. 228 (a), with the hollow block BDFS as the object to be represented, we have odes for its horizontal jjrojection, r'd'e'f for its vertical projection, f"e"s".v" for the side elevation; then on rotating the plane H clockwise on G. L. into coincidence with Y, and the profile jilane P about Q R until the pvo^ection- f" e" s" x" reaches f"'e"'s"'.r"', we would have tliat location of the views which has just been described. The lettering shows that each projectiim re}iresents that side of the oliject whicli is toward the plane of projection. 384. The same grouj)ing can Ije arrived at liy a different conception, which will, to some, have advantages over the other. It is illustrated l\v Fig. 228 (b), in which the same oliject as before is *For the convenience of those who have to take up this subject without previous study of Descriptive Geometry the Third-Angle section of tliis chapter is made complete in itself, hy the re-statement of the principles involved and which have been treated at somewhat greater length in the previous chapter: although a review of such luatter may be by no means disadvantageous to those who have already been over the fundamentals. 132 THEORETICAL AND PRACTICAL GRAPHICS. sujoposed to be smTdumled by a system of mutually -peqiendieular transparent ]ilanes, or, in other words, to be in a box having glass sides., and on each side a drawing made of what is seen through that side, excluding the idea, as before, of jierspective ^new, anil representing each point by a per- X^ig-- 233, iiPiii||l||[r|[i(B«p™R|^ pendicular from it to the j)lane. The whole sj^stem of box and planes, in the wood -cut, is rotated 90° from the position shown in Fig. 228(a), liringing tliem into the usual position, in which' the observer is looking perpendicularly toward the vertical plane. ^"igr- 22Q. 385. In Fig. 229 we may illustrate either the First or the Third Angle method, as to the toi> view of the object; ndes in the upper plane being the plan hy the latter method, and a^d^e^s^ by the former. nRTnnnRArHTc projectiox of solids. 133 Disregarding QTXX we have the object and planes iUustrating the first -angle method through- out, the lettering of each ])rt}jection showing that it represents the side of the object farthest from the plane, making it the exact reverse of the third- angle sj'stem. In tlic ordiiinri/ representation the same object -n'ould be reiDresented simply by its three views as in Fig. 280. In the elevations the short -dash Hnes indicate the in^^sible edges of the hole. The arcs show the rotation which carries the profile view into its proper place. 231. Fig. 3. 'r- e" i Ac I 1 C" Fig. 8. t r^L . -i_ '■'Ic: 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 system, while all below HK represents the same oliject by the Third Angle method. When looking at Figures 1, 2, 3 and 4 the observer queries: AVhat is the object, in space, whose front is like Fig. 1, top is like Fig. 2, left side is Hke Fig. 3 and riciht side like Fig. 4? For the view of the left side he might imagine himself as having Ijeen at first between G and H. looking in the direction of arrow X, after which both himself and the object were turned, together 134 THEORETICAL AND PRACTICAL GRAPHICS. to the right, through a ninety -degree arc, when the same side would be presented to his view in Fig. 3. Similarly, looking in the direction of the arrow M, an eqiial 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 M. His mental queries would then be answered about as follows: Evidently a cubical block with a rectangular recess — r'v'd'c' — in front; on the rear a prismatic projection, of thickness p /( and whose height equals that of the cube; a short cylindrical ring projecting from the right face of the cube; an angular j^rojecting 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 oliserver from direction M. To illustrate the third angle method by Fig. 231 we ignore all above the line H K. In Fig. 5 we have the same front elevation as before, but above it the view of the top; below 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'b', until the under side became the front elevation. Fig. 7 may as readilj' be imagined to be obtained by a shifting of the object as by the rotation of a plane of projection; for bj' translating the object to the right, from its jDOsition in Fig. 5, then rotating it to the left 90° about b'n', its right side would appear as shown. 387. For convenient reference a general resume of terms, abbreviations and instructions is next presented, once for all, for use in both the Third jungle and First Angle methods. (1) H, V, P the horizontal, verftcnl and profile planes of projection respectively. (2) H - projector the projecting line which gives the horizontal projection of a point. (3) V- projector the projecting line giving the projection of a point on V. (4) Projector -plane the profile plane containing the projectors of a point. (5) h. p the horizontal projection or plan of a point or figure. (6) v. p the vertical projection or elevation of a point or figure. (7) h. t horizo7ital trace, the intei'section of a line or surface with H. (8) V. t vertical trace, the intersection of a line or surface with V. (9) H- traces, V- traces plural of horizontal and vertical traces respectively. (10) G-. L ground line, the line of intersection of V and H. (11) V-parallel a line parallel to V and lying in a given plane. (12) A horizontal any horizontal line lying in a given plane. (13) Line of declivity the steepest line, with respect to one plane, that can lie in another plane. (14) Eabatment revolution into H or V about an axis in such plane. (15) Counter -rabatmenl or revolution . restoration to original position. 388. For Problems relating solely to the Point, Line and Plane. Given lines should be fine, continuous, black ; required liyies heavy, continuous, black or red ; construction lines in fine, continuous red, or short-dash black; traces of an auxiliary pdave, or invisible traces of any plane, in dash - and ■ For Problems relatinq to Solid Objects. (1) Pencilliiir/. Exact ; generally completed for the whole drawing before any inking is done ; the work usually from centre lines, and from the larger — and nearer — pai-ts of the object to the smaller or more remote. (2) Inlcinr) 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 ensvire uniformity. On tapering shade lines see Art. 111. (3) Shade Lines. In architectural work these would be drawn in accordance with a given direction of light. In American machine-shop practice the right-hand and lower edges of a plane surface are made shade lines if they separate it from invisible surfaces. Indicate curvature by line-shading if not otherwise sufficiently evident. (See Fig. 288). THE CONSTRUCTION AND FINISH OF WORKING DRAWINGS. 135 (4) Invisible lines of the objecf, black, invariably, in dashes nearly one-tenth of an inch in length. (5) Inking of lines other than of the object. "When no colors are to be employed the following directions as to kind of line are those most frequently made. The lines may preferably be drawn in the order mentioned. Centre lines, an alternation of dash and two dots. Dimension lines, a dash and dot alternately, with opening left for the dimension. Extension lines, for dimensions placed outside the views, in dash-and-dot as for a dimension line. Ground line, (when it cannot be advantageously omitted) a continuous heavy line. Construction and other explanatory lines in short dashes. (6) ]i'hen using colors the centre, dimension and extension lines may be fine, continuous, red; or the former may be blue, if preferred. Constructioti 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." Ink ground from the cake is also preferable to bottled ink. 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 rend 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 number. Extended Gothic, Koman, Italic Roman and Reinhardt's form of Condensed Italic Gothic are the best and most generally used types for dimensioning. The arrow- 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 first the equilateral triangle ah c for the plan of the base, making its sides of the pre- scribed length. If we make the edge a b perpendicular to ^^s- sss. the profile plane, 1, the face v a b will then appear in profile view as the straight line v"b". Being a right pyra- mid, with a regular base, we shall find v, the plan of the vertex, equally distant from a, b and c; and v a, vb, vc for the plans of the edges. Parallel to G. L. and at a distance apart equal to the assigned height, x, draw mv" and no" as upper and lower limits of the front and side elevations; then, as the h. p. and V. p. of a point are alwaj^s in the same perijendicu- lar to G. L., we project r, a, b and c to their respective 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 i)rojected to their proper levels, giving the .s/V/e deration, v"b"c". As the actual length of an edge is not shown in either of the three views, we employ the fol- E'ig-. 233- is the plane area which, folded on lowing construction to ascertain it: Draw vv^ ijeri^endicular to vb, and make it equal to x; 'v,b is then the real length of the edge, shown by rabatment about v b. The develojmient of the pyramid (Fig. '233) may be obtained by drawing an arc ABCA^ of radius = Ui 6 (the true length of edge, from Fig. 232) and on it laying off the chords A B, B C, CA^ equal to ah, he, ca of the plan; then V-ABCA^ VC and V B, wi.iuld give a model of the pyramid represented. im THEORETICAL AND PRACTICAL GRAPHICS. 390. Wm-klng drawing of n semi -cylindrical pij)e: outer diameter, x; inner diameter, y; height, For the plan draw concentric semi -circles aed and b s c, of diameters ing their extremities by straight lines a b, c d. At a distance apart of z inches draw the upper and lower limits of the elevations, and project to these levels from the points of the 2)lan. In the side view the thickness of the shell of the cylinder is shown liy the distance between e"f" and s"i" — the latter so drawn as to indicate an invisible limit or line of the object. The lino shading would usually be omitted, the shade lines generally sufficing to convey a clear idea of the fomi. 391. Half of a hollow, hexagonal prism. In a semi-circle of diameter ad step off the radius three times as a chord, giving the vertices of the plan a h c d of the outer surface. Parallel to b c and at a distance from it equal to the assigned thickness of the prism, draw ef temiinating it on lines (not shown) and y resj)ectively, join- ^ig-. 335. From e and / draw e h ccl. Drawing a' c" and drawn through b and c at 60 ° to a d. and fg, parallel respectively to a b and 7n't" as upper and lower limits, f)roject to them as in preceding problems for the front and side elevations. 392. Working drawing of a hollow, prismatic block, standing obliquely to the vertical and profile planes. Let the block be 2"x3"xl" outside, with a square oijcn- ing 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 jilan asxb, 3"xl", on a scale of 1:2. The inch -wide opening through the centre is indicated by the short- dash lines. For the elevations the upper and lower limits are drawn 2" ^igr- a3S. apart, and o, 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 lettering will enabile the student to recognize the three views of any point without difficult}'. 393. In Fig. 237 we have the same object as that illustrated by Fig. 236, but now represented as cvit by a vertical j^lane whose horizontal trace is V y. Tlie parts of the Itlock that are actually cut by the jjlane are shown in section -lines in the elevations. This is done here and in some later examples merely to aid the beginner in understanding the views; but, in engineering prac- tice, section -lining is rarely done on views not perpendicida/r to the section plane.. 3C fV-— .... c \V...A. ■ ^ ^^^^ %± --,. \ 1 \ yv^ : 1 \ \ \ \ \ \ \ \ ' ' 1 1 1 ' ,y^ ^ -., v^ l" j 1 i ' L 1 i ' ! b 'a 1 1 : \x ,s' \(l 1 ^! 1 1 is" 1 cj ^ 111 1 ] m" I £_j. e' d 'L._ h 1 i t. PROJECTION OF S OLIDS. — WORKING DRAWINGS. 137 it is customaiy to omit the X-lg-. 237. SECTIONAL VIEW 394. Sujypression of the Ground Line. In machine drawing ground line, since tlie fonni< of the various views — which alone concern us — are independent of the distance of the object from an imaginary horizon- tal or vertical plane. We have only to remember that all elevations of a point are at the same level; and that if a ground line or trace of any vertical plane is wanted, it will be perpendicular to the line joining the plan of a point with its projection on such vertical plane. 395. Sections. Sectional Views. Although earlier defined (Art. 70), a re-statement of the distinction between these terms may well precede problems in which they will be so frequently employed. When a plane cuts a solid, that portion of the latter which comes in actual contort with the cutting plane is called the .section. A sectional view is a view perpendicular to the cutting plane, and showing not only the section liut also the object itself as if seen through the plane. When the cutting plaiie is vertical such a view is called a sectioned elevation; when horizontal, a sectioned plan. 396. Working clraicing of a regidar, pentagoncd pyramid, hollow, truncated by an oblique plane; cdso the development, or "poHfcn," of the outer surface below the cutting plane. For data take the aUitude at 2"; inclination of faces, 6° (meaning any arl)itrary angle); inclinatiim of section plane, 30°; distance between inner and outer faces of pyramid, ^". (1) Locate v and v' (Fig. 238) for the plan and elevation of the vei-tex, taking them sufficiently apart to avoid the overlapping of one view upon the other. Through V draw the horizontal line .S' T, regarding it not only as a centre line for the plan but also as the h. t. of a central, vertical, refer- ence plane, parallel to the ordi- nary vertical jilane of projection. 138 THEORETICAL AND PRACTICAL GRAPHICS. SECTIONAL VfEW (The student should note that for convenience Fig. 238 is repeated on this page.) On the vertical line w' (at first indefinite in length) lay ofT v' s' equal to 2", for the altitude (and axis) of the pj'raniid, and through s' draw an indefinite horizontal hne, which will contain the V. p. of the base, in both front and side views. Draw v'h' at 6° to the horizontal. It will represent the v. p. of an outer face of the pyramid,, and b' will be the v. p. of the edge ah of the ham. The base abcde is then a regular pentagon circumscribed about a circle of centre v and radius y i = .s' 6'. Since the angle avb is 72° (Art. 92) we get a starting corner, a or b, by drawing va or vb at 36° to ST, to intercept the vertical through //. The plans of the edges of the pyramid are then v a, vb, v c, vd and v e. Project d to d' and draw v' d' for the elevation of (' d ; similarly for v e and v e, which hajipen in this case to co- incide in vertical projection. For the inner surface of the pyramid, whose faces are at a perpendicular distance of \" from the outer, begin by drawing g' V parallel to and \" fi-om the face projected in b' v' ; this will cut the axis at a p(jint t' wliieh will be the vertex of the inner sur- face, and g' t' will represent the ele\-ation of the inner face that is parallel to the face avb — v'b'; while gh, vertically above <"/' and included between va and vb, will be the plan of the lower edge of this face. Complete the pentagon gh---k for the jjlan of the inner base; project the corners to b' cV and join with t' to get the ele- vations of the interior edges. The. Section. In our figure let (?' H' be the section plane, sit- uated perpendicular to the ver- tical plane and inclined 30° to the horizontal. It intersects v' d' in j>', which projects upon vd at p. Similarly, since G' H' cuts the edges v' c' and v' e' at points projected in o', we project from the latter to vc and ve, obtaining and q. A like construction gives m and n. The polygon mnopq is then the plan of the outer boundary of the section. The inner edge g' t' is cut by the section plane at /', wliich projects to both vh and vg, giving the parallel to mn through /. The inner boundary of the section may then be completed either by determining all its vertices in the same way or on the principle that its sides will be parallel to those of the outer polygon, since any two planes are cut by a third in parallel lines. The line vi' p' is the vertical projection of the entire section. PROJECTION OF S OLIDS.— WORKING DRAWINGS. 139 (2) The side elevation. This might be obtained exactlj- as in the five preceding figures, that is, hy actually locating the side vertical, or lyrofile, plane, projecting ujion it and rotating through an arc of 90°. In engineering practice, however, the method now to be described is in far more general use. It does not do away with the profile plane, on the contrary presupposes its existence, but instead of actually locating it and drawing the arcs which so far have kejDt the relation of the views ■constantly before the eye, it reaches the same result in the following manner : A vertical line 6" T' is drawn at some convenient distance to the right of the front elevation ; the distance, from iS T, of any point of the plan, is then laid off horizontally from S' T', at the same height as the front ■elevation of the point. For, as earlier stated, S T was to be regarded as the horizontal trace of a vertical plane. Such plane would, e-^-idently, cut a profile plane in a vertical line, whicli we may •call S'T', and let the S' T' of our figure represent it after a ninety - degree rotation has occurred. The distances of all points of the object, to either the front or rear of the vei-tical plane on S T, would, obviously, he now seen as distances to the left or right, respectively, of the trace S' T', and would be directly transferred with the dividers to the lines indicating their level. Thus, e" is on the level of e', but is to the right of S'T' the same distance that e is above (or, in reality, behind) the plane ST; that is, e" d" equals eu. Similarly d"b" equals ib; n"x" equals nx. It is usual, wliere the object is at all s.ymmetrical, to locate these reference planes centrally, so that tlieir traces, used as indicated, ma}' bisect as many lines as possible, to make one setting of the •di\'iclers do double work. (3) T)-ve size of the Section. Sectioned View. If the section plane G' H' were rotated directly about its trace on the central, vertical plane S T, until j^arallel to the paper, it would show the .section m'p' — mnopq in its true size; but such a construction would cause a confusion of lines, the new figure overlapping the front elevation. If, however, we transfer the plane G' H' — keeping it parallel to its first position during the motion — to some new position S"T'', and then turn it 90° •on that line, we get m,7i,o,p, g,, the desired Adew of the section. The distances of the vertices of the section from S" T" are derived from reference to S T exactly as were those in the side ■elevation; that is, v} ^x^ = mx = m"x". We thus see that one central, vertical, reference plane ST is auxihary to the construction of two impoi-tant views ; S' T' represents its intersection with the profile ■or side vertical plane, while S" T" is its (transferred) trace upon the section plane G'H'. For the remainder of the sectional views the points are obtained exactly as above described for the section; thus c'c^c, is perpendicular to S" T" ; e,ii^ equals eu, and c,?'-! equals cv. (4) To determine the actual length of the vanous edges. The only edge of the original, uncut pyramid, that would require no construction in order to show its true length, is the extreme right- hand one, which — being parallel to the vertical plane, as shown by its plan vd being horizontal — is seen in elevation in its true size, v'd'. Since, however, all the edges of the pj'ramid are equal, we maj^ find on r'd' the true length of any portion f)f some other edge, as, for examjale o'c', b}^ taking that part of r'd' which is intercepted between the same horizontals, viz.: o"'d'. Were we compelled to find the true length of o'c', oc, independently of any such convenient relation as that just indicated, we would apply one of the methods fully illustrated hj Figs. 183, 184 and 187, or tlie follomng "shop" modification of one of them: Parallel to the plan oc draw a line y z, their distance apart to be equal to the difference of level of o' and c', ■which difference may be ■obtained frf)m eitlier of the elevations. From the plan o of tlie higher end of the line draw the -common perpendicular of and join f with c, obtaining the desired length fc. (5) To shoiv the exact form of any face of the pyramid. Taking, for example, the face o c d pj, revolve o j) about the horizontal edge cd until it reaches the level of the latter. The actual distance 140 THEORETICAL AND PRACTICAL GRAPHICS. of from c, and of p from d will be the same after as before this revolution, while the paths of o and p during rotation will be projected in lines or and pio, each perpendicular to cd; therefore, with c as a centre, cut the j^erpendicular or by an arc of radius fc — just ascertained to be the real length of oc, and, similarly, cut p%o by an arc of radius du) = 'p'd'; join r with c, iv with d, draw 10 r and we have in cdwr the form desired. (6) The development of the outer surface of the truncated pyramid. With any point F as a centre (Fig. 239) and with radius equal to the actual length of an edge of the jiyramicl (that is, equal to v'd', Fig. 238), draw an indefinite arc, on which lay off the chords DC, C B, B A, A E, ED, equal respectively to the like -lettered edges of the base abcde; join the extremities of these chords with V: then -on Z» F lay off DP=d'p'; make C 0= EQ=^ d'o'" = the real length of c'o'; also BN= A M = d' m'" = the actual length of a'7n' and b'n'; join the jDoints P, 0, etc., thus obtaining the development of the outer boundary of the section. The laattern A^B^CDE^ of the base is obtained from the plan in Fig. 238, while NMq.^p.,o., is a duplicate of the shaded part of the sectional vieW' in the same figure. (7) In making a model of tire pyramid the student should use heavy Bristol board, and make allowance, wherever needed, of an extra width for overlap, slit as at x, y and z (Fig. 239). On thi.s I^ig-- 233. overlap put the mucilage which is to hold the model in shape. The faces will fold better if the Bristol board is cut half way through on the folding edge. 397. For convenient reference the characteristic features of the Third Angle Method, all of which have now been fully illustrated, may thus be briefly summarized : (a) The various views of the object are so grouped that the plan or top view comes above the front elevation; that of the bottom below it; and analogously for the projections of the right and left sides. (b) Central, reference planes are taken through the various views, and, in each view, the distance of any point from the trace of the central plane of that view is obtained by direct transfer, with the dividers, of the distance between the same point and reference plane, as seen in some other view, usually the plan. 398. To draio a truncated, pyramidal block, having a rectangular recess in its top; angle of sides, 60°; lower base a rectangle 3" X 2", having its longer sides at 30° to the horizontal; total height •5%"; recess 1^" X ■^", and \" deep. (Fig. 240.) The small oblique projection on the right of the plan shows, pictorially, the figure to be drawn.. PROJECTION OF SOLIDS. — WORKIXG DRAWINGS. 141 The plan of the lower base will be the rectangle abde, 3" X 2", whose longer edges are inclined 30° to the horizontal. Take AB and ma as the H- traces of auxiliary, vertical planes, perpendicular to the side and end faces of the Ijlock. Then the sloping face whose lower edge is d e, and which is inclined 60 ° to H, will have d,?/ for its trace on plane mn. A jjarallel to mn and ■^" from it will give -Sj, the auxiliary projection of the upj^er edge of the face sved, whence sv — at first indefinite in length — is derived, parallel to de. Similarly the end face btsd is obtained by projecting db upon AB at 6,, drawing b^z at 60° to AB and terminating it at Sj by CD, drawn at the same height (xVO "^^ before. A parallel to bd through s.^ intersects vs^ at s, giving one corner of the plan of the upper base, from which the rectangle stuv is completed, with sides parallel to those of the lower base. As the recess has ^-ertical sides we may draw its plan, o p q r, directly from the given dimen- sions, and show the depth by short -dash lines in each of the elevations. The orcUnary elevations are derived from the plan as in preceding problems; that is, for the front elevation, a'u's'd', by verticals through the plans, temiinating according to their height, either on a' d' or on lo's', ■^" above it. For the side elevation, e"v"t"b", with the heights as in the front elevation, the distances to the right or left of s" equal those of the plans of the same points from .5 i, regarding the latter as the h. t. of a central, vertical plane, parallel to V. The plane ST of right section, jjerpeyidicular to the axis KL, cuts the block in a section whose true .size is shown in the line -tinted figure g^h^kj^, and whose construction hardly needs detailed treatment after what has preceded. The shaded, longitudinal section, on central, vertical plane K L, also interprets itself by means of the lettering. 142 THEORETICAL AND PRACTICAL GRAPHICS. The true size of any face, as a u v e, raixy be shown by rabatment about a horizontal edge, as a e. As V is actually yV above the level of e, we see that v e (in space) is the hypothenuse of a triangle of base ve and altitude -^". Construct such a triangle, v^'^e, and with its h^ypothenuse v.^e as a radius, and e as a centre, obtain v ^ on a perpendicular to a e through v and representing the joath of rotation. Finding u^ similarly we have a^i^v^e as the actual size of the face in question. If more views were needed than are shown the student ought to have no difficulty in their construction, as no new principles would be involved. 399. To draw a holloiv, pentagonal prism, 2" long; edges to be horizontal and inclined 35° to V; base, a regular pentagon of 1" sides; one face of the prism to be inclined 60° to H; distance lietween inner and outer faces, \". In Fig. 241 let H K h% parallel to the plans of the axis and edges; it will make 35° with a horizontal line. Perpendicular to HK draw mn as the h. t. of an auxiliary, vertical plane, upon which we may suppose the base of the prism projected. In end view all the faces of the prism would be seen as Ihm, and all the edges as points. Draw a,i6i, one inch long and at 60° to mn, to represent the face whose inclination is assigned. Completing the inner and outer pentagons, ■allowing \" for the distance between faces, we have the end vie^^' complete. The plan is then PROJECTION OF S OLIDS. — W RKIN G DRAWINGS. 143 obtained by drawing parallels to H K through all the vertices of the end view, and terminating all by vertical planes, a d and g h , 2)arallel to m n and 2 " apart. The elevations will be included between horizontal lines whose distance ajjart is the extreme height z of the end view ; and all points of the front elevation are on verticals through their plans, and at heights derived from the end view. The most expeditious method of working is to draw a horizontal reference line, like that of Fig. 243, which shall contain the lowest edge of each elevation; measuring upward from this line laj^ off, on some random, vertical line, the distance of each point of the end view from a line (as the parallel to ma through 6, in Fig. 241, or .c/y in Fig. 243) which repre- sents the intersection of the plane of the end view by a horizontal plane containing the lowest point or edge of the ol^ject; horizontal lines, tlirough the points of division thus ol^tained, will contain the projections of the corners of tlie front elevation, which may then be definitely located by vertical lines let fall from the plans of the same points. For example, e' and ./'', Fig. 241, are at a height, s, above the lowest line of the elevation, equal to the distance of e, from the dotted line through ij; or, referring to Fig. 243, which, owing to its greater complexity, has its construction given -more in detail, the distance upward from M to line G is ccjual to i/^y.. on the end view; from M to Q equals qiq,, and similarly for the -rest. Since the profile plane is omitted in Fig. 241 we take M' N' to represent the trace upon it of the auxiliary, central, vertical plane whose h. t. is M N; as already explained, all j^oints of the side elevation are then at the same level as in the front elevation, and at distances to the right or left of M' N' equal to the perpendicular distances of their plnm from M N. For example, e" s" equals e -s. The shade lines are located on the end view on the assumption that the observer is looking toward it in the direction HK. 400. Projections of a hollow, pentagonal prism, cut by a vertical plane obli to b^G^, will, however, evident^ hold the same relation to the plan as it stands, and transferring such new jjlane forward to O'Pj we then obtain the points of the new (third) ele^'ation by letting fall perpendiculars to 0' R^ from the vertices of the second plan, and on them laying off heights abo-\'e O'P, equal to those of the same points above MN in the second elevation. Thus j'9 equals /'/; TF'6 in the fourth equals IF" 6 in the second. The fourth elevation is a view in the direction of arrow No. 5, giving the equivalent of a ninety - degree rotation of the object from its last position. To obtain it take a reference line r r through some point of the second plan, and parallel to O'R,; then R R' rejDresents the vertical jjlane on r r, transferred. From R R' lay off — on the levels of the same points in the third elevation — distance 1 C" = c.X\; 4 TF" = H-^jT'F], as in j^receding analogous constructions. THE DEVELOPMENT OF SURFACES. 405. The developjment of surfaces is a topic not altogether new to the student who has read Chapter V and the earlier articles of this chapter;* so far, however, it has occurred only incidentally, but its importance necessitates the following more formal treatment, which naturally precedes problems on the interpenetration of surfaces, of which a develoi^ment is usually the jDractical outcome. *The following articles should be carefully reviewed at this point: 120; 191; 314-6; 3S9, and Case 6 of Art. 3%. THE DEVELOPMENT OF SURFACES. 151 A development of a surface, using the term in a practical sense, is a i^iece of cardboard or, more generally, of sheet -metal, of such shape that it can be either directly rolled up or folded into a model of the surface. Mathematically, it would be the contact- area, were the surface rolled out or unfolded upon a plane. The "shop" terms for a developed surface are "surface mi the flat" "stretch-out," "roll -out"; also, among sheet -metal workers it is called a pattern; but as pattern -making is so generallj^ under- stood to relate to the patterns for castings in a foundry, it is best to emijloy the cjualifying words sheet-metal when desiring to avoid any possible ambiguity. 406. The mathematical nature of the surfaces that are capable of devel(5pment has Ijcen already discussed in Arts. 344-346. Those most frec^uently occurring in engineering and architectural work are the right and ol)lique forms of the pyramid, prism, cone and cylinder. 407. In Art. 120 the development of a right cylinder is shown to be a rectangle of base equal to '2-n-r and altitude A, where h is the height of the cylinder and r is the radius of its base. 408. The development of a right cone is proved, by Art. 191, to be a circtdar sector, of radius equal to the slant height R of the cone, and whose angle & is found by means of the projjortion R : r :: 360° : 0; r being the radius of the base of the cone. 409. The development of a right pyramid is illustrated in Art. 389 and in Case 6 of Art. 396. 410. ^^'e next take up right and oblique prisms, and the oblique pyramid, cone and cylinder; while for the sake of completeness, and departing in some degree from what was the plan of this work when Art. 345 jjassed through the i^ress, the regular solids will receive further treatment, and also the developable helicoid. 411. The development of a right prism. Fig. 247, represents a regular, hexagonal prism. The six faces being equal, and e b c f showing their actual size, we make the rectangles A BCD, Bi BEFC, etc., each equal to ebcf; then AA^ f '^ equals the perimeter of the upper l)ase, and we have the rectangle A.A.^B^D for the development sought. 412. The deveJop/ment of a right prism, beloiv a cutting plane. last article develop first as if there were no section to be taken into account. This gives, as before, a rectangle of length A A, and of altitude a d, divided into six equal jiarts. Then project, from each point where the jilane cuts an edge, to the same edge as seen on the development. 413. Right Section. Rectified Curve. Developed Curve. A plane perpendicular to the axis of a. surface cuts the latter in a right section. The bases of right cones, pyramids, cylinders and p)risms fulfil this condition and require no special construction for their determination; but the development of an o):ili(jue i'orm usually involves the construction of a right section and tlien the laying off on a straight line of a length equal to the jjerimeter of such section. Should the right section Ije a curve its equivalent length on a straight line is called its rectification, which should not be confounded with its develojmient, the latter not Ijeing necessarily straight. 414. The development of an oblique pjrisni, when the faces are equal in tvidth. In Fig. 249, an oblicjue, hexagonal prism is shown, with x for the width of its faces. Since the perimeter of a right section would evidently equal 6 x we may directly lay off x six times on some iierfjendicular .,=!„„- a A B E H t s»== ~S! d ^ t'l I 3 C F C 3 Ai Taking the same prism as in the a f 152 THEORETICAL AND PRACTICAL GRAPHICS. :Fig-. 3-^9. E'ig-- SSO. to the edges, as that through a. The seven i^arallels to a b , drawn at distances x apart, will contain the various edges of the prism as it is rolled out on the plane; and the position of the extremities are found by perpendiculars from their original positions. The initial position o j 6 , is parallel to but at anj^ dis- tance fi'om a b. 415. The development of an oblique prism whose faces are unequal in ividth. In Fig. 250 c' d' h' g' is the elevation of the prism; n p a plane of right section. To get the true shajDe, 1-2-3-4, of the right section, we require abhfe — l^lan of the i:)rism.' Assuming that to have been given, imagine next a vertical reference - plan standing on ab. The right section plane np cuts the edge c' d' at n, which is at a distance x in front of the assumed reference - plane. Make n 2 = x. Similarly make oo = y, and p4 = 2; then 1-2-3-4 is the right section, seen in its true size after being revolved about the trace of the right-section plane upon the assumed reference -plane. Prolong jj n indefinitely, and on its extension make l'-2'=l-2; 2' 3' = 2-3, etc. Parallels to c' d' through the points of division thus obtained will contain the edges of the developed larism, and their lengths are definitely determined by perijendiculars, as ■"-,^ h'h", ff", from the extremities of the orig- inal edges. 416. The development of an oblique cylinder. Let am'n'k, Fig. 251, be an oblique cylinder with circular base. Take any plane of right section, as a'k'. Draw various elements, as those through b', c', etc., and from their lower extrem- ities erect perpendiculars to ak, as cc,, terminating them on the arc af^k, which represents the half base of the cylinder. On c c' make c'c" = cc^; on e e' take e' e" = ee^, and similarly obtain other points on the elements, through which the curve a' c" e" y" I' can be drawn, this being one -half of the curve of right section, shown after revolution about its shorter diameter. Making KA equal to the rectified semi -ellipse just obtained, lay off A B = am ah^; i? C = arc b-^Ci, etc., and through the points of division thus obtained on KA draw indefinite parallels to the axis of the cjdinder. These will represent the elements on the development, and are limited by the dotted lines drawn perpendicular to the original elements and through their extremities. The area a.^k.,NM is the development of one -half of the cylinder, the shaded area representing all between a' h' and the base al\ * In the iuterest of compactness the "Fii-st Angle" position of the views is ;employe(l in Figs. '250, 253 and THE DEVELOPMEST OF SURFACES. 153 417. The development of an oblique pyramkl. The development will evidently consist of a series of triangles having a common vertex. To ascertain tlie length of any edge we may carry it into or parallel to a plane of projection. Thus in Fig. 2.52 the edge vb is carried into the vertical jjlane at vh". Its true length is evidently the hypothenuse of a right- angled triangle of base o /j = n J ", and altitude v o . In Fig. 253 a pj^ramid is shown in plan and elevation. Making V c to length a" = v a we have (■'((" for the actual lengtli of edge v'a', a construction in strict >'^ analogy to that of Fig. 252. The ])lan V b being jjarallel to the base line shows that v'b' is the actual length of that edge. By carrying vci, where it becomes parallel to V, and then projecting Cj to of edge v'c'. x'lg-- sss. E"ig-. 2S2. ?" we get v'c" for the true To illustrate another method make vv^ = v'o; then v^d bv rabatment into H. is the real length of r' ed regular solid, is illustrated graphically by Fig. 266, l.)ut may l3e otherwise expressed as follows: x'ig-. ass. d : e d : c when V M\ _-_ A^ \ "m / w \, ■,/\ J\ OCTAHEDRON. DODECAHEDRON :: n/3 : ^/ 2 ; for the cube d : e :: >/ 3 : 1 : : •>/ 2 : 1 ; " " dodecahedron e = the greater segment of the edge the latter has been medially divided, that is, in" extreme and mean ratio. For the icosahedron e = the chord of the arc whose tangent is d; i. e., the chord of 63° 26' 6". Reference to Figs. 256-260 and the use of a set of cardboard models which can readily be made bj' means of Figs. 261-265 will enable the student to verify the following statements as to those ordinary ^-iews whose construction ^^■n\\\d naturally precede the solution of problems relating to these surfaces. E-ig-. SS5. In all liut the tetrahedron each face has an ecjual, opposite, parallel face, and excej^t in the cube such faces have their angular points alter- nating. (See Figs. 260, 267, 268.) The tetrahedron projects as in Fig. 256, upon a plane that is par- allel to either face. The ciibe projects in a square upon a jjlane parallel to a face, while on a plane perjoendicular to a body diagonal it projects as a regular hexagon, with lines joining three alternate yei-tices with the centre. The octahedron, which is practicallj' two equal square jDyramids with a common base, projects in a square and its diagonals, ujjon a plane perpendicular to either body diagonal; in a rhombus and shorter diagonal when the plane is parallel to one body diagonal and at 45° with the other ^S-S- 2ST- X^ig-. 2SS- X'ig'. 3S3. ICOSAHEDRON two; and (as in Fig. 267) in a regular hexagon with inscribed triangles (one dotted), when it is projected ujjon a plane parallel to a face. The dodecahedron projects as in Fig. 268 whenever the plane of projectioii is j^arallel to a face. 156 THEORETICAL AND PRACTICAL GRAPHICS. IFigr- STO. Fig. 260 represents the icosahedron i^rojected on a jjlane parallel to a face, and Fig. 269 when the projection -plane is perpendicular to an axis. 420. The Devdopahle Hclicoid. When the word heUcoid is used without qualification it is under- stood to indicate one of the ivarped helicoids, such as is met with, for example, in screws, spiral staircases and screw propellers. There is, however, a developable helicoid, and to avoid confusing it with the others its characteristic property is always found in its name. As stated in Art. 346 it is generated 1:),V moving a straight line tangentially on the ordi- nary helix, which cur^-e (Art. 120) cuts all the elements of a right cylinder a;t the same angle. Fig. 209 illustrates the completed surface pictorially ; Fig. 270 shows one orthographic jjrojection; and in Fig. 271 it is seen in process of generation by the hypothenuse of a right-angled triangle that rolls tangentially on a cylinder. The construction just mentioned is based on the property of non- plane curves that at any point the curve and its tangent make the same angle with a given plane; if, therefore, the helix, beginning at o, crosses each element of the cylinder at an angle equal to ob p in the rolling triangle, the hypothenuse of the latter will evidently vaoxe not only tangent to the cylinder, l^ut also to the helix. The following important properties are also illustrated by Fig. 271 : (a) The involute* of a helix and of its hori- zontal projection are identical, since the point !i is the extremity of both the rolling lines, o b and p.b. (b) The length of any tangent, as 'm 6, is that of the helical arc m a on which it has rolled. (c) The horizontal projection b q of any tan- gent b m equals the rectification of an arc a q which is the projection of the helical arc from the initial point a to the point of tangency to. The development of one nappe of a helicoid is shown in Fig. 273. It is merely the area between a circle and its in\-olute; but the radius p, of the base circle, equals r sec'- ^,t in which r is * For full treatment of the involute of a circle refer to Arts. 186 and 187. t This relation is due to considerations of curvature. At any point of any curve its curvature is its rate of departure froni its tangent at that point. Its radius 0/ curvature is that of the osculatory circle at that point. (Art. 380.) Now from the nature of the two uniform motions imposed upon a point that generates a helix (Art. 120) the curvatui-e of the latter must be uniform; and if developed upon a plane by vieans of its curvature it must become a circle — the oiily plane curve of uniform curvature. The radius of the developed helix will, obviously, be the radius of curvature of the space helix. Following Warren's method of proof in establishing its value let a, b and c (Fig. 272) be three equi-distant points on a helix, with & on the foremost element; then a' c' is the ■elevation of the circle containing these points. One diameter of the circle a' b' c' is projected at b'. It is the hypothenuse of a right-angled triangle having the chord 6 c, 6'c', for its base. Let 2 p be the diameter of the circle a'b'e'; 1r = bd, that of the cylinder. Using capitals for points in space we have B~C' =2pXbe; also iTc^ = 2rXbe; whence, dividing like members and substituting trigonometric functions (see note p. 31), we have p = rsec = /3, in which p is the angle between the line £C and its projection. Let 9 be the inclination of the tangent to the helix at b'. If, now, both A and C approach B, the angle 3 will approach as its limit; and when A, B and C become consecutive points we will have p = r3ec=e = the radius of the osculatory circle = the radius of curvature. For another proof, involving the radius of curvature of an ellipse, see Olivier, Cours de Geometric Descriptive, Third Ed., p. 197. T^ig-. THE INTERSECTION OF SURFACES. 157 ^ig-. ST-i- m n / / % the radius of the cyUnder on which the heUx originall}' lay, and 6 is the angle at which the x-ig- avs. helix crosses the elements. To de- termine p draw on an elevation of the cylinder, as in Fig. 274, a line a h, tangent to the helix at its fore- most point, as in that position its inclination 6 is seen in actual size ; then from 0, where a b crosses the extreme element, draw an in- definite line, OS, par- allel to c d, and cut it at m by a line a m that is perpendicular to a 6 at its intersection with the front element cf of the cylinder; then m = p = r sec.- 6. For we haA-e oa = 7) sec 9 = r sec ; and o a (= )•) : oa :: o a : o m ; whence o m = r sec ' 6 = p. The circumference of circle p equals 2 ir r sec 0, the actual length of the helix, as may he seen by developing the cylinder on which the latter lies. The elements which were tangent to the hehx maintain the same relation to the developed hehx, and appear in their tme length on the development. The student can make a model of one nappe of this surface by wrapping a sheet of Bristol board, shaped like Fig. 273, upon a. cylinder of radius r in the equation r sec' e = p; or a two- napped hehcoid by superposing two equal circular rings of paper, binding them on their inner edges Tvith gummed paper, making one radial cut through both rings, and then twisting the inner edge into a helix. THE INTERSECTION OF SURFACES. 421. When plane-sided surfaces intersect, their outhne of interpenetration is necessarily composed of straight hnes; but these not being, in general, in one plane, form what is called a tioisted or toarped polygon; also called a gauche polygon. 422. If either of two intersecting surfaces is curved their common line will also be curved, except under special conditions. 423. AMien one of the surfaces is of uniform cross section — as a cylinder or a prism — its end \iew will show whether the surfaces intersect in a continuous line or in two separate ones. In Cases a, b, c, d and (/ of Fig. 275, ^\heve the end ^-iew of one surface either cuts but one limiting line of the other surface or is tangent to one or both of the outhnes, the intersection will be a continuous line. Two separate curves of intersection will occur in the other possible cases, illustrated by e and /, in which the end view of one surface either crosses both the outlines of the other or else hes wholly lietween them. A cylinder will intersect a cone or another oj^linder in a plane curve if its end view is tangent to the outlines of the other surface, as in d and x-ig.. st-s. ^, Fig. 275. Two cones may also intersect in a plane curve, but as the conditions to be met are not as readilv illustrated they will l)e treated in a special prolilem. 158 THEORETICAL AND PRACTICAL GRAPHICS, ■Fi-s- SVS- Referenee Line 424. In general, the line of intersection of two surfaces is obtained, as stated in Art. 379, by passing one or more auxiliary surfaces, usually planes, in such manner as to cut some easily con- structed sections— as straight lines or circles— frOm each of the given surfaces; the meeting - points of the sections lying in any auxiliary surface will lie on the line sought. The application of the principle just stated is much simplified whenever any face of one or other of the surfaces is so situated that it is projected in a line.' This case is amply illustrated in the problems most immediately following. The beginner will save much time if he will letter each projection of a point as soon as it is determined. 425. The intersection of a vertical triangular prism by a horizontal square prism; also the developments. The vertical prism to be 1|" high and to have one face parallel to V; bases equi- lateral triangles of 1" side. The horizontal prism, to be 2" long, its basal edges f", and its faces inclined 45° to H; its rear edges to be parallel to and i" from the rear face of the horizontal prism. The elevations of the axes to bisect each other. Draw e i horizontal and 1" long_ for the plan of the rear face of the vertica} prism. Complete the triangle egi and project to levels 1^" apart, obtaining e'f, g'h',-i'j', on the elevation. Construct a^r^end view g" i" j" h", using t" j" to represent the reference line et, transferred. The end view of the horizontal prism is the square a" b" c" d", having its diagonal horizontal and upper and lower bases of the other prism, and with its corner from i" j". The plan and fi'ont elevation of the horizontal rived from the end view as in preceding constructions. Since the lines eg and gi are the plans of vertical faces their intersection by the edges a, b, e and d of the hori- n, m, I, p, q, r — aird project to the elevations of the same edge a a meets the other prism at o and h, Avhich project o' and h'. Similarly for the remaining points. The development of the vertical prism is shown in the shaded rectangle IJ, of length Sgri and altitude e'f. (See Art. 411). The openings 0, j^i g, r, [and h^l^m^n^ are thus found: For p^, which represents p', make Ppi = xp', and G P = g p, the true distance of p' from g' h' ; similarly, qiO = q' y, and G = o g. !>= midway between the b " one - eighth inch "= prism are next de- '^- of one prism we note zontal prism — as at "' edges. Thus the to the level of a" at THE IXTEBSECTIOX OF SURFACES. 159 The right hidf uf the horizontal iirism, u' a' c' q', is developed at r .,h .^b ./r ^ after the method of Art. 412. 42(3. The iiitcrscrthiii uf tiru jiri> il' f i ■s".s- s7-e- d X a y b z c d 1 / ■ 1 .'^ ■ ^ u "^-YVV s-xgr- ST"©. 278. As akeady a X (of Fig. 277) ; a; 5' = \-ertical 160 THEORETICAL AND PRACTICAL GRAPHICS. Although not required in shop work the draughtsman will find it an interesting and valuable x-ig-. seo. exercise to draw and shade either soHd after the removal of the other ; also to draw the common [solid. The former is illustrated by Fig. 279; the latter l)y Fig. 280. 427. The intersection of two prisms, one vertical, the other oblique but with edges parallel to V. Let abcd....a'r' (Fig. 281) be the plan and elevation of the vertical prism. Let the oblique prism be inchned 30° to H; its faces inclined 60° and 30° respectively to V; its base a rectangle 1^" X f", and its rear edge yV back of the axis of the vertical prism. Through some point o' of the edge e' o' draw an indefinite line, o' f, at 30° to H, for the elevation of the rear edge, and //, also indefinite in length at first but ^" back of s, for the plan. ^igf. 2S1. ,,'-'' Take a reference plane MN through s and as in Art. 397 (b) construct an auxiliary elevation on 71/ iV, transferring it so that it is seen as a perpendicular to o'/', thus obtaining the same view of the prisms as would be had if looking in the direction of the arrow. To construct this make o" f" equal to A") draw f'i" at 60° to MN and on it com- plete a rectangle of the given dimensions; after which lay off the points of the pentagonal prism at the same distances from MN in both figures. Project back, in the direction of the arrow, from /", g", h" and i" to the front elevation, and draw g' i' and the opposite base at equal distances each side of o'. For the intersection we get any point n' on an oblique edge, as g', by noting and projecting from THE IXTERSECTIOX OF PLAXE-SIDED SURFACES. 161 E-ig-. 2Sa. E'ig-- 2S3. X 71 where the plan gg meets the foce c d. For a vertical edge as c' m' look to the auxiliary elevation of the same edge, as c", getting /" and m" which then project back to I' and m'. The development need not again be descrilied in detail Imt is left for the student to construct, with the reminder that for the actual distance of any corner of the intersection from an edge of either prism he must look to that projection which shows the base of that prism in its true size : thus the distance of /' from the edge h' is /*"/". 428. The intersection of pyramidal surfaces by lines and planes. The iirinciple on ■i\-liirh tlie inter- section of pyramidal surfaces by plane -sided or single curved surtaces would be obtained is illustrated by Figs. 282 and 283. (a) In Fig. 282 the line a b. a'h', is supposed to intersect the given pyra- mid. To ascertain s' and t' — its entrance and exit points — we regard the elevation a' b' as representing a plane perpendicular to V and cutting the edges of the pyramid. Project m'. where one edge is cut, to hi, on the plan of the same edge. Obtaining n ami o similarly we have /// n o as the plan of the section made by plane a'h'. The plan n Ji meets mno at « and t which tlien project back to a' b' at s' and t', the points sought. As ab, a'b', might be an edge of a pyramid or prism, or an element of a corneal, cylindrical or warped surface, the method illustrated is of general appli- cabilitj'. (b) In Fig. 283 the auxiliary planes are taken vertical, instead of perpendicular to A' as in the last case. Tlie plane MX cuts a pyramid. To find where any edge r'o' pierces tlie plane MX jjass an auxiliary vertical plane x z through the edge, and note .'• and z, where it cuts the limits of MX; project these to .t' and z'; draw .i:' z', which is the elevation of the line of intersection of the original and auxiliary planes, and note s', where it crosses r'o'. Project -s' back to .s on the plan of v' o'. If a side elevation has been drawn, in which the plane in cpiestion is seen as a line M" X", the height of the points of intersection can be obtained therefrom directly. 429. The intersection of fun qvadrangidar jj/jramids. Let one pyramid be vertical; altitude r' z' ; base efgh. having its longer edges inclined 30° to V. The oblique pyramid. Let s' y', the axis of the oblique pyramid, be parallel to \' liut inclined 6° to H, and be at some small distance (approximately v h) in front of the axis of the vertical pyramid ; then -^ c will contain tlie \Aa\\ of tlie axis, and also of the diagonally opposite edges sa and sc, if we make — as we may — the additional requirement that a' e', the diagonal of the base, shall lie in the same vertical ijlane with the axis. Instead of taking a sej^arate end view of the olilique pyramid we may rotate its liase on the diagonal a' c' so that its foremost corner appears at //" and tlie rear corner at d" . whence b' and d' are derived by perpendiculars b" b' and d" d', and then the edges -s'^' and s' d'. For the plans b and d use sc as the trace of the usual reference plane, and offsets equal to h' h" and d'd", as i^reviously. The angle a' c' d', or , is the inclination of the shorter edges of the base to V. The intersection. "Without going into a detailed construction for each point of the outline of interpenetration it may be stated that eacli method of the ])receding article is illu.strated in this 162 THEORETICAL AND PRACTICAL GRAPHICS. ^"ig-. as-i problem, and that there is no special reason why either should have a preference in any case except where by properly choosing between them we may a^-oid the intersection of two lines at a very acute angle — a kind of intersection which is always undesirable. In the interest of clearness only the vhible lines of the inter- , section are indicated on tlie plan, (a) Auxiliary plane perpendicular tn V. To find 'm, the intersection of edge .if d with the face v h e, take s' d' as tire trace of the auxiliary plane containing tlie edge in question; this cuts the limiting edges of the face at i' and n' wliich then project back to the plans of the edges at i and II. Drawing ni we note m, when it crosses s d, and project ii) to m' on s' d'. Had ni failed to meet s d within the limit of the face v h e we would conclude that our assumption that .s d met tliat face was incorrect, and would tlien proceed to test it as to some other face, unless it was evident on ins})ection that the edge cleared the other solid entirely, ^>„ as is the case with s h, ,,--" / s'h', in the present in- stance. By using s' h' as an auxiliary plane the student will get a graphic proof of failure to intersect. (b) Auxiliary plane vertical. This case is illustrated by using v g as the trace of an auxiliarj' vertical plane containing the edge vg,r'(/'. Thinking this edge may possibly meet the face sba we proceed to test it on that assumi^tion. The plane vg crosses » a at I, and sb at p; these project to /' on s' a' and to p' on »' b' ; then p'l' meets v' g' at be carried around a bend by a four-piece elbow; the procedure would still closely resemble that of the last problem. Instead of one joint or curve of intersection there would Vie three, one less than the number of pieces in the pipe. Let oq'< sliow the size of the cylinders employed, and be at the same time the plan of the 164 THEOBETICAL AND PRACTICAL GRAPHICS. vertical piece o' s' n' a'. Until we know where a' n' will lie we have to draw o' a' and s' n' until they meet the elements from