Okt. O '|C^AA.i-» '^c^/iaX^ Ua^J2/vOi- THE PRACTICAL DRAUGHTSMAN'S BOOK OP INDUSTRIAL DESIGN, AND MACHINIST'S AND ENGINEER'S DRAWING COMPANION: FORMING A COMPLETE COURSE OP Jlttjatitcal, ^Kgiiweriitg, m)i ^xt^ikdmil graining. TRANSLATED FROM THE FRENCH 0» M. ARMENGAUD, THE ELDER, PROFESSOR OF DESIGN IN THE CONSERVATOIRE OF ARTS AND INDUSTRY, PARIS, AND MM. ARMENGAUD, THE YOUNGER, AND AMOUROUX, CIVIL ENGINEERS. REWRITTEN AND ARRANGED, WITH ADDITIONAL MATTER AND PLATES, SELECTIONS FROM AND EXAMPLES 0? THE MOST USEFUL AND GENERALLY EMPLOYED MECHANISM OF THE DAY. BY WILLIAM JOHNSON, Assoc. Inst., C.E , EDITOR OF "THE PRACTICAL MECHANIC'S JOURNAL." PHIL AD E LP II 1 A: H E N K Y CAREY B A I R D, INDUSTRIAL PUBLISHER, No. 4 6 WALNUT STREET. 1871. I o ^ '=\ E 7 / ■o V PREFACE. Industrial Design is destined to become a universal language ; for in our material age of rapid transition from abstract, to applied, Science — in the midst of our extraordinary tendency towards the perfection of the means of conversion, or manufacturing production — it must soon pass current in every land. It is, indeed, the medium between thought and Execution ; by it alone can the genius of Conception convey its meaning to the skill which executes — or suggestive ideas become living, practical realities. It is emphatically the exponent of the projected works of the Practical Engineer, the Manufacturer, and the Builder ; and by its aid only, is the Inventor enabled to express his views before he attempts to realise them. Boyle has remarked, in his early times, that the excellence of manufactures, and the facility of labour, would be much promoted, if the various expedients and contrivances which lie concealed in private hands, were, by reciprocal communications, made generally known ; for there are few operations that are not performed by one or other with some peculiar advantages, which, though singly of little importance, would, by conjunction and concurrence, open new inlets to knowledge, and give new powers to diligence ; and Herschel, in our own days, has told us that, next to the establishment of scientific institutions, nothing has exercised so powerful an influence on the progress of modern science, as the publication of scientific periodicals, in directing the course of general observation, and holding conspicuously forward models for emulative imitation. Yet, without the aid of Drawing, how can this desired reciprocity of information be attained ; or how would our scientific literature fulfil its purpose, if denied the benefit of the graphic labours of tlie Draughtsman ? Our verbal interchanges would, in truth, be vague and barren details, and our printed knowledge, misty and unconvincing. Independently of its utility as a precise art. Drawing really interests the student, whilst it instructs him. It instils sound and accurate ideas into his mind, and develops his intellectual powei's in compelling him to observe — as if the objects he delineates v/ere really before his eyes. Besides, he always does that the best, which he best understands ; and in this respect, the art of Drawing operates as a powerful stimulant to progress, in continually yielding new and varied results. A chance sketch — a rude combination of carelessly considered pcncillings — the jotted memoranda of a contemplative brain, prying into the corners of contrivance — often form the nucleus of a splendid invention. An idea thus preserved at the moment of its birth, may become of incalculable value, when rescued from the desultory train of fancy, and treated as the sober olVspring of reason. In nice gradations, it receives the refining touches of leisure — becoming, first, a finislied sketch, — then a drawing by the practised hand — so that many minds may find easy access to it, for their joint counsellings to improvement — unlil it linally emerges from the workshop, as a practical triunipli of mechanical invention— an illustrious I'Mmipio oi' a happy PREFACE. corabination opportunely noticed. Yet many ingenious men are barely able even to start this train of production, purely from inability to adequately delineate their early conceptions, or furnish that transcript of their minds which might make thdr thoughts immortal. If the present Treatise succeeds only in mitigating this evil, it will not entirely fail in its object ; for it will at least add a few steps to the ladder of Intelligence, and form a few more approaches to the goal of Perfection — "Thou hast not lost an hour whereof there is a record; A written thought at midnight will redeem the livelong day." The study of Industrial Design is really as indispensably necessary as the ordinary rudiments of learning. It ought to form an essential feature in the education of young persons for whatever profession or employment they may intend to select, as the great business of their lives ; for without a knowledge of Drawing, no scientific work, whether relating to Mechanics, Agriculture, or Manufactures, can be advantageously studied. This is now beginning to receive acknowledgment, and the routines of study in all varieties of educational establishments are being benefited by the introduction of the art. The special mission of the Practical Draughtsman's Book of Industrial Design may almost be gathered from its title-page. It is intended to furnish gradually developed lessons in Geometrical Drawing, applied directly to the various branches of the Industrial Arts : comprehending Linear Design proper ; Isometrical Perspective, or the study of Projections ; the Drawing of Toothed Wheels and Eccentrics ; with Shadowing and Colour- ing ; Oblique Projections ; and the study of parallel and exact Perspective ; each division being accompanied by special applications to the extensive ranges of Mechanics, Architecture, Foundry-Works, Carpentry, Joinery, Metal Manufactures generally, Hydraulics, the construction of Steam Engines, and Mill-Work. In its compilation, the feeble attraction generally offered to students in elementary form has been carefully considered ; and after every geometrical problem, a practical example of its application has been added, to facilitate its comprehension and increase its value. The work is comprised within nine divisions, appropriated to the different branches of Industrial Design. The first, which concerns Linear Drawing only, treats particularly of straight lines — of circles — and their application to the delineation of Mouldings, Ceilings, Ploors, Balconies, Cuspids, Rosettes, and other forms, to accustom the student to the proper use of the Square, Angle, and Compasses. In addition to this, it affords examples of different methods of constructing plain curves, such as are of frequent occurrence in the arts, and in mechanical combinations — as the ellipse, the oval, the parabola, and the volute ; and certain figures, accurately shaded, to represent reliefs, exemplifying cases where these curves are employed. The second division illustrates the geometrical representation of objects, or the study of projections. This forms the basis of all descriptive geometry, practically considered. It shows that a single figure is insufficient for the determination of all the outlines and dimensions of a given subject ; but that two projections, and one or more sections, are always necessary for the due interpretation of internal forms. The third division points out the conventional colours and tints for the expression of the sectional details of objects, according to their nature ; furnishing, at the same time, simple and easy examples, which may at once interest the pupil, and familiarise him with the use of the pencil. In the fourth division are given drawings of various essentially valuable curves, as Helices, and different kinds of Spirals and Serpentines, with the intersection of surfaces and their development, and workshop applications to Pipes, Coppers, Boilers, and Cocks. This study is obviously of importance in many professions, and clearly so to Ironplate-workers, as Shipbuilders and Boiler-makers, Tinmen and Coppersmiths. PREFACE. The fifth division is devoted to special classes of curves relating to the teeth of Spur Wheels, Screw? and Racks, and the details of the construction of their patterns. The latter branch is of peculiar importance here, inasmuch as it has not been fully treated of in any existing work, whilst it is of the highest value to the pattern maker, who ought to be acquainted with the most workmanlike plan of cutting his wood, ami eflecting the necessary junctions, as well as the general course to take in executing his pattern, for facilitating the moulding process. The sixth division is, in effect, a continuation of the fifth. It comprises the theory and practice of drawing Bevil, Conical, or Angular Wheels, with details of the construction of the wood patterns, and notices of peculiar forms of some gearing, as well as the eccentrics employed in mechanical construction. The seventh division comprises the studies of the shading and shadows of the jn/incipal solids — Prisms, Pyramids, Cylinders, and Spheres, together with their applications to mechanical and architectural details, as screws, spur and bevil wheels, coppers and furnaces, columns and entablatures. These studies naturally lead to that of colours — single, as those of China Ink or Sepia, or varied ; also of graduated shades produced by successive flat tints, according to one method, or by the softening manipulation of the brush, according to another. The pupil may now undertake designs of greater complexity, leading him in the eighth division to various figures representing combined or general elevations, as well as sections and details of various complete machines, to which are added some geometrical drawings, explanatory of the action of the moving pai-ts of machinery. The ninth completes the study of Industrial Design, with oblique projections and parallels, and exact perspective. In the study of exact perspective, special applications of its rules are made to architecture and machinery by the aid of a perspective elevation of a corn mill supported on colunms, and fitted up with all the necessary gearing. A series of Plates, marked a, b, &c., are also interspersed throughout the work, as examples of finished drawings of machinery. The Letterpress relating to these Plates, together with an illustrated chapter on Drawing Instruments, will form an appropriate Appendix to the Volume. The general explanatory text embraces not only a description of the objects and their movements, but also tables and practical rules, more particularly those relating to the dimensions of the principal details of machinery, as facilitating actual construction. Such is the scope, and such are the objects, of the Practical Draughtsman's Book of Industrial Design. Such is the course now submitted to the consideration of all who are in the slightest degree connected with the Constructive Arts. It aims at the dissemination of those fundamental teachings which arc so essentially necessary at every stage in the application of the forces lent to us by Nature for the conversion of her materials. For ■' man can only act upon Nature, and appropriate her forces to his use, by coniprohcnding her laws, and knowing those forces in relative value and measure." All art is the true apitlication of knowledge to a practical end. We have outlived the times of random construction, and ihc mvvc lu^apinL; together of natural substances. We must now design carefully and delineate accurately before we prooood to execute — and the quick pencil of the ready draughtsman is a proud possession for our purpose. Let the youthful student think on this ; and whether in the workshop of the Engineer, the studio of the Architool. or the factory of the Manufacturer, let him remember that, to spare the blighting of his fdmlosi hoiu's. auil the marring of his fairest prospects — to achieve, indeed, his higher a8i)iratioiis, and vei-ity his loftier thoughts. which point to eminence — he must give his days and nights, his business and his leisure, to the study of 5 n I) u 5 1 1 i a I SI c s i 9 ti . ABBREVIATIONS AND CONVENTIONAL SIGNS. In order to simplify the language or expression of arithmetical and geometrical operations, the following conventional wfijns are used : — The sign + signifies plus or mo/v, and is placed between two or mure terms to indicate additior. Example : 4 + 3, is 4 plus 3, that is, 4 added to 3, or 7. The sign — signifies minus or less, and indicates subtraction. Ex. : 4 — 3, is 4 minus 3, that is, 3 taken from 4, or 1. The sign X signifies multiplied by, and, placed between two terms, indicates multiplication. Ex. : 5 X 3, is 5 multiplied by 3, or 15. When quantities are expressed by letters, the sign may be suppressed. Thus \ve .write, indifierently — a X b, or ab. The sign : or (as it is more commonly used) -j-, signifies divided by, and, placed between two quantities, indicati»s division. Ex. 12 12 : 4, or 12 4- 4, or — , is 12 divided by 4. The sign = signifies equals or equal to, and is placed between two expressions to indicate their equality. Ex. : 6 + 2 = 8, meaning, that 6 plus 2 is equal to 8. The union of these signs, I ;: I indicates geometrical proportion. Ex. : 2 : 3 :: 4 : 6, meaning, that 2 is to 3 as 4 is to 6. The sign \ ~ indicates the extraction of a root ; as, ■v/ 9 = 3, meaning, that the square root of 9 is equal to 3. The interposition of a numeral between the opening of this sign, y', indicates the degree of the root. Thus — {/27= 3, expresses that the cube root of 27 is equal to 3. The signs z. ^^^ 7 indicate respectively, smaller than and greater than. Ex. : 3 / 4, = 3 smaller than 4 , and, reciprocally, 4 7 3, = 4 greater than 3. Fig. signifies figure ; and pi., plate. ' FRENCH AND ENGLISH LINEAR MEASURES COMPARED. French. English 1 Millimetre = •0394 Inches. 11) Millimetres == 1 Centimetre = •3937 " 10 Centimfetres = 1 Decimetre = 3-9371 " 10 Decimetres = 1 Metre = • 3-2809 Feet. 1-0936 Yards. 10 Metres = 1 Decametre = 1-9884 Poles or Eods. 10 Decametres -= 1 Hectometre = 19-8844 " 10 Hectometres = 1 Kilometre -i 49-7109 Furlongs. 6-2139 MDes. 10 Kilometres = 1 Myriametre 62-1386 " Bullish. 12 Inches 3 Feet 53 Yards 40 Poles 8 Furlongs ) 1760 Yards 1 Inch. = 1 Foot = 1 Yard = 1 Pole or Rod = 1 Furlong = 1 Mile French. . ( 25-400 Millimetrea. ' \ 2-540 Centimetres. : 3-048 Decimetres. : 9-144 5-029 Metres. 12 Decametres. : rblO Hcctometrea C N T E N T S. 1 Preface, .... Abbreviations and Conventional signs, VJidB iii vi CHAPTER I. LINEAR DRAWING, Definitions and Problems : Plate I. Lines and surfaces, - . . - . Applications. Designs for inlaid pavements, ceilings, and balconies : Plate II., . . . . . Sweeps, sections, and mouldings : Plate III., Elementary Gothic forms and rosettes : Plate IV., Ovals, Ellipses, Parabolas, and Volutes : Plate V., Rules and Practical Data. Lines and surfaces, - - . . . CHAPTER IT. THE STUDY OF PROJECTIONS, - Elementary Principles : Plate VI. Projections of a point, - - - Projections of a straight line, ... Projections of a plane surface, ... Of Prisms and Other Solids : Plate VII., - Projections of a cube : Fig. /^, - - Projections of aright scpiare-based prism, or rectan- gular parallclopipod : Fig. [g, - Projections of a quadrangular pyramid; Fig. (g, - Projections of a right prism, partially hollowed, as Fig. ©, Projections of a right cylinder : Fig. [1, Projections of a right cone : Fig. [^, - . Projections of a sphere : Fig. (g, - Of shadow lines, . - . . . Projections of grooved or fluted cylin(h'rs and ratchet-wheels: rt.Ai'E VUL., . . . The elements of architecture : Plate IX., - Outline of the T'uscan order, ... Rules and Practical Data. The measurement of solids, - - . . ib. 11 13 14 15 19 22 lb. 23 tb. 24 ib. 25 tb. ib. ib. ■ib. 26 ib. 27 28 29 30 CHAPTER in. ON COLORING SECTIONS, WITH APPLICA- TIONS. Conventional colors, .... Composition or mixture of colors: Plate X., Continuation of the Study of Projections. Use of sections — details of machinery : Plate XL, Simple applications — spindles, shafts, couplings, ■wooden patterns : Plate XII., . . . Method of constructing a wooden model or pattern of a coupling, ..... Elementary applications — rails and chairs for rail- ways : Plate XIII., .... Rules and Practical Data. Strength of materials, .... Resistance to compression or crushing force, Tensional resistance, .... Resistance to flexure, .... Resistance to torsion, . ^ . . Friction of surfaces in contact, ... 35 ib 36 39 40 41 42 ib. 43 44 46 49 CHAPTER IV. THE INTERSECTION AND DEVELOPMENT OF SURFACES, WITH APPLICATIONS, - . 49 The Intersections of Cylinders and Cones ; Plate XIV. Pipes and boilers, - - - . - 50 Intersection of a cone with a sphere, . . ib. Develoi'ments, - - - - - - ib Development of the cylinder, . - - 51 Development of the cone, - . . - ib The Delineation and Devklovment of IIelioks, Screws, and Seki-entines : Plate XV. Helices, - - - - - - 62 Dovclopiiu'iit of the helix, . . - . 5U Screws, ... . . ,7,_ Inlcrniil screws, ..... fi.j Sci'pciililii'S. -.-... |7). .\]>plicaliiin of the helix the construction of a staircase: I'l.Mir \\'l., - - - - 5^ Iv CONTENTS. The intersection of surfaces — applications to stop- cocks : Plate XVII., . - - . EuLES AND Practical Data. Steam, ...... Unity of heat, . . . . . Heating surface, . - . . . Calculation of the dimensions of boilers, Dimensions of firegrate, - - . . Chimneys, .-.--. Safety-valves, . . . . . CHAPTER V. THE STUDY AND CONSTRUCTION OF TOOTHED GEAR, Involute, cycloid, and epicycloid : Plates XVIII. and XIX. Involute : Fig. 1, Plate XVIII., - Cycloid: Fig. 2, Plate XVIII., - External epicycloid, described by a circle rolling about a fixed circle inside it : Fig. 3, Plate XIX., Internal epicycloid : Fig. 2, Plate XIX., - Delineation of a rack and pinion in gear : Fig. 4, Plate XVIII., - . - . . Gearing of a worm with a worm-wheel : Figs. 5 and 6, Plate XVIII., Cylindrical ok Spdr Gearing : Plate XIX. External delineation of two spur-wheels in gear : Fio- 4 ..... . Delineation of a couple of wheels gearing internally : Fig. 5, - _ Practlcat delineation of a couple of spur-wheels : Plate XX., The Delineation and Construction of Wooden Pat. terns for Toothed Wheels : Plate XXI., Spur-wheel patterns, .... Pattern of the pinion, . Pattern of the wooden-toothed spur-wheel, - Core moulds, ..... Bulks and Practical Data. Toothed gearing, ..... Angular and circumferential velocity of wheels. Dimensions of gearing, .... Thickness of the teeth, .... Pitch of the teeth, ..... Dimensions of the web, .... Number and dimensions of the arms, Wooden patterns, ..... CHAPTER VI. CONTINUATION OF THE STUDY OF TOOTHED GEAR. Conical or bevil gearing, . , . Design for a pair of bevil-wheels in gear : Plate XXII., Construction of wooden patterns for a pair of bevil- wheels : Plate XXIII., . . . . Involute and Helical Teeth : Plate XXIV. Delineation of a couple of spur-wheels, with involute teeth : Figs. 1 and 2, - . - - Helical gearing : Figs, 4 and 5, - Contrivances FOE Obtaining DifferentialMovements. 56 58 59 60 ib. 61 ib. 62 63 ib. 64 65 tb. ib. 67 ib. 68 69 70 ib. 71 ib. 72 74 75 ib. 76 77 ib. - ib. ib. 80 82 83 The delineation of eccentrics and cams: Plate XXV., 85 Circular eccentric, - ... ib. Heiirt-sliaped cam : Fig. 1, - - . . ib. Cam for producing a uniform and intermittent movement : Figs. 2 and 3, ... ih. Triangular cam : Figs. 4 and 5, - - -86 Involute cam : Figs. 6 and 7, - . . ib Cam to produce intermittent and dissimilar move- ments : Figs. 8 and 9, - - - .87 Rules and PnACTicAi, Data. Moclianical work of effect, . . - - 88 The simple machines, - - . - 91 Centre of gravity, - - . - - 93 On estimating the power of prime movers, - - ib. Calculation for the brake, .... ib. The fall of bodies, - - - - - 95 Momentum, - . . - - ib. Central forces, - - - - - ib. CHAPTER VII. ELEMENTARY PRINCIPLES OF SHADOWS, - Shadows of Prisms, Pvramids, and Cylinders : Plate XXVI. Prism, - - * - Pyramid, - - - - Truncated pyramid, . . . . . Cylinder, --.... Shadow cast by one cylinder on another, - Shadow cast by a cylinder on a prism, Shadow cast by one prism on another. Shadow cast by a prism on a cylinder, Principles of Shading : Plate XXVII., Illumined surfaces, - - . - . Surfaces in the shade, . . . . Flat-tinted shading, - . . . . Shading by softened washes, . . . Continuation of the Study of Shadows : Plate XXVIII. Shadow cast upon the interior of a cylinder. Shadow cast by one cylinder upon another, Shadows of cones, ..... Shadow of an inverted cone. Shadow cast upon the interior of a hollow cone. Applications, . . . . . Tuscan Order : Plate XXIX. Shadow of the torus, - - - - Shadow cast by a straight line upon a torus, or quarter round, . . - - . Shadows of surfaces of revolution, ... Rules and Practical Data. Pumps, .... Hydrostatic principles, .... Forcing pumps. ..... Lifting and forcing pumps, - . . . The hydrostatic press, .... Hydrostat-ical calculations and data — discharge of water through different orifices, - Gauging of a water-course of uniform section and fall, Velocity of the bottom of water-courses, Calculation of the discharge of water through rect- angular orifices of narrow edges, - 96 ib. 97 98 ib. ib. ib. 99 ib. 100 ib. ib. 101 102 103 ib. ^^. 104 105 ib. ib. i07 ib. 108 ib. ib. 109 ?"&. ?7)., Ill)* ib . HI CONTENTS. Calculation of the discharge of water through over- shot outlets, ..... 114 To determine the width of an overshot outlet, - tb. To determine the depth of the outlet, - - ib. Outlet with a spout or duct, - - - -116 TO TOOTHED CHAPTER VIII. APPLICATION OF SHADOWS GEAR: Plate XXX. Spur-wheels : Figs. 1 and 2, - Bevil-wheels : Figs. 3 and 4, - - ■ Application of Shadows to Screws : Plate XXXI. Cylindrical square-threaded screw : Figs. 1, 2, 2" and 3, ----- • Screw with several rectangular threads : Figs. 4 and .5 Triangular-threaded screw : Figs. 6, 6°, 7, and 8, ■ Shadows upon around-threaded screw : Figs. 9 and 10, 119 Application of Shadows to a Boiler and its Furnace : Plate XXXII. Shadow of the sphere : Fig. 1, - - - Shadow cast upon a hollow sphere : Fig. 2, Applications, . . . . - Shading in Black — Shading in Colours : Plate XXXIIT., ib. 117 118 ib. ib. tb. ib. 120 ib. 122 CHAPTER IX. THE CUTTING AND SHAPING OF MASONRY : Plate XXXIV., . - . . The Marseilles arch, or ari-ilre-voussure : Figs. 1 and 2, - - - - Rules and Practical Data. Hydraulic motors, - - . . - Undershot water-wheels, with plane floats and a circular channel, - - - - - Width, Diameter, ...... Velocity, . . . . ^ - Number and capacity of the buckets. Useful effect of the water-wheel, . . . Overshot water-wheels, - r - - Water-whecis, with radial floats, . . - Water-wheels with curved buckets. Turbines, ...... Realakks ON Machine Tools, . . , . - 123 ib. 126 ib. ib. 127 ib. ib. ib. 128 129 130 ib. 131 OHAPTEU X. THE STUDY OF MACHIN ERY AND SKETCHING. Various applications and combinations, - - 133 The Sketching of Macuinkry : Plates XXXV. and XXXVI., ib. Drilling Machine, - - - - - ib. Motive Machine.s. Water-wheels, . . . - . i;!.'j Construction and setting up of water-wheels, - ib. Delineation of watcr-wliecls, ... i;i() Design for a water-wheel, - - - - 1117 Sketch of a water-wheel, . . - - ib. (Wkrshot Water-Whei;i,s : Fig. 12, - - - ib. Delineating, sketching, and designing overshot watcr-whcels, - - - - - 13H Water-Pumps : Plate XXXVII. Geometrical delineation, - - - - 138 Action of the pump, . - - . - 139 Steam Motors. High-pressure expansive steam-engine : Plates XXXVIII., XXXIX., and XL., - - - 141 Action of the engine, .... 142 Parallel motion, - - - . ib. Details of Construction. Steam cylinder, - - - 143 Piston, .... • ib. Connecting-rod and crank, - - - > ib. Fly-wheel, - - - - - • ib. Feed-pump, ----- ib. Ball or rotating pendulum governor, - 144 Movements oftheDistribution and Expansion Valves, ib. Lead and lap, - . . . . 145 Rules and Practical Data. Steam-engines : low pressnre condensing engine without expansion valve, - - - - 146 Diameter of piston, . - - . - 147 Velocities, .----- 148 Steam-pipes and passages, - - - - ib. Air-pump and condenser, - - - ■ ib. Cold-water and feed-pumps, - - - . 149 High pressure expansive engines, - - - ib. Medium pressure condensing and expansive steam- engine, ...... 151 Conical pendulum, or centrifugal governor, - 153 CHAPTER XI. OBLIQUE PROJECTIONS. Application of rules to the delineation of an oscilla- ting cylinder : Plate XLL, CHAPTER XII. PARALLEL PERSPECTIVE. Principles and applications : Plate XLIL, - 154 .55 CHAPTER Xin. TRUE PERSPECTIVE. Elementary principles : Plate XLIIL, - - 11)8 First problem — the perspective of a hollow prism : Figs. 1 and 2, - - - - - ib. Second problem — the perspective of a cylinder: Figs. 3 and 4, - - - - - ir>a Third problem — the perspcclivc of a regular solid. when the point of sight is situated in a piano passing through its axis, and perpendicular to the plane of the picture : Figs. 5 and 6, - - ItiO Fourlh problem— the perspective of a bearing brass, phiced with its axis vertical: Figs. 7 and 8, - il>. Fifth jirobleni — the perspective of a stojnuH'li with a splieric-al boss : Figs. 9 and 10, - - - ib. Sixtli problem — the i)erspectivo of an object iilaced in any i)osition with regard \i.t the ])lane of tlie ]>icture : Figs. 11 and 12. - - - - lOl A])plications — Hourinill driven by belts: Plavks XLIV. and Xl-V. nescription of the mill, - - - - ib. CONTENTS. Representation of the mill in perspective, - Notes of recent improvements in flour-mills, Schicle's mill, .... Mulliu's " ring millstone," ... Barnett's millstone, . - - - Hastie's arrangement for driving' mills, Currie's improvements in millstones, Rur,ES AND Practiciai- Data. Work performed by various machines. Flour-mills, - - - - - Saw-mills, - - . - . Veneer sawing machines, - . - Circular saws, . . - . PAGE 163 164 ih. 165 166 ib. ih. 168 170 171 ib. CHAPTER XIV. EXAMPLES OF FINISHED DRAWINGS OF MA- CHINERY. Example Plate [^, balance water-meter, - - 172 Example Plate [g, engineer's shaping machine, - 174 Example Plate ©, [o), 1^, express locomotive engine, 178 Example Plate [^, wood planing machine, - - 180 Example Plate (g, washing machine for piece goods, 182 Example Plate [X], power-loom, ... ih. Example Plate 0, duplex steam boiler, - - 183 Example Plate (J), direct-acting marine engines, - 184 CHAPTER XV. DRAWING INSTRUMENTS, - 186 INDEX TO THE TABLES. ench and English linear measures compared, ultipliers for regular polygons of from 3 to 12 sides. Approximate ratios between circles and squares, - Comparison of Continental measures with French millime- tres and English feet, . . . - - Surfaces and volumes of regular polyhedra. Proportional measurements of the various parts of the (modern) Doric order, . . . - . Proportional measurements of the various parts of the Tuscan order, ...--- Weights which solids, such as columns, pilasters, supports, will sustain without being crushed, . . . Weights which prisms and cylinders will sustain when submitted to a tensile strain, - - - - Diameters of the journals of water-wheel and other shafts for heavy work, ------ Diameters for shaft journals calculated with reference to torsional strain, ------ Ratios of friction for plane surfaces, - - - Ratios of friction for journals in bearings, - Pressures, temperatures, weights, and volumes of steam, - Amount of heat developed by one kilogramme of fuel. Thickness of plates in cylindrical boilers, - Dimensions of boilers and thickness of plates for a pressure of five atmospheres, - - - - - Diameters of safety-valves, . - - - - Numbers of teeth, and diameters of spur gear. Pitch and thickness of spur-teeth for different pressures, - PAGE 6 19 20 21 30 33 34 42 44 46 49 ib. 58 59 ib. 60 •62 73 77 PA3B Dimensions of spur-wheel arms, - - - - 78 Average amount of mechanical effect producible by men and animals, - - - - - - 89 Heights corresponding to various velocities of falling bodies, 94 Comparison of French and English measures of capacity, - 110 Discharges of water through an orifice one mfetre in width, 111 Discharge of water by overshot outlets of one mfetre in width, 113 Discharge of water through pipes, - - - - 115 Dimensions and practical results of various kinds of turbines, 131 "Velocity and pressure of machine tools or cutters, - 132 Diameters, areas, and velocities of piston in low pressure double-acting steam engines, with the quantities of steam expended per horse power, - - - - 147 Force in kilogrammfetres given out with various degrees of expansion, by a cubic mfetre of steam, at various pres- sures, ------- 150 Proportions of double-acting steam-engines, condensing and non-condensing, and with or without cut-off, the steam being taken at a pressure of four atmospheres in the con- densing, and at five atmospheres in the other engines, - 152 Proportions of medium pressure condensing and expansive steam-engines, with two cylinders, on Woolf 's system — pressure four atmospheres, - - - - ib. Dimensions of the arms, and velocities of the balls of the conical pendulum, or centrifugal governor, - - 153 Power, quantity of wheat ground, and number of pairs of stones, with their accessory apparatus, required in flour- mills. 16S PRACTICAL DRAUGHTSMAN'S BOOK OF INDUSTRIAL DESIGN. CHAPTER I. LINEAR DRAWING. I N Drawing, as applied to Mechanics and Architecture, and to the Industrial Arts in general, it is necessary to consider not only the aiere representation of objects, but also the relative principles of action of their several parts. The principles and methods concerned in tliat division of the art which is termed linear drawing, and which is the foundation of all drawing, whether industrial or artistic, are, for the most part, derived from elementary geometry. This branch of drawing has for its object the accurate delineation of surfaces and the con- struction of figures, obtainable by the studied combinations of lines ; and, with a view to render it easier, and at the same time more attractive and intelligible to the student, the present work has been arranged to treat successively of definitions, principles, and problems, and of the various applications of which these are capable. Many treatises on linear drawing already exist, but all these, considered apart from their several objects, seem to fail in the due development of the subject, and do not manifest that general ad- vancement and increased precision in details which are called for at the present day. It has therefore been deemed necessary to begin with these rudimentary exercises, and such exemplifications have been selected as, with their varieties, are most frcqiioutly met with in practice. Many of the methods of construction will be necessarily such as are already known ; but they will be limited to those which are absolutely indispensable to the development of the principles and their applications. DEFINITIONS. OF LINES AND SURFACES. PLATE I. In Geometry, space is described in (he terms of its three dimen- sions — length, oreadth or thickness, iiiul height or (lc|)th. The combination of two of these dimensidiis rciircsi^iils siirfnce, tnd one dimension takes the form of a lltw. Lines. — There are several kinds of lines used in drawing — straight or right lines, curved lines, and irregular or broken lines. Right lines are vertical, horizontal, or inclined. Cmred lines are circular, elliptic, parabolic, <^c. Surfaces. — Surfaces, which are always bounded by lines, are plane, concave, or convex. A surface is plane when a straight-edge is in contact in every point, in whatever position it is applied to it. If the surface is hollow so that the straight-edge only touches at each extremity, it is called concave ; and if it swells out so that the straight-edge only touches in one point, it is called convex. Vertical lines. — By a vertical line is meant one in the position which is assumed by a thread freely suspended from its upper ex- tremity, and having a weight attached at the other ; such is the line A B represented in fig. A. This line is always straight, and the shortest that can be drawn between its extreme pomts. Plumb-line. — The instrument indicated in fig. A is called a plumb-line. It is much employed in building and the erection of machinery, as a guide to the construction of vertical lines and surfaces. Horizontal line. — When a liquid is at rest in an open vessel, its upper surface forms a horizontal plane, and all lines drawn upon such surface are called horizontal hnos. Levels. — It is on this' principle that what are called fiuid levels are constructed. One description of fluid level consists of two upright glass tubes, connected by a pipe comiiuinicating with the bottom of each. Wlien the instrument is partly filled with water, the wafer will stand at the same height in both tubes, and thereby indicate the true level. Another form, and one more generally used, denominated a spirit level — spirit being usually eniployeil — consists of a glass tube (fig. ©) enclosed in a metal case, <», attached by two supports, b, to a plate, c. The tube is almost filled \vith liquid, and the bubble of air, </, which remains, is always exactly in the centre of the tube wlion any surface, c P, ou which the iiistrnmeiit is placed, is perfectly level. Masons, carpenters, joiners, and other mechanics, are in iJio habit of using the instrument represented in fig. ©, consisting sinqily of a ])lniiil) line attached to the point of junction of the two inclined side pieces, ab, be, of equal lengtli, and connected iic:ir their free ends hy the cross-piece, a n, which has a marl; :il [if THE PRACTICAL DRAUGHTSMAN'S cuiitiv. VVIiLMi the plumb lino coincidus with this mark, the object, c u, on which the instrument is placed, is exiictly horizontal. Perpendiculars. — If the vertical line, a b, fig. 1, be placed on (he horizontal line, c d, the two lines will be perpendicular to, and Torm right angles with, each other. If now we suppose these lines to be turned round on the point of intersection as a centre, always preserving the same relative position, they will in every position 'x! perpendicular to, and at right angles with, each other. Thus the line, i o, fig. 5, is at right angles to the line, e f, although neither of them is horizontal or vertical. Broken lines. — It is usual to call those lines broken, which con- sist of a series of right lines lying in different directions — such as tiie lines b, a, e, h, f, n, fig. 14. Circular lines — Circumference. — The continuous line, e f g H, fig. 5, drawn with one of the points of a pair of compasses — of which the other is fixQd — is called the circumference : it is evidently equally distant at all points from the fixed centre, o. Radius. — The extent of opening of the compasses, or the dis- tance between the two points, o, f, is called a radius, and conse- quently all lines, as o e, o f, o a, drawn from the centre to the circumference are equal radii. Diameter. — Any right line, l h, passing through the centre o, and Ihnited each way by the circumference, is a diameter. The di'imeter is therefore double the length of the radius. Circle. — The space contained within the circumference is a jdane surface, and is called a circle : any part of the circumference, e r F, or F L G, is called an arc. Chords. — Right lines, e f, f g, connecting the extremities of arcs, are chords ; these lines extended beyond the circumference oecome secants. Tangent. — A right line, a b, fig. 4, which touches the circumfer- ence in a single point, is a tangent. Tangents are always at right angles to the radius which meets them at the point of contact, b. Sector. — Any portion, as b o h c, fig. 4, of the surface of a circle, comprised within two radii and the arc which connects their outer extremities, is called a sector. Segment. — A segment is any portion, as e f i, fig. 5, of the surfiice of a circle, comprised within an arc and the chord which subtends it. Right, continuous, and broken lines, are drawn by the aid of the square and angle ; circular lines are delineated with compasses. Angles. — We have already seen that, when right lines are per- pendicular to each other, they form right angles at their inter- sections: when, however, they cross each other without being perpendicular, they form acute or obtuse angles. An acute angle is one which is less than a right angle, as f c d, fig. 2 ; and an obtuse angle is greater than a right angle, as g c d. By angle is generally understood the extent of opening of two intersecting anes, the point of intersection being called the apejc. An angle is rectilinear when formed by two right lines, mixtihnear when formed by a right and a curved line, and curvilinear when formed by two curved lines. Measurement of angles. — If, with the apex of an angle as the centre, we describe an arc, the angle may be measured by the por- tion of the arc cut off by the lines forming the angle, with reference to the whole circle ; and it is customary to divide an entire circle into 360 or 400* equal parts, called degrees, and mstrumests called protractors, and represented in figs. ©, g, are constructed, whereby the number of degrees contained in any angle are ascertain- able. The first, fig. [6), which is to be found in almost every set of mathematical instruments, being that most in use. consists of a semicircle divided into 180 or 200 parts. In making use of it, its centre, b, must be placed on the apex of the angle in such a manner that its diameter coincides with one side, a b, of the angle, when the measure of the angle will be indicated by the division intersected by the other side of the angle. Thus the angle, a be, is one of 50 degrees (abbreviated 50°), and it will always have this measure, whatever be the length of radius of the ai-c, and conse- quently whatever be the length of the sides, for the measuring arc must always be the same fraction of the entire circumference The degree is divided into 60 minutes, and the minute (or 1') into 60 seconds (or 60") ; or when the circle is divided into 400 degrees, each degree is subdivided into 100 minutes, and each minute into 100 seconds, and so on The other protractor, fit. E, of modern invention, possesses the advantage of not requiring access to the apex of the angle. It consists of a complete circle, each half being divided on the inner side into 180 degrees, but externally the instrument is square. It is placed against a rule, R, made to coincide with one side, c e, of the angle — ^the other side, d c, crosses two opposite divisions on the circle indicating the number of degrees contained in the angle. It will be seen that the angle, dee, is, one of 50°. Oblique lines. — Right lines, which do not form right angles with those they intersect, are said to be oblique, or inclined to each other. The right lines, g c and f g, fig. 2, are oblique, as referred to the vertical line, k c, or the horizontal line, c j. Parallel lines. — Two right lines are said to be parallel with each other when they are an equal distance apart throughout their length ; the lines, i k, a b, and l m, fig. 1, are parallel. Triangles. — The space enclosed by three intersecting lines is called a triangle ; when the three sides, as d e, e f, and f d, fig. 12, are equal, the triangle is equilateral; if two sides only, as g h. and G I, fig. 9, are equal, it is isosceles ; and it is scalene, or irregular, when the three sides are unequal, as in fig. 6. The triangle is called rectangular when any two of its sides, as d l and l k, fig. 10, form a right angle ; and in this case the side, as d k, opposite to, or subtending the right angle, is called the hypothenuse. An instrument constantly used in di-awing is the set-square," more commonly called angle; it is in the shape of a rectangular triangle, and is constructed of various proportions ; having an angle of 45°, as fig. ©, of 60° as fig. [nl, or as fig. 0, having one of the sides which form the right angle at least double the length of the other. Polygon. — A space enclosed by several lines lying at any angle to each other is a polygon. It is plane when all ttie lines lie in one and the same plane; and its outline is called its perimeter. A polygon is triangular, quadrangular, pentagonal, hexagonal, tiepta- gunal, octagonal, ^c, according as it has 3, 4, 5, 6, 7, or 8 sides A square is a quadrilateral, the sides of which, as a b, b c, c d, * As another step towards a decimal notation, it was proposed, in 1790, to divide the circle into 400 parts. The suggestion was again revived in 1840, and actually adopted by several distinguished individuals. The facility afforded to calculators l»y the many sul)niultiples possessed hy the number 360, however, accounts (or the still vary general use of the ancient system of division. BOOK OF INDUSTRIAL DESIGN. aai D A, fig. 10, are equal and perpendicular to one another, the angles consequently also being equal, and all right angles. A rectangle is a quadiilateral, ha\ing two sides equal, as a b and F N, fig. 14, and perpendicular to two other equal and parallel sides, as a j" and b n. A parallelogram is a quadrilateral, of which the opposite sides and angles are equal ; and a lozenge is a quadrilateral with all the sides, but only the opposite angles equal. A trapezium is a quadiilateral, of which only two sides, as h i and ivi L, fig. 9, are parallel. Polygons are regular when all theii- sides and angles are equal, and are otherwise irregular. All regular polygons are capable of being inscribed in a circle, hence the great facility with which they may be accurately delineated. OBSERVATIONS. Wi; have deemed it necessary to give these definitions, in order to make our descriptions more readily understood, and we propose now to proceed to the solution of those elementary problems with which, from then- frequent occurrence in practice, it is important that the student should be well acquainted. The first step, how- ever, to be takon, is to prepare the paper to be drawn upon, so that it shall be well stretched on the board. To effect this, it must be slightly but equally moistened on one side with a sponge ; the moistened side is then applied to the board, and the edges of the paper glued or pasted down, commencing with the middle of the sides, and then securing the corners. When the sheet is dry, it wilJ be uniformly stretched, and the drawing may be executed, being first made in faint pencil lines, and afterwards redelineated with ink by means of a drawing pen. To distinguish those lines which may be termed working lines, as being but guides to the foimation of the actual outlines of the drawing, we have in the plates represented the former by dotted lines, and the latter by full continuous lines. PROBLEMS. 1. To erect a perpendicular on the centre of a given right line, as c D, Jig. 1 . — From the extreme points, c, d, as centimes, and with a radius greater than half the line, describe the arcs which cross each other ill a and b, on either side of the line to be divided. A line, A B, joining these points, will be a perpendicular bisecting the line, c D, in G. Proceeding in the same manner with each half of the line, c G and G D, we obtain the porpendicuiars, i K and l im, dividing the line into lour equal parts, and we can thus divide any given light lino into 2, 4, 8, I(i, &c., equal parts. This problem is of constant application in drawing. For instance. In order to obtain the principal lines, vx and yz, wiiich divide the sheet of paper into four equal parts; with the points, rsiw, taken as near the edge of the jiaper as possible, as centres, wo describe the airs which intersect each other in p and q; and with these hist as centres, describe also the arcs wiiich cut each other in y, z. Tiie right lines, v X kiuI y /, drawn Ihrnugh Hk; poinis, r, (j, ;ind ij. z. respectively, are perpendicular to each oilier, and servo as guides in drawing on diflerent parts of the paper, and are merely pencilled in, to bo afterwards effaced. 2. To erect ajierpnuUciildr on anu given jmini, as ii, in the line c i), Jig. 1 — Mark oflT on the line, on each side of the point, two equal distances, as c h and h g, and with the centres c and g describe the arcs crossing at i or k, and the line drawn through them, and through the point h, will be the Une required. 3. To let fall a perpendicular from, a point, as l, apart from the right line, c d. — With the point l, as a centre, describe an arc which cuts the line, c d, in g and d, and with these points as centres, describe two other arcs cutting each other in M, and the right line joining l and M will be the perpendicular required. In practice, such perpendiculars are generally drawn by means of an angle and a square, or T-square, such as fig. H". 4. To draw parallels to any given lines, as v x and T z. — For regularity's sake, it is well to construct a rectangle, such as r s tu, on the paper that is being drawn upon, which is thus done : — From the points v and x, describe the arcs r, s, t, u, and apphing the rule tangentially to the two first, draw the line r s, and then in the same manner the line t u. The lines r t and s u are also obtained in a similar manner. In general, however, such parallels are more quickly di-awn by means of the T-square, wiiich may be slid along the edge of the board. Short parallel lines may be drawn with the angle and rule. 5. To divide a given right line, as a b. Jig. 3, into several equal parts. — We have already shown how a line may be di\ided into 2 or 4 equal parts. We shall now give a sunple metliod for dividing a line into any number of equal parts. From the point a, dra^ the line a c, making any convenient angle with a b ; mark off on a c as many equal distances as it is wished to divide the line a b into ; in the present instance seven. Join c b, and trom the several points mai-ked off on a c, draw parallels to c b, using the rule and angle for this purpose. The line a b will be divided into seven equal parts by the intersections of the parallel lines just drawn. Any line maldng any angle with a B, as A J, may be employed in- stead of A c, with exactly the same results. This is a very usefui problem, especially applicable to the formation of scales tor the reduction of drawings. 6. A scale is a straight line divided and subdivided into feet, inches, and parts of inches, according to English measures; or into metres, decimetres, centimetres, and millimetres, according to French measures ; these divisions bearing the same proportion to each other, as in the system of measurement from w hich they are derived. The object of tlie scale is to indicate the proportion the drawing bears to the object represented. 7. To construct a scale. — The French scale being the one adopted in this work, it will be iioccssary to state that the rnitre ( = 39-371 English inches) is the unit of nieasurouK'nt, and is. divided into 10 decimetres, 100 coiitiuii'lros, and 1000 niillinu'triw*. If it is intended to execute the drawing to a scale of i or J ; Uie metro is divided by 4 or 5, one of the divi.sions being tlio length of a mitre on the reduced scale. A lino of #this length is drawn on the pMjier, and is divided into reduced dccimetros, &c., just :is the ineti-e is itself. Fig. 7 is pai't of a scale for reducing a drawing to one-liflli. Ill this .scale an e.xtra division is placed to the loi^ of /eio, which is subdivided, to fncilitalo the oblainmont of any re- quired measure. For oxamplo, if we want a length correspondini.- to 32 ceiitiint>tres, we place one point of liio oonipassos on tin- division marked 3 to the right of zero, and tlie other on fho second 10 THE PRACTICAL DRAUGHTSMAN'S division to the left, and the length comprised between these points ^\'ill be 3 decimetres, 2 centimetres, — 32 centimetres. The diagonal scale. — When very minute measurements are re- quired, greater precision is obtained witli a diagonal scale, such as fig. 8. It is thus constructed : — Ha\ing drawn a line and divided it, as in fig. 7, draw, parallel and equal to it, ten other lines, as c, d, e,f, &c., at equal distances apart, crossing these with perpen- diculars at the decimetre divisions. From one of the smaller divisions to the left of zero, di-aw the diagonal, b i, and draw parallels to it from the remaining centimetre dinsions, 1', 2', 3', &e. From the division corresponding to 1 decimetre, draw a diagonal to the point on the extreme parallel, i 4, cut by the zero perpen- diculai-, and di-aw also the parallel diagonals, 1 — 2, 2 — 3, and 3 — 4. It will be evident, that as in the space of the ten horizontal lines, the diagonal extends one di\asion to the left, it will intersect each intermediate line, as the 1st, 2d, 3d, &c., at the distance of 1, 2, 3, &c., tenths of such division, in the same direction, so that the diagonal line, 2', will cut the 5th line at a point 2^ of a diWsion distant from zero. Thus, one point of the compasses being placed on the point Z, and the other on the intersection of the same horizontal line with the perpendicular of the decimetre di\ision 3, the measure comprised between them wUl be 3 decimetres, 2 cen- timetres, and ^, or 5 millimetres = 325 millimetres. 8. To divide a given angle, as y c D,Jig. 2, inlo two equal angles. — With tlie apex, c, as a centre, describe the arc, h i, and with the two points of intersection, H, i, as centres, describe the arcs cutting each other in j ; join j c, and the right line, j c, will divide the angle, f c d, into two equal angles, h c j and j c i. These may be subdivided in the same manner, as shovvn in the figm-e. An angle may also be divided by means of either of the protractors, 9. To draw a tangent to a given circle, o b d h, fig. 4. — ^If it is required to draw the tangent through a given point, as D, in the circle, a radius, c d, must be di-awn meeting the point, and be pro- duced beyond it, say to e. Then, by the method already given, draw a line, f g, perpendicular to c e, cutting it in d, and it will be the tangent required. K, however, it is required to draw the tangent through a given point, as a, outside the circle, a straight line must be drawn joining the point, a, and the centre, c, of the circle. After bisecting this line in the point, o, with tliis point as a centre, describe a circle passing through a and c, and cutting the ^ven circle in b and h; right lines joining ab and A H will both be tangents to the given circle, and the radii c b and c H will be perpendiculars to a b and a h respectively. 10. To find the centre of a given circle, or that loiih which a given arc, as E F G, fig. 5, is draion. — With any three points, e, f, g, as centres, describe arcs of equal cu'cles, cutting each other, and ttu-ough the points of intersection draw right lines, i o and l o ; o, the point of intersection of these two lines, is the required centre. 11. To describe a circle through any three points not in a right line. — Since only one circle can pass through the same tlu-ee points, and since any circle may be described when the centre is found and a point in the circumference given — this problem is solved in exactly the same manner as the preceding. 12. To iriscribe a circle in a given triangle, as ab c, fig. 6. — A eh'cle is said to be inscribed in a figure, when all the sides of the latter are tangents to it. Bisect any two of the angles by right lines, as A o, B o, or c o ; and from the point of intersection, o, let fall perpendiculars to the sides, as o e, o f, and o g. These per- pendiculars will be equal, and radii of the required circle, o being the centre. 13. To divide a triangle, as gk i,fig. 9, into two equal parts. — ^If the parts are not required to be similar, bisect one side, as g i, in the point, 0, with which, as a centre, describe the semicii-cle, g k i, of which G I is the diameter. This semicucle wll be cut in the point, K, by the perpendicular, k o ; mark off on g i a distance, g l, equal to G K, and draw the line, l m, parallel to h i. The triangle, g l M, and the trapezium, h i l m, will be equal to each other, and each equal to half the triangle, g h i. If the given triangle were g n i, it would also be divided into two equal parts by the line, l m. 14. To draw a square double the size of a given square, a b c d, fig. 10. — After producing from diflerent corners any two sides which are at right angles to each other, as d a and d c, to H and l, with the centre, d, and radius, d b, describe the quadrant or quarter of a circle, fee; and through the points of intersection, f and e, -nith the lines, d a and d l, di-aw parallels to d l and d a respectively, or tangents to liie quadrant, fee; the square, f g e d, will be double the area of the given square, a b c d ; and in the same manner-a square, held, may be drawn double the area of the square, F G e D. It is evident that the diagonal of one square is equal to one side of a square twice the size. 15. To describe a circle half the size of a given circle, as acbd, fig. 11. — Draw two diameters, a b and c d, at right angles to each other; join an extremity of each, as a, c, by the chord, a c Bisect this chord by the perpendicular, e f. The radius of tha required circle will be equal to e g. It follows that the annulai space shaded in the figure is equal to the smaller circle within it. 16. To inscribe in given circles, as in fig. 12, an equilateral triangle and a regular hexagon. — ^Draw any diameter, g f, and with G, as a centre, describe the arc, doe, its radius being equal to that of the given circle ; join d e, e f, and e d, and d e f will be the triangle required. The side of a regular hexagon is equal to the radius of the cu-cumscribing circle, and, therefore, in order to in- scribe it in a circle, all that is necessary is to mark off on the cir- cumference the length of the radius, and, joining the points of in- tersection, as K I L H M J, the resulting figure will be the hexagon required. To inscribe figures of 12 or 24 sides, it is merely ne- cessary to divide or subdivide the arcs subtended by the sides obtained as above, and to join the points of intersection. It is frequently necessary to draw very minute hexagons, such as screw- nuts and bolt-heads. This is done more quickly by means of the angle of 60°, [H] , which is placed against a rule, (^, or the square, in different positions, as indicated in fig. 12. 17. To inscribe a square in a given circle, as acbb, fig. 13. — Draw two diameters, as a b, c d, perpendicular to one another, ana join the points of intersection with the circle, and acbd vv-iU be the square required. 18. To describe a regular octagon about a circle having a given radius, as o e, fig. 13. — Having, as in the last case, drawn two diameters, as e f, g h, draw other two, i j, k l, bisecting tne angles formed by the former ; through the eight points of interseo- BOOK OF INDUSTRIAL DESIGN. II tinn with the circle draw the tangents, e, k, g, j, f, l, i — these tan- gents will cut each other and form the regular octagon required. This figure may also be drawn by means of the square, and angle of 45°, ©. 19. To construct a regular octagon of wMch one side is given, as A. ^,Jig- 14. — Draw the perpendicular, o d, bisecting a b ; draw a f parallel to o d, produce a b to c, and bisect the angle, c a f, by the line E A, making e a equal to a b. Draw the line o a, perpendi- cular to, and bisecting e a. o g will cut the vertical, o d, in o, which wUl be the centre of the circle circumscribing the required octagon. This may, therefore, at once be drawn by simply mark- ing otf arcs, as E H, H f, &c., equal to a b, and joining the points, E, H, F, &c. By dividing and subdividing the arcs thus obtained we can draw regular figures of 16 or 32 sides. The octagon is a figure of frequent application, as for dravvdng bosses, bearing brasses, «&c. 20. To construct a regular pentagon in a given circle, as a b c d f, also a decagon in a given circle, as e r T!d,Jig. 15. — The pentagon is thus obtained ; draw the diameters, a i, e j, perpendicular to each other ; bisecting o e in k, with k as a centre, and k a as radius, describe the arc, a l ; the chord, a l, will be equal to a side of the pentagon, which may accordingly be drawn by making the chords .which form its sides, as a e, f d, d c, c b, and b a, equal to A l. By bisecting these arcs, the sides of a decagon may be at once obtained. A decagon may also be constructed thus : — Draw two radii perpendicular to each other, as o m and o r ; next, the tan- gents, N M and N R. Describe a circle having n m for its diameter ; join R, and p the centre of this circle, the line, R p, cutting the eircle in a ; R a is the length of a side of the decagon, and applying tt to the circle, as r b, &-c., the required figure will be obtained. The distance, r a or r c, is a mean proportional between an entire radius, as r n, and the difference, c n, between it and the radius. A mean proportional between two lines is one having such relation to them that the square, of which it is the one side, is equal to the rectangle, of which the other two are the dimensions. 21. To construct a rectangle of ichich the sides shall he mean pro- portionals between a given line, as a c, fig. 16, and one a third or two-lhirds of it. — a c, the given line, will be the diagonal of the required rectangle; mth it as a diameter describe the circle ab c d. Divide a c into three equal parts in the points, m, n, and from these points draw the perpendiculars, m d and n b ; the lines wMch join the points of intersection of these lines with the circle, as a b, a d, c B, c D, will form the required rectangle, the side of which, c d, is a mean proportional between c m and c a, or — Cm: CD::CD :CA; that is to say, the square of which c d is a side, is equal to a rec- tangle of which c A is the length, and c m the height, because CDxCD = CmxCA* In like manner, a d is a mean proportional between c A and m a. This problem often occurs in practice, in measuring timber. Thus the rectangle inscribed in the circle, fig. 16, wiiich may bo con- sidered as representing the section of a tree, is Uio loriii of tlu^ b(?am of the greatest strength which can be obtained IVoni (lui tree. * S«o the noted mid rulcn givim at the end of thin rhniitor APPLICATIONS. DESIGNS FOR INLAID PAVEMENTS, CEILINGS, AND BALCONIES. PLATE II. The problems just considered are capable of a great variety of applications, and in Plate II. will be found a collection of some of those more frequently met \vith in mechanical and architectural constructions and erections. In order, however, that the student may perfectly understand the ditferent operations, we would recommend him to draw the various designs on a much larger scale than that we have adopted, and to which we are necessarily limited by space. The figures distinguished by numbers, and showing the method of forming the outlines, are drawn to a larger scale than the figures distinguished by letters, and representing the complete designs. 22. To draw a pavement consisting of equal squares, figs. ^ and 1. — Taking the length, a b, equal to half the diagonal of the required squares, mark it off a number of times on a horizontal line, as from a to b, b to c, &c. At a erect the perpendicular I H, and draw parallels to it, as d e, g f, &c., through the several points of division. On the perpendicular, i h, mark off a number of distances equal to a b, and draw parallels to a b, through the points of division, as h g, i f, &,c. A series of small squares will thus be formed, and the larger ones are obtained simply by di-aw- ing the diagonals to these, as shown. 23. To draw a pavement composed of squares and interlaced rectangles, figs. [|3 and 2. — Let the side, as c d, of the square be given, and describe the circle, l m q b, the radius of which is equal to half the given side. With the same centre, o, describe also the larger eircle, k n p i, the radius of which is equal to half the side of the square, plus the breadth of tlie rectangle, a b. Draw the diameters, a c, e d, perpendicular to each other; draw tan- gents through the points, a, d, c, e, forming the square, j H f g ; draw the diagonals j f, g, h, cutting the two circles 'n the points, I, B, K, L, M, N, p, Q, through which draw paraUels to thv.^ diagonals. It will be perceived that the lines, a e, e c, c d, and d a, are exactly in the centre of the rectangles, and consequently serve to verify their correctness. The operation just described is repeated, as far as it is wished to extend the pattern or design, many of tlie hues being obtained by smiply prolonging those already dra^^^l. In inking this in, the student must be very careful not to cross tlie lines. This design, though analogous to the first, is somewhat different in appearance, and is applicable to the construction of trellis-work, and other devices. 24. To draw a Grecian border or frieze, figs. © and 3. — On two straight lines, as a b, a c, perpendicular to each other, mark ofl', as often as necessary, a distance, a i, representing the w idtli, c/, of tlie ribbon forming the iiattorn. Through all the points ot' di\ isioii, diaw paralk'ls to A B, a c — thus forming a series of small sciuairs, guided by wiiich the pattern may bo at once inked in, (•([ual ilistances being maintained betweeu tiio sets of linos, as in fig. ©. This ornament is frequently mot with iu arcJutooturo, biiiig used lor ceilings, cornices, railings, and b!iloonios ; jUso in cabinet work and maciiinory for borders, and for wood and iron gratings. 25. '/') draw a \mvcincnt comixyscd of squarts <iful rn;ular i^Utt- 12 THE PRACTICAL DRAUGHTSMAN'S gnns,figs. [D) and 4. — With a radius, e o, equal to half tlie width, E F, of tlie octagon, describe a circle, e g f h, and, as was shown in reference to fig. 13, Plate I., draw the octagon circumscribing it — the squai-e, a b c d, being first obtained, and its diagonals, A c, B D, drawn cutting the cu-cle In the points, i, J, k, l, tangents being then drawn through these points. The octagon may also be formed by marking off' from each corner of the square, a, b, c, d, a distance equal to a o, or half its diagonal — and thereby will be obtained the points of junction of the sides of the octagon. The pattern is extended simply by repeating the above operation, the squares being formed by the sides of four contiguous octagons, which are inclined at an angle of 45° to the horizontal lines. This pattern is generally produced in black and white marble, or in stones of different colours, whereby the effect is distinctly brought out. 26. To draw a fakement composed of regular hexagons, figs. [1 and 5. — With a radius, a o, equal to a side, a b, of the hexagon, describe a circle, in which inscribe the regular hexagon, a b c d e f. The remaining hexagons wiU readily be obtained by producing, in different directions, the sides and diagonals of this one. In fig. d, the hexagons are plain and shaded alternately, to show their aiTangement ; but in practice they are generally all of one colour. 27. To draw a pavement composed of trapeziums, combined in squares, figs. ^ and 6. — Draw the square, a b c d ; also its diago- nals, A c, b d; construct the smaller square, abed, concenti'ic with the fii'st. On the diagonal, b d, mark the equal distances, or, cf, and through e and/ draw parallels to the diagonal, a c; join the points of intersection of these with the smaller squares by the lines, k I, m n, which ■nill give all the lines required to form the pattern, requiiing merely to be produced and repeated to the desired extent. Very beautiful combinations may thus be formed in different kinds of wood for furniture and panels. 28. To draw a panel design composed of lozenges, figs. © and 7. — On a straight line, a b, mark off the length of a side of the lozenge twice ; construct the equilateral triangle, a b c ; draw the line c d, perpendicular to a b ; and draw a e and b f parallel to d c, and E F parallel to a b. Construct the equilateral triangle, e d f, cut- ting the triangle, a b c, in g and h, and join g h. In this manner are obtained the lozenges, a g h d and e g h c, and by continuing the lines and drawing parallels at regular distances apart, the remainder of the pattern will be readily constructed — this being repeated to any desired extent. 29. To draw a panel pattern composed of isosceles triangles, figs. [L, and 12. — Kin the last-mentioned fig. ©, we draw the longitudinal diagonal of each lozenge, we shall obtain the type of the pattern [L,. We will, however, suppose that the base, ab, of the triangle is given, instead of the side of the lozenge. Mark off this length twice on the line, a b, and construct the equilateral triangle, a c d, just as in the preceding case ; also the second similar triangle, D E F, thus obtaining the points g and h. Join a h, g b, e h, and G F, &c., and each point of intersection, as i, l, &c., vAW be the apex of three of the isosceles triangles. The pattern, [L,, is pro- duced by giving these triangles various tints. Tlie patterns we have so far given are a few of the common arrangements of various regular polygons. An endless variety of patterns may be produced by combining these different figures, and these are of great use in many arts, particularly for cabmet inlaid mosaic work, as well as for pavements and other ornamental constructions. 30. To draw an open-work casting, consisting of lozenges and rosettes, figs. ^ and 8. — The lozenge, a &c<^, being given, the points, a, b, c, d, being each the centre of a rosette, draw and indefinitely produce the diagonals, ac,bd, which must always be perpendicular to each other. Through the points, a, b, c, d, draw parallels to these diagonals, also an indefinite number of such parallels at equal distances apart. The intersections of these lines will be the centres of rosettes and lozenges alternately, and the former may accordingly be drawn, consisting merely of circles with given radii. The centres of the rosettes are joined by straight lines, and to right and left of these, at the given distances, fg, fh, parallels to them are drawn, thereby producing the concentric lozenges com- pleting the pattern. 31. To draw a pattern for a ceiling, composed of small squares or lozenges, and irregular but symmetrical octagons, figs. Q and 9. — The rectangle, a b c d, being given, its corners forming the centres of four of the small lozenges, draw the Unes, e f, g h, dividing the rectangle into four equal parts ; next mark off the semi-diagonals of the lozenges, as a i, a o, and join i and o. The centre lines of the pattern being thus obtained, the half-breadths, fg, fh, are marked on each side of these, and the appropriate parallels to them drawn. In extending the pattern by repetition, the points con-es- ponding to i and o will be readily obtained by di-awing a series of parallel lines, as 1 1 and o o. By varying the proportions between the lozenges and the octagons, as also those between the different dimensions of each, a number of patterns may be produced of very varied appearance, although formed of these simple elements. 32. To draw a stone balustrade of an open-work pattern, com- posed of circular and straight ribbons interlaced, figs. S and 10. — Construct the rectangle, a b c d. its corners being the centres of some of the required circles, which may accordingly be di-awn, with given radii, as Ab, cd; after bisecting a b in e, and dra\ving the vertical E g, make e f equal to e a, and with f as a centre, draw the cu-cle having the radius, Fg, equal to a J, drawing also the equal circles at c, b, e, &c. Draw verticals, such as g h, tangents to each of the circles,- which wUl complete the lines required for the part of the pattern, S, to the left. The rosettes to the right are formed by concentric cu'cles of given radii, as e e, e/. The duplex, fig. S, may be supposed to represent the pattern on the opposite sides of a stone balustrade. Where straight lines are rim into parts of circles, the student must be careful to make them join well, as the beauty of the drawing depends gi-eatiy on this point. It is better to ink in the circles first, as it is practically easier to draw a straight line up to a circle than to draw a cu'cle to suit a straight line. 33. 7'o draw a pattern for an embossed plate on- casting, composed of regular figures combined in squares, figs. K and 11. — Two squares being given, as a b c d and f g h i, concentric, but with the diagonals of one parallel to the sides of the other, draw first the square, abed, and next the inner and concentric one, efgh. The sides of the latter being cut by the diagonals, a c and b d, in the points, i,j, k, I, through these di'aw parallels to the sides of the square, a b c d, and finally, with the centre, o, describe a smaV BOOK OF INDUSTRIAL DESIGN. 13 cin:le, the diameter of which is equal to the width of the indented crosses, the sides of these being di-awn tangent to this circle. Thus are obtained all the lines necessary to delineate this pattern ; the relievo and intaglio portions are contrasted by the latter being shaded. In the foregoing problems, we have shown a few of the many varieties of patterns producible by the combination of simple regular figures, lines, and cu-cles. There is no limit to the multi- plication of these designs ; the processes of construction, however, being analogous to those just treated of, the student will be able to produce them with every facility. SWEEPS, SECTIONS, AND MOULDINGS. PLATE III. 34. To draw in a square a series of arcs, relieved by semicircular mouldings, figs. A and 1. — Let a b be a side of the square ; draw the diagonals cutting each other ia the point, c, through which draw parallels, d e, c f, to the sides ; with the corners of the square as centres, and with a given radius, a g, describe the four quadrants, and with the points, d, f, e, describe the small semicncles of the given radius, n a, which must be less than the distance, n b. This completes the figure, the symmetry of which may be verified by drawing circles of the radii, c g, c h, which should touch, the former the larger quadrants, and the latter the smaller semi- circles. If, instead of the smaller semicucles, larger ones had been drawn ndth the radius, d b, the outline would have formed a perfect sweep, being free from angles. This figure is often met wth in machinery, for instance, as representing the section of a beam, connecting-rod, or frame standard. 35. To draw an arc tangent to two straight lines. — First, let the radius, a b, fig. 2, be given ; with the centre, a, being the point of intersection of the two lines, a b, a c, and a radius equal to a b, describe arcs cutting these lines, and through the points of intersectton di-aw parallels to them, b o, c o, cutting each other in 0, which will be the centre of the required arc. Draw perpen- diculars from it to the straight lines, a e, a c, meeting them in u and E, which will bo the points of contact of the required arc. Secondly, il a point of contact bo given, as n, fig. 3, the lines being a b, a c, making any angle with each oilier, bisect the angle by the straight line, a d ; draw b o perpendicular to a b, from the point, B, and the point, o, of its intersection with a d, \\ill be the centre of the required arc. If, as in figs. 2 and 3, we draw arcs, of radii somewhat less than o b, we shall form conges, which Btand out from, instead of being t^ingents to, the given straight lines. This problem meets with an application in drawing fig. |3, which represents a section of various descriptions of castings. 3G. To draw a circle tangent to three given straight lines, wlilch make antj angles tvilh each other, fig. 4. — Bisect the angle of the lines, a b and a c, by the straight lino, a e, and the angle formed by a D and c a, by lluj line, c f. a e and c f will cut each other in the point, o, which is at an equal distance from each side, and is cons('(incntly tim centre fif the r<'(|nir('(l circle, wiiicli may !)(' drawn willi a i-adiiis, i'{|ii:il lo a lirn- IVuiii tlic |i(iiiil, o, |iri|H'M(li- cular to any of the sides. This |)iiil)l(iii is necessary lor llic cuni- pietion of tig. y. 37. To draw the section of a stair rail, fig. ©. — This gives riso to the problems considered in figs. 5 and 6. First, let it be re- quu-ed to draw an arc tangent to a given arc, as a b, and to the given straight line, c d, fig. 6 — d being the point of contact with the latter. Through d draw e f perpendicular to c d : make f d equal to o b, the radius of the given arc, and join of, thiough the centre of which draw the perpendicular, g e, and the point, e, of its intersection with e f, will be the centre of the required arc, and E D the radius. Further, join o e, and the point of intersec- tion, B, with the arc, a b, will be the point of junction of the two arcs. Secondly, let it be required to draw an arc tangential to a given arc, as a b, and to two straight lines, as b c, c d, fig. S. Bisect the angle, b c d, by the straight line, c e ; v.ith the centre, c, and the radius, c h, equal to that of the given arc, o a, describe the arc, a G ; parallel tone di-aw i h j, cutting e c in j. Join o j, the line, o j, cutting the arc, h g, in g ; join c g, and draw o K pa- rallel to c G ; the point, k, of its intersection with e j, will be the centre of the required arc, and a line, k l or k m, perpendicular to either of the given straight lines, will be the radius. 38. To draw the section of an acorn, fig. ©. — This figure calls for the solution of the two problems considered in figs. 9 and 10. Fiist, it is required to draw an arc, passing through a given point, A, fig. 9, in a line, a b, in which also is to be the centre of the arc, this arc at the same time being a tangent to the given arc, c. Make a d equal to o c, the radius of the ^ven arc ; join o d, and di-aw the perpendicular, f b, bisecting it. b, the point of inter- section of the latter line, with a b, is the centre of the required aic, A E c, a B being the radius. Secondly, it is required to di-aw an arc passing through a given point, a, fig. 10, tangential to a given arc, BCD, and having a radius equal to a. With the centre, o, of the given arc, and with a radius, o e, equal to o c, plus the given radius, a, draw the arc e ; and witli the given point, a, as a centre, and with a radius equal to a, describe an arc cutting the former in E — E wall be the centre of the requued arc, and its point of contact with the given arc will be in c, on the line, o e. It will be seen that in fig. ©, these problems arise in drawing either side of the object. The two sides are precisely the same, but reversed, and the outline of each is equidistant from the centre line, which should always be pencilled in when drawing similar figures, it being ditli- cult to make them symmetrical without such a guide. This is ;m ornament frequently met with in macliinery, smd in articles of various materials and uses. 39. To draw a wave curve, formed by arcs, septal and tangent lo each other, and parsing through given points, a, b, their radius being equal lo half the distance, A B, figs. E and 7. — Join a n, and draw tlie pri|)enilicular, E F, bisecting it in c. ^V'itll the centres, A luid c, and radius, A c, describe arcs cutting each other in G, and with the centres, b and c, other two cutting each other in ii; g imd h will be the centres of tlio required anis, forming the curve or sweep, A c B. This curve is very oonuuon in architecture, and is styled the ci/ma recta. 40. To draw a similar curve to the preceding, but formed by arcs of a given radius, as a. i.figs. [? ami 11. — Divide tlio stniigh* line into fo\ir eiiual parts by the pi'i-jKiudiculiirs, K F, g h. mui c D ; then, with the centre, a, and given radius, a i, which uuist idways be greater tiuui tlio quarter of a u, describe tho urv 14 THE PRACTICAL DRAUGHTSMAN'S cutting CD in c ; also with the centre, b, a similar arc cutting G H in H ; c and h will be the centres of the arcs forming the re- quired curve. Wliatever be the given radius, provided it is not too small, the centres of the arcs will always be in the lines, c d and G H. It will be seen that the arcs, c i and h l, cut the straight lines, c d and g h, in two points respectively. If we take the second points, k l, as centres, we shall form a similar curve to the last, but with the concavity and convexity transposed, and called the ajma reversa. The two will be found in fig. [F, the first at a, and the second at b. This figure represents the section of a door, or window frame — it is one well known to carpenters and masons. The little instrument known as the " Cymameter," affords a convenient means of obtaining rough measurements of contoiu^ of various classes, as mouldings and bas-reliefs. It is simply a light adjustable frame, acting as a species of holding socket for a mass of parallel slips of wood or metal — a bundle of straight wires, for example. Previous to applying tiiis for taking an im- pression of measurement, tlie whole aggregation of pieces is dressed up on a flat surface, so that their ends form a perfect plane, like the ends of the bristles in a square cut brush ; and these component pieces are held in close parallel contact, vvith just enough of stiff friction to keep them from slipping and falling away. The ends of the pieces are then applied well up to the moulding or surface whose cavities and projections are to be mea- dui'ed, and the frame is then screwed up to retain the slips in the position thus assumed. The surface thus moulds its sectional contour upon the needle ends, as if the surface made up of these ends was of a plastic material, and a perfect impression is there- fore carried away on the instrument. The nicety of delineation is )bviously bounded by the relative fineness of the measuring ends. 41. To draw a baluster of a duplex carUour, fgs. @ and 8. — ^It is here necessary to draw an arc tangent to, or sweeping into two known arcs, a i and c d, and having its centre in a given hori- zontal, e i. Extend e i to h, making i h equal to g d, the radius of the arc, c d. Join g h, bisecting g h by a perpendicular ; this will cut e H in the point, e, which is the centre of the required arc — e i being its radius. A line joining e g cuts c d in c, the point of contact of the two arcs. The arc, d f, which is required to be a tangent to c r», and to pass tlu-ough the point, f, is drawn with the centre, o, obtained by bisecting the chord, d f, by a perpendicuJiir which cuts the radius of the arc, c n. This curve has, in fig. @, to be repeated both on each side of the vertical line, m n, and of the horizontal line,/g'. 42. To draw the section of a baluster of simple outline, as fig. ^. — ^We have here to draw an arc passing through two points, a, b, fig. 12, its centre being in a straight line, b c; this arc, moreover, requiring to join at d, and form a sweep with another, d e, having jts centre in a line, f d, parallel to b c. Joining b a, a perpendicu- lar bisecting b a, will cut b c in o, which will be the centre of the first arc, and that of the second may now be obtained, as in problem 37, fig. 6. 43. The base of the baluster, fig. frO, is in the form of a curve, termed a scolia. It may be di-awn by various methods. The following are two of the simplest — according to the fii'st, the wrve may be formed by arcs sweeping into each other, and tan- gents at A and c to two given parallels, a, p, c d, fig. 13. Through A and c draw the perpendiculars, c o and a e, and divide the latter into three equal parts. With one division, f a, as a radius, describe the first arc, a g h ; make c i equal to f a, join i f, and bisect I F by the perpendicular, o k, which cuts c o in o. o will be the centre of the other arc required. The line, o h, passing through the centres, o and f, will cut the arcs in the point of junc- tion, H. It is in this manner that the curve in fig. [>[] is obtained. The second method is to form the curve by two arcs sweeping into each other and passing through the given points, a b, fig. 14, their centres, however, being in the same horizontal line, c d, parallel to two straight lines, e f and b c, passing through the given points. Through a, draw the perpendicular, a i. i, its point of intersection with c d, is the centre of one arc, a d. Next draw the chord, b d, the perpendicular bisecting which, will cut c d in o, the centre of the other arc, the radius being o d or o B. This curve is more particularly met with in the construction of bases of the Ionic, Corinthian, and Composite orders of arcliitecture. With a view to accustom the student to proportion his designs to the rules adopted in practice in the more obvious applications, we have indicated on each of the figs. ^, [B, ©, <Sic., and on the corresponding outlines, the measurements of the various parts, in millimetres. It must, however, at the same time be understood, that the various problems are equally capable of solution with other data ; and, indeed, the number of applications of which the forms considered are susceptible, wiU give rise to a considerable variety of these. ELEMENTARY GOTHIC FORMS AND ROSETTES. PLATE rv. 44. Having solved the foregomg problems, the student may now attempt the delineation of more complex objects. He need not, however, as yet, anticipate much difficulty, merely giving his chief attention to the accurate determination of the principal lines, which serve as guides to the minor details of the drawing. It is in Gothic architecture that we meet with the more numer- ous applications of outlines formed by smoothly joined circles and straight lines, and we give a few examples of this order in Plate IV. Fig. 5 represents the upper portion of a window, composed of a series of arcs, combined so as to form what are denominated cuspid arches. The width or span, a b, being given, and the apex, c , joining a c, c b, draw the bisecting perpendiculars, cutting a b in D and e. These latter are the centres of the sundiy concentric arcs, which, severally cutting each other on the vertical, c f, form the arch of the window. The small interior cuspids are drawn in the same manner, as indicated Ln the figure ; the horizontal, G H, being given, also the span and apexes. These interior arches are sometimes surmounted by the ornament, M, termed an ceil-de- iceuf consisting simply of concentric circles. 45. Fig. 1 represents a rosette, formed by concentric circles, the outer interstices containing a series of smaller circles, forming an interlaced fillet or ribbon. The radius, a o, of the cu'cle, con- taining the centres of all the small circles, is supposed to be given. Divide it into a given number of equal parts. With the points of division, 1, 2, 3, &c., as centres, describe the circles tangential to BOOK OF INDUSTRIAL DESIGN. 15 each other, forming the fillet, making the radii of the alternate ones in any proportion to each other. Then, with the centre, o, describe concentric circles, tangential to the larger of the fillet circles of the radius, Ab. The central ornament is formed by arcs of circles, tangential to the radii, drawn to the centres of the fillet circles, their convexities being towards the centre, o ; and the arcs, joining the extremities of the radii, are drawn with the actual centres of the fillet circles. 46. Fig. 6 represents a quadi-ant of a Gothic rosette, distin- guished as radiating. It is formed by a series of cuspid arches and radiating muUions. In the figure are indicated the centre Unes of the several arches and mullions, and in fig. 6^, the capital, con- necting the mullion to the arch, is represented drawn to double the scale. With the given radii, a b, a c, a d, a e, describe the different quadrants, and divide each into eight equal parts, thus obtaining the centres for the trefoil and quadrefoil ornaments in and between the different arches. We have drawn these orna- ments to a larger scale, in figs. 6*, &", and 6°, in which are indi- cated the several operations required. 47. Fig. 4 also represents a rosette, composed of cuspid arches and trefoil and quadi-efoil ornaments, but disposed in a different manner. The operations are so similar to those just considered, that it is unnecessary to enter into further details. 48. Fig. 7 represents a cast-iron grating, ornamented with Gothic devices. Fig. 7» is a portion of the details on a larger scale, from which it will be seen that the entire pattern is made np simply of arcs, straight lines, and sweeps formed of these two, the problems arising comprehending the division of lines and angles, and the obtainment of the various centres. 49. Figs. 2 and 3 are sections of tail-pieces, such as are sus- pended, as it were, from the centres of Gothic vaults. They also represent sections of certain Gothic columns, met with in the architecture of the twelfth and thirteenth centuries. In order to draw them, it is merely necessary to determine the radii and centres of the various arcs composing them. Several of the figures in Plate FV. are partially shaded, to indicate the degree of relief of the various portions. We have in this plate endeavoured to collect a few of the minor difficulties, our object being to familiarize the student to the use of iiis instru- ments, especially the compasses. These exercises will, at the same time, qualify him for the representation of a vast number of forms met with in machinery and architecture. OVALS, ELLIPSES, PARABOLAS, VOLUTES, &c. PLATE V. 50. The ove is an ornament of the shape of an egg, and is formed of arcs of circles. It is frequently employed in architecture, and is thus drawn: — The axes, ab and c d, fig. 1, being given, oorpondicular to each olhcr; with Iho point of intersection, o, as a centre, first describe the circle, cade, half of whicli forms tlio upper portion of the ove. Joining b c, make c f equal to b e, the difference between the radii, o c, o b. Bisect F n by the per- pendicular, G n, cutting c d in Ji. ii will be the centre, and n c the radius of the arc, c j ; and i, the point of intersection of h c with A n, will bo the centre, and i b the radius of tho anialler arc, I B K, which, together with the arc, h k, described with the centre, L, and radius, l d, equal to h c, form the lower portion of the required figure ; the lines, g h, l k, which pass through the respective centres, also cut the arcs in the points of junction, J and K. This ove will be found in the fragment of a cornice, fig. ^- A more accurate and beautiful ove may be diTiwn by means of the instrument represented in elevation and plan in the annexed engraving. The pencil is at a, in an adjustable holder, capable of sliding along the connecting-rod, b, one end of which is jointed at c, to a slider on the horizontal bar, D, whilst the opposite end is similarly jointed to the crank arm, e, revolving on the fixed centre, f, on the bar. By altering the length of the crank, and the position of the pencil on the connecting-rod, the shape and size of the ovo may be varied as required. 51. The oval differs from the ove in having the upper portion symmetrical with the lower ; and to di-aw it, it is only necessary to repeat the operations gone thi'ough in obtaining the curve, l b y fig. 1. 52. The ellipse is a figure which possesses the following ptv» perty : — The sum of the distances from any pomt, a, fig. 2, in Oio circumference, to two constant points, b, c, in the longer axis, is always equal to that axis, d e. The two points, b, c, are termed foci. The curve forming the ellipse is symmetric wit/ rsfereuc* both to the horizontal line or iLxis, d e, and to the vertical line, i g, bisecting the former in o, the centre of the ellipse. Linos, as B A, c A, b F, c F, &c., joining any point in the chcumfcrenco Willi the foci, B and c, are called rectors, and any pair proceeding from one point are together equal to the longer axis, d e, wliich is lalled the transverse, f g being tho conjugate axis. There are diflerent methods of drawing this curve, which we will proceed to in dicato. 53. First Method. — This is based on the definition given above, and requires that tho two axes bo given, as d e and f o, fig. 2. The foci, b and c, are first obtained by describing an :uv. \\ iiii tlie extremity, g or f, of the conjugjite axis as a centre, and with a radius, f c, equal to half the transverse axis; the an- will cut tho latter in the points, n and c, llu' I'oii. It" now we divide n c inuMiu.'iUy in H, and with the radii, n u, v. n, and the foci as ceiitn};!, wo describe arcs .severally cutting each otlior in i, J, K, A ; the ii four points will lie in the circumference. If, I'urtlier, we «g;uB iinequ.Mlly divide n E, siiy in i,, we can similarly ohtjiin t\uir ot'iei 16 THE PRACTICAL DRAUGHTSMAN'S poin Is in the circumference, and we can, in like manner, obtain any number of points, when the ellipse may be traced through tliem by hand. The large ellipses which are sometimes required in con- structions, are generally drawn with a trammel instead of compasses, the trammel being a rigid rule with adjustable points. — The gardener's ellipse: To obtain this, place a rod in each of the foci of the required ellipse ; round these place an endless cord, which, when stretched by a tracer, will form the vectors ; and the ellipse ■will be drawn by carrjing the tracer round, keeping the cord always stretched. 54. Second Method. — Take a strip of paper, having one edge straight, as d b, and on this edge mark a length, a b, equal to half the transverse axis, and another length, b c, equal to half tlie conju- gate axis. Place the strip of paper so that the point, a, of the longer measurement, lies on the conjugate, f g, and tlie other point, c, on the transverse axis, d e. If the strip be now caused to rotate, always keeping the two points, a and c, on the respective axes — the other point, b, vsill, in every position, indicate a point in the circumference which may be marked %\-ith a pencil, the ellipse being afterwards traced through the points thus obtained. 55. Third Method, fig. 3. — It is demonstrated, in that branch of geometry which treats of solids, as we shall see later on, that if a cone, or cylinder, be cut by a plane inclined to its axis, the resulting section will be an ellipse. It is on this property that the present method is based. The transverse and conjugate axes being given, as a b and c d, cutting each other in the centre, o, draw any line, a e, equal in length to the conjugate axis, c d, ajid on A E, as a diameter, describe the semicircle, e g a. Join E B, and through any number of points, taken at random, on e a, as 1, 2, 3, &c., draw parallels to e e. Then, at each point of division, on b a, erect perpendiculars, 1 a, 2 5, 3 c, &c., cutting the semicircle, and, at tlie corresponding di\Tsions obtained on A B, erect perpendiculars, as 1' a', 2' b, 3' c, &c., and make them equal to the corresponding perpendiculars on e a. A line traced through the various points thus obtained, that is, the extremities, a', b', c, &c., of the lines, wiU form the required ellipse. 56. Fourth Method. — On the transverse axis, a b, and with the centre, o, describe the semicircle, a f b, the axis forming its dia- meter ; and with the diameter, h i, equal to the conjugate axis, describe the smaller semicircle, h d i. Draw radii, cutting the two semicircles, the larger in the points, i^j, k, I, &c., and the smaller in the points, i',j', k', T, die. It is not necessary tliat the radii should be at equiangular distances apart, though they are di-awn so in the plate for regularity's sake. Through the latter points draw parallels to the ti-ansverse axis, a b, and thi'ough the former, parallels to the conjugate axis, c d, the points of intersection of these lines, as q, r, s, t, &.C., 'will be so many points in the required ellipse, which may, accordingly, be traced through them. It follows from this, that, in order to draw an ellips-e, it is sufHcient to know either of the axes, and a point in the cu-cumference. Let the axis, A b, oe ^ven, and a point, r, in the circumference, which must always lie within perpendiculars passing through the extre- mities of the given axis. Through r draw a line, rj', parallel to A. B, and a line, rj, perpendicular to it; with tlie centre, o, and •^us, o A, equal half the given axis, describe the arc, cutting J ui J ; join J o, and the line, j (\, will cut rj' in j' : oj' will be equal to half the conjugate axis, on. If the conjugate axis, c d, be given, proceed as before ; the arc, however, in this case, ha\ing the smaller radius, OD, and cutting rj' 'mj'; then join oj,' pro. ducing the line till it cuts rj, which will be in _;", and oj will equal half the transverse axis, a b. It has already been shown how to describe an ellipse, when the two axes are given. We may here give a method invented a short time back by Mr. Crane of Bii-mingham, for constructing an ellipse with the com- passes. Tliis method applies to all proportions, and produces as near an approximation to a true ellipse, as it is possible to obtain by means of four arcs of circles. By appljing compasses to any true ellipse, it will be seen that certain parts of the curve approach very near to arcs of circles, and that these parts are about the vertices of its true axes ; and by the natm-e of an ellipse, the cm've on each side of either axis is equal and simOar; consequently, if arcs of circles be drawn through all the vertices, meeting one another in four points, the opposite arcs being equal and similar, the resulting figure will bo indefinitely near an ellipse. Four circles, described from four different points, but with only two difi'erent radii, are then required. These four points may be aU within the figure ; the centres of the two greater circles may either be within or without, but the centres of the two circles at the extremities of the major axis must always be within, and, consequently, the w^hole four points can never be without the figure. Again, the proportions of the major and minor axes may vary infinitely, but they can never be equal ; therefore, any rule for describing ellipses must suit all pos- sible proportions, or it does not possess the necessary requirements. Moreover, if any rule apply to one certain proportion and not to another, it is evident that the more the proportions differ from that one — whether crescendo or diminuendo — the greater will be the difference of the result from the true one. From this it follows, that if a rule applies not to aU, it can only apply to one propor- tion ; and also, that if it apply to a certain proportion and not to another, it can only be correct in that one case. Let A B be any major axis, and c d any minor axis ; produce them both in either direction, say towards f and h, and make a f equal to c G ; then join c a, and through f draw f h parallel to c a. Set off B I, A J, and c k, equal to h c ; join j K, and bisect it in k, and at k erect a perpendicular, cutting c d, or c d produced, at M ; then make g e equal to g m ; j, e, i, m, wiU be the centres of the four circles required. Through the points, j and i, draw m n, m o E p, E Q, each equal to m c ; then M n and e p will be the radii of the greater circles, and j k, x o, of the less : the points of contact will therefore be at n, o, p, q, and the fig-ure drawn through a, h c, 0, B, Q, D, p, will be the required ellipse. BOOK OF INDUSTRIAL DESIGN. n T\ Several instruments have been invented for drawing ellipses, many of them very ingeniously contrived. The best known of tlii.'se contrivances, are those of Farey, WUson, and Hick — ^^the l;ist of which we present in the annexed engraving. It is shown as in working order, with a pen for drawing ellipses in ink. It con- sists of a rectangular base plate, A, having sharp countersunk points on its lower surface, to hold the instrument steady, and cut out to leave a sufficient area of the paper uncovered for the traverse of the pen. It is adjusted in position by four index lines, setting out the trans- verse and conjugate axes of the intended ellipse — these lines being cut on the inner edges of the base. Near sne end of the latter, a vertical pillar, b, is screwed down, for the (jurpose of carrying the traversing slide-arm, c, adjustable at any height, by a milled head, d, the spindle of which carries a pinion in gear with a rack on the outside of the pillar. The outer end of the arm, c, terminates in a ring, with a universal joint, e, through which the pen or pencil-holder, f, is passed. The pillar, B, also carries at its upper end a fixed arm, g, formed as an ellip- tical guide-frame, being accurately cut out to an elliptical figure, as the nucleus of all the varieties of ellipse to be drawn. The centre of this ellipse is, of course, set directly over the centre of the universal joint, e, and the pen-holder is passed through the guide and through the joint, the flat-sided sliding-pieee, h, being kept in contact with the guide, in traversing the pen over the paper. The pen thus turns upon its joint, E, as a centre, and is always held in its proper line of motion by the action of the slider, h. The dis- tance between the guide ellipse and the universal joint determines the size of the ellipse, which, in the instrument here delineated, ranges from 2| inches by 1|, to -pV ^Y I '"'^'i- ^'^ general, how- over, these instruments do not appear to be suflicienlly simple, or convenient, to bo used with advantage in geometrical drawing. 57. Tangents to ellipses. — It is frequently necessary to deter- mine the position and inclination of a straight lino which shall be a tangent to an elliptic curve. Three cases of this nature occur : when a point in the ellipse is given ; when some external point is ^ven apart from the ellipse ; and when a straight lino is given, to which it is necessary that the fangcnl, should \h' parallel. Fii'st, then, let the point, A, in the ellipse, fig. 2, bo given ; draw the two vectors, c a, n a, and produce the latter to m ; liiscct the angle, M a c, by the stniight line, n r ; this lino, N p, will Iki Vie tangent re(|uiri^d; that is, it will Idmi-Ii the (•urv(^ in the point, A, and in that point alone. Secondly, let the point, l, be given, apart from the ellipse, fig 3. Join L with i, the nearest focus to it, and with l as a centre, and a radius equal to l i, describe an arc, M i n. Next, with the more distant focus, h, as a centre, and with a radius equal to the transverse axis, a b, describe a second arc, cutting the first in m and N. Join m h and n h, and the ellipse wiU be cut in the points V and X ; a straight line drawn through either of these points from the given point, l, ^vill be a tangent to the ellipse. 68. Thndly, let the straight line, q r, fig. 2, be given, paralle to which it is required to di-aw a tangent to the ellipse. From the nearest focus, b, let f;vll on q k the perpendicular, s b ; then with the further focus, c, as a centre, and with a radius equal to the transverse axis, D E, describe an arc cutting b s in s ; join c s, and the straight line, c s, will cut the ellipse in the point, t, of contact of the required tangent. All that is then necessary is, to draw through that point a line parallel to the given line, q e, the accuracy of which may be verified by observing whether it bisects the line, s b, which it should. 59. — Tlie oval of five centres. Jig. 4. — As in previous cases, the transverse and conjugate axes are given, and we commence by obtaining a mean proportional between their halves; for tliis purpose, with the centre, o, and the semi-conjugate axis, o c, as radius, we describe the arc, c i k, and then the semi-ckcle, a l k, of which A K is the diameter, and further prolong o c to l, o l being the mean proportional required. Next construct the parallelo- gram, A G c 0, the semi-axes constituting its dmiensions : joining c A, let fall from the point, g, on the diagonal, c a, the per- pendicular, G H D — which, being prolonged, cuts tiie conjugate axis or its continuation in d. Havmg made c m equal to the mean proportional, o l, with the centre, d, and radius, d m, describe an arc, a m b ; and having also made a n equal to the mean pro- portional, o L, with the centre, h, and radius, h n, describe tlie arc, N a, cutting the former in a. The points, h, a, on one side, and ii', b, obtained in a similar manner on the other, together with the point, d, will be the five centies of the oval ; and straight lines, R H a, s h' b, and p a d, q i d, passing through the respective centres, wll meet the curve in the points of junction of the various component arcs, as at R, p, Q, s. This beautiful curve is adopted in the construction of many kinds of arches, bridges, and ■s'aults ; an example of its use is given in fig. ©. 60. The parabola, fig. 5, is an open curve, that is, one wiiich does not return to any assumed starting point, to however great a length it may bo extended ; and wiiich, consequently, can never enclose a space. It is so constituted, that any point in it, r>, is at an equal distance from a constant point, c, termed the focia, and in a perpendicular direction, from a straight line, a b, called the (llrrr/rix. The straight line, r c, iH'r[)t'ndicular to the directrix, A 1!, and jiassing tlinnigh the focus, r. is the a.vis o\' the curve, which it divides into two synunetricil portions. The point, .\. midway between F and c, is the apex of the curve. There are several methods of drawing this curve. (il. First method: — This is based on the dotinition just given ,111(1 rciiiiires that the foeua and directrix be known, as c, and .v D 'I'aKc any points on the directrix, A n, as a, k, h, i, and throu-ji liiem (haw jiarallols to iJio 'Lxis, K u, as also tlio straight line* • c 18 THE PRACTICAL DRAUGHTSMAN'S A c, E c, H c, I c, joining them with the focus. Draw pei-pen- dicular^ bisecting these latter lines, and produce them until they cut the corresponding parallels, and the points of intersection, /), ^, D, e, will bo in the required cur\e, wliich may be traced through them. 62. The sti-aight lines which were just drawn, cutting the parallels in different points of the curve, are tangents to the curve at the several points. If, then, it is required to di-aw a tangent througli a given point, c, it is obtained smiply by joining c c, making h a equal to c c, and bisecting the angle, h c c, by the straight line, c d, which will be the required tangent. K the point gix'en bo apart from the curve, the procedure will be the same, but the line corresponding to h c will not be parallel to the axis. 63. Second method: — We have here given the axis, a g, the apex, a, and any point, Z, in the ciu-ve. From the point, I, let fall on the axis the perpendicular, I g, and prolong this to r, making c e equal to I a. Divide I g into any number of equal parts, as m the points, i, J, k, through which draw parallels to the axis ; divide also the axis, a g, into the same number of equal parts, as in the points, f, gih; through these draw lines radiating from the point, e, and they will intersect the parallels in the points, m, n, o, which are so many points in the curve. 64. If it is required to draw a line tangent to a given parabola, and parallel to a given line, J k, we let fall a perpendicular, c l, on this last ; this perpendicular will cut the dii-ectrix in p, and p n drawn parallel to the axis will cut the curve in the point of con- tact, n. We find frequent applications of this curve in constructions and machinery, on account of the peculiar properties it possesses, which the student will find discussed as he proceeds. 65. The objects represented in figs, [o), [o)', are an example of the application of this curve. They are called Parabolic Mirrors, and are employed in philosophieal researches. The angles of incidence of the vectors, a h, a c, a d, are equal to the angles of reflection of the parallels, b b', c c', d d'. It follows from this pro- perty, that if in the focus, a, of one min-or, b /, the flame of a lamp, or some incandescent body be placed, and in the focus, a', of the opposite mirror, &'/', a piece of charcoal or tinder, the latter will be ignited, though the two foci may be at a considerable dis- tance apart; for all the rays of caloric falling on the muTor, b f, are reflected from it in parallel lines, and are again collected by the other mirror, b' f, and concentrated at its focus, a'. 66. To draw an Ionic volute, fig. 6. — The vertical, a o, being given, and being the length from the summit to the centre of the volute, divide it into nine equal parts, and with the centre, o, and a radius equal to one of these parts, describe the circle, abed, which forms what is termed the eye of the volute. In this circle (represented on a larger scale in fig. 7) inscribe a square, its dia- gonals being vei-tical and horizontal ; through the centre, o, draw the lines, 1 — 3, and 4 — 2, parallel to the sides, and divide the half of each into three equal parts. With the point, 1, as a centre, and the radius, 1 a (fig. 6), di-aw the arc, a e, extending to the norizontal line, 1 e, which passes through the point, 2. With this iatter pomt as a centre, and a radius equal to 2 e, draw the nex-t arc, extending to the vertical line, 2 /, which passes through the pomt, 3, tne next centre. The points, 4, 5, 6, &c., form the sub- sequent centres ; the arcs in all cases joining each other on a line passing through then- respective centres. The internal curve is drawn in the same way; the points, 1', 2', 3', &c., fig. 7"'% bemg the centres of the component arcs. The first arc is drawn with a radius, 1' a', a ninth less than 1 a, and the others are consequently proportionately reduced, as manifest ui fig. 6. The application of the volute will be found in fig. [U. 67. To draiu a curve tangenlially joining two slraigM lines, a b and B c,fig. 8, the points a and c being the points of junction. — Join a c, and bisecting a c in d, join d with b, the point of intersection of the lines, a b, b c. Bisect b d in e', which will be a point in the curve. Join e c, e a, and bisect the lines, e c, e a, by the per pendiculars, a b, c d; make e/and e' f equal to a fourth part of E D ; /and/' will be other two points in the curve. Proceed in the same way to obtain the points, g h and g' h', or more if desir able, and then trace the curve through these several points. Thi« method is generally adopted by engineers and constructors, and A^il] be met with in raOways, bridges, and embankments, and wherever it is necessary to connect two straight lines by as regular and per- fect a curve as possible. It is also particularly applicable where the scale is large. RULES AND PRACTICAL DATA. LINES AND SURFACES. 68. The square metre is the unit of surface measurement, just as the linear m^tre is that of length. The square mfetre is sub- divided mto the square decimetre, the square centimetre, and the square millimetre. WhOst the linear decimetre is a tenth part of the metre, the square decimetre is the hundredth part of the square metre. In fact, since the square is the product of a niunber mul- tiplied into itself, O'lm. X O'lm. = O'Ol square metres. In the same manner the square centimetre is the ten-thousandth part of the square metre ; for 0-01 m. X 0-01 m. = 0.0001 square metres. And the square mOlimetre is the millionth part of the square m^tre ; for, 0-001 m. X 0-001 m. = 0-000001 square metres. It is in this way that a relation is at once determined between the units of linear and sm-face measurement. Similarly in English measm-es, a square foot is the ninth part of a square yard ; for 1 foot X 1 foot =: \ yard X ^ yard = ^ square yai-d. A square inch is the 144th part of a square foot, and the 1296th part of a square yard ; for 1 inch X 1 inch = ^ foot X ]\ foot = -plj square foot, and 1 inch X 1 inch =: ^ yard x 3*5 yard = y-g^ square yard. This illustration places the simplicity and adaptability of the decimal system of measui-es, in strong contrast with the complexity of other methods. 69. Measurement of surfaces. — The surface or area of a square, as well as of all rectangles and parallelogi-ams, is expressed by the product of the base or length, and height or breadth measured BOOK OF INDUSTRIAL DESIGN. 19 perpendicularly from the base. Thus the area of a rectangle, the base of which measures 1-25 metres, and the height "75, is equal to 1-25 X -75 = -9375 square metres. The area of a rectangle being known, and one of its dimensions, llie other may be obtained by dividing the area by the given dimension. Example. — The area of a rectangle being -9375 sq. m., and the base 1-25 m., the height is •^^ = -75 m. 125 This operation is constantly needed in actual construction ; as, for instance, when it is necessary to make a rectangular aperture of a certain area, one of the dimensions being predetermined. The area of a trapezium is equal to the product of half the sum of the parallel sides into the perpendicular breadth. Examjile. — The parallel sides of a trapezmm bemg respectively 1-3 m., and 1-5 m., and the breadth -8 m., the area will be 1-3 +1-5 Q , ,„ — X -8 = 1-12 sq. m. The area of a triangle is obtained by multiplying the base by tialf the perpendicular height. Example. — The base of a triangle being 2-3 m., and the perpen- dicular height 1-15 m., the area will be 2-3 X — = 1-3225 sq. m. 2 ^ The area of a triangle being known, and one of the dimensions given — that is, the base or the perpendicular height — the other dimension can be ascertained by dividing double the area by the ^ven dimeusion. Thus, in the above example, the division of (1'3225 sq. m. x 2) by the height V\b m. gives for quotient the base 2-3 m., and its division by the base 2-3 m. gives the height ri5 m. 70. It is demonstrated in geometry, that the square of the hypothenuse, or longest side of a right-angled triangle, is equal to the sum of the squares of the two sides forming the right angle. It follows from this property, that if any two of the sides of a right-angled triangle be ^ven, the third may be at once ascertained. First, If the sides forming the right angle be given, the hypo, thenuse is determined by adding together their squares, and extracting the square root. Example. — The side, a b, of the triangle, a b c, fig. 16, PL I, being 3 m., the side b c, 4 m., the hypothenuse, a c, will be A c = V32-1-42 = 4/9 + 16 = ^2b = 5 m. Secondly, If the hypothenuse, as a c. be known, and one of the other sides, as A b, the third side, b c, will be equal to the square root of the difference between the squares of a c and A b. Thus assuming the above measures — BC= ■^'25 — 9 = V16 = 4m. The diagonal of a square is always equal to one of the sides mul tiplied by ■>^2; therefore, as V2 = 1-414 nearly, the diagonal is obtained by multiplying a side by 1-414. Exajnple. — The side of a square being 6 meti-es, its diagonal = 6 X 1-414 = 8-484 m. The sum of the squares of the four sides of a parallelogram is equal to the sum of the squares of its diagonals. 71. Regular polygons. — The area of a regular polygon is obtained by multiplying its perimeter by half the apothegm or per pendicular, let fall from the centre to one of the sides. A regular polygon of 5 sides, one of which is 9-8 m., and tne perpendicular distance from the centre to one of the sides 5-6 m., will have for area — 9-8 X 5 X —= 137-2 sq.m. 2 ^ The area of an irregular polygon will be obtained by dividing it into triangles, rectangles, or trapeziums, and then adding togetlier the areas of the various component figures. TABLE OF MULTIPLIERS FOR REGULAR POLYGONS OF FROM 3 TO 12 SIDES. Names. .Multipliers. B D Area I side = 1. Internal Anglo. F Apothegm or Perpendicular. Triangle, . . Square, . . . . Pentagon, . , Ht^xagon, . . Heptagon, . , Oclagon, . , Krmcagon, . DccMgon, . , (Indccagon, IJuodecagon, 3 4 5 6 7 8 9 10 11 12 2-000 1-414 1-238 1-156 1.111 1-080 1-062 1-050 1-0 10 1-037 1-730 1-412 1-174 radius. -867 •765 -681 -616 •561 -516 -579 •705 •852 side. M60 1-307 1-470 1-625 1-777 1-940 -433 1-000 1-720 2-598 3-634 4-8-28 6-182 7-694 9-365 11-196 60" 0' 90° 0' 108° 0' 120° 0' 128° 34'!; 135° 0' 140° 0' 144° 0' 147° 16',-', 150° 0' -2SS6751 •6000000 •6881910 -8660254 1-0382607 1-2071069 1-37373S7 1-53SS4I8 1-7028436 1-8660254 By means of (his liibli^ we cnn easily .solve iiiaiiy inlrrcsliiig problems connected with regular polygons, fi-oiii iJin triuiigln \\\> (o the duodecagon. Sutth are the following : — First, The width of a polygon being given, to find tlie radius of (he circumscribing circle. — Wlion the number of sides is oven, the width is understood a-s tho iierptindicular diatanco between two opposite and i)arallel sides; when the nuinl)(<r is unin-en, it is twice tiio perpendicular distiuice from tiio contru to one side. Rule. — l\liilti|ily h;ilt' llic widlii of the iioly;;-.in l>y llie l-ii-ti.r in (•(iliinin A, corrcspoiniiiig to the nuiiiher tit' .sides, !ind the prodiu-l will he, the reciiiircd radius. E.vamiilc. — I-et 18-5 in. lie tiie width of nil oi-lai^nni ; then, ^^J' X 1-08 =9-99ni.; or say 10 iniXres, tlie radius of tiie ciiriiinM-i-ibing cirpJe. so THE PRACTICAI. DRAUGHTSMAN S Second, The radius of a circle being given, to find the length of the side of an inscribed polygon. Rule. — iliiltiply the radius by the factor in colximn B, corre- sponding to the number of sides of the required polygon. Example. — The radius being 10 m., tlie side of an inscribed octagon will be — 10 X -"Go = 7-65 m. Third, The side of a polygon being given, to find the radius of the circumscribing circle. Rule. — Multiply tlie side by the factor in column C, corre- sponding to the number of sides. Example. — Let T'eS m. be the side of an octagon ; then 7-65 X 1-307 = 10 m., nearly. Fourth, TTie side of a polygon being given, to find the area. Rvle. — Multiply the given side by the factor in column D, corresponding to the nimiber of sides. Example. — The side of an octagon being 7'65 m., the area will be— 7-65 X 4-828 ~ 36-93 sq. m. THE CmCtlMFERENCE AND AJIEA OF A CIRCLE. 72. If the oircumferenee of any circle be divided by its diame- ter, the quotient will be a number which is called, the ratio oj the circumference to the diameter. The ratio is found to be (ap- proximately) — 3-1416, or 22 : 7; that is, the circumference equals 3-1416 times the length of the diameter. It is expressed, in algebraic formulas, by the Greek letter ji (jn). Thus, if C represents the circumference of a circle, and D its diameter, the following formula, C = jt D, or C = 3-1416 x D, expresses the development of the circumference. Thus, if tlie diameter of a circle, or D, = 2-7 m., or the radius R = 1-35 m., the circumference will be equal to — 3-1416 X 2-7, or 3-1416 x 1-35 x 2 = 8-482 m. The circumference of a circle being known, its diameter, or radius, is found by dividing this circumference by 3-1416 for the former, or 6-2832 for the latter. Thus, the diameter, D, of a circle, the circimiference of which is 8-482 m., is — 8-482 3-1416 := 2-7 m. and the radius, R, is- 8-482 1-35 m. 6-2832 The area of a circle is found by multiplying the circumference by half the radius. — Tliis rule is expressed in the following formula : — R The area of a circle = 2 « R x . R= This term, Tt R-, is merely the simplification of tlie formula. The number 2 being both multiplier and divisor, maj- be can- celled, and the product of R Into R is expressed by R-, or the square of the radius. It follows, then, that the area of a circle is equal to the square of the radius multiplied by the circumference, or 3-1416. Example. — The radius of a circle being 1-05 m., the area ■will be— 3-1416 X 1-05 X 1-05 = 3-4635 sq. m. The area of a cu-cle being known, tlie radius is determined by dindiag the area by 3-1416, and extracting the square root of thti quotient. Example. — The area of a circle being 3-4635 sq. m., the radius y 3-4635 = 1-05 m. 3.1416 The area of a circle is derived from the diameter ; thtis— Area = ,-t D X D rt D2 then, since H or ^lHi^= -7854, 4 4 the formula resolves itself into Area ;= -7854 x D-. That is to say, if we multiply the fraction, -7854, by the square of the diameter, the product will be the area. Example. — The area of a circle, the diameter of which mea- sures 2-1 m., is — -7854 X 2-1 X 2-1 = 3-4635 sq. m. It follows from this, that if the area of a square is kno^ii, that of an inscribed circle is obtainable, by multiplying by -7854 ; that is, the area of a square is to the area of the inscribed circle, as, 4 : 3-1416, or 1 : -7854. TABLE OF APPKOXniATE KATIOS BETWEEN CIECLES AND SQUARES. The diameter of a circle The circumference of a circle, . . The diameter, The circumference, The area of a circle, The side of an. inscribed square. 8. The side of a square, This table aifords a ready solution of the following amongst other problems : — First, The diameter of a circle being -125 m. or 125 ^1^ (mUli- metres), the side of a square of equal area is 125 X -8862 = 110-775"/„. X X X X X X X X X -8S62 ) _ •2821 S ~ -7071 / -2251 f -6366 1-4142 4-4430 1-1280 3-5450 the side of a square of equal area. the side of the inscribed square. the area of the inscribed square. the diameter of the circumscribing circle. the circumference of the circumscribing circle. the diameter of an equal circle. the circumference of an equal circle. Second, The circTimference of a circle being SGC^/n, the side of the inscribed square is 860 X -2251 = 193-586°/„. Third, The side of a square being 215-86'°/„, the diameter of the circumscribing circle is 215-86 X 1-4142 =305-27"/„. BOOK OP' INDUSTRIAL DESIGN. 21 'ITie radii and diameters of circles are to each other as the cir- cumferences, and vice, versa. The areas, therefore, of circles are to each other as the squares of their respective radii or diameters. It follow? hence, that if the radius or diameter be doubled, the eircumference will only be doubled, but the area will be quadrupled ; Ihus, a drawing reduced to one-half the length, and half the ()readth, only occupies a quarter of the area of that from which it Is reduced. 73. Sectors — Segments. — In order to obtain the area of a sector n( segment, it is necessary to know the length of the arc subtend- ing it. This is found by multiplying the whole circumference by the number of degi-ees contained in the arc, and dividing by 360°. Example. — The circumference of a circle being 3'5 m., an arc of 45° wdll be 3-5 X 45 360 •4375 m. The length of an arc may be obtained approximately when the chord is known, and the chord of half the arc, by subtracting the chord of the whole arc from eight times the chord of the semi-arc, and taking a third of the remainder. Example. — The chord of an arc being -344 m., and that of half the arc -198, the length of the arc is •198 X 8— -344 .,„„ = '4133 m. 3 The area of a sector is equal to the length of the arc multiplied into half the radius. Example. — The radius being -169 m., and the arc -266; •266 X -169 2 = ^0225 sq. m., the area of the sector. The area of a segment is obtained by multipljing the wdth ; that is, the perpendicular between the centre of the chord, and the centre of the arc, by -626, then adding to the square of the pro- duct the square of half the chord, and multiplpng twice the square root of the sum by two-thirds of the width. Example. — Let 48 m. be the length of the chord of the arc, and 18 m. the width of the arc, then we have 18 X ^626 = 11-268, and (11-268)2 = 126-9678; whilst ■ (?)' 576; therefore, 2 x V' 126-9678 + 576 x -^ = 636-24 sq. m., the area of the segment. The area of a segment may also be obtained very approximately by dividing the cube of the width by twice the length of the chord, and adding to the quotient the product of the width into two thirds of the chord. Thus, with the foregoing data, we have 18 = 48 x 2 60-7 and, 48 X 2 X 18 = 576-0 Total, 636-7 sq. m. A still simpler method, is to obtain the area of the sector of wliich the segment is a part, and then subtract the area of the COMPARISON OF CONTINENTAL MEASURES, WITH FRENCH MILLIMETRES AND ENGLISH FEET. Country. Austria, . Spain, . . . Pupal StatcB, Frankfort, llunilnirg, . Hanovor, . . llessc, .... Designation of Measure. (Vienna) Foot or Fuss = 12 iiiclies =: 144 lines, (Bohemia) Foot, (Venice) Foot, Foot (Palme) " Foot (Architect's Meas.) (Carlsruhe) Foot (new) = 10 inches = 100 lines, (Munich) Foot = 12 inches = 144 lilies, (Augsburg) Foot, (Brussels) Ell or Aune = 1 m6tre, Foot, (Bremen) Foot = 12 inches = 144 lines (Brunswick) Foot =: 12 inches = 144 lines (Crucow) Foot, (Copeiiliageii) Foot, (iMadii(l) Foot (according to Loli- miiM, <Justiliaii Vara( " Ijiscar), (iravaiia) Vara= ;5 Madrid toot, (Uoiiio) Koot, Arohiteot's Span = I foot, . . . Aiioieut Foot, Foot, l<'oot = 8 spans = 12 inches =: 96 parts, (Ilaiiovor) Foot ^ 12 inches 111 lines, (Daniistiidt) l''()ol,= 10 inciios = 100 linos Value in Millimetres. 31G-103 2'JU-416 435-185 347-398 396-500 300-000 291-859 296-168 1,000-000 285-588 289-197 285-362 356-421 313-821 282-655 835-906 817-965 297-896 223-422 294-216 284-61(> 286-490 291-995 300-000 Value in Feet. 1-037 -970 1-460 1-140 1-301 •984 •958 •972 3-281 -937 •949 •936 1-169 1-029 •927 2-742 2-782 •977 •783 •965 •938 •940 •958 •981 Country. Holland,. Lubeck, Mecklenburg, Modena, Ottoman Empire Parma Designation of Measure. Poland, . .. Portugal, . Prussia, . . Russia, . . . Sardinia, . Saxe, Sicilies, . . S-wedon, Switzerland, 1 usonny Wiirtoniburg,.,. (Amsterdam) Foot = 3 spans = 11 inches, (Rhine) Foot (Lubeck) Foot, Foot, (Modena) Foot, (Reggio),. (Constantinople) Grand pie,. . Ai-ms-]ongth= 12 inches = 1728 atonii (Varsovic) Foot = 12 inches ^ 144 lines, (Lisbon) Ft. (Arohitoot'sMoasnre) Vara = 40 inohos,. . (Berlin) Foot =12 iuoiios, . . . (St. Petersburg) Russian Footj " Arcliino, (Oiigliari) Span (Woiniar) Foot, Span ==12 iiulus (oimoes = 60 niinuti), (Stockholm) Foot, (Bale and Ziirioli) Fool, (Borne and Jsoiit'oliiMol) Foot = 12 iiK'hos, (Gonovii) Foot, (l.iiusaniio) Foot = 10 inolios = loil linos (I.uooriioiind ulluT I'lmtons) I't., l''oot, ... Foot = 10 inches = 100 linos,.. Value in Millimetres. 283-056 313-854 291-002 291-002 523-048 530-898 669-079 544 670 297-769 838-600 1,008-368 809-726 538-151 711-480 •202-578 281-972 208-670 296-888 804-587 298-2.'>S 487-900 30(1-000 8L'iS54 5481 67 280 4 90 Value in Foot. ■928 1-080 •954 •964 1-716 1-742 2^195 1787 -977 Mil 8-686 1-016 1-705 2884 •604 •926 •865 •974 •999 •902 1 •OOti •984 1-080 1-798 •940 22 THE PRACTICAL DRAUGHTSMAN'S trianele constituting the diiference between the sector and segment. To find the area of an annular space contained between two concentric circles, multiply the sum of the diameters by their difference, and by the fraction -7854. Example. — Let 100 m. and 60 m. be the respective diameters ; then, (100 + 60) X (100 — 60) x -7854 = 5026-56 sq. m. the area of the annular space. The area of a fragment of such annular space will be found by multiplying its radial breadth by half tlie sum of the arcs, or, more correctly, by the arc which is a mean proportional to them. CraCUMFERENCE AND AREA OF AN ELLIPSE. 74. The circumference of an ellipse is equal to that of a circle, of which the diameter is a mean proportional between the two axes ; therefore, it will be obtamed by multipljing such mean propor tional by 3-1416, the ratio between the diameter and circumferenco of a circle. Example. — Let 10 m. and 6-4 m. be the lengths of the respec- tive axes ; then, Via X 6-4 X 3-1416 = 25-13-28 m. The area of the ellipse is obtained by multiplmg the product of the two axes by -7854, the ratio between the diameter and the area of the circle. Example.— \0 X 6-4 X -7854 = —50-2656 sq. m. These rules meet with numerous applications in the indus- trial arts, and particularly in mechanics, as will be seen further on. The examples given will enable the student to understand the various operations, as well as to solve other analogous problems. CHAPTER n. THE STUDY OF PROJECTIONS. 75. To indicate all the dimensions of an object by pictorial deli- neation, it is necessary to represent it imder several different aspects. These various views are comprehended imder the general denomination of Projections, and usually consist of elevations, plans, and sections. The object, then, of the study of projections, or descriptive geometry, is the reproduction on paper of the appeai-- ances of all bodies of many dimensions as viewed from different positions. It is customary to determine the projections of a body on two principal planes, one of which is distinguished as the horizontal plane, and the other as the vertical plane, or elevation. These two planes are also called geometric projections or plans. They are annexed to each other, the horizontal plan being the lower ; the line intersecting them is called the base line, and is always parallel to one of the sides of the drawing. It is of great importance to have a thorough knowledge of the elementary principles of descriptive geometry, in order to be able to represent, in precise and determinate forms, the contours of all kinds of objects ; and we shall now enter upon such explanatory details as are necessary, commencing primarOy with the projections of a point and of a line. ELEMENTARY PRINCIPLES. THE PROJECTIONS OF A POINT. PLATE VI. 76. Let A B c D, figs. 1 and 1°, be a horizontal plane — repre- senting, for example, the board on which the drawing is being made, or perhaps the surface of a pavement. Also, let a B E f be a vertical plane, such as a wall at one side of the piece of pave- ment; the straight lice, which is the intersection of these two planes, is the base line. Finally, let o be any point in space, the representation of which it is desired to effect. If, from this point, 0, we suppose a perpendicular, o o, to be let fall on the hori2ontal plane, the point of contact, o, or the foot of this perpendicular, will be what is understood as the horizontal projection of the given point. Similarly, if from the point, o, we suppose a per- pendicular, o 6, to be let fall on the vertical plane, a b e f, the point of contact, 6, or foot of this perpendicular, will be the vertical projection of the same point. These perpendiculars are reproduced in the vertical and horizontal planes, by drawing lines, 6 n and n o, respectively parallel and equal to o o and o o. 77. It follows from this construction, that, when the two pro- jections of any point are given, the position in space of the point itself is determinable, it being necessarily the point of intersection of perpendiculars erected on the respective projections of the point. As in di-awing, only one surface is employed, namely, the sheet of paper, and we are consequently limited to one and the same plane, it is customary to suppose the vertical plane, a b e f, fig. 1, as forming a continuation of the horizontal plane, a b c d, being turned on the base line, a b, as a hinge, so as to coincide vfith it — just as a book, half open, is fully opened flat on a table. We thus obtain the figure, d c e f, fig. 1°, representing on the paper the two planes of projection, separated by the base fine, a b, and the points, o, 6, fig. 1% represent the horizontal and vertical pro- jections of the given point. It will be remarked, that these points lie in one line, perpen- dicular to the base line, a b ; this is because, in the turning down of the previously vertical plane, the line, n 6, becomes a prolonga- tion of the fine, n o. It is necessary to observe, that the fine, n 6, measures the distance of the point from the horizontal plane, whilst n o measures its distance from the vertical plane. In other words, if on o we erect a perpendicular to the plane, and BOOK OF INDUSTRIAL DESIGN. 23 measure the distance, n o', on this perpendicular, we shall obtain the exact position of the point in space. It is thus obvious, that the position of a point in space is fully determinable by means of two projections, these being in planes at right angles to each other. ' THE PROJECTIONS OF A STRAIGHT LINE. 78. In general, if, from several points in the given line, perpen- diculars be let fall on to each of the planes of projection, and their points of contact wdth these planes be joined, the resulting lines will be the respective projections of the given line. When the line is straight, it will be sufficient to find the pro- jections of its extreme points, and then join these respectively by straight lines. 79. Let M o, fig. 2, represent a given straight line in space, which we shall suppose to be, in this instance, perpendicular to the horizontal, and, consequently, parallel to the vertical plane of projection. To obtain its projection on the latter, perpendiculars, M m', o o', must be let fall from its extremities, m, o ; the straight line, m' o', joining the extremities of these perpendiculars, will be the required projection in the vertical plane, and in the present case it will be equal to the given line. The horizontal projection of the given line, M o, is a mere point, m, because the line lies wholly in a perpendicular, m 711, to the plane, and it is the point of contact of this line which consti- tutes the projection. In drawing, when the two planes are converted into one, as indicated in fig. 2°, the hoiizontal and ver- tical projections of the given right lines, m 0, are respectively the point, m, and the right line, m' 0'. 80. K we suppose that the given straight line, M 0, is horizon- tal, and at the same time perpendicular to the vertical plane, as in figs. 3 and 3", the projections will be similar to the last, but transposed ; that is, the point, (/, will be the vertical, whilst the straight line, m 0, will be the horizontal projection. In both the preceding cases, the projections lie in the same perpendicular line, m m, fig. 2% and 0' 0. fig. 5°. 8L When the given straight line, m 0, is parallel to both the horizontal and the vertical plane, as in figs. 4 and 4", its two pro- jections, m and m' 0', will be parallel to the base line, and they will eiich be equal to the given line. 82. When the given straight line, M o, figs. 6 and 5", is parallel to the vertical plane, a b e f, only, the vertical projection, m' o', will be parallel to the given line, whilst the hoiizontiil projection, m o, will be parallel to the base line. Inversely, if the given straight line be parallel to the horizontal plane, its horizontal projection will be i)arallel to it, whilst its vertical projection will bo parallel to the base line. 83. Finally, if the given straight line, m o, figs. 6 and 6", is inclined to both planes, the projections of it, m 0, m' o', will both be inclined to the base line, a b. These projections are in all cases obtained by letting fall, fiuiii each extremity of the line, per- pendiculars to each plane. The projections of a straight line being given, its position in sjjace is determined by erecting porpcndiculars to the liorizonliil plane, from the extremities, 7/1 o, of the jjrojcctcd lini^ and nialiing llicrn (•(lual to the verti('als, ?i in' .miicI /; n'. The same result riillows, ir fidiii IIk^ piiiiils, //(', o', ill till' vcitiiMl |il:ni(', we erect perpendiculars, respectively equal to the horizontal distances m n and p o. The free extremities of these perpendiculars meet each other in the respective extremities of the line in space. THE PROJECTIONS OF A PLANE SURFACE. 84. Since all plane surfaces are bounded by straight lines, as soon as the student has learned how to obtain the projections of these, he will be able to represent any plane surface in the two planes of projection. It is, in fact, merely necessary to let fall perpendiculars to each of the planes, from the extremities of the various lines bounding the surface to be represented; in other words, from each of the angles or points of junction of these lines, by which means the corresponding points will be obtained in the planes of projection, which, being joined, will complete the repre- sentations. It is by such means that are obtained the projections of the square, represented in different positions in figs. 7, 7*, 8, 8", and 9, 9'. It will be remarked, that, in the two first instances, the projection is in one or other of the planes an exact coimterpart of the given square, because it is parallel to one or other of the planes. 85. Thus, in fig. 7, we have supposed the given surface to be parallel to the horizontal plane ; consequently, its projection in that plane will be a figure, m p q, equal and parallel to itself, whilst the vertical projection will be a straight line, p' 0', parallel to the base line, a b. 86. Similarly, in fig. 8, the object being supposed to be paralle. to the vertical plane, its projection in that plane wOl be the equa. and parallel figure, m! d p' q', whilst that in tlie horizontal plane will be the straight line, mo. When the two planes of projection are converted into one, the respective projections will assume the forms and positions represented in figs. 7°, 8°. 87. If the given surface is not parallel to either plane, but yet perpendicular to one or the other, its projection in the plane to which it is perpendicular will still be a straight line, as p' o', figs. 9 and 9", whiist its projection in the other plane will assume tlie form, mop q, being a representation of the object somewhat fore- shortened in the direction of the inclination. The cases just treated of have been those of rectangular sur- faces, but the same principles are equally applicable to any poly- gonal figures, as may be seen in figs. 12 and 12', wiiich will b* easily understood, the same letters in various ch;u-acters indicating corresponding points and perpendiculars. Nor does the obtain- ment of the projections of surfaces bounded by curved lines, aa circles, require the consideration of other principles, as we shall proceed to show, in reference to figs. 10 ;i8(l 1 1. 88. In the first of these, fig. 10, the circular tlisc, M r q, is su]v posed to be parallel to the vertical plane, A B e f, and its projec- tion on that plane will be a circle, m' o' p' q', equal luid fwiniilel to itself, whilst its projection on the horizontal pUine, a b c d, will be a straight lino, q m o, equal to its (Uainoter. If, on the other hand, the disc is parallel to the horizontal plane, jus in fig. 11, its vortical projection will be the straight lino, /)' (Z /«', whilst its horizontal projection will be (lu^ circle, op m q. It' the given circular disc bo inclined to oillu'r piano, its projoc tiiin in lli.'il |)l:uu' will bo an ellipse ; and if it Is inoliuod to both phinrs, Imlli prcijocliiins will l>o olli[isos. This will bo niadooxi- 24 THE PRACTICAL DRAUGHTSMAN'S dent ty obtaining the projections of various points in the circum- ference. 89. When constructing the projections of regular figures, it facilitates the process considerably if projections of the centres and centre lines be first found, as in figs. 10, 11, and 12. In general, the projection of all plane surfaces may be found, when it is known how to obtain the projections of points and lines. And, moreover, since solids are but objects bounded by surfaces and lines, the construction of then- projections follows the same rules. PRISMS AND OTHER SOLIDS. PLATE VII. 90. Before entering upon the principles involved in the repre- sentation of soKds, the student should make himself acquainted with the descriptive denominations adopted in science and art, with reference to such objects ; and we here subjoin such as will be necessary. Definitions. — A solid is an object having three dimensions; that is, its extent comprises knglh, width, and height. A solid also possesses magnitude, volume, or capacity. There are several forms of solids. The polyhedron is a solid, bounded by plane surfaces ; the cone, the cylinder, and the sphere, are bounded by curved surfaces. Those are termed solids of re- volution, which may be defined as generated by the revolution of a plane about a fixed straight line, termed the axis. Thus, a ring, or annular torus, is a solid, generated by the revolution of a circle about a straight line, lying in the plane of the cucle, and at right angles to the plane of revolution. A prism is a polyhedron, the lateral faces of which are parallelograms, and the ends equal and parallel polygons. A prism is termed right, when the lateral ^es, or facets, are perpendicular to the ends; and it is regular, when the ends are regular polygons. A prism is also called a parallelopiped, when the ends are rectangles, or parallelograms ; and when it is formed of six equal and square facets, it is termed a cube, or regular hexahedron. This solid is represented in fig. ^. Other regular polyhedra, besides the cube, are distinguished by appropriate names ; as, the tetrahedron, the octahedron, and the icosahedron, wliich are bounded externally, respectively, by four, eight, and twenty equilateral triangles; and the duodecahedrcm, which is terminated by twelve regular pentagons. A pyramid 13 a polyhedron, of which all the lateral facets are triangles, uniting in one point, the apex, and having, as bases, the sides of a polygon, which is the base of the pyramid, as fig. ©. The prism and ppamid are triangular, quadrangular, pentagonal, hexagonal, &c., according as the polygons forming the bases are triangles, squares, pentagons, hexagons, &c. By the height of a pyramid is meant the length of a perpendi- cular let fall from the apex on the base ; the pyramid is a right pyramid when this perpendicular meets the centre of the base. A truncated pyramid, or the frustum of a pyramid, is a solid wliich may be described as a pyramid ha\dng the apex cut off by a plane parallel, or inclined to the base. A cylinder is a solid which may be described as generated by a straight line, revolving about, and at any given distance from, a rectilinear axis to which it is parallel. A cylinder wliich is genc^ rated by a rectangle, revolving about one of its sides as an axis, is said to be a right cylinder ; such a one is represented in fig. [E. A cone, fig. [p", is a solid generated by a triangle, revolving about one of its sides as an axis. A truncated cone is one which is terminated short of the apex by a plane parallel, or inclined to tlie base. This solid is also called the frustum of a cone. A cone is said to be right when its base is a circle, and when a per- pendicular let fall from the apex passes through the centre of the base. A sphere is a solid generated by the revolution of a scniiciicle about its diameter as an axis, as fig. @. A spheric sector is a solid generated by the revolution of a plane sector, as o' l e', about an axis, a h, which is any ladius of llie sphere of which the sector forms a part. When the axis of revolution is exterior to the generating sector, the spheric sector obtained vvill be annular or zonic. The zone described by the arc, L e', is the base of the spheric sector. The zone becomes a spheric arc when the axis of revolution is one of the radii forming the sector. A spheric wedge, or ungula, is any portion, as i H g y, fig. T, comprised between two scmicucular planes inclined to each other and meeting in a diameter, as i g, of the sphere. That portion of the surface of the sphere which forms the base of the ungula, is termed a gore. A spheric segment is any part of a sphere cut off by a plane, and may be considered as a solid of revolution generated by the revolution of a plane segment about its centre line. The plane surface is the base of the segment. Wlien the plane passes through the centre of the sphere, two equal segments are obtained, termed hemispheres. A segmental annulus is a solid generated by the revolution of a plane segment, d' b' k, fig. 7, about any diameter, a b, of the sphere, apart from the segment, d' k is the chord, and m n, its projection on the axis, is the height of the segmental annulus. A zonic segment of a sphere is the part, l n k d', of a sphere comprised between two parallel planes. A spheric pyramid, or pyramidal sector, is a pyramid of which the base is part of the surface of a sphere, of which the apex is the centre ; the base may be termed a spheric polygon. THE PKOJECTIONS OF A CUBE, PIG. ^. 91. A cube, of which two sides are respectively parallel to the planes of projection, is represented in these planes by equal squares, a b c d, and a' b' e' f', figs. V and 1. This is indeed but a combination of some of the simple cases already given. We have seen that when a side, such as a b e f, fig. A, is parallel to the vei-fical plane, its projection on the hori- zontal plane is reduced to a straight line, a b, fig. P, its projection on the vertical plane being a figure, a' b' e' f', fig. 1, equal to itself. Similarly, the side, a b c d, which is parallel to the horizontal plane, is projected on the vertical plane in the line, a' e', *ig. 1, and by the figure, a b c d, fig. 1°, in the horizontal plane, llie sides, A D H E and b c g e, fig. A, which ai-e perpendicular to both the horizontal and the vertical plane, are represented in both oy sti-aight lines, as a n and b c, fig. 1, and a' f' and b' e', fig. 1, BOOK OF INDUSTRIAL DESIGN. 25 hp.ing respectively in the same straight lines perpendicular to the base line, l t. It will also be perceived, that the base, f e g h, fig. A> cannot be represented in the horizontal projection, nor the side, D c G H, in the vertical, since they are respectively immedi- ately behind and hidden by the sides, a b c d and a b e f, repre- sented, in the projections by the squares, a b c d, fig. 1°, and a' b' e' f', fig. 1. They are, however, indicated in the planes to which they are perpendicular, by the straight lines, f' e' and d c. 92. It will be evident from these remarks, that in order to design a cube so that a model may be constructed, it is sufficient to know one of the sides, for all the sides are equal to each other. When the plans are intended to be used in the actual construc- tion of machinery or buildings, the objects should be represented in the projections as having then- principal sides parallel or per- pendicular to the horizontal and vertical plane respectively, in order to avoid the foreshortening occasioned by an oblique or in- clined position of the object with reference to these planes, because the actual measurements of the different parts cannot be readily ascertained where there is such foreshortening. To obtain, then, the projections of the cube, fig. ^, a square must be constructed, as a b c d, fig. 1°, having its sides equal to the given side or edge, the sides a b and d c being disposed parallel to the base line ; next, the square must be reproduced as at a' b' e' f', fig. 1, on the prolongations of the sides, A d and b c, which are perpendicular to the base line. THE PROJECTIONS OF A RIGHT SQUARE-BASED PRISM, OR RECTANGULAR PARALLELOPIPED, FIG. [B. 93. The representation of this solid is obtained in precisely the same manner as that of the cube, the sides being supposed to be parallel or perpendicular to the respective planes of projection. The base of the prism being square, its horizontal projection is necessarily also a square, a b c d, fig. 2°; but its vertical pro- jection will be the rectangle, a' b' e' f', fig. 2, equal to one of the sides of the prism. For the construction of these projections, the same datum as in the preceding case is required ; namely, a side of the base, and in addition, the height of the parallelepiped, or prism. the projections of a quadrangular pyramid, fig. ©. 94. This pyramid is supposed to be inverted, and having its base, A b c d, parallel to the horizontal plane : it follows upon this assumption, that its horizontal projection is represented by the square, A b c d, fig. 3". The axis, or centre line, o s, wliich is supposed to bo vertical, and consequently passes through the centre of the base, is projected on the horizontal plane as a point, o, fig. '<', and on the vertical plane as a line, o' s ; drav/mg througli tlio point, o', of this line, the horizontal line, a' b', equal to a side of llu! base, wiiicii is supposed to be parallel to the vortical plane, wo shall obtain the vertical projeclion of the base ; and joining a's, ii's', that of tho whole pynuiiid, tiio points a' and d' may bo found by prolonging the parallels, A D, B c, fig. 3". This may bo conveniently done with tho square, and the operation is usually termed squaring over a measurement — that is, from one projection to another. Tho lateral f'acels, s n c arid sab, are rcprosenttid in the vortical ])ro- jection by tho straight linos, a's, n' s, lig. 1, since Ihoy are per- pendicular to the vertical plane ; and the projection of the facet. D s c, is identical with a' s b', that of the front facet, a s b, imme- diately behind which it is. Since each of the inclined converging facets is liidden by the base, they cannot be dra\vn in sharp lines in the horizontal projection ; we have, however, indicated their positions in faint lines, fig. S". Were these lines full, the projec- tion would be that of a pyramid with the apex uppermost, or of a hollow, baseless pyramid, in the same position as fig. ©. THE PROJECTIONS OF A RIGHT PRISM, PARTIALLY HOLLOWED, AS FIG. ©. 95. The vertical and horizontal projections of the exterior of this solid, are precisely the same as those of fig. \B ; they are re- presented respectively by the square, a b c d, fig. 4°, and the rec- tangle, a' b' e( f', fig. 4. It will be perceived, that the internal surfaces of this figure are such as may be supposed to form some of the sides of a smaller prism ; the sides, g h i j and k l m n, are parallel to the vertical plane, and g k n j and h i m l perpendicular to it, and it follows that the projections of this lesser figure wUl assume the forms, g' h' i' j', fig. 4, and g h l k, fig. 4*. The lines, k g, l h, are faint dotted lines, instead of being sharp and full, as being hid by the base, a b c d, of the external prism. These lines wll be found to be different to the projection lines, or working Unes. The latter are composed of irregular dots, whilst those which indicate parts of the figure which actually exist, but are hidden behind more prominent portions, are composed of regu- lar dots. This distinction has been adhered to throughout the entire series of Plates. 97. On examining the examples just treated of, it vdU be ob- served, from the horizontal projections, that the contour, or out- line, is in every case square, whilst, from the vertical projections, it will be seen that each object is different. This demonstrates that one projection is not sufficient for the determination of all tlie dimensions of an object ; and that, even in the simplest cases, two different projections are absolutely necessary. It \\ill, moreover, be seen, as we advance, that in many cases, three, and at times more, projections are required, as well as sections through two or more pKanes. the PROJECTIONS OF A RIGHT CYLINDER, FIG. d. 98. The axis, o M, of this cylinder is supposed to be verfio^xl, and its bases, a b, e f, will consequently be horizontal. Its pro- jections in figs. 5 and 5° are represented by tho rectangle, a' b' e' f', on the one hand, and the circle, a c b d, on tho otlier. It is evi- dent, that to draw (hose figures, it is quite sutliciont to know the radius, o a, of the base, and tho height, o iM ; with the giver, radius, wo describe the circle, a c b d, which is tho horizontal {>i<. jection of the whole cyliiulor; then making the vertical, o' m, equju to tiic given lu'iglit, .Mild squaring o\or by iiicans of tho panUlfls, a a', 1! n', the diameter of tho circle, we draw, through o' and M, the liori/outals, a' b', e' f', com[)letiiig tho ivinillelognuu. a' li' r.' i', which is the vertical projection of the cyliiulor. Tin; puoJKCTiONs of a right cone, Flu. [?. !>!>. The projections of a right coiu' ditVor from tluise of tho cviiii- dor solely as far as reganls tho vertical piano. Thus it will be .>ieou, 1> 26 THE PRACTICAL DRAUGHTSMAN'S m figs. 6 and 6', that the horizontal projection of the cone, s a b, is exactly the same as that of a cylinder having an equal base ; but the vertical projection, s' a' b', in place of being a rectangle, is an isosceles triangle, of which the base is equal to the diameter of the circle, forming the horizontal projection, whilst tlie height is that of the cone. Similarly to the preceding case, in order to construct these projections, it is sufficient to know the radius of the cu'cular biLse and the height. THE PROJECTIONS OF A SPHERE, FIG. @. 100. A sphere, in whatever position it may be with reference to the planes of projection, is invariably represented in each by a circle, the diameter of which is equal to its own ; consequently, if the two projections, o and o, figs. 7" and 7, of the centre be given, we have mei'ely to describe circles with these centres, with a radius equal to that, o a, of the ^ven sphere. It would seem from this, that one projection should be sufficient to indicate that the object represented is a sphere ; but on referring to figs. 5°, and 6°, and T, it wiU be seen that a circle is one pro- jection of three very different solids — namely, the cylinder, the cone, and the sphere. This is a further illustration of the in- adequacy of one projection to give a faitliful representation of any solid form. It is true, that by shading the single projection, we approach nearer to the desired representation ; but still, such shaded projection would equally represent that of a cylinder with a hemispherical termination. The same remark applies to the shaded projections of cylinders and cones, and, indeed, to all solid bodies. OF SHADOW-LINES. 101. To distinguish and relieve those parts of a drawing which are intended to indicate the more prominent portions of the object represented, it is customary to employ fine sharp fines for that part of the outline on which the fight strikes in fuU, and strong and heavy lines for the parts which are at the same time in refief and in the shade ; the latter description of fines are called shadow- lines. For the maintenance of uniformity, it is obviously expedient to suppose the light to strike any object in some constant and particular direction. The assumed direction should be inclined, m order that some parts of the object may be thrown into shade, whfist others are more strongly iUuminated. Hitherto, a uniform method has not been generaUy adopted with regard to the assumed direction of the rays of fight. Some authors have recommended that it should be, as it were, paraUel to that diagonal, a g, of the cube, fig. ^, of which the projections are a c and a' e', figs. 1 and :"; others, however, cause the ray of light to take the direction a' e' in the vertical, and d b in the horizontal plane of projection, and some liive proposed that the ray should strike the object in a direction perpendicular to either of the planes. We have adopted the fu-st mentioned system, and we shall shortly indicate in what points it is superior, and on what account it is preferable, to any other. The line which we take as the diagonal of the cube, is that which extends from the comer, a, of the front facet of the cube, fifif. ^ tn the extreme and opposite corner, g, of the posterior facet. The projections of this straight fine in the representiitiuns of the cube, figs. 1 and 1°, are respectively the lines, a c and a e', lying ea«h at an angle of 45° with the base line. Thus, in gen- eral, in our dravvdngs the objects are supposed to receive the light in the du'ection of the arrows, r and r', in fig. 8, according as the projection is in the vertical or horizontal plane. 102. We must observe, that the actual incUnation of the straight fine thus adopted, is not that of 45° with respect to either plane of projection ; the angle of incUnation is in fact somewhat less, and may be determined by means of the diagram, fig. 9. For this purpose, it is necessary to suppose the perpendicular plane iu which the ray or line fies, as turned or folded down upon the ver- tical or horizontal plane, the turning axis being perpendicular to the base line. Let us, in the first place, suppose the two pro- jections, E and r', of the ray, to meet in the point, o, in the base line, L T ; taking any point in this ray, as projected in the horizontal plane at a, and in the vertical at a', with the point, o, as a centre, and radius, a o, describe the arc, a c a', cutting the base fine in the point, c; through this point draw the perpendicular, b b', limited each way by the lines, a b, a' b', di-awn paraUel to the base fine through the points, a, a . Joining o b and o b', the lines thus obtained indicate the position and inclination of the ray, when folded down, as it were, on either plane of projection ; and on applying a protractor, it will be found that the actual angle of in- cfination is one of 35° 16' nearly. Having, then, fixed upon the direction of the rays of fight, which ai-e, of course, supposed to be paraUel amongst themselves, it wUl be easy to determine which part of an object is illuminated, and which is in the shade. It will be perceived, for example, in figs. 1 and 1°, that the illumi- nated portions are those represented by the lines, a b and a d, on the one hand, and a' b' and a' f' on the other ; and that those in the shade are represented respectively by the fines, b c, c d, and b' e', f' e'. It must be observed, that according to this system, whatever part of the object is represented as Ulumtnated in one projection, is equaUy so in the other; the shaded parts corre- sponding in a simUar manner. What has just been said with reference to the cube, is equaUy appficable to all prisms or sofids boimded by sharp definite outlines, care being taken to employ heavy shadow-lines only on the outlines of parts which are both promident and in the shade — such shadow-fines separating the facets which are Uluminated, from those which are not. 103. With regard to round bodies, the projections of the lateral portions being bounded by fines which should not indicate prominent and sharply defined edges, so fuU a shadow-line should not be employed as that forming the outfine of a plane and pro- minent surface. Thus, in figs. 5, 6, and 7, the lines, b' e', s' b', and c' b' d', are not nearly as strong as the corresponding lines, b' e', in figs. 1 to 4. Nevertlieless, these lines should not be as fine as those on the Uluminated side of the object, but of a medium strength or thickness, to indicate the portion of the object which is in the shade. In other words, a sharp fine fine indicates the fully Uluminated outfine, a fuUer line the portion in shade, and a shadow-line still stronger that portion which is both in the shade, and has 9 prominent sharply defined edge. The straight lines, f' e' and a' b', figs. 5 and 6, wiU necessarily be fuU shadow lines, as representing the edges of planes entirely in the shade. BOOK OF INDUSTRIAL DESIGN. 27 In the liorizontal projection of the cylinder, fig. 5°, the illumi- nated portion corresponds to the semi-circle, adb, whilst that in the shade is the other semi-cu'cle, a ch; the points, a, h, of separation of the two halves, are obtained by drawing through Uie centre, o, a diameter, a b, perpendicular to the ray of light, d o, or by drawing a couple of tangents to the circle parallel to this ray. The straight line, a b, is inclined to the base line at an angle of 45°. Great care is necessary in producing the circular shadow-line, acb, and the nibs of the drawing-pen should be gradually brought closer as the extremities, a and b, of the shadow-line are approached, so that it may gi-adually die away into the thickness of the illuminated line. By inclining the drawing-pen, or by pressing it sideways against the paper, the desired effect may be produced ; the exact method, however, being obtained rather by practice than by following any particular in- structions. A very good effect may also, in some instances, be produced, by first dravdng the entire circle with the fine line, and then retracing the part to be shadow-lined with a centre slightly to one side of the first centre, and repeating this untQ the desired strength of the shadow-line is obtained. 104. In the plan of a cone, fig. 6°, the part in the shade is always less than the part illuminated; but it requires an especial con- struction, which will be found in the chapter treating of Shadows, for the determination of the lines of separation, s e ; and it is sel- dom that such extreme nicety is observed in outline drawings, the shadow-line of the plan of the cone being generally made the same as that of a cylinder, or perliaps a little less, according to tlie judgment of the artist. Yet, if the height of the cone be less than the radius of the base, the whole conical surface will be illuminated, and consequently its outUne should have no siiadow- Jine. 105. In explanation of the motives which have guided us in the adoption of the diagonal of a cube, as projected in the lines, R, r', fig. 8, as the direction of the rays of light, in preference to the other systems proposed, we shall proceed to point out some of the inconveniences attending the latter. In the first place it must be observed, that if we adopt, as the direction of the rays of light, the diagonals projected in a' e' and D B, figs. 1 and r, that part of the object which is represented in the plan as illuminated, does .not correspond with the part repre- sented as illuminated in the elevation : in such case, the shadow- lines would be AB and b c in the horizontal projection, and f' e', d' e' in the vertical, so that there is no distinction made betwec^n the plan and the elevation; whereas, according to the system adopted by us, it is at first sight apparent wliich is (he plan, and whicli the elevation, from (he mere .shadow-lines, whidi are in Ihc latter at (ho lower [)arts of (ho object; whilst, in (lie Coi-nier, (licy arc, on tiie contrai-y, at (ho up|)cr j)mi'Is. K Is nol nntin-al, moic- ovcr, ((• snpposc, lliat in Hk; I'l-prcscnlnlidii iifinn- (ilijccf, llic light can li(! made to come as it were from behind llu^ ol)jc<'(, tor in (liat case (he side nearest (he spec(a(()r would evidently be in (he miaiiu; and y(^t it is only on such a siii)posilii]n that the prdjci'llnns ot the ray of light can be such as u u and a' k'. Thus i\ dimble inconvenience may bcs urged against (his system. If, on the otIi'T hand, (h(i rays oC light are Hni)]M)scd (o be pir- pondioilar to either plane, such confusion will resuK as (o rendei- it impossible to ascertain, by any reference to the shadow-fines what is, or what is not, illuminated, and thus the object of employ- ing shadow-lines would be lost sight of. Thus let us suppose, for example, that the light is perpendicular to the vertical plane, whence it follows that the whole of the anterior facet, figs. 1 to 4, is fully illuminated ; but, at the same tune, all the facets perpen- dicular to the vertical plane are equally in the shade, and it would consequently be necessary to use shadow-lines all roimd, or else not at all ; and whichever plan was adopted, would be quite unintelligi- ble. Besides this, it is unnatural to suppose that the spectator should place himself between the light and the object. Indeed, it is unquestionable that the most appropriate du-ection to be given to the ray of light is as before stated, that of the diagonal of a cube, of which the facets are respectively parallel to the two planes of projection ; and the projections of this diagonal are, consequently, inclined to the base fine at an angle of 45°, but proceeding from above in the vertical projection, and from below in the horizontal projection, as shown by the arrows, e and e', fig. 8. PROJECTIONS OF GROOVED OR FLUTED CYLINDERS AND RATCHET WHEELS. PLATE VIII. 106. The various diagi'ams in this plate are designed principally with the view of making the student practically conversant with the construction of the projections of objects ; and, besides teach- ing him how to deUneate their external contours, to enable him to represent them in section, that their internal structure may also be recorded on the drawing. Figs. 1 and 1° are, respectively, the plan and elevation of a right cylinder, which is grooved on its entire external surface. The grooves on one-half of the circumference are supposed to be pointed, being formed by isosceles triangles of regular dimen sions, and may represent the rollers used in flax machinery, in apparatus for preparing food for animals, and in many other machines. The other half of the circumference is formed into square or rectangular grooves, the lateral faces of which are cither parallel to the centre lines which radiate from the centre, or are (hemsehx's radiating. 107. To construct the horizontal projeelina nt' this cylinder, that is, as seen from above, wo must first ascertiiin how many grooves are contained in the whole circumference ; then di-awing a circle with a radius, a o, wliich should idways bo gi-eater tlian thM of the given cylinder, divide it into twice as many equal pjirts a.s (here are grooves. If (he student will refer back to the section treating of linear drawing, illusVated in Plate 1.. he will liml simple methods of dividing circles into '2, 3, 4, (!, 8, and I'J equal parts, and, further, of subdi\iding these. Thus, as the cylinder, lijj. 1, contains 24 grooves, its cireumforcnco must bo divided into 48 equal par(s. To obtain these, begin by drawing two dimnoters, a n, c i>. perpendicular to each odier ; then, from e.'icli oxtromily, mark oil" tJio liii'jth of (ho radius, a o, thus <)b(aining the ("our points numocreil s on one side, and (lu> points nundiered I on the otiior — making, wiih tlu' points of intersection iif the two dianiotors with the cir- enmCerence, in all, I'J points. It remiiiiiii simply to bisivt ojid) spac(>, as A — t, U — I, or 4 — 8, &c.. as \\\'\\ as the leaser sivaces I 28 THE PRACTICAL DRAUGHTSMAN'S thus found; tliis will give the 48 divisions required. Tlirough the points of division draw a series of radii, whieli ^\ill divide the inner circle described, with the radius, o f, into the same number of equal parts. Tlie depth of the grooves is limited by the circle described with the radius, o E, whilst the outside of the intervening ridges is defined by the cu-cle of the radius, o f. All the operations which we have so far indicated, are called for in the construction of both the triangular and rectangular grooves. In proceeding, we must, in the former case, join the points of in- tersecti ~>n, a, b, c, d, which are in each circumference alternately ; whilst ir the latter ease we require no fresh lines, but have simply to ink in alternate portions of the two circles, as well as the radial lines joining these. 108. To draw the vertical projection, fig. 1', it is necessary that the depth should be given, say m' n' = 54. First set out the two horizontals, m' p', n' q', limiting the depth of the figure ; then, to obtain the projection of the grooves and ridges, square over each of the points, e,f,g,-'h,&.c., and di-aw parallels through the po'iits thus found in fig. 1°, as e',f',g',h'. This completes the elevation, and represents the whole exterior of that part of the cylinder below the horizontal, m p. 109. It has already been observed, that two projections are not always sufficient to form a complete representation of an object ; thus it wiW be evident, from a consideration of figs. 1 and 1", that a third view is necessary to explain the interior of the cylinder. The radius, o g = 42, of the central circular opening, is not ap- parent in fig. 1", it is only to be found in the plan; whereas we have already seen, that, to determine its exact position, it should be represented in two projections. From figs. 1 and 1°, it is im- possible to see if the opening exists throughout the depth of the cylinder, or if its radius be the same dowTi to the bottom ; and the same remark applies to the key-way. In consequence of this, it is expedient to draw the object as sectioned — for example, through the centre line, m p — ^by a plane parallel to the projection. Such a section is represented in fig. 1° ; and from it, it is at once manifest that the central eye or opening, as well as the key-way, extend equally throughout the depth of the cylinder. The outline of these parts is formed by the verticals dra\vn through the points, g', m\ v', h', obtained by squaring over the corresponding points, G, m, n, H, in the plan. This ^■iew also shows that the external grooves are equal throughout their depth, as Lodlcated by the ver- ticals drawn through m', l', e', p'. When the outlines of the in- terior of an object are few and simple, they may be indicated in an elevation, such as 1", by dotted lines. But if the outUnes are numerous or complex, too great a confusion would result from this method ; and it is far better, in such case, to give a sectional \iew. That portion of the solid mass of the cylinder, through which the sectional plane passes, is indicated in fig. 1', by a flat^tinted sihading, so as to distinguish it from the parts which the plane does not meet: this is the plan generally adopted to show the parts in section ; the strength of the shade, or sectioning, is varied according to the nature of the material. Thus, for cast-iron, a darker shade is used, whilst a lighter one indicates wood or stone ; and as an example of tliis distinctive use of various degrees of snadc, we have to point out that the sectioning in fig. 1', indicates the object to be made of copper, wliUst thi^t in fig. 2' corresponds to cast-iron, and in fig. 3' to wood or masonry. 110. It must be observed, that the section lines, of whatever description they may be, are always inclined at an angle of 45" with the base line ; this is to distinguish the sectioning from flat tints frequently employed in elevations, to show that one surface is less prominent than another: this latter flat-tinting is generally produced by perpendicular or horizontal lines. The line, t j', which indicates the base of the mternal cylinder, g mn, should not be a shadow-line equal in strength to the bases of the sectional parts, for the latter are more prominent. This point is seldom attended to as it should be ; gi-eater beauty and effect, however, would result if it were. This remark appUes equally to all projections of objects, of which one portion is more prominent than another. Thus, in figs, l", 2", S", the vertical lines passing through f are considerably more pronounced than those passing t'li-ough p' q' and lying in a posterior plane. It is the more important to observe these distinctions in representations of complex objects, so as to assist as much as possible a comprehension of the dravring. After the preceding consideration of fig. 1 on this plate, figs. 2 and 3, representing ratchet wheels and fluted cylinders, will be quite intelligible to the student ; such operations as are additional, being I rendered quite obvious by the views themselves. THE ELEMENTS OF ARCHITECTURE. PLATE IX. 111. Columns of the different orders of architecture are fre- quently employed in buildings, and also in mechanical constructions, as supports, where it is desired to combine elegance with strength. The ancient orders of architecture number five ;* as, 1. The Tuscan. 2. The Doric. 3. The Ionic. 4. The Corinthian. 5. The Composite. A sixth order is sometimes met svith, denominated the Poestum- Doric. 112. Each order of architecture comprises three priacipal parts: the pedestal, the column, and the entablature, in all ftie ordei-s, the pedestal is a third of the length of the shaft in height, and the depth of the entablatm-e is a fom-th of the shaft. The proportion between the diameter and height of the column varies in each order. The height of the Tuscan column is seven times the diameter at the lowest part ; the Doric, eight times ; the Ionic, nine times; the Corinthian and Composite, ten times. The pedestal is frequently altogether dispensed with. All the differ- ent parts, in the various orders, bear some proportion to a module, which is half the diameter of tlie lower part of the column. This module may be termed, the imit of proportion. It is di\-ided * We have adhered to the classification wnich, from being of more ancient date, is supported by superior authority ; but we do not profess, in this work, to decide which carries more reason with it. Reason frequently runs counter to authority Modern architects say there are only three orders — the first comprising Ancient, Modern, and Tuscan Doric ; the second, Greek, Roman, and Modern Ionic ; and the third, Corinthian and Composite. BOOK OF INDUSTRIAL DESIGN. 29 into 12 parts, in the Tuscan and Doric orders; and into 18 parts, in the Ionic, Corinthian, and Composite. The whole height of the Tuscan order is 22 modules 2 pails, apportioned as follows : — The column is 14 modules; the pedestal, 4 modules 8 parts ; and the entablature, 3 modules 6 parts. The whole height of the Doric order is 25 modules 4 pai-ts — the column being 16 modules; tlie pedestal, 5 modules 4 parts ; and the entablature, 4 modules. The whole height of the Ionic order is 28 modules 9 parts — the pedestal, 6 modules; the colmnn, 18 modules ; and the entablature, 4 modvdes 9 parts. The whole height of the Coiinthian and Composite orders is 31 modules 12 parts — of which 6 modules 12 parts form the pedestal, 20 modules the cohmin, and 5 modules the entablatmx'. As we do not propose to treat especially of arcliiteuture, we Iiave not given drawings of all the various orders, but have con- fined ourselves to the Tuscan, as being the simplest, as well as the one more generally adopted in the construction of machinery. At the end of this Chapter, will be found tables of the dimensions of the various components of the Tuscan order, and we also there give a similar tabla for the Doric order. OUTLINE OF THE TUSCAN ORDER. 113. The whole height being given, as m n, the proportions of the different parts may always be determined. Let this height be, for example, 4 metres 272 millimetres, fig. 7. First, divide it into 19 equal parts, then take 4 such parts for the height of the pedestal, 12 for that of the column, and the remaining 3 for the entablature. Then,' according to the order which it is intended to follow, the height, m n, of the column, is divided into 7, 8, 9, or 10 equal p.arts, and the diameter of the lower part of the column \vill be equal to one of these divisions : thus, in the Tuscan order, the diameter, a b,is ^ of the height, ?« n ; the half of this diameter, or the radius of the shaft, is the unit of proportional measurement, called the module, and with which all the components of the order are measured : it follows then, that in tlie Tuscan order this mo- dule is jj- of the height of the column, in the Doric -^, in the Ionic Jj, and 4^ in the Corinthian and Composite. - 114. The three members of an order are each subdi\ided into three divisions. Thus the Pedestal is composed of the Socle, or lower Plinth, a ; of the Dado, b ; and Cornice, c : the column consists of the Base, or Plinth, d ; the Shaft, e ; and the Capital, f ; and in the entablature are the Architrave, g ; the Frieze, ii ; and the Cornice, l 115. Bofjre proceeding to delineate these different parts, and the mouldings witli whicli they are ornamented, it is expedient to set off a scale of modules, detennined in the manner just stated, the module being, of course, subdivided into 12 equal parts. To make the mouldings and various details more intelligible, we have drawn tlic various portions of the order, sepai-ately, to a larger scale. Thus the socio and pedestal of the column are ropre- .lentc'! in elevation in fig. 2, and in plan in (ig. 3, to a scale 2.'- times tiiat of the comj)leto view, fig. 1, and the modulo will, of coui'se, be proportionately larger. All the numbers indicated on these figures, give the exact measurements of each part and eiu'h moulding, so that they may bo drawn in i)ert'ect ac^cordance with tJio scale given. Ft conduces considerablj to the symmetry and cxactitudu of the drawing, to set off all the measurements from the axis or centre line, c d. The module being but an aibitrary measmement, it ia necessaiy, in practically caiTying out any design, to ascertain the different measures in metres and parts of metres ; and for this reason we have given additional scales in metres, to correspond to those in modules ; and we have also expressed in millimetres, on each figure, the measxu-ements of the various details, placing the metrical in juxtaposition with the modular ones. And, to give a distinct idea as to the degrees of prominence or relief of the various members, a part of the elevation is shown as sectioned by a plane passing through the axis of the shaft, this part being sufficiently distinguishable from the sectional flat-tinting. In the horizontal projection, fig. 3, are also represented portions of sections in two different planes, one being at the height of the line, 5 — 6, and the other at that of 7 — 8. The first shows that the shaft is round, as well as the fillet,/, and the torus, g, wliilst the base, 7(, and cornice, i j, are square : the second section shows, in a similar manner, that the dado, b, the socle, a, and its fillet, f. are square. The flat-tintings sufficiently indicate the parts in section. Fig. 4 represents the entablature and tlie capital of the column La elevation and in section. Fig. 5 is a horizontal section of the column with its capital, as it were inverted, and is supposed to be half tlii-ough the line, 1 — 2, and half through 3 — 4. The whole is what is termed a false section, the parts in section being in paraUcl, but not identical planes. The different measurements are given in modules and metres, as in the other figures ; they indicate the respective distances from the axis, c' d'. 116. The execution of this design offers little or no dilficidtv ; but all the operations required, as well as the pai-ts to which the measurements apply, are carefully indicated. It is, therefore, unnecessary to enter into further detaOs, except as far as relates to such parts as involve some peculiarity ; the shaft of the column, for example, and one or two of the mouldings. Referring, in the first place, to the column, it is to be observed that it is customary to make the shaft cylindi'ical for one tliird of the height, that is, of equal diameter tlu-oughout tliat extent: above that point, however, it diminishes gradually in diiuneter up to the capital. This taper is not regular tiiroughout, being scarcely perceptible at the lower part, and becoming more and more convergent towards the top. Its contour is consequently a curve, instead of a straight line. This curvature constitutes what is termed the eniasis, and is employed to correct the appaient narrowness of a recti]inc;u- column at the middle. Such defooti\o appearance only takes i)lace when the cyliniirical piece, or column, is between a pedestal and an entablature having jilano surfaces A cylinder, or sphere, always seems to occupy less sjwce tiian a plane surface equiil to its greatest section. Thus llie outline of a cylinder or sphere, appeai-s to grow less wiien it is shaded. Now, where the column is in contact wilii the plane surface of the pe- destal or entablature, it cannot appear less in proportion, the proximity "f the latter correcting sucii appeanince, whilst th.u influence is loss felt at the central [kwU wliich is furthest from lli» I)edestal and entablature. A true cyliiuler, therefore, in sucn ])()silion, appi'ars to bo thinner at the middle, and this is corrected by the entasis, or curved contour. 30 THE PRACTICAL DRAUGHTSMAN'S But many autJaorities consider tliis a fastidious nicety, and it IS frequently disregarded, particularly in designing short tliick columns for machinery, and also where the other extreme is reached, and thf' columns become mere rods. \Vlaat may be termed the mechanical entasis, is, moreover, em- ployed in beams, levers, and connecting-rods of all descriptions ; tlie object of this convexity, and Lucreased width in tlie middle part, in such cases, being to obtain strength and rigidity, whilst it undoubtedly adds to the beauty of form. To determine the amount of the entasis in the Tuscan column, divide the line, c d, fig. 6, which represents two-thirds the height of the shaft, iato any nimiber of equal parts, say six. With the point, d, as centre, and a radius, d e, equal to one module, draw an are of a circle ; next, having made c i; equal to 9^ parts, draw through V a line, v x, parallel to the axis, c d\ this parallel will cut the arc in the point, x ; divide the ai'c, e x, into six equal parts, and then tlirough the points, 1, 2, 3, &c., thus obtained, draw par- allels to the axis. These parallels \vill intersect the horizontal lines drawn through the divisions, q, r, s, t, of the axis, respectively, in the poiats, 1', 2', 3', &c., and through these will pass the re- quked curve, forming the contour of the shaft. This curve, being symmetrically reproduced on the opposite side of the axis, c d, will complete the outline of the shaft. In the entablatm-e and pedestal will be found two similar mouldings, termed cymatia ; they are both examples of the cyma reversa, discussed in reference to Plate 3. The slight peculiarities in their construction, will be easOy understood from the enlarged view, fig. 8. Tlie quarter rounds and accompanying minor mould- jQgs belonging to the capital and entablature, are also represented separately, and on a larger scale, in figs. 9 and 10. RULES AND PRACTICAL DATA. THE MEASUREMENT OF SOLIDS. 117. We have already seen that the volume or solidity of a body, »8 the extent of space embraced by its three dimensions — length. Avidth, and height ; tlie last of these being frequently termed depth, or thickness. The volume of a solid is deteixnined when it ia ascertained what relation it bears to, or how many times it contains, any cube which is adopted as the unit of the measurement. Such a unit is the cubic metre, just as a square metre is employed to measure surface, and a linear metre length. The subdivisions of the cubic metre are the cubic decimetre, the cubic centimetre, and the cubic millimetre. The relations these bear to the linear subdivisions will be obvious from the following comparison. Whilst 1 metre = 10 decimetres = 100 centimMres = 1000 millimetres. 1 Cubic metre = (10"^ x 10" x 10"- =) 1000 cubic deci- metres — (100'=- X 100'- X lOO'- =) 1,000,000 cubic centimetres = ( 1000 "/^ X 1000 ■"/„ X 1000"/„ = ) 1,000,000,000, cubic millimetres ; consequently, 1 cubic decimetre — "001 or j^,'^,, cubic meti-e, the cubic centimetre = -0000001 or -yovWot cubic m^tre. SimOaiiy, we measure volume by cubic yards, feet, or inches, just as we measure surface by square, and length by linear yards, feet, and inches. A cubic foot is -^^ of a cubic yard, for — 1 cubic foot = i yard x J yard X J yard = ^V J^^ > and an inch, or Jj foot X 7^ foot X ^2 foot = -^jL^ foot. 119. Parallelopipeds. — The volume of a parallc'.opiped is equal to the product of its base multiplied into its height. Example.— Fig. ©. PI. 7. Let A F = 2 feet, F E = 1-4 feet, and F H = 1-4 feet. Then the base = 1-4 X 1-4 = 1-96, and 1-96 X 2 = 3-92 cubic feet; or more simply, 1-4 X 1-4 X 2 = 3'92 c. ft. A cube itself, having all its dimensions equal — ^it? volume is expressed by the third power of the measure of one of its sides; that is, by the product of one side three times into itself. Thus the cube, fig. A, of which one side measures, say 1-4 feet, contains r4 x 1*4 X 1*4, or 1-4' = 2-744 cubic feet. In general, the volume of a right prism, whatever be its base, is equal to the product of the base into the height. table of surfaces, and volumes of regular polthedea. Number of Sides. Name. Surface. Voldme 4 Tetrahedron, . 1-7820508 •1178519 6 Hexahedron, or Cube, 6-0000000 1-0000000 8 Octahedron, 3-4641016 -4714045 12 Dodecahedron, 20-6457788 7-6631189 20 Icosahedron, 8-6602540 2-1816950 120. Pyramids. — The volume of a polygonal pyramid is equal to its base multiplied into a third of its height. Example. — Let S O, fig. © = 2 inches, A B and A D e:ich = 1-4 inches; the cubic contents of the pyramid are — 1-4 X 1-4 X 2 „ „„ , . . , X = 1-3066 cubic mches. Thus the volume of a pyramid is one-third of that of a right •irism. having an equal base, and being of the same height. The volume of a truncated pyramid, with parallel bases, is equal to the product of a third of the height, into the sum of the two bases added to the square root of tlieu- product. Thus, if V represent the volume of a truncated pyramid, of which the height, H, = 3 feet, the lower base, B,= 6 square feel^ the upper, B', =: 4 square feet ; we have — V = 1^ X (B + B' + VWB') = — X (6 s. f. + 4 s. f. + -^6 X 4) = 14-898 sq. feet. BOOK OF INDUSTRIAL DESIGN. 31 111 practice, when there is little difference between the areas of the bases, a close approximation to the volume is obtained by taking the half of the sum of the bases, multiplied into the hdio-ht. Thus, with the preceding data, we have /B -t- B'\ V = Hx(^^ — J = 15 sq.ft. 121. Cylinders. — The cubic contents of aHy cylinder, as fig. [^, is equal to the product of the base mto the height. Thus, in the case of a cylinder of a circular base, we have B = rt R^ (72) ;* consequently, the volume, V, = n; R^ X H. First Exam-pie. — What is the volume of a cast-ii-on cylinder, of which the radius, R, = 20 inches, and the length, H, = 108 inches ? ¥= 3-1416 X 20- X 108 = 135,717 cubic inches. The volume may also be derived from the diameter of the cylin- der, in wliich case we have — V= fl^xH; or, 4 V = -7854 X 402 X 108 = 135,717 cubic in. The convex surface of a right cylinder, when developed, is equal to the area of a rectangle, having for base the rectilinear develop- ment of the circumference, and for height that of the cylinder. It is therefore obtained by multiplying the circumference into the height or length. With the data of the preceding case, the convex surface is expressed by the formula — S = 2rtRxH, orrtD XH= 3-1416 X 40 X 108 = 13,571-7 cubic inches. The volume of a hollow cylinder is equal to the difference between that of a solid cylinder of the same external radius, and that of one whose radius is equal to the internal radius of the hollow cylinder. Or, it is equal to the product of the sectional area into the height, such area being equal to the difference between two circles of the external and internal radius, respectively. Example. — It is requu-ed to find the volume, V, and the internal surface, S', of a steam-en^ne cyhnder, including its top and bottom flanges in the volume. Let the following bo the dimensions : — External diameter, D, = 56 inches; internal diameter, D', = 60 inches; length or height, H, = 120 inches; external projection of the flanges, F, = 5 inches, and their thickness, E, = 4 inches. Then, for the internal surface, wo have — S' = 3-1416 X 60 X 120 = 18,850 sq. in. For the volume of the body of the cylinder, we have — rt562 ^502 V' = — ^ -^ X 120= (-7854 X 66=)— (-7854 x 50-) x 120 = 60,000 cubic inches. And for the additional volume of (lie flanges — r„ _H (56 + 10)- _rt. 'J6 V" = ^ X 4 X 2 = (-7854 X 66-) — =: 7666 cubic inches. 4 4 (•7864 X .66=) X Whence the whole volimio — V + V" = 67,666 cubic inches, • Whon wo wi»h to rol'or tho «tU(!on(. U> uny riilo or priiiriplo ulrondy given, nre do »n by iiiimiiih of tho iiumbor of llui |mragru|ih RontiiiiiliiK kuoIi rulo or priiioi- ple. Ill the ]>rvBciil inalnuco, whut is rofurrud to will bu found nt jiugii 30. 122. Cones. — Tlie cubic content of a cone is equal to the pro- duct of its base into a third of its height ; or, V = B X 5. 3 In the right cone, fig. P", of which the base is circular— V;t = R^xH_lD3>,H; 3 "~ 4 3 and as rt, or 3-1416 -;- (4 x 3) := -2618, the formula resolves itself into — V = -2618 x D2 X H. Example. — What is the volume of a right cone, of which the height, H, = 24 inches, and the diameter of the base, or D, = 17 inches 1 We have — V=-2618 X 172 X 24= 1816 cubic mches. As we shall demonstrate, at a more advanced stage, the development of the convex surface of a right cone is equal to ine sector of a circle, of which the radius is the generatrix, and tlie arc the circumference of the base of the cone — consequently, the conical surface is equal to the product of the circumference of the base into the half of the generatrix: whence is derived tlie following formula : — G S = 2rt R X — =rtR X G. With the data of the foregoing example, and allomng the generatrix to be equal to 25| inches, we have — S = 3-1416 X 8-5 X 25-5 = 681 cubic inches. 123. Frustum of a cone. — The volume of the frustum of auuno may be obtained in the same manner as that of the truncated pyramid (120). The convex smface of a truncated cone is equal to the product of half the generati-ix of the frustum into the sum of the circumferences of the bases, and is expressed in the follow- ing formula : S=h X 2rt (R + R') = L X rt (R + R'). Example. — Let the length, L, of the generatrix of tho couio frustum, = 14 inches; tho radius, R, of the lower base, = 8-5 inches ; tho radius, R', of the upper base, = 3-8 inches ; then tlio convex surface — S = 14 X 3-1416, X (8-5 + 3-8) = 54 square-inches. 124. Sphere. — Tho volume of a sphere may be asoerfjiinod a* soon as its radius is known. Its surface is equiU to four times that of a circle of equal diameter. This is expressed by the formula — S = 4rt R3 = rt D=' = 3-1416 X D^ or tho square of tho dimnetor multiplied by 3-1416. The volume is t'nual to the iiroduct of tiio surface into ono-third of the radius, as in the I'ormula — V = 4rt R=' X ^=4 ^ "* ^^''' *"■ ^' = •'■'^'^ ^ ^^ '• or, if we enq>loy tlie di:iini'tral ratio — V = rtD^ D = -6236 X D' 32 THE PRACTICAL DRAUGHTSMAN'S Example. — We would know what is the surface and the volume of a sphere, of which the diameter measures 25 inches. The surface — S= 25- X 3-1416 = 1963-5 sq. inches Tlie volume — V = -5236 X 25=' = 8181-25 cubic inches. To find the radius or diameter of a sphere, of which the volume is Known, it is sufficient to invert the preceding operations, the formulas becoming as follows — 3V V . 4-188 ' R= 4rt 3 Wlience, Similarly, whence. R V 4-188 D3 V •5236' ^ = V -.5236 ' which, with the preceding data, gives R = 12-5 inches, and D = 25 inches. Tlie radius is derived from the surface by means of the follow- ing formula: — S R- = whence, '4x3-1416' 1-5664 D^ = S 3-1416' whence, " = \/3 S 3-1416* 125. Spheric sectors, segments, and zones. — The surface of a zone or spheric segment, is equal to the product of the circiun- ference of a circle of the sphere, luto the height of the zone or segment; or, S = 2rt R X H. Example, — The height, H, of a spheric segment being 1-5 inches, and the radius, R, of the sphere, 7-5 inches, the surface — S = 2 X 3-1416 X 7-5 X 1-5 = 70-686 sq. inches. The volume of a spheric sector is equal to the product of the sur- face of its spherical base, into one-third the radius of the sphere of wMch it is a portion. The corresponding formula is therefore^ R 2 V=2rtRxHx - = -rtxR2H= 2-094 x R^ x H. o o Example. — The volume of the spheric sector, whose spheric base is equal to the surface considered in the pre%'ious example, is — V = 2-094 X 7-52 X 1-5 = 176-68 cubic mches. The volume of a spheric segment is equal to the product of tne arc of the circle of which the chord is radius, into one-sixth of the height of the segment; or, V = rt r= X - = -5296 x r^ x H. 6 Example. — Let r = 6-5 inches, and H 1-5 inches; the tuen volume — V = -5296 X 6-52 X 1-5 = 33-56 cubic inches. The volume of a spheric ungula is equal to the product of thts gore, which is its base, into a tliird of the radius. The formula b — V=f A X R2; 3 where A — the area of the gore. The volume of a zonic segment is equal to half the sum of its bases, multiplied by its height, plus the volume of a sphere of which that height is the diameter ; whence the formula — \={ =1 )xH+- IP 6~" 126. Observations. — The volumes of spheres are proportionaj to the cubes of their radii, or diameters. Let V = 14-137 cubic inches, and v = 4-188 cubic inches. It will be found that the respective radii are — R = 4-188 and _ -^ /14-137 _ 5. ~V "4088-^^- '•=VfT88=V^ ^ 4-188 4-188 = 1; and, consequently, D = 3 and d = 2. The cubes of these numbers, that is, 27 and 8, have the same ratio to each other as the volumes given ; that is to say — 27 : 8 :: 14-137 : 4-188. When of equal height, cylinders are to each other, as well as cones, as the squares of the radii of their bases. When of equal diameter, these solids are to each other as their heights. First, then, we have — V = rt R2 X H, and v = n r^ x H ; whence, V : D :: R2 : 7-2 And, secondly, V = rt R2 X H, and D = rt R2 X ft; whence, Y:v:'.H:h. The volume of a sphere is to that of the circimiscribed cylinder as 2 to 3. A sphere is said to be inscribed in a cylinder, when its diameter is equal to the height and diameter of the cylinder. The volume of an annular torus, or ring, is equal to the product of its section into the mean circumference. We have pointed out (90) that an annular torus is a solid, generated by the revolu- tion of a circle about an axis, situated in the plane of the circle, and at right angles to the plane of revolution. Let R be the radius of the generating circle, and r the distance of its centre from the axis, we have — V = rt R2 X 2rt r = 19-72 R^ x r. BOOK OF INDUSTRIAL DESIGN. 3a PROPORTIONAL MEASUREMENTS OF THE VARIOUS PARTS OF AN ENTIRE ORDER. THE (modern) DORIC ORDER. Designations of the Members and Mouldings constituting the Order. Pi < Eh &3 Cornice, 'Reglet, . . Cavetto, . Fillet, . . Cymatium, Corona, . Fillet, . . Mutules, . Guttae, . . FUIet, . . Cymatium, Capitals of the Triglyphs, A ( Taenia, ARCHITRAVE, jp^j^^' o Reglet, . Cymatium, , ^ Abacus, . Capital,., -j e.^i^^; _ Shaft, , Base, . H W Cornice, . Dado, Bas£,. Three Annulets, ^Necking, .... f Beading or Astragal, Cincture, ' ' I Sh£ Shaft Proper, 'Fillet,. . Beading, Torus, . Plinth, . •i f Reglet, Quarter-Round, Fillet, Corona, .... Cymatium, . . . fFillct, Beading, Cyina Reversa, . . . Plinth Sul)-l'linth or Socio, Total height of the Order, Measurements according to Vignoles, in Modules of 12 Parts. Amount of Projection from the Axis of the Shaft. M. P. 2 10 2 7 6i 6' 5 4i 2" n 3" 1 Oi lU 11" 10 111 10 3^ 2^ 2 lU loi 10 1 11^ 10' 1 1 U 1 2 1 5 1 5 1 11 1 10? 1 9| 1 9 1 6,V 1 6J 1 5 6 7 7 81 9 9.V Heights. M. P. 1 3 li 4 2 2 1 6 2 10 M. P. 1 6 - 4 1 6 1 13 lOJ ■ 14 1 Vie 1 X IJ 4 I l" 2 oi "•J 4 4 10 )■ 5 4 Measures in Decimals The Module = 1. Amount of Projbc.'on from the Axis of the Shaft. 2-833 2.583 2-542 2-500 2-417 2-375 2-167 2-125 1-250 1-083 1-042 •959 •917 •833 •959 •833 r292 1-271 1-188 1-167 1-146 -959 -875 -833 1-000 -959 •833 rooo M04 1-167 1-417 1-417 1-017 1-889 1-806 1-750 1-512 1-459 1-417 1-500 1-583 1-708 1 -750 1-792 Heights. 13 083 250 042 125 333 042 042 209 042 166 166. 500 167 833 y 1-500 1-500 1-000 4-000 042" 083 209 209 124 333 083" 042 875 1-000 14-000 0831 083 1 334 f 500 J 1-000 ► 16 000 •042 " ■083 •042 •209 . -500 •124 •000 4-000 V 1(1(100 •042 1 •083 •1()6 •209 •333 8-333 25-338 34 THE PRACTICAL DRAUGHTSMAN'S PROPORTIONAL MEASUREMENTS OF THE VARIOUS PARTS OF AN ENTIRE ORDER. THE TUSCAN ORDER. Designations of the Members and Mouldings constituting the Order. Amount of Projection from the Axis of the Shaft. Pi <: < o Quarter-Round, . . Beading, Fillet, Larmier or Corona, Fillet, Cymatium, Fkheze, Architrave O H i Q CL, •1 listel, Facia, H W P5 {Listel, Abacus, Echinus, or Quarter-Round, . . Fillet, Necking, l^^"^" ISul',::; Shaft Proper, (Fillet,. < Torus, ( Plinth, o O ( Listel, 1 Cymatii; Dado, . j Listel, Socle or Plinth, Measurements according to Vignoles, in Modules of 12 Parts. Measurements in Decimals. The Module = 1. M. P. 2 3i 2 1 lU 1 10| 1 7i 10" 9l 1 2i 1 n 1 r lOi 9l 11 lOi 9i 1 O' 1 U 1 41 1 4i 1 8i 1 8" 1 6 4^ 6i Heights M. P. 4 1 I 1 4 1 2 2 10 1 4 1 2 h »J I 3 6 1 1 3 3 1 4 1 11 10 II 1 M4 8 I 3 8 13 8 6 ► 4 8 Amount of Projection from the Axis of the Shaft. 2-292 2-000 1-959 1-875 1-625 1-125 -833 •792 •959 •792 1-709 1-667 1-417 1-375 1-542 1-709 Heightg. •333 "~1 •083 •042 •500 •042 •333 1^167 •167 •833 1-333 1-167 1-000 3-600 1-209 •083] 1-125 •250 1-083 •250 -875 •083 •792 •334 J •917 -083' -042 •875 •792 1-000 \ 11-875 1-125 •083 1-375 •417 1-375 •500 1-000 12-000 1-000 Y 14000 •500 •167 1 I -333 \ 3-661 I 3-667 •083 ) ..„„ •417 f ^0^ 4-667 Total height of the Order, 22 2 22-167 With the help of these tables we can easily determine the proper measurement for any member or moulding, in feet, inches, or fnetres, when the height of the whole order is given. For this • When two measurements are given, the first applies to the tipper portion, the ■"wnd lo the lower. purpose the given height must be divided by the decimal measure- ment in the tables for the total given height ; the quotient is the measurement of the module proportioned to such height. Then that of any required member is found by multiplying this module into the decimal in the table con-esponding to such member. First Example. — It is required to know what is the diameter of BOOK OF INDUSTRIAL DESIGN. 35 the lower part of the shaft according to the Tuscan order, the height of the entire order being 15 feet. The height of the entire order being 22-167 when the module 22-167 :1. we have- 15 = 1-4778 the module, and 1-4778 x 2 = 2-9556 feet, the diameter of the lower part of the shaft. Second Example. — \Vhat is the height of the socle or lower plinth according to the Tuscan order, supposing the module to be 1-4778 feet 1 We have 1-4778 x -417 = -616. In like manner the dimensions of all the other details may be easily determined according to the Tuscan or Doric order. CHAPTER m. ON COLOURING SECTIONS, WITH APPLICATIONS. CONVENTIONAL COLOURS. 127. Hitherto we have indicated the sectional portions of objects by means of linear flat-tinting. This is a very tedious process, whilst it demands a large amount of artistic skill — only obtainable by long practice — to enable the draughtsman to pro- duce pleasing and regular effects ; and although, by vai;ying the strength or closeness of the lines, as we have already pointed out, it is possible to express approximately the nature of the material, yet the extent of such variation is extremely limited, and the dis- tinction it gives is not sufficiently intelligible for all purposes. If, however, in place of such line sectioning, we substitute colours laid on with a brush, we at once obtain a means of rapidly tinting the sectional parts of an object, and also of distinctly pointing out the nature of the materials of which it is composed, however numerous and varied such materials may be. Such colours are generally adopted in geometrical drawings ; they are conventional — that is, certain colours are generally understood to indicate par- ticular materials. In Plate X. we give examples of the principal materials in use, with their several distinctive colours ; such as stone and brick, steel and cast-iron, copper and brass, wood and leather. We propose now to enter into some details of the composition of the various colours given in this plate. THE COJMPOSITION OR MIXTURE OF COLOURS. PLATE X. 128. Stone. — Fig. 1. This material is represented by a light dull yellow, which is obtained from Roman ochre, with a trifling addi- ton of China ink. 129. Brick. — Fig. 2. Alight red is employed for this material, and may 1)0 obtained from vermilion, which may Hometimi's be brighten(HJ by the addition of a little carmine. A pigiiirnt fiiuinl in most colour-boxes, and tcniieil LIfrhi lied, may also be use.! w In ii great |)urity and brigiitness of tint is not wanted. If it is desired to distinguish firebrick from the ordinary kind, since the former is lighter in colour and inclined fo yellow, somo gamboge mu.st bo mixed vvifh the vermilion, the wlioh^ being laid on more faintly. In (external vi(^ws it is customary to iiidieate the outlines of the Individual bricks, but in the section of a mass of brickwork lliis "efinoHK^iit may he dispensed with, except in ciisi's where it is intended to show the disposition or method of building up. Thus, in furnaces, as also in other structures, the strength depends greatly on the method of laying the bricks. When vermilion is used in combination with other colours, the colour should be constantly mixed up by the brush — as, from its greater weight, the vermilion has a tendency to sink and separate itself fi-om the others ; and if this is overlooked, a varying tint of unpleasing effect will be imparted to the object coloured. 130. Steel or Wrought Iron. — Fig. 3. The colour by which these metals are expressed is obtained from pm-e Prussian blue laid on light — being lighter and perhaps brighter for steel than for wrought- iron. The Prussian blue generally met with in cakes has a con- siderable inclination to a greenish hue, arising from tlic gum with wliich it is made up. Tliis defect may be considerably ameuded by the addition of a little carmine or crimson lake — the proper proportion depending on the taste of the artist. 131. Cast-iron. — Indigo is the colour employed for this metal; the addition of a little carmine improves it. The colours termed Neutral Tint, or Payne's Gray, are frequently used in place of the above, and need no further mixture. They are not, however, so easy to work with, and do not produce so eipiable a tint. 132. Lead and Tin are represented by similar means, the colour being rendered more dull and gray by the ;uklition ol' ("hiiii ink and carmine or lake. 133. Cupper. — Fig. 5. For tliis metal, pure carmine or crimson lake is ])roper. A more exact imitation of the reidity may be ob- tained by the mixture, with either of these colours, of a little China ink or burnt sieiuia — the carmine or lake, of course, considerably predominating. 134. Brass or Bronze. — Fig. 6. These are expressed by an orange colour, the former being the brighter of the two; burnt Ri>man ochre is the simplest pignu'iit for producing this colour. Where, however, II very bright tint is desire^l, a mixture should be made of gamboge with a little vermilion — i-ire being taken to keep il constantly agitated, as before recomiuended. Many draughtsmen use simi)le gamboge or other yellow. i:!5.— »\'i'(«/.— Fig. 7. It will be observable, from l>reee.iini; e\;iliiplrs, (liMt the tints li:i\i' been elio-.ell W illl iel'er<nee lo tlu) actual eoloms of the materials wliieli they are iutcmled to express — carrying out tlii> same jiriueiple, we should ha\e a very wide niUjiri) ill the e.ise ivf wood. 'I'lu' eolour i,'i'iu rally used, however, U 36 THE PRACTICAL DRAUGHTSxMAN'S burnt umber or raw sienna ; b it the depth or strength with which it is laid on, may be considerably varied. It is usual to apply a, light shade first, subsequently showing the graining with a darker lint, or perhaps with burnt sienna. These points are sus- ceptible of great variation, and very much must be left to the judgment of the aiiist. 136. Leaiher, Vulcanized India-Rubber, arid Gutla Percha. — ?ig. 8. These are all represented by very similar tints. Leather Dy light, and gutta percha by dark sepia, whilst vulcanized india- rubber requires the addition of a little indigo to that colour. We may here remark, that if the student is unwilling to obtain an extensive stock of colours, he may content himself with merely a good blue, a yellow, and a red — say Prussian blue, gamboge or yellow ochre, and crimson lake. With these three, after a little experimental practice, he may produce all the various tints he needs; but, of course, with less readiness and facility than if his assortment were larger. 137. The Manipulation of the Colours. — We have seen by what mixtures each tint may be obtained, and we shall proceed to give a few hints relative to their application. It may be imagined that it is an easy matter to colour a geometrical drawing — that is, sunply to lay on the colours; but a little attention to the following observations will not be misplaced, as the student may thereby at once acquire that method which conduces so much to regularity and beauty of effect, and which it might otherwise require some practice to teach. The cake of colour should never be dipped in the water, as this causes the edges to crack and crumble off, wasting considerable quantities. Instead of this, a few drops of water should be first put in the saucer, or on the plate, and then the required quantity of coloiu' rubbed down, the cake being wetted as little as is absolutely necessary. The strength or depth of the colour is obtained by proportioning the quantity of water, the whole being well mixed, to make the tint and shade equable throughout. When large surfaces have to be covered by one shade, which it is desired to make a perfectly even flat tint, it is well to produce the required strength by a repetition of very light washes. These washes correct each other's defects, and altogether produce a soft and pleasing effect. This method should generally be employed by the beginner, as he will thereby more rapidly obtain the art of producing equable flat tints. The washes should not be applied before each preceding one is perfectly dry. When the drawing- paper is old, partially glazed, or does not take the colour well, its whole surface should receive a wash of water, in which a very small quantity of gum-arabica or alum has been dissolved. In proceeding to lay on the colour, care should be taken not to fill the brush too full, whilst, at the same time, it must be replenished before its contents are nearly expended, to avoid the difference in tint which would otherwise result. It is also necessary first to try the colour on a separate piece of paper, to be sure that it will produce the desired effect. It is a very common habit with water-coloiir artists to point the brush, and take off any super- rtuous colour, by passing it between their lips. This is a very bad and disagreealjle haoit, and should be altogether shunned. Not only may tne colour which is thus taken into the mouth be injurious to health, but it is impossible, if this is done, to produce a fine even shade, for the least quantity of saliva which may be taken up by the brush has the effect of clouding and altogether spoiling the wash of colour on the paper. In place of this un- cleanly method, the artist should have a piece of blotting-paper at his side — the more absorbent the better. By passing the brush over this, any superfluous colour may be taken off, and as fine a point obtained as by any other means. The brush should not be passed more than once, if possible, over the same part of the draw- ing before it is dry ; and when the termination of a large shade is nearly reached, the brush should be almost entirely freed from the colour, otherwise the tint will be left darker at that part. Care should be taken to keep exactly to the outline ; and any space contained within definite outlines should be wholly covered at one operation, for if a portion is done, and then allowed to dry, or become aged, it will be almost impossible to complete the shade, without leaving a distinct mark at the junction of the two portions. Fmally, to produce a regular and even appearance, the brush should not be overcharged, and the colour should be laid on as thin as possible ; for the time employed in more frequently replenishing the brush, because bf its becoming sooner exhausted, will be amply repaid by the better result of the work under the artist's hands. CONTINUATION OF THE STUDY OF PROJECTIONS. THE USE OF SECTIONS — DETAILS OF MACHINERY. PLATE Xr. 138. We have already shown, when treating of the illustrations in Plate VIIL, that it is advisable to section, divide or cut through, various objects, so as to render their internal organiza- tion clearly intelligible ; and we may now proceed to demonstrate, with the aid of sundry examples, brought together in Plate XI., that in particular cases sections are indispensable, and even more necessary, than external elevations. It is with this object that, in many of our geometrical drawings, w"e have given representa- tions of objects, cut or sectioned through then- axes or centres, so as to accustom the student to this description of projections, the impoilance and utility of which cannot be overrated. 139. Footstep Bearing. — Figs. 1 and 1° are the representations, in plan and elevation, of a footstep, formed to receive the lower end of a vertical spindle or shaft. This footstep consists of seve- ral pieces, one contained within the other ; and it is evidently impossible to say, from the external views, what their actual entire shape may be, although a part of each is seen in the hori- zontal projection, fig. 1. If, however, we suppose the whole to be divided by a vertical plane in^^the line, 1 — 2, fig. 1, we shall be enabled to form another vertical projection, fig. 1', showing the internal structure, and which is termed a vertical section, or sec- tional elevation. This figure shows, first, the thickness of the external cup-piece, or box, a, as also the dimensions of the open- ing, a, which is made in its base, for the introduction of a pin, to raise the footstep proper, b, when necessary ; secondly, the thick- ness and internal depth of the footstep, b, as also the internal vertical grooves, b. which serve for the introduction of the key, c; thirdly, the form and manner of adjustment of the centre-bit, c, which sustains the foot of the vertical spindle or shaft. This BOOK OF INDUSTRIAL DESIGN. 37 centre-bit, which, of coiu-se, should not turn with the spindle-foot, is prevented from doing so by means of the key, c, which fits into a cross groove in its under side, the key itself being held firmly by the grooves, b, into which its projecting ends are made to fit. Of these details, the cup-piece, a, is of cast-iron, the footstep, B, of gun-metal or brass, the centre-piece, c, of tempered steel, and the small key, c, of wi-ought-iron. Therefore, bearing in mind what has already been said, we may indicate these various materials in the sections, either by line-shading, of different strengths, as in the figure, or by means of colours, corresponding to those employed in Plate X.; and we may here remark, that where line-sectioning is used, brass, gun-metal, or bronze, is fi-equently expressed by a series of lines, which are alternately full and dotted. There are, besides, many ways of varying the effect produced by line-shading. For example, the spaces between the lines may be alternately of different widths, or the lines may be alternately of different strengths. Strictly speaking, figs. 1 and 1' are all that are necessary for the representation of the object under discussion. The cup- piece, A, however, which is externally cylindrical, has, at four points, diametrically opposite to each other, certain projectmg rectangular plane surfaces, d, which are provided to receive the thrust of the screws which adjust the footstep accurately in the centre. The width of these facets is shown in the plan, fig. 1 whilst their depth is obtainable from the elevation, fig. 1°. If, instead of these facets, d, being, as they are, tangential to the cylinder, a, they had projected, in the least, at their centres, their depth would necessarily have been given in the section, fig. l*, and in such case the elevation, fig. l", might have been altogether dispensed with. Whilst referring to the representation of the projecting facets, in connection with the cylinder, a, we may remark, that when a cyb'nder is intersected by a plane, which is parallel to its axis, the line of intersection is always a, straight line, as ef, figs. 1 and 1°. 140. SluJfi.ng-box cover, or gland. — In pumps and steam-engine cylinders, the cover is furnished, at the opening through which the piston-rod passes, vdth a stufiing-box, to prevent leakage. The hemp, or other mateiial used as packing, is contained in an enlargement of the piston-rod passage, and is tightly pressed down by a species of hollow bush with flanges, as rcoresented in plan in fig. 2, and in elevation in fig. 2°. In this instance, the neces- sity of a sectional view is still more obvious than in the case of the footstep already treated of. In the vertical section, fig. 2\ it is shown, that the internal diameter is not unilbnn throughout, and that there is a ring or ferule, b', lot in at the lower part of the interior. The cylindrical opening, a, of the gland, coincides exactly with llie dianuiter of the piston-rod ; the internal diameter of a i)ortion, li, of lii<^ ring, b', is also the same. Tlu^ [)art, c, aowever, comprised lictwccii llicso two, is gi-catcr in (liiuncter, so us to lessen Iho (jxiciit of Murfiico in frictional cipiiImcI wilji the piston-rod, and it also serves for (ho lodgiiiciil of luliiic'iting tiialtcr. It is furllicr disceriiihic in tli(> section, that the (laiigcs ov lugs, (/, which project on either side of tlu^ n|i|)er portion of (Ik^ gland, have e:ic-li a eylimh-icnl (ipening, i; IhiMiunleiut, their whuli' ili'|pth. 'i'heso arc the hoU^s for the holts, wiiich forces down the el.uul, and Hccuro it to the corresponding tlaiiges, or lugs, on tho stulling-box. The aimular hollowing out, /, at the upper ard internal part of the gland, acts as a reservoir, into which the lubricating oil is first poured, and whence it gradually oozes ^to the interior. The ring, b', is forcibly fitted into the bottom the gland, and terminates below in a wedge, in the same manner as the gland itself, the double wedge jamming the packing against the piston-rod and the sides of the stuffing-box, and thus forming a steam-tight joint. The ring, b', is generally made of brass, both with a view to lessen the friction, and to its being replaced with facility when worn, without the necessity of renewing the whole gland. The latter is generally made of cast-iron, though the whole is sometimes made of brass or gun-metal. 141. SpliericalJoirU. — In some cases, a locomotive receives water from its tender by means of pipes which are fitted with spherical joints, as a considerable play is necessary in consequence of the engine and tenddr not being rigidly connected together, and also to obviate any difficulty of attachment from the pipes in the locomo- tive not being exactly opposite to those in the tender. This speeit-s of joint, represented in plan and elevation in figs. 3 and 3", gives a free passage to the water, in whatever position, within certain limits, one part may be with respect to the other. For its con- struction to be thoroughly understood, the vertical section fig. 3' is needed. This view, indeed, at once explains tho various compo- nent parts, consisting — first, of a hollow sphere, a, of the same thickness as the pipe, b, of which it forms the prolongation ; and, second, of two hemispherical sockets, c, d, which embrace the ball, a, and which are firmly held together by bolts passing through lugs, a, a. When this species of joint is used of a small size, as at the junction of a gas chandelier wth the ceiling, the two half- sockets are simply screwed togetlier — this method, indeed, being adopted in many locomotives. It must be borne in mmd, tliat our object in this work is simply to instruct Uie student to accurately represent mechanical and other objects, and for this purpose Ave employ both precept and example ; but such examples do not necessarily comprise the latest and most improved or efficient forms. The half-socket, c, forms part of the continuation, e, of the feed-pipe, whilst the half-socket, d, is a detached piece, neces- sarily moveable, to allow of the introduction of the spherical pjirt, A. This half-socket, d, is partially cut away at tho lower part, and does not fit closel}' to the neck of the ball. This allows the pipe, B, to move to a slight extent from side to side in any direction ; and tho upper end of tho ball, A, is cut away to a corresponding extent, to prevent any diminution of tho opening into tho pipe, e, when tho two portions are thus inclined to each other. The pi[H', K, with its half-socket, c, is an example of the coud>ination of a cylinder with a s|iliere, and gives us occasion to observe, that tho intersection fornieil by the nu'eting or junction of these solids is always a circle in one projection, and a straight line in the other. The subject of sueli iiiteisietioi\s will be discussed nioni -n det.iil in reference to Plate .\l\'. The sockets, and i), are liirnied wiili four external nigs, or evi"- pieees, /i. for connection by bolts, as l<*>fon< stated. The eiirved onijines of these lugs, which glide taiigeiitially inio (hat of tho JMidy of the socket, give rise to the solution of a probloiu wliicli may he thus jiut : To draw with ii niirn rndiiis an arc Iniiiirnlial In Iwii giivn tiirs. The solution is (Inis dhtainiHl: with tho eendes, 38 THE PRACTICAL DRAUGHTSMAN'S o, o, and nidii equal to tlioso of the respective arcs given, plus th.it of the required arc, describe arcs at about the position in which, the centres of the required ares should be ; the intersections of these arcs will give the exact centres, as f, &c., and the lines joining f, with the centres, o, o. give the points of junction of the «ic, G H, with tl;e other two. This spherical joint, which requires gr<;at accuracy of adjustment of the different parts, is generally •jast in brass, being finished by turning and grinding. 142. Safely-Valve. — To insure, as far as is practically possible, the safe and economical working of steam boilers, they are usually titted with pressure gauges, level indicators, alarm whistles, and safety-valves. The object of the safety-valve is to give an outlet to the steam as soon as it reaches a greater pressure than has been determined on, and for which the valve is loaded. Figs. 4. 4°, and 4', respectively, represent a horizontal section, elevation, and ver- tical section of a safety-valve in common use. This apparatus consists of two distinct parts : first, the cast-iron seat, a, per- manently fixed to the boiler-top by three or four bolts, the joint being made perfectly steam-tight by means of layers of canvas and cement ; second, the valve itself, b, which is sometimes cast-iron, sometimes brass. The valve-piece, b, is cast with a central spindle, c, hollowed out laterally into the form of a triangle with concave sides, for the purpose of giving a passage to the steam, and, at the same time, of lessening the extent of frictional contact of the spindle with the sides of the passage— some contact, however, being neces- sary for the guidance of the valve. The method of drawing the horizontal section of this valve-spindle is similar to that given for fig. A, Plate III. (34), with tliis difference, that it is drawn in an equilateral triangle, mstead of in a square. The base of the valve- piece, or the part by which it rests on the seat, a, consists of a very narrow annular surface ; the upper edge of the seat is bevelled off internally and externally, so that the surface on which the valve- piece rests exactly coincides with that of the latter. The upper external surface of the valve-piece is hollowed out centrally to receive the point of the rod through which the weighted lever acts upon the valve ; this lever is adjusted and weighted to corre- spond with the pressure to which it is deemed safe to submit the boiler, so that, when this pressure is exceeded, the valve rises, and the steam blows off as long as relief is necessary. 143. Equilibrium or Double-heat Valve. — Steam engines of large dimensions, such as those for pumping, met with in Cornwall, as well as marine engines, are often furnished with a species of double-beat or equilibrium valve in place of the ordinary D slide. An example of this description of valve is given in figs. 5, 6", and 5'. It possesses the property of giving a large extent of opening for the passage of the steam, with a very little traverse, and very little power is required to work the valve. The valve here represented consists of a fixed seat, a, of cast-iron or brass, and forming part of the valve chamber ; and a bell-shaped valve-piece, b, also in brass, fitted with a rod, c, by means of which it is moved. The contact of the valve with its seat is eflTected at two places, a and b, which are foiTOed into accurate conical surfaces — one, a, being mternal, and the other, b, external. When the valve is closed, these surfaces coin- cide with similar ones on the seat, and when it is lifted, as in fig. 5', two annular openmgs are simultaneously formed, thus giving a double exit to the steam — which issues from the upper opening. through the central part of the valve-piece, b. The rod or spindle of this piece is fixed to a centre-piece cast in one with the valve^ piece, and connected to it by four branches, c. The seat is simi- larly constructed. The external contour of this valve presents a series of undulations, involving the following problems in tlieii delineation : — To draw the curved Junction of the body of the valve with the upper cylindrical part. This is similar to the one treated of in reference to fig. 5, Plate III. (37), and may easily be drawn with the assistance of the enlarged detail, fig. 5' (Plate XI). Next, for the junction of the branch, c, with the more elevated boss, we require : To draw an arc tangent to a given straight line, and passing through a given, point. The solution of this is extremely simple : we have merely to erect a perpendicular on the line, ef, fig. 5°, r.l the point of contact, e, of the tangential arc ; to join e and the given point, g, through which the arc is to pass ; on the centre of the line, e g, to erect a perpendicular, h i, and the point, i, of inter- section of this line with the perpendicular, i e, wOl be the centre of the arc sought, eh g, and i e, the radius. The central leaves or feathers of the seat, a, are drawn accord- ing to a problem already discussed (38). The student will now see the imperative necessity of internal views or sections for the perfect intelligibility of the construction and action of various pieces of mechanism. With reference also to the examples collected together in this plate, a little considera^ tion will show that the internal formation could not generally be sufficiently indicated by dotted lines ; for, besides the complication and confusion that would result from such a method, many such lines would confound themselves with full ones representing some external outline. We have not thought it necessary to enter more into detail respecting the methods of constructing the various outlines, being persuaded that the dotted indications we have given will be quite sufficient for the student who has advanced thus far, the more so since the requisite operations bear great resemblance to those treated of in reference to Plate III. SIMPLE APPLICATIONS. SPINDLES, SHAFTS, COUPLINGS, WOODEN PATTERNS. PLATE XII. 144. For the conveyance of mechanical action, under the form of rotatory, or partial rotatory motion, details, technically kno%vn as shafts or spindles, of ^vrought and cast-iron and wood, are used. Shafts of the latter description, namely, cast-iron and wood, are employed chiefly in hydraulic njotors, water and wind mills, and in all machines where a great strain has to be transmitted, render- ing considerable bulk necessary. Of these two kinds, wooden shafts, being more economical, have been preferred in some cases, particularly when the length is great, since they will better sustain severe shocks. Wrought-iron shafts are employed for the trans- mission of motion in factories and workshops, and for the main paddle-shafts of steam-vessels. Wrought-iron has the advantage of being less brittle than cast, and of possessing gi-eater tenacity and elasticity. 145. Wooden Shaft. — Figs. 1, 4, 5, and 6, represent f^ifFerent BOOK OF INDUSTRIAL DESiGN. 39 projections of a woodeu shaft, such as is used for a water-wheel. Fig. 4»^hows, on one side, a lateral elevation of the shaft, furnished witii its iron ferules or collars, and its spindle ; and at the same tim(!, on the other side, a vertical section, passing through the centre of the shaft, giving the ferules in section, but supposing the central spindle, with its feathers, to be in external elevation, (xenerally, in longitudinal sections of objects enclosing one or more pieces, the innermost or central piece should not be sectioned, unless it has some internal peculiarity — ^the object of a section be- ing to show and explain such peculiarity where it exists, and being quite unnecessary where the object is simply solid. In the same manner, it is not worth while sectioning the various minutias of machinery, such as bolts and nuts, simple cylindrical shafts and rods and screws, unless these are constructed with some intrinsic peculiarity. Fig. 5 is a transverse section through the middle of the shaft, and merely shows that it is solid, and that it has the external con- tour of a regular octagon. Fig. 6 is an end view of the same shaft, showing the fitting of the spindle, with its feathers, into the socket and grooves, formed in the end of the shaft to receive them, and the binding of the whole together by the ferules or hoops. These views are what are required to determine all the vaiious parts of the shaft. It manifestly consists, in fact, of a long prismatic beam of oak, a, of an octagonal section, and of which the extremities, b, are rounded, and .slightly conical. The spindles, b, which are let into the ends, are each cast with four feathers, c, and a long tail-piece, d, uniting and strengthening them. Some en^neers construct the spindles without the addi- tional tail-piece, d. Though this simplifies the thing considerably, it is an arrangement which does not possess so much strength as when the spindle is longer. The beam-ends are turned out and grooved to receive these spindles, the grooves for the feathers being made rather wider than the feathers themselves. When the spindles are introduced into the sockets, b, thus formed for them, the whole are bound together by means of the iron hoops, f, which are forced on whilst hot. After this, hard wooden wedges are jammed in on each side of the feathers, thus tightening and solidifying the whole mass. In addition to this, iron spikes, g, are sometimes hammered in, to jam up the fibres of the wood still closer. Fig. 1, which is a shaded and finished elevation of one end of the shaft, givfts an accurate idea of its ai)pc'arance when complete and ready for adjustment. 146. Cast-iron Shaft. — There are several descriptions of cast- iron shafts. Some are cast hollow, otiicrs (juito solid, and cylin- drical or prismatical in cross section. Such as ai'c intended to sustain very great strains, arc generally strengthened by the addi- tion of feathers, which project more towards the middle. These pve groat rigidity to the piece. A shaft of this description is represented in elevation in fig. 7, half being sectioned tiirougli the irregular lino, 1 — 2 — 3 — 4, and lialf in external elevation. Fig. 8 is an end-view of it ; and fig. 9 a transverse section through the lino, !) — G, in fig. 7. In pr;icticc, it is not considered al)s(iliit('ly necessary that a section should follow a straight line. Ficipiciitly a much greater amount of c.xiil.in.ilion inny bo given in one view, by supposing the obj(!ct sciclioncMJ liy portions of plnncs at diU'cr- ent jyarts, and solid and easily (^ornpreluindcd portions are geneiiilly shown in elevation, as the feathers of the shaft, in the present in- stance, or the spokes of a spur-wheel or pulley. The shaft under consideration is such a one as is employed for hydraulic motors. The body, a, is cylindrical and hollow, and it is cast with four feathers, b, disposed at right angles to each other, and of an ex- ternal parabolic outline, so as to present an equal resistance to torsion and flexure throughout. Near the extremities of these feathers, four projections are cast, for the attachment of the bosses of the water-wheel. These projections are formed with facets, so as to form the corners of a circumscribing square, as shown in fig. 8 ; and they are planed to receive the keys, i, by which they are fixed and adjusted to the bosses or naves, which are grooved at the proper places to receive them. The spindles, d, which terminate the shaft at each end, are cast with it, and are afterwards finished by turning. The shaft thus consists of only one piece, or casting. 147. Although we have already shown the method of dra%\nng a parabola, in Plate V., the outline of the shaft feathers affords a practical exemplification, which it will be useful to illustrate. We here also give the method generally adopted — ^because of its sim- plicity — when the curve is a very slow or obtuse one, such as is given to the feathers of shafts, beams, side-levers, connecting-rods, and similar pieces. It is understood, in these cases, that two points in the curve are given ; the one, a, fig. 7, being at tlie summit, and at the same time in the middle of the piece, and the other, />, situated at the extremity. In the present instance, we suppose the heights, a c and b d, from the axial line, m n, of the shaft to be given. This line, m n, may also be taken as the centre line of a beam, or connecting-rod. After having di-awn through tlie point, b, a line, e b, parallel to the axis, divide the perpendicular, a e, into any number of equal parts, and transfer these divisions to the line, b i, the prolongation of the line, db; then draw lines from the points, 1, 2, 3, to the summit, a. Further, divide also the length, c d, into the same number of equal parts as the perpentlicular, and, through the divisions, 1', 2,' 3', di-aw other perpendiculars, the respective points, /,«-, h, of intersection of these with the lines already drawn, will be points in tlie required curve. As the lower feather is an exact counterpart of tlio upper one, the perpen- diculars may be prolonged downwards, and corresponding distances, as 1'^', 2' — g', 3 — li, set olTon them. To draw, also, the half of each feather to the left, it is merely necessary to erect perpen- diculars of corresponding lengths, at corresponding points in the axis. A different method of dramng this curve is soiiiotiiiios adopted ; namely, the one which wo have already given in Plato IX., for the entasis of the Tuscan column. As, iiowever, it does not possess the advantages of the true parabolic fonu, and as the curve becomes too sudden towards the extremities, wo think the method given in Plato XII. is to be preferred. . Fig. 3 represents a portion of the shiitY just discussed, shaded and finished, the lines running in ditlerent directions, the bettor to distinguisii the tl.il iVoni llu' round surfaces. 1 IS. Shall (^tuiiliiiii. — 111 extensive factories, and other works, will re consi(liM!ible lengths of .sliafling lire necessary, they havo lo lie conslniiii'd in sr\iT:il pii'ces, and coupli'tl tt>gether. Theso couplings are genenilly ot' ciist-iron, and toniied of one or inoro pieces, according to their size. One form consists of a spivios ^>f 4U THE PRACTICAL DRAUGHTSMAN'S cylindiiciil socket, aceuralely turned internally, which receives the ends of the two shafts to be connected, these being scarfed or halved into each other, so as to be bound well together, and re- volve like one continuous piece. According to another form, two sockets are employed, of increased diameter at the part where they meet, and formed at this part into quadrant-shaped clutches, gearing witli each other. The coupling represented in side eleva- tion, in fig. 10, is of this Ivind ; and in front elevation, as separated, in fig. 11. This coupling was designed for a shaft, of which the diameter at the collars was 28 centimetres. The socket, a, of the coupling, is adjusted on the end of the first part of the shaft, c. The other socket, a', is similarly adjusted on the end of the other part, c', of the shaft. These two socket-pieces gear with each other, when brought together, by means of the projections or clutches, B and b', concentric portions, however, being adapted to fiit the one into the other, to insure the coincidence of their centres. Fig. 11, which is a front view of the socket- piece, a, shows the exact shape and dimensions of these projecting clutches, each occupying a quadi-ant of the circle on the face of the socket-piece. Those of the second piece, a', are precisely the same, occupying, however, the intervals of those on a, so that the two may fit exactly into each other, as shown in fig. 10. The perfect and accurate union of these coupling-pieces with the two portions of shafting is obtained, in the first place, by means of two keys, a, diametrically opposite to each other, and let half into the shaft, and half into the socket-piece ; and secondly, by screws, h, one of which is visible in fig. 10. The keys, a, are for the pui-pose of fixing the coupling-pieces to the two portions of shafting, making them solid therewith ; and the screws, b, prevent the longitudinal separation of the two halves of the coupling. THE METHOD OF CONSTRUCTING A WOODEN MODEL OF. PATTERN OF A COUPLING. 149. After the design for any piece of mechanism which it is proposed to cast has been decided on, it is generally necessary to construct a model or pattern in wood, by which to form the moulds for the casting. The proper formation of such a pattern is no easy matter, and requires considerable skill on the part of the pattern-maker, as also a knowledge of the kind of wood most ap- propriate, and of the various precautions needed to insure success when the mould comes to be prepared. It is customary to construct the patterns of deal, because of its cheapness. Sometimes, however, plane-tree or sycamore or oak is used ; and for small patterns, and such as require great precision, mahogany, box, or walnut. Whatever kind of wood is used, it should be perfectly dry, and well seasoned. The pattern is made solid, or holj£w and built up, according to the dimensions of the object. For a drum, for instance, a column of any considerable width, a steam-engine cylinder, or for a coupling of large size, such as that represented in figs. 12, 13, and 14, the pattern is generally hollow, for economizing the wood, and reducing the ■veight of the piece. Also, if built up, there is less risk of warp- ing or alteration of form, from changes of temperature. In fig. 13, the pattern of a coupling-piece is represented, pai-tly in exter- nal side elevation, and partly in longitudinal section, being cut by a vertical plane, passing thi-ough the axis. Fig. 12 is a frout elevation, showing the projecting clutches. It is easy to see from these views, that the pattern is formed of two boards, d d', round the circumferences of which are fitted a series of staves, e, secured to the boards by screws. The wood for these staves is first cut up into pieces of the requu-ed thickness, and the sides are then bevilled off, to coincide with the radii, c d, c e, fig. 14. They are then fitted to the boards, d d', and at this stage present the ap- pearance of the left-hand portion of fig. 14. The di-um is after- wards put into the lathe, and the cu'cumference is reduced to a cylindrical surface, Uke the right-hand portion of the same figure. On one of the ends, d, of this drum, is fixed the projecting clutch-piece, b, which has been previously cut out of a board of greater thickness, so as to present the outline of fig. 12. On the opposite end, d', of the drum, are fixed several discs, or thicknesses, of wood, F, which are turned down to a diameter proportionate to the central socket which it is intended to form in the coupling- piece. After having been turned where necessary, the pattern is treated with sand-paper to make the surface as smooth as possible, and to prevent the adherence of the loam of which the mould is formed. Such patterns, particularly when of small size, are more- over coated wdth black-lead, well rubbed in, to ^ve a polish and hardness to the surface. The diameter of the core-piece, f, is less than that of the shaft to which the coupling is to be fitted, so as to leave some margin in the casting for turning and giinding dovm the socket to the exact dimensions. The core itself, which gives form to the socket, is a cylinder of loam placed in the centre of the mould, fitted into the recess formed for that purpose by the piece, F. As in the present example, the mould would be con- structed on end, and the core is very short, it would not require further support ; but where a core is very long, or placed Ln a horizontal position, it requires to he supported at both ends, and, further, to be strengthened by wires or rods passing through its centre. For this reason, a core-piece, as f, is only attached to one end of the drum. It wiU be observed that this is slightly conical ; the drum is so also, but to a less extent ; the core itself, however, is quite cylindrical. 150. Draw, or Taper, and Shrink, or Allowance for Con- traction. — In order that the pattern may be lifted from the mould without bringing away portions of the sides, it is necessary to form its sides with a slight taper, or draw, a.i it is technically called. For example, the diameter of the core-piece, f, as well as that of the drum itself, must be less at the lower extremity, or at the part first introduced into the mould, than at the opposite extremity. A very slight difference of diameter is sufBcient for the purpose. Cast-iron, as is the case with all the metals of the engineer, is of less bulk when cold than when in a state of fusion, and, because of this contraction, it is necessary to make the patterns of somewhat larger dimensions than the casting is to be when finished. It follows, then, that when the pieces to be cast have afterwards to be planed, turned, gi-ound, or grooved, it is necessary to bear in mind, in constructing the wooden pattern, not only tke after re- duction due to the contraction, or, as it is termed, the shrink, of the metal, but also that whicb is occasioned by the reducing pro- cesses involved in finishing the article. In general, grey iron requires an allowance for shrink of from 1 to 1 jV per cent. ; whiU BOOK OF INDUSTRIAL DESIGN. 41 iron, however, requires a much larger allowance. The allowance to oe made for the reduction caused by the finishing processes, depends entirely on their nature. When, with a view to avoid the expense of constructing a pattern, the mould is formed from the actual object which is to bo reproduced or multiplied, the mould-makers obtain the necessary margin by shifting the model slightly during the formation of the mould. This, of course, can only bo done with advantage when the piece is not of intricate shape. ELEMENTARY APPLICATIONS RAILS AND CHAIES FOR RAILWAYS. PLATE XIII. 151. In railways, the two iron rails on which the trains ran are placed at the distance apart, or gauge, of li metres, and are generally formed of lengths of 4| to 5 metres. In England, the gauge is generally 4 feet 8i inches, and the rails are rolled in lengths of from 12 to 15 feet. These rails are supported by cast-iron chairs, placed at from 9 to 10 decimetres asunder, and adjusted and bolted on oak sleepers, lying across the rails, imbed- ded even with the surface. Those chairs which occur at the junctions of the lengths of rails are made wider at the base, and of greater length, so as to embrace the end of each length of rail, and render their rectilinear adjustment and union as perfect as possible. In Plate XIII. we give details of a very common form of rail and chair. There are many different forms in use ; but the method of drawing or designing each will be similar, and may be thoroughly understood from the exemplification here given. Figs. 1 and 2 represent the elevation and plan of a chair, with a portion of the rail which it supports. Fig. 3 is a vertical section through the line, 1 — 2. in the plan ; but supposing the chair to be turned round, or to belong to the right-hand rail — showing, in connexion with fig. 1, the relative positions of the two lines of rails, with their respective chairs. Fig. 4 is a side view of the chair alone, and fig. 5 is an end view of a length of rail. This chair, which is de- signed wdth the view of combining solidity and strength with economy of material, consists of a wide base, a, by which it is seated on the sleeper, and of two lateral jaws, b, b', strengthened by double feathers, c, c'. The base, b, is perforated at a — the holes being cylindrical, and slightly rounded at their upper edges. These holes are for the reception of the bolts which secure tho chair to tho sleeper. Tho space between tho jaws of tho ciiair is for tho reception of tho rail, d, and the wooden wedge, e, whii'ii holds it in position. In this oxiimplo, (ho vertical section of the rail, d, presents an outline which is synimetiical with rclereiico both to tho vortical centre lino, & c, and also to the liori/oiil.'il lino, (/c, (ig. 6. This permits of tho niil being turncid when one of the running surliu'cs ■s worn. Tho section of the wedge, k, is also syninictrieal with reference to Its diagonals, so that it is immaterial which way it 18 introduced, whilst it also fits equally well to tho rail wiien the latter is reversed. The outline of the rail is composed of straight lines and arcs, which are geometrically and evenly joined, as shown in fig. 5. The necessary operations are fully indicated on the drawing itself. These operations are, for the most part, but the repetition and combination of the problems treated of in the first division of the subject. We have, moreover, given some of the problems de tached, and on a larger scale, in figs. 6, 7, 8. Fig. 6 recalls the problem (35), which has for its object the dra\ving of an arc, ij k. tangent to the straight lines, fg and gh^ the radius, o/r, being given, equal to 31-5"'/„. (fig. 2, Plate III.) This problem meets with an application at fg h, fig. 5. In fig. 7 we have the problem (37), which requires that an arc, Imn, be dra\vn tangent to a sti-aight line, n p, and to a given arc, qr I, the point of contact, n, being known (fig. 6, Plate III.) This meets with its application at Imn, fig. 5. The problem illustrated by fig. 8 is, to draw a tangent, g^/', to two given circles of radii, s t and o' k^, respectively. In this problem (9), we require to find a common point, u, on the line, OS, which joins the centres of the two circles. To effect tliis, we di-aw through the centres, o and s, any two diameters, v x and ■»' x', parallel to each other. Join two opposite extremities of each, as v and x', by the straight line, v x', which will cut the line, o' s, in the point, u. The problem then reduces itself to the draw- ing of a tangent to any single circumference (fig. 4, Plate I.), from a given point, u. The tangents obtained, in the present instance, will lie in one straight line, and be the line required — tangent to both circles. The application of this problem is at xk^t in fig. 4. On fig. 9 we have also indicated the solution of a problem (41), which is — to draw an arc, y z, of a given radius, a' b', tangent to two other ai-cs, having the radii, c' d' and e'/" (fig. 8, Plate III.) This problem is called for in drawing the outline of tiie jaw of tho chair, where it runs into the base, a, near the edge of the bolt- hole, a, fig. 3. To complete the outline of the chiiir, it remains for us to show how to determine the lines, g' h', wliich represent the intersections of portions of cylindrical surfaces, as will be gathered from tigs. 1 to 4. To avoid a confusion of lines, wo have reproduced this portion in figs. 10, 11, and 12, wiiich represent — tlie two former, vertical sections of each cylindrical portion, and tlio latter, the lino of intersection in plan. We must first determine on fig. 12, which corresponds to lig. 2, the horizontal projection, i', of any point, as i, taken on tho arc, g' h', in fig. 10; letting fiill from this point, on tho b:iso line, L T, a perpendicular, i i', and also drawing IVom it a iiorizontal line, i i^. This latter line meets tho cylindrical outline, g' h', fig. 11, in i*. Project t' in i' on the biiso line, triuisferring it to ii lino at riglit angles to tlie hasu line, by means of a quadrant of a oia-lo, and (Ir.iw through the point thus obtainoil a line pjirallel to the b;iso line, and meeting the line 1 1' in i', which will he a point in tlio curve required ; other points, as ;', ii', are I'onml in a simihir manner. It. must be observed, lli;it when iJie two i'\ lindriciil portions !UX> of ciinal di:nneter.s, their intersection with each other, j;' A', ns will lie (l.inonslr.iteil hereatler, is projwied hori/.outjUly as a stniight lini ; the e,,;iler Ijie ilitVerenoo botwoon iJio two cylinders, the more 42 THE PRACTICAL DRAUGHTSMAN'S cun'ed \vill tlie line of their intersection be, as is apparent in figs. 10, 11, 12. The outlines of the feathers, c and c', glide into that of the base. A, with a curve which, in the plan, is projected in the arc, k' v. The operation necessary to determine these curves, is quite analogous to that treated of in reference to the preceding figures, and will be found sufficiently explained in figs. 13, 14, and 15. We should here remark, that we have given explanatory diagrams of all the sweeps or combinations of curves, both that the student may be well exercised in many of the problems already discussed, and also vrith a view of collecting, in one plate, several of the diflfi- culties which more frequently meet the draughtsman in the course of his practice. In such objects as that chosen for exemplification, very little of the nicety here carried out is observ^ed, and the curves are generally obtained by measurements with callipers from the object itself, or are formed of arcs determined by the eye. The rails are not adjusted in their chairs perpendicularly, but are inclined sUghtly towards each other, in such a manner that their centre lines, c 5, form a slight angle with the vertical, c IP: this inclination is given to counteract any tendency that the carriages may have to run off the rails, as is the case more particularly in curves, from the effort made by the wheels to run in a straight line. The expedient of laying the outer or convex rail at a level slightly higher than the other, is also resorted to in quick curves, for the like purpose of keeping the ti-ains on the line. RULES AND PRACTICAL DATA. STRENGTH OF MATERIALS. 152. The various materials employed in mechanical and other constructions, differ considerably in their several natures, both vtdth reference to the amoimt of force they ■nlll bear or resist uninjm-ed, and the description of force or mode of applying it, to which they offer the greatest resistance. Such forces are termed, according to the mode in which they are applied — tension, compression, flexure, and torsion. A series of practical rules have been deduced from often re- peated experiments, which serve as guides for readily calculating the dimensions of any piece of mechanism, with reference to the description and degree of force to which it will be subjected. RESISTANCE TO COMPRESSION OR CRUSHING FORCE. 153. Compression is a force which strives to crush, or render more dense, the fibres or molecules of any substance which is sub- mitted to its action. According to Rondelet's experiments, a prism of oak, of such dimensions that its length or height is not greater than seven times the least dimension of its transverse section, will be crushed by a weight of from 385 to 462 kilogrammes, to the square centimetre of transverse section, or a weight of from 5,470 to 6,547 per square inch of transverse section. In general, with oak or cast-iron, flexure begins to take place in a piece submitted to a crushing force, as soon as the length or height reaches ten times the least dimension of the transverse section. Up to this pomt, the resistance to compression is pretty regular. Wrought-iron begins to be compressed under a weight of 4,90f) kOog. per square centimetre, or of nearly 70,000 per square uich, and bends previously to crushing, as soon as the length or height of the piece exceeds three times the least dimension of the trans- verse section. We show, in the following table, to what extent per square inch we may safely load bodies of various substances. Table of the Weights which Solids — such as Columns, Pilasters, Supports — will sustain without being crushed. WOODS AND METALS. Proportion of 1 ength to least dimension. Description of Material. Up to 12. Above 12. Above 24. Above 48. Above 60. lb. lb. lb. lb. lb. Sound oak, .... 426-'750 355-625 213-375 71-125 35-562 Inferior oak, . . . 210-215 119-490 7M25 " " Pitch pine, .... 533-437 440-975 266-007 106-687 " Common pine, . 137-982 116-645 69-702 " « "Wrought-iron, . 1422O-000 11877-875 7 112-501. 2375-575 1194-900 Cast-iron 28450-000 23755750 14225-000 4741-666 2375-575 Rolled copper, . 11707-175 '■ *' " " STONES, BEICKS, AND MOKTAKS. Description of Material Basaltic Marble, Swedish and Anvergnese, Granite from Normandy, " green, from Vosges, " grey, from Bretagne, " " from Vosges, " ordinary, Marble, hard, " white and veined, Freestone, hard, " soft, Stone from Chatillon, near Paris, Very hard freestone, or lias, from Bagneux, near Paris, A softer stone, from the same place, Stone from Areueil, near Paris, " ' from Saillancourt, near Pontoise, best quality, . " from Conflans, much used at Paris Hard calcareous stone, Ordinary calcareous stone, Calcareous stone from Givry, near Paris, " " ordinary, from the same place, An inferior stone, termed Lambourde, Bricks, very hard, " inferior, " hard and well-baked, " red, A soft stone, Lambovrde vergetee, Plaster, mixed with water, " mixed with lime-water, Mortar, best, eighteen months old, " ordinary, eighteen months old, " " of lime and sand, Cement, " Eoman, or Neapolitan Lenfftli being less than 12 times least dimension. lb. 2845 996 882 925 697 569 1422 427 1280 6 242 726 185 355 199 128 711 427 441 171 33 171 57 213 85 85 71 104 57 36 50 68 53 Rule. — To find, by means of this table, the greatest compress- ing weight to which any piece may be submitted ^\ith safety : — Multiply the transverse sectional area of the piece by the number ta BOOK OF INDUSTRIAL DESIGN. 43 the table, corresponding to the material, and to the proportionate 'ength of the piece. And inversely, from the weight which a piece is to support, its smallest transverse section may be determined. By dividing this weight, expressed in pounds, by the number in tlie table corresponding to the material, and to the proportionate length. First Example. — What weight can be pat vrtth safety upon a pillar constructed of ordinary bricks, the pUlar being of a rectan- gular section, of 50 inches by 60, and the height being below 12 times the length of this cross section ? We have 50 x 60 = 3000 square inches of transverse sectional irea. Then, according to the table, we have — 3000 X 67 = 171,000 lbs. Second Example. — What must be the transverse sectional area .)f a square post of sound oak, 19 feet 8 inches in height, and rt'hich will safely bear a load of 60,000 lbs. ? According to the table, if we suppose the length to be not more than 12 times the least cross section, the number or coefficient of compression, in pounds per square inch, is 426-75. Then, and 60,000 426-75 ^140 square inches; ■•^140 = 11-8 inches, the length of the supposed side. Comparing tins 11-8 inches with the given height, we find that 19 ft 8 in. 11-8 in. 236 11-8 20. Tliis shows — and we have constructed the example with this riow — that in this mstance the proportionate length has not been sorrectly estimated ; and therefore, instead of taking the number 426"75, as in the first column, we must take that in the second column, for a proportionate length of between 12 and 24 times the cross section. The calculation wOl, consequently, have to be rectified thus — 60,000 „„ . . 155^625 = ^^^'^ '^"^'■^ "'*''' and VlSS-l = 13 inches nearly, the proper dimension for the cross section of the post. Third Example. — Wliat is the greatest load that can be borne with safety by a solid cast-iron column, 3 inches in diameter, and 12 feet in height? It is, in the first pl.ace. evident that the ratio of the diameter to the height is 12 feet, or 144 inches, -f- 3 inches = 48. Consequently, the section -785 x 3^ x 4741-666 = 33,.500 lbs. In shops and warehouses, builders employ solid caaf-iron columns, instead of brick pillars, so as to take up less .space. These columns arc generally calculated to support loads of above 33,000 lbs. each. They are usually about 3 inches in diameter, and 12 feet high. In which case, supposing a cubic foot of cast- iron weighs 452 lbs., they will weigh (3 inches being equal to -25 foot)— •785 X •26=» X 12 X 452 = 266 lbs. If, instead of these columns being massive or solid, wo employ, m place of two of them, a hollow one, to support tho proportional o load of 66,000 lbs., and lu'lng 6 iiiclics in (iiainctcr, this incrc;is(i in the diameter makes the ratio of the length to it 24, instead of 48 ; and the coefficient to be taken from the table will conse- quently be 14,225, mstead of 4742. Now, 66,000 -^ 14,225 = 4-64 square inches, would be the cross section of a soUd pOlar, equivalent to that of which tho thickness is sought. Since, however, the diameter of the latter is 6 inches, its section of solidity would be — •785 X 62 = 28-26 square inches. Then, deducting from this area 4-64 square inches, as above determined, we have 28-26 — 4-64 = 23^62, for the cross sectional area of the central hollow. From this we deduce the Latemal diameter, thus — ^23-62 /23-62 V "^785 = 5-485 inches. And, finally, the thickness of the column will be — 6 — 5-485 2 = -2575 inches. The weight of such a column, if 12 feet in height, will be — 4-64 -^^ X 12 X 452 = 174-77 lbs. 144 This result shows very markedly how great an economy results from the employment of hollow in place of solid cast-iron columns. The thickness, determined as above, of -2575 inch, is theoretically sufficient, but in practice we seldom find such castings under half an inch thick. In the above examples, too, the mouldmgs usually added to tlie columns are not taken into the account. With these, the weight will be a tenth or so more, according to the description of moulding. TENSIONAL RESISTANCE. 155. A tensQe force is one which acts on a bod}' in the direc- tion of its length, tending to increase the length, and when carried to a sufficient extent, to cause rupture. As vidth reference to compression, many experiments have been made to determine the sectional area to be given to bodies of various materials submitted to a tensile strahi, so that they may safely resist a given force. First Example. — Required the sectional area for four square tension rods of wrought-iron, to connect the top and bottom of a hydraulic press, in winch tho force which teuds to separate tliese two ends, and consequently to rupture tlie rods, is equal to 500,000 lbs. Each rod must be capable of resisting 500,000 ,, „ „ — = 125,000 lbs. According to the fjible, the best wrought-iron may be sjifoly subjected to a strain of 14,225 lbs. per squiu-o inch of cross section. We have, consequently, 125,000 „„ . , — — = 8-79 square inches. 14,225 for the area of tim cross section ; and V 8-79 = 2-964, or iicirly 3 iiiohos, for a side of tho square rod. If tho rod were round, wo sliould havo— • /8-79 3-346 inches, lor ilio diwuotor. 44 THE PRACTICAL DRAUGHTSMAN'S In tlie same manner, the diameter proper for steam-engine piston-rods may be calculated, when the pressure on the piston is known. Table of Weights which Prisms and Cylinders will sustain when submitted to a Tensile Strain. Description of Material. Oak, Deal, Woods. .,, ,, . ( sound, . . with the eram, ■< -,■ ^ ( ordmarj', across the grain, ■with the grain, . across the grain. Ash, with the grain, . . . Elm, Beech, " . . . Per square inch of cross section: Metals. Wrought or ( PP^rior and select samples, bar iron 1 'if6rior,indiscriminatelyselected ' ( medium. Sheet J in the direction in which it was rolled, iron, ( in the direction perpendicular to thi; Hoop iron, soft ' Be Laigle, -009 inch, or -23 ■"/ in diameter, inferior, and of considerable diameter best, from -02 to -04 inch, or -o to 1 "/„ in diameter, . . medium quality, -04 to -12 inch, _ or 1 to 3 '°/n, in diameter Ii'on. wire rope, T„ „„ui„, ( ordinary, oblong links, . . . Iron cables, -^ , ,,•'' j > ° . ' ( strengthened by stays. Unannealed iron wire. Grey cast iron, | ™n yert,ically, strongest kind. r un horizontally, inferior, cast or wrought, selected, Steel i ^'^^^^}°\' hadly tempered, taken indis- 'I criminately, 1^ medium, Gun metal, average, r rolled lengthwise Copper, \ f superior quality, ^^ ' I hammered, Unannealed copper wire, l_ cast, 'superior, under -04 inch, or 1"/ in diameter, medium, from -04 to -08 inch, or 1 to 2 ""/„ in diameter, inferior, Yellow copper, or fine brass Unannealed ( superior, under -04 inch, or 1-/, brass wire, j ^^^ diameter, ' ( medium, Platinum ( hardened unannealed, -0045 inch, wire 1 °'' ^"^ ^^ diameter, ' ( hardened, annealed Cast tin Cast zinc Sheet zinc, Cast lead, Sheet lead, Cordage. Hawsers and cables of Strasburg hemp, "5 to -6 inch, or 13 to 14 ""/o, in diameter, Do. of Lorraine hemp Do. of Lorraine or Strasburg hemp, -9 inch, or 23 "/„ in diameter, Do. of Strasburg hemp, 1-5 to 2 inch, or 40 to 54 ■"/„ in diameter, Old rope, -9 inch, or 23 "/„ in diameter, . . Black leather bands, lbs. 1,138 853 228 1,138 to; 1,280 \ 60 1,707 1,479 1,138 14,225 5,917 9,474 9,957 8,535 1,069 21,337 11,850 18,996 Per square centimetreof cross section. kilog. 80 60 16 80 to 90 4-2 120 104 80 1,000 416 666 700 600 750 1,500 833 1,333 14,225 1,000 7,112 500 5,690 400 7,587 533 3,201 225 3,087 217 23,702 1,667 8,535 600 17,781 1,250 5,448 383 4,979 350 6,117 433 5,932 417 3,314 233 16,600 1,167 11,850 833 9,488 667 2,987 210 20,143 1,416 11,850 833 27,511 1,933 8,066 567 711 50 1,422 100 1,185 83-3 303 21-3 320 22-5 6,259 440 4,623 325 4,267 300 3,912 275 2,987 210 284 20 Second Example. — Required the amount of tensile or tractive force which can safely be resisted by a carriage draw-shaft, made of ash, and having a cross-sectional area of 15'5 square inches. According to the table, we have — 1707 lbs. X 15-5 = 26-460 lbs. 155. Pulley Bands. — The following simple formula may gene rally be employed in practice, to determine the dimensions proper for pulley bands : — 20 X H P . V in which L is the width of the band in inches ; H P, the force in horse power ; and v, the velocity in feet per minute. By horse -power is meant a force equal to 33,000 lbs., raised 1 foot high in a minute. French engineers call it 75 kilogrammes, raised a meti-e high per second, which is very nearly the same as the English measure. To suit the French system, the above for mula would be — 1500 X H P Li — , V V, in this case, signifying the velocity in centimetres per second, and L, the width in centimetres. The thickness of the leather is supposed to be that of strong ox-liide, say about -2 inch, or 5'°/n,. The above formula gives rise to the following rule : — Multiply the horse power by the constant multiplier, 20, and divide the pro- duct by the velocity in feet per minute, and the quotient will be the loidlh of the band in inches. Example. — Let H P = 2 horse power, d = 10 feet per minute, then, 20 X 2 L = 10 4 inches. This formula satisfies the following conditions — that tne band do not slide round the pulley which it embraces ; that it be not liable to increase perceptibly in length ; and that it be capable of resisting the strain transmitted by it. It is found advisable never to make the respective diameters of two pulleys, coupled together, in a greater ratio to each other than 1 : 3. RESISTANCE TO FLEXURE. 156. The resistance of a piece of any material to flexure, is the effort which it opposes to all strain acting upon it in a direction perpendicular to its length, as in the case of levers, beams, or shafts. Bodies may be submitted to the strain of flexure in several ways. Thus, the piece may be tu-mly fixed in a wall by one end, whilst the straining weight or force is applied at the other ; or it may be securely fixed at both ends, and the weight apphed in the centre ; or it may be supported at the centre, and have the weight applied at both extremities. We shall first consider the case of a piece fixed by one end, and subjected to a strain at the other. Let W be the weight in pounds, placed at a distance, L, in inches from the wall, in which the piece under experiment is fixed ; C, a coefficient varying vidth the material ; a, the horizontal dimension in inches of cross section; b, the vertical dimension, similarly expressed ; — then the greatest weight that the piece wiL BOOK OF INDUSTRIAL DESIGN. 45 bear, without undergoing alteration, will be determiaabie by the following formuia, Inv piece oeing of rectangular section, and fixed at one end, and weighted at the other. Now, C = 8,535, for wrought-iron ; 10,668, for cast-iron ; 854, for oak and deal. Substituting these values of C, in the preceding formula, we shall have, for pieces of rectangular section, according to the material — For Wrought- Iron, u 8535 Xab^ . , 1422-5 Xab^ P = ^T^ , or more simply, = . TX" For Cast-Iron, n 10,668 X a 6-, • , P — — '- — -~ -', or more simply, 6 L 1778 xab^ For Wood. 854 X a &2 142 X ffl b^ P = — -f •, or more simply, = ^ . These formulae lead us to the follovwng rule for pieces of rect- angular section: — Multiply the horizcmial dimension in inches of cross section, by the square of the vertical dimension in inches, and by a coefficient depending on the material : then divide the product by the length in inches, and the quotient will be the wp.ight in pounds, which the piece will sustain without alteration. This rule is derived from the fact, that the transverse resistance of pieces submitted to a deflective strain is inversely as their length, directly as then* width, and as the square of their vertical thickness. According to this, pieces fixed at one end, and intended to bear a strain at the other, should be placed on edge ; in other woras, the greatest cross section should be parallel to the direction of the strain. First Example. — What weight can be suspended, without causing deflection, to the free end of a wrought-iron bar, fixed horizontally into a wall at one end, and projecting 5 feet {— 60 inches) from it; the bar being of a rectangular cross section, having its horizontal dimension, a = 1-2 inches, and its vertical dimension, & = 1-6 mclies 1 We have 1422-5 X 1-2 X 1-6= P = 60 =: 72-8 lbs. This result is obtained on the supposition that the bar is placed on edge ; but what would bo the weight, other things being equal, supposing the bar to bo placed on its side — that is, when 1-6 inches is its horizontal dimension, a, and 1-2 its vertical diinciision, i? We have, in this case, 1422-5 X 1-6 X 1-23 P = 60 64-6 lbs. This inferior result shows the advantage of placing the bur on edge. When the pi(H-,o under oxperiiuont is of s(iiiui-o instead of oblong section, a nocossariiy = b, and a //•* bccoinos //', and this is conse- quently to bo substituted in the fonnual for Mio lornior. If, however, the piece is cylindrical, the formula wOl be — D representing the diameter, 854 X D3 For wrought-iron, P = For cast-iron, P = For wood, P = L 1066 X D^ L 85 X D3 T ■ In each of the cases just referred to, the transverse sectional dimensions of pieces fixed at one end, and submitted to a strain at the other, are determined by the following formulas : — Material. Wrought-iron, . . Cast-iron, Wood, FoEM OF Section. Rectangular. a J- a b~ a Z)- PL 1,422-5 PL 1,778 PL 142 Square. PL 1,422-5 PL 1,778 PL OiiLular. 142 PL 854 PI 1066 PL 85 The rule derivable from these formulae for the determination of the transverse section, whether rectangular, square, or circular, of a bar or beam fixed by one end and loaded at the other is thus stated : — Multiply the weight in pounds by its distance in inches from the sup- port ; divide the product by a coejjicient varying wilh the material and form of section ; and extract the cube root, which mil give in inches the vertical dimension, the side of the square, or the diameter of tJie circle, according as the bar or beam is rectangular, square, or circular in cross section. First application : WHiat should be the transverse section of a rectangular wrought-iron bar, intended to carry at its free end, and at a distance of 5 feet from its support, a weight of 728 lbs., Ino bar being supposed to be placed on edge ? We have hero, 72-8 lbs. X 60 in. 1,422-5 then, if a bo taken — 1-2 inches. =3-071; /3-071 1-6 indu's, the vortical dimension. Second application : ^Vhat should the side of the cross swliou measure, of a square bar, under sunilar circumstances otJierwiso ? b^ = 72-8 X 60 ~Ti22-5 = 3-071, and J = V3-071 = 1-454 in., the thickness of the bar. 157. Obsf.kvations. — W'Iumi the bar, or beam, under oxjH'n nu-nt possesses in itsclt' any wiight cajwblo of inllui-iu-ing its resistance ; or, besides the \\ I'i^^lit suspended or acting at one end, lias a weight equally distributed throughout itslengili; the trans- verse-sectional dinu iisimis are, in the lirst [ijace. deterniir.vHl \\iih. 46 THE PRACTICAL DRAUGHTSMAN'S out taking the additional weight into consideration. This is, then, calculated approximately, and the half of it added to the suspended load, a fresh calculation being made with this sum as a basis. A bar, or beam, fixed by one end, and loaded at the other, has always a tendency to break off at the shoulder, or point of junc- tion, with its support, because it is on that point that the weight or strain acts with the greatest leverage. When, therefore, the transverse section of the piece has been determined, in accordance witli the formulae given above, which are calculated for the dimensions of the piece at the shoulder ; the section may be bene- ficially diminished towards the free extremity, thereby economis- ing the material, and lessening its own overhanging weight. The curve proper to give the outline is the parabola, as described in reference to Plates V. and XII. It may also be obtained in the following manner, for the particular case imder consideration : — Calculate the transverse section for different lesser lengths of the piece, the other data remaining as before, and the required cm-ve wUl be one which passes through the outline of each section, when they are placed at distances from the load equal to the lengths for which they are calculated. This curve is also given to bars, beams, or shafts, fixed at both ends and loaded in their middle, or sustaining a uniform weight throughout their length. The cast- b'on shaft represented in Plate XII. may be taken as an example of this. Steam-engine beams and side levers are also formed with feathers of this shape, as it gives them a uniform resistance throughout, so tliat they are not liable to break or give way ir any one point rather than another. A bar, or beam, supported in the centre, and loaded at either end, win support double the weight capable of being canied by one of similar dimensions, supported at one, and loaded at the other end; it is, indeed, e\ident that each weight will only act ^\ith half the leverage, being only half the distance from the point of support. Similarly, a bar, or beam, freely supported at both extremities, and loaded in the centre, will support a weight double that sus- tained by a piece of the same dimensions ILxed at one, and loaded at the other end. Therefore, in calculating the proportions for these two last-mentioned cases, it is necessary simply to double the coefficient, c, given for the first case. A bar, or beam, firmly and solidly fixed by both ends, will sup- port a load four times as great as one of the same dimensions fLxed at one end, and loaded at the other extremity. It will, con- sequently, be necessary to quadruple the above coefficient in this case. For calculating the diameters of the spindles, or journals, of cast-ii-on shafts for hydraulic motors, which are intended to sus- tain great weights, the following particular formula may be employed : — D= VWx -1938, where D expresses the diameter in inches, and W the weight to be sustained in pounds. TABLE 01 THE DIAMETERS OF THE JOUKNALS OF WATEE-AV^HEEL AITD OTHEE SHAFTS FOR HEAVY WORK. Diameter of Journal in Inches. Diameter of Jonmal in Inches. Total load in Total load in Pounds. Pounds. Cast-Iron. Wrought-Iron. Cast-iron. Wrought-Iron. 17*2 * •4315 70343 8 6-9040 137-4 1 •8630 84373 H 7-3355 463-7 n 1-2945 100156 9 7-7670 1099-0 2 1-7260 117793 H 8-1985 2146-7 2i 2-1575 137388 10 8-6300 3709-5 3 2-5890 158604 m 9-0615 6890-5 H 3-0205 182864 11 9-4930 8805-6 4" 3-4520 208950 111 9-9245 12619-5 H 3-8835 237296 12 10-3560 17176-5 5 4-3150 268012 121 10-7875 22858-0 51 4-7465 311666 13' 11-2190 29676-0 6 5-1780 338026 m 11-6505 37730-0 6i 5-6095 376993 14 12-0820 43873-0 7 6-0410 418845 14i 12-5135 68915-7 7i 6-4725 463685 15 12-9450 According to this formula, the diameter of the cast-iron spindle, or journal, is found by extracting the cube root of the weight, or strain, in pounds, and multiplying it by the constant, -1938, the product being the diameter in inches. The diameter for wrought-iron spindles, or journals, may be derived from that for cast-iron, by multiplying the latter by -863 ; or, directly, by employing the multiplier, -1673, in the above for- mula. Example. — Of what diameter must the spindle of a water- wheel shaft be, the total strain being equivalent to 70,000 lbs. 1 Here, D= ^70,000 X -1938 =7-987, or 8 inches nearly. A ^^^-ought-u-on spmdle of (7-987 X -863 =) 6-9 inches, will answer the same purpose. RESISTANCE TO TORSION. 158. When two forces act in opposite directions, and tangen- tially to any solid, tending to turn its opposite ends in different BOOK OF INDUSTRIAL DESIGN. 47 directions, or to twist it, it Ls said to be subjeoted to torsion, and offers more or less resistance to tliis action according to its form and composition. Taking, for example, the main shaft of a steam- engiae, at one end of which the power acts through a crank, set at right angles to it, and at the other the load, by means of wheel gear — the resistance which this load presents, on the one hand, and on the other, the power applied to the crank, represent two forces which tend to twist the shaft, subjecting it to the action of torsion. In machinery, all shafts and spindles which communicate power by a rotatory, or partially rotatory, movement on their axes, are subject to a torsional strain. Those which sustain the greatest torsional efforts are those shafts denominated first movers, the first recipients of the power. Such are the fly-wheel shafts of land engines, and the paddle-shafts of marine engines. In these the action is fui'ther complicated and heightened by the irregularity with which, in reciprocating engines, the power is communicated to them. Such shafts as carry very heavy toothed gearing, but receive and transmit the power in an equable manner, and without a fly-wheel, are termed second movers ; and finally, such as carry only pulleys, or comparatively small toothed wheels, are comprised in the class of third movers. Such shafts, again, as meet v^th an intermittent resistance, as is the case with all cam movements, require increased strength to meet this irregularity of action. In constructing formulae for the determination of the diameters of shafts, regard must always be had to the class to which they belong, and also to the description of work they have to perform. As the journals are the parts of a shaft on which the greatest strain is concentrated, it is obviously to the determination of their dimensions that our investigation should be directed. The prac- tical formula, for ascertaining the diameter proper for the journal of a cast-iron first-mover shaft, is — = ^- HP R X 419. Here, d = diameter of journal in inches. H P = the horse power transmitted by the shaft. R = the number of revolutions of the shaft per minute. This formula is expressed in the following rule. 1.59. To determine the diameter at the journals of a cast-iron first-mover shaft : — Divide the horse poiver of the engine by tlie numoe.r of revolvXions of the shaft per minute, muUiply the quotient by the constant, 419, and extract the cubic root, which will be llie diameter required in incites. For the journals of cast-iron shafts wliich are second movers, the formula is — VHP V-R- X206; and for third movers — s/Tj-p ThcHO are, in fiu't, siiiiilar to the formula given for first tnovors, willi tlio (^\•coption, that for tlicso tho (constant Mniltii>li('r is 11!), whilst for tho latter it. is 'MHi and 10(i, rospcctivoiy. 160. For tlio journals of wrouj^'lit iroii sliafis tho saino foriniiho are (employed, tho iiiu]li|)!i('rs only l)(Mng changod ; llicse are iJI9 for first movers, 1.34 for secornl tnovcis, and ()7'li lor third movers. If, v(ath a view of suppressing the radical sign in the above for- mulae, we raise both sides of the equation to their third or cubic power, and further express the multiplier by m, we have J3 HP from which formula it wtR be seen, that the cube of the diameter of the jom-nal is proportional to the force transmitted. Similarly, the resistance of a journal is proportional to the cube of its dia- meter. In other words, one journal, of which the diameter is double that of another, is capable of sustaining a strain eight times greater, since the cube of 2 is 8. 161. As, in consequence of the necessity of extracting cubic roots, the calciriation, according to these formulae, becomes very tedious and complex, we have rendered it much simpler by means of the table on the next page. We may, however, first observe, that the formula. d= = HP -^ Km, may be put in the form- d^ m HP ~ R ' or again , reversing the terms. m 17^ R ~HP If now we divide the coefficient, m, by the cubes of the series, 1, 2, 3, 4, &c., representing the diameters of the journals in inches, we shall obtain a series of numbers corresponding to R HP" Thus, if 419 be successively divided by the cubes, 1, 8, 27, 64, &c., the numbers in the second column of the table will be obtained; and by dealing with fho other multipliers in hke manner, the numbers in the 3d, 4th, 5th, 6th, and 7th columns, will be found. Rule. — ^When the table is used, the rule for determining the diameter of the journal of a shaft is thus stated : — Divide tlie number of revolutions per minute of the shaft by the liorse power, and find (lie number in the table which is nearest to the quotient thus obtained, bearing in mind the class and 7naterial, ami the corresponding number in the first column will be tlie diamcttr required. First Example. — \Vliat should be tho di.Tmctcr at (ho journals of a ca.st-iron first^motion shaft, for an engine of 20 horse powei, the shaft in question to make 31 rovolutions per minute? We have — R HP - i' - 1-55 - 20 ~ ^ ''''• It will bo observed that this quotient is tho nearest to tiio number 1-526 in the second colunm of Uio table, and that 1-526 is opposite to 6^ ; tho diameter, d, of the shaft journal should ooii- so(iU('nlly bo 6.'. inches in dianiotor. If ;i sIkH'I lor tlio siuno purpose .is tho above bo niado of w roui^ht-iiMn, we must look in the liClh cohinin for ll>e nunilnT to wliiih 155 approaches nearest. It will l>i> observed that it lios bolwi'on tho numbers 1-99'J and \W1, respeelively >>pp(>site to 5 and 5,'| inelios ; the iiiMin(>tor of tho journal should consoquciitly lio lielween these — siiy about 5 J inches. 4S THE PRACTICAL DRAUGHTSMAN'S TABLE OF DIAMETERS FOR SHAFT JOURNALS, CALCULATED WITH REFERENCE TO TORSIONAL STRAIN. Journals of Cast-iron Shafts. Journals of Wrought-Iron Shafts. Diameter in laches. First Movers. Second Movers. Third Movers. First Movers. Second Movers. Third Move: s. i 3352-000 1568-000 848-000 1981-200 1072-000 540-800 1 419-000 206-000 106-000 249-000 134-000 67-600 H 124-133 61-037 31-408 73-778 39-704 20-030 2 52-375 25-750 13-250 31-125 16-750 8-450 2* 26-816 13-190 6-790 15-872 8-576 4-327 3 15-519 7-630 3-922 9-222 4-963 2-504 3i 9-773 4-805 2-475 5-808 3-123 1-577 4 6-547 3-219 1-656 3-891 2-094 1-563 44 4-598 2-266 1-163 2-732 1-475 •742 6 3-352 1-648 •848 1-992 1-072 •541 54 2-519 1-239 •637 1-497 •806 •406 6 1-940 •954 •491 1-153 •620 •313 6} 1-526 •750 •386 •906 •488 •246 7 1-222 •601 •309 -726 •391 •197 7i 1-002 •493 •253 •595 •325 •162 8 •838 •402 -207 •487 •261 •130 8i •682 •335 •173 •405 •218 •110 9 •575 •282 •145 •341 •184 •093 9i •489 •240 •124 •290 •156 •079 10 •419 •206 •106 •249 •134 •068 10^ •362 •178 •092 •215 •116 •058 11 •314 •155 •079 •187 •101 •051 111 •275 •135 •069 •163 •089 •044 12 •242 •119 •061 •144 •078 •039 12^ •214 •105 •054 •127 •068 •034 13 •191 •094 •049 •114 •061 •031 13i •170 •084 -043 •101 •054 •027 14 •153 •075 •038 •091 •049 •024 144 •137 •067 •035 •082 •044 •022 15 •124 •061 •031 •074 •039 •020 1 2 3 4 6 6 7 Second Example. — ^We require to ascertain the diameter proper for the journals of a shaft of the second class, intended to trans- mit a force, equal to 15 horse power, at the rate of 40 revolutions per minute. Here, A _ 40 _ 2-67 j^___267. This quotient lies between the numbers 3-219 and 2-266 in the third column, and between 3-123 and 2-094 m the sLxth. It follows, then, that if the shaft is to be of casl^u-on, its journals must be between 4 and 4^ mches in diameter ; or, if it is to be of wrought- iron, between 3i and 4 inches, there being about half an inch of difference between the two materials in this instance. Third Example. — A shaft, intended for a third mover, is to transmit a force equal to 6 horse power, at a velocity of 50 revo- lutions per minute, what should be the diameter of its journals in cast or wrought-iron 1 Here, -5:- = ££ = 8-333. HP 6 This number, in the third column, lies between 13-25 and 6-79; therefore the diameter for cast-iron should be between 2 and 2J, say, 2 1 inches. For wrought-iron the diameter should be 2 inches, as the number, 8-45, opposite to this in the seventh column, almost coincides with the quotient above obtained. The length of the journals and their bearings should always be greater than their diameter. For large sizes, it should be l-2d to l-Ad, and for smaller sizes, Vbd to 2d. Thus, the length of a vn-ought-iron journal, 1-5 inches in diameter, should be from (1-5 X 1-5 =) 2-25 to (1-6 x 2 =) 3 mches. When shafts have to resist both a torsional and a lateral or transverse strain, the diameter of their journals should be determined with reference to that strain which is the greatest, or which of itself would require the greatest dimensions. When shafts are not of any great length — 3 to 6 feet, for example — their diameter need not be above a tenth greater than that of their journals. Solid cast-iron shafts of above six feet in length should have a diameter one-fifth, or even one-fourth greatei than that of their journals. BOOK OF INDUSTRIAL DESIGN. 49 FRICTION OF SURFACES IN CONTACT. 162. Friction is the resistance which one surface offers to another — movbg or sliding on it. Friction may be distinguished ^is sliding friction, and the friction of rotation. The former is that which arises from the simple rubbing of one surface upon another ; the latter, from the rotation of one surface upon another. The friction caused by the rubbing of plane surfaces is inde- pendent of the extent of surface or velocity of movement ; it depends essentially on the weight of the body, or, more accurately, the pressure binding the two surfaces together. It may therefore be said, that the fiiction is in proportion to the pressure. Similarly, the friction of a journal in its bearings is independent of the length of these, but is proportional to the diameter and to the pressure. We give tables for each of those classes of friction, indicating the ratio of the friction to the pressure, and consisting of a series of coefficients, whereby the pressure must be multiplied in order to ascertain the amount of resistance due to friction. Table of the Ratios of Friction for Plane Surfaces. Description of Materials Disposition of the Fibres. Condition of the Surfaces. Ratio of Friction to Pressure. At Starting. In Motion. Oak on oak, ...... Parallel. Do. 1 Across. Do. Endwise (on one piece). Parallel. Do. Do. Do. Do. Flat or on edge. i Flat. Dry. Lubricated with dry soap. Dry. Wet with water. { Dry. Do. Do. Do. Do. Wet with water, j Do. ( Oiled or greased. Dry. Do. Do. 1 1 62 44 44 71 43 38 53 80 62 65 62 12 47 28 16 14 •48 •16 •34 •25 •19 Ash, lieech, or deal on oak, . . JEIempen cord on o;ik Wrought-iron on oak, •38 •52 •49 •22 Pump leather on cast-iron,. . . Belt i "" oaken dru ms, t on cast-iron pulleys,. . . Cast-iron on cast-iron •27 •15 •19 Example. — What power is necessary to raise an oaken flood- gate weighing 30 lbs., and against which a pressure is exerted equal to 700 lbs. 1 We have (•71 X 700 =) 497 -f- 30 = 527 lbs. at starting, and (-25 x 700 =) 175 -f 30 = 205 when in motion, supposing the pressure continues the same. Table of the Ratios of Friction for Journals in Bearings. Description of Materials. Cast or wrought-iron journals, in cast or wrought-iron, brass, ■ or gun-metal bearings, ' Cast-iron in cast-iron Cast-iron in brass or gun-metal. Cast-iron in lignumvitse, - Wrought-iron in brass or gun- i metal 'i Wrought-iron in lignumvitiB, . ■ Condition of the Surfaces. Lubricated with oil } or lard, '^ Similarly lubricated ) and wet with water f Lubricated with or- j dinary oil, and wet > with water, ) Greased, Greased and wet, . . . . Unlubrieated, Lubricated with oil and lard Lubricated with a pre paration of lard and plumbago, Greased, Greased and wet, . . Badly lubricated, . . Lubricated with oO and lard Lubricated with ordi nary oil Ratio of Fric- tion to Pres- sure, with re- gular Lubrica- tion. •07 to -08 •08 ■14 •16 •19 •18 •10 •14 •09 •19 •25 •11 •19 Rule. — To determine the frictional pressure, f, acting on tno bearings of a journal, always bearing in mind the weight of the shaft and the gear carried by it, the power transmitted, as also the resistmg load : Multiply the product of these, p, by the coejicient, c, to obtain the amount of friction ; next multiply this by the constant •08, and by the diameter d., in inches, (or by -OSd.,) m obtain the amount per revolution; and, finally, multiply this by the number of revolutions in a minute, which will give the amount of power con- sumed by friction during this unit of lime. Example. — What amount of power, a, is absorbed by the friction of the journals of a cast-iron shaft revolving in bearings ; also, of cast-iron, under the following conditions ? The diameter at (ho journals = 5 inches. The pressure of the shaft and gear = 20,000 lbs. The velocity = 5 revolutions per minute. According to the table, the coefficient, c, is -076. Here we have — F=-08d X a xP, = -08 X 5 X ^075 X 20,000 = 3,000 lbs. CHAPTER IV. THE INTERSECTION AND DEVELOPMENT OF SURFACES, WITH APPLICATIONS. Ifi3. Nowhere is descriptive geometry moro useful, in its appli- cation to the industrial arts, than in the dotcrinination of the lines of intersection, or junction, of the various solids, whether the in- tersection bo that of two similar solids with each other, as a fVlindof with a cylinder ; or of dissimilar ones, as a cylinder with a sphere or a cone. With the sud, however, of this branch ot geometry, wo can dotormino, in tho most o.\act nnuinor, the pro- portions of all tho curves — of double as of single cun'ature — which may bo produced by tho intcrsottions of surfaces of revolution, the constructive or generative data of which are knowu. 50 THE PRACTICAL DRAUGHTSMAN'S The apijlieations of forms in which such curves occur are exceed- ingly numerous; they abound in the works of the brazier, the tinsmith, the joiner, the carpenter, the builder, and the architect, IS well as in all descriptions of machinery. This treatise would, indeed, be incomplete, were we not to render the delineation of these curves quite familiar to the student. Intimately connected also with this branch of the subject, is the development of curved surfaces ; that is, the determination of the dimensions of such surfaces when extended in a plane, so that the workman may be able to cut out such pieces with the certainty of their taking the form, and fitting the place assigned for them. The study of projections, moreover, comprises the methods of delineation of such curves as helices, spirals, and serpentines, which frequently occur in mechanical and architectural con- struction. THE INTERSECTIONS AND DEVELOPMENT OF CYLINDERS AND CONES. PLATE XIV. PIPES AND BOTLEES. 164. The intersections of cylindrical or conical surfaces may be curves of either single or double curvature. A curve is said to possess a double curvature, when it is not situated wholly in one plane. The problem to be considered in representing the lines of intersection, reduces itself to the determination in succession of the projections of several points in the curve, and the completion, by tracing the projections of the entire curves through the points thus found. The principle generally to be followed in these cases is, to imagine the two cylinders, for instance, to be cut by one and the same plane — their intersection in that plane being a point in the line sought. Thus, to obtain a point in the line of intersection of two right cylinders, a and b, represented in figs. 1 and 2, draw any plane, c d, parallel to the axes of both cylinders. This plane cuts the vertical cylinder, a, as projected horizontally in the points, e'/, and as projected vertically through the lines, e' e^, and/'/^. The horizontal cylinder, b, is cut in c rf in the horizontal projection, and in c' d' in the vertical projection, which latter is obtained as follows : — The semi-base, g i, of the cylinder, b, is drawn in plan as in fig. l"; then prolonging c d until it cuts this plan in c^, itwUl give the distance, c^ h, of the cutting plane from the axis of the cylinder ; this distance is then transferred to i c', fig. 2, and through c the line, c' d', is drawn as required. The points, e-, /-, of Intersection of this line, c' d', with the lines, e' e-,/'/-, are points in the intersecting line respectively on each side of the vertical cylinder, a. By repeating this operation vidth another plane, as m n, parallel to the first, other two points will obviously be found, as P and o-. Tlie extreme points, a', k'. are naturally determined by the inter- section of the outlines of the cylinders. As for the point, b', it is found by means of the imaginary plane, g p, tangential to the ver- UcaI cylinder, a, and also to the horizontal one, b, when, as in this wse. the two are of equal diameter. As many points having been found as constitute a sufficient guide, according to the scale or size of the dramng, and the pro- portion between the intersecting solids, their reunion into on<" continuous line completes the delineation of the curve of intersec- tion. It will be observed that, in fig. 2, the vertical projection of this curve is a straight line, as a' b', or b' k', these two being at right angles to each other ; this results from the fact, that the cylinders, a and b, chosen for this illustration are of equal diameter, and have their axes situate in the same plane, and at right angles to each other, in such a manner that the curves of intersection, which are elliptical, lie in a plane at right angles to the plane of the vertical projection. It follows, then, that this peculiarity being known, all that is necessary in similar cases — that is, when two equal right cylinders, having their axes in the same plane, cut each other — is simply to draw lines from the extreme points, a' and k\ to the summit, b', the projection of the point of intersection of their axes — the operations above described being, in such cases, altogether dispensed with. 165. When the cylinders are of unequal diameters, the curve of intersection becomes one of double curvature, notwithstanding that the axes of the cylinders may lie in the same plane. Thus, figs. 7 and 8, which represent two intersecting cylinders, a and b, of very different magnitudes, show that the curve of intersection, a' b' k', drawn according to the method before given, is one of double curvature, becoming the more flattened at b', according as the diameters of the cylinders differ more. To render it quite plain that the operation is the same, we have indicated the various points obtained, by the same letters which mark the cor- responding points in figs. 1 and 2. We must further remark, that in figs. 7 and 8 the curve is determined with the assistance of two elevations, taken at right angles to each other ; whilst, in "figs. 1 and 2, an elevation and a plan were employed, similarly, at right angles to each other. We show the application or exemplification of this cui-ve iv figs. 4 and 5, which represent a steam-engine boiler, c, seen partly in elevation and partly in section. The tubular piece, d, which is a species of man-hole, is supposed to be cylindiical, and is attached to the body of the boiler by means of a flange, which gives rise to the external intersectional curves, a b, c d, and the internal one, ef. THE INTERSECTION OF A CONE WITH A SPHERE. 166. Whenever a cone is cut by a plane parallel to its base, the section presents an outlme simOar to that of the base ; ihen, when the cone under consideration is a right cone of a circular base, all such parallel sections are circles. Thus, in figs. 3 and 3°, repre- senting a right cone, a' b' s', the plane, a' b', parallel to its base, a' b', cuts the cone so as to present a circle, of which the diameter is exactly contained within the extreme generatrices, a' $', b' s'. If, then, with the centre, s, and radius, a s = a' b'-~ 2, we describe a circle, it will be the outline- of that part of the cone intercepted by the cutting plane. The section of a sphere, c, by any plane whatever, is also a circle. When this cutting plane, a' b', for instance, is perpen- dicular to the plane of projection, it is necessarily projected as a straight line, as in fig. 3", and as a circle, as in fig. 3, in the plane BOOK OF INDUSTRIAL DESIGN. 5» of projection to which it is parallel. It follows, from the existence of thebe respective properties, that we have at hand a very simple method for determining the curve of intersection of a cone with a sphere, whatever may be the relative position of their axes. This method consists ia supposing a series of parallel planes to cut both the cone and the sphere, so as to produce circular sections of both — the intersections of the outlines of which will conse- quently be points in the curve sought, as indicated in fig. 3. The iutoi'section, a' b', is a circle, the diameter of which is limited by the extreme generatrices, s' a', s' b, of the cone, where they encounter the gi-eat circle of the sphere, c. The same method holds good when the cone is cut by any plane, a' g, inclined to the base, the outline of the section being ia this case an ellipse, which is projected in the plan, fig. 3, by the line, a i" g' n', the resultant of the various intersections m the planes adopted in the construc- ts iii and obtainment of the curve. The same occurs with the intersection of a cone, a' b' s', with a cylinder, a'b'df; and when their axes lie in the same straight line, the intersection, a b', is also a circle, the diameter of which is equal to that of the cylinder. 167. If their axes are parallel, though not in the same straight line, the intersection of these two surfaces becomes a curve of double curvature, which may be determined either according to the method adopted in reference to figs. 1 and 3, or by supposing a series of planes to cut the cone parallel to its base, and conse- quently at right angles to the generatrix lines of the cylinder; by this means circular sections will be produced, those of the cylinder being always the same, but those of the cone varying according to the distance of the planes from its apex. The points of intersec- tion of the various circles representing, respectively, sections of the cone and cylinder, wOl be points in the curve of intersection sought. DEVELOPMENTS. 168. By this term is meant the unrolling, extending, or flatten- ing out upon a plane, of any curved surface, in order to ascertain its exact superficial measurement. The more generally used surfaces or forms which are capable of development in this manner, are — the cylinder, the cone, prisms, pyramids, and tho frusta, or fragments of these solids. Tin and copper-smiths and boiler-makers, who operate upon metals which come into their hands in the foi-m of thin sheets, have continually to transform these sheets into objects which are analogous in form to ihese solids. To do their work with skill and exactitude, and not by mere guess, and also to avoid the cutting of the material to waste, tiny should make plans of the whole or part of tho object as finished, so that they may calculate the exact development of the surface, both as to form and size, and cut it at once from tho sheet of nielal with all possible precision. THE DEVELOPiVIENT OF THE CYLINDER. 1()9. Here, taking fig. 2, which wo have on a former occasion considered as a coui)le of solid cylinders, to roprescnit, in the present case, two pipes or hollow cylinders formed of thin sheet motiil, let us set about ascertaining what should he the sliapi' .•uui size of the pieces of metiil as extended out Hat, of which Ihc m' two cylinders are to be formed. It is to be observed, in the first place, that the rectification or development into a straight fine of a circle, is equal to its diameter multiplied by 3'1416, &c. ; whence the development of the base, p q, of the right vertical cj linder, a. fig. 2, of which the diameter measures "322 metres, is obviously equal to -322 x 3-1416 = 1-012 m. This length, then, 1-012 m., is set off on the line, m m, fig. 10, and the circumference having been divided into a number of equal parts, as was done to obtain the curve of intersection of the two cylinders in fig. 1 ; the line, m m, is divided into the like number of equal divisions. Through each of these points of division, perpendiculars are erected upon the line, m m, representing so many generatrix lines corresponding to those of the cylinder, a, fig. 2 ; and for the sake of greater intelligibility, we have marked the corresponding lines by the same letters. Ne.xt, on each of these are set oft' distances, m b', e' e^, I' P, p d,p p, o' o-, Q k', <Sic., equal to the respective distances in fig. 2. By this means are obtained the points, b', e', I', &c., in fig. 10, through which the curve passes which forms the contour of intersection corresponding to tliat portion of the semi-cylinder, b' a b", fig. 1, which is intersected by tlie horizontal cyhnder, b ; and as the other half of the cylinder is precisely the same, the cur\ e has simply to be repeated, as shown in fig. 10. It is unnecessary to detail the method of finding this develop- ment of the horizontal cylinder, b, as it is identical hi principle to that just discussed. It may be gathered from the above exemplification, that tlie principle generally to bo followed in obtaining the development of cylindrical surfaces, is, first, to unfold it in a direction at right angles to one of its generatrices, or in the direction the generatrix takes in the construction of the solid, and theu to set oft" from the straight line thus produced, at equal distances apai-t, any number of distances previously obtained from the projections of the out- line or line of intersection when the cylmder is joined to anotlier, Of of its section when cut by any plane. The curve of this out- line is finally obtained by tracing a line through the extremities of the generatrices, di-awn perpendicular to the base. THE DEVELOPMENT OF THE CONE. 170. As in the case of the cyUndcr, so likewise, in order to find the development of the cone, do wo unfold it, as it were, in Uie direction of motion of its generatrix. Now, as lUI the generatrices of a right cone are equal, and converge to one point, the apex, ii follows that, when tho conical surface is d^'veloped upon a iilane, these generatrices will form radii ol' a poiiion of a circle ; coiise- (juently, If with one of tho generatrices, as a railius, we describe a circle, and cut otVas much of the circumference as sli:ill be equal to that of the base of the developid cone, wo sliall obtain .1 sector of a circle eqiuil in area to the lateral s\irl'ace of the cone, as developed upon a plane. Fi"-. 9 represents Ilu> development of tlie Irustum, or truucateo cone, a' b', a' r,', as projected in lig. 3*. and of which the :i|h>\ would be s', were the cono ontiu>. The operation is as fol- lows: — We shall suiiposc the coni> to be developol in the direction lak.'M by Iho jMiuiadix, s' a', lig. 3*; therefor.', wiih a radius egual 52 THE PRACTICAL DRAUGHTSMAN'S to s' a', and with the centre, s', describe the fragment of a cu-cle, a' b' a-, fig. 9. Having divided the circular base, a b, fig. 3, of the cone into some arbitrary nmnber of equal parts, say 16, and having drawn the respective generatrices, 1 s, 2 s, 3 s, &c., set oflf on the arc, A' B A-, fig. 9, an equal number of arcs, each equal to the respec- tive arcs obtained by the subdivision of the circle, a b, fig. 3. From the points thus obtained, 1' 2' 3', &c., fig. 9, ivaw the radii, 1' s', 2' s', 3' s, dz-c, representing generatrices corresponding to those projected in fig. 3. By this operation we obtain the development of the entire cone, and find that it produces the figure s' a' b' a-, fig. 9, the circular perimeter of which is equal to that of the base of the cone itself The cone, however, under consideration, is divided by a plane, a' h', fig. 3, parallel to its base, which reduces it to a frustum of a cone ; the development of the conical surface of which is equal to the space contained between the arc, a' b' a^, corresponding to the base of the cone, as just determined, and the arc, a c i, of lesser radius, drawn with the same centre, s', and with a radius equal to the generatrix, s' a', of the portion of the cone taken away. The development, then, of the truncated cone, is the fragment of an annular space, distinguished in fig. 9 by a flat tinted shade. 171. In the case of a truncated cone, of which the dividing plane is in(;liiied to the base, as a' g, fig. 3°, instead of being parallel, or if the cone is joined to a cylindrical or spherical body, and the line of intersection is curved in any way, the development of this edge of the conical surface will no longer take the form of the arc, a c b, fig. 9. It* true representation will be obtainable by setting off", on the several radii, fig. 9, lengths corresponding to the respective generatrices as intercepted by the plane, or curve, of intersection. In order to obtain the lengths of the respective generatrices, which can be done from the vertical projection, fig. 3°, each intermediate one, as 4' t, &c., must be squared across to an extreme one, as at a' i', and indicated by the horizontal lines ; this will give the exact length of each — being otherwise, as projected, considerably fore- shortened. Thus, the division of the cone by the inclined plane, a' g, fig. 3", produces an ellipse, the development of which, in fig. 9, takes the form of the curve, a i g b. In the ccnstructicn of the boiler, represented in figs. 4 and 5, wliich is formed of several pieces of sheet metal, we shall find extensive applications of the principles just discussed. It must be borne in mind that, in calculating the development, or the size and shape of the component pieces, an allowance must be made for the lap of the pieces over each other, for the purpose of joining them together, as indicated in the drawing. 172. In cylindrical steam boilers, the extremities are generally constructed in the shape of hemispheres — this form offering the greatest resistance to internal or external pressure. As the sphere cannot be developed upon a plane, these hemi- spherical ends cannot be made in a single piece, unless cast or forged. In practice this difficulty is overcome, by forming this portion in from 5 to 8 gores, according to the size of the boiler, these being sunnounted by a central cap-piece. After being cut out, these several pieces are hammered to give them the necessary sphericity. In fig. 6, we give a practical method of approximately determining the development of one of such gores; this consists in drawing with the centre, o', an arc, m n, corresponding to the radius of the hemisphere. On this arc, from m to n, set off the circular length of the gore ; set off", also, the length, p q, corresponding to one of the six divisions, as seen in the end view, fig. 5. On the arc, m n, fig. 6, mark an arbitrary number of points, at equal distances asunder, as 1, 2, 3, 4, 5 ; draw through these horizontal lines, cut- ting the vertical, m o', thereby giving the various radii, o' 1", (/ 2", o' 3", &c., with which arcs are di-awn as indicated ; the rectifies^ tion, or development, of these arcs, contained between the radii, p o', g o', are then obtained, and transferred to perpendiculars di-awn through the points, 1', 2', 3', &c., on the line, o' n', which is the rectification, or development, of the arc, m n, with its series of divisions. Thus, from the arc, p q, is obtained the line, p' q', and similarly the entire figure, p' n' q', which is an approximation to the surface of the gore, as supposed to be flattened out. The necessary allowance for lap is superadded, as shown by the flat tinting in the drawing, fig. 6. The gore cut to this outline in sheet metal is then hammei-ed to a proper form upon a mandril, or anvU, with a spherical surface. THE DELINEATION AND DEVELOPMENT OF HELICES, SCREWS, AND SERPENTINES. PLATE XV. 173. That curve is called a cylindrical helix, which may be said to be generated by a point caused to travel round a cylinder, having, at the same time, a motion in the direction of tlie length of the cylinder — this longitudinal motion bearing some regular prescribed proportion to the circular or angular motion. The dis- tance between any two points which are nearest to each other, and in the same straight line parallel to the axis of the cylinder, is called the pitch — ^in other words, the longitudinal distance tra- versed by the generating point during one revolution. This definition at once suggests a method of drawing the lateral projection of this curve, when the two projections of the cylinder and the pitch are known. This method consists in dividing the circumference of the base of the cylinder into any number of equal parts, and drawing parallels to the axis thi'ough the points of division projected on the vertical plane ; at the same time a portion of the axis, equal to the pitch, must be divided into the like number of equal parts, and as many lines must be di-awn perpendicular to the axis. The intersections of the cor- responding lines of each set will be points in the curve. Let A and a', figs. 1 and 2, be the horizontal and vertical pro- jections of a right cylinder, and a' — a^ the length of the pitch of the helLx, generated by the traverse, as already defined, of the point projected in a and a'. The circle described with the radius, a o, and representing the base of the cylinder, is divided into 12 equal parts, starting from the point, a. Through each of the points thus obtained, a ver- tical line is di-awn. The pitch, a' a-, is similarly divided into 12 equal parts, and a corresponding number of horizontal lines are drawn to cut the vertical ones in the points, 1', 2', 3', &c. ; thea* BOOK OF INDUSTRIAL DESIGN. 53 points are next connected by the contmuous line, a', 1', 3', 6', 9', a^, which forms the vertical or lateral projection of the helix. Half of this curve is indicated by a sharp full line, as being on the front surface, a, 3, 6, of the cylinder, whilst the other half is in dotted lines, representing the portion of the curve which is on the other side, a, 9, 6. The number of divisions of the circumference of the cylinder is a matter of indifference as regards the accurate delineation of the curve, and it is therefore natural to choose a number that calls for the simplest operations — an even number, for example, as 6, 8, or 12; and in the present instance, wherein the starting point, a, lies in the horizontal diameter, a 6, of the base, it will be observed that two points occur in the same vertical line, as 2 — 10, which gives the points, 2', 10', in the vertical projection. The operations wOl be similar, if the given starting point be diametrically opposite to a, as 6', the pitch, b^ b-, being equal to 174. The conical helix is different from the cylindrical one, simply in that it is described on the surface of a cone instead of on that of a cylinder, and the operation consequently differs very slightly from the one before described; the horizontal and vertical projections of the cone are given, and also the pitch. Fig. 3 is the vertical projection of a truncated cone, c, the bases of which, a' b', c' d', are represented in the plan, fig. 1, by the concentric circles described, with the respective radii, a o and c o. The outer circle having been divided as already shown, radii are dravra to the centre, o, from all the points of division, 1, 2, 3, &c., which cut the inner circle in the points, e,f,g, &c. These points are then projected upon the upper base, c' d', in fig. 3, those on the outer circle being similarly projected on the lower base, a b' ; the respective points in each base are next joined, thus forming a series of generatrices of the cone, as 1- — e^, 2- — /-, o' — o-, &c., which would all converge in tlie apex, if the cone were complete. These lines are cut by horizontals drawn through a corresponding number of divisions in the length of the given pitch, a' c', and the points of intersection thus obtained lie in the curve which it is required to draw. The horizontal projection of the curve is then obtained by letting fall from the points of intersection last obtained, a series of verticals which cut the respective radii in the plan, fig. 1. This produces a species of spu-al, or volute, e^,/^, g^, h^, 2^, &c. By following out the same principles, helices may bo represented as lying upon spheres, or any other surfaces of revo- lution. THE DEVELOPMENT OF THE HELI.X. 175. It will be recollected that a cylinder, and also a cone, are capable of being developed upon a plane surface, and that the base of either, when rectified, or converted into a straight line, is equal to the diameter multiplied by 3'1416. Lot, then, a 6, fig. 4, bo a portion of the development of the base of the cylinder, A, figs. 1 and 2; to obtain the development of (he helix drawn upon this cylinder, we must first divide it off into ienglhs, corrc- sponding and equal to the arcs obtained by the division of llie circle, (I (>. On each of the divisions thus obtained, as 1, 2, 3, &<•., we then erect perpendiculars, making thoni equal rospecliveiy to the distances from the starting point, a, of tho several divisions of the pitch. The extremities of these perpendiculars, as 1', 2', 3', &c., will be found to lie in the same straight line, a 6', which consequently represents the development of a portioD of the helix. In general, the development of a helLx is a straight line, forming the hypothenuse of a right-angled triangle, the base of which is equal to the circumference of the cylinder, and the height to the pitch of the helix. Several helices di-awn upon the same cylinder, and having tho same pitch, or a helix which makes several convolutions about a cylinder, is represented by a series of parallel curves, the distance between which, measured on any fine parallel to the axis, is always equal to the pitch. The development of the conical helix may be obtained by means of an operation analogous to that employed for the development of tne cone (Plate XIV.) ; and in this case the result will be a curve, instead of a straight line. We meet with numerous applications of the helical curve in the arts, for all descriptions of screws ; and staircases, and serpentines. 176. Screws are employed in machinery, and in mechanical combinations, either for securing various pieces to each other, so as to produce contact pressure, or for communicating motion. Screws are formed with triangular, square, or rounded threads or fillets. A screw is said to have a triangular thread, when it is generated by a triangle, isosceles or not, the three angles of which describe helices about the same given axis, situate ui the same plane as the triangle. Figs. 5 and 5° represent the projections of a triangular- threaded screw, such as would be generated by the helical move- ment of the triangle, a' b' c', of which the apex, a', is situate on a cylinder of a radius equal to a o, and of which the other angles. b', c', are both situate on the internal cylinder, having the radius, b o, which is called the core of the screw, and is concentric with the first. The difference, a b, between the radii, a o and b o, indi- cates the depth of the thread. When, as in the case taken for illustration, the screw is single- threaded, the pitch is equal to the distance between the two point.s, b' and c' ; that is, to the base of the triangle. The screw is one of 2, 3, 4, or 5 threads, according as tho pitch is equal to 2, 3, 4, or 5 times the base of the generating triangle. From what iins already been discussed, tho method of delineating the triangular- threaded screw will be easily comprehended; for all tliat is nives- sary is to draw the helices, generated by tho three angular points, in the manner shown in reference to figs. 1 and 2. \\"e liavo, notwithstanding, given tho entire operation for one semi-convolu- tion of tho thread, in figs. 5 and 6*. When one of tlio curves, as a' 3' 6', is obtained, it is repeated as many tinie< as there are cor. volutions of tho thread on tlie length of the screw. To do thir, with facility, and without reiieating the entire operation, it is. customary to cut out a i)atlern of the curve in iianl cuii-buani, or, by picl'erenoe, in veneer wood ; then sotting liiis pattern to tlio points of ilivision, (/'(•'/', ])reviously set (tlV, tho curves arc easilv (Irnwn parallel to one another. Tho s;nuo may bo done \\il!i lhv> inner liolioiil curves. It must be obsorvod, that, to comploli' tiio outline of the .vrow, 04 THE PRACTICAL DRAUGHTSMAN'S tlioso vai-ious curves require to be joined by the portions, h> d', d' V, vvhicii, thouijli in fig. 5° they are dj-awn as simple straight lines, should, if it is wished to bo precise, be shown by lines slightly curved and tangential to the curves passing througli the points, a and h', as in fig. 5'. These curves are the result of a series of helices, traced by the component points of the Imes forming the sides, a' b', a' c', of the generating triangle. In practice, this nicety is disregarded, and simple straight lines are employed. 177. A screw is termed square-threaded when it is generated by a square or by a rectangle, the parallel sides of which lie in right concentric cylinders, and the angles or corners of which describe helices about the axes of these cylinders. Figs. 6 and 6", represent projections of a square-threaded screw — the thread being generated by the square, a' c' b' d'. The horizontal side, a' c', determines the depth. The height, a' d', marks the width of the thread, and d' d is the width of the interval, which is generally equal to that of the thread. When the screw is single-threaded, the pitch, d e', is equal to the sum of the wdths of the thread and of the interval, or, in the case before us, to twice the side of the squai-e. Of course, when the pitch is equal to 2, 3, or 4 times a' e, the screw is 2, 3, or 4^ threaded, in all cases having as many intervals as threads. The opei'ations called for in delineating the screw of a single square thread, are fully indicated upon the figures. The delineation of a screw of several threads does not possess any additional points of difficulty. INTERNAL SCREWS. 178. An internal screw, or nut, is a screw in intaglio, cored out of a solid body — instead of being in relief, and having the material cut away from it — in such a manner that its more indented portions correspond to the more elevated portions of the common or external screw, whilst the more indented portions of the latter correspond to the more elevated portions of the former. In order to represent the helical fillets or threads of the internal screw, it is necessary to section it by a plane passing through its axis ; it is in this manner that we have represented the nuts, m n p q, in figs. 6° and 6°, the first having a triangular thread fitting to and embrac- ing the screw, d, which is represented as just introduced into it ; the other has a square thread similarly adapted to the screw, e. It follows, from these nuts being represented in section, that we only see the half of each corresponding to the posterior portion of tlieir respective screws, d and e ; and in consequence of this, the helical curves are inclined in the opposite direction to those repre- senting the anterior portions of the screws. Those screws are distinguished as right-handed screws, of which the thread in relief rises from the left to the right, as in the screws, r, e ; and as left-handed when the thread takes the direction from right to left — that is, for example, in the direction taken by the curves representing the nuts, d, e. SERPENTINES. 170. Serpentine is the name given in practice to a pipe or tube bent to the helicoidal form ; but, in geometry, it is the term given to the solid generated by a sphere, the centre of which traces a helicoidal path. This form is often employed, whether hollow, as for pipes, such as the worm of a distilling apparatus, or solid, as for metal springs The first thing to be done in delineating this solid, is to deter mine the helix traced by the centre of the generating sphere, ita pitch, and the radius of the cylinder on which it runs, being given Tlie helix having been di-awn, a series of circles are described witj the radius of the sphere, and with various points of the curves asi centres ; curves drawn tangential to these cu'cles, will then form the outline of the object. Figs. 7 and 7° represent the plan and ele- vation of a serpentine formed in this manner. The circle di'awn with the radius, a o, is the base of the cylinder, on which lies the helix generated by the traverse of the point projected vertically in a'. Next is given the radius, a' b', of the generating sphere, and the pitch, a* a^, of the helix. Tliis helix is then projected accord- ing to the operation indicated on the di-awing, and already described, by the curve, a', 1', 2', 6', 9', and a^ ; it may be con- tinued indefinitely, according to the number of convolutions desired. With diiferent points of tliis curve as centres, and with the radius, a' b', are then described a series of circles pretty near to each other, and two curves are drawn tangential to these, as shown on fig. 7'. In going over this figure with ink, it is of importance to limit these curves to the portion of the outline, which is quite apparent or distinct ; thus, for the anterior portion, a, 1,2, 3, 6, fig. 7, of the serpentine, the lower curve, c efg, ends at the point of contact, c, with the circle whose centre is a~, whOst the upper one, hid, ends in the point, d, on the circle described with the centre, 6'. The posterior portions of these curves are limited by the points, g and i, where the bend of the serpentine goes behind, and is hid by the anterior portion. The horizontal projection of the serpentine is always comprised within two concentric circles, the distance asunder of which is equal to the diameter of the generating sphere, as in fig. 7. Fig. 7- represents a tubular serpentine, which is supposed to be divided by a plane, 1 — 2, in fig. 7, passing through its axis. It is, consequently, the posterior portions that are visible, and they are inclined from right to left ; the section at the same time shows tue thickness of the tube, or pipe. In the arts, we also see serpentines, both solid and hollow, generated by conic, or other helices ; of this description are the springs employed in the moderator lamps, and the forms of distil- lery worms are sometimes varied in this way. 180. Observation. — The curves representing tlie outline of screws and serpentines, the rigorously exact delineation of which we have just explained, are considerably modified when these objects come to be represented on a very small scale ; thus the triangular- threaded screw may be represented, as shown in fig. 8, by a series of parallel straight, instead of curved, lines — these being inclined from side to side to the extent of half the pitch. These lines should be limited by two parallels to tl e axis on each side, mark- ing the amount of relief of the thread. When the scale of the drawing is still smaller, and greater simplicity desirable, the draughtsman is content with a series of parallel lines, as in fig. 9, limited by a single line on each side parallel to the axis. For the square-threaded screw, the helical curves may similarly be replaced by straight lines, as in fig. 10. The same also applies to the serpentine, as shown in fig. 11. BOOK OF INDUSTRIAL DESIGN. 55 THE APPLICATION OF THE HELIX. THE CONSTEUCTION OF A STAIRCASE. PLATE XVI. 181. The staircases, which afford a means of communication oetween the various floors of houses, are constructed after various systems, the greater number of which comprise exemplifications of the helix. The cage, or space set apart for the staircase, varies m form with the locality. It may be rectangular, circular, or elliptic. Figs. 1 and 2 represent a staircase, the cage of which is rec- tangular this space being provided for the construction of the main frame of the stair, with its steps and balustrade, and with a central space left sufficient for the admission of Ught from above. In the case of a cylindrical cage, the curve with which the steps rise is helical from bottom to top ; but in a staircase within a rec- tangular cage, the steps rise for some distance in a straight line, and only take the helical twist at the part forming the junction between the rectilinear portions running up alternate sides of the rectangle. Stairs are sometimes made without this curved part, a simple platform, or " restbg-place," connecting the two side portions. For the division of the steps, we take the line, efg h i, passing through the centre of their width, and taking exactly the direction it is intended to give the stairs. The first or lowest step, a, which lies on the gi'ound, is generally of stone, and is larger and wider than the others. For the stairs, as for the helLx, the pitch or height, say 3-38 m., from the basement to the floor above, is divided into as many equal parts as it is wished to have steps. The centre line, efg h i, is also divided into a like number of equal pai-ts. In general, the number of steps should be such, that the height of each does not exceed 19 or 20 centunetres. The larger the staircase is, the more may this height be reduced — say, perhaps, as low as 15 or 16 centimetres. The width, 1 — 2, of the step should not be under 18 to 20 centimetres. If, for example, in the given height of 3-38 m., we wish to make 21 steps, we must divide this height into 21 equal parts, and draw a series of horizontal lines through the points of division, which will represent the horizontal sui-fiiccs of the steps. For those steps which lie parallel to each other, it is simply requisite to erect verticals upon the points of division on the centre line in the plan. The points of intersection of these with the horizontals above, as 1, 2, 3, 4, fig. 2, indicate the edges of these steps. For the turning steps, however, or winders, as those steps are called which are not parallel to each other, a particular opera- tion is necessary, termed the balancing of the steps, the object of which is to make the steps as nearly equal in widtli as possitilo, without, at the same time, making thcin very narrow on the inner edges, or rendering the twist or curve too sliarp or sudden. Where' fho stairs are narrow, as in the case we have illustrated, the balancing should conuiience a step or two before rcacliiiig the curved portions. This balancing may bo obtMijicil in the follow- ing manner: — A i)art of the rectilinear ])<)rtioii, /) /, (([ual to three «t«ps, is d(;veloii<(l, and then a part of the lurvcd pmlion, / ;;/ n, equal to three more steps. On the vertical, I q, fig. 3, set ofi' the heights of the first three steps; and through the point, q, draw the horizontal, q 4, representing the development of the widths of the steps in a straight line. Also, on the prolongation, i q'. of the vertical, t q, set off" the heights of other three steps. Through the point, q', draw a horizontal, and make q' 10 equal to the arc, I m ru, in the plan, fig. 1, as rectified. The straight Une, t 10, will then be tlie development corresponding to the curve of the framepiece. At the latter point, n, erect a perpendicular on this line, and at the point, 6, a perpendicular to the straight line, t 4. The point ot intersection, o, of these two perpendiculars, gives the centre of the arc, p k n, which is drawn tangentially to these lines. Then, through each point of division on the vertical, q </, draw hoiizon- tals, meeting this curve in the points, _;', k, I, m, through which diaw parallels to q q'. Then transfer the respective distances. J 6, k t, I 8, m 9, comprised between the arc and the two straight lines, I 4 and t u, on the line of the framepiece, p k n,in the plan, fig. 1, as at _;' k, k I, I m, m n. Next, draw straight lines through the points,y, k, I, m, and through the respective points of division, 6, 7, 8, 9, on the centre line, which will give the proper incli- nation for the steps as balanced. The second half of the curved portion is obviously precisely the same as the first in plan, and may easUy be copied from it. Having thus determined the position of the steps in tlie hori- zontal projection, they must next be projected on the vertical plane, by means of a series of verticals, wliich cut the respective horizontals drawn through the points of division, 1, 2, 3, 4. As in fig. 2, the anterior wall of the cage is supposed to be cut away by the line, a 6' 10', in the plan, the outer edges of the steps are seen and are determined by erecting verticals on the correspond ing points, 6', 7', 8', 9', &c. The perpendicular portions, v v', of the steps, which are over- hung by the horizontal portions, and consequently invisible in the plan, fig. 1, are, nevertheless, indicated there in dotted lines, parallel to the edges of the steps. To render them quite distinct, however, and at the same time to show the manner in which tliey are fitted into the framepiece, we have represented diem, in fig. 4, without the actual steps, supposing them to be cut in successioiij horizontally, through then- middles. The framepiece is the principal piece in the staircase. It is situated m the centre of the cage, and sustains each step, and, consequently, must bo constructed very accurately, for upon it, in a great measure, depends the strengtli and solidity of tJio staircase, For a staircase of proportions, like those of the one represented ir. the plate, the framepiece is generally made of oak, in three pieces ; the middle piece, c, corresponding to the curved portion, whils* the other two, B and d, jouied to that one, form the rectilinea' portions. A special set of diagrams is necessary, to determine tlie shape and proportions of the various jiart-s of this fr^iopioce. The method lure to be followed is, in the first place, to draw the joints, by w liith the vertical portions of llio steps are attached to the frainepiece. These can easily be obtained by squaring tiiem over from tig. 4 to tig. 5, in which hust are the iiori/.ontal dixision lines, corresponding to tig. 2. It will be observed, tliat tiie joints referred to are bevilled ofl", so as not to be npjifirent externally, The faces on the IVuniepicco arc seen on tig. 6, at llie jwuts, b. c. 50 THE PRACTICAL DRAUGHTSMAN'S unci tlio method of obtaining them is sufficiently indicated by the dotted lines. The framepiece has a certain regular depth through- out, and is cut on the upper side, to suit the form of the steps, and below, to the curvilinear outline, d b' c' d' e' f, which is nothing but the combination of a helix with a couple of straight lines. These straight lines, a' h' and e' f, are naturally parallel to the curve passing through the edges of the steps, 1, 3, 5, 13, 16, 19. The curved part, 6' cS which corresponds to the anterior face, h- c", of tlie framepiece, is drawn in precisely the same manner iis a common cylindrical helix. It is the same with the part, d' e', which corresponds to the interior portion, d- e-. If, in order to better indicate the space occupied by the framepiece, it is wished 10 construct an end view of it, this may be done, as in fig. 6, from the data furnished by figs. 4 and 5. In figs. 12, 13, and 14, we have given, on a larger scale, the different views of the curved part, c, of the framepiece, so as to show its construction more plainly, as well as the form of the joint connecting the three parts of the framepiece together. Each of these figures is inscribed in a rectangle, indicated by dotted lines, and representing the rectangular parallelepiped, in which the piece may be said to be contained. To strengthen the junction of this piece with the portions, e and d, they are connected by iron straps or binding-pieces let in, or by bolts passing through, the entu-e thickness of the wood. Figs. 7 and 8 represent, in plan and elevation, the details of the landing-stage, which forms the top step of one flight of the stairs, and is on a level with the upper floor. It is with this piece that the upper portion, d, of the framepiece is connected, by a joint similar to that uniting the other portions. Fig. 9 is a sec- tion, through the centre of this step-piece, through the line, 1 — 2, in the plan ; whilst fig. 10 is another section, through the line, 3 — 4 ; and fig. 11a tliird, through the line, 5 — 6. The form, dimensions, and joint of tliis piece, are all fully mdicated in this series of figures. The shaft of the staircase, or the open space left in the centre of the cage, is partially occupied by a balustrade, formed by a number of rods of iron or wood, attached at their lower extremities to the framepiece, as shown in fig. 15, and united above by a flat bar of iron, surmounted by a hand-rail of polished furniture-wood, of the form given in Plate III., fig. ©. The position of the rods, as given in the plan, fig. 1, is sufficient for the determination of theur vertical projection or elevation. THE INTERSECTION OF SURFACES. • APPLICATION TO STOPCOCKS. PLATE XVII. 182. We have already discussed several examples of intersec- tions of surfaces, as in pipes and boilers, and we shall now pro- ceed to give some others, which are pretty generally met with, p;vrticularly in the construction of stopcocks ; and for that reason we take one of these contrivances as an illustration. A stopcock is a mechanical arrangement, the function of which Is to establish or interrupt at pleasure the communication through pipes, for the passage of gases or liquids. It consists of two dis- tinct parts, one called the cock, and the other — adjustable and moveable in the first — ^termed the key or plug. Stopcocks are generally made of brass, compositioii-metal, or cast-iron, and the cock is formed with or without flanges, for attachment to vessels or piping. The key is generally conical, so as to fit better in its seat. The degree of taper given to the key varies with different constructors. We have shown in dotted Tmes. in fig. 2', various degrees of taper to be adopted, according as greater tightness or greater facility of movement is wanted. The part of the cock which receives the key is, of course, turned out to a corresponding conical surface. This portion of the cock is connected to the tubular portions by shoulders, of a slightly ellip- tical contour. A stopcock answering to this description is repre- sented, in plan and elevation, in figs. 1 and V. In these figures wOl be seen the conical part, a, which embraces the key, the cylindrical portions, b, united to the former by the shoulders, d, and terminated by the flanges, c. The conical key, e, adjusted in the cock, is surmounted by a handle, f, by means of which it is turned. The key is retained in the cock by a nut, g, working on a screw, formed on the lower projecting end of the key. Fig. 1' represents an end view of this cock, and fig. 2' is a vertical projection of the key alone. Fig. 2* is a view of the key, looking on the lower end. Fig. 2 is a hori- zontal section through the line, 1 — 2, in fig. r ; and fig. 2° is a vertical section through the line, 3 — 4, fig. 1. It will be easy to see, from these various views, that, in order to represent the stopcock with exactitude, it has been necessary to find, on the one hand, the projection of the intersection of the elliptic shoulder, d, with the external conical surface of the central part. A, of the cock ; and on the other, that of the cylindrical sur- faces of the handle, f, of the key, as well when this is placed in a position parallel to the vertical plane, as in fig. 2°, or inclined to this plane, as in fig. 1°. We have, moreover, to determine the intersection with the external surface of the key of the rectangular opening, h, provided for the passage of fluids ; and also the inter- section of a prism with a sphere, which occurs in the shape of the nut, G, which secures the key m its place. The various opera, tions here called for are indicated on the figures, which we shall proceed to explain. Figs. 3 and 3" show the geometrical construction of the inter- section of the horizontal cylinder, f', of the handle, with the verti- cal cylinder, e', of the key. The curve is obviously obtained according to the method already described in reference to figs. 1 and 2, Plate XIV. We have, however, repeated the operations, the exemplifications being a variation from that previously given. 183. When the horizontal cylinder, f', figs. 3' and 3°, becomes inclined to the vertical plane, its curve of intersection with the vertical cylinder, e', assumes a different appearance as projected in this plane. Its construction, however, is precisely the same, as follows : — To obtain any point in the curve, we proceed just as in the preceding example, drawing the vertical plane, d' e', parallel to the axes of the cylinders, this plane cutting the vertical cylinder through the lines, d f and e g. This same plane cuts the horizon- tal cylinder, as projected in plan, at d' e', whence the vertical pro- jection is obtained, after drawing the semi-cylindrical end view, as m fig. 8. The distance, i V, being set off' on h h', the horizontal BOOK OF INDUSTRIAL DESIGN. 57 iinc, d e, drawn through the point, W, represents the mtersection of the plane with the horizontal cylinder, ft h', being, of course, measured from its r xis. It will be seen that the line, d e, is cut by the vertical lines, d f and e g., in the points, d, e, which lie in die curve sought ; and the same construction will apply to every other point in the curve, dbec- 184. Figs. 4 and 5 represent the intersection of an elliptical cylinder wth a right cone of circular base, corresponding to the external conical surface, a, of the cock, at its junction with the shoulders, d. Fig. 5 is a plan, looking on the cock from below, and which shows the horizontal projection of the intersectional curves. The solution of the problem requiring the determination of these curves consists in appljing a method already given — namely, in taking any horizontal plane which cuts the cone, so as to present a circular section on the one hand, and the cylinder in two straight lines on the other — the points of intersection of these straight lines with the circle representing the section of the cone. Thus, by drawing the plane, c d, fig. 4, we obtain a circle of the diameter, c d, which is projected horizontally, as with the centre, o, fig. 5 ; we have also two generatrices of the cylinder, both projected in the vertical plane in the line, a b, and in the horizontal plane in the lines, a h' and a- h-. Hanng dra\vn the semibase of the cyhnder, d e, as at d'fe', and having taken the distance,/^, fig. 4°, and set it off, in fig. 6, from the axis, as from g> to a', and to a-, we thereby obtain the generatrices, a' V and a- b^, which cut the circle of the dia^ meter, c' d', in the four points, h' i', which are squared across to, and projected in, the vertical plane in the points, h, i. lu the same manner we obtain any other points, as m, n; k I being the plane taken for this purpose. The extreme points of the curve are obtained in a very simple and obvious manner, as /, g, r, s, being the points of intersection of the extreme generatrices, or the outlines of the two solids. With regard to the points, t, u, which form the apices of the two curves, their position may be obtained from the diagram, fig. 4°, by drawing from the point, s, which would be the apex of the entire cone, a tangent, s i, to the base, d' fe', of the cylinder, and projecting the point of contact, t, in the line, x y, representing a plane cutting the cone, which must be projected in the horizontal plane. Then, making g' x', fig. 5, equal to t V, fig. 4% and drawing horizontals through x', their intersections, i' u', with the circular section of the cone, will be the points sought, which are accordingly squared over to fig. 4. The operations just described are analogous, it will be observed, to those employed in obtaining the intersection of two cylinders. If, in the case of the cone and cylinder, the latter h.id been one of circular instead of elliptical base, as is frequently the case, still the constructior, as a little consideration will show, must be pre- . isely the .same, and the resulting curves would be analogous — tliat is. when the diameter of the cylinder is less than that of the cone at .he part where it meefs the lowest generatrix of the cylinder ; tne curves, however, assume a different appearance when the dia- meter of the cylinder exceeds this, as is .siiown in figs. 6 and 7. In this case the intersections are represented by the cur\-es, s I r :ind p 11 q; the method of obtaining these is fully indicated on the diii'Tiims. 185. The opening or slot, h, cut through the key of the stop- cock, is generally rectangular, rather than circular, or similar to the tubular portions of the cock. The object of this shape is to make the key as small as possible, and yet retain the required extent of passage. This rectangular opening gives rise, in fig. 2', to the intersectional curves, a b, c d, which are portions of the hyperbola, resulting from the section made by a plane, cutting the cone parallel to its axis. The operations whereby they are deter- mined are indicated in figs. 10, 11, and 12. To render the character of the curve more apparent, we have, in these figures, supposed the generatrices of the cone to make a greater angle with the axis than in fig. 2'. The line, a b, repre- sents the vertical plane in which the curve of intersection lies. It is evident that, if we delineate a series of horizontal planes, as c d, ef, ghfik, fig. 10, we shall obtain a corresponding series of circles in the horizontal projection, these circles cutting the plane, a b,m the points, I', m', n', p', &c. These points are squared over to the vertical projection, fig. 10, giving the points, l,m,n,p; and the apex, o, of the curve is obtained, by drawing in the plan, fig. 12, with the centre, s, a circle tangent to the plane, a b, and then projecting this on the vertical plane, fig. 10, as shown. From these diagrams, it is easy to see that the opening, h, will be partly visible when the key is seen from below% as in fig. 2'. 186. Figs. 8 and 9 represent the vertical and horizontal pro- jections of the nut, g, which secures and adjusts the key in its socket. This nut is hexagonal, being terminated by a portion of a sphere, the centre of which lies in the axis of the prism. Each of the facets of the prism cuts the surface of the sphere, so as to present at their intersection portions of equal circles, which should be determined in lateral projection. The diameter of the sphere is generally three or lour times that of the circle circumscribing the nut, but, to render the curves more distinct, we have adopted a smaller proportion in the case under exammation. The sphere, y, is represented by tw'o circles of the radius, o a ; and the nut by an hexagonal prism, the a.xis of which passes through the centre of the sphere. The anterior facet, a! b', of this nut, cuts the sphere, so as to show a circle of the diameter, c' d'. This circle, projected vertically on fig. 8, cuts the straight lines, a e and h f, of the prism, in the points, a and b ; and the portion of the circle com- prised between these two points, consequently, represents the intersection of this facet witii the sphere. The other two facets, a g' and b' h', which are inclined to the vertical piano, also cut the sphere, .so as to produce, at their intersection with the surface, arcs of equal radii with tiiat of tlie facet, a b. From their incli- nation, these arcs become slightly elliptical, being comprised, on the one hand, between the points, a and g, and b, h, on tlie other. The summits of these ellipses are obtained by drawing horizontal lines tangential to the arc, a b, and cutting it in the points, V, /, by peqK'ndiculars drawn through the middle of the lateral fa«'-ts. In practice, it is quite sutlicicnt to descrila' cia'ular arcs. pas,sing through the points, g, k, a, and b, I, h. Wo have already seen, in reference to Plato XIV., that the intersection of a right cylinder witJi a sphoro, througli tlio centre of which its axis passes, gives a circle pr<>joctod laforally !lh a straight line. Thus, the opening o', which pas-scs throiigli tlio nut, being cylindrical, produces, by Ha intersection with the sphcrr, a I J 58 THE PRACTICAL DRAUGHTSMAN'S .Mivlo of the diameter, m' n', in tlie plan, projected vertically in the straight line, m n. Fig. 8° indicates the analogous operations required to determine the same intersections when the nut is seen with one of the angles in the centre, and only two facets visible, as represented in fig. V. The elliptic curve, b^ P h-, corresponding to the one, b 1 h, must obviously be comprised between the same two horizontal lines passing through these points, and an arc is drawn through them as before. We may here observe, that the proticient draughtsman will, doubtless, deem it unnecessary, except in extraordinary cases, to enter into such minute details of construction for the various intersectional curves as those wc have discussed, being guided simply by his own judgment, and the appearance presented by ditferent experimental proportions. All draughtsmen, however, will find that some practice in obtaining the exact representation of the various curves, according to the methods here given and rules laid down, will be of immense advantage to them — enabling them, from possessing a tliorough theoretical knowledge of the rela- tions of the various forms of solids to each other, to approach much nearer truth, when, at a more advanced stage, they relinquish tlie aid of such constructive guides. RULES AND PRACTICAL DATA. STEAM. 187. All liquids become changed to vapour when their tem- perature is sufficiently elevated. When water, contained in a close vessel, is elevated to a temperature of 212° Fahrenheit, it produces steam of a pressure or force equal to that of the atmosphere. The pressure of the atmosphere is a force capable of sustaining, in a vacuum, a column of water 33 feet high, or a column of mer- cury 30 inches high. This force is equal to a weight of about 15 lbs. per square inch. Thus, taking the square inch as the unit of superficial measure- ment, the pressure or tension of the steam at 212° Fahrenheit is also equal to 15 lbs. When the containing vessel is hermetically closed, as in a boiler, if the temperature be increased, the steam becomes endued with more and more expansive force — this increase of force, however, not being directly proportionate to the increase of temperature. The tension or expansive force of steam, as also of gases generally, is inversely as the volume ; thus, at the pressure of one atmosphere, for example, the volume of the steam or gas being one cubic foot, the same quantity of steam would occupy only half the space at a pressure of two atmospheres, and reciprocally. TABLE OF PRESSURES, TEMPERATURES, WEIGHTS, AND VOLUMES OF STEAM. Pressure in Pressure in Inches Pressure per Square Temperature Temperature Weight of a Cuhic Foot Volume of a Pound Atmospheres. of Mercury. Inch. Fahrenheit. Centigrade. of Steam. of Steam. Degrees. Degrees. Lb. Cubic Feet. 1-00 30-0 15-00 212 100-0 -3671 2-7236 1-25 37-6 18-75 224 106-6 •4508 2-2183 1-50 45-0 22-50 234 112-4 •6332 1-8852 1-75 62-5 26-25 243 117-1 •6144 1-6281 2-00 60-0 30-00 250 121-5 •6936 1-4433 2-25 67-5 33-75 258 125-5 •7729 1-2938 2-50 75-0 37-50 264 128-8 -8491 1-1777 2-75 82-5 41-25 270 132-1 •9284 1-0771 3-00 90-0 45-00 275 135-0 1-0058 •9942 3-25 97-5 48-75 280 137-7 1-0826 -9237 3-50 105-0 52-50 285 140-6 1-1582 ■8634 4-00 120-0 60-00 294 145-4 - 1-3086 •7642 4-50 135-0 67-50 301 149-1 1-4572 •6862 5-00 150-0 75-00 308 153-3 1-6033 •6236 5-50 165-0 82-50 314 156-7 1^7494 •5716 6-00 180-0 90-00 320 160-0 1^8946 •5273 6-50 195-0 97-50 326 163-3 2^0359 •4912 7-00 210-0 105-00 331 166-4 2-1777 •4583 8-00 240-0 120-00 342 172-1 2-4562 •4071 1 With the assistance of this table, we can solve the following problems : — First Example. — What is the amount of steam pressure acting on a piston of 10 inches diameter, corresponding to a temperature of 275 degrees? It will be seen that the pressure corresponding to 275° is equal to three atmospheres, or to 45 lbs. per square inch. The area of a piston of 10 inches in diameter is equal to 102 X .7854 = 78-54 sq. inches; consequently, 78-54 X 45 = 3534-3 lbs. Thus, to solve the problem, we look in the table for the pressure corresponding to the given temperature, and multiply it by the area of the piston expressed in square inches. Second Example. — Wliat weight of steam is expended during each stroke of the piston, the length of stroke being 3 feetl We first obtain the volume expended, 78-54 12 X 3 = 19-635 cubic feet. BOOK OP INDUSTRIAL DESIGN. 59 At a pressure of three atmospheres, a cubic foot of steam weighs I'OOoS lb.; consequently, 19-635 X 1-0038 = 19-75 lbs. To sc)he this problem, then, we ascertain the volume expended in cubic feet, and multiply it by the weight corresponding to the gi\en temperature, or pressure — the product is the weight in pounds. UNITY OF HEAT. 188. With a \iew to facilitate various comparisons connected with the subject of steam, the French experimentahsts have adopted the term calorie, or unity of heat. This is defined as the amount of heat necessar}' to raise the temperature of a kUogramme (= 2-203 lbs.) of water, one degree centigrade. Thus, a kilogramme of water at 25° contains 23 unities of heat ; ■Mi'l, in the same manner, 60 kilog. of water at 30' contains 30 X 60 = 3000 unities. The number of unities of heat is obtained by multiplying its weight in kilogrammes by the temperature in degrees centigi-ade. The amount of heat developed by different descriptions of fuel varies according to their quality, and according to the construction of the furnaces. According to M. Pdelet, the mean quantity of heat developed by a kilogramme of coal is equal to 7300 calories, or unities of \ieat. According to M. Berthier, that developed by a kilogramme of wood charcoal varies from 5000 to 7000 unities. In the following table will be found the results of experiments with different descriptions of fuel : — Table of ihe Amount of Heat developed hy one Kilogramme of Fuel. Description of Fuel. Number of unities of heat developed by 1 kilo^. Quantity of steiim practically obtainable from 1 kilojf. Wood Charcoal Coke Medium Coal *^ . . . Dry Turf, Common Turf, containing 20 per cent, j of water, ] Inferior Turf^ Dry Wood of all descriptions, Common wood, containing 20 per i cent, of water, Turf Charcoal 6000 to 7000 6000 7500 4800 3000 1500 3600 2800 5800 Kilog. -6 to 6 " 8 •75 " 7 •8 " 2 3-7 2-7 1-8 to 3 In the last column of this table, we have given the quantities of steam produced by the combustion of one kilogramme of fuel, being such as are practically obtainable in apparatus most com- monly met with. Example. — What is the quantity of coal necessary for the sup- ply of a furnace intended to produce 250 kilog. of steam ? The average produce of 1 kilog. of coal being 65 kilog., wo have 250 6-5 — 84 kilog. of coal. of the shape represented in figs. 4 and 5, Plate XIV.— that is, cylindrical, and terminated by hemispheres. They are frequentlv accompanied by two or three tubular pieces in connection with the main portion of the boiler by pipes. BoOei-s answering to thLs description are termed French boilers, being of French origin ; they are found very effective, and are much used in the manufacturing districts of England. These boilers are made of plates of wrought^ iron, the thickness of which varies, not only according to the size of the boilers, but also according to the pressure at which it is intended to produce steam. The proper thickness for the plates of cylindrical boilers may be determined by the following formula, which is the one adopted by the French Government in their police regulations : — _ 18 X d X p 10 + 3; where 189. The boilers in which the .steam is to be produced, may bo T = thickness in millimetres; d = diameter of boiler in metres ; p = pressure in atmospheres, less one. The rule derivable from the formula is — ■ To multiply the eff'ective pressure of the steam in atmospheres by the diameter of the boiler, and by the constant 18, dividing the product by 10, and augmenting the quotient by 3, which will give the thickness in millira^tres. To simplify these calculations, we give a table showing the thickness proper for boiler plates, calculated up to a diameter of 2 metres, and to a pressure of 8 atmospheres above the atmo- sphere : — Table of Thicknesses of Plates in Cylindrical Boilers. Diameter of Bailer. Pressure of Steam in Atmospheres. 2. 3. 4. 6. 6. 7. 8. Metres. Millim. Millim. Millim. Millim. Millim. Ml lim. MiUim. -50 3-9 4-8 5-7 6-6 7-5 8-4 9-3 •55 4-0 5-0 6-0 7-0 7-9 8-9 99 •60 4-1 51 6-2 73 8-4 95 105 •65 4-2 53 6-3 7-7 8-8 10-0 112 •70 4-3 55 68 8^0 9-3 10-5 11-8 •75 4-3 5^7 7-0 8-4 9-7 11-1 12-4 •80 4-4 3-9 7-3 8^8 10-2 11-6 13-1 •85 4-5 6-1 7-6 91 10-6 12-2 13-7 •90 4-6 6-2 7-9 9-5 111 12-7 14-3 •95 4-7 6-4 8-1 9-8 11-5 13-3 160 1-00 4-8 6-6 8^4 10-2 12-0 13-8 15-6 MO 5-0 7-0 8-9 10-9 12-9 14-9 " 1-20 5-2 7-3 9-5 11-6 13-8 160 u 1-30 6-3 7-7 100 124 14-7 il " 1-40 5-5 8-0 10-6 131 16-6 U u 1-50 5-7 8-4 HI 13-8 i( 11 u 1-60 5-9 8^8 11-6 145 11 it u 1-70 6-1 91 12-2 15-2 *^ u It 1-80 6-2 95 13-7 160 11 u u 1-90 6-4 9-8 13-3 (i " u u 2-00 6-6 10-3 13-8 U " 11 u To suit English measures, the fomiuln is- 18_x_d_><ji 10,000 1182. 60 THE PRACTICAL DRAUGHTSMAN'S And liere, T = thickness in inches ; d = diameter in inches ; f = pressure in atmospheres, less one. HEATIN& SURFACE. 190. In priictice it is generally calculated that a square m^tre of heating surface will produce from 18 to 25 kilog. of steam per hour, whatever be the form of boiler, whether cylindrical, with or without additional tubes, or waggon-shaped. The amount of heating surface, per horse power, generally adopted, is from 1 to \b sq. m. In this surface is not only included that which is directly exposed to the action of the foe, but also that which receives heat from the smoke and gases which traverse the flues ; this last being, of course, mucli less eftective in the production of steam. According to circumstances, one-half or a third of this surface may be exposed to the du-ect action of the foe, which will give, for the whole heating surface, two-thirds of the entire smface of the boiler. In the following table we give the principal dimensions, corre- sponding to given horses power of boilers of the French descrip- tion — that is, cylindrical with two tubes or smaller cylinders below. Table of Dimensions of Boilers and Thickness of Plates for a Pressure of five Atmospheres. Horses Power. Length of Boiler. Len^h of ihe two Tubes. Diameter of Boiler. Diameter of the Tubes. Thickness of Plates for the Boiler. Thickness of I*lates for the Tubes. jr. M. M. M. "/m. "/.. 2 1-65 1-75 -66 •28 8 8 4 2-10 2-20 •70 •30 8 8 6 2-VO 2-85 •75 •35 9 10 8 3-40 3-60 •80 •35 9 10 10 4-10 4-30 -80 ■38 10 10 12 4-80 5-00 •80 •38 10 10 15 5-60 5-80 •80 •45 10 10 20 6.60 6-80 •85 •50 10 10 25 8-00 8-20 •85 •50 10 10 30 8-30 8-50 1-00 •60 10-5 10 35 9-50 9-70 1-00 •60 11 10 40 10-00 10-30 1-00 •70 11 10 In cylindrical boilers, without additional tubes, the water should occupy two-thirds of the whole space, and in boilers, vidth the tubes, it should occupy about one-half the main cylindrical body of the boiler, in addition to the tubes. In order that the steam may not carry along with it small quantities of water, which action is tei'med '^priming," the boiler is surmounted by a cylindrical chamber, or dome, in which the steam collects, and from the highest part of which it makes its exit, quite out of reach of the water thrown up by the ebullition. CALCULATION OF THE DIMENSIONS OF BOILERS. First Example. — What is the proper length for a cylindrical boiler, without additional tubes, capable of supplying an engine of 6 horses power, supposing the diameter to be '8 m., and the heat- ing surface 1-3 sq. m., per horse power? We have ] •S x 6 = 7^8 sq. m., total heating surface. Now, as the heating surface should be two-thirds of the entire surface of the boiler, it follows that 2 ^ 4rt R 7-8 sq. m. = L X 2 rt R X — = L X — - — , where L represents the length sought, and R the semi-diametei, = •4 m.; so that, substituting for rt and R their respective numerical values, we have — 4 7-8 sq. m. =:: L X 3^14 x -4 x ^ = ^ x r676 ; whence, L = _T8_ r675 X 4-65 m. As it may be well to know the capacity of the boiler for watc; and steam, this may be ascertained according to the rules pre- viously given for the contents of cylinders, spheres, &c. (121 — 124^. The boiler being terminated by hemispheres, the length of the cylindrical portion will be equal to 4-65 — 04 X 2)= 3^85. We shall have, then, for the volume corresponding to the cylindi-ical portion — V= Y X 3^14 X •i- X 3^85 — 1^29 cub. ra. and for that of the hemispherical ends — 2 4 — X — X 3^14 X •4=' = •ng cub. m. The whole volume of water is, consequently, r29 -1- -179 = 1^469 cubic metres. The remainder of the volume, which is occupied by the steam is obviously 1-469 = -734 cubic metres. and the contents of the entire boiler, 1^469 + -134: — 2-203 cubic metres. This result might have been obtained by the following general formula : — 4 L X rt R^ + -.R3= 3^85 X 3^14 X -42 -t- — X 3^14 x 2^203 cubic metres. 191. We here quote the portion of the regulations enforced by the French Government, relating to the steam-boilers, as showing what conditions are deemed necessary in France for the insurance of public safety, and also as forming a good basis for calcula- tions : — " (33.) The boUers are divided into four classes. " The capacity of the boiler, including that of the tubes, if there be any, must be expressed in cubic metres, and the maxi- mum steam pressure must be expressed in atmospheres ; and these two quantities multiplied into each other. " If the product exceeds fifteen, the boiler is of the first clasSy It is of the second class, if the product is more than seven, but does not exceed fifteen. Of the third, when more than three, and not exceeding seven. And of the fourth, when not exceeding three. " If two or more boilers are arranged to work in concert, and have any communication with each other, direct or indii-ect, the BOOK OF INDUSTRIAL DESIGN. 61 term taken to represent the capacity must be the sum of the capacities of each. " (34.) Steam-boilers of the fost class must be stationed outside of all dwelling-houses or workshops. " (35.) Nevertheless, in order that the heat, which would other- wise be dissipated by radiation, may be better economised, the officer may allow a boQer of the first class to be stationed inside a workshop, provided this does not form part of a dwelling-house. "(36.) Whenever there is less than 10 metres in distance between a boUer of the first class and a dwelling-house or public road, a wall of defence must be built, in good and solid masonry, and 1 m6tre thick. The other dimensions are specified in article 41. This wall of defence must, in all cases, be distinct from the masonry of the furnaces, and separated from it by a space of at least 50 centimeti-es in width. It must, in a like manner, be separated from the intermediate walls of the neighbouring houses. " If the boiler be sunk into the ground, in such a manner that no part of it is less than 1 metre below the level of the gi-ound, the wall of defence shall not be required, unless the boiler is within 5 metres of a dwelling-house or of the public road. " (38.) Steam-boilers of the second class may be stationed inside a workshop which does not form part of a dwelling-house, or of a factory or establishment consisting of several stories. " (39.) F a boiler of the second class be within 5 mefres from a dwelling-house or the public road, an intermediate wall of defence shall be erected, as prescribed in article 36. " (41.) The authority given by the inspecting-officer for boilers of the first and second class, shall indicate the situation of the boiler, its distance from dwelling-houses and the public roads, and shall determine, if there be space enough, the direction to be given to the axis of the boUer. " This authority shall also specify the situation and dimensions, as to length and height, of the wall of defence, where this is niquired, in conformity with the above regulations. " In determining these dimensions, regard must be had to the capacity of the boiler, to the pressure of the steam, as well as to all circumstances tending to make the boiler more or less danger- ous or inconvenient. " (42.) Steam-boilers of the thii-d class may be stationed in workshops which do not form parts of dwelling-houses, and it is not necessary to erect a wall of defence. •* (43.) Steam-boilers of the fourth class may be stationed in aoy workshop, even if this forms part of a dwelling-house. " In this case, the boilers must bo furnished with an open manometer. " (44.) The furnaces of steam-boilers of the third and fourth class shall be entirely separated, by a space of at least 50 centi- metres, from any dwelling-house." According to these regulations, a boiler of the dimen.sions taken for illustration, and supposing the maximum pressure to bo 3 atmospheres, would be in the second class, for — 2*203 x 3 = •i-609, which is below 7. Second Example. — What should be the (liiiiensions of a cyliii- Jrical boiler, with two additional tubes, intended to supplv an engine of 16 horses power, the diameter of the main body bcmg ■9 m., and that of the tubes -45 m. ? Assuming 1-2 sq. m., per horse power, for the heating surface, we shall have 1-2 x 16 = 19-2 sq. m., for the entire heating sur- face. Half of the surface of the main cylinder of the boiler, and three-fourths of that of the tubes, is the best disposition of this heatmg surface. These data give rise to the following formula : — ,.„ 2rtRxL ^ , 3 19-2 sq. m. = -fr27i;xLx2x — = 2 4 rt R L -f Srt r L — . L here represents the length of the boiler and tubes, which is the only unknown term. Substituting for R and r their numerical values, -45 and -225, we have — 19-2 sq. m. = 3-14 x -45 x L -|- 3 x 3-14 x -225 x L; or 19-2 sq. m. = L (3-14 x -45) -f 3 x 3-14 x •225 — L (1-413 -f 2-12); whence, 19-2 19-2 L = = 5-43 m. 1-413 -f- 2-12 3-533 Thus, the total length of the boiler is 5-43 m., but the ends being hemispherical, the length of the cylindrical portion is equal to — 5-43 — -9 = 4-53 m. The tubes usually project in front of the main body, to a dis- tance of about 50 centimetres; but, for convenience in constructing the return flues, they do not extend as far back, so that they aro of about the same length as the main body. 192. In distillery boilers, a horse power is understood to mean the capability of evaporating 25 kilogi-ammes of water in an hour. Thus, a boiler of 10 horses power should be capable of evaporating 250 kilog. of water in that time. Now, assuming 1-12 sq. m. of heating surface, per horse power, for a steam-engine, we should only have an evaporation of from 18 to 20 kilog. per hour, per horse power, and per square metre of heating surftice. DIMENSIONS OF FIRE-GRATE. 193. In practice, 1 square metre of grate will burn from 40 to 45 kilog. of coal per hour. Thus, a boiler intended to produce 280 kilog. of steam per hour, will require, for this purpose — assuming that 1 kilog. of coal produces 6-65 of stoam — 280 "TTTTT = 43 kilogrammes of coal ; 6-65 •= and the fui-nace of this boiler should have a grate, measurin;,' 1 square metro. The grate-bars aro generally of cast-iron, of from 30 to 35 millimetres in width, but having between lluni a siuico of only 7 or 8 millimetres, so that tho intervjJs only t>i'cnpy a fourth or a filth of tho whole area. It has been found that greater strength and dnrahility i.-» obtained by making tho bars straight above, and strengthened by ])arab(ilic featliers below. CHIMNEYS. 191. 'riu> luit^ht of i-liiuMieys is very varialile, and oannol Ih» subjected to ;tny fixed rule. 'I'lie cross section at the sunuiiit depenils upi»n the size of iJio grille luul i.s generally iduuil u sixth of 62 THE PRACTICAL DRAUGHTSMAN'S this. In the followmg application will bo found calculations respecting chimneys, and examples of the various rules we have just given. APPLICATION. We propose calculating the dimensions of the furnace of a boiler, with its chimney, for an engine of 8 horses power, for example, to be worked on the high-pressure system, consuming, as a maximum, 5 kilogrammes of coal, per horse power, per hour, the amount of heating surface being taken at 1-52 sq. m., per horse power. For 8 horses power, the heating surface will be — 1-62 X 8 = 12-16 sq. m. Each square metre of heating surface producing, at an average, 18 kilogrammes of steam, we have — 12-6 X 18 = 218-88 kilog. of steam. As 5 kilog. of steam are produced by 1 kilog. of coal, then — 218-88 „,., — - — — 43-8 kilog., representing the quantity of coal consumed per hour. The grate area, corresponding to this consumption, assuming that one square decimetre is sufficient for 1-2 kOog. per hour, will be— 43-8 1-2 = 36 square decimetres, supposing a fourth of this area to be free to the passage of air. It now only remains to calculate the cross sectional area of the chimney. With reference to this we must remark, that 18 cubic metres of air are required for the consumption of 1 kilog. of coal ; therefore, 43-8 kUog. will require 43-8 X 18 = 788-4 cubic metres. This air, in traversing the fire, relinquishes a portion of its oxygen, which is partially replaced by carbonic acid gas and steam. If the gases escape from the chimney at a mean temperature of 300° centigrade, the volume being, according to M. Peclet, at the rate of 38-54 cubic m. per kilog. of coal, will be 43-8 x 38-54 = 1 688 cubic metres per hour. If we divide this by 3600, we shall obtain the quantity which escapes per second ; namely, 1688 3600 = -4689 cubic m. K we assume, as is usually the case with boilers of the propor- tions here discussed, that the chimney is 22 metres high, the external atmosphere being at a temperature of 15° centigi'ade, the rate of exit of the gases may be obtained by the following formula : — V = V2g H a (t'—t). In the case under consideration, H =: 22 m., a is the constant multiplier, -00365, t' = 300°, t = 15°, and 2g — 19-62. Substi- tuting, then, for the letters their numerical values, we have V = i^ 19-62 X 22 X -00365 x (300 — 15) = 21. This signifies that the gas will escape from the chimney-top at the rate of 21 metres per second, if it meets with no resistance from the lateral surfaces of the flues and chimney; the actual rate, however, is only 70 per cent, of this — or, 21 X -7 = 14-7 m. If we divide the volume of gas which escapes per second by ^Jie rate at wliich it escapes in that time, as just determined, we shall obtain the cross sectional area proper for the upper part 01 the chimney ; as thus — •4689 „„ J. • .^ =3-2 square decmaetres. Thus the chimney, which is supposed to be square, will only require to measure, internally, something less than two deci- metres each way at the point of exit; this, however, is a mini- mum dimension, and it vidll be advisable to give it gi-eater dimen- sions than these. Thus it might be made 25 centimetres square, or even 30 or 35 centimetres, if there is any likelihood of the power of the boiler being increased afterwards, such mcrease being frequently called for in manufactories. A damper, however, should always be provided at the base of the chimney, by means of which the draught may be suited to the requu-ements. SAFETY VALVES. Table oj Diameters of Safety Valves. Pressu re in Atmosph sres. Extent of Henting Surface. H 2 H 3 3* 4 5 6 7 8 Sq. M. "/n. "/m. ""L "/m. «/m. "/m. "/m "/m. -"■/n,. "/m 1 25 21 18 16 15 14 12 11 10 9 2 35 29 25 23 20 19 17 15 15 13 3 48 86 31 29 26 24 21 19 17 15 4 50 41 36 32 29 27 24 22 20 19 5 56 46 40 36 33 30 27 24 22 21 6 61 60 44 39 36 34 30 27 25 23 7 66 54 48 43 39 36 32 29 27 2S. 8 70 58 51 46 42 39 34 81 29 27 9 75 62 54 48 44 41 36 33 30 28 10 79 65 67 51 47 43 38 35 32 80 11 83 68 60 54 49 45 40 36 33 31 12 87 71 62 56 51 47 42 38 85 33 13 90 74 65 58 53 49 44 40 86 34 14 98 77 67 60 55 51 45 41 37 35 15 96 80 70 62 57 53 47 42 38 36 16 100 82 72 65 59 55 48 44 40 38 17 103 85 74 67 61 56 50 45 42 39 18 106 87 76 68 63 58 51 47 43 4C 19 109 90 78 70 64 60 53 48 44 41 20 111 92 80 72 66 61 54 49 45 42 21 114 94 82 74 68 63 56 50 46 43 22 117 97 84 76 69 64 57 51 47 44 23 119 99 86 77 70 66 58 53 48 45 24 122 101 88 79 72 67 59 54 49 46 25 125 103 90 81 74 69 60 55 50 47 26 127 105 91 82 75 70 62 56 51 48 27 129 107 93 84 77 71 63 57 52 49 28 132 109 95 85 78 73 64 58 53 50 29 134 111 97 87 80 74 65 59 54 51 80 136 113 98 88 81 75 66 60 55 52 32 140 116 100 90 82 76 67 62 57 53 34 145 119 104 94 86 79 69 64 59 55 36 149 122 107 96 87 82 71 65 61 57 88 151 125 110 97 90 83 74 66 62 58 40 156 130 113 101 92 86 75 69 64 59 45 167 137 119 107 97 91 80 73 68 63 50 174 1« 125 113 104 96 84 76 70 67 55 184 151 132 119 107 101 88 80 75 70 60 193 158 137 121 118 106 94 84 78 73 195. Steam-engine boilers are always provided with various accessories, as safety valves, manometers, floats, alarm whistles. The manometer is an instrument which serves to indicate the pressure of the steam inside the boUer in atmospheres, and frac^ tions of atmospheres. These instruments are constructed aftei various systems. BOOK OF INDUSTRIAL DESIGN. 63 The float serves to indicate the level of the water, and the whistle to give the alarm when the water is much below the pro- per level. The safety valve pro\ides an exit for the steam when the pres- sure is too high. We have given a drawing of one at fig. 4, Plate XI. Their diameters vary with the dimensions of the boilers and the pres- sure of the steam. The regulations of the French Government contain the following rules and the above table for their determination. To find the proper diameter for the safety valve, the heating surface of the boUer, expressed in square metres, must be divided by the maximum pressure of steam intended to be mamtained, expressed in atmo- spheres, previously diminished by the constant -412: the square root of the quotient being extracted, is to be multiplied by 2-6, and the product will be the diameter sought, expressed in centi- metres. This rule may be put as a formula, thus : — d= 2-6 V 71 — -412 ' where d is the diameter of the \alve in centimetres, s the heating surface of the boiler, including both fire and flue surface, e.xpressed in square metres, and n the number expressing the pressure Id atmospheres. CHAPTER V. THE STUDY AND CONSTRUCTION OF TOOTHED GEAR. 196. Toothed gear is a mechanical expedient, universally employed for the transmission of motion. It is met with of all proportions, from the minute movements of the watch, to the ^gantic fittings of manufacturing workshops. Toothed gear is generally constructed with a view to the following principle of action — that the lateral acting-surfaces develop the same arc during the same duration of contact, whilst their angular velocities vary inversely as their diameters. By the angular velocity of any body, turning about a centre, is meant the angle passed through by the body in a unit of time ; whilst the real or linear velocity of any point is the space passed through by this pomt, whether the direction of motion be rectilinear or circular. Thus, various points on a crank, taken at different distances from the centre of the shaft, have all the same angular velocity, whilst their actual velocity differs considerably, because of their respective distances from the centre. It is the same with a pendulum, which vibrates through an angle, or has an angular motion about its centre of suspension. The angular velocity of a body is greater, JUS the angle passed through m the same time is greater. Two points may have the same angular velocity, although the space passed through by each may be very different. Thus, all the points in the pendulum are affected with an equal angular motion, wliilst their actual velocities, or the course traversed by each, vary as the distance from the centre of motion. This description of gear consists of a series of projections, or teeth, regularly arranged on straight, cylindrical, or conical sur- fswes, termed webs, and disposed so as to act on each other during a limited time. In order, however, that the gearing action may take jilaco in a regular, even manner, it is indispensably necessary that tiic sur- faces of the teeth should bear upon each other (aiigcntiaiiy, throughout the entire duration of their contact ; and lor this pur- pose, far from being arbitrarily designed, their I'niin should b(( determined with the utmost geometrical exactitude, lor on their form entuely depends their accurate and easy working. It is. therefore, obviously incumbent on the student to give particular attention to the delineation of these teeth. The curves generally adopted in practice for the outline of teeth, are the involute, the cycloid, and the epicycloid. It is useful to Investigate the nature and construction of tnese curves, both on account of their application to the teeth of wheels, and also because of their employment in several other mecnanical contrivances. INVOLUTE, CYCLOID, AND EPICYCLOID. PLATES XVIII. AND XIX. INVOLUTi:. Figure 1.— Plate XVIII. 197. \Vlien a thread is unwound from the circumference of a circle, and is kept uniformly extended, its extremity will describe the curve known as the involute. This definition serves as a basis for obtaining the geometrical delineation of the involute. Let a b c be the given ciri-le of the radius, a o, and a the extremity of a tlire.id wound upon it. Starting from the point, a, mark off, at equal distances apart, several points, as a, h c, so near to each other, that the interven- ing arcs may be taken for straight lines without sensible error. Through each of these points draw tangents to the cin'le, or jH>r- pendiculars to the corresponding radii ; and ou these t.angents sot ofi' distances, equal to the rectifications of tlio respective arcs, a a, A b, A c, &c. ; by which means are obtained the points, a', b\ r', &o., and the curve passing through these points is a portion at' the tnolulc. Hy continuing Uio development or unwinding of the hrcad, the curve may bo extended to a series of convolutions, lUTcasing more and more in radius, and becoming a spwies of spiijil. .M'ler one complete evolution of the cin-uniference, the shortest distance between two oonsivutivo convolutions is always the sjune, and eipinl to Uie development or rectification of tJio 64 THE PRACTICAL DRAUGHTSMAN'S i-ia-umference of the generating circle, which forms the nucleus of tlie curve. The points, a, i, c, being taken at equal distances apart on the circumference, the tangents are respectively double, triple, >&c., that of tlie first, a a ; and if, as we directed, these points are suffi- ciently near to each other, the curve may be dra\vn, with closely approximate accuracy, by describing a succession of arcs, having these tangents for radii. Thus, witli the point, a, as centre, and radius, a a', the first arc, a a', is di-awn ; and ^vlth the centre, h, and radius, b b', the second arc, a' b', in like manner; and similarly with the rest. We shall show the application of the involute to toothed gear, worm wheels, and also for cams and eccentrics. FiGUKK 2.— Plate XVIIL 198. When a circular disc is rolled upon a plane surface in a rectilinear direction, any point in the circumference of this disc generates the curve called the cycloid. Thus, any point taken on the outside of a locomotive wheel in motion, describes as many repetitions of the curve as the wheel makes revolutions. In order that the curve may be perfect and true thi'oughout, it is necessary that the motion should take place without any slid- ing upon the plane ; in other words, the length of the straight line forming the path of the disc should be equal to the portion of the circumference which, during the motion, has been applied to, or in contact TOth, the plane throughout that length. We propose to delineate the cycloid generated by the point, a, of a circle of the given radius, a o, and rolling upon a given straight line, b c. There are several methods of sohing this problem. Isi Solution. — Set off on the circumference, starting from the point, A, a number of distances equal to a a, so small that the arcs so divided may be taken as straight lines. Set off the same dis- tance a like number of times along the straight line, a c, and at the points, a b c, erect perpendiculars, cutting the line, o o', gene- rated by the centre of the rolling circle, and parallel to the given straight line, b c. Li this way are obtained the points of inter- section, o, o^, 0-, which are the centres of the circle when in the positions corresponding to the points of contact, a, b, c, d. With each of these points as centres, then describe portions of circles, on each of which successively set off the lengths of the arcs, a a', A b', A d, &.C., from a to a", 6 to a^, and from c to b^, and so on throughout. The curve, a a" a- b^ c-, passing through the points thus obtained, is the c)-cloid required. 2d Soluiion. — The points in this curve may also lie obtained by drawing horizontal lines through the points of division, a' b' c', of the original circle, and then intersecting these by the arcs drawTi with the respective centres, o, o*, o-. 3d Soluiion. — In place of dravring arcs of circles -nith the various centres, as indicated on the right-hand side of fig. 2, the curve may be obtained by setting off successively from the vertical, a o, on the horizontals, as before drawn, distances equal to those respectively contained between the original circle and the perpendi- culars thi-ough the several corresponding positions of the centre ; thus, the distances, e e'.,ff,gg', h h', &c., are set off from 1 to a- 2 to b", 3 to C-, &c. To avoid confusion, we have constructed the diagram appertain ing to this last solution to the left-hand side of fig. 2, which showa a portion of a second cycloid similar to the first. When the generating circle has made half a revolution, the sum- mit of the curve is obtained, as at d', the point corresponding to the diameter, a d. The length, a c, of the given straight line, is obviously equal to the rectification of the semi-circumference of the generating circle, whose radius is a o. By continuing the construction, a complete curve may be obtained, having equal and s3"mmetrical portions on either side of the vertical, c d, and havmg for its base a line double the lengtli of A c, and cons3quently equal to the rectification of the entire circumference of the generating circle. The cycloid is the curs'e more generally given to the teeth of wheel gear and endless screws. EXTERNAL EPICYCLOID. FiGUSE 1. — Plate XIX. 199. The epicycloid only differs from the cycloid, in that the generating circle, instead of rolling along in a straight line, does so ai-ound a second cu-cle, which is fixed. When the two circles are in the same plane, the point taken generates a right or cylindri- cal epicycloid ; when the two circles are situate in different planes, but maintaining a uniform ang-le to each other, the generated curve becomes a spheric epicycloid; in this case the generating circle is supposed to revolve about a fixed centre, at the same time rolling along the circumference of the stationary circle. 1st Soluiion. — For the delineation of the right epicycloid, the methods of construction to be adopted are analogous to that ^ven for the cycloid. Thus, let a o be the radius of the generating circle, and a c the radius of the fixed circle ; divide the former into a number of equal parts in the points, a', b', c', d', &c., and on the latter divide off as many arcs equal to the arcs of the former, start- ing from A, as at a, b, c, d, &c. Through these latter points of divi- sion draw radii, ca,cb,cc, and prolong them so as to cut a circle, the radius of which is c o ; this circle being generated by the centre of the moving one during its rotation about the stationary one ; in this way are obtained the points, o, o', o-, o^, which are the succes- sive positions of the centre of the generating circle, as during its rotation it is successively in contact at the points, a, b, c, d, of the fixed circle. Then, with these points as centres, describe the several arcs of equal radii \\'ith the generating ch-cle, making them severally equal to the corresponding arcs, a a', a 6', a c', as from a to a-, b to b-, cto c-, &c. The curve passing through the points, a-, b~, C-, is the epicycloid requu-ed. 2d Solution. — The points of this curve may also be determined by drawing, with the centre, c, arcs passing through the points of division, a', b', c', d', and cutting the arcs described with the various centres, o, oS o-, o^, in a-, b^, c", d-, which are so many points in the epicycloid. 3d Soluiion. — The curve may also be delineated by transferring the distance between the points, e, f, g, &c., of the generating cucle in its original position, and the radii, c ^ c i, c k, passing BOOK OF INDUSTRIAL DESIGN. 66 through the different points of contact on the stationary circle, measured upon the arcs described with the centre, c, to the same arcs, but so that the extremities of the whole may lie in the pro- longation of the radius, c b. Thus the distances, e e',ff, gg\ &c., are set off, from 1 to a-, 2 to J -, 3 to c-, &c. The diagram refer- ring to this construction forms the right-hand portion of fig. 1. When the generating circle has made an entire revolution, the em've obtained is an entire epicycloid, a d b, comprising two equal md symmetrical portions on either side of the line, d e, which is »qual to the diameter of the moving circle. EXTERNAL EPICYCLOID DESCRIBED BY A CIRCLE ROLLING ABOUT A FIXED CIRCLE INSIDE IT. Figure 3.— Plate XIX. 200. For this diagram, which is analogous to the preceding one, the radii of the circles are given, c a being that of the fixed circle, and B A that of the moving one. Divide the first circle into any number of equal parts, in the points, a, i, c, d, &c., and divide off, on the larger circle of the radius, b a, a like number of arcs, equal to those on the other circle, as from a to a', a' to b, &c. Then with the point c as centre, and with the radius b c, describe a circle, cutting the radii, c a, c a, c ft, c c, in the points, b, b', b', b^, and with each of the last as centres, and with the radius, a b, describe arcs, which will be tangents to the tixed circle, at the different points of contact, a, b, c, in succession. Then, with the centre, c, describe arcs, passing successively through the points, «', b', c', d', on the moving circles, as in its first position. These Vist vvill cut the arcs tangential to the given circle, in the points, t^, 6', c^, d^, and the curve passing through these points is the ipicycloid sought. The other two methods given, of drawing the common epicy- ' )■ J, are also applicable to this last case. INTERNAL EPICYCLOID. Figure 2. — Plate XIX. 201. The epicycloid is termed internal, when the generating circle rolls along the concave side of the cu-cumference of a fixed circle. Let c A be the radius of the fixed circle, and b a that of the generating circle. As in preceding cases, so also here, wo commence by dividing the moving circle into a certiiin number of equal parts, and then dividing the fixed circle correspondingly, so that the arcs thus obtained in each may be equal. We then pro- ceed as in the case of the external epicycloid, according to which- ever of the three solutions we propose adopting, all being alike applicable. The operations are fully indicated on fig. 2, and the same distinguishing letters are employed as in fig. 1. When the generating circle is equal to half of the fixed circle, the epicycloid generated by a point in the circunif'eronco is a straight line, equal to the diameter of the fixed circle. Thus, in fig. 3, Plato XVIII., the epicycloid generated by (ho point, a, of the moving circle of the radius, A c, after a semi-revolution, coin- i'i(l(^H exactly willi the diameter, a b. II', wilh circles of the same i)roi)or1i(ins iis those in fig. 3, Plate XVIII., we take a point, d, outside tlu^ generating circle, Inii preserving a constant distjinco from it, the epicycloid generated by it will be the ellipse, d f e g, having for its transverse axiS the line, D E, equal to the diameter, a b, of the fixed cu-cle, augmented by twice the distance, d a, of the point, d, from its extremity ; and for conjugate axis, the line, g f, equal to twice the same distance, D A, alone. If it is wished to determine this curve according to its properties as an epicycloid, and without having recourse to the methods given in reference to Plate V., and proper to the ellipse, it may be done by adding the distance, a d, to that of the radius, c A, in each successive position occupied by the generating circle during its rotation. If the generating point be taken mside the moving circle, the curve produced wUl also be an ellipse. The epicycloid is the curve most employed for the form of the teeth, whether of external or internal spur or be\'il wheels. Toothed gearing may be divided generally into two categories ; namely, right, cylindrical, or " spur " wheels, and conical, angular, or " bevil " wheels. In the first are comprehended the action of a rack and pinion, that of a worm or tangent-screw with a worm- wheel, and finally, that of two wheels. We may remark, that in all these modes the teeth are so formed and arranged, as to act equally well whichever of each couple be the driver, and in which- ever dii-ection the motion takes place. THE DELINEATION OF A RACK AND PINION IN GEAR. Figure 4.— Plate XVIII. 202. A rack is a species of straight and rigid rod or bar, formed with teeth on one side, so as to take into or gear with the teetn of a right wheel, generally of small diameter, and in such case tenned a pinion. Such a rack is represented at a b in the figure. In proceeding to construct this design, as well as for all kmds of toothed gear, it is necessary to have determined beforehand the thickness, a b, of the teeth, as this dimension varies accord- ing to the power or strain to be transmitted; and rules ;md tabies, for this purpose, will be found at the end of the chapter. When the rack and pinion are made of tlie same met;xl, tiie thickness of the teeth should be the same in both. The spaces or intervals between the teeth ought also to be equal in such case. Theoretically speaking, the intervals should be equal to the thick- ness of the teeth ; but in practice, they arc made a little mder, to admit of freer action. 203. The pitch of the teeth comprises the width of tin liuitli and that of the interval. In a wheel this pitch is measured upon a circle of a given radius, termed the pi'imilire or pitch circle, and in the rack on a straight line tangent to the pitch I'ircle of the pinion, and also called the priiniliie or pitch line. 204. Let o c be the radius of the pitch circle of a pinion L>v;ir- ing with a rack, of whicii the jtitch line is a b. Wo i>roposo, ni tho first place, to detei-inino the curve of the teeth of the pinion, so as to gear with and drive the rack, and we shall subsequently deterniino the curve of the teeth of the r;u'k, enabling it to gear with and drive tlu^ pinion. 'J'lie (i|)eiati()ns consist in rolling the straight line, A c, uiugon- tially to the i>itch circle, o c : during this movement, tiio ponu, c. will "■enerato an involute, c n, whicii may be drawn in liio niannor' inilicaled in lig. 1 — a constrnctioii wliifli is t'uitlier tvpeatcd ut ii' (/'. (Ml one of the teeth of the pinion, lii,'. 4. This cuiN e possesses this property, that if tiio toetli are t'ornu-d t 66 THE PRACTICAL DRAUGHTSMAN'S to it, and the pinion be turned on its axis, the point of contact, c, uill always be in the straiglit line, a, b, traversing this line at pre- cisely the same velocity as the pinion at that distance from the ccnti-e, that is, at the pitch circle; consequently, we divide this pitch circle into as many equal parts as there are to be teeth and intervals in the pinion, and at each of the points of division repeat the involute curve, c d, which will, of course, fulfil the same con- ditions at the various positions ; then, each of these divisions recti- fied is set off on the pitch line, a b, of the rack, as many times as is necessary. For each tooth the curves are placed symmetrically with reference to the radius which passes through their centres, as indicated at o df, so that the pinion may act equally well when turning in one direction as in tlie other. 205. Smce the teeth cannot have an indefinite length, they may be limited as far as is compatible with the following considera- tions: — The tooth of the wheel, which is the di-iver, should not relinquish contact with the one upon which it acts, until the tooth immediately succeeding it has taken up its original position, which, in the working of two wheels, corresponds to the line joining the centres, and in that of a pinion and rack, to the radius, o c, per- pendicular to the pitch line, a b. Thus, supposing the pinion to move in the direction indicated by the arrow, the tooth, e, which is acting on the tooth, h, of the rack, should continue to impel it until the following tooth, g, shall have taken its place, when it will itself have taken the place of the tooth, F, having made the tooth, h, of the rack traverse to i. It will be observed that the curved part of the tooth is in contact at the point, c, on the pitch line, a b ; the tooth might be cut away at this point; but in practice, in order that the pinion teeth may act thi-ough a somewhat greater interval, and to avoid the play results ing from wear, they are truncated at a little beyond this point, c, a circle being described with the centre, o, cutting the cm-ves of all the teeth at equal distances from the centre. To allow of the passage of the curved portion of the teeth of the pinion, the rack must be grooved out, so as to present bearing surfaces, which are determined simply by the perpendiculars, bf, c d,g b, to the pitch line, a b, and passing through the points of division already set out on this line. These perpendiculars, at the same time, form the sides or flanks of the rack teeth. Rigorously speaking, the depth of the intervals on the rack should be limited by the straight line, m n, tangential fo the exter- nal circle of the pinion ; but, to prevent the friction of the teeth against the bottom, it is preferable to augment the depth of the tioUows by a small quantity, joining the sides of the teeth ^vith the bottom by small quadi-ants, which, avoiding sharp angles, £rives greater strength to the teeth. 206. As in practice, toothed gear is constructed so as to drive, or be driven, indifferently, we require yet — to complete the design under consideration — to give to the teeth of the rack such cxu-va- ture as is necessary to enable them to drive the pinion with which they are m gear in their turn, always fulfilling the conditions of a -egular and uniform motion, both of the rack at its pitch Hue, and if the pinion at its pitch circle. With a view to the determination of this curve, we may remark, w at if. with the radius, o c, as a diameter, we describe the circle, o L c, and cause it to roll along the straight line, a b, the point of contact, c, will generate a cycloid, c k, which may be constructed according to the methods indicated in fig. 2. If the same circle is made to roll along the mterior of the pitch circle, g c j, of the pinion, the same point, c, will generate a right epicycloid, coinciding with the radius, o c, as has been seen in reference to fig. 3. Then, if we give to the teeth of the rack the curve, c k, and to the flanks of the pinion teeth the straight line, c o, the arrange- ment will exactly fulfil the condition sought ; that is to say, that, in impelling the pinion teeth from right to left, the curve, c k, of the rack teeth will constantly apply itself to the straight Ime, o c, being always tangential to it. For example, suppose the curve, c k, to be traversed to the position, c' l, the radius, o c, will then be in the position, o l ; then, if from the point, l, the straight line, l c, be drawn, the angle, o l c, will be a right angle ; that is to say, the line, o l, will be perpendicular to l c, and, consequently, tangential to the curve, L c', in the point, l. If, therefore, the motion of the rack is regular and uniform, that of the pinion will be equally so. The same curve, c k, is drawn at each of the points of division of the pitch line of the rack, as was already done for the teeth of the j pinion. To find the proper length to give to the teeth, all that is neces- sary is to place, in the generating circle, o l c, a chord, l c, equal to twice c &-, and through the point, l, thereby obtained, to draw a straight line, m n, parallel to a b. If, through all the points of division in the pitch circle of the pinion, are drawn radii con- verging in the centre, o, they will give the flanks of the teeth, as i J, k I, &c., which are limited by a cu'cle described with the centre, o, and tangential to the straight line, m n ; for the same reason as that assigned in the case of the rack, however, the spaces between the teeth are made a little deeper, and the sides of the teeth are joined to the bottoms by quarter circles, the circle in which the bottoms lie being described with a radius somewhat less than that of the circle last drawn. As it would be a tedious process to repeat the operations for determining the curves in the case of each individual tooth, it is a convenient plan to cut a piece of card or thin wood to the curve, so as to form a pattern or template, by the application of which to each of the points of division, the sides of the teeth may be drawn, care being taken to make the two sides of each perfectly symme- trica] with reference to the centre line of the tooth. Even the labour of making a template or pattern is often dis- pensed with, and, in place of the ciirve, a simple circular arc is employed for the side of the tooth, the arc being of such a radius as to approximate as near the true curve as possible. With this view the arc should be timgential to the side of the tooth, and passing through the external comer. Thus, supposing it is wished to substitute an arc for the true curve of the rack teeth, such as o r of the tooth, p, since this arc has to pass through the point, r, corresponding to l, and obtained by making r' r equal to L q, and to be a tangent at o, to the vertical, o p, draw the chord, o r, and bisect it by the perpendicular, s t, and its point of inter- section, s, with the pitch line, a b, wOl be the centre of the required arc, and the sides of all the teeth may afterwards be drawn BOOK OF INDUSTRIAL DESIGN. 67 VFith the same raoius, care being taken to Keep the centres in the line, A B. An analogous operation will give the proportions of the arc, suhstituting the curve of the pinion teeth. THE GEARING OF A WORM WITH A WORM-WHEEL. Figures 5 and 6. — Plate XVIII. 207. This system of gear is constructed on the same principles as that of a rack and pinion, which method requires that, in the first place, the worm and worm-wheel be supposed to be sectioned by a plane passing through the axis of the former, and at right angles to that of the latter. The representation of this section becomes analogous to the diagram, fig. 4 ; that is to say, the pitch circle, g c j, of the worm-wheel being given, and also the straight pilch line, a b, of the worm tangential to this circle, and parallel to the axis of the worm, the involute curve, c d, is sought for the teeth of the wheel, and the cycloid, c k, for those of the worm. The lengths of these curves are limited, as in the preceding e.\am- ple, and when the whole is complete, an outline will be produced similar to the tinted portions of fig. 6. It is in this manner that the gearing of the worm and worm-wheel is made to depend upon the same principles as that of a rack and pinion, and the same method may be employed in construction in determining the outline of the teeth, as we h.ave shown. To represent the worm and worm-wheel geometrically in exter- nal elevation, instead of a section of the teeth alone, it is necessary to know the diameter and pitch of the worm on the one baud, and the thickness of the worm-wheel, fig. 5, on the other. Let m' a' be the distance of the pitch line, a b, from the axis, m' n, of the worm, and a b the width of the wheel. When the worm is single-threaded (177), the pitch of the helix is the same as that of the teeth, and, therefore, the thickness of a tooth, added to the width of an interval. In this case, each revolution of the worm turns the wheel to the extent of one tooth, and this is the arrangement represented in the figures. If the worm, however, is double or triple-threaded, its helical pitch will be correspond- ingly two or three times the pitch of the teeth; and in such case, each revolution will turn the wheel to the extent of two or three teeth. The worm-wheel being of a certain thickness, and requiring to gear with the convolutions of the worm, must necessaiily have its teeth inclined to correspond with the obliquity of the worm-thread. It is further to be observed, that the sides of the wheel-teelh being simply tangential to the worm-thread, contact cannot, rigorously speaking, take place in more than one point of each tooth and convolution. This point constantly changes wilii liic motion, but always lies in the plane, o' m', of the section. In delineating the convolutions of the worm-thread, helices have to be drawn passing tiu'ough the external corners, d, e, and internal corners, /, g-. Wo have repeated these points to the left- hand side of fig. 6, where the required ojjeralions are fully indi- eated, in connection with tho i)r()jeclion, fig, 0, and in accordMiicu vvilh Iho principles already explained (173). The corresponding points in tiio two figs. (5 and (i) are distinguishod by llu' s.-iiuc k'tters and numbers. 208. For the representation, in external elevation, of the teeth of the worm-wheel, it is required to develop a portion of the cylindrical surface generated by the revolution of the pitch-line, A B, about the axis of the worm, and containing the portion, A ik I m, tor example, of the helix, described by the central point of contact, A. To obtain this, make the line, e' a', fig. 7, equal to the semi-circumference, a' m e^, rectified. At the point, e', erect the perpendicular, c' e', and make it equal to c e, fig. 6, or half the pitch, and join e' a', \vhereby will be obtained the actual inclination of the worm-thread. On each side of the point, m, on e' a', mark distances, m a' and m b', equal to m' a and m' b, fig. 5, and through these points draw parallels to c' e', and tLe poi-tion, ■p q, of the enclosed line compiised within them, wiU serve to deter- mine the width and inclination of the teeth of the worm-wheel. Through the points, p, r, draw p I and r s parallel to e' a', and mark oflT the distances, t s and s q, which are equal, on the pitch circle of the wheel, fig. 6, from s to i and q, after having drawn through the points, s, but only m faint pencil or dotted lines, the contours of the teeth as sectioned at f and g'. It is then sufficient to repeat these outlines through the points, i and q, limiting their length by the same internal and external circles. Finally, the edge view of the worm-wheel, fig. 5, being the lateral projection of the teeth, is determined by squaiing across the points, w, r, a;, to m', 1)', a;', which give the interioi-s of the teeth ; and the points, u-, v^, x", being squared over to w^, v^, x^, give their exterior edges. Worm-wheels are sometimes constructed with the form of the teeth concave, and concentric with the axis of the worm, with the view of their being in contact with the convolutions of the worm- thread throughout a certain extent, in place of only touching at single points. This arrangement, which requires a particular operation for its construction, is generally adopted when great precision is required, and when it is wished to avoid, as much as possible, any play between the teeth and the woi'ui-lhread during the transmission of motion. CYUNDRICAL OR SPUR (iEARING. PLATE XIX. THE EXTERNAL DELINEATION OF TWO SFUR-WHEELS IN GEAli. FlUUUE 4. 209. Spur-toothed wheels are sucii as have fheir teetli parallel, and lying upon a cyliiuirical surface or web. When a couple (if such wheels are of unc(|ual size, the smaller one is generally called a pinion, and the larger one a spur-wheel. Two wiieols, which are intended to gear together, cannot work satislactorily in concert, unless their radii or pitcli circles are e.xai-tly proportional to the number of teeth contained by each. Consequently, in onlir to coMstiiict ilesigns for couples of tootliod wheels, it In necessary to know — the niunher of teeth of each, and the radius of one or other of tluiu ; or the radii or diameters of both, and tl»o uuuiher of teeth iil'oni'; or the distanct> between their COntres, and the radius or lUMulur ol' teeth i>i' olio; or linally. the nuinbor of revolutions of catli in the .>iunu> lime, and liie distance beiwoon 68 THE PRACTICAL DRAUGHTSMAN'S their centres, or the radius and number of teeth of one of them. In the rules and data at the end of this chapter, mil be found the solution of the several problems involved in these various cases. If we assume the followng data, A B = 240, and B C = 400, these being the respective radii of the pitch cu-cles of two right wheels, and n = 24, the number of teeth of the pinion— we at once ascertiiin the number of teeth, N, of the spur-wheel, by the following proportional formula : — A B : B C :: n : N, or 240 : 400 :: 24 : N = 40. Then describe the pitch circles of the radii, a b and b c, and divide tliem respectively into 24 and 40 equal parts, thereby ibtainiug the pitch, or the central point of each tooth, wliich is exactly the same on both pitch circles. Next subdivide the pitch into four equal parts, to obtain the centres of the intervals, and, at the same time, the points through which the flanks of the teeth pass. K, with the line, a b, on the line of the centres, a c, as a diameter, we describe a circle, the centre of which is at o, and suppose this circle to roll round the pitch circle, d b e, of the spur-wheel, the point b, at present in contact, \v\l\ generate an epicycloid, b f, as shown previously in reference to tig. 1 ; and this curve is the one proper to give to the side of the teeth of the spur-wheel, and it is accordingly repeated symmetrically on each side of the several teeth, as shown in the diagram. If, fm'ther, we suppose the same circle of the radius, o b, to roll round the interior of the pitch ciicle, g b h, of the pinion, we shall obtain the internal epicycloid (sometimes called hypocycloid), b o, as already explained in reference to fig. 3, Plate XVIU., and a por- tion, B a, of this, forms the flank of the pinion tooth. Supposing the curve, b c, to form a part of the wheel, turning about the centre, c, in the direction of the arrow, i, it will fulfil the condition of impelling the flank, b a, which forms part of the pinion, so as to turn about the centre, a, in the like uniformity. In other words, the space passed through by the point, b, on the pitch circle, g b h, shall be exactly the same as that passed through oy the same point, b, considered as belonging to the spur-wheel, on the pitch circle, e b d. 210. In proceeding to the determination of the length to give to the tooth, it is first to be observed that the epicycloidal curve should be sufficiently long to bear upon the side of the tooth, through an extent of circumferential movement equal to the leno-th of the pitch from the line of centres ; that is to say, until the flank, at present in the position, e a, shall have arrived to the position, c. d. At this moment, it will be observed that the curve, b f, has reached the position, hf, and is in contact with the flank of the pinion tooth in the point,/, on the circumference of the generatino- circle of the radius, a o. It will thus be ob\ious that the pomt,/, may be obtained by simply cutting off, on the generating circle, an arc, b/, equal to the length of the pitch. Through this point,/, describe a circle having c for its centre, and it will cut all the teeth at the proper length. The depth of the intervals is theoretically determmed by describing, with the centre, a, a circle tangential to the fij-st; but in practice, as it is necessary to leave a slight space between the ends of the teetli and the bottoms of the intervals into which these work, the circle in question is described witi; a somewhat smaller radius, a& A a. 211. Hence it is manifest, on the supposition that the spur-wheel is intended always to be the driver, without being diiven at any time by the pinion, the teeth of the spur-wheel would only require to be of the form indicated at J, and those of the pinion, like the portion of a tooth, K, slightly tinted for the sake of distinction ; but generally, and for obvious reasons, all spur gear is so constructed as to act reciprocally, and equally well, whichever be the driver, and we must, therefore, shape the teeth of the pinion, so that it may, in turn, perform tliat function. With this view, describe a circle with the centre, o', of the radius. B c, taken as a diameter ; and suppose tliis ckcle to roll round the pitch circle, h b g, of the pinion, the point, b, at present in con- tact, will generate the epicycloid, b l, which is the proper curve to be given to the teeth of the pinion. The same point, b, con- sidered as on the spur-wheel, will, as we have seen, generate a straight line, b' o', when rolling in the same manner round the I interior of the circle, e b d, and this line forms the flank of the tooth of the spur-wheel. The operation proceeds in the same manner as for the pinion, the length of the teeth of which is deter- mined by making the arc, b/', equal to the length of the pitch, and describing, with the centre, a, a cii'cle passing through the point,/'. The depth of the intervals of the spur-wheel is, in like manner, limited by a circle described with the centre, c, and radius, c g, which is somewhat short of being a tangent to the external circle of the pinion, so as to allow a little play to the teeth in their passage, as already explained. In this manner are obtained tlie complete forms of the teeth, which are regular, symmetrical, and similar to each other, and satisfy the conditions of reciprocal gearing. in the graphic operations here discussed, we have supposed the intervals between the teetli to be exactly equal in width to the teetli themselves; but as, in practice, it is necessary to allow of some play between the teeth, in order that they may work into each other with facility, this object is attained by reducing the thickness of the teeth a little ; and in the drawing, when the scale is not very large, it will be sufficient to delineate the ink lines just within the thickness of the pencil lines. Where it is wished to be more precise, this allowance may be calculated at about Jjth or Jjth of the pitch. To give strength to the teeth, the interior angles of the intervals aie roimded, as shown at each tooth in fig. 4. WTien the pinion is but of small diameter, the web, m, which carries the teeth, is c;ist soUd with the boss, the interval being filled up with a disc ; but when the wheel is larger, as in the case of the spur-wheel, the web, Ji', is attached to the boss, p', by arms, Q, which are strengthened by feathers, rounded in at the angles, as represented in fig. 4. delineation of a couple of "WHEELS GEARING INTERNALLY. Figure 5. — Plate XIX. 212. The principles observed in determining the relative num- bers of the teeth, with reference to the example just discussed, apply in like manner to the case before us ; that is, such numbers must be in the exact ratios of the diameters of the pitch circles. The curvature of the teeth is also determinable by means of the BOOK OF INDUSTRIAL DESIGN. 69 same operations, modified to suit the diiferent positions of the parts with respect to each other. Thus the curve, b l, of the pinion tooth, is generated by the rolling round the pitch circle, & B H, of the circle described with the centre, o, and radius, o b, equal to the half of b c, the radius of the pitch circle, d b e, of the larger wheel. This is an application of the operations explained in reference to fig. 3. The flanks, b a, or the sides of the teeth, are obtained by simply drawing radii, or lines converging in the point, c. In the same manner, the curve, b f, of the teeth of the large wheel, is generated by rolling along the interior of its pitch circle, B p E, a circle described from the centre, o', and radius, b o', equal to half the radius, b a, of the pitch circle, g b h, of the pinion. These curves being obtained, the outlines of the teeth are com- pleted in the manner explained in reference to fig. 4. It may, however, be observed that, in the diagram, fig. 5, though the teeth might be cut otf by a circle passing through the point, /, and described with the centre, a, they are prolonged beyond that, so that the teeth remain longer in contact, and a greater number of teeth are, consequently, engaged at one time, allowing the strain to be distributed over a greater number of points. It is the fact of the curvatures of the two lines of teeth being in the same direc- tion, which admits of a greater number of teeth being engaged at once, without that increase of friction, and other disadvantages, which would result from such an arrangement with wheels like fig. 4. THE PRACTICAL DELINEATION OF A COUPLE OF SPUR-WHEELS. Plate XX. 213. In the cases treated of in the preceding sections, which comprehend the general principles involved in rack and wheel gearing, we have assumed that the rack and pinion, or pinion and spur-wheel, are constructed of the same material, and in this case tlie thickness of the teeth is the same in any two working together. It very often happens, however, in actual construction, that one of the two has wooden, and the other cast-iron teeth, or of other dissimilar material. When this is the case, the thickness of the one description must necessarily be greater than that of the other, to compensate for the difference in the strength of the materials. The pitch, however, will still be the same for both wheels ; for, since the intei-vals on one wheel correspond to the teeth on the other, a tooth and an interval on one must obviously be equal to an interval and a tooth on the other. A couple of wheels of this description are represented in plan and elevation, in figs. 1 and 2. We here assume the wheels to be in the ratio to one another of 3:4; whence, giving the pinion 36 teeth, the spur-wheel must have 48. After dividing the pitch circle of the spur-wheel, drawn with the radius, c b, into 96 equal parts, \\w points of division representing the centres of the teeth and of (he intervals, and the |)ilcli circle of the pinion drawn with the radius, A b, likewise, into 72 equal parts — with the centres, o and o', doscribo the circles wliicn generate the epicycloidal curves, b f and b l. Take \\ of the pitch, b c, for the thickness of the wooden tooth, d e, and >,", for that of the cast-iron tooth, allowing the remaining ..', for (lie play in working. Next draw a .series of radii, to indicate the flanks of the teeth, both of the pinion and spur-wheel, and at the point of their junction with the pitch circle, di-aw the curved por- tion of each, with the aid of a small pattern or template, cut to the curves, b l and b f ; and, finally, limit the lengths of the teeth and the depths of the hollows in the manner already pointed out, in reference to Plate XIX. As draughtsmen are generally satisfied with representing the epicycloidal curves by arcs of circles which almost coincide with them, and nearly fulfil the same conditions, such arcs must be tangential to the radial sides of the teeth at their points of inter- section with the pitch circle. They are determined in the follow- ing manner: — Let fig. 10 represent one of the pinion teeth, drawn to a larger scale. Through the point of contact, b, draw a taa- gent, B o, to the pitch circle ; then bisect the chord, b n, whieh passes through the extremities of the curve, by a perpendicular, which wUl cut the tangent, b o, in the point, o. This is the centre of the arc, b m n, which very nearly coinc.des with the epicycloidal curve. The same arc is repeated for each side of all the teeth of the pinion, the radius, b o, being preserved throughout. .\n analogous operation determines the radius of the arc to be substi- tuted for the curve in the teeth of the spur-wheel. It is generally advisable to make wooden teeth about three- fourths as long as the pitch, and cast-iron teeth about two-thirds as long. In no case, however, should the lengths of the teeth in the two wheels geared together be less than those obtained by calculation, and determined by the points, /, f, situated on the circles described with the centres, o, o', by which the epicycloids are generated. The ratio of the curved external portion, n vi, «»f the tooth to the flank, n ^, is 4 : 5. In other words, tlie wluilo height or length of the tooth being divided into 9 equal parts, 4 of these are to be taken for the length of the curved portion, :md 5 for the rectilinear flanks. When the teeth are of cast-iron, the thickness, p q, of the web should be equal to the tliickness, r s, of the tooth. Sometimes it is made only Jths of this ; but in that case it is strengthened by a feather on the interior. For wooden-toothed wheels, since it is necessary that the tenon, t, of the tooth be firmly secured, the web is made of a thickness, p q, often double that of the tooth. The tenons of the teetli must be adjusted very carefully and accurately in the web. They are made with a slight taper, and are secured on the interior of the web either by iron pegs, as at u, passing through them, or by a series of wooden keys or wedges, v, driven in between them, and forming strong dove-tiiil joints. Those two methods of fixing the teeth are shown at diflerent parts on fig. 1, and moie in det;ul in fig. 7. There is a third modification, which also possesses some advantages. We have represented it at t, tig. 3, whence it will be seen that it con.sists in forming the teeth wiih a couple of •shoulders, z, which allow of the tenons, /, being niaiie nuicli stronger, and also tjiko away llurcby some of the weight of nutai, two ol)j(!cts of great imporlance. 'J'he width, .i' y, of (ho (ee(li is equal (o two or three limes (heir pilch. In \\liriU enlii<'l\ of I'Mst-iroii, the web is of the sjunc width as Ihe tci lb; bui ii is nuich broader wiien (lie teeth jl-o ot wood, for it requires to be morti.sod, to rocoivo tlio tenons of liui (I'tlli, and sluiuld have n width equ;d to that t.A' tiu> leetli, jilus ;iii amount equal to once and n half or twice their '.iiickMes,s. Wu 70 THE PRACTICAL DRAUGHTSMAN'S have already mentioiiecl, that in wheels of moderate size, the web, M', is attached to the boss, p', by arms, q. The number of these arms varies, 4, 6, or 8 being used according to the diameter. In the present case the wheels have sLx arms ; this number, amongst otlier reasons, being more particularly convenient, because the number of teeth are divisible by 6. Wlience it follows, that the feathers wiiich strengthen the arms on either side of the wheel, can be made to lie between two of the teeth, at each of the six points of attachment to the web. The feathers are joined to the body of each arm by cavetto or concave quarter-round mouldings, with or without fillets, as indi- cated in figs. 5 and 6, which reoresent sections of the arms taken through 1—2, 1—2, in fig. 1 At other times the feathers are united to the body of the arm by plain chamfer portions, as shown in fig. 8; or, even more sin^ply still, and without filling up the angle formed, as in fig. 9, ihe feathers being united, as it were, to the body of the arm without any additional moulding. In all cases, however, these feathers are made wth a taper, being thicker at their point of union with the bod}-, and gradually decreasing in thickness outwardly. Figs. 3 and 4 represent cross sections of the wheels, taken through the irregular line, 3 — 4 — 5, on fig. 1. We may observe, in reference to these sections, that at the upper part of each the plane of section is supposed to be parallel to the arm, or the arm is, as it were, turned so as to be parallel to the plane, c c', or a a', fig. 1, in order that it may be projected in the sectional view with- out foreshortening. At the low-er parts of these views, however, tlie arms are projected, as in the oblique position represented in fig. 1. In this description of drawings, these oblique projections are generally dispensed with, and are, indeed, avoided, as they do not readily give the exact measurements of the parts represented. The operations indicated on the figures complete the general design of Plate XX., whether of the plan, elevation, or sections. THE DELINEATION AND CONSTRUCTION OF WOODEN PATTERNS FOR TOOTHED WHEELS. PLATE XXI. SPUR-WHEEL PATTERNS. 214. If, as we have already endeavoured to impress upon the student, great care is required in the construction of wooden pat- terns in general, above all is this care and extreme accuracy called for in the execution of the patterns of toothed wheels, because of the great exactitude absolutely needed in the proportions of the various parts — as tliat, for example, between their diameters and numbers of teeth. The pattern-maker must make allowance, not only for the shrink- ing of the cast-iron, but also for the quantity of metal to be taken away by turning and finishing afterwards. Moreover, the pattern, w^hich is necessarily in many pieces, must be joined together so strongly and solidly, that it may not run the risk of changing its shiipe during the construction of the mould. For wooden-toothed wheels, the web must be pierced with a number of openings or mortices to receive the tenons of the teeth. But in place of producing these mortices on the wooden patterns — which system, besides weakening it, would render the formation of the mould much more difficult — small projections corresponding to the teeth are fixed externally to the web. These projections form sockets in the mould, in which the actual loam cores are fixed, which foi-m the mortices w^hen the piece is cast. Bearing in mind these various considerations, we may proceed to the construction of the patterns for two spur-wheels, such as are represented in Plate XX. PATTERN OF THE PINION. 215. Figs. 1 and 2 show a half plan and a vertical section of tlie wooden pattern of the pinion. It Ls composed of many prin- cipal pieces — namely, the web, or crown, and its teeth ; the boss, witli its core-pieces ; and the arms, or spokes, with their feathers. We shall proceed to examine these various parts in succession. WEB OR CROWT*. The pattern-maker takes planks, of from 25 to 30 millimetres in thickness, and cuts out of it a series of arcs, a, of a uniform radius, corresponding to tliat to be given to the pinion, with the addition of the allowance for shrinkage and loss from finishing. These arcs are built up like brickwork, the joints of one layer, oi series, being opposite to solid portions of the contiguous layers, as shown in fig. 3. This arrangement prevents the liability to warp or change the fonn, from variation in the humidity of the atmosphere, as would be the case were the crowTi made of a single piece. This piece being finished and glued together, and the joints quite dry, is put into a lathe, and there turned quite true, both externally and internally. The two surfaces are here made per- fectly parallel, and the whole is reduced to the exact dimensions determined on, and shown upon a large working drawing of the actual size, previously prepared, generally by the pattern-maker himself At this stage, the external surface of the crown, is divided ofl" by lines, showing the positions of the teeth, which are then some- times sunply screwed or nailed on. It is, however, much prefer- able, and conduces very much to the soUdity of the wiieel, to cut out grooves of a trifling depth on the periphery, into which the teeth are fixed, being formed with a dovetail for that purpose, as shown at b, in fig. 1. BOSS. The boss is made in two pieces, each one solid block of wood, D, except when the w heel is of a large size, in which case the boss requires to be built up of several pieces. These blocks are each turned separately to the exact dimen- sions given in the plans, and they secure between them the thick- ness of the body part of tlie arm. ARMS OR SPOKES. The body of each arm, c, fig. 4, is also cut out of planks of a uniform thickness, being formed not only to the external contour of that part of the arm which is afterwards the only part visib.o BOOK OF INDUSTRIAL DESIGN. 11 in the casting, but also comprising, above and beyond this, the projections by which, in the pattern, it is attached to the boss on tlie one hand, and to the crown on the other. The extremity, a, of the boss end of the arm is in the form of a sector, correspond- ing to a sixth part of the cu-cle of the boss, the pinion having six arms ; the lateral facets, b, of this part are grooved out, to receive small tongue-pieces, or keys, c, fig. 1, so as to form a strong joint when glued together. The other extremity, d, of the arm is cut circularly, to the form of the crown, or web, into which it is fitted, penetrating to a slight extent, the crown being previously formed with a socket to receive it. Next, the feathers have to be attached to the body, c, of the arm. These feathers, b, are each cut out in sepai-ate pieces, to (he shape indicated in fig. 5: they have supplementary projec- tions, e and /, at their opposite extremities, whereby they are fixed into the crown and boss. When all these feathers are in their place, and the arms glued into the crown, the two portions, d, d, of the boss are fixed to them, the grooves for the reception of the ends of the feathers being glued, as well as the other parts, to give greater solidity. Finally, the boss is surmounted by the conical projecting pieces, f, f, which serve to produce in the mould the cavities, or sockets, which retain the loam core in position, the core being provided to produce the eye of the wheel, into which <Jie shaft is fitted. To give compactness and strength to the whole, a bolt, g, is passed through the centre; and this method of securing permits of the core projections, f, f, being changed for larger or smaller ones, if desired, without having to pull the entire wheel to pieces. [f, to add to the elegance of the shape of the wheel, it is wished to ornament the arms with mouldings, as at i, these are applied at the angles of junction of the feathers with the body of the arm. These are simply glued or naOed on. The sectional view, fig. 6, she. .s the form and position of these mouldings. It is to be observed that, in wheels of a moderate size, when cast-iron teeth are to work on cast-iron, they are at once cast to the exact shape, and the pattern is constructed accordingly ; but it is almost always indispensable, where cast-iron and wooden t«eth have to work together, to finish and reduce the former after being cast ; and the projections, b, on the pattern answering to them, must consequently be made of larger proportions every way, to provide for the quantity of metal taken away in the finisliing process. PATTERN OF THE WOODEN-TOOTHED SPUR-WIIEEr.. 216. Figs. 7, 8, and 9 i-epresent, in elevation, plan, and vertical section, the wooden pattern of the spur-wheel, which gears with the pinion just described. It consists, like that wheel, of the crown or web, the boss, and the arms ; and these various parts, which are designated by letters corresponding to those employed in the preceding example, are constructed exactly in the same manner. There is, however, an essential difTerencc in the exterior of the crown : in place of this carrying the projections, b, cut to the shape of the teeth, and such as will actually be iJifxhiccd on Ihe castii.g, it has other projections, b', of a simpler fortii, intended to produce in the mould the sockets for receiving the corc-pioccs which form the mortises in the casting, to receive the tenons of the wooden teeth. These projections are let into the crown, or simply applied thereto, and fixed by nails, as at Z, or by screws, aa at m, the latter method being preferable, as it has the advantage of pei-mitting the nmnber of teeth to be changed without injury to themselves or to the crown. In the wooden pattern, the length of the projections, b', is carried to the edge of the face of the crown, on that side which descends into the lower half of the mould-frame, to allow of the more accurate adjustment of the core- pieces, and also to facilitate the recovery of the pattern from the mould. These core-pieces, however, are so formed as to make the moi-tises no wider than is necessary, and to leave a sufficient thick- ness of metal for the strength of the crown, as already pointed out in reference to Plate XX. CORE-MOULDS. 217. The core-pieces for the mortises should not only be piaced at equal distances apart throughout the circumference of the crown, but they must all also be of precisely the same form and dimen- sions throughout, so that the mortises may be perfectly equal. With this view, a wooden core-box or mould is made ; and there are several methods of doing this. Thus, fig. 10 represents a face view, and fig. 11a horizontal section, through the line 3 — 4 in fig. .10, of one form of core-mould, consisting of a single piece. iTie portion, n, of the cavity corresponds to the projecting core-piece, b', outside the crown, and the portion marked o, to the mortise, or hollow socket, in the crown : this last has the same section as the crown in the width of the cut-out part. The moulder fills the cavity of the core-mould with loam, pre%"iously prepared, and after pressing it well in, levels it off" with a straight-edged doctor or scraper ; he finally inverts the mould, thus releasing the core com- plete. The operation is repeated as many times as there are teeth ; and when the cores are all dry, they are placed with great care in the mould, their supplementary projections, b', being let into the sockets formed to receive them — thereby insuring the accuracy of their adjustment. Figs. 12, 13, and 14, show another construction of wooden core- mould, formed in two separate pieces, h and i. These have be- tween them the cavity, n o, corresponding to that in the one just described. In this last case, the surface of the core which re- quires to be levelled off with a scraper, is only at one of the extre- mities instead of on the lateral faces, as in the other, and the cores are released by separating the two pieces, h, i, which are rendered capable of accurate adjustment to each other by moans of marking- pins, k. To return to the wheel itself: when it is of very large dimen- sions, the blocks, d, of the boss are secured together by two or more bolts, g, in place of one. The mould for the wheel is in two i>iecos, the lower frame, or "drag," being let into the ground in the moulding shop; the uppor frame or top part, is moveable, and it will bo obvious that very great caro is required to lift this oflFthe |witteni, .so !us not to injure the regnlarity and ,shar|inoss of llio impression ; and for this i)ur- pose, sullicieut " draw " or taper must bo ifivon to tlio vjirious ixarta. as the crown, the boss, and tin- fe.ntluTs on (ho amis, as ftln'.idv pointed out. 72 THE PRACTICAL DRAUGHTSMAN'S WTien the patterns are hea^y, two screw-staples, or "draw- plates," L, fig. 1, 8, and 15, of iron or brass, are countersunk into the crown, and into these draw-handles are screwed, by which the pattern is lifted out of the mould. In figs. 1, 2, 8, and 9, are combined, in smgle views, several different projections, to avoid repetitions of the diagrams, and to simplify the whole drawing, and bring it into a small space. This system is very much used in dra^\ings, or plans, made for actual ( onstruction. RULES AND PRACTICAL DATA. TOOTHED GEARKG. 218. It has been already laid dowTi, as a fundamental rule, that in order to work well, all toothed wheels coupled together must have the same ratio between the numbers of their teeth as between their diameters. It follows from this principle, that when we know the radii of the pitch cu-cles of t^o wheels, and the number of teeth of one of them, we can determine that of the other, and reciprocally. Thus, putting x to represent the number of teeth of a wheel of the radius, e ; and n to represent the number of teeth of a wheel of the radius, r, we have the direct proportionals, a : n : : r : r ; whence we can, at any time, ascertain any one of the terms when the other three are known. First Example. — Let the radius of the pitch circle of a spur- wheel be 12 inches, and the number of teeth on it 75, what should be the number of teeth on a pinion gearing ^^ith it, the radius of the pitch circle of which is 8 inches 1 We have 75 : n :: 12 : 8; whence 75 X 8 12 = 50 teeth. Second- Example. — Let 75 and 50, respectively, be the number of the teeth of a spur-wheel and pinion, and 12 inches the radius of the pitch circle of the former, the radius of the pitch circle of the latter may be found by means of the proportion — 75 : 50 :: 12 : r; whence, 50 X 12 „ . , r = — i^^ — = 8 mches. 7o 219. The velocities of rotation, or the numbers of revolutions of the shafts of a spur-wheel and pinion in gear with each other, are in the inverse ratio of the respective diameters, radii, or num- bers of teeth of the two. Consequently, putting V to represent the velocity of rotation of the pinion shaft, the radius of the pitch circle of which equals r, and the number of the teeth n, and putting v to represent the velo- city of the spur-wheel shaft, of which the pitch circle radius equals R, and number of teeth N, we have the inverted proportions — V : r :: r : R, and V : t; :: 71 : N. In either of these proportions, we can determine, as in the former axample, any one term when the three others are known. First Example. — A spur-wheel, the pitch circle radius of wliioh is 10 inches, has a velocity of 25 revolutions per minute ; whit is the pitch circle radius of a pinion to gear with it, and make 60 revolutions in the same time 1 By the inverse proportion, 25 : 60 :: r : 10; whence, 25 X 10 60~^ 4g- inches, the pitch circle radius of the pmion. A spur-wheel has 60 teeth, and is requhed to run at 25 revolu- tions per minute, and at the same time to drive a pinion at the rate of 75 revolutions per minute, what should ^e the number of teelii of the latter ? Here, 75 : 25 : : 60 : n ; whence, 25 X 60 n= — —— =20, the number of teeth the pinion must havt* These principles apply equally to puUeys or drums put in com- munication with one another by cords or belts, and kno\vn as belt- gearing. Sometimes, in systems of geared spur-wheels, aU that is known is the distance apart of their centres, the number of teeth which they are to carry, or the number of their revolutions in the same time. In this case we have, on the one hand, an inverse proportion between the distance of their centres, the sum of then- revolutions, and between their respective radii and revolutions ; and, on the other hand, a direct proportion between the distance of the centres, the sum of the teeth on both wheels, and their respective radii, oi the ntunber of teeth of each. Let D be the distance apart of the centres of a spur-wheel and pinion of the respective radii, R, r, and number of teeth, N, n, or the reciprocal velocities, r and V; we have first the following inverse proportion, D: V+i::R: V; and, secondly, the direct proportion, D:N + 7i::N:R. First Example. — Let 45 inches be the distance between the aentres of a spur-wheel and pinion, the former of which is to make 22 revolutions per minute to the other's 15i; what should be their respective radii? We have, first, 45 : 22 + 15-5 :: R: 22; whence, „ 45 X 22 R = _____ = 26-4 mches, and whence, 22 + 15-5 45 : 22 + 15-5 :: r : 15'5; 45 X 15-5 22 + 15-5 ■ 18'6 inches. ^Vhen the pitch cu-cle radius of one of the wheels is ascertameO, it is evidently unnecessarj' to search for the other radius by means BOOK OF INDUSTRIAL DESIGN. 73 iiT tlie second proportion, for it is sufficient to subtract the one found from the sum of both; thus, 45 — 26-4 =: 18-6; or, 45—18-6 = 26-4. Second Example. — The distance, d, between the two centres oeing known =: 45 inches, and one wheel carrying 31 teeth and She other 44, what are their respective radii 1 We have here, in the first place, 45 : 31 + 44 :: R : 44; whence, 45 X 44 "-31 + 44-^""' and 45 : 31 +44 :: r : 31; whence, 45 X 31 '•=31 + 44 = ^«-«' or, more simply, r = 45 — 26-4 = 18-6 inches In like manner, the respective radii of a spur-wheel and pinion, to gear together, may be determined geometrically, when the dis- tance between their centres is known, as well as the numhera of revolutions of each, by the following rule : — Divide the distance into as many equal parts as there are of any measure contained exactly in the sum of the velocities, such mea- sure being also contained exactly any number of times in each of the velocities alone. Then, for the pinion radius, take as many of these measures as are contained in the lesser velocity, and for the radius of the spur-wheel, the remainder of them. Example. — Let 16 inches be the distance between the centres of a spur-wheel and pinion which make 6 and 4 revolutions re- spectively, or any equi-multiples or equi-submultiples of these, as 12 and 8, or 3 and 2. Divide the distance into 10 equal parts, and take 4 of these for the pinion radius, and 6 for the spur-wheel radius. This rule is of very simple application when the ratios of the numbers of revolutions are whole numbers, such as 1 : 4, or 2 : 5 ; for all that is necessary is to add the two together, to divide the distance between the centres to correspond, and to take the re spective numbers of measures for each wheel. The following table will be of great assistance in the solution of various problems connected with systems of gearing, when tho number of teeth, the pitch, or the radius are known. TABLE FOR CALCULATING THE NUMBERS OF TEETH AND DL&.IHETERS OF SPUR GEAR, FROM THE PITCH, OR VICE VERSA. Number. Coefficient. Nnmber. Coefficient. Number. Coefficient. ^fumber. Coefficient. Number. Coefficient 10 3-183 39 12-414 68 21-644 97 30-875 126 40-106 11 3-501 40 12-732 69 21-963 98 31-193 127 40-424 12 3-820 41 13-050 70 22-281 99 31-512 128 40-742 13 4-138 42 13-369 71 22-599 100 31-830 129 41-061 14 4-456 43 13-687 72 22-917 101 32-148 130 41-379 15 4-774 44 14-005 73 23-236 102 32-467 131 41-697 16 5-093 45 14-323 74 23-554 103 32-785 132 42016 17 5-411 46 14-642 75 23-872 104 33-103 133 42-334 18 5-729 47 14-960 76 24-191 105 33-421 134 42-652 19 6-048 48 15-278 77 24-509 106 33-740 135 42-970 20 6-366 49 15-597 78 24-827 107 34-058 136 43-289 21 6-684 50 15-915 79 25-146 108 34-376 137 43-607 22 7-002 51 16-233 80 25-464 109 34-695 138 43-925 23 7-321 52 16-552 81 25-782 110 35013 139 44-244 24 7-639 53 16-870 82 26-100 111 35-331 140 44-562 25 7-957 54 17-188 83 26-419 112 35-650 141 44-880 26 8-276 55 17-506 84 26-737 113 35-968 142 45-199 27 8-594 56 17-825 85 27-055 114 36-286 143 45-517 ■ 28 8-912 57 18-143 86 27-374 115 36-604 144 45-835 29 9-231 58 18-461 87 27-692 116 36-923 145 46-153 30 9-549 59 18-780 88 28-010 117 37-241 146 46-472 31 9-867 60 19-098 89 28-329 118 37-559 147 46-790 32 10186 61 19-416 90 28-647 119 37-878 148 47-108 33 10-504 62 19-734 91 28-965 120 38-196 149 47-4-27 34 10-822 63 20-053 92 29-284 121 38-514 150 47-745 35 11-140 64 20-371 93 29-602 122 .38-833 1 151 48-063 36 11-4.59 65 20-689 94 29-920 123 39-151 152 4S-3S2 37 11-777 66 21 -DOS 95 30-238 124 39-l(:9 153 48-700 38 12-095 67 21-32() 96 30-557 125 39-788 154 49020 RULES CONNECTED WITH THE PRECEDING TABLE. [. To find tho diameter of a spur-whoel, when the number and pikh of the toeth are known. Mulli/ilij Ihc ciMffirirnl in l)w lahh; corrfspomliitg to Oie tiutiiber if teeth, by the given pilch in feet, imhes, mdtrcs, or other measures, and the product will he llw di'imeler in feel, inches, or jnttres, to corresponiL 74 THE PRACTICAL DRAUGHTSMAN'S First Example. — What is the diameter of a spur-wheel, of 63 teeth, having a pitch of H inches? Opposite the number 63, in the table, we find the coefficient, 20053. Then— 20-053 X 1-5 = 30-08 inches, the diameter of tlie spur-wheel. Second Example. — \Vliat are the diameters of two wheels, of 41 and 150 teeth respectively, their pitch being | inch? On the one hand, we have 13-05 X -75 = 9-7875 inches, the diameter of the pinion of 41 teeth; and on the other, 47-745 X -75 = 35-8 inches, the diameter of the spur-wheel of 150 teeth. n. To find the pitch of a spur-wheel, when the diameter and number of teeth are known. Divide the given diameter by the coejicienl in the table correspond- ing to the number of the teeth, and the quotient will be the pitch sought. First Example. — What is the pitch of a wheel of 30-08 inches diameter, and of 63 teeth ? Here — 30-08 : 20-053 = 1-5 inch, the pitch required. Second Example. — It is required to construct a spur-wheel, of 126 teeth, to work with the preceding, what must be its diameter? Here— 1-5 X 40-106= 60-159 inches, the diameter of a wheel of 126 teeth, and of the same pitch. in. To find the number of teeth of a wheel, when the pitch and diameter are known. Divide the given diameter by the given pitch, the number in the table corresponding to the quotient will be the number of teeth sought. If the quotient is not in the table, take the number correspond- ing to that nearest to it. First Example. — The diameter of a spur-wheel is 30-08 inches, and the pitch of the teeth is 1-5 inch, what number of teeth should the wheel have ? 30-8 : 1-5 = 20-53; which quotient corresponds to 63 teeth. Second Example. — What should be the number of teeth of a pinion, the diameter of which is 875 miUimetres, and which is intended to gear with a rack, of which the pitch is 25 millimetres ? 875 : 25 = 35. The number most nearly corresponding to this is 110, the number of teeth to be given fo the pinion. ANGULAR AND CIRCUMFEKENTIAL VELOCITY OF WHEELS. 619. When it is known what is the angular velocity of the shaft of a fly-wheel, spur-wheel, or pulley, the circumferential velocity may be found by the following rule : — Multiply the circumference by the number of revolutions per minute, and the product will give the space passed through in the same time ; and this product being divided by 60, will give the velo- ii.y of the circumference per second. Fxample. — Let the diameter of a wheel be 4 feet, and the number ot its revolutions per minute 20, what is the velocity a* the circumference ? The circumference of the wheel =4 x 3-1416 = 12-5664; then 12-5664 X 20 = 251-328 feet, the space passed through per minute by any pomt m the circum- ference ; and 251-328 60 :4-2, the velocity in feet per second. When the velocity at the circumference is known, the angulai velocity, or the number of turns in a given time, may be ascer- tained by the following rule : — Divide the circumferential velocity by the circumference, and ine quotient will be the angular velocity, or number of revolutions in the given time. In the preceding case, 4-2 feet being the circumferential velocity per second, and 4 feet the diameter, we have 4-2 _ 4 X 3-1416 ~ ■^^'*' the angular velocity per second ; and •334 X 60 = 20, the number of revolutions per minute. In practice, it is easy to ascertain the velocity of a wheel, the motion of which is uniform. With this view, a point is marked with chalk on the rim of the wheel, and note is taken of how often this point passes a fLxed point of observation in a given time ; then this number of revolutions is multiphed by the circumference described by the marked point, and the product divided by the duration of the observation expressed in seconds. The result will be the velocity of the circumference of the wheel. Every other point on the wheel will have a diiferent velocity, propor- tioned to its distance from the centre of motion. Example. — A wheel, 2 feet in diameter, having, according to observation, made 75 revolutions per minute, what is its circum- ferential velocity (per second) ? "5 X 3-14 X 2 60 = 7-83 feet. circumferential velocity of the wheel. Reciprocally, when the circumferential velocity (per second) is known, the number of revolutions per minute is found by means of the formula — V X 60 ^~3-14xD' or, with the data of the preceding case. N = 7-83 X 60 = 75 revolutions per minute. 3-14 X 2 When several spur-wheels or pulleys are placed on the same shaft, the circumferential velocity of every one of them is found in the same manner, by multiplying the number of revolutions by the respective circumferences, and dividing the products by 60. Example. — Three wheels or pulleys, a, b, c, are fixed on one shaft; the radius of the pulley, a, is equal to 1-1 feet; that of the pulley, b, 1-6 feet; and that of the pulley, c, 2-15 feet; and the shaft makes 12 turns per minute, — what is the circimiferential velocity of these three pulleys ? BOOK OF INDUSTRIAL DESIGN, 76 For the pulley, a, we have — 6-28 X M X 12 V = ;::: = 1-38 feet per minute; 60 for the pulley, ,,, 6-28 X 1-6 X 12 V' = — = 2 feet; 60 and for the pulley, c — 6-28 X 2-15 X 12 V" = — = 2-7 feet oO DIMENSIONS OF GEARING. 220. In designing tooth-gearing of all descriptions, it is neces- sary to determine — first, the strength and dimensions of the teeth ; second, the dimensions of the web which carries the teeth ; and, third, the dimensions of the arms. THICKNESS OF THE TEETH. 221. The resistance opposed to the motion of the wheel or the load, may be considered as a force applied to the crown, to pre- vent its turning, and the power, during its greater strain, as applied to the extremities of the teeth. The teeth then should be con- sidered as solids fixed at one end, and loaded at the other ; and the equation of equilibrium for then; is — Fxh^^kxfxw; m which formula, P signifies the pressure in kilogrammes at the extremity of the tooth ; h, the amount of projection of the teeth from the web in centimetres ; k, a numerical coefficient ; t, the thick- ness of the teeth in centimetres ; w, their width in centun^tres. In this formula, the numerical coefficient, k, which is calculated with reference to the motion of toothed geaiing, varies with the material of which the teeth are constructed. From Tredgold's experiments with well-constructed cast-iron wheels, this coefficient has been calculated to be 25 for that metal ; and adopting it, the preceding formula will then become P X h — 25 X {^ X w; wTience, _ 25 X ^w ^- h ■' a lormula in which three dimensions are variable. The following ratios usually exist between these quantities : — w varies between 3 t and 8 t. h = 1-2 t to 1-5 t. Let, then, w = 5t, and h = 1-2 t, so that, substituting these values in the equation, it becomes — „ lb X 5 X t xC P= 1-2 X. =104x.'; whence, P «» = -— and / = -098 -^/p] 104 It me above ratio between the thickness, t, and width, ir, !»■ adopted for all proportions ; for low pressures or small loads, we shall have teeth much too thin and small ; and for hii,'li pres- sures, on the other hand, the defects of too great lliickncss jind pitch. To retain, then, the thicknesses within convenient lunits, it is well to vary the ratio of t to w, according to the pressures ; and in order that the pitch may not be too great, the width of the teeth is determined at the outset, according to the pressure or load which they have to sustain, in the follovraig manner : — L For 100 to 200 lb., make 2^ = 3 i; when Z = -126 ^F n. « 200 300 " M=3-5; " t — -\\l\'Y III. " 300 400 " w= At " < = -110^P' IV. " 400 500 " w = A-5t " t = -104 ^P" V. " 500 1,000 « w= ot " t — -098 ^ P" VL " 1,000 1,500 « w = 5-5t " t — -093.^ F VII. " 1,500 2,000 " M! = 6 i! « t = -089 ^P" Vm. " 2,000 3,000 " w = 6-5t " <=-084*'P" IX. " 3,000 5,000 " w= It " < = -082 ^'¥' X. " 5,000 and upwards, " w= 8i " t = -Oil V'p' The height, or projection, h, should be comprised between 1-2 t and 1-5 t, the latter applicable to low powers or loads, and the former to high ones. For teeth of wood, which are ordinarily made of beech or ehn, the coefficient should be augmented by a third in each of the last given formulae, which become — L t = -168 W making w = 3-0 t. IL t—-156VP' « w=3-5t. in. i—-lnVP' " w = 'l-Oi. IV. i^-lSdW " w = 4-5t. V. t — -13\W " w=o-Ot. VL t = -124:VF " w = 5-5t. VIL t=z-n9VP' " w=6-0t. vm. t = -U2VP' " w=6-5t. IX. i — -109 W " w = 7-0 1. X. t = -103\^ " w; = 8-0^ All these formulae are constructed on the supposition that, although there are generally several teeth in contact at the same time, yet each should be capable of sustaining the whole strain as if there were only one in contact, and they should be strong enough to compensate for wear, and sustain shocks and irregularities ui the strain for a considerable length of time. The pressure, P, on the teeth may be determined according to the amount of power transmitted by the wheels per second at tho pitch circumference. This pressure is obtained by ditiding the strain to be transmitted, expressed in kilogranwuirr, by the lelncily ]>tr sirond of the pilch circumference. A kilognunmetro is a term corrt'spondiiig to llut English expression, " one pound raised one foot liigJi per minute." A kilogrammetro is equal to one kilognunme raised one mt^tru high per second : it is written sliorlly thus — k. ni. First Example. — A spur-wlicol is intended to tnmsinit n forto equal to a power acting nt tho pitch oirxMunference of 600 kilo- gramnietres, at the rate of 209 ni. per second, wluit pressure huvo tho teeth to sustain .' 76 THE PRACTICAL DRAUGHTSMAN'S Here, 500 k. m. 2-09 : 239 kiloff., the strain that each tooth must be capable of resisting without risk of breakage, even after considerable use and wear. Second Example. — A spur-wheel, 2 metres in diameter, transmits a force equal to 20 horses power, and makes 25 revolutions per minute, what is the pressure on the teeth ? We have, in the first place, 20 H.p. = 75 X 20 = 1500 kilogrammetres, and V = 3-14 X 2 X 25 60 =: 2-62 m. per second ; whence, 1500 ^^^ , ., -^^g^ = 573 kilog.. the pressure on the tooth. When the power that a wheel has to sustain at its circumference is known, the thickness proper for the tooth may be calculated by one of the preceding formulas, according to the material of which it is constructed. Thus, in the former of the last two examples, in which P = 239 kilog., the thickness of the tooth, if of cast-iron, should be making w = S-5t: t — -lin V239 = 1-8 cent. = 18 millimetres. And, in the second example, where P = 573 kil., the thickness will be, supposing the teeth to be of beech, and w = 5t, i =: -131 4/573 = 3-23 c, or 32-3 millimetres, w = 5 X 32-3 = 161-5 millimetres. Third Example. — A water-wheel of 4-2 metres diameter makes 4^ revolutions per minute, and transmits a force equal to 25 horses power by means of a spur-wheel, the radius of which is 1-65 m., it is required to determine — ^first, the pressure on the teeth of this spur-wheel ; and, secondly, the thickness of their teeth. In the first place, 25 X 75 = 1875 kilogrammetres, ana V = 1-65 X 2 X 3-14 X 4-5 60 = -777 m. ; whence, 1875 •777 2413 kilogrammetres ; consequently, making w = 6-5 /, the thickness of the tooth will be t — -084 V2413 = 3-7 c = 37 millim., and w = 37 X 6-5 = 240-5 millim. Fourth Example. — The cast-iron pinion of a powerful machine is 1-06 m. in diameter, it is fixed on a shaft which should transmit an effective force of 200 horses power, at the rate of 45 revolutions per mmute, what is the pressure on the teeth and their dimensions ? The power transmitted is 200 X 75 = 15,000 kilogrammetres, •ind _ 1-06 X 3-14 X 45 » — ^7j = 2-37 metres per second. The pressure on the teeth is — consequently, making w = 8 l; we have, for the thickness of the teeth, in cast-iron, i = -077 4/6333 = 61-2 "■/„. and w — S x 61-2 = 489-6 ■"/„. For a pinion of the above proportions, actually constructed, the thickness was made 75 millim., and the width 525 millim. PITCH OF THE TEETH. 222. It will be recollected (203) that the pitch of cast-iron spur-wheel teeth, measured on the pitch circumference, comprises the thickness, t, of the tooth, and the width of the interval, which last is, in ordinary cases, made equal to i, augmented by one-tenth; this gives, ^ = 2-1 1. Thus, with the data of the preceding examples — In the 1st, ^=2-1x18 = 27-8 ■"/„. 3d, ^ = 2-1 X 37 = 77-7 "/^ 4th, p = 2-l X 61-2 = 128-5 "/^ When the spur-wheel is intended to carry wooden teeth, as in the second of the preceding examples, it will generally be coupled with a pinion, having cast-iron teeth, which should be of about three-fourths the thickness of the wooden ones ; in this case the pitch will be equal to t + -151 + -11= I + ■85t= 1-85 t. Thus, in this example, we should have — p=32-3 X 1-85 = 59-8°/^. After this is done, that is, when the pitch is ascertained, which, as has already been observed, should be precisely the same on the pitch circles of any two wheels working together, the number of teeth of one of the wheels may be obtained by the following formula — P where N signifies the number of teeth of the spur-wheel ; R, the radius of the pitch circle ; and p, the pitch, measured on this circle. First Example.— What is the number of teeth on a spur-wheel of two metres diameter, and the pitch of which is -0278 metres ? Here — ,, 2 X 3-14 X 1 ^^ -0278 =225 teeth. It vAW be easily understood, that the fraction arising from the operation must be neglected, since we cannot have a part of a tooth. In cases, therefore, where there is a fraction, the pitch must be slightly increased. Thus, in the example xmder consi- deration, the pitch becomes — 2rt R 6-28 ^==T = -225=-02^9'"-' instead of 0-278 m. Second Example. — It is required to determine the number of wooden teeth to be carried by a spur-wheel of two metres diameter^ the pitch being -0598 m. Here, T.T 3-14 X 2 ^ = ^059^ = ^«^- BOOK OF INDUSTRIAL DESIGN. 77 Wlien a spur-wheel is to have wooden teeth, it is necessary that the number of these be some multiple of the number of arms of the wheel, in order that they may be conveniently attached to the web ; thus, in the present example, if the wheel is to have 6 arms, the number of teeth must be 102 or 108, to be divisible by that number ; and if the former be adopted instead of 105, the pitoh will be slightly augmented in consequence. To obviate the necessity of making long and tedious calcula- tions, a table is subjoined, showing the thickness and pitch of teeth of spur-wheels, in which is adopted the coefficient '105 of M. Morin, which makes the formula, t = -105 V^ for cast-iron teeth, and ^ = •145 VT for wooden teeth : the width being constantly equal to nearly 4-5 the thickness. TabU of the Pilch and Thickness of Spur Teeth far different Pressures. Of Cast-Ii-on. Of Wood. Pressure in Kilogrammes. Thickness of Teeth in Millimetres. Pitch in Millimetres. Thickness of Teeth in Millimetres. Pitch in Millimetres. 5 10 15 20 30 40 50 60 70 80 90 100 125 150 175 200 225 250 275 300 350 400 500 600 700 800 900 1000 2-3 3-3 4-0 4-6 5-7 6-6 7-4 8-1 8-7 9-4 9-9 10-5 11-6 12-8 13-8 14-8 15-7 16-6 17-3 18-2 19-6 21-0 23-4 25-7 27-7 29-7 31-5 33-2 4-9 6-9 8-5 9-7 12-0 18-9 15-6 17-0 18-4 19-7 20-8 22-0 24-4 26-9 29-1 31-1 33-0 34-8 36-3 38-1 41-2 43-2 49-1 54-0 58-2 62-4 66-1 69-6 3-2 4-7 5-6 6-4 7-9 9-1 10-2 11-2 12-1 12-9 13-7 14-5 16-1 17-7 19-1 20-2 21-7 22-9 23-9 251 27-1 29-0 32-4 35-5 37-2 41-0 43-8 45-8 5-9 8-7 10-4 11-8 14-4 16-9 18-9 20-8 22-4 23-9 25-3 26-8 29-8 32-7 34-8 37-4 40-1 42-4 44-2 46-4 50-1 53-6 69-9 65-7 69-1 75-8 83-0 84-7 With the assistance of this table, and the preceding rules, wo can always determine, not only the thickness and pit<'h of the teeth, but also their height and width, since these are in proportion to their thickness. DIMENSIONS OF THE WEB. 223. The width of the web is ordinarily equal to that of the teoth when the whole is of cast-iron. Nevertheless, in some cases — such as. for example, where very groat irregularities in the pressure and speed, and reiterated shocks have to bo borne in the heavy nia- ininerv in engine shops — the web ia made wider than the teeth, projecting also on either side of the teeth, so that these are whoUy or partially imbedded, which increases their power of resistance very considerably. These lateral webs are generally each made of about half the thickness of the tooth. The thickness of the web, or crowTi, is never made less than three-fourths that of the tooth, and very frequently it is further strengthened by an internal feather, as already mentioned. 213. When the teeth are of wood, the web is much thicker, to give sufficient hold to the tenons of the teeth ; it is generally made about 1-5 to 2- times the thickness of the tooth. NUMBER AND DIMENSIONS OF THE AEMS. 224. The number of arms, or spokes, which a spur-wheel ought to have, has not, up to the present time, been precisely ard scientifically determined. According to general expenence, up lo a diameter of 1 metre, or about 3 feet, four arms are sufficient; from 1 metre to 2 metres, or 3 feet to 6 or 7 feet, sLx are necessary and sufficient ; beyond 2-5 m., or 8 feet, eight arms aie used: and for 5 m., or 16 feet, ten are given; it is seldom this last number is exceeded, except for wheels of extraordinary di- mensions. The section of the arms of the wheel is always in the form ot a cross, the stronger portion of which lies in the plane of the cir cumferential strain, whether these arms are cast in one piece with the boss and the crown, as is the case with wheels ol small diameter — that is, of such as have not a greater radius than 2 m. or 64 feet ; or whether they are cast in separate pieces, and after- wards fitted together. The thicker part, then, of the arm must be strong enough to bear the circumferential strain. Experience has shown, that when a spur-wheel is in motion, and acted upon by a considerable force, this strain has a tendency to make the arms assume a twisted shape, and produce on them a lateral inflexion. It is to obviate and prevent this, that the arms arc strengthened by feathers. The power acts with greatest eflfect near the boss of the wheel, so that it is necessary to make them wider at this part than near the crown, so as to approximate to the form which presents au equal resistance throughout. This will be observed in the figures in Plate XX. Tlie boss must have such a thickness as will allow of the wheels being solidly fixed on the shaft. A thickness of 5 inches may be considered a maximum for the bosses of moderately-sized wheels. The dimensions of the arms should be in proportion fo the widtii of the web or crown, their thickness being ordinarily about J that of the crown. Tiiis proportion is n good one for wheels under 6i feet in diameter. For larger sizes, i the width of the web is considered sufficient. The lateral feathers sliould have, at tlie very most, only tlu' thickness of the arm. GeiicniUy. the wiilili ol' ilic arm near llio web is made about 5 of its width near the boss. The lollowing table, calculated from Tredgold's experiments, shows the pro|)or- tions to bo given to the arms or spokes of spur-wheels, noeonliiig to the strain acting at their circumferences; supposing the dia- meter of tiio wheels to bo 1 in., ami the nunibi>r of anus 6. their tlii<'kiu'ss being taken equal to \ the widlii of the crown. The liiiiicnsiiuis triveii are the averages, or those to he applied to tho :inM. hall-wnv belwi'cn the boss and the crown. 78 THE PRACTICAL DRAUGHTSMAN'S Table cf the Dimensions of Spur-wheel Arms. Timgentia\ Strain on the Width of the Arm in Width over all, of the Whtel iu kilog. centimetres. Feathers in centimetres. 10 4-20 1-21 10 6-00 2-00 80 8-00 3-00 158 8-50 3-90 >244 9-70 4-85 «36 10-67 6-30 430 11-64 6-80 680 12-12 8-25 730 13-10 8-73 870 13-80 9-70 1100 14-50 10-67 1210 15-50 11-64 1500 16-00 12-60 1750 16-50 13-68 2200 17-00 14-06 2300 17-50 16-50 2660 18-00 17-00 2840 18-50 17-95 3220 19-00 19-50 3500 19-50 19-40 To apply the numbers in this table to wheels of other diameters, they must be multiplied by V R, R being the radius of the wheel for which the dimensions are to be calculated. WOODEN PATTERNS. 225. When a casting has not to be turned, oi otherwise re- duced, about 1 per cent, must be allowed in the dimensions oi tne pattern, and if the piece has to be turned, a little more than this. It is, however, impossible to give any rule in this last case, as the allowance to be made depends entirely upon the nature and desti- nation of the piece. The larger the piece, the greater should be the per centage given. No piece can be cast with mathematical precision — ^whether it is, that, on the one hand, the pattern loses its true shape, and lines, which have been made perfectly straight or circular, become twisted, notwithstanding that every precaution has been taken in perfecting it; or, on the other hand, that, in lifting it from the loam, the moulder is forced to move it laterally, to some slight extent, so that the casting becomes larger at one part, or twisted at another ; or, again, that the metal does not shrink equally at all parts. With regard to the last-mentioned source of error, it has often been found that the diameter of a wheel, measured through the line of the arms, is sensibly less than as measured across the centres of the spaces between the arms. This diflFerence is indeed so great, that in wheels of 10 to 15 feet diameter, it reaches an eighth or a sixth of an inch. It is manifest, that all these considerations must be borne in mind when constructing "ivooden patterns for castings ; otherwise, errors of considerable magnitude will arise. CHAPTER VI. CONTINUATION OF THE STUDY OF TOOTHED GEAR. CONICAL OR BEVIL GEARING. 226. Cylindrical or spur-wheels are only capable of transmitting motion between shafts which are parallel to each other ; and when the shafts are inclined, or form any angle with each other, the wheels require to be made conical, and are then called bevil- wheels. In order that this description of gear may be capable of working well and regularly, and of transmitting considerable power when needed, as with spur gear, it is essential that the shafts or axes of any pair working together be situated in the same plane ; in this case, the axes will meet in a point which is the apex common to the two wheels. Formerly, when it was required to transmit power through shafts intersecting each other at right angles, a species of lantern- wheel was employed for one of the wheels, consisting of a couple of discs with cylindrical bars for teeth, passing from one to the other parallel to the axis ; and the wheel to gear with this one was formed with similar teeth, also parallel with the axis, but projecting up from a single disc or ring. This form of gearing is still to be found m old mills ; but it is very defective, and very inconvenient when any speed is required. Sometimes, as for some descriptions of spinning machinery — the cotton-spmner's fly or roving-frame, for example — bevil-wheels are used, in which the axes are not situate in the same plane ; these are tei-med " skew bevils," from the teeth having a hyperbcloidal twist in order that they may act properly on each other. This kind of wheel does not work well, and is seldom employed, except where the size is very small, or where a small power only has to be trans- mitted; the peculiar form of their teeth also renders them very difficult to construct. Their use is so limited, that further details respecting them are uncalled for. Indeed, they ought rather to be avoided, since there are very few cases in which common bevil- wheels cannot be substituted for them with advantage. The teeth of bevil-wheels are made of wood or metal, similarly to spur-wheel teeth, and their geometrical forms are determined on the same principles. DESIGN rOR A PAIR OF BEVIL-WHEELS IN GEAR. PLATE XXII. 227. We propose, in the present example, to give the largei wheel wooden teeth, and the smaller ones cast-iron ones, as was done with the pair of spur-wheels last described. Let A B and a c, figs. 1 and 2, be the axes of the two whe'^ls assumed here to be at right angles to each other ; though we BOOK OF INDUSTRIAL DESIGN. must observe, that what follows will apply equally well to the i.onstruction of a couple of wheels, the axes of wMch make any angle with each other, acute or obtuse. Let B D = -220 m., and e f = -440 m., the radii of the pitch i-ircles of the two wheels. It is, in the first place, necessary to determine the position these circles should occupy on their respec- tive axes. With this view, on any point, b, taken on the axis, a b, erect a perpendicular, b d, and make it equal to the radius of the smaller wheel, and through the extremity, d, draw a line, d l, parallel to this axis ; in the same way, at any point, e, taken on the axis, a c, erect the perpendicular, e f, equal to the radius of the larger wheel, and through the extremity, f, draw f h parallel to A c. The point of intersection, g, of these two lines, f h and p L, is the point of contact of the two pitch circles, the radii of which are g i and g k. Make i h and k l, respectively, equal to the radii, and join the points, h g l, to the common apex, a, thereby determining what are termed the " pitch " cones, a h g and a g l, of the two wheels, the straight lino or generatrix, a g, being the line of contact of the two cones. These pitch cones possess the same properties as the pitch circles, or, more correctly, pitch cylinders, of spur-wheels ; that is to say, their rotative velocity is in the inverse ratio of their diameters, and their diameters are pro- portional to the respective numbers of their teeth. The proportions of the pitch cones being thus obtained, with the centres, o and o', figs. 2 and 3, taken on the prolongation of the ^ven axes, describe the pitch circles, a h' i' and g' k' l'. Divide these circles into as many equal pai-ts as there should be tooth ; (hat is to say, in the present case, 24 and 48, respectively, which operation will give the pitch ; each part is then bisected to obtain the centres of the teeth and of the intervals, and on each side of the centre lines are set off the demi-widths of the teeth, regard bemg had to the difference to be made between the wooden and cast-iron teeth, as already explained (213). The external contours of the teeth will be situated in cones, the generatrices of which are perpendicular to those of the pitch cones ; they are obtained by drawing through the point of contact, g, on the line, a g, a perpendicular, b c, meeting the axis of the smaller wheel in b, and that of the larger one in c ; the points, b and c, are the apices of the two cones, b h g and c g l. If these last-mentioned cones be developed upon a plane, it will be easy to draw upon it the exact forms of the teeth. Now, we have seen (170) that the development of a cone on a plane surface takes the form of a sector of a circle, which has for radius the generatrix of the cone, and for arc the development of the base of the cone. As it is unnecessary to develop the entire cone in the present case, it is sufficient to describe with any point, b', fig. 4, with a radius equal to b g, an arc, a e h, on which, starting from the point, c, are divided off distances — one, c d, equal to the thick- ness of the tooth of the smaller wheel, fig. 3, and the other, c e, to that of the tooth of the larger wheel, fig. 2. The same operation IS performed for the larger wheel ; that is, with tlio jiaiiit, c', situated on the prolongation of b' c, and with a radius equal to c g, describe the arc, f c g, on which are meiusurod the distances, respectively, equal to the former ones, e (/ and c e. This done, the outlines of the tccrtli are obtained liy means of precisely the «>ame operations as those oxpl.iincii in refcronco to the spur-wheels. Thus, on the radius, b' c, considered as a diameter, describe a circle, i cj, which, in rolling round the circle, fcg, considered as the pitch circle of the larger wheel, determines the epicycloid, e h, which gives the curvature of the teeth of the larger wheel ; in the same manner, the circle, k e I, described on the radius, e c', as a diameter, and rolling round the circle, a c b, gives the epicycloid, c m, which is taken for the curve of the teeth of the smaller wheel. After having repeated these curves sym- metrically on each side of the teeth, these are limited by drawing chords in the generating circles from the point, e, each equal to the pitch of the teeth, as c n, c k, and then with c' and b' as centres, describe circles passing, one just outside the point, n. and the other just outside the point, k ; and to indicate the line of the web, describe a second couple of circles, nearly tangents to the preceding. Then project the points, o and p, which indicate the depth and extre- mities of the teeth, over to the line, b c, in o and p' ; through these last draw straight lines to the apex, a, which wBl represent the extreme generatrices of the teeth, as in vertical section. As all the teeth converge in one point, it is obvious that thtj contour of the inner ends of the teeth cannot be the same as that of the outer ends ; the difference is the greater, according as the width, G r, on the generatrix line of contact is itself greater, ii; proportion to the entne cone, and to the greater or less angle formed by the extreme generatrices. In other respects, this contour is determined in the same manner as the first. Thus, through the point, r, is drawTi the straight line, s /, perpendicular to a g, wliich cuts the two axes, and gives the proportions of the two cones, on the surface of wliich lie the contours of the inner ends of the teeth. Continuing the opera, tion as above, portions of the cones are developed, arcs being described with the points, s' and t', as centres, and radii equal to r s and r i. The diagram, fig. 5, which is analogous to fig. 4, fully explains what further is to be done. What has been said so far, has referred only to one tooth of each wheel. In proceeding with the execution of the design, after cutting out templates to the form of the teeth as obtained by means of them, the outline is repeated, as often as is necessiiry, on the external cones, the generatrices of which are b g and g c, for the outer ends of the teeth, and on the internal cones, the genera- trices of which are r s and r t, for the outlines of the inner ends of the teeth. At the same time, and in order that the oix-ration may be performed with regularity, a series of lines should bo drawn through the points, o, p, of the two wheels, lying on the surface of tbe external cones, a H g, a g l, and uniting at tlie apex, a, by means of a "false square," of a form analogous to that represented at X, in fig. 3, Plato XXIII., for the smaller wheal, and like that rei)resented at T, fig. 4, of the sjmio Plate, for the larger wheel. The forms of the teeth being thus obtained, the partial .section, fig. 1, of the two wheels is drawn, tlio radii of the slial^s Ix-ing given, as well as the thickness of the bosses and webs, tlie jiropor tions employed in the present example being indicated on the drawings. It will be observed that those teeth wliicii are of wooa nr(^ adjusted in liie weh of the larger wheel, in the sjune niannoi as in the spur or cylindrical wheels, the forms of the tenons U-iiiv nuidilied, so thai their .sides nil incline to the common !1|h«x, a 80 THE PRACTICAL DRAUGHTSMAN'S The sections, together with the developments, figs. 4 and 5, are sufficient for the purposes of construction, as all the required mea- surements can be obtained from them ; but when It is desired to produce a complete external elevation of the two wheels, it will be necessary to find the projections of the teeth and other parts. With this view, the teeth are first actually drawn upon the planes of projection parallel to the bases of the wheels, as showTi in figs. 2 and 3. It will be recollected, that divisions have already been made on the pitch cii-cles, a h' i', and l' k' g', indicating the centre lines, as well of the teeth as of the intervals, and marking the positions of the flanks; and, consequently, all that remains is to draw the external outlines and the curved portions. For the bmaller wheel, the operation consists in projecting to y, in fig. 3, tlie point, p, fig. 1, which limits the lower and outer edge of the tooth, and in describing with the centre, o, and radius, o p', a circle hmiting the whole of the teeth externally, and cori-espondlng to the section of the cone in which the points, p p', fig. 1, lie. In this circle, also, terminate the curves of the outer portions of the teeth, and tlieir exact points of intereection are obtained by measuring on each side of the centre lines, r o, distances, v u, v p, equal to the corresponding distances, t' u', i' p', in fig. 4. Then, through the points. It, p, draw a series of lines, converging to the centre, o ; and through the points, e, e, found in a similar manner, draw similarly converging lines, indicating the inner angles of the intervals. Further, find the circular arc to represent the epicycloidal curve, passing through the points, w, p, and tangential at the same time to the lines, o e, at the points, e, e. The method of doing this is shown in fig. 3 : it consists in draw- ing through the point, e, a line, e z, at right angles to the radius, o e, and in bisecting the chord, e w, by a perpendicular cutting, e z in z, which will be the centre of the required arc. Arcs of the same radius, e z, are employed for the curves of all the teeth on the smaller wheel, and the outline of these is completed by determin- ing, in a similar manner, the arcs for the corresponding curved portions of the inner ends of the teeth, after ha\Tng projected and drawn circles through the points corresponding to r and p'', of fig. 1. Finally, the lines of the web between the teeth — that is, the bottom lines of the intervals — are drawn, the projections of the points, y and y', being found, and circles described with the centre, o, passing through them. It will be observed that, on a portion of fig. 3, is represented a view of a quarter of the lower and inner side of the wheel, whilst the other portion of the figure is an external view, showing the teeth as in plan ; in the former case, the outline resembles that of a spur-wheel, for, as it is the larger ends of the teeth and web on which we are looking, the narrower and mclined portions are hid behind. The lateral projection of the teeth of the small wheel, fig. 1, is obtained, first, by successively projecting or squaring over, from the plan, fig. 3, the points, e, e, to the pitch line, g h; and secondly, by similarly squaring over the points, p p, to the gxteri'al line, p p'. Through the points, e, e, draw a series of lines eonverging at the apex, b, and representing the flanks of the teeth, and liinited by the line, y y; then draw curves tangential to the.'^e flanks at the points, w, p, making them pass through the extreme points, e, e. Where the scale of the drawing" is very large, and it is wished to be particularly precise in delineating these curves, points intermediate between m, p, and e, e, may be obtained by describing intermediate circles in fig. 3, representing sections of the cone, projected in straight lines in fig. 1, over to which are projected the points of intersection of the curves, \vith the circles in fig. 3. Through the points, u p, draw straight Unes, converg- ing in the apex, a, and find the lateral projection of the inner end? of the teeth, supposing planes to pass through the points, r, y' and p' ; these points in the circular projection of the planes, fig. 3, being squared over to the corresponding rectilinear projections in fig. 1. The inner ends of the teeth are then completed by drawing the flanks, e', y', all converging in the apex, s, and joined by arcs passing through the points, e', p'. The upper left-hand quadi-ant, ivi, of fig. 2, is a face view of the teeth of the larger bevU-wheel, with wooden teeth, the whole being drawn in the same manner as in fig. 3. The method of find- ing the centre, z, of the arc, which is substituted for the cuE\-ed portion of each tooth, is shown in fig. 2°. From this view (fig. 2) are obtained the various points required to produce the lateral projection of the teeth in fig. 1. The operations are precisely the same as those just described in reference to fig. 3, and the smaller wheel ; the same distinguishing letters are also used to point out the simOarity. The same figure (2) also comprehends at n a second quadrant of the wheel, drawn as seen from the under side, so as to show a face view of the tenons of the wooden teeth, the sides of which all converge in the point, o'. A third quadrant, p, gives a view of the outer side of the web or crown, the teeth being supposed to be removed, so that the mortises are seen. The last quadrant, e. gives a back view of the web, also without the teeth. Fig. 6 is a section of one of the arms of the larger bevil-wheel, made through the line, 1 — 2, in fig. 2. Fig. 7 is a section of the web made through the line, 3 — 4, fig. 2, passing through the centre of the mortise ; and fig. 8 comprehends a lateral projection and two end views of one of the wooden teeth. BevU, as well as spur-wheels, are fixeci on their shafts by means of keys, and pressure screws, v, are often added to insure their perfect adjustment centrally. The measurements given in the diagrams will enable the student to form an accurate idea of the actual proportions of the variou.-- parts. the coksteuction of wooden patterns for a pair of bevtl-wheels. Plate XXTTT. 228. The observations we have already made with reference to the patterns of spur-wheels, are evidently equally applicable to the construction of patterns for bevD-wheels ; stUl, at the same time, the difference in the form of the latter calls for further details, more especially appertaining to them. PATTERN OF THE SMALLER BEVIL-WHEEL. 229. F^gs. 1 and 2 represent the two projections of the pattern of the smaller of the tw"o wheels in the preceding Plate. Fig. 3 is a vertical section through the line, 1 — 2, of fig. 2, showing on one side the layers of wood put roughly together, and intended tn BOOK OF INDUSTRIAL DESIGN. 8} form the crown ; and on the other, a view of the same as finished, with the arm and its feathers. It will be seen from these figures that the crown is built up in tlie same manner as that of the pinion in Plate XXI. ; the layers of wood are, however, in steps, increasing in diameter dovmwards, so as to give the required conical form when turned. When these pieces are glued together, the whole is turned externally and internally in such a manner as to conform exactly to the full-sized drawing, previously made on a board planed smooth for the pur- pose. " Squares," also, should be made from the dravidngs, to serve as guides in producing the correct conical inclination. After turning the top face, V b', perpendicular to the axis of the cone, the pattern-maker proceeds to turn the external conical surface, a' V, of the web or crown. As a guide in doing this, he takes a "false square," t, fig. 4, of which one side, h b, corresponds to the plane face, b' b', and the other, a b, to the inclination of the conical generatrix, a' b' : it is very easy with this to take off just as much of the wood as is necessary, without the liability of going too far. It is also necessary to determine the inclination of the generatrix, b a', of the outer cone, perpendicular, it will be recollected, to the contact generatrix, g r, by means of the square, X, fig. 3, the side, a b, of which is applied exactly to the conical surface, a' b', and the side, a c, then gives the inclination of the conical surface, a' c' ; and the same square being turned round wdll give the inclination of the internal conical surface, b' d', the gene- ratrix of this, the smaller cone, being s r, parallel to b g, that of the larger one. Finally, the thickness at a' c' and b' d', is measured on the wooden web, so as to obtain the proportions of the internal conical surface, c' d', to be turned out in a similar manner. Mortises have now to be cut in the crown to receive the ends of the arms, c, and their feathers, e. As the wheel under considera^ tion is of very small diameter, the number of arms is limited to four ; these arms are so placed inside the crown that the feathers are all on one side, and towards the wider end of the cone. Their attachment to the web is by means of a circular groove or mortise, seen at e'/', fig. 2, and at g' d', fig. 3, and they are united at the centre to each other and to the boss, in the same manner as the arms of the pinion, described in reference to Plate XXI. The arras are not placed in the middle of the boss, as in the spur-wheel and pinion, but are simply applied to the base of the boss, which may, consequently, bo of a single piece ; and the feathers are let into a groove extending their whole length, and are fixed into the boss and (M'own at either extremity. The boss is slightly coned, so as to give the " draw " necessary in the construction of the mould. Its outer edges are indicated by the lines, mn,mn, whilst the other lines, o p, which are, on the contrary, parallel to the axis, «how the depth of the grooves cut to receive the feathers of the arm. Those last, as shown in the section, fig. 10, ;uo tliicker near the arms. The core pieces, e, are added on either end of the boss, and lln^ whole is held firmly together by means of a central bolt. The pattern being so far advanced, the external conical surface if* divided into as many equal parts as there are to bo teoth and uitervals, and, with the assistance of the " false sciuaro," T, lines wlmii represent generatrices of the cone, are drawn through the points of division, to indicate the positions of the teeth or of the grooves to receive them. Each tooth is cut out separately according to the full-size draw ing made, as already mentioned, which, besides containing the ver- tical section, fig. 3, should also show the exact form of each end of the tooth, b', and of the dovetail joint attaching them to the web Fig. 5 shows a portion of this drawing for the larger ends of the teeth. PATTERN OF THE LARGER BEVIL WHEEL. 230. Figs. 6 and 7 represent the elevation and plan of the pattern of the larger bevU wheel, with wooden teeth, represented in Plate XXJI. Fig. 8 is a vertical section through the axis of the wheel, showing on one side the arrangement of the pieces of wood built up upon one another, and forming the crown, a, and on the other side, the same piece, turned and finished, attached by the arm, c, to the boss, d. Fig. 9 represents the false square, T, employed as a guide for giving the proper incUnation to the external conical surface, a' b', of the crown. Fig. 11 is a transverse section of one of the arms, or spokes, taken through the line 7 — 8 in fig. 7. Whatever explanations are called for regarding the construction of the crown, a, the arms, c, and the boss, d, as well as the uniting of these parts with each other, have already been given in reference to preceding examples. We have distinguished all corresponding parts and working lines by the same letters. The only difference between this last and the preceding example consists in tlie disposition of the tooth pieces, b', placed on the out- side of the crown, to form the sockets in the mould for receiving the core pieces for the mortises, into which the wooden teeth are to be fixed after the piece is cast. It must be observed, in the fii'st place, that these projections must be shaped so that the end, k I, is inclined to the surface, b' a', instead of being perpendicular to it. This inclination must be sufliicient to allow of the easy disengagement of the piece from the mould. This disposition is necessary, because the lower half of the mould takes the hnpression of the outside of the crown, with the tooth pieces and the upper portions of the arms, wliilst the top part of the mould takes the inside of the crown, the feathers of the arms, and the boss, the position of the whole being the reverse of that in which they are represented in the drawing. The core pieces for the teeth are formed by the moulder in core boxes, similar to those described in reference to figs. 10 mid 11, Plate XXI., which wo have reproduced in Plato XXIII., figs. 12, 13, and 14, as modified to suit the dillerent form of tooth. Fig. 12 is a face view, and figs. 13 and 14 are sections made through the lines 9—10 and 11—12 otMig. 12. It will be observed that, at the larger (Mil (it llic tonlli. the part to project is foniieci with an iiuTniatioli corrt'sponding to /,■ /, in tig. 7, already reforn-d lo Jis rciiuircd in this case. The operations (■•■illcd for in dclincMling those piiftorns are all fully indicated, and are analogous to those in the pn>ceding plalos. Tli<« observations, also (It!) and 214), already made, with referenco to calculating the allowance to bo made for shrinking, and for tJio turning and tiuishing proces.ses, are eiiually jipplicablc to Iho awo before us. h9, THE PRACTICAL DRAUGHTSMAN'S INVOLUTE AND HELICAL TEETH. PLATE XXIV. DELISEATIO.N OF A COUPLE OF SPUR-WHEELS WITH INVOLUTE TEETH. FlGUBES 1 AND 2. 231. In the various systems of gearing just discussed, wherein epicyeloidal teeth have been employed, it will have been observed — 1st. That the outline of the teeth of one wheel depends on the diameter of the other wheel with which it is in gear. 2d, That the distance between the centres of any couple of wheels cannot be altered in the slightest degree without deteriorat- ing the movement. 3d. That the distance from the respective centres of the point of contact varies throughout the duration of the contact; from which must obviously result irregularity in the action and inequality in the amount of friction. The practical defects arising from these causes have induced a search after other forms, and amongst these a modification of the involute has been tried. The form in question possesses the fol- lowing advantages : — Ist The form of the teeth of such a wheel is quite independent of the diam.ter of the wheel with which it is to gear. 2d. The distance between the centres of the wheels may be varied without disadvantage. Some authors also attribute to this form the property of trans- mitting the pressure uniformly throughout the duration of the eon- tact. This, however, cannot be the case altogether, for the distance of the point of contact from the centres of the wheels is constantly varying — ^the variation not being accurately proportional in the two wheels. This system of gearing is constructed on the following principles : — Let the centres, o and o', fig. 1, of the two wheels be given, and the radii, o a and a o', of the respective pitch circles ; also, let o B be the radius of any circle described with the centre, o ; to the circumference of this last draw a tangent, a b, passing through the point, a, and prolong it indefinitely in either direction. From the centre, o', let fall on this line a perpendicular, o' c, on which will accordingly be the radius of a second circle tangent to the same line. These circles of the radii, o b and o' c, are those from which are derived the involute curves, a b and c d, forming the outline of the teeth. For the rest, the wheels are drawn just as in Plate XVIH. (197.) It must be observed that the curve, a b, which is the involute of the circle of the radius, o b, is that for the tooth of the spur- wheel, the centre of which is the same, and the radius, o a : and, in like manner, the curve, c d, the involute of the smaller circle, is that for the teeth of the pinion of the radius, o' a. It thus follows that the form of the teeth of the spur-w^heel is quite independent of the diameter of the pinion, whilst that of the pinion teeth is independent of the diameter of the spur-wheel. From which it follows, that wheels constructed in this manner may be set to gear with any wheels whose teeth are formed on the same principle, and whose pitch is the same, whatever may be their respective ntameters. The epicyeloidal system does not admit of this, although, when the wheels are large, and there is not much dif- ference between their 4iameters, a slight de\iation from strict mathematical proportions is not found practically inconvenient. The mvolute curves, a b and c d, are repeated sjinmetrically on either side of the division lines representing the centre lines of the teeth. If we now suppose the two involutes, a b' and a c', to be in contact at the point, a, on the line of centres, o o', and we measure off" on the common tangent, a B, a distance, a e, equal to the pitch, /g', as measured at the pitch circle, and then, with the centre, o', describe a circle passing through the point, e, this circle will be the external limit of the pinion teeth. In like manner, if, on the other portion, a c, of the tangent, we measure a distance, a e', also equal to the pitch, /^, and with the centre, o, describe a circle passing thi-ough tlie point, e', it will be the limi t of the spur-wheel teeth. It is further obnous, that circles passing a little within the point, e', on the one hand, and e, on the other, will determine the depth of the intervals, or the line of the web of the pinion and spur-wiieel respectively. Fig. 3 is a diagram to show — first, how that the point of con- tact of the two involute curves is always in the line of the common tangent, b c. Thus, referring again to fig. 1, and supposing the pinion to turn in the direction of the arrows, the point of contact, as A of the involute, a 6, is grad'ially removed away from the centre, o', of the pinion, whDst it approaches nearer and nearer to the centre, o, of the spur-wheel. Returning to fig. 3, it is showTi, in the second place, that the distance between the two centres, o, o', may be varied without its being necessary to alter the cunes ; but, in such case, the inclination of the tangent \viU be different, becoming, for example, as b c', when the two centres are brought nearer together. In practice, instead of determining the radius, o B, arbitrarily, and then deri%ing the other radius, o c, from it, or vice versa, the circles which sene for generating the involutes may be found, as w'ell as the inclination of the tangent, by the following method : — On one of the pitch circles, that of the pinion, for example, take an arc, a ?', equal to the pitch of the teeth ; draw the radius, o' i, and on it let fall a perpendicular, a m, from the point, a ; o m will then be the radius of the generating circle for the involute curve f i of the teeth of the pinion, and by prolonging ot a to n, wliich is, in M fact, the common tangent, and drawing the radius, o tj, perpendi- cular to it, or, what is the same thing, parallel to o' m, o n will bo the radius of the generating circle for the involute of the teeth of the spur-wheel. If this rule is applied to wheels of large diameters, it will give curves dififeiing very slightly from epicycloids. By taking for the generatrng circles, as in the first case, radii, o B and o' c, sensibly less than the radii of the pitch circles, the inclination of the common tangent to the line joining the centres is greater, and the resulting form of tooth possesses greater propoi- tionate width and strength at the roots, which is desirable foi gearing intended to transmit great or irregular strains. It will be observed further, that, according to this system, tho rectilinear portion of the flank of the tooth is almost reduced to nothing, indeed the cur^-e may be continued down to the line of the web with advantage, as the tooth will, in consequence, be much stronger near the w-eb, which is not the case with the epicyeloidal BOOK OF INDUSTRIAL DESIGN. 83 toeth, for in these the flanks all converge towards the centre of the wheel, and the tooth is, in consequence, narrower at the neck, close to the web, than at the pitch circle. Fig. :3 is a fully shaded elevation, or vertical projection of the spur-wheel separated from the pinion. The portions of these wheels not particularly referred to, are constructed on the same general principles as those previously discussed. helical gearing. Figures 4 and 5. 232. If to a worm-wheel we apply, instead of a worm, a pinion with teeth helically inclined to correspond to the similarly inclined teeth of the worm-wheel, we shall have a spur-wheel and pinion constructed on the helical principle. This system, invented in the seventeenth century by Hooke, but reproduced since by White and others, claims to possess two properties which have been often thought to be incompatible with each other — ^namely, uniformity of angular velocity, and freedom from other than rolling friction between the teeth. In other words, the arcs described by driver and follower will be equal in equal times, and the contact between the teeth will resemble that of circles rolling on planes. Added to these properties, and consequent to them, are the advantages of a constant contact, and of an insusceptibility to the play between the teeth, which invariably exists more or less palpably in gearing constructed according to the systems before described. The form of the helical teeth, as taken in a sectional plane at right angles to the axis of the wheel, may be derived either from a couple of epicycloids, or a couple of involutes ; it is only the sides which, in common spur-gearing, are parallel to the axis that here follow the inclination of a succession of helices coming in contact one after the other. The arrangement is such that the contact of each tooth commences at one side of the wheel and crosses over to the other, and does not cease until the following tooth shall have commenced a fresh contact. Thf helicoidal system may 1)0 applied either to wheels having their axes parallel, as spur-wheels, or intersecting, as bevil-whcels, or again inclined, but not intersecting, as skew bevils. In figs. 4 and 5 are represented, in face and edge view, a spur- wheel and pinion, constructed according to this system of Hooke's, this being its simplest iind most common application : — Let a o an.l a' o be the radii of the respective pitch circles of the two wheels, these radii being, of course, in the same ratio as the numbers of the teeth, as in common gearing. The radii are supposed to lie in a vertical plane, b' c', and it is on this plane, as turned at rigiit angles, that the operations represented in fig. 4 are sujjposed to bo performed. Tiiese operations have for their ohjoet the ohtainnient of the outline of the teeth, and are i)recisely the same as for any oilier epicycloidal system of gearing. Thus, the curves, a I> and a c, are derived from the generating circles, o n a and a n' o', as also fho flanks, a (/ and a /;; hut it is uiineccssai-y to repeat a, dclailecl explanation of the proceeding. Supposing, thi^n, the outline of the (cetli lo jx? .Irauii as nn lln^ plane, b' c', representing say the anterior face or base of the wheels, next draw the line, e f, (fig. 5,) representing the opposite face, and parallel to the first, limiting also the breadth of the wheels. To proceed methodically, the teeth should also be drawn as seen on this plane, e f being behind the outlines of the anterior ends of the teeth, a distance equal to a a', or rather more than the pitch. These last outlines need only be represented in faint dotted or pencil lines in fig. 4, as the parts they represent are not actually seen in that view when complete. Thus, starting from the point, a', on the pitch circle of the spur-wheel, and from the point, a', on the pitch circle of the pinion, we repeat the contours of the teeth, as obtained at e a i and d a n, respectively. As the result of this disposition, it will be observed, that if the curve, A i, of the tooth, a, of the spur-wheel is in contact, at the pitch circle, with the flank, g d, of the tooth, g, of the pinion at the anterior face, b' c', and if the wheels be made to turn to a cer- tain extent in the direction of the arrows, the curve, a' i', on the opposite face, e f, wUl in time be found to be in contact with the corresponding flank, g' d', of the pinion. In other words, if the space between the curves, a i and a' i', be filled up by a helicoidal surface, as also the space between the flanks, g d and g' d', all the points of one such surface will be in contact successively with the corresponding points on the other ; so that when, for example, the curve A i', shall have reached the position, a^ P; that is, when it shall have passed through a distance equal to a a', the posterior curve, a' i, will have assumed the position held originally by a t ; or rather, a position directly behind this in the plane passing through the axis, and the point of contact between a' i' and a' d' will then obviously be in the line of centres, o o'. It thus follows, that any'tvv'o teeth which act on each other will be constantly in contact on the line of centres throughout a space equal to a a'. This space, a a', is, as before stated, somewhat greater than the pitch of the teeth, so as to allow a following couple of teeth to act on each other, and be in contact on the line of centres before the couple in advance shall be quite free, and thus a constant contact on the line of centres is preserved throughout the entire revolu- tion. In order to delineate the lateral projection, fig. 5, it w ill bo necessary to find the curves which i'oiin the outline of the helicoidal surfaces of the teeth. The princi])h', according to which this is to be done, is precisely what has already been explained (208). In the present case, however, as we have but IVaginonts of helices to draw, in place of finding the pilch ut llic lirlix. and then (li\idiii^ it and the circnnd'erfnce proportionately, it will be sullieient to divide tho width, b' e, of the wheels, into a certain nundier of equal parts; and through the points of liivision, to draw lines parallel to b' c'. Further, tho arcs, a a', v c', / /', must be divided into a like number of equal parts. To render the diagram dearer, these divisions are Iranst'orrcil to 1, 2, 3, 4, &c., and 1', 2', 3', 4', &c. (fig. 4.) Each point, 1,2, 3, I, beinf squared over, in succession, to tiio corresponding linos m fi,r. 5 — namely, the lines of division first obtained, and lying ]>arallcl to the faci-s of thi< wheels, the operation will give tho iMirvc, 1, 3, 6, 6, (fig. ft,) corresponding to tho ontlino of tho exter- nal edge, extending from i to i'. The curve. 1', 3', !i\ fi', siiniliirly gives the other edee. it is alse oli\ ious thai ihe lino of junction 84 THE PRACTICAL DRAUGHTSMAN'S of the tooth with the web will be represented by the helical curve, a a\ (fig. 5), having the same pitch as the last, but lying on a cylin- der of a somewhat smaller diameter. The lateral projections of all the teeth are determined in the same manner, but they will, of course, assume various aspects, from the different positions in which they lie with respect to the vertical plane. 233. In construction, in order to determine the exact inclination of the teeth, the following proportional formula is employed. The four terms of the formula being, the radius of the wheel, its width, the given circumferential distance, corresponding to a a', and the pitch of the helix ; that is, a a' : a o : : b' e : x, x being the heli- cal pitch for the spur-wheel, or the quantity sought. It may be obtained geometrically, simply thus : — Make the straight line, M n, (fig. 6,) equal to the arc, a a', as developed ; at the extremity, n, of this line, erect a perpendicular, n l, equal to the width, e' e, of the wheels ; join l m, which will give the mean inclination of the tooth, corresponding to the pitch cu-cle. Then make n i equal to the arc i i', rectified, and n j equal to the arc, e e', rectified, which will give the inclinations, l i and l j, of the helices, passing through the extremity, i, of the tooth, and the line, e, of junction of the tooth to the web. It will be understood that the helices of the pinion-teeth will have the same inclination as those of the spur-wheel teeth, with which it is in gear, and the helical pitch is, in consequence, differ- ent ; for, the radius is smaller, and the corresponding proportional formula becomes a a' or a g' : a o' : : b' e : x. The motion of wheel-work, constructed according to the helical system, is remarkably smooth, and free from \ibratory action, but it has the defect of producing a longitudinal pressure upon the axes, from the obliquity of the surfaces of contact to the plane of rotation. This, however, may be obviated, and the longitudinal action balanced, by making the wheels duplex ; that is, as if two wheels, on each axis, were joined together — the inclination of the helices being in contrary directions, or right and left handed. Such wheels, though duplex, need not be wider or thicker, in proportion, than simple ones ; for the arrangement would permit of a much greater obliquity of the teeth, the only limit, indeed, to the degree being the tendency to jam, which would arise were the inclination very great. When the wheels are placed on axes which are inclined to each other, as in common bevil-wheels, the helices become such as are described upon conical surfaces, and require to be drawn in the manner already shown (174), the form of the tooth being previously determined, for each end, by means of the developed planes of the opposite faces of the wheels. Besides the epicycloid and involute and their various combina- tions, other and more complex curves have at different times been proposed for the forms of wheel teeth. The most worthy of notice amongst these is that derived from the "hour-glass " curve, the properties of which have lately been investigated in a very scientific manner by Professor Sang of the Imperial School at Constantinople. If a couple of discs, with their pitch circles touching, be made to revolve at a rate proportionate to the required number of teeth in each, a point may be imagined as travelling along a curve, returning upon itself in such a manner that it will describe the forms of the respective teeth on each disc. In the system of teeth alluded to, this point is made to travel along the "hour- glass " curve, a curve similar to that described by the piston-rod attachment in Watt's parallel motion, and also exhibited by the vibration of a straight wire, whose breadth is double its thickness. The form of tooth obtained in this manner is demonstrated by its inventor to be theoretically superior to all others yet known. The chief advantage appears to be, that whilst according to the epicy- cloidal and involute systems, the form of the entire tooth is made up of two ciu-ves of different natures, whose junction cannot, in consequence, be perfectly smooth or fluent, the point of inflexion or passage from one curve to the other, occurring, moreover, precisely where the best action would otherwise be. The " hour- glass " curve produces one continuous analytic curve for the entire outline of the wheel, thereby avoiding all sudden transitions, such outline, at the same time, allowing of the interchange, in any way, of wheels of the same pitch. The great exactness and nicety obtainable by and called for in the construction of teeth on this system, is, however, far beyond the requirements of ordinary machinery. Indeed the practical engineer and machinist will not be at the trouble of emplo}ing even epicycloidal or involute curves, but contents himself with arcs of circles approximating pretty nearly to these curves. The method generally pursued in determining the best proportions for the radii of these substitutive ares is as follows : A pair of templets or thin boards are cut to the curvature of the pitch circle and generating circle, respectively, of the whe^l, the shape of whose teeth is sought. The generating templet carries a point which is made to describe the outline of the tooth on an additional board, by rolling its edge on that of the pitch templet. The operator then finds by trial with a pair of compasses, a centre and radius which will give an arc agreeing as nearly as possible with the curve traced by the templet. Through the centre thus found he describes a circle concentric with the pitch circle, and in which the centres for the arcs of all the teeth will ob%iously lie, and retaining the radius, he steps from tooth to tooth in both directions, until all the teeth are marked out. A very ingenious and useful scale was invented some years ago by Professor Willis, which renders unnecessary this preliminary operation for obtaining the radii and centres. This scale, termed the " Odontograph," is now largely emplo}'ed, and is found to give very excellent forms of teeth. Its application is very convenient. A graduated side of the instrument has a certain inclination to another, which is first made to coincide with a radius of the wheel, whilst its point of intersection with the first is placed in the pitch circle. The graduated side gives the direction in which the centres lie, whOst the lengths of the radii are obtained from tables calculated for the purpose, indicating the respective distances on the graduated scale, and corresponding to the given pitch and number of teeth. Wheels with teeth formed according to this scale are capable of being interchanged, which is not the case with those in which the arcs are determined according to other rules. After going through the explanations given, and rules laid down in the last few sections, the student should be quite competent BOOK OF INDUSTRIAL DESIGN. 85 to design practical arrangements and combinations of toothed gear according to whichever of the systems may be preferred. CONTRIVANCES FOR OBTAINING DIFFERENTIAL MOVEMENTS. THE DELINEATION OF ECCENTRICS AND CAMS. PLATE XXV. 234. Eccentrics and cams are employed to convert motion, whilst toothed-wheel work is for the simple transmission of it. Endued themselves \vith a continuous circular movement, they are so constructed as to give to what they act upon, an alternate rectilinear movement, or an alternate circular movement, as the case may be, the motion so produced being obtainable in any desired direction. CIRCULAR ECCENTRIC. 235. There are several descriptions of eccentrics. The simplest and most generally employed, consists of a circular disc, completely filled up, or open and with arms, according to its size, and made to turn in a uniform manner, being fixed on a shaft which does not pass through its centre. Such eccentrics are represented in Plate XXXIX. The stroke of such a piece of mechanism is always equal to twice the distance of its centre from that of the shaft on which it turns ; that is to say, to the diameter of the circle described by its centre during a revolution of the shaft. The motion of the piece acted upon is uninterrupted during either back or forward stroke, but it is not uniform throughout the stroke, although that of the actuating shaft is so ; the velocity, in fact, increasing during the fii'st half of the stroke, and decreasing during the second half. heart-shaped cam. Figure 1. 236. When it is required to produce an alternate rectilinear motion which shall be uniform throughout the stroke, the shape of the eccentric or cam is no longer circular; it is differentially curved, and its outline may always be determined geometrically when the length of the stroke is known, together with the radius of the cam, or the distance of its centre from the nearest point of contact. An example of this form of cam is represented in the figure. Let a a' be the rectilinear distance to be traversed, and o, the centre of the shaft on which the cam is fixed, it is required to make the point, a, advance to the point, a', in a uniform manner during a semi-revolution of the siiaft, and to return it to its original position in the same manner during a second semi-revolution. With the centre, n, and radii, o a, and o a', describe a couple of circles, and divide them into a certain number of equal jiarts by radii passing through the points, 1, 2, 3, 4, &c. Also divide the length, a a', into half as many equal parts as the circles, as in the points, 1', 2', 3', &.C. D(^scribe circles passing through these points, and concentric with tiie first. Th(^so circles will succc^s- sively intersect the radii, o 1, o 2, o 3, &.C., in the points, b, c, </, c, &<!., and the continuous curve passing through those points vvili be the theoretical outline of the cam, which will cause the poLntj a, to traverse to a', in a uniform manner, for the equal distances, a' V, V 2', 2' 3', &c., passed through by the point, a, correspond in succession to the equal angular spaces, a' 1, 1 — 2, 2 — 3, &c., passed through by the cam during its rotation. As it is not possible to employ a mathematical point in prac- tice, it is usually replaced by a friction roller of the radius, a i, which has its centre constantly where the point should be ; and it will be seen, that in order that this centre may be made to travel along the path already determined, it wDl be necessary to modify the cam, and this is done in the following manner : — With each of the points, b, c, d, &c., on the primitive curve as a centre, describe a series of arcs of the radius, a i, of the roller, and draw a cun-e tangent to these, and such eurv'e will be the actual outline to be given to the cam, b. It will be seen from the drawing, that the curv'e is symmetrica], with reference to the line, a e, which passes through its centre ; in other words, the fii-st half which pushes the roller, and conse- quently the rod, a, to the end of which the roller is fitted, from a to a', is precisely the same as the second half, with which the roller keeps iij contact during the descent of the rod from a' to a. Thus the regular and continuous rotation of the cam, b, produces a uniform alternate movement of the roller, and its rod, a, which is maintained in a vertical position by suitable guides. In actual construction, such a cam is made open, and with one or more arms, like a common wheel, or filled up, and consisting of a simple disc, according to its dimensions ; and it has a boss, by means of which it is fixed on the shaft. When it is made open, it is cast with a crown, of equal thickness all round, and strength- ened by an internal feather, curved into the boss at one side, and into the arm or arms at the other. Examples of the heart-shaped cam are found in an endless variety of machines, and particularly in spinning machinery. cam for producing a uniform and intermittent movemesl. Figures 2 and 3. In certain machines, as, for example, in looms for the " picking motion," eases occur where it is necessary to produce a uniform rectilinear and alternate motion, but with a pause at each extremity of the stroke. The duration of tiie pause may bo equal to, or greater, or less, than that of the action. Fig. 2 represents tho plan of a cam designed to produce a movement of this description ; and in this case the angular space passed through by the cam, in making the point, a, traveree to the position, a', is supposed to bo equal to half the angular space described by it, whilst the point, a, is stationary, whether in its position nearest to tlie centre, or its furthest, a', from it. For this reason, tlie cin-les of tho ra(hi, o a, and o a' are caili (li\ idiil into six (•<|ual ]iarts in iho points, a', 1, /, ^'S h, and J. Of these portions, tfii' two opposit.-. 1 /"and j h, corresi)ond to the eccentric curves, b f and / A, which produce the niovenu'Ml, whilst the otlu'r portions correspond to tliu l)auses. After liavini,'' diawn tlu' dianu'lers, 1 /i, and /" /, the ecceniru' curves, b f, and / /i, an- deterniined in precisely tlie sjuno manner as tile contiinious enrve :ihi'ady discussed, and represented in fiy. 1. 'J'liat is t(i s:iy. Hie aii's I /', and j /i, are U- he ilivided into « 80 THE PRACTICAL DRAUGHTSMAN'S certain number of equal parts by radial lines; and the line, a a', being divided into a like number of equal parts in the points, 2', 3', 4', &c., concentric cu-cles are to be di-a\\-n through those points, which will be intersected in the points, c, d, e, by the radial division lines. Lines passing through these points of intersection will be the curves sought, bf, and I h. The arcs, b a I, and fg h, which unite the extremities of the curves, are concentric with the shaft, and consequently, as long as the point remains in contact with these arcs, it will continue with- out motion, although the cam itself continue its rotation. The observation made with reference to the preceding example of a cam, applies equally to the one, c, under consideration — that is, with regard to the actual shape to be given to it, which is derived from the substitution of a friction roller of the radius, a i, for the mathematical point, a. The operation is fully indicated on the diagram. This eccentric not being intended to overcome any great resist- ance, is made very light, a considerable portion of the metal being cut away, and merely a couple of arms left for stiffhess. The crown, arms, and a great part of the boss, are, in fact, all of a thickness, as will be more plainly seen in fig. 3, which is simply a section made through the line, 1 — 2, in fig. 2. Fig. 3 also shows the proportions of the roller, and its spindle. When the moving point, or the roller, is constrained to move through an arc, instead of a straight line, being, for example, at the end of a vibratory lever, the curves of the cam are no longer s3Tnmetrical, but the operations by which they are determined are stOl the same, the difference arising from the divisions of the arc, which takes the place of the straight line, a a'. TEIANGIILAR CAM. FiGUKES 4 AXD O. 238. A species of cam, in the form of a curvilinear equilateral triangle, is sometimes employed in the steam-engine, to give mo- tion to the slide valve. This valve is generally of cast-iron, of a rectangular form, as at t, figs. A and ©. It is hollowed out in its inner side, to form a passage, and it applies itself, with its inner planed edges, to a surface, a b, on the cylinder, d, also planed true, and called the valve face. Its function is to allow the steam to pass alternately to the upper part of the cylinder, by the port, c, or to the lower part, by the port, d, whilst the hollow part of the valve forms a communication alternately between either of these ports, c, d, with the escape pipe, e. To obtain the desired effect, it is necessary that the slide valve be actuated with an alter- nate rectilinear reciprocatory movement; for this purpose it is attached to a vertical rod, i, passing through a stuffing-box in the valve casing, and connected to the rod, w, represented in fig. ©, and forming one piece, with the rectangular frame, f, inside, which works the triangular cam, g. It is the last piece which has to effect the raising and lowerinor of the valve a certain distance and intermittently, in such a man- ner that the port, c, for example, may be open to the entering steam for a certain time, whilst the other, d, is in communication with the escape pipe, and reciprocally. Let e, fig. 4, be the whole stroke of the valve, or the dist<mee Uirough which it traverses; with the centre, o, and with this distance, o e, for a radius, describe a circle, and divide it into six equal parts, in the points e, 1, 2, 3, 4, and 5. With any two adja- cent points, as 1 and 2, and with the same radius, o e, describe two arcs, o 2, and o 1, so as to form the curvilinear triangle, o — 1 — 2, which is exactly the outline of the eccentric, g, each side of which . is equal to a sixth of the circumference. Draw the parallels, 5 — 1 and 4 — 2, tangential to the two side." of the triangle, g, and we shall thus obtain the upper and lower internal surface of the frame, f. The cam is made of steel, as well as the two sides of the frame, F, which bear upon it. It is adjusted and secured by the screw- bolt, h, to the disc, h, keyed on tlie shaft, j, as shown in the hori- zontal section, fig. 5, taken through the line, 3 — 4, in fig. 4. It wOl be easOy conceived, that if the shaft turns in the direc- tion of the arrow, the curved side, o 1, of the cam, acting against the upper side of the frame, will cause it to rise, carrying with it the rod, u, in such a" manner, that when the point, 1, shall have reached the position, e ; that is, when the cam shall have made a sixth of a revolution, this side of the frame will occupy the posi- tion, m n. thereby indicating that the slide-valve has been elevated to a distance equal to half o e, and that, in consequence, the port, d, is uncovered, so as to allow the steam to enter the lower part of the cylinder (fig. (B) ; W"hilst, on the other hand, a communica- tion is established between the upper port, c, and the escape orifice, E, so that the steam can pass out from the upper end of the cylin- der. If the movement of the cam be continued during a second sixth of a revolution, the slide-valve will remain in the same posi- tion, because the arc, 1 — 2, w-hich is concentric with the axis, does not change the position of the frame, as long as it is in contact with its side, m n. As soon, however, as the point, 2, of the cam reaches the position, e, the side, o 1, will be in the position, o 5, and it will, in consequence, be in contact with the low"er side of the frame, w"hich is in the position of the horizontal centre-line, 3 — 4. The further revolution of the cam, therefore, makes the frame descend from its pressure on the lower side, until the side, o 1, of the cam, occupies the position, o 3, w^hen the lower side of the frame will occupy the position, m' n', corresponding to the position of the valve, represented in fig. [B. It follows from this aiTangement, that the valve will remain stationary when it arrives at each extremity of its stroke, and the pause each time wiU be of a duration corresponding to one-sixth of a revolution of the cam shaft. The upward and downw'ard movements each take place during a third of a revolution, and the velocity of the valve will not be uniform, although the rotation of the cam-shaft is so. In actual construction, the angles of the cam are slightly rounded off, to avoid a too sudden change of motion, and to prevent the too rapid wear of the sides of the frame. involute cam. Figures 6 axd 7. 239. In certain industrial arts, an instrument is employed for pounding, crushing, and reducing substances, such as plaster or tanbark, for example, in which the direct-acting force is the w-eight of the instrument itself brought into play by its descent through a determined height. The mechanical forge-hammer is a well known working application of this expedient. BOOK OF INDUSTRIAL DESIGN. 87 In these cases, the stamp, or hammer, has to be raised or tilted up p.eparatory to each succeeding stroke, and it is obvious that this may be most economically done in a gi-adual manner. It is fcnci-ally eflt'ected by a cam, the outline of which is the involute c'.ir.e already described ; this form being preferable on account of iiie uniformity of its action. The office, then, which the cam under consideration has to fulfil, is the raising of the stamp, or load, to a certain height, and t'len the letting it fall, without impediment, upon the object sub- n itted to its action. The diameter of the cam-shaft being predetermined, as well as Ir at of the generating circle, which last is usually the same as that ol the boss of the cam, the design is proceeded with as follows : — Letting a be the cam-shaft, and taking a o as the radius of the ge lerating circle, whilst a a! is the height to which the projection, K, r.g. ©, formed on the stamp, c, is to be raised, develop the circuiiiference (197) of the circle of the radius, a o, by means of a seiies of tangents which give the points, c, d, e, &c., the curve passing through which forms the involute, hfi. The inner por- tion, h o, is not a continuation of the involute, but simply joins the boss with a circular turn, because the stamp projection, b, does not approach the cam-shaft, a, nearer than the point, a, to which 6 'oriesponds. Through the point, a, draw the vertical, a a', and make it equal to the height to which the stamp has to be raised; then with the centre, a, and a radius equal to a a', describe the arc, a' m i, which will cut the involute in the point, i, and this pomt is consequently the outer limit of the cam. A little con- sideration will show that if the cam-shaft, a, be turned in the direction of the arrow, supposing that it is originally placed, so that the point b, coincides with a, it must necessarily raise the litting-piece, b, the lower side of which is indicated by the line, ni a, and will carry it by equal increments up to the position, m' a'. 'J'lie point, i', will then have attained the position, a', and the rota- tion continuing, the next moment it will pass it, when the cam will be entirely clear of the lifting-piece, b, and this last being unsup- ported, must fall by its own weight. The involute might have been derived from a generating circle of the radius, a a, and had this radius been adopted, the resulting curve would have been shorter, notvv'ithstanding that it would give the same extent of lift. The angular space passed over would also be less, and this would admit of a higher velocity of the cam- shaft, and the strokes might be given in more rapid succession, whilst on the other hand, a greater power would be required to raise the same weight. The cam we have represented in fig. [o), is such as is employed to actuate the chopping stamp of mills for reducing oak, or other bark, for the preparation of tan. The bark is placed in a kind of wooden trough, e, solidly fixed* into the floor. The stamps are armed with a series of cutters, n, in the form of crosses. The side of the trough next to the stamp is vertical, whilst the opposite side is elliptical in shape, and the matter under operation ha.s, con- sequently, always a tendency to fall under the stump. The stamps are kept vertical bv slides in which they work. They arc generally from 450 to 700 pounds Weight, and fall through a height of from 16 to 20 indicH. Fig 7 is ,'i pl.-iii i.l' llic cMrii as seen Cri in hVl.jw, .■ii'il t'lilly indicates the width of the rim, and of the boss, and the thickness of the feather or disc uniting the two. A series of such cams are frequently employed in different planes on the same shaft, actuating a corresponding series of stamps, and in such case they are arranged in steps so as to come into action one after the other. Two or more are also sometimes employed in the same plane, and working a single stamp. In this latter case, the generating circle requires to be of much larger diameter in proportion, but the principle of construction Ls how- ever the same. cam to produce intermittent and dissimilar movements. Figures 8 and 9. 240. In certain examples of steam engines, the valve movement is obtained from a species of duplex cam, which bemg formed of two distinct thicknesses, affords a means of adjustment whereby the valve may be made to move intermittently and at different rates, the proportions of which are variable at pleasure. The object of this is to form and shut off the communication between the cylinder with the steam-pipe, at any required point of the stroke. In other words, the arrangement permits of the working of the machine on the expansive principle, and of varying the " cut-off " point at pleasure within certain limits. We shall see, at a more advanced period, what is to be understood by the fore- going expressions. In designing cams of this class, we primarily determine the radius o a, of the cam boss, and the entire length, b c, of the stroke to be given to the valve-rod. This distance, which in the present mstance we shall take as equal to three tunes the height of the port, must not be traversed at one movement. On the contrary, a third only of this is at first passed through, with some rapidity, and the remaining two-thirds are traversed at a later period, in a continuous manner : in other words, after a third of the stroke has been traversed, a slight pause takes place before the remainder is traversed, and a second pause also occurs before the commencement of the return stroke. After describing a couple of concentric circles with the respec- tive radii, o a, and o c, and having determined tlie angular spaces, a d, and f g, corresponding to the times during which the valve is to remain stationary, and the spaces, g h, and of, corresponding to the duration of the movements ; divide the whole stroke, b c, into three equal parts in the points, i,j, through which describe circles concentric with the preceding. Through the points,/, g, h, draw radii, and produce them to/, g', and h'. As the cam will act on two friction rollers, g, diamotrioally opposite to each other, their radius is determined, -.is a e: one is di-awn with its centre, e, on the radius, o a produced, iuul tangen- tial to the circle described with that radius: the other, witli its centre, e , on the radius, o c produced, is, in like mnnnor, tangentia' to the circle described wth this railius. Between the two points, d and A-, and oomprisod within tiiei given angle, g o h. a curve, k I rf, is drawn and united by tangcnfiai arcs at either oxtieniily with tho cirilos of liio radii. </ o and » \, respectively, in such a iiiMniier as to avoid any sudden changi> of direction. Next di\ ide tho aiv, i,-' /i, into a certdn nuinlH-r of equiU parts in Ilie imiiils. 1, 2, \-c., and cnriy tiio radii jicross to 1'. "J , ^•i'. ; Hull, on .Mcli (if llu'se ladii, as a centre line, dosi>ritH> an are 88 THE PRACTICAL DRAUGHTSMAN'S corresponding to the radius of the roller, g, and tangential to the curve, k I d. By this means will be obtained the points, t s I, indicating the successive positions of the centre of the roller on the line, e e', when unpolled by the curve, kid. If, then, starting fi-om these several points, we measure on the con-esponding cross 'ines, 1 — r, 2 — 2', &c., distances equal to e e', which is obviously constant, we shall obtain the positions, r', s', l!, of the centre of the opposite roller, g', corresponding to those of the first. Then, with these points, r, s, t, as centres, describe arcs of the radius of the roller, and draw the curve, d' V f, tangential to them, and unite them to the circles of the i-adil, c o and/ o, in a similar manner to the opposite curve. The curve, d' I' f, will obviously, from its construction, be in contact with the roller, g', whilst the fii-st, dlk, is in contact with the other roller, g. In order that the rollers, g, g', may maintain then- relative position, and move in the same rectilinear du-ection, they are carried m bearings, h, forming, with four tierods, i, a frame which em- braces the cam and cam shaft, the middle of the rods being planed to rest and slide upon the latter. To one end of the frame is bolted the cast-iron connecting rod, J, fig. @, jointed to the be!I-crank lever, K. This last vibrates on the centre, u, and by its second arm actuates the link, v, con- nected to the rod, x, of the valve t, fig. H, above. In the position given to the cam and roller frame, in fig. 8, the valve is not cover- ing the upper part, c', and this remains open whilst the cam rotates through the angle, a o d, because the arc, a d, and its oppo- site, c d, are both concentric with the axis of the cam shaft, o. When, however, the point, d, shall have arrived at the position, a, supposing the cam shaft to continue to turn in the direction of the arrow, the cam will shortly pass through the angle, dog, and the projecting curve, dlk, will push the roller, g, to the right, and the opposite roller, g', being drawn in the same direction, will roll along the corresponding curve, d' I' g'. This movement will cause the valve to be raised to the extent of a third of its stroke, cor- responding exactly to the width of the port, c'. This port will, in fact, be completely closed when the radius, o k, of the cam shall have reached the position, o e. At this point, the valve is required to remain stationary for a short time, during which the cam, in continuing to revolve, describes the angle, g of. As soon, how- ever, as the radius, of, reaches the position, o e, the valve, and its actuating gear', will again move, and continue to do so, until the lower port, c', be completely open. This movement wOl take place whilst the cam describes the angle, foe, and is caused by the curve, a' b' c', which pushes the roller, g, and the frame still further to the right. The curve, a' b' c', is united by a gradual turn to the circles of the radii, o k and o c, in the same manner as the curves previously di-awn. The opposite and corresponding curve, a mn, is obtained in the same manner as d' I' g', opposite to, and derived from, the first curve, d I k. The operations in both cases are fully indicated on the diagram, and it has merely to be borne in mind that the object is to keep the two rollers, g and g', in contact with the cam in every position of the latter. After the cam has passed through the angle, foe, the valve, with its gear, remains stationary during another interval, in which the angle, c o d", is traversed, and then the first curve will com- mence to act upon the roller, g', and cause it, with the frame, to return from right to left, and the movements and intervals will take place in the same order as to time as in the up stroke of the valve already described in detail ; but the direction will be reversed — that is, the valve will perform its return stroke — until it reaches its original position, as represented in fig. [1. To proceed : it is easy to conceive the cam, as constructed in two pieces, precisely alike in all respects, and laid upon one another, as M and m', fig. 9, one M being fast to the shaft, whilst the other, m', is capable of being adjusted to the first in any relative position. Since the rollers, g, g', are long enough to be in contact with both, it wOl follow they will, in any given position, be acted upon by that half of the cam which projects most at that particular point; so that, if the curved portion, d I k, of one is turned slightly in advance, it will come into action sooner, and, by consequence, will cause the valve to shut off the communication between the steam pipe and the cylinder at an earlier period of the stroke. In this manner is obtained a means of varying the rate of expansion at which the engine is worked. Fig. 9 is a horizontal section, showng the two halves, m m', of the eccentric, and the arrangement of the details of the friction rollers, g g', and frame, h i. Fig. i?" is a front view of the valve face of a steam-engine cylin- der, showing the disposition of the ports. An innumerable variety of movements may be produced by the agency of cams ; but the principles of their construction are mostly the same as those just discussed, and the examples given will be a sufficient guide in designing others. RULES AND PRACTICAL DATA. MECHANICAL WORK, OR EFFECT. 24 L To work, considered in the abstract, is to overcome, during any certain period of time, a continuously replaced resistance, or series of resistances. Thus, to file, to saw, to plane, to draw bm-dens, is to work, or produce mechanical effect. Mechanical work is the effect of the simple action of a force upon a resistance which is directly opposed to it, and which it continuously destroys, giving motion in that direction to the point of application of the resistance. It follows from this definition, that the mechanical work or effect of any motor is the product of two indispensable quantities, or terms : — First, The effort, or pressure exerted. Second, The space passed through in a given time, or the velocity. The amount of mechanical work increases directly as the increase of either of these terms, and in the proportion compounded of the two when both increase. If, for example, the pressure exerted be equal to 4 lbs., and the velocity 1 foot per second, the amount of work ■wUl be expressed by 4 x 1 = 4. If the velocity be double, the work becomes 4 x 2 = 8, or double also ; and if, with the velocity double, or 2 feet per second, the pressure be doubled as well — that is, raised to 8 lbs. — the work will be, 8 x 2 = 16, or the quadruplicate of its original amount. In the term " velocity," " time " is understood ; so that, in fact, just as space or solidity is represented in teims of three dimensions, BOOK OF INDUSTRIAL DESIGN. 89 length, breadth, and depth, so also is mechanieal effect defined by the three terms representing pressure, distance, and time. This analogy gives rise to the possibDity of treating many questions and problems, relating to mechanical effects, by means of geometri- cal diagrams and theorems. The unit of mechanical effect (corresponding to the geometrical cubical unit) adopted in England, is the horse power, which is equal to 33,000 lbs. weight, or pressure, raised or moved through a space of 1 foot in a minute of time. The corresponding unit employed in France is the kilogrammetre, which is equal to a kilogi'amme, raised one metre high In a second. Thus, supposing the pressure exerted be 20 kilog., and the distance traversed by the point of application be 2 metres in a second, the mechanical effect is represented by 40 k. m. ; that is, 40 kilog., raised 1 metre high. This unit is much more convenient thau the English one, from its lesser magnitude. Indeed, when small amounts of me- chanical effect are spoken of in English terms, it is generally said that they are equal to so many pounds raised so many feet high. That is to say, this takes place in some given time, as a minute, for example. The time must always be expressed or understood. It is impossible to express or state intelligibly an amount of mechanical effect, without indicating all the three terms — pressure, distance, and time. It is to the losing sight of this indispensable definition, that we may attribute the vagueness and unintelligibility of many treatises on this subject. The French engineers make the horse power equal to 75 kilogrammetres ; that is, to 75 kilog., raised one metre high per second. The motors generally employed in manufactures and industrial arts are of two kinds — living, as men and animals ; and inanimate, as air, water, gas, and steam. The latter class, being subject only to mechanical laws, can continue their action without limit. This is not the case with the first, which are susceptible of fatigue, after acting for a certain length of time, or duration of exeilion, and require refreshment and repose. What may be termed the amount of a day's work, producible by men and animals, is the product of the force exerted, multiplied into the distance or space passed over, and the time during which the action is sustained. There will, however, in all cases, be a certain proportion of effort, in relation to the velocity and duration which will yield the largest possible product, or day's work, for any one individual, and this product may be termed the maximum effect. In other words, a man will produce a greater mechanical effect by exerting a certain effort, at a certain velocity, than he will by exerting a greater effort at a less velocity, or a less effort at a greater velocity, and the proportion of effort and velocity which will yield the maximum effect is different in different individuals. TABLE OF THE AVERAGE AMOUNT OF MECHANICAL EFFECT PRODUCIBLE BY MEN AND ANIMALS. Nature of the Work. Mpin weight elevated or effort exerted. Velocity or distance per second. Mechauical effect per second. Duration per diem. Mechanical effect per diem. A man ascending a slight incline, or a stair, without a burden, his work consisting simply in the elevation of his own weight, A labourer elevating a weight by means of a cord and pulley, the cord being pulled downwards, A labourer elevating a weight directly, with a cord, or by the hand, A labourer lifting or carrying a weight on his back, up a slight incline, or stair, and returning unladen, A labourer carrying materials in a wheel-barrow, up an incline of 1 in 12, and returning unladen, A labourer raising earth with a spade to a mean height of five feet,. . ACTION ON MACHINES. A labourer acting on a spoke-wheel, or inside a large drum. At the level of the axis, Near the bottom of the wheel, A labourer pushing or pulling horizontally, A labourer working at a winch handle, ; • • •. A labourer pushing and pulling alternately in a vertical direction,. . . A horse drawing a carriage at an ordinary pace, A horse turning a mill at an ordinary pace, A horse turning a mill at a trot, An ox doing the same at an ordinary iiaco, Amnio do. do. \n ass do. do. Lbs. 143 40 44 143 132 60 Feet. •50 •65 •66 •13 •06 •13 132 •60 26 2-30 26 1-97 17i 3^4 11 3-61 154 2-95 90 2-95 66 6-56 M3 1-97 66 2-96 31 2-62 Lbs. raised 1 foot high. 71-5 26-0 24-6 18-6 8-5 7^8 66-0 59-8 51-2 43-0 39-7 454-3 292-0 4330 281-7 194-7 81-2 Hours. 8 6 6 10 10 8 8 8 8 10 8 8 8 Lbs. raised 1 foot high 2,059,201' 561,600 531,360 401,760 306,000 280,000 1,900.800 1,722,240 1,474.560 1,238,400 1,143,360 16,354,800 8,409,600 7,014,600 8,ir.i,960 5,607,360 2,338,560 It may be gathered from this table that a laboui-cr turning a winch handle can make its extremity pass through a distance of 2-46 feet per second, or 60 x 2-46 = 147-6 feet per minute. Then, supposing the handle has 13,? inches, = 1-147 feet radius, vhich corresponds to a cin-^imferenco of 6-28 x 1-147 = 7-2 feet at (he |)oint of aitplication, the labourer is cjiiwblo of JUi !ivi>ra£f«» velocity of 1476 7-2 = 20 turns (nearly) \h'v niinufo. Also, (he sjinu' labiMtror exerting ii fon-o equal to 171 Hw- ^^l** 90 THE PRACTICAL DRAUGHTSMAN'S the velocity of 2*46 feet per second, will produce a mechanical effect equal to 17i X 2-46 = 43 lbs. raised 1 foot high per second, or of 43 X 60 = 2580 per minute, and 2580 X 60 = 154,800 lbs. raised 1 foot high per hour. And as he can work at this 8 hours per diem, the total mechanical effect during this time will be, as indicated in the table, equal to 1,238,400 lbs. raised 1 foot high. We may then calculate that, as a day's work, a labourer tiM-ning a winch-handle can elevate in a continuous manner 17^ lbs. 2-46 feet high per second ; when, however, the labourer has only to apply his strength at intervals to a crane, a windlass, or a capstan, he can develop a much greater force for a few minutes. According to experiments tried in England with a discharging crane, a man can in 90' raise a load of 1048'6 lbs. to a height of 16| feet. Now, to compare this with the tabulated quantities, we must multiply the weight raised, 1048-6 lbs., by the height, 16i feet, and divide the product by the duration of the action, or 90"; the quotient, 192, indicates the number of pounds raised 1 foot high in a second, to which the mechanical effect is equal. It has been proved by experiments, that under the most favourable circumstances, an Irish labourer of extra strength can, by great exertion, raise to the same height, 16^ feet, a load of 1474 lbs. in 132', which is equal to a mechanical effect of 1474 X 16-5 „ .,,,., — = 184-25 lbs. raised 1 foot high per second. A man can evidently only exert such a force during a very limited period ; we cannot, therefore, compare this kind of labour with that which continues through several consecutive hours. Although the load and velocity as given in the table are those most conveniently proportioned to each other, still, when the case requires it, they might be altered to some extent ; thus, if it is necessary to apply a force of 25 lbs. to the extremity of the winch-handle instead of 17|, then the velocity would be reduced, and would become — := 1-72 feet per second, instead of 2-46. It has been ascertained from actual observations, that a horse, going at the respective rates of 1, 3, 5, and 10 miles per hour, cannot exert a greater tractive force than the corresponding weights, 194, 143, 100, and 24 lbs., and cannot draw an/thing appreciable when going at the rate of 15 miles per hour. Thus, when it is \vished to increase the force exerted, a decrease takes place in the velocity ; and reciprocally, when it is wished to gain time and speed, it can only be done at the expense of the load. Thus, in the case of the winch-handle, the two factors must always produce an effect equal to 43 lbs. raised 1 foot high per second, whatever ratio they may have to each other. In all cases of the direct action of forces, a certain velocily is impressed, for without movement there could not be the action of a force. There are two kinds of motion — uniform and varied motion. 243. Unitorm Motion. — A body is said to have a uniform motion when it passes through equal distances in equal times. Thus, for example, if a body traverses 5 feet in the first second, 6 feet in the second, and so on throughout, its motion is uniform. Putting D to represent the distance, V the velocity, and T the time, the formula, D = V x T, indicates that the distance Ih equal to the velocity multiplied by the time. First Example. — The velocity of a body subject to a uniform motion is 3 feet per second, through what distance will it havo passed in 15 seconds ? D = 3 X 15 = 45 feet. D From the preceding formula, D = V x T, is obtained V = Tj^; that is to say, the velocity per second is equal to the distance divided by the time. Second Example. — The distance passed through in 15 seconds is 45 feet, what is the velocity ? 15 3 feet. The wheel-gear of machinery, as well as many other instru- ments of transmission, is, for the most part, actuated in a uniform manner. 244. Varied Motion. — When a body passes in equal times through distances which augment or decrease by equal quantities, the motion is called uniformly varied. The distance in motion uniformly varied is equal to half the sum of the extreme velocities multiplied by the time in second First Example. — What is the distance passed through in 4 seconds by a body in motion, the velocity of which is 2 feet per second at starting, and 6 feet per second at the termination ? T^ 2 + 6 D = — - — X 4 16 feet. Second Example. — What is the distance passed through in 4 seconds by a body in motion, which at starting has a velocity of 6 feet per second, but which is gradually reduced to 2 feet 1 6 + 2 D: X 4 = 16 feet. It will be seen from these two examples, that, with like condi tions, the total distance is the same for motions uniformly acce lerated or retarded. The velocity at the end of a given time, in uniformly acceler- ated motion, is equal to the velocity at starting, plus the product of the increase per second into the time in seconds. First Example. — What velocity will a body have at the end of 8 seconds, supposing the initial velocity = 1, and that it increases at the rate of 3 feet per second 1 V = 1 + (8 X 3) = 25 feet. The velocity which, at the end of a given time, a body uniformly retarded should have, is equal to the initial velocity minus the product of the diminution per second, multiplied into the time in seconds. Second Example. — A body in motion starts with a velocity of 22 feet per second, and its velocity decreases at the rate of 2 feet per second, what will be the velocity at the end of 10 seconds? V = 22 — - (2 X 10) = 2 feet. 245. The motions of which the various parts of machines are capable are of two principal kinds — continuous, and alternate or back and forward motion. BOOK OF INDUSTRIAL DESIGN. 91 These two kinds of motion may take place eitlier in straight or curved lines, the latter generally being circular. In the actual construction of machinery, we find that, from diese principal descriptions of motions, the following combinations are lerived : — ( Continuous rectilinear. Continuous rectilinear motion is converted into < Continuous circular. ( Alternate circular. Alternate rectilinear motion is converted into Continuous circular motion is converted into Alternate circular motion is converted into Continuf)Us rectilinear. ' Continuous circular. ( Alternate circular. r Continuous rectilinear. J Alternate rectilinear. ] Continuous circular. L Alternate circular. C Alternate rectilinear. < Continuous circular. ' Alternate circular. THE SIMPLE MACHINES. iJ46. This term is applied to those mechanical agents which enter as elements into the composition of all machinery, whether their function be to elevate loads, or to overcome resistances. The simple machines are generally considered to be six — the lever, the wheel and axle, the pulley, the inclined plane, the screw, and the wedge. A much more scientific and comprehensive arrangement of the elementary machines is that lately suggested by Mr. G. P. Ren- shaw, C.E., of Nottingham. According to his system, the elemen- tary machines, or mechanical powers, are five — namely, the lever, the incline, the toggle or knee-joint, the pulley, and the ram. The wheel and axle, of the first system, is evidently but a modification of the lever, and the screw and wedge are modifi- cations of the inclined plane ; whilst no mention is made of tlie toggle-joint and ram — the last so well represented by the hydro- Btatie press. All these machines act on the fundamental principle, known as that of virtual velocilies. According to this principle, the pressure or resistance is inversely as the velocity or space passed through, or that would be passed through, if the piece were put in motion. The nwmenlum of the power and resistance is equal when the machine is in equilibrio. By momentum is understood the pro- duct of the power by the space passed through by the point of application. Time is occupied in the transmission of all mechanical force. In any mechanical action we do not see the effect and the cause at the same instant. Thus, in continuous motion, in which the time expended is not apparent at first sight, each succeeding por- tion of the motion is due to a portion of the impelling action exerted a certain time previously. This will be more obvious on observing the commencement and termination of any motion. The motion does not commence at the instant that the power is applied, nor does it cease at the exact moment of the power's cessation. The fiction of the vis inerliic has been invented to account for these latter observed facts, but it explains them very awkwardly. Thus, bodies are said to possess a certain force which is opposed to a change of stiito, whether from rest to motion or motion to rest. If such a resistive force existed, it wduld require an effort to overcome it, in addition to what is actually accounted for by the motion. If it is said that this is again given back at the termination of tlie motion, another fiction is required to account for it in the meantime, that is, during the continuation of the motion. Moreover, there is nothing analogous to it throughout the entire range of physical science. The facts are described in a much more simple and philosophi- cal manner, when they are said to arise from the time taken in the transmission of motive force. Why there should be this expen- diture of time is a more abstruse question. It probably arises from the elasticity of the component particles of bodies and resisting media, and is regulated by the laws which govern the relation to time of the vibrations of the pendulum. In all machines, a portion of the actuating power is tost or misapplied in overcoming the friction of the parts. 247. The Lever. — The lever, in its simplest form, is an inflexible bar, capable of oscillation about a fixed centre, termed the fulcrum. A lever transmits the action of a power and a resistance, or load ; the distance of the power, or load, from the centre of oscillation, is called an arm of the lever. There are two kinds of power levers, distinguished by the posi- tion of the fulcrum as regards the power and the resistance. These become speed levers, by transposing the power and resist- ance. By a power machine, is meant one which gives an increase of power at the expense of speed, and by a speed machine, one that gives an increase of speed at the expense of power, and all the simple machines are one or the other, according to the relative position of the power and resistance. In all cases of the lever, the power and the resistance are in the inverse ratio to each other of their distances from the centre of oscil- lation. That is to say, that when, in equilibrio, the momentum of the power, P x A, or the product of this power into the space described by the lever arm. A, is equal to the product, R x B, of the resistance, into the space described by the lever arm, b : whence the following inverse proportion : — P : R:: B: A; Any three of which terms being known, the f(»urth can be found at once. 248. The wheel and axle is a perpetual lever. As a jiowor. the advantage gained is in the proportion of the radius of the circum- ference of the wheel to that of the axle. That is to sjiy, the power, r, is to the resistance, R, as the radius, b, of the axle, is to the radius, a, of the wheel, or the length of the winch handle — in the simpler form of this machine, consisting of an axle and a winch handle. The same rules and formuke obviously apply to this, as to the first described form of lever. Thus, multiply the resistance l>y the radius of tJie a.\lc, and divide by that of the handle, and the quotient will bo the power. In windlasses and cranes, consisting of a .system of wlioel-work, the power is applied to a handle fixed on the spiiulle of a pinioi., which transmits the power to a spur-wheol, fixed in tlio s|)indlo ot the barrel, about which the cord, or rope, carrying the load to be raised, is wound. Where there arc scxcnil pairs ol" such wlu'cls, it is neocss;iry to include in the caKulalions tiio ratios of the pinions to the spui . 92 THE PRACTICAL DRAUGHTSMAN'S Tlie proportional formula will, in tliis case, be the same as for a system of levers : — F : R :: b X b' X b' : a X a X a" ; Or, the power is to the resistance, as the produst of the radii of the pinions and barrel is to the product of the radii of the spur- wheels and handle. From this we derive the following rules : — I. MuUiply the load to be raised by the product of the radius of the barrel into the radii of the pinions, and divide the sum obtained by ike product of the radius of the handle into the radii of the spur- wheels, and the quotient will be the power, which, when applied to the handle, irill balance the load. n. Multiply the power applied by the radius of the handle, and by the radii of the spur-wheels, and divide the prodvct by the radius of the barrel, and by the radii of the pinions, and the quotient will be the resistance ichich will balance the poicer. in. Multiply the radii of the pinions and barrel, and divide the prodjici by the radii of the handle and spur-wheels, and the quotient will be the ratio of the power to the resistance. 249. The Isclexe. — Wlien a body is raised up a vertical plane, its whole weight is supported by the elevating power, and this power is consequently equal to the weight elevated. When a body is drawn along a horizontal plane, the tractive power has none of the weight of the body to sustain, but merely to overcome the friction of the surface. If, however, a body is drawn up an inclined plane, the power required to elevate it is proportionate to the inclination of the plane, in such a manner, that If the power ads parallel to the plane, the length of the plane will be to the load as the height is to the power. The advantage gained by the use of the inclined plane, as a power, is the greater the more its length outmeasures its height; it is then the ratio of the length to the height which determines that of the power to the resistance, whence we obtain the follow- ing rules : — I. The resistance, multiplied by the height and divided by the length cf the 'plane, is equal to the power required to balance a body on the inclined plane. II. The power, multiplied by the length of the plane and divided by the height, is equal to the resistance. ni. The resistance, multiplied by the height of the plane and divided by its length, is equal to the load on the plane. The wedge and the screw are noticeable modifications of the incline. An incline wrapped round a cylinder generates a screw. \Vlien used as a power-machine, it is generally combined with a lever, as in presses. The advantage gained depends upon the length of the lever and the pitch of the screw. Multiply the actuating force by the cu-cumference described by the end of the arm or lever, and divide the product by the length of the pitch of the screw ; the quotient, minus the friction, which is very con- siderable m these machines, will be the pressure exerted by the screw, and the velocity will, of course, be in the mverse ratio of the theoretical pressure to the actuating force. 250. The Toggle. — This is met with chiefly in punching presses. Deflected springs and rods are also examples of it, and also the twisted cords used by carpenters to stretch their saws in frames. As in the other machines, the resistance is to the power, as the space passed through by the latter is to the space passed through by the former. 251. The Pullet. — There are two kinds of pulleys, the one turning on fixed centres, the other on traversing centres. The pulley, which turns on a fixed centre, serves simply to change the direction of the motive force, without altering the relations of power and velocity. It is, in fact, only the moveable pulleys which can be classed amongst the elementary machines. A single moveable pulley, acting as a power, doubles it at the expense of the speed ; thus, if a weight of 10 lbs. be suspended to one extremity of the cord, it will balance 20 lbs. hung to the axis of the pulley. This arises from the fact that, from the arrange- ment of the cord, the pulley only rises through half the height passed through by the motive force ; thus, if the latter pass through 6 feet, the pulley will only rise 3 feet, and the resulting momentum of the power, 10 x 6, WHl be equal to that of the resistance, or 20 x 3, so that the two will be in equilibrium. Though the stationary pulley cannot be considered as a mechanical power, yet, in changing the direction of the motion, it affords great facilities in the application of force ; thus, it Ls easier to pull downwards than upwards, as the labourer brings his weight to bear in the former case. WTien several pulleys or sheaves are placed on one axis Lq a suitable frame, it is called a block. Where two or more blocks are employed, it is only the moveable ones which increase the power, and this increase is equal to double the number of sheaves, or pulleys, in the block or blocks. The mechanical advantage of the block, as a power, arises from the fact, that the space traversed by the motive power is equal to the sum of the doublings of the cord round the pulleys, whilst the load is only elevated to a distance corresponding to this space divided by the number of these doublings. A clock line and weight, in which the line goes round a pulley fLxed in the weight, is an example of a speed-pulley, that is, one in which the power, or resistance, is transposed, for the weight, or motive power, causes the moveable end of the cord to pass through twice the space it passes through itself. 252. The Ram. — This is the most economical tugmentor of power that we have. It is freer from friction and other disad- I vantages than the other simple machines, and it ia, in its action, very closely allied to the pulley. Each derives its advantage from the division of the points of support, for tlie proportionate area of the piston in the Bramah press represenfej the number of points over which the pressure, or resistance, is dilused. 253. Remarks. — ^It is essential that, to avoid illusive mistakes, the student should perfectly understand, that wl.en, in emplojincr mechanical forces, the effect of the power appLed is augmented, the distance passed through by the resistance, or load, is dimin- ished, with reference to that passed through by the power, in exactly the same ratio that this is increased. This is true in all cases, and may be stated thus : What we gain in force, by means of machinery, we lose in speed, and reciprocally. It follows from this, that the true object of maehinery cannot be to augment the work performed by the motive agent, but ta convert any primary action in a manne.- appropriate to the BOOK OP INDUSTRIAL DESIGN. 93 circumstances in which the power is to be used. Thus we can make a very small force, as that of a man, elevate an enormous weight, but with a speed proportionately slow. Finally, The mechanical effect developed in a given time by a given force, through the instrumentality of machinery, must alivays equal the useful effect obtained, plus the amount lost in overcoming frictional and other resistances ; and the useful effect of machinery will be the greater, according as the causes of these resistances are diminished. CENTRE OF GRAVITY. 254. All bodies are equally subjected to the action of weight. Gravity, or weight, is the action of that universal attraction which draws all bodies towards each other, and by which, in the case of bodies on the surface of the earth, these are drawn towards its centre. The power, of whatever nature it may be, which balances this action, is equal to the weight of tne body. The curvature of the surface of the earth being quite inap- preciable for small distances, gravity is considered as acting in parallel lines, and its direction is given by the plumbline. The centre of gi-avity is that point in any body in which the action of its entire weight may be said to be concentrated. If the body be suspended by this point it will be in equilibria, in whatever position it is put. The position of the centre of gravity depends upon the nature and form of any body ; it may generally be found in the follow- ing manner : — Suspend the body by a thread attached to any point whatever in it; when the body is motionless, the line of the suspension thread will pass directly through the centre of gravity. Suspend the body by any other point, and the centre of gravity will also be in the continuation of the line of the thread, so that the actual centre must be at the point of intersection of the two lines thus obtained. This simple expedient reminds us of the application of the square to the finding of the centres of circles — the unknown centre on the endface of a shaft, for example — where the inter- section of any two lines, drawn along the blade of the square, when the head is laid against the periphery of the shaft in two different positions, gives the required point of centre. The centre of gi-avity of regular bodies, as spheres, cylinders, prisms, is in the centre of their configuration. The centre of gi'avity of an isosceles tiiangle is one third up the centre line which bisects the base. The centre of gravity of a pyramid, with a triangular or poly- gonal base, is one fourth up the line which joins the summit with the centre of gravity of the base. It is the same with a cone. The centre of gravity of a hemisphere is situated three-eighths up the radius at right angles to the base. The centre of gravity of an ellipse is in the point of intersection of the axes. When a body is placed in a vertical or inclined position on a plane, it is necessary, in order that it may rest upon it in that po- sition without falling, that the vertical lino passing through the centre of gravity shall fall within the external outline of the side in contact with the plane. This limit, however, allows of consi- derable deviation from the vertical in the geiu^ral contour of bodies, as is instanced in the case of leaning or inclined cdiliccs. The stability of bodies increases as the extent of their bases is g^reater in comparison with their height, and also, as the vertical line, passing through the centre of gravity, meets the plane on which the body rests nearer to the centre of the base. A body is said to be more stable when it requires a greater force to overturn it. A cone is more stable than a cylinder of the same height and base. The stability of walls depends greatly on the kind of foundations given to them, and on the proportionate extension of their bases. ON ESTIMATING THE POWER OF PRIME MOVERS. 255. As we shall see further on, the power of prime movers may be calculated from the dimensions of the various parts of the engine. Still, the many different modes of construction tend to modify considerably the actual useful effect, and engineers have endeavoured to construct an apparatus, by means of which the actual power, or useful effect of engines, may be measured with exactitude. Prony's brake, which is the instrument most generally used for this purpose, acts on the principle of the lever, and consists of a cast-iron pulley in two halves, united by screws. This is fixed on the main shaft of the prime mover, the force of which it is wished to measure. It is embraced by two jaws, which may be tightened down upon the pulley by screws. To the lower jaw is attached a long lever, from the end of which is suspended a scale for weights. If it is known what power the engine was designed to possess, it is simply necessary to put into the scale the weight corresponding to this power, that is, the weight which, by the action of the lever, will give a pressure equal to the supposed power of the machine. Having fixed the apparatus on the engine, and provided means of efficiently lubricating the frictional surface of the pulley with soap and water, and having balanced the apparatus in such a manner that it will not be necessary to take into the calculation anything but the weight placed in the scale, the steam may bo gradually let on. The engine will perhaps shortly acquire a greater velocity than that for which it was designed ; if this is the case, the jaws are gradually screwed closer and closer upon the pulley. As the friction thereby increases, the velocity will dimi- nish, and full steam may be let on. After a short time, and when the friction is so great that the lever is raised slightly above the horizontal line, and the engine is going at its proper velocity, and the pressure of the steam at its correct point, so that i\w power of the engine balances the load on the lever, it may be con- cluded that the engine develops the i)ower lor which it was in- tended. If the lever rises considcnihly, it will he necessary to increase the weight in the scale, so as to obtain the actual uiaxi- mum power of the engine ; and, on the contrary, if the engine does not appear to have the desired power, the weight must be reduced, by which nu\nis its actual power will be ascertainable. CALCULATU)N 1-OK THIv BKAKK. 25(). The weight which will balance the foive of a macliinc may be cuk'ulaled when ihc len^lli of llie Icmt arm is known, or. more conectly, the radius tVoin the centre of the shalt to the jioinl of s\ispension of the weight, and the nominal lior^o-pon i<r, by U»o I'dllowing rule : — 94 THE PRACTICAL DRAUGHTSMAN'S Multiply the nominal hmse-power by 33,000, ami divide the pro- duct by llie circumference described by the end of the leitr, and by the number of revolutions per minute, and tlie quotient will be the ■weiglU sought. Let us take, for example, the main shaft of a steam-engine of 16 horse-power, which runs at the rate of 30 revolutions per ruinutc, the radius of the brake being nine feet — 16 X 33,000 Here we have w ^ ■ = 311-4 lbs. 6-28 X 9 X 30 Such is the net weight to be suspended from the end of the lever, the brake being previously balanced by being suspended on its centre of gravity. The actual power, or maximum effect of an engine, may like- wise be calculated by means of the following rule : — Multiply the circumference described by the lever, by the number of revolutions of the shaft per minute and by the weight in the scale, and divide the product by 33,000 and the quotierd will be the actual force of the engine hi horses power. For example, let us suppose that the main shaft of a steam- engine makes 30 revolutions per minute, that the radius of the lever is 9 feet, and that the net weight in the scale is 311-4 lbs., what is the maximum force of the engine ? ^ 6-28 X 9 x30 X 311-4 ^= 33:000 -^^H-P- TABLE OF HEIGHTS CORRESPONDING TO VARIOUS VELOCITIES OF FALLING BODIES. Velocity. Height. Velocity. Height. Velocity. Height. Velocity. Height. Velocity. Height. Inches Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. Inches. •1 •0001 5-7 •165 16-5 1^388 44-5 10-094 72-5 26-794 •2 •0002 5-8 •171 17-0 1-473 45-0 10-322 73-0 27-164 •3 •0005 5^9 •177 17-5 1-561 45-5 10-553 73-5 27-538 •4 •0009 6-0 •184 18-0 ^651 46-0 10-786 74-0 27-914 •5 ■0013 6^1 •190 18-5 1-745 46-5 11-022 74-5 28-292 •6 -0019 6^2 •196 19-0 1-840 47-0 11-260 75-0 28-673 ■1 •0026 6^8 •202 19-5 1-938 47-5 11-501 75-5 29-057 •8 •0034 6^4 •209 20-0 2-039 48-0 11-744 76-0 29-443 •9 •0043 6-5 •215 20-5 2-142 48-5 11-990 76-5 29-832 1-0 •0051 6-6 •222 21^0 2-248 49-0 12-239 77-0 30-223 1-1 •0062 6-7 •229 21^5 2-356 49-5 12-490 77-5 30-617 1-2 •0074 6-8 •236 22-0 2-467 50-0 12-744 78-0 31-013 1-3 -0087 6-9 •243 22-5 2-580 50-5 lS-000 78-5 31-412 1-4 •0101 7-0 •250 23-0 2-696 51-0 13-258 79-0 31-813 1-5 •0115 7-1 -257 23-5 2-815 51-5 13-520 79-5 32-217 1-6 •0131 7-2 -264 24^0 2-936 52-0 13-784 80^0 32-624 1-7 •0148 7-3 -272 24^5 8-060 52-5 14-050 80^5 33-033 1-8 •0166 7-4 •279 25^0 3-186 53-0 14-319 SVO 33-445 1-9 •0185 7-5 •287 25^5 3-315 53-5 14-590 81^5 33-859 2-0 •0204 7-6 •295 26^0 3-446 54-0 14-864 82-0 34-275 2-1 •0225 7-7 ■302 26-5 3-580 54-5 ] 5-141 82^5 34-695 2-2 -0247 7-8 •310 27-0 3-716 55-0 15-420 83^0 35-116 2-3 •0270 7-9 •318 27^5 3-855 55-5 15-701 83^5 35-541 2-4 •0294 8-0 •326 28^0 8-996 56-0 15-986 84-0 35-968 2-5 •0319 8-1 ■334 28^5 4-140 56-5 16-272 84-5 36-397 2-6 ■0345 8-2 ■343 29^0 4-287 57-0 16-562 85-0 36-829 2-7 -0372 8-3 ■351 29^5 4-436 57-5 16-854 85-5 37-264 2-8 -0400 8-4 ■360 SO^O 4-588 58-0 17-148 86^0 37-701 2-9 -0429 8-5 •368 30^5 4-742 58-5 17-445 86^5 38-141 3-0 ■0459 8-6 ■377 31^0 4-899 59^0 17^744 87^0 38-583 3-1 •0490 8-7 •386 31^5 5-058 59-5 18-046 87^5 39-028 3-2 •0522 8-8 •395 32^0 5-220 60-0 18-351 88^0 39^475 3-3 -0555 8-9 •404 32-5 5-384 60-5^ 18G58 88-5 39-925 3-4 •0589 9-0 •413 33^0 5-551 61-0 18-968 89^0 40-377 3-5 -0624 9-1 ■422 38-5 5-721 61-5 19-280 89-5 40-832 3-6 -0660 9-2 •431 34-0 5-893 62-0 19-595 90-0 41-290 3-7 -0697 9-3 •441 34^5 6-067 62-5 19-912 90-5 41-750 3-8 -0735 9-4 •450 35^0 6-244 63-0 20-232 91-0 42-212 3-9 -0775 9-5 •460 35^5 6-424 63-5 20-554 91-5 42-677 4-0 •0816 9-6 •470 36-0 6-606 64-0 20-879 92-0 43-145 4-1 ■0856 9-7 •480 36^5 6-791 64-5 21-207 92-5 43-615 4-2 ■0899 9-8 •490 37^0 6-978 65-0 21-537 93-0 44-088 4-3 ■0942 9-9 •500 37^5 7-168 65-5 21-869 93-5 44-563 4-4 ■0986 10-0 ■510 38^0 7-361 66-0 22-205 94-0 45-041 4-5 •1032 10-5 ■562 38-5 7-556 66-5 22-542 94-5 45-522 4-6 ■1078 11-0 •617 39^0 7-753 67-0 22-888 95-0 46-005 4-7 ■1125 11-5 •674 39^5 7-953 67-5 23-225 95-5 46-490 4-8 •1174 12-0 •734 40^0 8-156 68-0 23-571 96-0 46-978 4-9 •1228 12-5 •797 40-5 8-361 68-5 23-919 96-5 47-469 5-0 •1274 13-0 •861 41^0 8-569 69-0 24^969 97-0 47-962 5-1 •1325 13-5 •929 41^5 8-779 69-5 24-622 97-5 48-458 5-2 •1378 14-0 •999 42^0 8-992 70-0 24-978 98-0 48-956 5-3 •1431 14-5 vol 2 42^5 9-207 70-5 25-336 98-5 49-457 5-4 •1486 15-0 M47 43^0 9-425 71-0 25-696 99-0 49-960 ; 5-5 •1541 15-5 1-225 43-5 9-646 71^5 26-060 99-5 50-466 5-6 •1598 160 1^305 44-0 9-869 72^0 26-425 100-0 50-975 BOOK OF INDUSTRIAL DESIGN. 93 THE FALL OF BODIES. 258. When bodies fall freely of their own weight, the velocities wliich they acquire are proportionate to the time during which they have fallen, whilst the spaces passed through are as the squares of the times. It has been ascertained by experiment that a body falling freely from a state of rest, passes through a distance of 16 feet and a small fraction, in the first second of time. At the end of this time it has a velocity equal to twice this distance per second. From this it follows that if the times of observation are — • 1" 32 ft..., 2' 64 ft... 3" 4" 96 ft.... 128 ft. 16 16' --... 64 " ...48" .. 144 ' 80' 256 " 112" 3, 4 3, 4 9, 16 5, 7 The corresponding velocities will be The spaces passed through fi-om the ) coinmencement, J The spaces passed through during each ( second, ) That is to say, that the times are as the numbers, 1, 2, The velocities also as, 1, 2, The spaces passed through as the squares, 1, 4, And the space for each interval as the odd numbers, 1, 3, These principles apply equally to all bodies, whatever may be their specific gravity, for gravity acts equally on all bodies ; the effect, however, being modified by the resistance of the media through which the bodies pass, which is greater in proportion, as the specific gravity Is less. 259. The velocity which a body will acquire in a given time when falling freely, will be found by multiplying the time ex- piessed in seconds by 32 feet. Example. — Let it be required to ascertain the velocity acquired by a body falling during 12 seconds. V = 12 X 32 = 384 feet per second. When a body falls from a given height, H, the ultimate velo- city, or that acquired by the time the base is reached, will be given by the formula (g being the velocity gravity causes a body to acquire in the first second) V = VJgH, or V =3 4^64 x H, which leads to the following rule : — Multiply the ^ven height in feet by 64, and extract the square root, which will be the velocity in feet per second by the time the body shall have fallen through the height, H, not taking resistance into consideration. Example. — What will be the ultimate velocity of a body falling a distance of 215 feet ? V = 1^64 X 215 = 117-3 feet per second. From the above formula. we obtain V v = vTpr, 2g-H, then ya ya ^^2^ "^64' whence we have this rule : — Divide the square of the velocity in feet per second by 64, and the quotient will express the height through which a body must fall unimpeded, from a state of rest, in order to obtain that velocity. Example. — A body has acijuired a velocity of 117'3 feet per second, through what height must it have fallon ? 117:5' H: 64 = 215 feet, the height of the fall. To obviate the necessity of calculating the corresponding heights and velocities, we give a very extensive tabic, calculated for tenths of inches The niiinher.H, however, being (•<|nully correct as representing feet or yards, those of both columns bemg of the same denomination. MOMENTUM. 260. The force with which a body in motion strikes upon one in a state of rest, is equal to the product of the mass of the moving body multiplied into the velocity ; this product is termed its momentvun. If a body with a mass, m, is animated with a velocity, v, its momentum is equal to m v. The term, nt, however, may be taken as signifying the mechanical effect of a weight falling during a second of time, or through 32 feet, there- w fore, OT = —, that is, the weight in pounds divided by 32 feet, w y. V whence, mv ^^ • g What distinguishes the simple momentum or force of impact of a body from the mechanical effect of a prime mover is, that whilst the former is due to a single impulse, we have in the latter to consider the continuous action of the impelling force. 261. When a motive force Imparts continuously a certain velo- city to a body, the result of its action is what may be termed vis viva, or continuous momentum ; it is numerically the product of the (moving) mass multiplied into the square of the velocity im- parted to it. Putting M to represent the mass of a body, and V the velocity impressed upon it, WV» M V» or g is- the expression of the vis viva of the body. This force is double that developed by gravity. For, in fact, when a body of the weight, W, falls from a height, H, it acquires from its fall as ultimate velocity, V, which we have already shown to be equal to V2gH = ^g and the mechanical effect, W H, is consequently expressed by WV» ^g ' MV now, putting for P, its value, ^l g, the formula becomes — g— • Thus, the mechanical effect developed by gravity is equal to half the vis viva unparted to a body. CENTRAL FORCES. 262. When a body revolves freely about an axis, it is said to be subjected to two central forces ; vne one, termed " centripetal," tends to draw the body to the axis ; the other, termed " ccntri- fufral," or tangential, and duo to the tendency of bodies in motion to proceed in straight lines, strives to carry tlie body away from the centre. Tiiese forces are equal, and act trjmsvcrsciy to eai-h other. The centrifugal etVort exerted by a body in rotative motion, and which tends to separate the component p;irticlos, is expressed by the following formula :— WV F = gxR' in which W r( I'rescnt.s the weight of tlu- body : V, the verocify in 0(j THE PRACTICAL DRAUGHTSMAN'S teet per second ; and R, the radius, or distance of tlie centre of motion from the centre of the revolving body. Example.— Let a ball of the weight W = 23 lbs., attached to a radius, R, measuring 5 feet, rotate with a velocity, V = 40 feet per second, what is the centrifugal eflFort, or the pull of the ball on the radius 1 23 X 40 X 40 F = 32 X 5 ■ = 230 lbs. raised 1 foot high per second. CHAPTER Vn. ELEMENTARY PRINCIPLES OF SHADOWS. 263. We have already, when treating of shadow lines, laid it down as a rule to be observed generally, in mechanical or geome- trical drawing, that the objects represented shall be supposed to receive the light in parallel rays, in the direction of the cubic dia- gonal, running from the upper left hand corner of the anterior face of the cube, do\vn to the lower right hand corner of the posterior face. We have also shown that the horizontal and vertical projections of this cubic diagonal make angles of 45° with the horizontal or base line. The advantages of this assumption of the direction of the rays of light will, no doubt, have been appreciated. Amongst these, it has the merit of at first sight plainly pointing out the relative degrees of prominence of the various parts of an object, even with the aid of a single projection or view. 264. This point, then, being determined, on considering an ob- ject of any form whatever, as receiving in this way the parallel rays of light, it may be conceived that these rays will form a cylindri- cal or prismatical column, the base of which wUl be the illumined outline of the object. The part met by these rays of light will be fully illumined, whilst the portions opposite to this will be as entirely void of light. The absence of light on this latter part may be termed the shadow proper of the object — ^that is, its own shadow upon itself. 265. If, further, we suppose the Imninous rays surrounding the object to be prolonged imtil Intercepted by the surface upon, or adjacent to which it lies, a portion of such surface wUl be unUlu- mined, because of the interception of some of the rays by the object ; the outline of this unillumined portion will be limited by, and depend upon the contour of the object, and it is termed the shadow cast, or ihroum, by an object on any surface. The line which separates the UlumLned from the unillumined portion is termed the line of separation of light and shade, or the outline of the shadow. This is modified by the form of the reci- pient surface, as well as by that of the object which gives rise to it. It is always bounded by straight lines when the generating surfaces are planes ; and by curves when either or both are cylindrical, conical, spherical, or otherwise curved. 266. As a general rule, the determination of the outlines of shadows proper, and cast, reduces itself to the problem of finding the point of contact of a straight line representing a luminous ray, with a piane or other surface. The application, however, of this general principle, though apparently so simple, gives rise to many riifticulties it practice, from the variety of cases presented by the '•aVerent forms of objects and it is necessary to give several special examples, to explain the most simple and expeditious expedients which may be employed in such cases, always with a due regard to geometrical accuracy. We shall primarily choose for these applications objects of sim- ple form, and bounded by plane surfaces ; next, such as are wholly or partially cylindrical ; and we shall proceed, in succession, to objects of more complex forms. The objects which we have taken in preference, as examples, are such as are most frequently met with in machinery and architecture ; they will, notwithstanding, afford quite sufficient illustration in connection with what has to be said respecting the study of shadows. SHADOWS OF PEISMS, PYRAMIDS, AND CTLINDEES. Plate XXVI. PRISMS. 267. Let the figures 1 and 1 a be given, the horizontal and ver- tical projections of a cube, it is required to determine the form of the shadow cast by this cube on the horizontal plane. In the position given to this cube it is easy to see that the sides which are in the light are those represented by a d and a c, in the horizontal projection, and projected vertically in a' e' and a' c'. The opposite faces, b c and b d, fig. 1, and b' c', b' e', fig. 1 a, are consequently in the shade ; as, however, these latter faces are reduced to mere lines in the representations, the shadow proper can only be shown by a thick shadow line, produced by China ink in line drawings, and by a narrow stroke of the brush in water- colour drawings. These lines, which distinguish the UlumLned sides of the cube from those wliich are not so, are termed, as we have said, the lines of separation of light and shade. It now only remains to find the shadow cast by the cube on the plane, l t. 268. When the object rests on the horizontal plane, as supposed in this case, and is at a greater distance fi-om the vertical plane than is equal to its height, the entire shadow cast by it will be in the horizontal plane ; and to determine its outline here, it is merely requisite to draw straight lines from each comer of the cube, representing the rays of fight, as c c, B &, d d, parallel to R, and to find the points, c, b, d, in which these lines meet the plane. To effect this, through the points, c', and a', fig. 1 a, the pro- jections of the two first, b d, draw the rays, c' c', and a' b', paral- lel to r', and meeting the base line, l t, in c' and b'. If now, through these points, we draw perpendiculars to the base line, as c' c b' b, these will cut the first rays in c, b, and d. The contour of the shadow cast is, in consequence, limited by the lines cc, cb, b d, and d d. BOOK OF INDUSTRIAL DESIGN. 97 The face, e b', being that on which the cube rests, has no pro- minence, and cannot therefore cast any shadow. It follows, then, that the shadow, as above determined, is all that is apparent. It is generally represented by a fiat, uniform shade, laid on with the brush, and produced by a gi-epsh wash of China ink. 269. It will be observed that the lines, d b and h c, are parallel to the straight lines, d b and b c. This is because these are themselves parallel to the horizontal plane ; for when a line is parallel to a plane (82), its projection on this plane is a line paral- lel to itself; and hence we have this first consequence, that — When a straight line is parallel to the plane of prq/'eciion, it casts a shadow on the plane, in the form of an equal and parallel straight line. 270. It will also be observed, that the straight lines, v> d, b b, c c, which are the shadows cast by the verticals, projected in d, b, and c, are inclined at an angle of 45° to the base line ; whence we derive the second consequence, that — When a straight line is perpendicular to the plane of projection, it casts a shadow on the plane in the form of a straight line, parallel to the rays of light, and consequently inclined at an angle of 45" io the base line. 271. These observations suggest a means of considerably simpli- fying the operations. Thus, in place of searching separately for each of the points, c, b, d, where the rays of light pierce the hori- zontal plane, it is sufficient to determine one of these points, such as b, for example, and through it to draw the straight lines, b d, be, parallel and equal to the sides, d b and b c, of the cube and intersecting lines, inclined at an angle of 45° drawn from the points, D c. In the actual case before us, we may even entirely dispense v«th the vertical projection, fig. 1°, since it would bave been sufficient to prolong the diagonal, a b, to b, making b b equal to b a, or to make the inclined lines, b d, or b b, equal to the diagonal, a b ; because the vertical projection, c' c', and horizontal projection, c c, of the same ray of light, are always of the same length, which fol- lows from our having taken the diagonal of the cube for the direc- tion of this ray, the two projections, a b and a' b', of this diagonal being obviously equal. Whence follows the third consequence, that— If, through any point of which the two projections are given, we draw a straight line, representing the ray of light, and if loe ascer- tain the point in which this ray meets either plane, the length of the ray in the other plane of projection will be the same. 272. Finally, it is to be observed that the distance, b d, taken on the prolongation of the vertical line, c b, is equal to the entire lieight, c' b', namely, that of the cube ; and consequently, in place of employing the diagonal to obtain the various points, d, b, c, we may make the distance, B d, equal to the height of tho cube, and draw, through d, a. straight lino, d b, parallel and equal to d n, and through b a second, b c, parallel and c(|n;il to B c, and then join d V, c c. Thus the shadow cast on a piano by a point, is at a distance from the projection of tho )>oint, equal to IIk^ distance of the point itself from the plane. 273. Figs. 2 and 2" niprosent a prism of he.vagonal base, su])- |)o;icd to be elevated above the base line, hut at the same time at such a distance from the vertical plane, that all the shadow cast will be in the horizontal plane. It will be seen that the vertical faces, a b, b c, and a f, are illumined, whilst the opposite ones, e d, d c, and e f, are in the shade. Of these latter faces, c d is the only one visible in the vertical projection, fig. 2", and represented by the rectangle, c' d' h g, which should be shaded to a deeper tint than the cast shadows, to distinguish it. 274. The operation by which we determine the shadow cast upon the horizontal plane, is evidently the same a-s in the preced- ing ease ; still, since the lower base, j h, does not rest upon the horizontal plane, it will not be sufficient merely to draw the rays of light through the points, c, d, e, f, of the upper side ; it is, in addition, necessary to draw corresponding rays through the points, J, I, G, H, of the base. It is to be observed, as in the preceding case, that as these two faces are parallel to the horizontal plane, the shadow cast by each upon this plane will be a figure equal and parallel to itself; so that, in place of seeking all the points of the shadow, it would have been quite sufficient to obtain one of these points, as d, for example, of the upper side, and A:, of the lower side, and then, starting from them, to draw a couple of hexagons, parallel and equal to a b c d e f. It will also be understood, that as it is only the outside lines, those of the separation of the light and shade, which make up tho contour of the shadow, it is not necessary to determine the points which fall within this contour, and correspond to those points in the object itself which do not lie in the lines of separation of the illuminated and shaded parts. 275. Thus it is unnecessary to find the points, a, b, e,h; and generally, in making drawings, we do not seek the shadows cast by points fully illuminated, or within the borders of the shaded portion ; and the contour of the shadow is derived simply from points lying in the line of separation of *he light and shade on tho object. 276. From what we have already explained, it will be gaihored, that the projection in one plane of tho shadow, cast by a point, cau be obtained by drawing the diagonal of the square, a side of which is equal to the distance of the point from tho piano, as shown in the other projection. For example, the shadow, /,-, on the hoiizontal piano of tho point, tho two projections of which are f and i, figs. 2 and 2*, may 1)0 got by forming the square, F Ik, a side of which, F /, is equal to tho distance, i /', of tho point from tho horizontal plane. In the same way, wo have tho points, g, i,f, correspcmding te G, I, F, which are tUo same height as the first above tho piano. For th<:- points, <.•,</, ^,/, which corrosi)ond to iho upper side, a' 1)', of tlu^ prism, wi> draw the diagonal, n' </', of tho squ;iri\ having for a side llio lioiglit, n' ;;i, of tho jioi::!, 1)', iilovo tho horizontal piano, and sot out this diagonal fnun C to c, D to a E to (', &,o. 277. When several straight linos oonvorgo to a point, fno- shadows thoy oast on oithor jilaiio of projivtion must mvcssarilj THE PRACTICAL DRAUGHTSMAN'S also, converge to a point. Thus, in the pyramid, figs. 3 and 3°, the apex of which is projected in tlie points, s and s', the edges of all the sides Lemi; directed to this point, cast shadows on the norizontal plane, bounded by lines converging to the point, s, the shadow east by the apex on the same plane. In order, then, to find the shadow cast by a pyramid, on either of the planes of pro- jection, it is sufficient to draw tlie ray of light through the apex, and ascertain the pomt at which this ray meets the plane; then to draw lines to this point from all the angles of the base of the pyramid, if this rests upon the plane. If, however, tlie pyramid is raised above the plane, it will be necessary to find the shadows cast by the various angles of the base, and then draw straight lines from these to the shadow of the apex. TRUXCATED PYRAMID. 278. When we have only the frustum of a pyramid to deal with, and the apex is not given, it is necessary to find the shadows cast both by the angles of the base, and by those of the surface of truncation. Thus, the points, e, f, g, h, of the upper side, cast their shadows on the horizontal plane, in the points, e,/, g, h, which are obtained by drawing through each point, in the vertical projection, e', f', g', h', the rays, inclined at an angle of 45°, meet- ing the base line in the points e',/,^, h', which are squared over to the horizontal projection, so as to meet the corresponding rays, drawn through the points, e, f, g, h. Then, if we draw lines from the points, e,f, g, h, to the angles, a, b, c, d, situated in the horizontal plane, we shall obtain the shadows cast by each of the lateral edges of the pyramid. For the same reason that these edges are diversely inclined to the horizontal plane, the shadows cast by them on this plane have also difTerent inclinations to the base line : but the edges of the upper side or surface of truncation being parallel to this plane, cast a shadow, which in figure is equal and parallel to this side ; this would not have been the case had it been inclined to the plane. It is e\ident that, in the position in which the pyramid is represented with regard to the rays of light, the two faces, a e h d and A E F B, are in the light, whilst their opposites, d h g c and c G F B, are in the shade. This last, which is the only one visible in fig. 3", is there distinguished by a moderate shade of colour. CTLISDER. 279. A cylinder with a circular base being a regular solid, all that is wanted, in determining the lines of separation of light and shade, is, when the cylinder is vertical, to draw a couple of planes tangential to it, and parallel to the rays of light, as in figs. 4 and 4°. These tangential planes are projected in the horizontal plane, in the lines, a a, si, tangents to the circle, and inclined at the angle of 45°. By their points of contact with the circle, these tangents give the lines of separation of light and shade, which are projected vertically in a' c and b' d. One of these lines is appa- fcnt on this view, but the other is not. We have thus the portion, A E b, of the cylinder, in the Ught, and its opposite, a r b, in the shade. A very small portion of this last is seen in fig. 4°, and is there siightiy shaded. • 280. Wltn reference to the cast shadow, it is to be remarked, tliat for the \ erv reason that the lines of separation of light and shade are vertical, the shadows they will cast on the horizontal plane will be in two lines, c a and d b, with an inclination of 45', as already explained, these lines being identical v\ith the prolonga- tion of the tangential rays. The two bases of the cylinder being parallel to the horizontal plane, their shadows will be circles equai to themselves ; and all that is required is to find the shadows, n, o, cast by their respective centres, n and &, and with the points, n, o, as centres, to describe circles, with a radius equtxl to o a. The entire shadow cast by the cylinder is comprised between the two semicircles and the two tangents, c a, d h. SHADOW CAST BY ONE CYLINDER UPON ANOTHER. 281. Hitherto we have only considered the shadow cast by an object upon one of the planes of projection. It frequently hap- pens, however, that one body casts a shadow on another, or that the configuration of the body itself is such, that one part of it casts a shadow on another. Let fig. 6 be the vertical projection of a short cylinder, a, with a concentric cylindrical head, b. We have, in the first place, to find the line of separation of light and shade upon these two cylinders ; and for this purpose w"e require to draw a second ver- tical projection, fig. 6", at right angles to the first, and in the line of its axis. In this figure, the projection of the ray of light also makes an angle of 45° with the base line. We must, consequently, draw the two straight lines, c' c and d' d', tangential to the circles, a' and B , and project, or square over, the points of contact, c' and d', to fig. 6, drawing the lines, a b and d d, which separate the light from the shaded part of the objects. Instead of drawing these tangents, we can directly obtain both points of contact, by drawing the radius, o c' d', at right angles to the ray of light. 282. The shadow cast by the projecting head, b, upon the cyKnder, a, is limited to that due to the portion, d' c' h', of the circmnference. Different points in the outline of this shadow are determined, by first taking any points, c', e', f', g', upon the arc, d' c' h', and drawing through each of them lines, representing the parallel rays of light, and meeting the circumference of the cylin- der, a', in the points, c, e', f, g'. Having projected the first- mentioned points on the base, d h (fig. 6), draw through the points, c, E, F, G, a series of lines parallel to the first, and likewise representing the rays of light, and square over the points of con- tact, c',e',f,g', which will give the points, c,e,f,g, of the curve, which is the outline of the shadow upon the cylinder, a. As seen in a former example, instead of squaring over the points, c', e',f,g', w^e can obtain the same result by making the corresponding rays, c c, e e, f/, Gg, equal to the lines, c' c', e' e', shadow cast by a cylinder upon a prism. 283. Figs. 7 and 7" represent two vertical projections of a prism, A, of an octagonal base, having a cylindrical projecting head, b. As in the preceding case, draw the radius, o d', perpendicular to the ray of light, thereby obtaining the point of contact, d', and, in consequence, the line of separation, d d. of light and shade on the cylindrical head, b. The inclined facet, c i, of the prism, being in the direction of the ray of light, and, consequently, inclined at an angle of 45° wltli BOOK OF INDUSTRIAL DESIGN. 99 the vertical plane, is considered to be completely in the shade. The edge line, a b, fig. 7, is therefore the line of separation of light and shade on the prism-shaped portion of the object, and the surface, a bid, in consequently tinted. The shadow cast upon the prism by the overhanging head, b, reduces itself to that due to the portion, c' f' h', merely, of the circumference of the latter, and it falls upon the two faces, c'/' and/' h', of the latter. The lines indicated on the diagram, with their corresponding letters, when compared with those of the preceding example, will show that the operations are precisely the same m both cases, and, ic the latter, the curves, cef and fgh, are the resulting outlines of the shadow. In general, it >s unnecessary to obtain more than the extreme points of the o^rve, and another near the middle. Through the three points thu'; obtained, arcs of circles can then be drawn. The curves are bowever, in reality elliptical. SHADOW CAST BY ONE PRISM UPON ANOTHER. 284. Figs. 8 and f" represent a couple of vertical projections, at right angles to ea^ih other, of a prism of an octagonal base, sur- mounted by a similar and concentric, but larger prism. Although the operation." c.-xlled for in this case are precisely the same as in the kwo preceding', still it is axi exemplification which cannot be omitted; ind its chief use is to show, that 27(6 •ihadoio cast by a straight line upon a plane surface is inia- liubhj c, straight line ; and, consequently, it is sufficient to determine k<i extreme points, in order to obtain the entire shadow in any one ylane. Thus, the straight line, e' c', casts a shadow upon the plane facet, /c, which is represented by the straight line, ec. it is further obvious, that The shadow cast upon a plane surface, by any line parallel to it, must be parallel to that line. Thus, the straight line, e' g', of the larger prism, b, being jiaral- lel to the plane faccit, f g', of the prism, a, casts a shadow upon the latter, which is represented by the straight line,/g-, parallel to the line, f g, the vertical projection of the edge, f' g'. It is not, however, the same with the portion, ef because the corresponding portion, e' f', of tlie edge of the larger prism, is not parallel to the facet,/' e!. SHADOW CAST BY A PRISM UPON A CYLINDER. 285. Figs. 9 and 9" represent vertical projections, at right angles to each other, of a portion of an iron rod, a, surmounted by a con- centric head, b, of a hexagonal base. The main object of this diagram is to show, that When a right cylinder is parallel, or perpendicular, to a plane of projection, any straight line, which is perpendicular to tlie axis of the cylinder, and parallel to the plane of projection, casts a. shadoio upon the cylindrical surface, which is represented by a curve, similar to the cross section of such surface. If, therefore, the cylinder is of circular base or cross section, as we have supposed in the present case, the shadow cast upon i( will be a portion of a. circle, of the same radius as the cylindci-. Tl,\is. the straight line, »' f', sitiialcfd in a, piano, at right jiiiglcM In til" axis of the cylinder, a, and Ix^ing, at llu; same lime, parallel lo <lif' vortical plane, casts a shadow upon the cyliiulci-, which is re- presented by the portion, c ef, of a cu-cle, the centre, o', of which is obtained- by drawing through the point, o, a line, o i, representing the ray of light, and extending to the prolongation of the edge, d' f'. The line, o i, cuts the circumference of the cylinder in the point, i', which is squared over to i, upon the other projection, h i. fig. 9, of the ray, o i. The lower point, c, is obtained from the upper one, i, being symmetrical with reference to the axis of the cylinder. The ray, h i, being continued to the axis, cuts it in the point, o', which is, consequently, the centre of the arc, cei, the radius, i o' or c o', of wliich is equal to that, o i', of the cylinder. 286. The edge, f' h', although situated in a plane perpendicular to the axis of the cylinder, is not parallel to the vertical plane, and does not, therefore, cast a shadow of a circular outline upon the cylinder, but one of an elliptical outline, as fg h, which is obtain- ed by means of points, the operations being fully indicated on the diagrams. If the head, b, which casts a shadow upon the cylin- der, were square, instead of hexagonal, as is often the case, one of the sides of the square, as i h', fig. 9°, being perpendicular to the vertical plane, would cast a shadow on the cylinder, having for outline the straight line, h i, making the angle of 45° with the axis. Thus, ivhenever a straight line is perpendicular to the plane of pyrojection, not only is its shadow, as cast upon this plane, a straiglU line, inclined at the angle of 45°, but it is also the same on an object projected in this plane, no matter of what form. Observation. — In the four examples last discussed, we have only represented half views of the objects in the auxiliary vertical pro- jections, figs. 6", 7," 8°, and 9", this being quite sufficient for deter- mining the shadow, as it is only that produced by this half which is seen. It is obvious, that the same operations will answer the pur- pose, whether the axis of the object be horizontal or vertical. SHADOW cast by A CYLINDER IN AN OBLIQUE POSITION. 287. In figs. 5 and 5', we have given the horizontal niul vertical projections of a right cylinder, having its axis horizontal, but inclined to the vertical plane. As in this oblique jjrojcction we cannot obtain the points of contact of the luminous rays w itli the base in a direct manner, it becomes necessary to make an especia. diagram, in order to determine the lines of sejiaiation of light and shade, which are always straight lines, paiallcl lo Ihc axis of ilui cylinder. To this ert'ect, we shall make use of a general constriiclion, sus- ■ eeptible of ap|)lication to a variety of such cases. This construction consists in determining the projection of the luniinous ray, in any given plane, perpendicular to either of the geoinetriciU planes, w lienco may bo derived its form and aspect in either of the latter planes. It follows, that if we have any curve in (he given plane, wo can easily lind llio point of separation of tlu' light and shade situated upon this curve, by drawing a couple of tangents to it, [virallel to the ray of light projected in this plane, and transferred to the other plane of projection. Thus, let li o and i;' o' be the projections of the luminous ray . it is projwsed to liiul llie projccticui ,.l'lhis ray upon llic plane, d />, of the base of llie cylin.lrr. To obtain this, project the u.iliit. v. lo r, by means of a pi-rpciulicular lo a b. and r i icprosout.s tlio horizontal projcctitm i>l' the ray of li;^lit \\\w\\ (he plane, we , ftiuf 100 THE PRACTICAL DRAUGHTSMAN'S the vertical projection, r' o', is obtained by squaring over the point, o to 0', on the base line, and the point, r to r', on the horizontal, k' e', and then joining o' r . Next, draw tangents to the ellipses, which represent the vertical projections of the ends of the cylinder, fig. 5', making these tangents parallel to the ray of light, / o'. Their points of contact give, on the one hand, the first line, c' i\ of separation of light and shade, which is visible in the vertical pro- jection, and, on the other hand, the second line, e f\ which is not visible in that projection. By squaring over these points of contact, respectively, to the two ends, a b and g h, of the cyliuder, in the horizontal projection, we obtain the same lines of separation of light and shade, cd and/e, as in this projection ; the former of wliich lines is invisible, whilst the latter is visible. The same lines, c d and/e, fig. 5, can be obtained independently of the vertical projection, fig. o", in the following manner : — Draw an end view of the cylinder, as at a' b\ having its centre in the con- tinuation of the cylinder's axis. Upon this end view, also, draw the ray of light, as projected upon the base, after describing the circle, a' m b, with the radius, o a ; make r r" equal to the height of the point, r', above the bottom of the cylinder, thereby obtaining the line, o r^, representing the ray of light upon the end view of the cylinder. Next, draw a couple of tangents to the circle, a" m b\ parallel to o r^, and their points of contact, c\f^, will represent the end view of the lines of separation of light and shade, which are transferred to the horizontal projection, fig. 5, by perpendiculars drawn from them to the straight line, a b. 288. When the shadow proper, of the cylinder, has been thus determined, it \vill not be difficult to find the outline of its shadow cast upon the horizontal plane. In the first place, the shadows of the two bases are found, being in the form of ellipses; and next, those, c' d'' and /' e", cast by the lines, c d and fe ; namely, those of the separation of light and shade upon the object itself. These lines vnll necessarily be tangents to the ellipses, representing the shadows of the bases. It may be observed, that the transverse axes of the ellipses are parallel to the line, r* o. If the cylinder were inclined at an angle of 45° to the veitical plane, stUl remaining parallel, however, to the horizontal plane, the lines of separation of light and shade would, in the horizontal pro- jection, be confounded with the extreme generatrices, or outlines, of the cylinder, the visible semicylinder being wholly in the light, and the opposite semicylinder wholly in the shade. In the vertical projection, the line of separation would be in the line of the axis, and would divide the figure horizontally iiito two equal parts. PRINCIPLES OF SHADING. Plate XXVH. 289. Before proceeding to the further study of shadows, we must observe that shade ws^ proper and cast — which are simply represent- ed by flat tints, so as not to render the diagrams confused — should be modified in intensity according to the form of the objects, and the position of their surfaces with reference to the light. The study of shading carries us somewhat into the pr.)vince of toe non-mechanical pamter, who is guided by his taste rather than | b} mathematical rules : still, whilst we acknowledge the difficulty of lapng down an exact theory on this subject, we would recom- mend the following systematic methods, which will render the first difficulties of the study more easily surmountable. In painting, and in every description of drawing, the eifects of light and shade depend upon the following principles : — ILLU.MISED SURFACES. 290. When an illumined- surface Tias all ils poinis at an equal distance from ilte eye, il must receive a clear shade of uniform inten- sity throughout. In geometrical drawing, where all the visual rays are supposed to be parallel and perpendicular to the plane of projection, all sur- faces parallel to this plane have all their points equally distant from the eye : such is the plane and vertical surface, a b a d, of the prism, fig. A. 291. Of two such surfaces, disposed parallel to each other, and illumined in the same manner, thai which is nearer to the eye should receive a shade of less intensity. 292. Any illumined surface, inclined to the plane of the picture, having its points at varying distances from the eye, should receive a shade of varying intensity. Now, according to the foregoing principle, it is the most advanced portion of an object which ought to be the lightest Lq colour ; this effect is produced on the face, a dfe, which, as shown in the plan, fig. 1, is inclined to the vertical plane of projection. 293. Of two illumined surfaces, that which is more directly pre- sented to the rays of light should receive a shade of less intensity. Thus, the face, e' a', fig. 1, being presented more directly to the light than the face, a' b', is covered with a shade which, being gra- duated because of the inclination to the plane of the picture, is still, at the more advanced portion, of less intensity than that of tlie latter face. It is near the edge, a d, that the difference is more sensible. SURFACES IN THE SHADE. 294. When a suiface in the shade is parallel to the plane of p^u^ jection, or of the picture, il must receive a deep tint of uniform inie>i- sity throughout. An exemplification of this will be seen on the fillet, b, fig. ©, Plate XXVIII., which is parallel to the vertical plane : the differ- ence of shade upon this fillet, in comparison ^\•ith that upon the more projecting portion, a, which is parallel to it, but in the fight, distinctly points out the ditference between an illumined surface and one in the shade, in conformity with the two principles, 290 and 294. 295. Of two parallel surfaces in the shade, that nearer the eye should receive the deeper tint. Thus, the shadow cast upon the fillet, b, fig. ©, Plate XX\aiI., is sensibly deeper than that cast by it upon the vertical plane, which is more distant. 296. When a surface in the shade is inclined to the plane of the picture, the part nearest to tlie eye should receive the deepest tint. The face, bghc, fig. A, Plate XXVIT., projected horizontally in b' g', fig. 1, is thus situated. The shade is made considerably deeper neai- the edge, b c, than near the more distant one, g h. BOOK OF INDUSTRIAL DESIGN. lOi 297, When two surface!' in the shade are uneqiuzlly inclined, umh reference to the, direction of the rays of light, the shadow cast by any object should he deeper upon that which receives it more direcily. Thus, the shadow, a dfe, cast upon the face, f, of the prism, fig. [1, Plate XXVI., should be slightly stronger than that cast upon the face, g, because the first is more directly presented to the light than the second, as showTi by the lines, /' h' and /' c', fig. T. These first principles are exemplified in the finished figures on Plate XXVI., XXVIL, and subsequent ones. As, in order to produce the gradations of shades, it is important to have some knowledge of actual colouring or shading by means of the brush, we shall proceed to give a few short explanations of this matter. Two methods of producing the graduated shades are in use — one consisting in laying on a succession of flat tints ; the other, in softening off the shade by the manipulation of the brush. We have already said two or three words about the laying on of flat tints, when treating of representing sections by distinguishing colours. (137.) These first precepts may serve as a basis for tJie first method of shading, which is the less diflScult of the two for beginners. In fact, according to it, the gi-aduated shade is produced by the simple superposition of a number of flat tints. FLAT-TINTED SHADING, 298. Let it be required to shade a prism, A, Plate XVII., with flat tints : — According to the position of this prism, with reference to the plane of projection, as seen in fig. 1, it appears that the face, a' h'-> is parallel to the vertical plane, and is fully illumined ; it should, consequently, receive a clear uniform tint, spread over it by the brush, and made either from China ink or sepia, as has been done upon the rectangle, a, b, c, d, fig. A. When the surface to bo washed is of considerable extent, the paper should first be prepared by a very light wash, the full intensity required being arrived at by a second or third. (137.) The face, b' g', being inclined to the vertical plane, and com- pletely in the shade, should receive a tint (294) deepest at the edge, b c, and gradually less intense towards g h ; this is obtained by laying on several flat shades, each of different extent. For this purpose, and to proceed in a regular manner, we recommend the student to divide the face, b' g', fig. 1, into several equal parts, as in the points, 1', 2', and through these points to draw lines parallel to the sides, b c, g h, fig. A. These linos should be drawn very lightly indeed, in pencil, as they are merely for guides. A first greyish tint is then spread over the surface comprised botwocu tho first line, 1 — 1, and tho side, b c, as in fig. 2; when this is quite dry, a second like it is laid on, covering tho first, and extending from the side, b c, to tho line, 2 — 2, as in fig. 3. Finally, these are covered with a third wash, as in fig. Z^, extending to llie outer edge, g li, and completing the gniduatcnl sliado of the rectangle, bcgh. The number of washes by which tho gradation is expressed, evidently depends upon tho width of tho surface to bo shaded ; and it will be seen that ilw c:''oator the number of washes used, the lighter they should be, and the lines produced by the edges of each will be less hard, and a more beautiful effect will resdt. The student must remember to efface the pencilled guide-lines, as soon as the washes are sufficiently diy. 299. This method of overlaying the washes, and covering a greater extent of surface at each succeeding time, is preferable to the one sometimes adopted, according to which the whole surface, b g h c,is first covered by a uniform wash; a second being then laid over b 2 — 2 c; and finally, a third over the narrow strip, b ] — 1 c. When the shade is produced in this manner, the edges of the washes are always harder than when the washes are laid on as we recommend — the narrovi'est first — for the subsequent washes, coming over the edge of each preceding one, soften it to a consi- derable extent. The fiice, e' a', fig. 1, being likewise inclined to the vertical plane, but being wholly illumined, should receive a very light shade (292), being, however, a little bolder towards the outer edge, ef, fig. A. The shade is produced in the same way as that of the face, U g", but with much fainter washes. 300. Let it be proposed to shade a cylinder, fig. [B, with a series of flat tints : — In a cylinder, it is necessary to give the gradations of shade, both of the illumined and of the non-illumined portion. In reference to this, it will be recollected that the line of separation, a b, of light and shade, is determined by the radius inclined at an angle of 45°, as o a, fig. 4, perpendicular to the ray of light ; consequently, all the shadow proper, which is apparent in the vertical projection, fig. d, is comprised between the lino, a b, and the extreme generatrix, c d. Consequently, according to the principle already laid down (296), the shade of this portion of the surface should be graduated from a b to c d, as was the case with the inclined plane surface, b' g', fig. 1, the greater intensity being towards a b. On the other hand, all that part of the cylinder comprised be- tween the line, a b, and the extreme generatri.x,/g-, is in the light ; at the same time, from its rounded form, each generatrix is at a different distance from the vertical plane of projection, and makes different angles vvdth the ray of light. Consequently, this portiou of the surface should receive graduated sh.idos. (292.) To express the etfect in a projjcr manner, it is necessary to know what {Mrt of the surface is tho clearest and most brilliant : and this is cviilcnlly the part about tho generatrix, c i, \\g. ©, situated in the vertical plane of tho ray of light, r o, fig. 4. In consequence, however, of the visual rays being perpendicular to the vertical plane and parallel to tho line, v o, tho portion which a|)pears to tho eye to bo tho clearest will be nearer to this line, v o, and may be limited, on tho one hand, by the line, x o, bisecting the angle made by the linos, R o and v O, and on the other, by the line, k o; squaring over, then, tho points, c' and m', tig. I, and tlrawiii'j; the lines, c i and m n, fig. ©, we obtain the suilace, c / m n, wliirli is the most illumined. 301. This surface is bright, and rcni;iiiis white, wlun the cy.indej is iKilislu'd, as :i t iii iicd iron shaft, for e\,Mm|ile, or a marble coliuuii ■ it is covered willi n \\>j:Ui shade, being always clearer, however, timn the rest of liie suillu'e, when the eyliiuler is iinpulished, as a cast- iron l)i|ie. 302. .\tler these pieliniinary observations, we ni;iy prooeod t< 102 THE PRACTICAL DRAUGHTSMAN'S shade the cylimlcr, /' m' a! e', fig. 4, dividing it into a certain num- ber of equal parts, the more numerous according as the cylinder is •rrcater. These divisions are squared over to the vertical projection, and straight lines drawn lightly with the pencil, as limiting guides for the colour. We then lay a light gray shade on the surface, acdb, fig. 5, to distinguish at once the part in the shade ; when this is dry, we lay on a second covering, the line, a b, of separation of lio^ht and shade, and extending over a division on either side of it, as shown in fig. 6 ; we afterwards lay on a third shade, covering two divisions to the right and to the left, as in fig. 7 ; and proceed in the same manner, covering more and more each time, always keeping to the pencil lines. The diflferent stages are represented in figs. 8, 9, and 10. 303. We next shade the part,/e ig, laying on successive shades, but lighter than the preceding, as indicated in figs. 8, 9, and 10. The operation is finally terminated by laying a light wash over the whole, leaving untouched only a very small portion of the bright surface, e m n i, fig. [B. Tliis last wash has a beautiful and soften- mg effect. SHADING BY SOFTENED WASHES. 304. This system of shading differs from the former in producing the effects of light and shade by imperceptible gradations, obtained by manipulation with the brush in the laying on of the colour : this system possesses the advantage over the first, of not leaving any lines, dividing the dift'erent degrees of shade, which sometimes ap- pear harsh to the eye, and seem to represent facets or flutings, which do not exist. For machinery, however, the former system is very effective, bringing out the objects so shaded in a remarkable manner. In- deed, we recommend all machinery to be shaded in this manner, whilst architectural subjects will look better treated according to the second system. In this, the laying on of the shade is much more difficult, requii-- ing considerable practice, which will be aided by proceeding in the following systematic course. 305. Let it be proposed to shade a truncated hexagonal pyramid, fig. ©, Plate XXVII. The position of this solid, with reference to the vertical plane of projection, is the same as that of the prism, fig. A. Thus the face, abed, should receive a uniform flat shade of little intensity ; rigorously keeping to rules, this should be slightly graduated from top to bottom, as the face is not quite parallel to the vertical plane. The face, b g h c, being inclined, and also in the shade, should receive a deep shade, graduated from b c to g h; to this effect ap- ply a fiirst light shade to the side, b c, fig. 16, softening it off to the right, taking the line, 1 — 1, as a limiting guide in that direction: this softening is produced by clearing the brush, so that the colour may be all expended before the lighter side is reached ; and when the shade is wide, a little water should be taken up in the brush once or twice, to attenuate the colour remaining in it. By these means an effect will be produced like that indicated in fig. 15., care oeing tasec not to extend the wash beyond the outline of the object. When this first wash is well dry, a second is laid over it, produc- ed exactly in the same manner, and extending further to the right, covering the space, /; c 2 — 2, as shown in fig. 16. Proceeding in the same manner, according to the number of divisions of the face, we at length cover the whole, producing the graduated shade, bghc, fig. ©. The operations are the same for the face, e a df, which is nearly perpendicular to the rays of light, but is considerably inclined to the plane of projection. In rigorously following out the established principles, the shade on this face should be graduated, not only from e/to a d, but also from eatofd. Also, on the face, b g h c, in the shade, the tint should be a trifle darker at the base, c h, being graduated off towards b g. But for objects so simple in form as the one under consideration, this nicety may be neglected — at any rate, by the beginner — as only increasing his difficulties ; the proficient, on the other hand, is well aware how attention to these refinements assists in producing effective and truthful representations. 306. Let it be proposed to shade a cylinder with softened washes, fig. ©, Plate XXVII. By following the indications given in fig. 4, for the regular im- position of the shades, as explained with reference to the flat-wash shading, the desired effect may be similarly produced by substituting the softened washes. It is scarcely necessary to divide the circum- ference into so many parts as for the former method ; a first shade must be laid on at the line, a b, of separation of light and shade, and this must be softened off in both directions, as in fig. 11; a second and a third wash must then be applied and similarly softened off, and in this manner we attain the effects rendered in figs. 12, 13, and ©. We have not deemed it necessary to give diagrams of all the stages, as the method of procedure will be easily understood from preceding examples. The student should practise these methods upon different objects of simple form, and he will thereby rapidly acquire the necessary facility. 307. Wlien spots or inequalities arise in laying on a wash, from defects in the paper or other accidents, they should be corrected with great care. If they err on the dark side, they should, if pos- sible, be washed out ; the best means of doing this, in very bad cases, is to let the drawing become perfectly dry, and then slightly moisten the spots, and gently rub off the colour with a clean rag. Lights may be taken out in this way, where, from their minuteness or intricate shape, it would be difficult to leave them whilst laying on a flat shade, in the midst of which they may happen to be. A de- fect on the light side is more easily corrected, by applying more colour to the spots in question — being careful to soften off the edges, and to equalize the whole wash. Figs. ^, [U, ©, ©, [1, Plate XXVI., represent several shaded objects, the shadows of which have already been discussed, as indi- cated in figs. 1 to 9. These may serve as guides, also, in shading with washes of colour, although the shades in that plate are pro- duced by lines, whilst the figures in Plate XXVII. represent t^-- actual appearance of the wash-shading method. ' Finally, we have to recommend the adoption of a much larger scale for practice, as it is desirable to be able to produce largo washes with regularity and smoothness of effect. BOOK OF IXDUSTRL\L DESIGN. 103 CONTINUATION OF THE STUDY OF SHADOWS. Plate XXYIIL shadow cast tjpos the dtteeior of a ctirsdee. 308. When a hollow cylinder, as a steam-engine cylinder, a cast- ir )n column, or a pipe, is cut by a plane passing through its axis, we hare, on the one hand, a straight projecting edge, and, on the other, a portion of one of the ends, which cast shadows upon the internal surface of the cylinder. We propose, then, to determine the form, as projected, of the shadow cast upon its interior by a steam-engine cylinder, a, sec- tioned by a plane passing through its axis, figs. 1 and 1°. In the first place, we seek the position of the shadow cast by the rectilinear projecting edge, b c, which is, in fact, produced by the intersecting plane, b' a'. This straight line, b c, being vertical, is projected horizontally in the point, b', and casts a shadow upon the cylinder, as represented by the straight line, 6/, which is also vertical, and Ls determined by the point, h , of intersection of the ray of light, b' V, with the surface of the cylinder, e' V of. Thus, when a straight line is parallel to a generatrix of the cylinder, the shadow cast by it will he a straight line parallel to the axis. It b, therefore, evidently quite sufficient to find a single point, whence the entire shadow may be derived. 309. We next proceed to determine the shadow cast upon the interior of the cylinder by the circular portion, b' e' r', of the upper end. If we take any point, e', on this circle, and square it over to e in the vertical projection, and draw through this point a ray of light, e' e*, e e, it will be found to meet the cylindrical surface in the point- e', which is squared over to e, the length of the ray being equal in both projections, according to the well known rule. This applies to any point in the arc, e' f'. The extreme point on one side is obtained by a tangent to the circle in the point, f', ^ving the point, r, in the vertical projection ; the opposite extreme point, b, being already given as the top point of the straight edge, b c ; we have, therefore, the curve, i eb, for the upper outline of the shadow due to the circular portion, b' e' r'. 310. If , as in figs. 1 and 1', we suppose the piston, p, with its rod, T, to be retained unsectioned in the cylinder, we shall have to determine the form of the shadow cast by the projecting part of the piston upon the interior of the cylinder, and represented by the cune, dho. For this purpose we take any points, b', h', &, on the circumference of the piston, and draw through them, in both pro- jections, the rays of light which meet the surface of the cylinder, b' y 0, in the points, b' h' o', which are projected vertically in dho: the curve passing through these points is the outline of the shadow sought The curved portions of these shadows are elliptical. The piston-rod, t, being cylindrical and vertical, casts a shadow, of a rectangular form, upon the interior of the cylinder, the vertical sides, i). k Z, being determined by the luminar tangents, i' i", k' k', parallel to the axis. SHADOW cast by O.XE CYLI.VDER tTPOX A.NOTHEB. 311. Let figs. 2 and 2* be the projections of a convex scroi- cylinder, a, tangential to a concave semicylinder, b. funning a pat- tern often met with in mouldings. This problem, which consists in determining the shadow proper of a convex cylinder, together with that cast by it upon the surface of a concave cylinder, in addition to that cast by the latter upon itself, is a combination of the cases discussed in reference to figs. 4 and 4', Plate XX\T., and to figs. 1 and 1* in the present plate. The operations called for here are fully indicated on the diagram ; and we have merely to remark, that it is always well to start by deter- mining the extreme points, as c', d,' which limit the shadow proper c G, and cast shadow, v c g: these points may be obtained more exactly, as already pomted out, by drawing the radii, o c' and d' e, perpendicular to the luminous rays. SHADOWS OF COSES. 312. In this branch of the study, we propose to determine, first, the shadow proper, or the line of separation of light and shade upon the surface of the cone ; second, the shadow cast by the cone upon the vertical plane of projection ; and- third- the shadow cast upon the cone, and upon the vertical plane of projection, by a prism of a square base, placed horizontally over the cone. 313. First: We have laid it down as a general principle, that, in order to determine the shadow proper of any surface, it is neci-ssary to draw a series of parallel luminous rays tangential to this surface. When, however, the body is a solid of revolution generated by a straight line, as a cylinder or a cone, it is sufiicient to draw tangen- tial planes parallel to the Inminotis rays, to obtain the lines of sepa>- ration of light and shade. In the ca.=e of the cone represented in figs. 3 and 3*, and of which the axis, s t, is vertical, the operation consists in drawing from the apex, s and s', two lines, making angles of 45°, as s s and s' s', gi\Tng, in the point, s', the shadow cast by this apex uj>on the hori- zontal plane. From this point we draw a straight line, a s', tan- gential to the base, a' c' b', of the cone. This straight line repre- sents the plane, tangential to the cone, as intersecting the horizontal plane of the base ; and the contact generatrix is then obtained by letting fall from the centre, s', a radius, s' a', perpendicular to the line, a* s' ; and this line, s' d, is the horizontal projection of one of the lines of separation of light and shade. The vertical projection of this straight line is obtained by squaring over the p<^)int of con- tact, a',, to a, and then drawing the straight line, s a. The other line, s b, of the separation of light and shade, is similarly obtainea by means of the tangent, s' b'. Its vertical projection is, however, not apparent in fig. 3*. 314. Second : The shadow cast by the cone upon the vertical plane is limited, on the one hand, by the line of separation of light and shade, and, on the other, by the portion of the illummed Laso comprised between the two separation lines. Now, the straight line, s a, casts a shadow, represented by the straight ILae, s* a*, aa indicated in the diagram ; and the base, a' e' c' b', casts a shadow, represented by the elliptic cur%e, / e d a', which is detennintil by points, as in the ca.«e considered in reference to fig. 5, Plate X.Wl 315. Third: The shadow cast by the lower aide, o u, of iho rectangular prism, r, upon the convex surface of the cone, is found in oiTordance with the principle already onunciat*-*! — that when n stmight line Ls jwirillel to a plane of pr»jo»tion, it oAsts a sJuulow 104 THE PRACTICAL DRAUGHTSMAN'S upon this plane, which is represented by a straight line, equal and jMirallel to itself. It follows, then, that if we cut the cone by a plane, M N, parallel to its base, the shadow cast by the straight line, g h, upon this plane, will be found by drawing from the point, i, of the base, situated upon the axis of the cone, and projected horizontally in the point, s', a luminous ray, which meets this plane, m n, in the point, i, projected horizontally in the point, i', upon the projec- tion of the same ray. If, next, we make i' i^ equal to s' j', and through i' draw the straight line, g" h^ this last will be the shadow cast by the edge, g h, of the prism, upon the plane, M n. This plane, however, cuts the cone in a cii'cle, the diameter, m n, of which is comprised between the extreme generatrices, whilst the circle is projected horizontally in ivi' l' n'. The intersection of the straiglit line, g" h°, with the circle, gives the two points, i^ and i', which being projected vertically in i, i", upon the straight line, M N, constitute two points in the outline of the shadow cast upon the cone. Continuing the operations in this manner, and taking any other intersecting plane parallel to m n, any number of points may be obtained. It will be observed that these planes are taken at a convenient height, when the projections of the straight line, g h, cut the corresponding circles ; and with regard to this, much use- less labour may be avoided, by at tirst determining the limiting points of the curve. Thus, in the example before us, we get the summit, g, of the curve, by making i j, fig. 3°, equal to j' s', fig. 3. Through the point, j, we then di-aw a luminous ray, and the point, (r", at which it meets the extreme generatrix, a s, of the cone, is squared over to the generatrix, s t, by means of the horizontal,g-"^, whence g is the summit of the curve. We next obtain the extreme points, h, h', of the same shadow, by making s' g", fig. 3, equal to s' g', and Squaring over g" to g', in fig. 3°. Through g^ draw the straight line, g' h', parallel to the luminous ray, as situate in a vertical plane, passing through it ; the ray, as we have already seen, making, in this plane, an angle of 35° 16' with the base line. The point, h', at which this ray meets the extreme generatrix, s a, de- termines the plane, h' h, which is intersected by the luminous rays, making angles of 45°, and drawn through the points, i and g, in the points, h, h', the limits of the curve sought. The shadow cast by the prism, p, upon the vertical planes, pre- sents no peculiarity apart from the principles already fully explained. SHADOW OF AN INVERTED CONE. 316. When the cone, instead of resting upon its base, has its apex downwards, as is the case with the one represented in figs. 4 and 4", the rays of light illumine a less portion of its surface ; and the lines of separation of light and shade are determined by draw- ing from the apex, s s', lines at an angle of 45°, which are prolonged towards the light, until they intersect the prolongation of the plane of the base, a b. It will be observed, that the points of intersection, s, s', lie to the left, instead of to the right of the cone. Through the point, s', the horizontal projection of the point, s, draw a couple of lines, s" a', s' b', tangential to the circumference, a' b' d', of the base. The radii, s' a' and s' b', drawn to the points of contact, represent the horizontal projection of the two lines of separation of light and phade, and show that the illumined portion of the cone, consisting of the surface, b' g^ a' s', is smaller than the portion, b' d' a s', in the shade. In the case of the cone \vith its apex uppermost, the contrary would be observed, the portion in the shade being there less than that in the light ; and the method given of determining the proportion of shadow of the inverted cone is suggested by the consideration, that this proportion must be exactly the reverse of that for the cone with its apex uppermost. The first-mentioned line, s' a, is the only one apparent in the vertical projection, fig. 4°. It is found by squaring over the point, a', to a, and joining this last to the apex, s. As the cone is trun- cated by the plane, d e, the line of separation obviously terminates at the point, c, of its intersection with tliis plane. 317. The cone, thus inverted, is surmounted, moreover, by a square plinth, the sides of which, f g' and c' h', cast shadows upon its convex surface. The side, f g', as projected vertically in g', fig. 4", is perpendicular to the vertical plane, and consequently its cast shadow is a straight line, making an angle of 45°, as of. The extreme limit,/, is determined by proceeding as in previous exam- ples ; that is to say, by making i G^ in fig. 4", equal to s' g', in fig. 4, and then drawing through the point, g', the straight line, g^ h", parallel to the ray of light, as in the diagonal plane, that is, at an angle of 35° 16' to the horizon ; next draw the horizontal line, h" h, and it will be intersected by the straight line, g f, in the poiat,/, which is consequently the shadow cast by the corner, g. The following method, although more complicated, is of more universal application : — Draw the vertical projection of the outline of the body which receives the shadow, as sectioned by the verti- cal plane, in which the ray of light lies, which passes through the point whose shadow is sought; draw the same ray of light as projected in the vertical plane, and its intersection with the projec- tion of the sectional outline will be the projection of the shadow of the point. Thus, in the present instance, as the plane of the ray, g' s', passes through the apex of the cone, the latter will present a trian- gular section, the vertical projection of which may be obtained by squaring over the point, g^ fig. 4, to the base line, a b, fig. 4° ; then, if a straight line is drawn from the vertical projection of this point to the apex, s, it will represent the projection of the section of the cone, and it will be intersected by the luminous ray, g/, in the point, /, which is the point sought. If the plane passing through the point does not likevidse pass through the axis of the cone, the section will be a parabola, which may be di-awn according to methods already discussed. If the objeet is a sphere instead of a cone, the section wall be a circle, whether the plane passes through the centre or not, and the verti- cal projection will, in all cases, be an ellipse. As a good idea of the whereabouts of the point sought may always be formed on inspection, it will generally be sufficient to find one or two points in the parabola or ellipse, near the supposed position, when a suffi- cient length of the curve may be drawn to give the intersection of the luminous ray, as g/. As the plinth is square, the summit, g, of the curved outline, corresponding to the shadow of the front edge, g h, is obtained directly by the intersection in g" of the line, g/, with the extreme generatrix, a s, the horizontal line, g" g, being drawn through this • point. Any oth^v point in the curve, as i", is afterwards found BOOK OF INDUSTRIAL DESIGN, 105 by means of the sectional plane, m n ; g' h, fig. 4, is the shadow of Ihe edge, g h, in that plane, and it cuts the circle representing the section of the cone in the same plane in the point, i", which is ob- viously a point in the outline of the shadow. SHADOW CAST UPON THE INTERIOR OF A HOLLOW CONE. 318. Fig. 5 represents a plan of a hollow truncated cone, and tig. 5" is a vertical section through the axis of the object. It is required to determine the horizontal projection of the shadow cast upon the internal surface of the cone by the portion of the edge, a' b c, and the vertical projection of the shadow cast by the sec- tional edge, D s, and by the small circular portion, a' d', projected vertically in a d. It is to be observed, in the first place, that the straight line, d s, which is a genejatrix of the cone, casts a shadow upon the latter, in the form of a straight line, for the plane parallel to the ray of light, and passing through this line, d s, must cut the cone in a generatrix ; "we therefore draw through the point, d', the ray of light, d' d', making an angle of 45° with the base line, and from the centre, s', let fall the perpendicular, s' e', this straight line representing the horizontal projection of the intersection of the cone by the plane passing through the line, d' s', and at the same time parallel to the ray oi ngnt. By squaring over the point, e', to e, fig. 5°, and joining e s, we have the vertical projection of this line of intersection, and consequently the shadow cast by the line, D s. The diagonal ray of light, d d, drawn through the point, d, deter- mines the limit, d, of the shadow. The horizontal projection of the extreme points, a' and c, of the curved outline of the shadow, is also obtained by means of the tangents, s' a' and s' c, drawn from the point, s, in which the ray of light passing through the apex intersects the plane of the base of the cone. The determination of the central or symmetrical point, b', of the same curve, is derived from the straight line, d b, drawn from the point, D, parallel to the ray of light, s e", as in the diagonal plane, that is, as at s r" ; the point, b, in which this straiglit line meets the generatrix directly opposite to that passing through the point, D, is projected horizon- tally in the point, b', upon the prolongation of the diiigonal ray of light, s' s'. 319. The operation for finding any intermediate point in the curve, is based on principles already explained ; namely, that when a line or a surface is parallel to a plane, the shadow cast is also a line or a surface equal and parallel to the first. If, then, we draw a plane, M n, parallel to the base, d f, of the cone, the shadow cast by this base upon the plane, m n, will be a circle ; it will consequently be sufficient to draw through the centre, o, fig. 6°, a ray, o a, which will meet the plane, m n, in a, which must bo squared over to a', on the horizontal projection of the same ray. Next, with the point, a', for a centre, and with a radius (Hjual to D o, describe a circle, h' i j; this will represent the entire shadow that would bo cast by the base, D f, of the cone upon the plane, m n ; this plane, however, cuts the cone in the circle, of (vhich M n is the diameter and vertical projection, whilst h' m' j n' « (he horizontal projection ; this circle is cut by the former in the points, ii' and j, wliicli are consequently two points in the outlines of the shadow in fig. 5, and the one of these which is nwn in llic vertical |)rojection is squared over to H, upon the line, ivi n. apflications. 320. In this plate, as well as in Plate XXVI., we have given shaded and finished representations of several objects, which serve as applications of the several principles we have just pointed out, whether referring to shadows proper, or cast, or to graduated shad- ing. Thus, fig. A represents the interior of a steam-engine c\'linder with piston and rod. In this example, regard has been had to the general principle, that shadows are the stronger the brighter the surfaces on which they fall would be, if illumined — that is, when such surfaces are perpendicular to the rays of light, any shadow cast upon them will be most intense ; the shade is consequently made deepest about the generatrix, corresponding to g ft, in fig. 1°, and situate in the vertical plane of the rays of light passing through the axis of the cylinder : to the right and left of this line, the shade is softened oft'. 321. In the graduation of the shade, regard has also been had to the eflfects of the reflected light, which prevents a surface in the shade from being quite black. In a hollow cylinder, for the por- tion in the shade, it is the generatrix, f f', fig. r, which should receive the shade of least intensity, as it receives the reflected rays of light more directly. It will be recollected that the point, f', is obtained by means of the radius, t f', perpendicular to the ray of light. Fig. [B represents a portion of a common moulding, and shows how the distinction made between the shadow proper, and the cast shadow, tends to bring out and show the form of the object. Fio-. © is an architectural fragment from the Doric order, given as an application of shadows cast upon cones, as well as those cast by cones upon a vertical plane. This example also shows how necessary it is, in producing an effective representation, to make a diflference in the intensity of shadows cast upon planes parallel to the plane of projection, and at different distances from the eye ; and also to give gradations to such shadows when cast upon rounded surfaces. Fig. [B) is a combination of a cylinder with a couple of cones. with their apices in opposite directions, showing how differently the eflfects of light and shade have to be rendered upon each. There is less shadow upon the upper cone than uj)on the cylin- der, whilst there is more upon the lower cone ; the reasons of those differences have already been explained in reference to figs. 3' and 4°. Fig. H represents an invc^rtcd and truncated oono, showing the manner of shading the same, and the form of the shadow cast by t!lio square tablet above ; and tig. P is a view of a hollow cone, sec- tioned across the axis, presenting a further variety of combinations TUSCAN ORDER. Platk XXIX. SHADOW OF THE TOKUS. 322. In geometry, the torus is a solid, gonoratod by a circle, ro. volving about an axis, continuing constantly in the plane ol iJiia axis, ill such a iiiamur, that all sections made by phmes [ViUs-sinji thidiinh the axis are equal ciivles, and all sections by pianos perjH>n- iliiiilai- In the axis v.ill also be cin-les, but of variable diiunetciv 106 THE PRACTICAL DRAUGHTSMAN'S Wc have seen, that in architecture, the torus is one of the essen- tial parts of the base, and of tlie capital of the column, of each order. It nill, therefore, be useful to give the methods of deter- mining the shadows upon it, or cast b}' it, in the quickest and most accurate manner. Figs. 1 and 1° represent the two projections of a torus, a, sup- posed to be generated by the semicircle, af c, revolving about the vertical axis, o p ; namely, that of the column. 323. We propose to determine the shadow proper of this torus, (ir the line of separation of light and shade upon its external sur- face. It will be convenient, in the first place, to seek the principal points, which, for the most part, present little difficulty. Thus, by drawing, parallel to the ray of light, r o', a couple of tangents to the semicircles, afc^ which limit the contour of the torus in the vertical projection, we at once obtain the two extreme points, 6, rf, of the cuned line of separation. These points are more exactly defined by letting fall perpendiculars from the centres, o, o', of the semicircles, upon tlie tangents. Then, by drawing through the point, h, the horizontal, i e, the middle point, e, of the cm-ve will be obtained upon the vertical line, o p. To obtain the curve in the horizontal projection, square over the points, b, e, d, of fig. T, to b', e', d', fig. 1, which wUl lie in a circle, having b e ox o d for radius. An additional point, g', is obtained by drawing the diagonal, o' g', perpendicular to the ray, r' o' ; this radius cuts the outer circumference,/' ft/', of the torus. La the point, g'. This circxmiference,/' h /', is projected vertically in the hori- zontal line,//, passing through the centres, o, o , of the semicircles, and the point, g', is squared over to g, in fig. 1°. To find the point, ?', which seems to be the lowest in the curve, and which is situated in the vertical plane passing through the luminous ray, i o , of the horizontal projection, fig. 1 — 2, we pro- ceed as follows : — Suppose the vertical plane, i o', to be turned about the axis, o p, so as to coincide with the plane of projection, when the section of the torus by the plane, i' o', being obviously a semicircle, wiU coincide with the semicircle, afc, draw a tangent, k i'', to the last, parallel to the ray of light, as in the vertical plane, i o' — that is, at an angle of 35° 16', as has been already explained — the point of contact, ■?, is the one sought. But it has to be transferred to the original position of the vertical plane ; and for this purpose it is squared over to ^^ in the horizontal projection. Then o' i' is made equal to o' i^, and the point, i', again squared over to i, in the horizontal, t" i, drawn through f". It is generally sufficient to find five principal points, as b, i, e. g, and d, in the curved line of separation of light and shade ; but if, because of the large scale of the drawing, it is wished to obtain intermediate points, this may be done by drawing planes passing through the axis ; such, for instance, as o' b', which cuts the torus in a circle of the same radius as the generating circle. We then proceed to find the point of contact of the ray of light, according to the method indicated in figs. 5 and 5°, Plate XXVT. ; that is to say, we seek the projection of the luminous ray upon this plane, o' b'. For this purpose, we let fall up"n this plane a perpendicu- lar, r' i, from any point taken upon the luminous ray, r' o', and we obtain a face view of this ray, as projected upon the plane, o' b', by supposing tne latter to be turned about the axis, o', until it coincides with / o', r' then coinciding with r\ Then, as the height, r' r, of the point, r', above the horizontal plane, is equal to that of the point, r, the line joining r o will be the face \iew of the projection of the ray in the plane, o b'. In turning round tlie plane, o' b', the section of the torus will become coincident with the semicircle, afc, as in a previous operation. K, therefore, we draw a straight Ime, m n, tangential to this semicircle, and parallel to the ray, r o', the point of contact, n, will be the point of separation of light and shade, as in the plane, o' b'. Finally, we square n over to n', fig. 1 ; make o' n^ equal to o' n', by describing an arc with the centre, o', and radius, o' n', and cutting the line, o' b', in n*. This point, n\ we again square over to n^, upon the horizontal, n ri', in fig. 1°, and rf is the point sought. Or we might have drawn the vertical projection of the section of the torus by the plane o' b', which would have been an ellipse, similar to that in fig. 5°, Plate XXVI. ; and we might have proceeded, as shown in reference to that figure, the result being the same in both cases. If the circular arc be prolonged to beyond the radius, o' g', and upon it, g I be made equal to g" n\ another point, I', will be ob- tained, symmetrical with w', with reference to the radius, o' g , which is at an angle of 45° to the base line, and perpendicular to the luminous ray. This point, T, is to be squared over to I, in the vertical projection, and upon the horizontal, p I, drawn at a distance, q p, above the centre line, / /, equal to the distance, q s, of the horizontal passing through the point, n\ below it. 324. When the shadow proper of a torus is known, it is very easy to determine the shadow which it will cast upon the horizontal plane — the plinth or pedestal below it, for instance — by drawing, through any points in the line of separation of light and shade, a number of lines parallel to the luminous ray, and then finding the points at which these lines intersect the horizontal plane. Thus, in figs. 2 and 2°, a portion of the torus, a, casts a shadow upon the horizontal plane, b c, the outline of which is a curve ; but the por- tion, a' b' c', of this curve is all that is visible. Any point, b, b', in this curve is determined by the meeting of the ray of light drawn from the point, 1 1', with the plane, b c. The half of the line of separation of light and shade upon the posterior portion of the torus, fig. 1°, which is not seen in the front elevation, is similar to the anterior half; it is indicated in dotted lines, the portion, o b, being simDar to the front portion, e d, whilst d\s similar to e b. 325. When the torus is surmounted by a cylindrical fillet, the line of separation of light and shade upon the latter will cast a shadow upon the surface of the torus. Thus, in figs. 2 and 2°, this will be the case with the fillet, d, the line of separation of light and shade of which is. fh. This Ime, being vertical, casts a shadow, which is a straight line,/' ?', parallel to the luminous ray, and de- termined bv drawing through the point,/, a luminous ray,/ -u meet- ing the horizontal plane, a a, in i, which point, i, is squared over to the horizontal projection. It remains to determine the shadow cast by the circular portion,/'/', which is in the shade: this may be done according to the general method explained in reference to figs. 3, 4, and 5, of Plate XXVT., and which ^ve shall have occasion to repeat on figs. 3 and 3" of the present plate. This method is also applicable for the determmation of the shadow, nj, n'j', cast by the cylinder or shaft, e, upon the annular gorge, which unite^s this cvlinder with the fillet, d. BOOK OF INDUSTRIAL DESIGN. 107 SHADOW CAST BY A STRAIGHT LINE UPON A TORUS OR QUARTER-ROUND. 326. Fig. 3 represents the horizontal projection, as seen from below, of a fragment of a Tuscan capital, of which fig. 3" is the vertical projection, the object of these figures being to show the form of the shadow cast by the larmier, f, which is a square prism, ujxin the quarter-round, a, which is annular. We yet again recall the general principle, that when a straight line is parallel to a plane, its shadow upon this plane is a straight line parallel to itself For the rest, it will be sufficient to compare the operations indicated with those of figs. 3 and 3", Plate XXVIII., to see that they are precisely the same : thus, on the one hand, we have the diagonal, g /, for the shadow cast upon the quarter-round, where it is limited by the curve, b e I, the line of separation of light and shade upon this ; and, on the other hand, we have the curve, . i" g i', likewise limited by the same curve, for the shadow cast by the edge, g h, of the larmier upon the quarter-round. Figs. 3 and 3" complete what refers to the shadow of the capital of a column ; they show the operations necessary to determine the shadow cast by the line of separation of light and shade of the quarter-round upon a cylinder, as well as that east on the same jylinder by a portion of the larmier. The operation, in fact, sunply consists in drawing the luminous rays through various points, i" e, in a portion of the line of separation of light and shade upon the quarter-round, finding their intersection with the cylindrical surface of the shaft, e, by "means of tlie horizontal projection. There is no peculiarity or difficulty in this procedure, and the whole being fully indicated upon the diagrams, we need not pause to detail it further. To render the diagi-ams just discussed more generally applicable »nd intelligible, we have not given to the different paits the precise proportions prescribed by this or that architectural order ; such pro- portions, however, will be found in fig. /^, which represents the model fully shaded and finished, being the entablature and eolunm ©f the Tuscan order. A double object is intended to be gained by this beautiful example of drawing ; namely, to show the application of the principles laid down regarding shadows, and the distinctness and niceties to be observed in the various intensities of the washes, and in the general shading. SHADOWS OF SURFACES OF REVOLUTION. 3'27. It will be recollected, that a solid or surface of revolution is that which may be said to bo generated by a straight or curved line, caused to turn about a given fixed axis, and maintaining a uni- form distance therefrom ; thus, the cylinder, the cone, the sphere, the torus, are all sui'faces of revolution ; so, also, is the surfkco generated by the curve, a b c, revolving about the axis, a b, figs. 4 and 4". It follows, from the above definition, that every section iiiiido ])crpendicularly to the axis will bo a circle, and all such sec- tions will be parallel. Every section made l)y a piano passing through the axis will give an oulliiic ((lual to the generating curve, and which may be termed a mcridinii. 3'28. The shadow of a surface of rcvohilidii lUMy Ixi (lelrriiiincd in two differont ways : by drawing sectional plnncs |)erpi luliciihn- to the axis, and then considering the sections made by tluse planes aa bases of so many right cones ; or by imagining a sciirs of planes passing through the axis, and then projecting the ray of light upon these planes, so as to draw lines tangential to the dif- ferent parts of the outline, and parallel to the projections of the ray of light, the points of contact of which will be points in the line of separation of fight and shade sought. This latter method having been applied in the preceding figs. 1 and l", Plate XXIX., and figs. 3 and 4, Plate XXVIII., we deem it more useful, in the present instance, to explain the operations called for in the first method. Take, then, any horizontal plane, b d, figs. 4 and 4% cutting the surface of revolution in a circle, the radius of which is b e, and the horizontal projection, b' e' d', through the points, b and d, draw a couple of tangents to the generating curve which fonns the outline of the surface of revolution. These tangents will cut each other in the point, s, upon the axis, tiiis point being the apex of an im- aginary cone, s b d; through this apex draw a luminous ray, s / and a' b', meeting the horizontal plane of the section, b df,'mf,f'; from this latter point, the horizontal projection, draw two straight lines,/' g' and/' i', tangents to the circle, b' c' d' ; then the points of contact, g' and i', will be the two points of the line of separa- tion of light and shade intersected by the plane, b d, and they are therefore squared over to the vertical projection, fig. 4°, i, only being there visible. It is in a similar manner that the points, h and _;', are deter- mined, these points being situated in planes, c d and e f, parallel to the first. It is to be observed, however, that, in these two last cases, the imaginary cones will be inverted, and the luminous raj must consequently be drawn to the left instead of to the right, as has already been explained in reference to figs. 3" and 4°, Plate XXVIII. 329. Wlien the tangents to the generating curve are vertical, as is the case with the sectional planes, m n and a I, the points, m and n, of the line of separation of light and shade, are determined by lines, inclined at an angle of 45°, and tangential to the circular sec- tions in the horizontal projection, because these circular sections are the bases of imaginnry cylinders and not cones. When a sufficient number of points have been obtained in this manner, as in fig. 4°, a curved line is drawn through them all, which will give the visible portion, ?/( / 7i hj e, of the line of sejiaration of light and shade upon the surface of revtilution. This method is o-eneral, and may be applied to surfaces of rcxohition of any outline whatever. As it is well to determine directly the lowest point, /;, of this and similar curves, it may be done in the same numner as for the torus- figs. 1 and r, namely, by drawing the ray of light, k b, at the ssmio inclination to the base line, as it is in the diagonal aiul vertical plane, and then drawing parallel to it a tangcTit to the outline of the surface of revolution, llic projccliou I'oi- the niomoiil being supposed to be in ii plane parallel to the ray of light, k' a'; the (lislance of the jxiint of contact, A. from the axis, being thou mea- sured upon the horizontal projection, r' a', of the huninou8 my lijves the jioiiil, /.'. which is finally squared over to A', in the hori- zontal line in the virlicMl projection pa.ssing through the simie poinl ofconlJicl. A |)orlioM of llii-i cnrvi', n.uinly, the lower part, i: A /, Ci».Hl!« « .shadow upon llie cylindrical lillcl.co; lo .lelirniiue this shadow 108 THE PRACTICAL DRAUGHTSMAN'S it will, in the first place, be necessary to delineate the horizcmtal projection of the curve, e kj, and then to diaw luminous rays through one or two points in the latter, to meet the circle, c' o', the horizontal projection of the fillet. The points in which the luminous rays intersect the circle, are then to be squared over to the vertical projection of the same rays, whence is derived the curve, c p q. The various operation lines are not indicated on the figures, to avoid confusion, but the proceeding will be easily com- prehended. 330. Fig. 4° represents the vertical projection of a baluster, such as is often seen in balconies of stone or marble, and sometimes also in macliinery, serving as an isolated standard, or as a portion of the framing. Below the fillet, c o, is an annular gorge, upon the surface of which the base of the fillet casts a shadow. It is easy to see that this shadow is obtained in precisely the same manner as those occurring in figs. 3 and 3°, Plate XXVIIL, as well as in subsequent diagrams. Figs. © and © represent the shaded models of two descriplions of baluster, consisting of surfaces of revolution. We recommend the student to draw them upon a large scale, and to determine the outline of the shadows in rigorous accordance with the principles which we have laid down. Such balusters are generally made of stone, and are susceptible of various sizes and proportions. We have, however, supposed them to be drawn to a scale of one-tenth their actual size. Many forms and combinations, of which we have said nothing, will be met with in actual practice ; but our labours would be interminable were we to give them all. Our exemplifications involve all the principles that are needed, and each case will suggest the modification of operations applicable to it. RULES AND PRACTICAL DATA. PUMPS. 331. There are three kinds of pumps. I. Lifting pumps, in which the piston or bucket lifts the water, first drawing it up by suction. We engrave one of this kind in Plate XXXVn. II. Forcing pumps, in which the piston presses or forces the water to any distance. The feed pumps of steam-engines are of this class, and one is represented in Plate XXXIX. III. Lifting and forcing pumps, in which both the above actions are combined. HYDROSTATIC PRINCIPLES. 332. Whatever be the height at which a pump delivers its water • — whatever be the calibre or inclination of the suction or delivery pipe — the piston has always to support a weight equal to a column of water, the base of wWch is equal to the area of the piston, and the height is equal to the difference of level of the water below, from which the pump draws its supply, and the point of delivery above. Thus, putting H to represent the ditference in the level, D for 62-5 X H X 1-08; the diameter of the piston, and P for the weight or pressure on the piston — p _rtD^ ~ 4 To express this pressure in pounds, it must be multiplied by 62-5, that being the weight in pounds of a cubic foot of water ; the formula then becomes — rtD'H P = 62-5 lbs., 4 ' the measurement being expressed in feet. 333. Independently of this load, which corresponds to the useful effect of the machine, the power employed in elevating the piston has other passive resistances to overcome, namely — 1st. The friction of the piston against the sides of the pump. 2d. The fi-iction of the water itself in the pipes. 3d. The retardation of the water in its passage to the pump by the suction valve. 4th. The weight of this valve. These resistances can only be determined approximately. Still it follows, from the experiments of M. d'Aubisson, that the load to be overcome in raising the piston is equal to rtD^ 4 or, more simply, 52-5 D' H. It is sufficient to add to this the weight of the piston and rod. The power exerted in depressing the piston, being assisted by the weight of itself and the rod, is always less than that required to raise it. 334. In ordinary pumps, the volume of water delivered for each stroke of the piston, instead of being given by the formula, —^ X I, or -785 D- 1, where I is the length of stroke, is determined by an expression which varies between •6 D^ I and -7 D' I. The velocity of the piston generally ranges between a minimum of 60 feet and a maximum of 80 feet per minute. The diameter of the suction and discharge pipes is generally equal to f or f of that of the body of the pump. It may be remarked, that the height to which liquids rise in vacuo, by the pressure of the atmosphere, is in the inverse ratio of their specific gravities. Thus this pressure, which is equal to 15 lbs. to the square inch, makes water rise to 33 feet, whilst mercmy only rises to 30 inches, its specific gravity being 13-59 times that of water. If the atmosphere presses on a liquid lighter than water, it will cause it to rise higher in vacuo than 33 feet, in proportion to the difference of the specific gravity. In practice, more than 29 or 30 feet cannot be calculated on for the lift of the pmnp, because of the difficulty of obtaining a perfect vacuum. FORCING PUMPS. 335. What has been here said of lifting pumps, applies as well to forcing pumps. The resistance, however, to be overcome, is somewhat greater in the latter case — for instance, at the moment of opening the discharge valve ; and in general this occm-s v«"itli BOOK OF mDUSTRIAL DESIGN. 109 all valves having a great body of water above them, and with their upper surface greater than the area of the orifice above. LIFTING AND FORCING PUMPS. 336. A pump of this description ordinarily consists of a cylinder with a short suction pipe, a discharge pipe, a solid piston, termed a plunger, and suction and discharge valves. Two such pumps are frequently coupled together, in which case n single suction and discharge pipe serves for both. 337. The power necessary to work one or more pumps is ex- pressed by 52-3 D^ Hv; or, taking into account the force necessary to work the piston by itself, 55-7 D" H d ; v signifying the velocity in feet per minute. This velocity is obviously obtained by multiplying the number of strokes per minute by the length of stroke ; thus — V = 2n I, n being the number of back-and-forvvard movements per minute j consequently, the power required is equal to 55-7 D' H X 2nl = 111-4 D= H n Z; this product representing pounds raised one foot high per minute, the measurements being in feet. With these premises, we can solve such problems as the fol- lowing : — First : What force, F, is required to work a pump, having a piston 6 inches in diameter, a stroke of 18 inches, and a velocity of 15 double-strokes per minute ; the whole height between the well and the point of delivery being 70 feet ? The velocity v=2nl — S0 x 1^ = 45 feet. Then F = 53-7 D^ X H X i; = 53-7 X -23 x 74 x 43 = 46,997 lbs. raised one foot high per minute. To express this in horses power, we must simply divide it by 33,000 ; therefore, 46,997 F = „„ „ = li horses power, nearly. Second : What quantity will the same pump raise in ten hours 1 Assuming, according to the formula (333), the effective volume, V = -6 DW, or V = -6 X -23 x 1-3 = -223 cubic feet per stroke ; and the volume per minute, •225 X 15 = 3-375 cubic feet; and per hour, 3-375 X 60 = 202-5 cubic feet. The quantity of water raised in ten hours will consequently be 202-5 X 10 = 2,025 cubic feet. Third: What diameter should be given to the piston of a pump which raises 202-5 cubic feet of water per hour, the velocity being 45 feet per miimte, the length of stroke 18 inches, and the height to which the water is raised 75 feet ? The formula above, relative to the effective discharge per stroke, V = -6 D'^ X /, by transposition, becomes ^ -6 X /. Now, the volume, 202-5 cubic feet, discharged per hour, is, per minute, 202-5 This last again reduces itself to 2 X 1-5 X 3-375 43 = -225 cubic feet per stroke; consequently, Wlience, D= = •223 •6 X /. 6: = 3-376 ciHc feet. X. / '225 ^=V -6x 1-5 = 6 inches. THE HYDKOSTATIC PRESS. 338. This powerful machine is an application of the lifting and forcing pump. It consists of a bulky piston, or plunger, termed a ram, working in a cylinder to correspond, and communicating, by a pipe of small bore, with a small but very strong forcing pump. To the top of the large piston is fixed a table or platform, which compresses or crushes what is submitted to the action of the machine. The pressure exerted upon the water by the smaller piston, is, by means of the fluid contained in the pipe, transmitted to the base of the ram ; and as, according to the well-known hydrostatic law, the pressure is equal on all points, the total force acting on each piston vidll be in proportion to their area ; so that if, for example, the diameters of the pistons are to each other as 1 to 5, the prc-s- sure on the larger one, the ram, will be 25 times as great as that exerted by the pump-piston. Suppose a man can apply a force equal to 60 lbs. to the end of a lever 3 feet long, and that the poin* of connection with the piston-rod is only 1| inch from the fuli-rum, the leverage of the power will be 24 times as great as that of the resistance, and the pressure upon the ram will consequently be 24 X 25 X 60 = 36,000 lbs., an effort equal to that of 600 men acting at once. In the hydrostatic press, we have, consequently, to consider two mechanical advantages — that of the simple machine, tlie lever, and that of the ram : these advantiiges are, however, necessarily com- pensated for by the diminution in the velocity of the nun. On these principles, enormously powerful presses and lifting- machines have been constructed. The one capable of lifting 18,000 tons, at the Menai Tubular Bridge, is an unparalleled example. HYDROSTATICAL CALCULATIONS AND DATA — DISCHARGE OF WATER THROUGH DIFFERENT ORIFICES. 339. The discharge of a volume of water, in a given fiine. vanc.-^ according to Ihe velocity of the water, and dcpeiuls upon the area imd form of the discharge orifice. Surface Velocity. — The velocity of \\-ator at (ho surface of a wator- coui'so or river, of which it is wisiiod to asi-ortain the tliscliargc. is obtained by means of a lloal, whii'li is llnowii into ilu' p:irl wlicro the current is strongest. As the w iml, if there is .-iny. alVccts the ri'sult very considerably, the lloni must project above the surface tus little as possible. A distance of as great a length as coiuenieni is measured on the part of the stream where the current is most regu- lar, and the tinui occupied by the llont in passing that distance is noted by a secoiuls watch. The space passeii tiirougli is then diviiU'd by the time expressed in seconils, nud the quotient will be the sur face velocity per second. no THE PRACTICAL DRAUGHTSMAN'S It is usual to try several floats in different parts of the current. ExampZe.— Suppose the space passed through by each float is 150 feet in 35 seconds, what is the surface velocity 1 150 V = 35 • = 4-28 feet per second. If the velocity is not uniform throughout the len^fth of the canal, the velocity at any point may be obtsuned by means of a small paddle-wheel, the floats of which just dip into the water. The number of revolutions per minute of this instrument boing multi- plied by its mean circumference— that is, the circumference corres- ponding to the centre of the immerged part of the float — the pro- duct expresses the velocity per minute ; and, by dividing by 60, the surface velocity per second is obtained. Example.— SnpTpose that the wheel makes 120 revolutions pel minute, and that the mean circumference is equal to H foot, what i'5 the surface velocity of the current 1 120 X 1-5 „ ^ ^ , . = 3 feet per second. 60 ^ 340. Mean Velocity. — The velocity above obtained is only that at the surface ; now, the mean velocity, V, of the whole body of water, which is what is necessary to know for the gauging of the river or canal, is deduced fiom the first, by multiplying it by a coefficient, which varies in the following proportions : — For a surface velo- j city equal to ) Tlie ratio of V to j V is i •5 ft. 15 ft. •78 3 ft. ■81 5 ft. •83 6-5 ft. •85 8 ft. 10 ft. •87 ll-5ft 13 ft. Example. — ^What is the mean velocity of a current of which the surface velocity is 5 feet per second ? It is equal to -83 X 5 = 4^15 feet. The mean velocity of water in an open water-course or river of aniform cross-section is determined by the following formula : — /a H W_x -22 V p T. V = 56-86 x \ / — X -236. P L This formula requires the obtainment of the exact level of the surface of the water throughout a certain length, L, the greater the oetter ; the cross-sectional area, A ; the form of the immerged perimeter or profile of the bed ; and the height of the fall, H, cor- responding to the length, L. Example. — ^What is the mean velocity of the water in a water- course of uniform rectangular cross-section, having a width of 35 feet, a depth of 12 feet, and with a fall of -8 feet in a distance of 1400 feet? The cross-sectional area. A, = 36 X 12 = 420 square feet. The immerged profile. P, = 35 + (2 X12) = 59 feet. Then, V = 56-86 X V 420 sq. ft. -8 59 1400 •236 = 3-39 feet per second. Thus, according to this formula, it is necessary to extract the square root of the product of the quantities placed under the radi- ••al sign y ; next to multiply this root by the co-efBcient 56-86; and, finally, to subtract from the product "236 feet. When the measurements are in metres this last item is -072. COMPAKISON OF FRENCH AND ENGLISH MEASURES OF CAPACITY. The French litre is equal to a cubic metre, and therefore tc 10-76 cubic feet, or -220 gallon. The gallon is equal to 4-543 litres or cubic metres, and the cubic foot to -9929 litres or cubic metres. THE GAUGING OF A WATER-COURSE OF UNIFORM SECTION AND FALL. 341. When we know the mean velocity of a water-course of regular section and uniform fall, the discharge per second can be obtained by the following formula : — D = A x V , in which D signifies the discharge per second; A, the cross-sectional area and V, the mean velocity. Example. — What is the discharge of a water-course, the cross- section of which is 4-2 square metres, and the mean velocity 1-065 metres 1 D -- 4-2 X r065 = 4-473 cubic metres, or 4-473 litres per second VELOCITY AT THE BOTTOM OF WATER-COURSES. 342. The velocity of water at the bottom of water-courses ls> still less than the mean velocity. Putting V to represent the surface velocity, \ tne mean velo- city, and V the ground velocity, the relation of the three will be expressed by V = 2 V — V. That is to say, the velocity at the bottom of a canal is equal to twice the mean velocity minus the surface velocity. Example. — The surface velocity of a water-way is found to be 2 metres, and the mean velocity calculated to be 1-55 metres, what is the ground velocity 1 V = 2 X 1-55 — 2 = 1-10 metres. Too gi-eat a velocity at the bottom of a water-course tends to loosen and carry away the bed, undermining the sides and causing a great deal of damage ; too small a velocity, on the other hand, by allowing the matter suspended in the water to settle, is a cause of obstruction. The following table shows the limit of velocity accormng to the nature of the bed, which cannot be exceeded without danger : — Nature of the Bed. Soft brown earth, Soft claj' Sand, Gravel, Flint stones, Shingle, Agglomerated stones, soft schist, Rock fragments, Solid rock, Limit of the Velocity per second. Metres. -076 •152 -305 -609 -614 1-220 1-52S 1-830 3-050 Feet. -25 ■49 1-00 2-00 2^02 4-00 5-00 6-00 10-00 343. Prony's measure. — The produce of any source ma)-' als: be measured by damming up the entire width of the stream with thin planks pierced with holes of 20 mUUmetres in diameter, dis- posed in a horizontal line. These holes are at first covered, and are opened in succession, until the level of the water within them BOOK OF INDUSTRIAL DESIGN. in is maintained above their centres ; so that when this is effected, the discharge is calculated from the number of orifices which require to be open. The quantity of water discharged by each orifice of -02 m. in diameter, in a board -017 m. thick, and under a column -03 m. above the centre, is 20 cubic metres in 24 hours. Another method of gauging a stream of water, consists in setting up an under or overshot sluice-gate at a similar dam, the discharge being calculated accordiug to the following rules in reference to this subject : — CALCITLATION OF THE DISCHARGE OF WATER THROUGH RECTANGULAR ORIFICES OF NARROW EDGES. 344. As it is of importance, in a majority of circumstances, to bo able to calculate the discharge of water by sluice-gates, or by Iho vertical discharge-gates of hydraulic motors, so as to know the volume, and, consequently, the value of a stream of water, we snail commence by giving a table, which enables us to determine this discharge in a very simple manner, and places these operations within the capacity even of labourers and working mechanics. TABLE OF THE DISCHARGES OF WATER THROUGH AN ORIFICE ONE METRE IN WIDTH. Height of the Vol ame disch argeU in litres per second, corresponding to the heights : — orifices in centi- metres. •2 m. ■3 m. •4 m. ■5 m. •6 m •7 m. 1 •8 m. 1^0 m. r2m. r4m. rem. rsm. 2 m. 2-5 m. 3'0 m. 3^5 m. 4-0 m. 4 50 61 71 79 86 93 99 110 121 130 138 146 154 172 188 201 215 6 62 76 88 98 107 116 124 138 161 162 173 182 191 214 255 251 268 6 75 91 107 117 128 139 148 165 181 194 207 218 229 257 281 301 321 7 86 106 122 136 148 161 172 192 210 226 241 255 267 299 327 350 374 8 98 120 139 155 170 184 196 219 240 268 275 290 305 341 374 400 427 9 109 135 156 174 191 208 220 246 267 289 309 326 343 382 420 450 481 10 1-22 149 173 193 212 228 246 272 298 321 342 362 380 424 466 500 533 11 133 164 189 212 230 249 267 299 327 353 376 398 418 466 511 650 587 12 145 178 206 230 251 272 291 326 356 384 409 434 455 507 557 699 640 13 157 192 222 249 272 294 314 352 386 416 443 469 492 649 602 647 693 14 168 206 238 267 292 316 338 379 414 446 476 504 630 590 648 697 745 15 179 220 255 285 312 338 361 405 443 477 609 539 666 631 693 747 799 16 190 234 271 304 330 360 385 432 472 509 642 574 603 673 739 797 852 17 201 248 287 322 350 382 414 456 601 540 676 610 638 715 784 847 905 18 213 262 304 340 370 403 432 484 529 671 608 644 677 757 830 896 958 19 223 276 324 358 392 425 454 510 558 601 641 680 715 799 876 946 1011 20 235 291 337 377 414 447 ■ 486 636 686 627 675 715 763 841 922 996 1065 21 247 305 354 396 431 470 512 563 615 664 708 751 790 884 968 1046 1118 22 259 320 370 417 451 492 638 590 645 696 742 787 828 926 1014 1096 1171 23 271 334 388 434 472 615 550 616 674 726 776 823 865 968 1060 1146 1224 24 282 348 404 452 492 537 674 643 703 758 809 869 903 1010 1106 1195 1278 25 294 363 420 471 516 659 598 670 733 790 843 895 941 1062 1152 1245 1331 26 306 377 437 490 538 681 626 697 762 822 877 930 978 1094 1198 1295 1384 27 318 392 454 509 569 604 646 724 791 853 911 966 1016 1136 1245 1346 1437 28 329 406 471 527 573 626 679 740 820 886 944 1001 1054 1172 1291 1395 1491 29 340 421 487 546 602 649 693 777 850 916 978 1037 1092 1220 1337 1444 1544 30 353 434 504 564 624 670 718 804 880 948 1010 1073 1129 '1262 1385 1494 1597 31 364 449 521 683 635 • 694 741 831 909 980 1046 1109 1167 1305 1 129 1544 16.50 32 376 463 538 602 656 716 765 857 939 1011 1079 1144 1206 1366 1475 1594 1703 33 388 477 555 622 676 737 789 884 969 1043 1113 1180 1212 1389 1521 1644 1756 34 400 491 572 640 696 759 813 911 998 1074 1147 1216 1279 1431 1668 1693 1810 35 415 507 688 659 717 782 837 938 1027 1103 1180 1252 1317 1473 1614 1743 1 863 36 424 520 605 677 737 804 861 965 1057 1138 1214 1288 13.56 1516 1660 1793 1916 37 436 534 622 696 758 826 886 981 1086 1169 1248 1324 1392 1.567 1706 1843 1969 38 450 549 638 715 778 849 909 1018 1115 1201 1283 1359 1430 1599 17.52 1893 2023 39 462 664 663 734 798 872 933 1045 1145 1232 1315 1395 1 168 Kill 1798 1943 2076 40 484 577 671 753 819 894 957 1070 1174 1266 1351 1431 1.506 1683 1814 1992 2129 41 591 688 772 840 915 981 1097 1203 1298 1384 1467 1543 1725 1890 2012 21S2 42 " 606 705 790 860 936 1005 1124 1233 1329 1419 1503 1581 17(i8 1936 2092 2236 43 5* 620 722 809 881 961 1028 1151 1262 1361 1 163 1538 1618 1809 1982 2142 2289 44 " 635 737 828 901 983 1053 1171 1291 1393 1486 1671 1656 1851 2029 2 1 92 2313 45 " 619 754 847 920 1005 107(> 120 1 1321 1 124 1520 1609 1694 1894 2075 22 1 1 2391 4f) '' ()63 771 866 911 1 028 1 10(1 1231 1350 1 156 1554 1()36 1731 1936 2121 2291 2449 47 15 677 787 885 961 1050 1121 12,57 1380 1 188 1588 1681 1769 1978 2167 234 1 2504 48 »1 691 804 903 982 1072 11 18 1281 1409 1519 1622 1716 1807 2020 22 1 3 2391 2559 49 11 706 820 922 1002 1095 1172 1311 1438 1551 1656 1753 1845 2062 2339 24 10 26 1 4 50 11 719 836 940 1023 1115 1194 1337 1468 1583 1690 1789 1882 2104 239r) 2490 2669 112 THE PRACTICAL DRAUGHTSMAN'S This table has been calculated by means of the follo-vving for- mula : — T) = wh X ^ 2gH X 1000; in \vhich D represents the volume of water discharged in litres per second ; w, the width of the orifice in metres; /(, the height of the orifice ; H, the column, or the height of the pressure, in metres, measured from the centre of the orifice to the upper level of the re- servoir ; g, signifies the action of gravity, being equal to 9'81 metres; r, = V 2gli, the velocity due to the height H (see 258) ; and, finally, m is a coefficient, which varies in practice according to the heights, h and H, from -59 to -66, supposing the contraction of the orifice to be complete ; that is to say, it occurs on all four sides of the orifice. In the first column of the table we give the heights of the orifices in centimetres, and in the foUowng columns the results of the dis- charge effected, in litres per second, for various heights of the column of pressure, from -20 to 4 metres. By means of this table we can now determine, by a very simple operation, the volume of water discharged through a vertical flood- gate, or through a rectangular orifice, of which the edges are nar- row ; the level of the reservoir being above the top of the orifice, and the contraction complete. We have, in fact, sunply to find the number in the table corresponding to the given height of the orifice, and to the column of water acting at its centre, and then multiply this number by the given width. Example. — ^What is the volume of water discharged by the oi-ifice of a vertical water-gate, 1'5 metres wide, the height of the orifice being -25 m., and the height of the column, from the centre of the orifice to the upper level in the reservoii-, 2-5 m., and the contrac- tion being complete ? In the table, on a line \vith the height, 25 centimetres, and in the column corresponding to 2'5 m., wOl be found the number 1052. We have, therefore, 1"5 X 1052 = 1578 litres for the actual discharge per second. It will be equally easy to estimate very approximately the dis- charge of water, corresponding to data which do not happen to be in the table. First Example. — What is the volume of water discharged by a vertical sluice-gate, -8 m. in width, the height of the orifice being 16 centim., and the column upon the centre 2*75 m. ? This height of column, 2.75 m., is not m the table, but it lies between that corresponding to 2'5 m. and 3 m. ; consequently, the discharge for the height of orifice, 16 c, will be comprised between the numbers 673 and 739, and it will be about 706 ; therefore the discharge ^v^ll be 706 x -8 = 664'8 Utres per second. Second Example. — Suppose the height of the orifice to be 6'6 c, instead of 16, the other data remaining the same. As this height is comprised between 16 and 17 centimetres, the dis- charge effected will evidently be between the numbers 673 and 715, corresponding to a column of 2'5 m., and between the num- bers 839 and 784, for a column of 3 metres. It will therefore be very nearly a mean between these four numbers; 673 + 715 + 739 +784 or- • = 727-75 litres. i be Whence we obtain 727-75 x -8 = 582-2 litres for the effective discharge. 345. Incomplete Contraction. — When one or more sides ot the orifice are simply the prolongation of the sides of the reservoir or stream, the contraction is sensibly diminished, and the corre- sponding coefficient is consequently greater. In this case, in order to calculate the effective discharge, the numbers must be multiplied bv 1-125, if the contraction is only on one side. 1-072, " " " two sides. 1-035, " " « three sides. Example. — Required the volume of water discharged by an ori- fice of -25 m. in height, 1-3 m. in width, and with a column of -8 m.) measured from the centre of the orifice, the bottom of the openuig being in a Hne with the bottom of the reservoir ; that is to say, the contraction taking place only on the three sides 1 It will be found, according to the table, that the effective discharge is 598 litres for a width of one metre, and consequently 598 x 1-3 = 777 litres, is the discharge for 1.3 m., when the contraction is complete. We have, therefore, 777 x 1-035 = 804 litres, the actual discharge sought. 346. Inclined Sluicegate. — It very often happens that the sluicegate is inclined. In this case, if there is no contraction on the sides or bottom of the orifice, the coefficient needs to be considera- bly augmented. Thus, to calculate the effective discharge, it is necessary to multiply tlie numbers in the preceding table by 1-33, if the sluice is inclined at an angle of 45°, or with 1 m^tre of base to 1 in height, and by 1-23, if the inclination is 60°, or 1 metre of base to 2 in height. Example. — It is desired to know the volume of water discharged tiu-ough an orifice inehned at an angle of 45°, having 17 m. in height vertically, 1-25 m. in width, and at a distance of 1-2 m. below the surface of the reservoh ; the two vertical sides and the bottom being in a line with the sides of the reservoir. From the table we shall find 398 x 1-25 = 622-5 litres for the discharge with a vertical orifice and complete contraction ; conse- quently, 622-5 X I'S^ = 828 litres will be the effective discharge sought. 347. When vertical floodgates have their lower edges very near the bottom of the reservoir, as is generally the case, to determine the discharge, Multiply the numbers given in the table by 1-04. Example. — What is the volume of water discharged per second by a sluice, the orifice of which is opened to a height of -38 m., having -8 m. in width, and 2-5 m. being the distance from the centre to the upper level ? The table gives 1,599 litres for the discharge effected through an orifice of a metre in width. Whence 1,599 x -8 x 1-04 = 1,330-0 litres, the effective discharge sought. When two sluices are at not more than three metres distance from each other, and are open at the same time, the discharge will be obtained by BOOK OF INDUSTRIAL DESIGN. 113 Multiplying the numbers given in ike table by -9 15. Example. — K the orifices of two sluices, situated at a couple of metres distance from each other, have together a width equal to 1-5 m., and are both opened to a height of '45 m., the column of water upon their centres being 1-8 m., what will be the effective discharge of the two together per second ? In the table, we find that 1609 litres corresponds to a column of 1-8 m., and a width of 1 m. Therefore, 1609 x 1-5 a 915 = 2208-35 litres is the required discharge. TABLE OF THE DISCHARGE OF WATER BY OVERSHOT OUTLETS OF ONE METRE IN "WIDTH. Heights Discharge Heights Discharge Heights Discharge of the in of the in of the in reservoir litres per second. reservoir litres per second. reservoir litres per second. level above the bottom level above the bottom level above the bottom of the outlet. 1st Case. 2d Case. of the outlet. 1st Case. 2d Case. of the outlet. 1st Case. 2d Case. 6-0 20 21 28-6 259 283 52-0 639 698 6-5 23 24 29-0 266 290 62-6 648 708 6-0 26 27 29'5 273 298 63-0 658 718 6-5 29 31 30-0 280 306 53-5 667 728 7-0 32 34 30-5 287 313 64-0 676 738 7-5 36 38 31-0 293 321 54-5 685 748 8-0 40 42 31-5 301 329 55-0 694 758 8-5 43 46 32-0 309 337 55-5 704 769 9-0 47 60 32-5 315 344 66-0 713 779 9-5 51 64 33-0 323 353 66-5 724 790 10-0 56 69 33-5 330 361 57-0 733 800 10-5 60 63 34-0 338 369 57-6 743 811 11-0 64 68 34-5 346 377 58-0 753 822 11-5 68 73 35-0 353 385 58-5 762 832 12-0 72 77 35-5 360 393 59-0 771 842 12-5 77 82 36-0 368 402 69-5 781 853 13'0 82 87 36-5 376 410 60-0 791 864 13-5 86 92 37-0 382 419 60-5 801 875 14-0 92 98 37-5 392 428 61-0 811 886 14-5 97 103 38-0 399 436 61-6 821 896 15-0 101 108 38-5 408 445 62-0 831 907 16-5 107 114 39-0 416 453 62-5 841 918 16-0 111 119 39-5 423 462 63-0 851 929 16-5 117 125 40-0 431 471 63-5 861 940 17-0 121 130 40-5 439 479 64-0 871 951 17-5 127 136 41-0 447 488 64-5 882 963 18-0 132 142 41-5 465 497 65-0 892 974 18-5 138 148 42-0 463 506 65-5 902 985 19-0 143 154 42-6 472 615 66-0 912 996 19-5 149 160 43-0 481 625 66-5 922 1007 20-0 154 166 43-5 488 533 67-0 932 1018 20-5 160 173 44-0 497 543 67-5 943 1030 21-0 166 179 44-6 506 652 68-0 954 1042 21-5 171 185 45-0 614 661 68-5 965 1054 22-0 176 192 45-5 623 671 69-0 976 1066 22-5 182 199 46-0 631 581 69-5 987 1078 23-0 188 205 46-5 540 590 70-0 998 1090 23-5 194 212 47-0 549 599 70-6 1008 1101 24-0 202 219 47-5 558 609 71-0 1019 1113 24-5 207 226 48-0 667 619 71-6 1030 1125 25-0 212 233 48-5 576 629 72-0 1011 1137 25-5 220 240 49'0 584 638 72-5 1052 1149 26-0 226 247 49-5 593 648 73-0 1063 1161 26-5 233 254 60-0 603 658 735 1073 1172 27-0 2.39 261 60-6 612 668 74-0 1084 1184 27-5 245 268 51-0 621 678 71-6 1095 1196 28-0 253 276 51-5 630 688 76-0 1106 1208 114 THE PRACTICAL DRAUGHTSMAN'S CALCULATION OF THE DISCHARGE OF WATER THROUGH OVER- SHOT OUTLETS. 348. The practical formula employed by engineers to determine the quantity of water which escapes in a second of time, through an overshot or open-topped outlet, is the following : — D = W X H X V29H X »i X 1000; in which formula, D represents, as before, the discharge in litres per second ; W, the width of the outlet in metres ; H, the depth of the outlet, as measured vertically from its bottom edge, to the level of the water in the reservoir. The following table is calculated by means of this formula, it being supposed. First, That the wadth of the outlet is 1 metre. Second, That the heights of the outlet increase at the rate of •005 m., from -05 m. up to -75 m. These heights are expressed in centimetres in the first column of the table, the corresponding ve- locities being given in the table at page 94. Third, That the outlet is supposed to be narrower than the reser- voir, or water-course, in which case, MM. Poncelet and Lesbros give the following numerical values for the term m. : — For the height, H, of . . . The term, m, is m. m. m. m. m. m. m. •03 •04 •06 ■08 •10 •15 •20 •412 •407 •401 •397 •395 •393 •390 •22 •385 The corresponding discharges in this case are given in the second column of the table. They are expressed in litres per second. Or, fourth, that the outlet is virtually of the same wdth as the reservoir, or water-course, ha\ing its lower edge only a little, if anything, above the bottom. In this case, according to M. d'Au- buisson (M. Costal's experiments), the coefficient, m, is equal to -42 on the average. The corresponding discharges will be found in the third column of the table. Rule. With the aid of this table, the calculation for determin- ing the effective discharge of water by an overshot outlet, reduces itself to the following: — Multiply the width of the outlet, expressed in metres, by the number given in the second column, and corresponding to the height of the outlet in the first column, when the outlet is narrower than the water- course, and when the water b discharged freely into the air ; And by the number in the third column corresporuiing to the sam£ height, when the water-course is of the same width as the outlet, its depth, likewise, not being sensibly greater than that of the lower edge of the outlet. First Example. — It is necessary to determine the volume of water discharged per second by an overshot outlet, the width of which is 2-5 m., and the height of the overflow -22 m., the case being supposed of the first description. It will be seen from the second column of the table, that the dis- charge effected through an outlet of a metre in width, and of -22 m. in depth, is 176 litres per second ; whence we have 176 X -lb — 44» litres, the volume sought. Second Example. — Required to determine the discharge with the same data ; the case being supposed of the second description. In the third column, the number corresponding to the depth of •22 m. will be found to be 192 litres ; whence, 192 X 25 = 480 litres, the volume sought. Remark. — If the given height happen to fall between some of the numbers given in the table, it will be necessary to take a mean proportional between the two corresponding results, in order to obtain the actual discharge. Example. — What is the quantity of water discharged by an overshot outlet of 3 metres in width, and of a depth equal tt> •183 m.? In the first case, the discharge effected, for 1 metre in width, wil> be between 132 and 138 litres, the mean between which is very nearly 136. Consequently, 136 x 3 = 408 litres, the effective discharge per second. And in the second case, the discharge effected for 1 metre in width, being comprised between the numbers 142 and 148, will be about 146. Whence, 146 x 3 = 438 litres effective discharge. TO DETERMINE THE WIDTH OF AN OVERSHOT OUTLET. 349. W^hen the volume of water to be discharged per second is knowTi, and it is wished to calculate the width to be given to an overshot outlet, or sluice-gate, so as to effect the desired discharge wth a given height of water, this may be done in the following manner : — Take from the table the number corresponding to the given height (this number expressing the discharge for a width of 1 metre), and divide the given volume, expressed in litres, it will give the required 2tidth in metres. First Example. — What width must be given to an outlet, re- quired to discharge 600 litres per second, with a depth above the bottom edge of '12 m. 1 In the second column of the table, and opposite •12 m., will be found the number 72. We have, then — 600 -^ 72 = 8^33 m., the width sought. Second Example. — ^What width must be given to an open sluice, required to discharge 448 litres of water per second, wth a depth of -205 m. ? From the table, we find that 1 60 litres is the effectual discnarge, corresponding to a width of 1 m6tre. Wlience — 448 -4- 160 = 2-8 m., the width sought. TO DETERMINE THE DEPTH OF THE OUTLET. 350. Cases may occur where we are limited as to width. It is then necessary to ascertain the least depth necessary to effect tho required discharge, which may be done by means of the foUowing rule : — Divide the discharge expressed in litres per second, by the width in metres, and take the nujnber in the second column tohich is nearest to the quotient obtained, the number in the first column corresponding tvill give the depth sought, or very nearly so. BOOK OF INDUSTRIAL DESIGN. 115 Example. — With what depth of outlet will a discharge of 350 litres per second be effected, the width being limited to 2 metres 1 Wo have 350 -4- 2 = 175 litres. In the second column of the table will be found the number 176, corresponding to a height of -22 m. in the first column, which will therefore be the required height, witiiin a millimetre. 351. Observation. — When it is not possible to measure the depth, H, with exactness, the lesser depth, h, must be taken im- mediately over the lower edge of the outlet, and multiplied by rn?*, so as to obtain the actual value of H, corresponding to the num- bers given in the table, according as the outlet is narrower than the reservoir, or water-course, or equal to it in width. First Example. — Determine the discharge effected through an outlet, 4 metres wide, the depth, ft, immediately above the lower edge being equal to •!! m., the width being about four-fiftlis of that of the reservoir. We have '11 m. x 1'178 = -IS m., for the assumed neight, 11, of the reservoir level. Corresponding to this height, we have, in the second column, the quantity, 82 litres. Then 82 x 4 = 328 litres, the effective discharge sought TABLE OF THE DISCHARGE OF WATER THROUGH PIPES. Diameters of the Pipes. Mean velocity in •10 n. •15 m. •20 m. ■25 m. •30 ra. metres per Discharge Fall Discharge Fall Discharge Fall Discharge Fall Discharge Fall second. in per me ire in per metre in per metre in per metre in per metre litres in length litres in length litres in length litres in length litres in length per in per in per in per in per in second. nentimetres. second. centimetres. second. centimetres. second. centimetres. second. centimetres. 0-10 0-8 0-02 1-8 0-01 3-1 0-01 4-9 0-01 7-07 0-01 0-15 1-2 0-04 2-6 0-03 4-7 0-02 7-4 0-02 10-60 0-01 0-20 1-6 0-07 3-5 0-05 6-3 0-03 9-8 0-03 14-14 0-02 0-25 2-0 0-10 4-4 0-07 7-8 0-05 12-3 0-04 1767 0.03 0-30 2-3 0-15 5-3 0-10 9^4 0-07 14-7 0-06 2r20 0-05 0--35 2-7 0-19 61 013 11-0 0-10 17-2 0-08 24^74 0-07 0-40 31 0-25 7-1 0-17 12-6 0-12 19-6 0-10 28^27 0-08 0-45 3-5 0-31 8-0 0-21 141 0-16 22-0 0-12 31-81 010 0-50 3-9 0-38 8-8 0-25 15-7 0-19 24-5 0-15 35^34 013 0-55 4-3 0-46 9-7 0-30 17-3 0-23 27-0 0-18 38^88 0-15 0-60 4-7 0-54 10-6 0-36 18-8 0-27 29-4 0-22 42-41 0-18 0-65 51 0-63 11-5 0-42 20-4 0-32 319 0-25 45-95 0-21 0-70 5-5 0-73 12-4 0-49 22-0 0-36 34-4 0-29 49-48 0^24 0-75 5-9 0-83 13-2 0-56 23-6 0-42 36-8 0-33 5301 0^28 0-80 6-3 0-95 14-1 0-63 251 0-47 39-3 0-38 5655 031 0-85 6-7 1-06 150 0-71 26-7 0-53 417 0-43 60^08 0-35 0-90 7-0 M9 15-9 0-79 28-3 0-59 44-2 0-48 63-62 ©•40 0-95 7-5 1-32 16-8 0-88 29-8 0-66 46-6 0-53 67-15 044 1-00 7-8 1-46 17-7 0-97 31-4 0-73 491 0-58 70-7 049 MO 8-6 1-76 19-4 1-17 34-5 0-88 54-0 0-70 77-7 059 1-20 9-4 2-09 21-2 1-39 37-7 1-04 58-9 0-83 84-8 069 1-30 10-2 2-44 23-0 1-63 40-8 1-22 63-8 0-98 91-9 0^81 1-40 iro 2-82 24-7 1-88 44-0 1-41 68-7 113 98-9 094 1-50 11-8 3-24 26-5 2-16 47-1 1-62 73-6 129 106-0 108 1-60 12-6 3-68 28-3 2-45 50-3 1-84 78-5 1-47 113-1 1-22 1-70 13-3 4-14 30-6 2-76 53-4 2-07 83-4 1-66 120-2 1-38 1-80 141 4-64 31-8 3-09 56-5 2-32 88^3 1-85 127-2 165 1-90 14-9 516 33-6 3-44 59-7 2-58 93-3 2-06 134-3 172 2-00 1.5-7 5-71 35-3 3-80 62-8 2-85 98^2 2-28 141-4 1-90 2-10 16-4 6-29 371 419 66-0 3- 14 1031 2-51 1484 210 2-20 1T2 6-89 38-9 4-60 69-1 3-45 108-0 2-76 1555 230 2-30 18-0 7-53 40-6 5-02 72-2 3-76 112-9 301 1626 2-50 2-40 18-8 8-19 42-4 5-46 75-4 4-09 117-8 3-28 169-6 a-T* 2-50 19-6 8-88 44-2 5-91 78-5 4-44 122-7 3-65 176-7 a-9o 2-60 20-4 9fi0 45-9 6-40 81-7 4-80 127-6 3-83 183-8 3-20 2-70 21-2 10-34 47-7 6-89 84-8 517 132-5 4- 1 4 190-8 3-44 2-80 220 11-11 49-4 7-41 88-0 5-56 137-4 4-.i:. , 197-9 3-70 2-90 22-8 11-92 51-2 7-94 fll-l 5-95 142-3 4-77 205-0 3-97 3-00 23-6 12-74 530 8-50 94-2 6-37 147-3 5- 10 3121 4-25 116 THE PRACTICAL DRAUGHTSMAN'S Second Example. — ^With the like data, what would be the effec- tive discharge, supposing the outlet to be of the same width and depth as the reservoir ? We have, as before, -11 x M78 = -13 m. for the depth, H, to which 87 litres is the corresponding discharge, as in the third column. Whence — 87 X 4 = 348 litres, the actual discharge. OUTLET WITH A SPOUT, OR DUCT. 352. It may happen that a spout or duct, slightly inclined, or even horizontal, is fitted to the outlet, and that it is more contract- ed, both at the bottom and at the sides, than the reservoir. In such case, the discharge is sensibly different ; and to determine it, it is necessary to multiply the numbers in the second column of the table by •83, when the height is -2 m., or upwards ; by '8, when the height is -IS m. ; and by '76, when the height is only '1 m. PIPES FOR THE CONDUCTION OF WATER. 353. The formulas employed in calculating the proportions of a conduit for water of uniform section, consisting of cylindrical tubes, are the followins; : — V = 53-58 / dY y/-^- 0-025; and D = S V = Tid" xV. In which, V is the mean velocity ; D, the volume in litres; d, the internal diameter of the conduit : F, the fall per metre, or the length, L, of the conduit, divided by the difference between the levels at either extremity ; and S, the section of the conduit. In order to abridge the calculations, we jrive a table, with the aid of which, various questions relative to the laying down of water-ducts, formed by cylindrical tubes, may be solved very speedily. First Example. — What fall must be given to a conduit, -1 m. in diameter, in order that it may discharge 11 litres of water per second 1 From the table it will be seen, that the fall, in this case, should be -1 c, or 1 millimetre, per metre. SecoTid Example. — What diameter must be given to a conduit, 500 metres in length, in order that it may discharge 168 cubic metres of water per hour, the whole fall being -265 m. ? We have 168 cubic metres, or 168,000 litres, -t- (60 x 60) = 46-65 litres, discharged per second ; and -265 -^ 500 = -53 c, the fall per metre. It will be seen from the table, that the diameter necessary foi this discharge, and with this fall, is -25 m., or 25 centimetres. CHAPTER Vm. APPLICATION OF SHADOWS TO TOOTHED GEAR. PLATE XXX. SPUR "WHEELS. Figures 1 and 2. 354. We have already pointed out, that before shading an object in a finished manner, it is generally necessary to lay down the out- lines of all the shadows, proper and cast, which may happen to be occasioned by the form of each part. Thus, before proceeding to apply the finishing shades to the spur- wheel and pmion, fig. A, we must first detei-mine, separately, on each wheel, both the shadow proper of the external surface of the web, and the shadows of the teeth upon it, and also upon them- selves. The operations called for with one of the wheels are indi- cated in the figures. The external surface, a c, of the web, a, of the spur wheel, being cylindrical, the line of separation of light and shade wUl be obviously determined by a tangent parallel to the luminous ray, or better, by the radius, o d, at right angles to this. By squaring over the point of contact, d, in the horizontal projection, we obtain the line, d' e, in the vertical projection. Similarly, by squaring over the point, e, we get the straight line, f' g, for the line of separation of light and shade on the outer ends of the teeth. which are likewise cylindrical. A portion of the lateral surface of the teeth is also in the shade, as will easily be determined, by draw- ing lines through the extreme angles, as a, 6, c, &c., parallel to the luminous rays. Thus the surfaces, a d, b e, and cf, do not receive any light, and are, therefore, shaded in the elevation, as within the outlines, a' d' g h, b' e' ij, and c'/' k I. Each of these teeth, also, easts a shadow upon the cylindrical surface of the web; and as their edges, a' h, b' j, d I, are vertical, their shadows on the web are also vertical. These last are deter- mined by drawing the luminar lines through the points, a, b, c, and a', b', c', and then squaring over the points of contact, 7n, n, o, to m', n', d . To complete the shadows of the teeth upon the web, it is further necessary to obtain the outline corresponding to the edges, ad, b e, c /, &c. We already have the extreme points, d', d, f, and w', n', d, and in most cases these are sufficient. Where, however, greater exactness is required, it is well to find a few intermediate points. The lower edge of the tooth, also, casts a shadow upon the web, which is obtained in the same manner, by drawing luminar lines through the points, p, q, r, meeting the surface of the web in points projected vertically in r', d'. BOOK OF INDUSTRIAL DESIGN. ll"; Some of the teeth, also, cast shadows upon each other ; but as their surfaces are vertical, these shadows are simply determined by the contact of the luminar lines with them. Thus, the edges pro- jected in s, t, y, &.C., have for shadows the straight lines projected vertically in u' u', x' x\ z' z'. Finally, when we have drawn the horizontal projection of the wheel, as in the present example, we have to determine the shadow cast by the web upon the tenons of the teeth, and upon the arms, or spokes. All these surfaces being horizontal and parallel, the shadow cast upon each will be a circle equal to the one, h i l, which is the projection of the inner edge of the web. All that is necessary, then, is to draw through the centre, o, o', a line parallel to the luminous ray ; and to find the points of intersection, o^ and o', with the planes, m o" and n o"', in which lie the upper surfaces of the tenons and of the arms, and to describe arcs with the points, o" and o^ as centres, and with the common radius, o h (280). In the same manner we obtain the shadows cast by the boss of the wheel, and by the feathers upon the arms. When we have thus gone through the requisite operations for each wheel, we proceed vwth the shading, according to the prin- ciples laid down (289, et seq.), covering first the portions which require a more pronounced shade, and leaving the lighter parts to the last. The specimen, fig. A, which we recommend to be copied on a larger scale, indicates the various gradations of shade required to produce the proper effect, according to the different positions of the planes, and to the contour of the suifaces. These wheels are also supposed to be mounted upon their shafts, which are shaded as polished cylinders. bevil wheels. Figures .3 and 4. 355. The procedure here called for will be the same as in the preceding case — that is to say, we must first draw the outlines of the shadows, proper and cast, for each wheel. The figures repre- sent a horizontal and vertical projection of a bevil wheel with cast- iron teeth, the shadows being indicated on the different surfaces. The external surfaces of the teeth and of the web being conical, the shadows proper are determined in the same manner as for the cone, by drawing through the apex a plane parallel to the luminous ray, and finding the generatrix at which this plane touches the conical surface (313). It is in this manner that, for the outer ends of the teeth, we ob- tain the generatrix projected in o a, fig. 3, and for the outer surface of the web, that projected in o b. These generatrices, which are the lines of separation of light and shade, are projected vertically in the straight lines, o' a' ai\d d' b', converging to the apex of the cone ; since, however, these lines occur between two teeth in the present example, they are not apparent in fig. 4. Some of the teeth have their lateral faces in the shade, whilst all the lower conical surface corresponding to the wider ends of the teeth is in deep shade, as indicated in fig. 4 by a darker tint. Wo have, besides, merely to determine the shadows cast hy the outer edges, a d, b e, c f, and by the curved portions, d g, e h, and f i. Now, the outer edges, a d, b c, c f, cast Hhadows upon the conical surface of the web. which are n'pr(^s('nled hy straight lines coinciding with generatrices on this surface ; and therefore, to determine them, we must draw through the corresponding edges a series of planes parallel to the luminous ray ; the whole of these necessarily passing through the common apex, o, it is simply re- quisite, therefore, to find the shadow cast by any one point in these edges. Let us take, for example, the points, d, e, f, all situate in the same circle, e d f ; the operation, then, is to find the shadow of this circle upon the conical surface, and is the same as that which we have already indicated and explained several times ; it consists, in fact, in drawing any planes, g h and i j, per- pendicular to the cone's axis, and, consequently, parallel to the plana of the circle, e c? F. 356. We have seen that the shadow cast by the circle, n d f.. upon each of the planes, will be a circle equal to itself; and it ia, therefore, simply necessary to find the shadow cast by the centre, . o, o'. This shadow falls in o, o', on the plane, g h, and in o'^ o', un the plane, i i ; if, then, with the points, o and f/, as cenbes, and with the radii, o k' and o^ j', equal to the radius, o e, we drav- a couple of arcs, these arcs will cut the circles, g' k' h' and i' l' j'. Hi ; projections of the sectional planes, in the points, k' and j', wh'-j , being squared over to the vertical projection in the points, k apd , will give two points in the curve, j k M n, representing the shado v cast by the circle, e d f, upon the conical surface of the v/oj. Consequently, if we draw tb*^ luminar lines through the poinis, d', e',f, &c., the respective points of their intersection with the curve, as m, p, q, will represent their shadows cast upon the web surface. These points are squared over to lu', p', Q', in the h iri- zontal projection. The points, g, h, i, situated upon the upper base of the cone, obviously cast no shadows, the shadows of the teeth, however, springing from them ; if it is wished to determine any points be- tween these and those already found, it will be necessary to de- scribe an imaginary circle, such as g', k', h', passing between the points, d and g, the outer and inner angles of the teeth. Tlio curve, R s T, as projected in the elevation, will be found to repre- sent the shadow cast by this circle upon the conical surface of the web. As the edges, a d,b e,cf, cast shadows which coincide witl^ generatrices of the cone, they may be obtained simply by drawing straight lines through the several points, w', q', and p', converging in the apex of the cone in both planes of projection. Finally, the shadows cast by some of the outer edges of tl.o teeth, such as/ c, upon the teeth inunediately behind, are defined by drawing the luminar line,//, through the point,/, meeting the flank, I m, of the other tooth, wliich lies in a vortical plane. This point of contact is projected vertically in /', on the vertical projec- tion,/' /', of the luminar line. It now remains to draw a line, /' ;», through this point, /', and through the apex of the cone, and this lino will represent the shadow cast by the cdge,/c. 357. In the case where \W himinar lino passing through the extremity of the tooth— as that, for examiile, drawn through tlio point, f — falls upon a c\u-ved portion of the tooth behind, it is necessary, if great accuracy is reciuin ■il, to iiuagiuo a vortical plane jiassiug through this jioint and lluoui^li the luminar line. Miul then to find the intersection of this plane with the curved sui- face of llie tooth. This would letiuiu' a separate diagnuu ; hu» 118 THE PRACTICAL DRAUGHTSMAN'S the operatiiiii is very simple, and lias been explained in reference to previous examples (287). 358. The example, fig. [B, represents the application of finished shading to a bevil wheel with wooden teeth, in gear with a pinion on each side, each with cast-iron teeth. It is to be remarked, that altliough the shadows are not the same upon each of these wheels, because of their difl:'erent positions with regard to the light, these are, nevertheless, determined by means of the same operations as tliose which we have just explained. In shading this example, the principles and observations already discussed must be borne in mind, and note taken of the various lights and shades. It is also to be observed, that, from the posi- tions of the two pinions, the inner end of the shaft of one is com- pletely in the shade, whilst the inner end of tlie other is illumi- nated. APPLICATION OF SHADOWS TO SCREWS. PLATE XXXI. 359. It has already been shown, that a screw may be generated by a triangle, a rectangle, or by a circle, the plane of which passes through the axis of the screw, the generating movement being along a helical path. The screw is, consequently, called triangular, square, or round-threaded. In each of these cases, the outer edges cast shadows upon the core of the screw, or upon the twisted sur- face of the consecutive convolutions of the thread itself. If the screvif is surmounted by a head, there will be, in addition, the .shadow east by this upon the outer surface of the thread, as well as upon the other parts. We shall proceed to explain the methods of determining the various shadows upon these different kinds of screws. cylindrical square-threaded screw. Figures 1, 2, 2°, and 3. 360. The limit of the shadow proper upon the screw, is obtained in the same manner as that upon a right cylinder, by drawing the radius, o a, at right angles to the ray of light, r o, and then squar- ing over the point, a, to a' and a', and drawing a line through these parallel to the axis of the cylinder. In the same manner we obtain, by projecting the point, b, the line of separation of light and shade, b' B^ upon the surface of the core. The shadows cast by the outer edge of the threads upon the cylindrical surface of the core, are simply determined by means of the straight lines c c,r) d, drawn parallel to the luminous ray, r o, and meeting the circle, e (^ b, the projection of the core, in c and d; then, by squaring over the points, c d, to c' d', and drawing through the latter the straight lines, c c', d d', parallel to the ray of light, r', we obtain the points, c', d', for the shadows sought. We can, in the same manner, obtain as many points as ai-e necessary to complete the curved outline of the shadow. When the threads of the screw are inclined to the left, as in figs. 2* and 3, instead of being inclined to the right, as in figs. 1 and 2, the operations necessary for determining the curve of the shadows are still the same. This is rendered sufficiently plain by the em- ployment of the same letiers to represent similar and sjTiimetrical points ; it only requires to be observed, that the end view, fig. 3 is that of the right half of the screw, whilst that in fig. 1 is one of the left half, or, one may be supposed to be the turning over of the end of the screw to the right, whilst the other is to the left ; the ray of light is similarly respectively represented to the right and left ; this, however, does not make any difference, as it is the length of the line merely, as d d, which is required. The luminous ray, in both cases, makes an angle of 45° with the axis of the screw, vi'hich is horizontal. The polygonal head, f g h i, which separates the right-handed from the left-handed portion of the screw, casts a shadow upon a part of the latter, represented by curves, which will be easily de- termined, in accordance wth previous examples (285 and 286), and the principal points in which are/, g, in figs. 2° and 3. screw with several rectangular threads. Figures 4 and 5. 36 L The construction of the shadows of a rectangular-threaded screw is the same, whether it be in a horizontal or vertical position, or whether it be right-handed or left-handed. Thus, the screw with several rectangular threads, represented in figs. 4 and 5, has in the first place a shadow proper, limited by the vertical line, a' a'', as squared over from the point, a, and next, the shadow, c' d'f'j cast upon the core by the outer edge of the thread, c' d' e' ; there is, moreover, a portion of the shadow cast by the circular shoulder, G H I, upon the threads, and also upon the core. The outlines of these shadows are found in precisely the same maimer as those in figs. 1, 2, and 3 (361). TRLANGULAR-THREADED SCREW. Figures 6, C, 1, and 8. 362. When the screw is generated by an isosceles triangle, such as c a d, fig. 6, of which the height, a b, is greater than the half of the base, c d, there will be a shadow cast by the outer edge of the thread upon the twisted surface of the succeeding convolution. In proceeding to determine the outline of this shadow, in accordance with the general method, which consists in finding the points of contact of the luminous rays with the surface, we are led to seek, in the first place, the curve of intersection of this surface, with a plane passing through the luminous ray, and parallel to the axis of the screw. For this purpose, let e o, fig. 7, be the sectional plane ; its in- tersection with the outer edge, c g' p, of the screw-thread will be in the point, e, e', figs. 6 and 7, and similarly its intersection with the inside, a I g, will be in the point, r, r'. To obtain interme- diate points of the sectional curve, we must describe various circles with the centre, o, and radii, o m, o n, representing the projections of so many cylinders, on which lie the helices comprised between the inner one, a I g, and outermost, d b" s, being of the same pitch as these latter. We thereby obtain the points of intersection, h, i, fig. 7, which are to be squared over to li', i', in fig. 6, and then, by joining the several points, E^ h', i', r, we get the cur\'e of inter- section of the plane with the helical surface of the thread ; so that, if we draw a luminar line, e' e', through the point, e', in the same i BOOK OF INDUSTRIAL DESIGN, 119 plane, its intersection with the curve, e^ h' i' r, will give a point, e', in tlie outline of the shadow sought. In like manner, by drawing other planes, as f h and g i, parallel to tiie first, E 0, we shall obtain the intersectional curves, f^/' h' and ir'J' g I, and further upon these the points,/' and^', of the outline of the shadow. By proceeding thus, we can obtain as many points as may be deemed necessary for the construction of the shadow cast by the outer edge, c g' p, of the thread, and the curve obtained is, of course, repeated on the several convolutions of the thread. We would remark, that there is no shadow ca.st when the depth of the thread is such, only that a h, fig. 6, is less than the half of the base, c d, of the generating triangle. The diagrams, figs. 6" and 8, which represent a portion of a left- handed screw, will show that the operations required Lq this modi- fication, to determine the outlhies of the shadows, are precisely the same as those last explained. The core, n, which separates the two portions of the double screw, as well as the end, n', receives a shadow cast by the outer edge of the adjacent convolution of the thread. shadows upon a round-threaded screw. Figures 9 and 10. 363. These figures represent a species of screw generated by a circle, ah c d, the plane of which passes through the screw's axis, and of which each point describes a helix about the same axis. The intervals or hollows between the convolutions of the thread are also formed with a helical surface generated by a semicircle, d e f, tangential to the first. We have, then, to determine the limiting line of the shadow proper upon the screw, and the shadow cast by this line upon the hollows. The projecting thread being a species of spiral torus or serpen- tine, the determination of its shadow will be similar to that of the shadow of the ring (323). Thus, if the screw be sectioned by a vertical plane, g o, passing through its axis, itb intersection with the thread will evidently be a circle, as projected in^' V, fig. 9. This circle, being inclined to the vertical plane of projection, fig. 10, is projected therein in the form of an ellipse, the principal points,/, It, I, of which are obtained by squaring over the points, y, fe', V, respectively, upon the helices cor- respondmg to the points, a, b, c. If, then, upon the plane, g o, which we suppose to be reproduced at o g, fig. 10", we project the luminous ray, r o, it will be sufficient to determine the point of contact of this ray with the curve,/ k I; for this purpose, find the projection of the ray upon the vertical plane in g' &', fig. 10"; then draw a line, g^ o^, tangential to the ellipse,/ /,: I, and parallel to the straight line, g" o', its point of contact, m, with the ellipse will be a point in the line of separation of light and shade upon the outer surface of the screw-thread. By proceeding in this manner, any number of points in this line may be obtained. By continuing the sectional plane, g o, across the hollow of tlie screw, we .shall likewise obtain the elliptic curve, n o', the principal points in which are equally situated ui)on the helices .which pass througli the points, d, e,f; itMS sufficient to prolong the luininar line, g' o', until it cuts the ellipse, n o'' p, so as to obtain the point, o', which is the shadow cast by the corresponding j)uiiil, ?/(, of the line of separation of light and shade upon the hollows oi intervals between the convolutions of the thread. It is to be remarked, that the prolongation of the line of separa- tion of light and shade, s t, casts a shadow upon the outer surface of the convolution immediately below ; and, in the same manner, the shoulder above casts a shadow over the projection and hollow of the adjacent thread. APPLICATION OF SHADOWS TO A BOILER AND ITS FURNACE. PLATE XXXII. shadow of the sfheee. Figure 1. 364. It will be recollected, that a sphere is a regular solid, gene- rated by the revolution of a semicircle about its diameter. From this definition it follows, that its convex or concave surface, ac- cording as it is considered solid or hollow, is a surface of revolution, of which every point is equally distant from the centre of the gene- rating circle. To determine, then, the shadow proper, upon the surface of a sphere, we can proceed according to the general prin- ciple (328) ; but, in this particular case, the follomng will be the simpler method. Let us suppose the sphere to be enveloped in a right cylinder., having its axis parallel to the luminous ray ; this cylinder will touch the sphere at a great circle, which is, in fact, the line of separation of light and shade, and the plane of which is perpendicular to thp luminous ray, and, consequently, inclined to the planes of projec- tion ; it follows, therefore, that the projection of this line upon those planes will be an ellipse. Thus, let fig. 1 represent the horizontal projection of a sphtre, whose radius is o a, the projections of the extreme generatrices, b c and D E, of the cylinder, parallel to the luminous ray, touch tho external contour of the sphere in the points, c, e, which are dia- metrically opposite to each 'Other, and are the extremities of tho transverse axis of the ellipse. As, in general, this curve can be drawn when its two axes aro determined, it merely remains to find the length of its conjugate axis. To this effect let us imagine a vertical plane to pass through the luminous ray, R o, and let us take two lines tangent to tho section of the sphere in this plane, and parallel to tlie luminous ray ; if now we turn this plane about the line, r o, considered as an a.xis, so as to fold it over upon, and nuiko it coincide with, tho horizontal plane, the great circle, which is its section with thu sphere, will obviously coincide with tho cin-le dniwn with tlm radius, a o. The luminous ray will, as already seen (287), bt( turned over to r' o, making an angle of 35° 16' witli the lino, h o. It may also be obtained by making tho lino, r' r, jx^rj>ondicular to R 0, and equal to a side of tho square, .ns g k, and then joining r' o. Tho two luminous rays tangential to tho sphere will then coincide with tho straight linos, ii l and m n, luirallol to k'o, their points of contact with llio great circle will be tho oxtn-niitios of the diiunotor. l n, iHT|H'iHliculnr lo k' o. If now wo iiruigine the plane to l»e returned to its original j »lt' >n, tho iH>inl.s, l ;uui 1'20 THE PRACTICAL DRAUGHTSMAN'S N, will be projected in l' and n', thereby giving the length, l n, of the conjugate axis of the ellipse sought. 365. If, in place of constructing this ellipse by the ordinary methods, it is preferred to determine the various points by means of a series of analogous sections, with their subsequent operations, it will be sufficient, for example, to draw the plane, a b, parallel to R o, and then to turn over, as it were, the section of the sphere formed by this plane, so that it shall coincide with the horizontal plane, making its centre, at the same time, to coincide with the centre, o, of the sphere, the section in question being a circle, described with the radius, c o, equal to a c. Next draw the tan- gent, e d, parallel to r' o, and then project the point, d, on the diameter, l n, to d', upon the original line of section, and a point in the elliptic curve. In this manner, as many points may be ob- tained as are wished. The distance, c d\ being made equal to c d', d' will be the symmetrical point in the opposite, and now apparent part of the ellipse. If the projection of the sphere were supposed to be upon the vertical plane, the operations would be identical, only the transverse axis of the ellipse, instead of being in the direction, c e, would, on the contrary, be in the direction, a i, perpendicular to it, as seen on fig. [B, representing the hemispherical end of a boUer, shaded and finished. shadow cast upon a hollow sphere. Figure 2. 366. When a hollow sphere is cut by a plane passing through its centre, and parallel to the plane of projection, the inner edge of the section will cast a shadow upon the concave surface, the outline of which will be an elliptic curve, which may be determined in ac- cordance with the general principle of parallel sections, already explained (287), or by means of the simpler system of sections and auxiliary views adopted in the preceding example, and of which we shall proceed to give another instance, in fig. 2. This figure represents the projection of a hollow sphere upon the vertical plane, being, in fact, the section through the line, 1 — 2, of the boiler, represented in figs. 4 and 5. If this hemisphere be sectioned by a diametrical plane, a b, parallel to the luminous ray, the section represented in the auxiliary view, fig. 3, will be a semi- circle, a' c' b'. The luminous ray, lying in this plane, and passing through the point, a, a', will, in. fig. 3, be represented by the line, a' c', parallel to the line, r' a, obtained, as indicated in fig. 2, by the method already explained. This straight line, a' c', cuts the circle, a' c' b', in the point, c', wliich must be squared over to c, on the line, a b, fig. 2, when c will be the shadow cast by the point, A. In the same manner we obtain the points, h, d, by means of the sectional planes, ah, c d, parallel to a b, and cutting the sphere in semicircles, represented by a' b' e, in fig. 3. This semicircle is cut in the point, b', by the line, a' b', parallel to a' c'. The extreme pomts, D, E, are obviously situated at the extremities of the diame- ter, D E, perpendicular to the luminous ray, r o, and representing ihe transverse axis of the ellipse. These shadows are frequently met with in architectural and mechanical subjects ; as, for example, '0 mcJies, domes, and boilers. APPLICATIONS. 367. Fig. 4 represents a longitudinal section, at the line, 3 — 4, in fig. 5, of a cylindrical wrought-iron boiler, with hemispherical ends, and surmounted by a couple of cylindrical chambers, one of which serves for a man-hole, and has a cover fitted to it. Fig. 5 is a plan of the same boiler, looking down upon it, and showing the cylindrical chambers. Fig. 6 is a transverse section, made at the line, 5 — 6, in figs. 4 and 5. For this boiler, we have to determine — First, In fig. 4, the shadows, b d c and e J c, cast upon the spherical surfaces at either end of the boiler, as well as those, c gi anij k I, upon the cylindrical surface, together with the shadows cast on the interior of the cylindrical chambers. Second, In fig. 5, the shadows proper of the external cylindrical and spherical portions of the boiler, and the shadows cast upon _ these by the cylindrical chambers. I In fig. 7, we apply the same letters as to the analogous diagram, fig. 3, this view being drawn for the purpose of obtaining the elliptic curve, D (^ c, of the shadow cast by the circular portion, a c d, upon the internal spherical surface of the end of the boiler. In the same manner is obtained the portion, e J c Z, by means of the diagram, fig. 8, observing that the sections made parallel to the ray of light, above the line, a b, give semicircles, whilst those made below, such as f c, give the circular portion to the right of the line, a o, but an elliptical portion to the left of this line, m conse- quence of this portion of the plane cutting the cylindrical part of the boiler obliquely. It must be remarked, that the cylindrical chambers, situatea on tne top of the boiler, give rise to the inter- sections, I J K F, which cast shadows upon the interior of the boiler, instead of the rectiUnear portion, i f, of the extreme generatrix of the cylinder, which would have cast a shadow, had the cylindrical chambers not been there. The shadow cast by the edge of these intersecting surfaces is limited to the curves, J K f, which may be easily delineated with the aid of the section, fig. 6, by squaring over the points, j, k, l, f, to the arc, j' k' l' f', and then drawing a series of luminar lines through these points; that is, lines parallel to the ray of light. These vvtII meet the internal surface of the cylinder in the points, J', k', r, which are squared over again to the longitudinal section, fig. 4, by means of horizontals, intersecting the luminar lines, drawn through the corresponding points in the edges of each chamber, in the points, _;', k, I. The rectilinear portion, F i, of the uppermost generatrix of the cylinder, has for its shadow, on the internal sur- face thereof, the similar and equal straight line, I i, which coincides, in the projection, with the axis, o o (308). A part of the extreme left-hand generatrix, i n, of each cylindri- cal chamber, likewise casts a shadow upon the internal surface of the boiler, the outline of which is a curve, i mj, which is simply an arc of a chcle, described with the centre, o, and with the radius, o i, equal to that, c o, of the boiler. This shadow is chcular, because the straight line, i n, which casts it, is perpendicular to the axis of the cylinder; whilst the axis and itself lie in a plane, parallel to the plane of projection. We can, however, determine the points, i, m, j, of the curve. BOOK OF INDUSTRIAL DESIGN. 12i iiuleDendently, with the assistanne of the auxiliary projection, fig. 6, at nght angles to fig. 4. It is the same with the curve, n p q, which is likewise an arc of (I circle, because the straight line, n p, the edge of the cover which closes the top of the chamber, is at right angles to the axis of the latter, and at the same time parallel to the vertical plane of projec- tion. The edges, n r and r m, being vertical, have for shadows upon the internal surface of the chamber, a couple of vertical straight lines, parallel to themselves (309). The chamber to the right having a circaiar opening in the cover, has a shadow upon its internal surface, necessarily different from that in the other cham- ber. It is, however, easily obtained, and in the same manner as in figs. 1 and 1°, Plate XXVIII. It must be observed, however, that a portion, .s t w, of this shadow is due to the under edge, s t u, of the cover-piece ; whilst the other part, s v, takes its contour from the upper edge, v x, of the same piece. A comparison of figs. 4 and 5 will render these points easy of comprehension. There remains, finally, the curve, c e g h, and the rectilinear por- tion, h i, together extending from the first, a d c,to the straight line, i i, and which represents the shadow cast by the arc, a f, g h, of the hemispherical end of the boiler, and the straight part, h i, of the upper edge of the cylindrical portion. The whole curve, d d,c g i, representing the shadow cast by (be edge of the section of the boiler upon the internal surface of the latter; is precisely the same as that distinguished ib architecture by the name of the niche shadow. It is to be observed, however, that the position in this case is different, as the axis of the niche is vertical. We have now to draw the shadows, proper and cast, upon the outer surface of the boiler, as seen in horizontal projection, fig, 5. As for the shadow proper, it consists partly in that limited by the line of sepaiaiion of light and shade, d d, obtained by the tangential line, making an angle of 45° with the horizon, and touching the uircle in the point, c, and partly in that bounded by the elliptic curves, c d and d c e, upon the spherical ends of the boiler, the manner of determining which has already been thoroughly discussed in reference to fig. 1. 368. As to the shadows cast by the cylindrical chambers, either on their neck pieces, or upon the outside of the boiler itself, they are simply represented by lines inclined at an angle of 45°, as a' d', b' e', drawn tangential to the outsides of the cylinders, and which are prolonged in straight lines, as far as the line of separation of light and shade, upon the cylindrical portion of the boiler ; that is, in case they stand out far enough from the boiler surface. If, on the contrary, they do not rise very high, as exemplified in the end view, fig. 9, it will be necessary to determine the outline of the shadow cast by a portion of the upper edge, b' c', as lying either upon the cylindrical part of the boiler, or upon one of tlio spherical ends. To find the shadow in this last case, we have supposed an imaginary vertical plane to pass through tho luminous ray, r' o', fig. 5, producing an elliptical section of the cylinder, and a circular one of the spheiical part. This plan being repro- duced at V? o", fig. 9, and turned about, to coincide with the hori- zontal |tlano, we have the curve, f" g' h'^ representing the section in ()ucsti()n. Tho point of contact, b', being transferred to b', is mIso turned down, as it were, upon the horizontal phuui, to tho point, b'; so that if we draw a line, b' i', through this point, e', parallel to the luminous ray, r' o', similarly brought into the hori zontal plane, this line, b' i'', will cut the interscctionai curve in the point, 1^ ; the horizontal projection, i', of this point, upon the line, r' h^ being obtained by letting fall the perpendicular, i" I^ upon the latter. The corresponding point, i', in fig. 5, is taken at a dis- tance from b', equal to b" i^ in fig. 9. Proceed in the same manner with another sectional plane, parallel to the first, and passing through the point, c', in order to obtain a second point, c', of the shadow. The operations necessary for determining the intersec- tional curves are sufficiently indicated in figs. 5 and 9. 369. The cylindrical steam-boiler, represented in longitudinal section in fig. A, in end elevation in fig. ©, and in transverse section in fig. ©, conjoins the various applications of shadows, of which we have been treating, in reference to .spheres and cylin- ders ; whilst, at the same time, they serve as examples of shading, by lines or by washes, indicating the effects to be aimed at, and to be attained by the following out of the various principles already laid down. 370. We must remind the student, that, in order to produce tnese effects, he must not always confine himself to the representation of the shadows proper and cast merely. He must, further, show the gradations of the light or shadow upon each part, as has already been explained with reference to solid and hollow » vlinders. As upon a cylinder or a cone, there is always a line of pre-eminent brilliancy, so likewise, upon the surface of a sphere, will there be a point of greater brilliancy than the rest. This point is actually situated upon the luminous ray, passing through the centre of tho sphere, fig. 1. Since, however, the visual rays are not coincident with the luminous rays, the apparent posi- tion of this point is somewhat changed. Thus, if we bring the vertical plane, b o, fig. 1, into the horizontal plane, the luminous ray will coincide, as has been seen, with the line, r' o', and, conse- quently, its point of intersection with the sphere will coincide with the point, i. On tlie other hand, the visu.il rays which are perpen- dicular to the horizontal plane will coincide with parallels to o o, when brought into the horizontal plane. This latter line intersects the sphere in the point, c ; and as the light is reflected from any surface in the direction of the visual rays, so as to make the angle of incidence equal to the angle of reflection, if we divide the angle, i o c, into two equal angles, by tho line, n o, the point, 7i, will bo that which will appear to the eye most brilliantly illuminated. Tho positions, n' and i', in the vertical piano of the point.s, n and i", are obtained by letting fall (icrpiMulii-iilars u[)on llie line, o a, repre- senting this i)!an(!. In shading up a drawing it is preferalile to place llie bright or li'ditest part between the two points, n' and i", a more pleasing elVect being obtained thereby. When the si)here is polishctl, as a steel, brass, or ivory ball, a circular spot, of pure white, nuist bo left about the i)oint in (piestion. When, however, the body is rough, as i>) .supposed in fig. 03, (his part is always lighter than tho rest ; but, nl the same lime, it is covered by a faint wa.sh. In llie case of a hollow spiiere, figs. 2 and 3, wo have to liear in mind, not only to indiiate Iho position of the bright spot, which is |)roj('cted, in the same manner, upon the luminous niy, a ii. ••uul lies between tho points, n', i', but also tho point in tho oast sliadow, 122 THE PRACTICAL ])RAUGHTSMAN'S which should be the least prominent. This latter will be found to be at m, fig. 2, as determined by the radius, o' m, fig. 3, di-awn perpendicularly to the ray of light, a' c', as brought into the same plane as fig. 3. 371. The boiler is represented as placed in its furnace, which is built entirely of bricks, with a diaphragm passing down the middle of its length, to oblige the flames and gases issuing from the grate to pass along the flue to the left, then to return by that to the right, and passing through a third flue, before it reaches the chimney. In this third flue is placed an auxiliary boiler, full of water, and in communication with the main boiler by a pipe passing to the bottom of each. In this auxiliary boiler, the feed- water becomes heated before entering the main boiler, so as not to reduce the temperature of the latter to a serious extent, upon its introduction into it. The main boiler is represented as half full of water. It should generally be two-thirds full, but is delineated as but half full ; so that a greater portion of the shadow cast upon its interior may be visible. The remainder of the space, as well as the cylindrical chambers, is supposed to be filled with steam. The base of the chimney is of stone, whilst the stiilk is of brick. The foundations of the furnace are likewise of stone. Besides this present example of a boiler, we give a further ex- ercise for finished shading in Plate XXXIII., the objects in which we recommend the student to copy, on a scale two or three times as large, so as to acquire the proper skill and facility of treatment. SHADING IN BLACK.— SHADING IN COLOURS. PLATE XXXIII. 372. In a great number of drawings, and particularly in those termed working drawings, and intended for use in actual construc- tion, the draughtsman contents himself by shading the objects with China ink — sometimes, perhaps, covering this with a faint wash of colour, appropriate to the nature of the material. The shading, on the one hand, brings out the parts in relief, and ren- ders the forms of the object intelligible to the eye; whilst, on the other hand, the colours indicate of what material they are made. This duplex artistic representation makes th« drawing much more life-like, and more easily comprehended. A drawing may be coloured in several ways. The simplest plan is first to shade up the various surfaces with China ink, having due regard to the respective forces and gradations of tone, according to the lights and shades, as has been done in the preceding plates. The entire surface of each object is then covered with an especial wash of colour, the line of which is quite conventional. It must be laid on in flat washes, according to the instructions given in reference to Plate X. This first method of operating may suffice in many cases, but it leaves out much to be desired in the effective appearance of the drawing, its aspect being generally idthout vigour, cold and monotonous. A better result is obtainable by not carrying the China ink shading to so great a depth, and by covering the surfaces by two or three washes of colour, laid on in irradations, as was done with the China ink itself, so as to produce « sufficient strength of colour at the darker parts, whilst the light parts are left very faint ; and where the objects are polished, a puro white line or spot is left, which will add considerably to the bril- liancy of the whole. A softer and more harmonious effect can be produced by the use of a warm neutral tint, instead of the China ink, for the preliminary shading. This colour, however, is very difficult to mix, and to keep uniform. When a little practice has given some skill and facility in the preparation and combination of the colours, the di-aughtsman may proceed, at the outset, in a more direct and vigorous manner, leaving out altogether the preliminary shading with China ink, and laying on at once the successive coloured washes, rendering, at the same time, the effects of light and shade, and indicating the nature of the material. This last method has the merit of giving to each part of the dramng a richer translucence, more warmth, and a more satisfactory fulfilment of all desirable conditions. In general, all drawings intended to be shaded should be deli- neated with faint gray instead of black outlines, as for a simple outline drawing ; the faintness of such lines avoids the necessity of making them very fine, and their greater breadth affords a much better guide to the shading-brush. A black outline, however fine it may be, always produces a too sharp and hard appearance, whilst there is much gi-eater risk of overstepping it in laying on the washes. 373. In Plate XXXIII. we give a few good examples of objects shaded in colours, comprising the materials most in use in con- struction. Fig. 1 represents the capital of a Doric column in wood. Al- though the woods are naturally very different in colour, still, in mechanical drawings, a single tint is used indiscriminately : it is, as we have said, entirely conventional. In fixing upon these colours, the object in view has been to avoid confusion, and to employ a distinct and intelligible colour for the representation of each substance, without seeking to copy the natural colour in all its varieties. In colouring this wooden capital, after the preliminary opera- tions which we have mentioned, for determining the outlines of the shadows, proper and cast, it is first shaded throughout with China ink, and when this shading has reached a convenient depth, and is thoroughly dry, the whole surface is to be covered with a light wash, which may be a mixture of gamboge, lake, and China ink, or burnt umber alone. The colour, in fact, should be analo- gous to that of fig. 4, Plate X. ; it should, however, always be fainter than in that example, which represents the material in sec- tion, and is, therefore, stronger. This proceeding may be easily modified, and made to resemble the effect of the second method, by leaving certain parts of the object uncoloured, and by softening off the shade in those places where the light is strongest, with a nearly dry brush. If, however, the draughtsman has become somewhat familiarized with the use of the brush and the mixture of the colours, he may, as we have said, omit the preliminary shading in black, by modifying eacli shade as laid on, mixing the China ink directly with the colours, and then gradually bringing up the shades, either according to 1 he system of flat washes, or the more difficult one of softened shades. Care must be taken in laying on these shades to commence at the deeper parts, and then to cover these over again by the subse BOOK OF INDUSTRIAL DESIGN. 123 quent washes, which gradually approach the bright part of the object ; for in this way a more brilliant and translucent effect will bo obtained. When the objects are of wood, it is customary to represent the graining in faint irregular streaks, care being taken to make these as varied as possible. A general idea of the effect to be produced will be obtained from fig. 1. Following out these principles, the draughtsman may proceed to colour various other objects composed of different materials, merely varying the mixtures of colour according to the instructions given in reference to Plate X. Fig. 2 represents the top of a chimney of brickwork, the form being circular. In this external view, the outline of each brick is indicated ; and to render them more distinct from each other, a line of reflected light has been shown on the edges towards the light, near the brighter part of the chimney. Indeed, it is generally advisable to leave a narrow, pure white light at the edges of an object which are fully illuminated, as it gives an effective sharp appearance. Fig. 3 represents the base of a Doric column in stone, showing the flutings. This being an external view, the tint to represent the stone is not made nearly so strong as for the sectional stone-work, represented in Plate X. A yellowish grey may be used for it, made by mixing gamboge, the predominant colour, with a little China ink, adding a little lake to give warmth. These three examples of wood, brick, and stone, represent bodies with rough surfaces, and which, therefore, can never receive such brilliant lights as objects in polished metal ; no part, indeed, should be entirely free from some faint colour. Fig. 4 represents a nut or bolt-head of wrought-iron ; and, as we have supposed it to be turned and planed, and polished upon its entire surface, it has been necessary to leave pure white lights at the brighter parts, to distinguish the surfaces from those which are rough and dull. It is the same in the example, fig. 5, repre- senting the base of a polished cast-iron column, and in the lateral projection, fig. 6, of polished brass upper and lower shaft-bearmgs or brasses. We would hope that the principles of shadows and shading, ex- plained and exemplified in the last two chapters, may serve as sufficient guides for the various applications which may present themselves to the draughtsman — whether his skill be called forth to render the simple effects of light and shadow, or to produce the gradations of shade and colour due to roundness or obliquity of surface — to the various positions of the objects in their polished or unpolished state, and to the various materials of which they may be composed. Thus, it will be understood, that although two objects are pre- cisely alike in material and form, if they are situated at unequal distances from the spectator, the nearer one of the two must be coloured more strongly and brilliantly than the more distant, more force and depth being given to the darker shades. CHAPTER IX. THE CUTTING AND SHAPING OF MASONRY. PLATE XXXIV. 374. The operation of stone-cuttmg has for its object, the pre- paring and shaping stores in such manner that they may be built up into any desired form in a compact and solid manner ; great care and skill, as well as mathematical knowledge, is more parti- cularly required in the preparation of stones for arches, vaults, arcades, and such like structures. The study of the shaping of stones is based entirely upon descrip- tive geometry, being indeed but a particular application or branch <)f it, and in it have to be considered the genei'ation of surfaces, as well as their intersections and developments. In proceeding to adapt the stones to the position they are to occupy, the mason should prepare a preliminary drawing of the actual size of each stone, as well as a general view of the entire erection, indicating the joints of each stone ; these, according to the various positions to be occupied by them, are called liey sloncs, arch stones, &c. It is not our intention to give a complete treatise on the shaping of masonry; but, as this study seems to belong, in ])art, to geonic- trical drawing, we have thought it (|iiit(r vvilhiii llu' design of llic pres(nit work to give a few ap|)lic:ilions, siiflicicint to show (lie line of jnocedure to be followed out in operations of this naliuc. the marseilles arch, or arriere-vo^ssure. Figures 1 and 2. 375. We propose to prepare the designs for llie b:iy and arch of a door or window, to be built of stonework, the upper part being cut away, so as to present a twisted surface, analogous to that known as the arriere-voussure of Marseilles. This surface Is such as would be generated by a straight lino, c A, kept constantly upon the horizontal, c' k', projected vertically in the point, c, and moved, on lln' one hand, upon tiie semi-baso, BED, of a right cylind(>r, having c' k' for its axis : and, on tJio other, upon the circular arc, f k a, situated in a piano pimillol to that, of the base, bed. The lateral faces, f B l n and a r Q d, of tiio bay, are vortionl, and are projected horizontally in v' n' and a' d', fig. 2. Tiioso faces intersect the twisted surface at the curves, f h and .\ p, which w(! shall proceed to detorinino. For this purpose, the lirst thing to ho done is to seeil llio ]no. jections of tiic straight generator lino, o a. a.s oocupymg dilToroiu positions, so as to obtain tlioir points of iiitorsocuon wiui ouo «i the oblique planes. 1 124 THE PRACTICAL DRAUGHTSMAN'S We may remark, that if the are, f k a, be prolonged to the right, for example, of fig. 1, and a number of lines be diawn through the point, c, as c j, c a. c b, and c c, they will represent so many ver- tical projections of the generatrix, c a, in diftorent positions. These straight lines meet the semicircle, b e d, in the points, i, c, d, e, which are projected horizontally in the points, i', c', d\ e'. These same lines also cut the circular are, f a g, in the points, j, a, b, g, which are projected upon the line, f' g', the horizontal projection of this line, in the points, j', a', b', a'. By drawing lines thi-ough these last, and through those first obtained, i', c', d', e, we obtain the straight lines, c" j', c' a', c* b', c^ g', which are the horizontal projections of the generatrix, c a, and correspond to the vertical projections in fig. 1. These straight lines cut the plane, a' d', in the points, m'/' i', which are then projected vertically to m,/, i, upon the straight lines, c J, c a, c b ; the curve, a ni/i d, passing through each of these points, is the line of intersection sought, and it is reproduced sym- metrically at f b, to the left hand of fig. 1, so as to avoid the neces- sity of repeating the diagram. To obtain this line of intersection full size, it is necessary to bring the plane, d' a', into the plane of the picture, by supposing it to turn about the vertical, d q, projected horizontally in d', as an axis ; during this movement, each of the points, a', m', /', mil describe an arc of a circle about the centre, d', finally coinciding •with the points, M^ a", f, and -P. Through the corresponding points. A, M,/, i, in the vertical projection, we must draw a series of horizontal lines, a a", m m',//", upon which, square over the preeiiding points, m", A^_p, i^, by which means will be obtained the cm-ve, a" m" f d, representing the exact form or parallel projection of the line of intersection. 376. The preliminary design thus sketched out, gives nothing but the outline of the surface of the erection, and it now remains to divide it into a certain number of parts, to represent the indivi- dual stones of which it is to be built up. The number of divisions necessarily depends upon the nature of the material and sizes of stone at the mason's disposal ; the number snould, however, in all eases, be an odd one, so that a central space may be reserved for the principal piece, known as tlie key- stone. The di\dsions are struck upon the semicircle, b e d, by a series of radii converging in the centre, c ; it is these lines which repre- sent the divisions of the stones. Below the arched part, the regular pieces, as X, consist of a series of stones of equal dimensions, the joints of which are horizontal. The horizontal projections of each of the stones forming the arch are straight lines, because the joints lie in planes perpendicular to the vertical plane, whose intersection with the twisted surface is always a straight line correspondiag to a generatrix ; thus, the planes, o c and p c, of the joints on either side of the key-stone, e, are perpendicular to the vertical plane, and pass through the axis, c k' ; and the portions of them, n h and n I, comprised between the directing circles, bed and f k a, are represented in the horizontal projection by the straight lines, n' h' and o' I', which are the joints of the stones as seen from below — that is, the lines of their iQter- Bection with the tw isted surface. It IS the same wim the planes, m c and j c, in which lie the joints of the corner stones, t z ; the portions, k g and 1 1»., of tbt joints falling upon the twisted surface, are likewise projected hori- zontally at the straight lines. A' g' and m' i'. 377. The design of the erection being thus completed, the shaper should delineate each individual stone as detached from the arch, in such a manner as to represent all the faces of the joints, and he then takes for each, a stone of the most convenient dimensions from amongst those at his disposal, which are generally hewm out roughly in the shape of rectangular parallelopipeds; on each of these pieces he marks off the parts to be cut away, to reduce the stone to the requu'ed form and dimensions. Thus, supposing he commences with the key-stone, k, for ex- ample, detailed in front view' and vertical section through the middle, in figs. 3 and 4; he takes a parallelepiped, of which the base, p q r s, is capable of circumscribing the two parallel faces of the upper part of the key-stone, and of which the height is at lea.st equal to the length, I u'. After having cut and finished the two vertical faces, f d and u r, of the prism, as well as the horizontal face, i! u', he measures off upon the anterior face, fig. 3, the parallel and vertical sides, t o and u p, and then the oblique lines, o h and p I, which, it will be remembered, converge to the same point, c, the axis of the voussure. He next sets off upon a template the arcs, n o and h I, fig. 1, and reproduces them thence upon the paral- lelepiped, fig. 3, at n o and h I. After this preliminary marking out, the stonecutter reduces and takes away all the material upon the sides of the parallelopiped which lies outside the lines, o h and p 1 ; these faces being finished, the shape-designer lays out upon them the lines projected at nh and o I. In order that the form of this joint may be more easily comprehended, we have brought the face, p I, into the plane of the picture, representing it in fig. 4^', as parallel to the plane of projection. This view, it will be seen, is easily obtained ; for, on the one hand, we have the line, p' y, equal to k K^ representing the thick- ness of the wall or of the arch ; and, on the other hand, all the other dimensions, as projected horizontally in fig. 2, so that the inclination of the line, d I', can be determined with the most rigor- ous exactness. Tliis straight line, as well as the corresponding one on the op- posite face, o 7i, serves as a guide to the stonecutter in reducing the twisted portion of the surface of the key-stone, comprised be- tw^een them ; and as affording a means of verification, it may be remarked, that this surface should be cut in such a manner, that a rule or straight edge may be applied to all parts of it, being guided by the arcs, n o and h I ; the former of which springs from the point, o', and the latter from the point, I', on the face, p' x, fig. 4*. To determine the faces of the joints of either of the two corner pieces, z, represented in detail, and detached in figs. 5 and 6, but on which the faces are not represented in then- full dimensions, it is necessary to proceed in the same manner as before, bringing each face into the plane of projection — that is, delmeating auxiliary views of them, as if parallel to this plane. Thus, to obtain the actual dimensions of the face of the joint projected at o h, with the point, w, as a centr«, describe a series of arcs, with the respective radii, o w,n ic, h w, so as to reproduce thj points, o, ?i, K at 6\ 7^^ h\ upon the vertical, o^ w ; then, by settmg BOOK OF INDUSTRIAL DESIGN. 12A off, h^ h', equal to n' h', fig. 2, and joining tlie points, h' n', fig. 7, we get the inclination of the generatrix line, n" h', which is project- ed vertically at n h, fig. 1. The form of the joint face is completed by drawing the horizontal lines, o" 20 , h' z', y' v', and the verticals, u' v' and z' j/, which last are already given full size in fig. 6. It will be observed that fig. 7 is on the plate removed a little to the right of the vertical, o' w ; but this is a matter of no importance, and is merely done for convenience sake. The same system of auxiliary projections is applicable to the determination of the dimensions'of the other face of this piece — namely, that projected at m g, which is brought round to the hori- zontal, m' g\ and drawn with full dimensions in fig. 8 ; only, for this last face, it is necessary to bear in mind the portion of the line of intersection of the sides with the arched part which it contains, and which is obtained in its actual propoi-tions, as at a^ m^, by means of a template formed to the curve, a" m", in fig. 1. The stone, Y, beneath, of course, contains the remainder of the inter- sectional curve. The methods just explained, in regard to the shaping of the key- stone and one of the corner-stones, may be extended, without difli- culty, to the remaining portions of this Marseilles arch. In this application it has been necessary to determine the propor- tions of the twisted bay of the arch, as well as the faces of the joints ; but in the more general case of straight bays, such as that represented in fig. 1"', the operations are considerably simplified, and the designer has merely to attend to the form of the joint faces, making use, for this purpose, of the auxiliary projections, as above described. The delineation of the various parts of this figure pre- senting no new peculiarity, it need not further detain us. 378. Let it be proposed to delineate a circular vault with a full centering, bounded by two plane surfaces oblique to its axis, figs. 9 and 10. This example is taken from the entrance to the tunnel on the Strasbourg Railway, near the Paris terminus, and it is a form frequently met with in the construction of railways. In the representation of this vault, we have supposed one of the oblique planes to be parallel to the vertical plane of projection, and it consequently follows that the axis of the arch is inclined to this plane. Let A B be this axis, and c d the horizontal projection of a plane at right angles to it; with the point, b, as a centre, describe the semicircle, cad, representing the arch in its true proportions, as brought into the plane of the picture. Let us suppose this semicircle to be divided into some uneven number of equal parts, as in the points, a, h, c, d, e,/; through each of these points draw straight lines, passing also through the centre, b, and representing the joints, a g, b h, c i, of the arch stones, being, of course, normal to the circular curvature of the arch, and being limited in depth, as wo shall suppose, by the second outer semicircle, g i I, concentric with the first. Each of these joint faces intersects the centering of the arch in a straight line parallel with its axis, and the horizon- tal projections of these intersections, as seen from Ix'low, arc ob- tained simply by drawing through the points, a, h, c, d, lines parallel to the axis, b a ; those last extend as far as the vertical plane, a r, which bounds a portion of the vault. The external faces of (ho key and arch stones are limited by straight vertic:»' lines, such as in h, i n, oj, and horizontals, as m i and n o. We have now to obtain the projections on the vertical plane, fig. 10, of the intersection of each of the arch stones by the plane, A E. We may remark, in the first place, that since this plane is oblique to the axis of the cylindrical arch, it produces an elliptical section, having for its semi-transverse axis the length, c' a, and for its semi-conjugate axis, the length, a b , equal to the radius, a' b. As much of this ellipse as is required is drawn according to one or other of the many methods given (53, ei seq.) — say as at c' b^ b' fig. 9, which curve is reproduced at c' 0' a", in the elevation, fig. 10. If we, in like manner, obtain the projection of the semicircle, f i I, which limits the radial joints, we shall also obtain the portion of an ellipse, f" g" i", and we have further merely to project the points, a', b', c', upon the first ellipse, in a" b" c" ; as also on the second one, the points, g", h", i", corresponding to g h and i. The straight lines, f" c", g" a", h" b", i" c", represent the intersections of the faces of the arch stone joints, with the plane, a e. The vault being supposed to extend no furthc baok than the plane, c d, it will be necessary to represent the iniersection of this last with the arch stones which extend thus far upon this plane, c D. We have, therefore, to project the elliptic curves, c'" b'" c" and f'" ^" i'", corresponding to the quarter circles of the radii, F B and c B. As the arch stones cannot extend the entire length of the vault, they are limited by planes, m n, perpendicular to the axis, and, consequently, parallel to c d. so that the projections of these joints will be but repetitions of portions of the same elliptic curve : care is taken so to dispose the blocks of stone, that no two joints form a continuous line, the joints in one course being brought between those in the adjacent ones, as is customary in all brick and stone work. 379. We have now to determine the intersection of the oblique plane, a g, with the remaining half of the same circular vault, a id then to obtain the projection of this intersection upon the vertical plane. The plane, a g, also produces an elliptical section of the vau't; this is represented at g' gt, as brought into the picture in the auxiliary diagi-am, fig. 11, which gives its actual proportions; the semi-trans- verse axis, g' 0, of this ellipse is equal to G a, and its semi-oonjug:ito axis is equal to the radius, a' b, of the vault. After having divided this curve into a certain uneven number of equal parts, draw normals,* p u, q v, r x, s y, and t 2, through the points of division representing the joints of the arch stones, the remaining sides of the external faces of which arc limited by l.ori- zontals and verticals, as before. If the vault is supjiosed not to cxtciul beyond the piano, a g, the arch stones will have to be shaped as facing stones, and their joints will require to be set oil' upon the first ellipse, g' q t, and to be limited by the second, 11' (/' /^ ohiained from the interscotional plane, H i, drawn parallel (o ,\ c ; by drawing strmght lines from the points of division ohtniiinl u|i"n the ellipse, g'(//, to the cenlre. o, \\e obtain the points, ji', (/^ i'\ .•>'', of iniersection of these line^ uiuin the second ellipse, and thr slraii^lit linos, /. /-', q 7', r r\ s .v'. re- presenting the intersections of the ;neli >ion<s with the insi.lo of the • A Hm> Is Hiilit to 1)0 naniiiil to ii ourvi-. wlu-n il Is pi-rpctnllc»lnr lo n t iiwenl •• tlic curve piisslnit llirmmli Us point of lutiTbccUoii wllh the oiirvo n3). 126 THE PRACTICAL DRAUGHTSMAN'S vault ; these straight lines are projected horizontally in f j)", q' q", r' r", &c., fig. 9, where they are visible, because the diagram is supjiosed to be a projection of the vault, as seen from below ; the two diagrams, figs. 9 and 11, will render the determination of the vertical projection, fig. 10, very easy, the same lines there being designated by the same letters. To limit the arch facing stones, and unite them conveniently with the regular courses of the vault, they must be cut by planes, such as J K and l p, fig. 9, perpendicular to the a.xis, a b. The intersec- tions of these planes with the vault produce portions of circles, which are projected as ellipses in fig. 11, such as l' 5° v and u' r* x', for the inside of the vault, and j' q r' and l' s' p', for their outer ex- tremities, these various ellipses corresponding to the radii, c b and F B. The joints of these stones are finally completed by planes, such as k' p" s q, fig. 11, passing through the axis, a b, and through horizontal lines, y' z' and o s, in the vault, the latter and inner one of which only is visible in the elevation, fig. 10. 380. In constructing this vault, it is necessary to make detailed drawings of each particular stone, showing the dimensions of all the faces. In figs. 12 to 15, we have represented one of these arch stones, ©, in plan and elevation, as detached from the erection, fig. 10, and showing more particularly such lines as are not apparent in fig. 11. Thus, in these views may be distinguished: — 1st. The anterior face, v q r x, which is projected horizontally upon the line of the plane, a g ; this face, it will be remembered, intersects the vault at the elliptic curve projected at q r. 2d. The face of the joint, a; r r" w, of which the one edge, x r, is projected upon the same plane, g a, at x' r', whilst the opposite edge, w r^ is projected upon the line, v" r", parallel to g a ; and the lower edge, r r', the line of intersection of this face with the interior of the vault, is projected in the line, r' r\ whilst, finally, the upper and fourth edge, x w, is projected at x' v)f. 3d. The second joint face, v q(f w, is opposite to the first, and projected at v/ ({ 5* w. 4th. The face, q s z y, the horizontal projection of which is q' s' z' y' ; this face is situated in a plane passing through the axis of the vault, and is additionally represented in the diagram, fig. 14, on the radius, p q". 5th. The species of dovetail joint, q q^ y z r^ r, of which the edges, q q^ and r r\ are projected, as has been seen, at q q* and / r" ; whilst the sides, q r and z r', are similarly projected at </ r and z' r", and finally the side, q" y a.t q* y'. 6th. Lastly, the posterior face, q, of which it will be easy to render an account by means of the distinctive letters, which are invariably the same for the same points, although additional special marks are superadded to obviate confusion amongst the various figures. To render this vertical projection more intelligible, we nave added the subsidiary view, fig. 14, representing the projection of the block m the plane, l p, as brought into the plane of the picture ; we have thus the actual proportions of the faces projected In the planes, s' r" and p' q". To obtain the various points seen in lias view, it is sufficient to set oflT the vertical distances from the Ime, G o, of the elevation, figs. 11 and 13, obtaining in this man- ner, for instance, the points, r\ q\ v^, x\ &c., corresponding to r', (f, V, and X. '^'nd examples chosen for this plate (XXXIV.) combine the more difficult problems and applications met with in the shaping and arrangement of stonework, and will make the student acquainted with the operations upon which designs for these purposes ai-e based, as well as with the general methods to be adopted La obtain- ing oblique projections by the employment of auxiliary projections, taken, as it were, in planes parallel to the different surfaces, and then brought into the plane of the picture ; this system, at the same time, being of much use in ascertaining the exact proportions of various surfaces, such as the joints of masonry. RULES AND PRACTICAI. DATA. HYDRAULIC MOTORS. 381. The fall of a stream of water varies with the locality, and gives rise to the employment of different kinds of hydraulic motors, which are denominated as follows, according to their several pe- culiarities. First, Undershot water-wheels, which receive the water below their centres, and the buckets or floats of which pass through aa enclosed circular channel, at the part where the water acts upon them. Second, Overshot water-wheels, which receive the water from above. Thud, Wheels with vertical axes, known as turbines, and which are capable of working at various depths. Fourth, Water-wheels, with plane floats or buckets, receiving the water below their centres, and working in enclosed channels, through a portion of their circumference. Fifth, Similar wheels, with curved buckets. Sixth, Hanging wheels, mounted on barges, and suspended in the current. UNDERSHOT WATER-WHEELS, WITH PLANE FLOATS AND A CIRCULAR CHANNEL. 382. The most advantageous arrangement that can be adopted in the construction of an undershot water-wheel, with plane floats, and working in an enclosed circular channel, is that in which the outlet is formed by an overshot sluice-gate, and when the bottom of this outlet is -2 to '25 m., or about 8 inches, below the general level of the reservoir. Let it be required to determine the width of an undershot water- wheel, with the following data : — First, The discharge of water is 1,200 litres per second. Second, The height of the fall is 2'475 metres. Thii-d, The depth of the water at the sluice-gate is to be -23 m. WIDTH OF THE WHEEL. It will be seen, in the table at page 113, that, with an outlet of ■23 in depth, a discharge can be effected of 188 litres of water per second, for a width of 1 metre ; consequently, the width to be given to the sluice, to enable it to discharge 1,200 litres per second, should be — 1200 -T- 188 = 6-38 metres. BOOK OF INDUSTRIAL DESIGN. la: DIAMETER OF THE WHEEL. 383. The diameter to be given to a wiieel of this description has not been accurately determined, because it has not a direct influence u[>f.n the useful effect that may be obtained from it. Nevertheless, it is manifest that it should not be too small ; for in that case the water would be admitted too nearly in the horizontal line passing through the centre, or even above it, which would cause great loss of power. Neither should it be too great, for in that case the ex- aggerated dimensions would but involve an increased bulk and weight, and, consequently, a greater load and more friction, without any compensating advantage. In general, for a fall of from 2 to 3 metres, it is advisable to make the extreme radius of the wheel at least equal to the mean height of the fall, augmented by twice the depth of the water upon the edge of the outlet. Thus, in the case before us, the height of the fall being limited to 2-476 m., the outer radius of the wheel should not be less than 2-475 m. plus twice the depth of the overflowing body of water when at its fullest — say -6 m.; that is to say, in all, 3-075 m., which corresponds to a diameter of 6-15 metres. Water-wheels, on the same system, with a fall of water of from 2-6 to 2-7 metres, have often an extreme diameter no greater than tlus. VELOCITY OF THE WHEEL. 384. Theoretically speaking, the velocity which it is convenient lo give to an undershot water-wheel should be equal to half tiiat due to the height of the overflow of the water ; that is to say, equal to from r to 1-1 m. in the present case. Nevertheless, practice shows that this rule may be departed from without inconvenience, and the wheel may be made to attain a velocity of from r5 to 1 -6 m. per second at pleasure, which is a very great advantage in many circumstances. If the wheel makes three turns per minute, the mean velocity at the outer circumference, and at the edges of the floats, will be — 6-15 X 3-1416 X 3 60 = 1-021 m. per second. Thus, when the height of the overflow is -24 m., the correspond- ing velocity of the water being 2-17 m. nearly, as shown in the table at page 94, which gives the heights, 23-66 and 24-67 cent., iherefore, the ratio of the velocity of the wheel to that of the water is -47 : 1. If the height of the overflow were reduced to -15 m., which sup- poses that the discharge would only be 101 litres x 6-32 m. — 638 litres per second, the corresponding velocity of the water would not be more than 1-72; and in this case, the ratio of the velocity of the wheel, sup- posing it to be still the same, to that of the water, would be — •5^5 : 1. NCMBEF AND CAPACITV OF THE BUCKETS. 385. Although the number of buckets cannot bo delerniiiicd in accordance with any exact rule, it is, ncvcrtiielcss, of importance that their pitch should not be much greater than the depth, or thickness, of the overflowing body of water iicting upon them. It is also necessary that the number of the buckets should be divisible by that of the arms of the wheel, so that the whole may be put together conveniently. Now, since the outer circumference of the wheel is 6-15 X 3-1416= 19-32 metres, we can very conveniently give it 8 arms and 64 buckets ; and the pitch of these last will be -32 m. With this distance between the buckets, there should not generally be a greater depth of overflow than -25 or -26 m. ; because, at -27 m. the water will begin to choke, as it will not be admitted easily into the buckets, and will rebound against the interior of the channel, giving rise to a continual shak- ing action. Thus, then, in determining the number of buckets for an under- shot water-wheel, receiving the water from an overshot outlet, it is necessary to calculate the spaces between them, so as to be about a third, or at least a fourth, gi-eater than the depth of the water at the outlet, whilst their number must be divisible by the number of arms of the wheel. For water-wheels of from 3-5 to 4-75 metres in diameter, six arms for each rim or shrouding ; for wheels of 5 to 7 metres in diameter, there should always be eight arms for each ishrouding ; and the number of arms should obviously increase for wheels of greater diameters than 7 metres, of which, however, there are but few examples. With regard to the capacity of the buckets, and the channel, taken together, it should be equal to at least double the volume of water discharged. Therefore, on this basis, we can always easily determine the depth to be given to the buckets, when the maximum discharge is known. Thus allowing, in the present instance, the maximum discharge to be 1,340 litres per second, instead of 1,200, since the velocity at the outer circumference is 1-021 m. per second, the number of buckets contained in this space is equal to 1-021 ^-32= 3-19. Then— 1-340 H- 3-19 = -43 cubic metres nearly, the quantity which should be in each bucket during the revolutioa of the wheel. If, then, the capacity is to be double this, it will be equal to -86 cubic metres. The product, however, of the width, 6-38 m., of the wheel, multiplied by the space between two consecu- tive buckets, -32 m., is equal to 2-022 m. We have, then, -86 -r- 2-022 = -42 in., for the depth of the buckets. The distance between the buckets, however, is not tho same at the inside as at the extremities, and the capacity is also further diminished by the thickness of the sides of tho buckets, and by the inner portions, which make an angle of 15 ' with the outer portions. For these reasons, tho depth should be somewhat increased. When tho discharge of water is considorable, and w <> arc limited as to the width of the wheel, it is prel'erablo to do awav with the inner inclined iiortion of the buckets, as indicated in tho drawiii"-, Plate XXXVl., prolonging tlu'in i-onsiderahly toward" the centre of the wheel. USEFUL EFFECT OF THE WATKlt-WHEEL. 386. The absolute force of a stream of water is the product of the water discimrgcd per second, exiiressid in kilog;nunmes, t»* 128 THE PRACTICAL DRAUGHTSMAN'S the height of the fall expressed in metres, or the weight in pounds by the height in feet. Thus, when the discharge is 1300 litres or kilog. per second, and the total height of the fall 2-475 m., the product of 1300 kilog. by 2-475 m. expresses in kilogrammetres the absolute force ; this may be converted into horses-power by dividing the result by 75 : we have, therefore — 1300 X 2-475 = 3217 km., and 3217 -=- 75 = 43 horses-power. Undershot water-wheels, with plane-bottomed buckets and circu- lar channels, when well constructed, are capable of utilizing from 70 to 75 per cent, of the absolute force of a stream of water. OVERSHOT WATER-WHEELS. 387. Let it be proposed to construct a water-wheel to receive the water from above, under the following circumstance : — 1st. The effective vertical height of the fall, or the distance between the upper and lower level, is 4-56 m., without sensible variation. 2d. The quantity of water discharged per second is supposed to be almost uniform, and is measured by a vertical sluice-gate, with complete contraction at the outlet. 3d. The width of the sluice-gate is "5 m., the height of the open- ing -14 m., and the charge or height of the reservoir level above the centre of the outlet, -55 m. Solution. — From the table at page 111, of the discharges of water, we find that 280 litres per second is the quantity which escapes at an orifice -14 m. in height, by 1 metre wide ; and with a pressure upon the centre due to a height of -55 m.. We have, con- sequently, 280 X -5 = 140 litres. This discharge being known, if we are not limited to any parti- cular width of wheel, it may be constructed thus, for it gives as great a useful effect as can be expected in ordinary circum- stances. In such case, the velocity, v, should be regulated to about one metre per second at the circumference, because the advantage that might result from a less velocity would be counterbalanced by the consequent increase in the width of the wheel. If we adopt the velocity, v, of 1 metre, we find that V, of the water, at its point of escape from the outlet, should be 2 metres per second, to act with proper effect upon the buckets ; now it %vill be seen in the table, at page 94, that this velocity corresponds to a height of -205 m. above the centre of the orifice. This height has to be deducted from the total fall. For small discharges of water, it is advisable to make the height of the orifice as small as possible, so that the depth of the water may be trifling, which will permit of its entering the buckets much more freely : it may be taken at -06 m., or H ^ -06 m. The half of this height, or -03 m., must be added to the first -205 m., to give the whole height of the water in the duct behind the outlet ; that is, from the upper level to the lower edge of the orifice. Taking also -01 for the extent of the trifling fall of the small spov?* i-eafhing from the front of the outlet to the top of the wheel, and -01 m. for the play-space which may be supposed to flidst between the end of the spout and the latter ; after deduct- ing all these quantities from the total fall, namely, 4-56, we shall have remaining — 4-56 — (-205 + -03 +-01 4- -01) = 4-305 m. for the extreme diameter, d, of the wheel. The channel which conducts the water to the wheel, and the width of the outlet orifice, should be disposed as much as possible, so that it may not meet with contraction from the lateral or bottom edges of the sluice-gate. Referring again to the table on page 111, the discharge, 140 litres, must be divided by the number 76, corresponding to the height -06 m., and to the charge -2 m. ; it must also be divided by the coefficient, 1-125, when we shall have — 140 _ ~ 1-125= 1-66 m. 75 for the width of the outlet orifice. By adding -1 m. to this, we have 1-76 m. for the width of the wheel. The depth of the buckets is determined thus : — -140 m. d = 8 X = 214 m.; 3 X 1-78 m. X 1 m. consequently, the internal diameter, d, of the wheel becomes — d' — 4-305 — (-214 X 2) = 3-877 m. By augmenting this depth about a fifth, which will make it •257m., we get the distance to be allowed between the buckets; so that, as the internal circumference is equal to — 3-14 X 3-877 = 12-174 m., dividing this by -257, it becomes 12-174 , , .t,K„ = 47-3 ; say 47 buckets. For a water-wheel, however, of 4-305 metres in extreme diameter, there should be eight arms ; and if it is intended to make the shroudings of cast-iron, and in segments, it is advisable that the number of buckets be divisible by 8; it will, therefore, be convenient to have 48 instead of 47, and in this case the space allowed be- tween each will be reduced to — 12-174-^48 = -254m. It now merely remains to draw the wheel ; for this purpose, the concentric internal and external circles are described ■with the de- termined radii : the first is tlien to be divided into 48 equal parts, and radii are drawn through each point of division, as indicated in Plate XXXVI. ; on each of these, outward from the internal cu-cle, is marked off a distance equal to a little more than half the depth of the buckets, say -12 m., to indicate the bottoms of the buckets. The water-wheel, when constructed in this manner, may give off 79 or 80 per cent, of the absolute force of the fall of water. Now this force, expressed in horses-power, is equal to — 140 x 4-56 75 = 8-51 horses-power. Deducting 5 or 6 per cent, at the most, for the friction of water- wheel shaft in its bearings, we may still calculate, with certainty, that the power utilized and transmitted by this wheel will be equal to 74 or 75 per cent., or 8-51 X -75 = 6-38 horses-power. The number of revolutions which this wheel should make per minute is — BOOK OF INDUSTRIAL DESIGN. 129 60 -=- 4-305 X 3-14 = 4-44, since its velocity, v, is 1 metre per second, or 60 metres per minute. In tracing out the preceding solution, it will have been seen that the width to be given to the wheel is 1"76 m. ; a much less width might have been obtained, by making the wheel revolve faster, and by augmenting the velocity of the water also. Let us suppose, for example, that the question lias to be solved on the hypothesis, that the velocity of the water-wheel is to be 1'5, instead of 1 metre, per second; it wUl then be necessary, in order that the water may escape from the orifice at double this velocity, that it be equal to 3 metres per second. For this velocity, the height of the upper level, above the centre of the orifice, should be '46 m. Allowing "06 m. for the height of the open part of the sluice-gate, the whole height above the wheel will be •46 -1- -03 + -02 = -51 metres ; consequently, the outer diameter of the latter should be d = 4-56 — -51 = 4-05 metres, the width of the sluice-gate, or w = = I'll metres, •06 X 3 X -7 and consequently the width of the wheel = 1-11 4- -10 = 1-21 metres. This width, it will be seen, is considerably less than that first calculated. This wheel, however, which is narrower, and revolves at the rate of 1'5 metres per second, will not be capable of trans- mitting so great a useful eflfcct, by four or five per cent. Never- theless, it may be preferable in many circumstances to adopt this lesser width, either to render the wheel lighter and less costly in construction, or to avoid the necessity of much intermediate gear between the wheel and the machinery to be actuated. Thus, it is evident that this wheel should make (60 X 1-5) -r- (4-305 X 3-14) = 6-66 revolutions per minute, whilst the first wheel only made 4'44. The other parts of the wheel are proportioned according to the above rules ; they will, however, differ but slightly from those of the first wheel. The proportions of the water-wheel might still be otherwise modified; thus, the depth of water at the outlet might be allowed to be greater than that taken for a basis in the preceding cal- culations. Thus, the outlet might be opened to the height of •1 m. instead of only -06 m. : in this ease, the width of the outlet and of the wheel would be much less. But this arrangement would have many disadvantages, for it would be necessary to make the buckets more open ; that is to say, the angle made by the outer portion of the bottom of the bucket with the tangent to the circumference passing through its extremity, instead of being 15° or 16°, as is usual, would have to be 30^^ or 32°; the buckets would have to be deeper and more capacious ; they would empty thctnsiilvcs sooner: from all which causes would follow a decrease in the useful eH'cct given out, which might roach oven to 15 per cent. It la true, on tlio other hand, that the width of the outlet would bo reduced to 1 metre, supposing the wheel to revolve at the rate of 1 metre per second, and that it would not be more than •67 m. when the wheel revolves at the rate of 1-5 m. per second ; in which case, the depth of the buckets would be about -34, and the spaces between them -4 m. each. It will be easily conceived that such an arrangement cannot bo advantageously adopted, except where there is plenty of water to spare, and when the constructor is limited as to the width of the wheel. WATER-WHEELS WITH EADIAL FLOATS. • 388. In old mills we sometimes meet with water-wheels w^hicti have plane floats placed radially, working in straight inclined chan- nels, with a vertical outlet more or less distant from the centre of the wheel. These wheels generally give out 25 to 35 per cent, of useful effect of the absolute force of the stream. In them the floats are three or four centimetres clear of the sides of the channel ; when a greater space than this is allowed, the useful effect is sensibly diminished. Generally, the width of such wheels is equal to that of the outlet. At the present day, water-wheels are never constructed with plane floats arranged in this way. When a wheel is required to have a great velocity, it is preferable to construct it to work in an enclosed circular channel, and to receive the water from above, or from an orifice with a sufficient column above it, to give the proper tionate velocity to the water. Such wheels are constructed in the same manner and with the same care as undershot water-wheels; in fact, they do not differ from the latter, except in that these receive the water from an open topped or overshot duct. The useful effect given out by them varies from 40 to 50 per cent., according as the sluice-gate is more or less near to the upper level of the water. Thus, the nearer the channel approaches to the upper level, the more like the wnoel becomes to a common undershot one, and, in consequence, the use- ful effect is greater. In the construction of a water-wheel of this kind, the same rules are followed as are already laid down for conimou undershot w heels with open outlets. Thus, let it be proposed to construct a wheel for a i;ill of 1-75 metres, and with a discharge of 440 litres of water per second : lot the centre of the outlet orifice bo at '4 m. below tlie upper level, and the height of the orifice itself, 'IS m. By referring to the table on page 111, it will be scon that t\\r discharge of water through an orilice, under those circunistancos, is 255 litres per second for a width of one metre, and it w ill thorel'oro be evident that the wlieol should have 440 ■rr-p = 1-72 metres in wultli. 255 The velocity of the water at the sluico-gale, corresi)ondin;j to the column of '4 in., is 2802 per second ; consequontly, if wo uiake the Ncldoily of ihi' whorl oiiual (o •,').') liinos tiiat of tho water, it will bo 'J'S02 X '55 = rSl nu^lros por second. The ili.-unolor of ilio wliool is of itsolf a mailor of inditVorenco ; it .slmuM lio rocluroil ;is imioli as possililo, so ii.s to lessen tho cost of consli lution ; iiolw iilistuudiiig, it should never bo loss liiiui iw ioo 130 THE PRACTICAL DRAUGHTSMAN'S 2-7 metres. llie wholi) height of the fall ; thus, in the present example, it should not be less than 4 metres. It has often been asserted that the power is increased by in- creasing the diameter; it seems incontrovertible, however, that the power transmitted must be in proportion to the height of the fall, and to the quantity of water discharged. If the diameter of the wheel is increased, the angular or rotative velocity is diminished, and, consequently, the momentum and actual force communicated remain the same. Taking the diameter at 4 metres, we have ■ • = 7'2, the number of revolutions per minute. 4 X 3-1416 ^ If a wheel of this diameter were adapted to an open-topped or overshot duct, with a depth of water at the sluice-gate equal to •2 m., the velocity of the water being then reduced to 1-981 metres, The velocity at the circumference of the wheel would not be more than 1-981 X -55 = 1-09 m., and, consequently, the number of turns only 1-09 X 60 4 X 3.1416 = 5-2 per minute. But then, as the discharge in such case, at an overshot outlet of •2 m. in depth and 1 metre in width, is 166 litres per second (see table, page 113), the width of the wheel must be made equal to 440 166 Thus, it will be seen that the water-wheel which revolves more rapidly is narrower than the one with the same discharge of water by an open outlet, and it is, consequently, less costly in construc- tion ; but then, it only gives out, as useful effect, about 50 per cent, of the absolute force of the stream, whilst that given out by the other description may reach, as we have seen, as much as 70 per cent. With regard to the other dimensions of the wheel, we have merely to refer to what has been said about the common under- shot water-wheel. WATER-WHEELS WITH CURVED BUCKETS. 389. These wheels are fitted with inclined ducts for the water, the inclination being equal to a base of 1 metre for every 1 or 2 metres in height — that is to say, to 45 to 600 ; they are enclosed for a short distance Ln a circular channel, and between two side walls. They are seldom constructed except for low falls of from -5 to 1-3 metres, and when a great velocity is requii-ed ; the useful effect they give out varies from 45 to 65 per cent. It is of importance that the water duct be brought as close up to the circumference of the wheel as possible, and that, at its lower part, it should have an enlargement of 10 or 15 centimetres, to facilitate the disengagement of the water, and render its action freer ; this enlargement should commence at a distance from the vertical line passing through the centre of the wheel, equal to the space between two consecutive buckets. The velocity of the wheel should be from -5 to -55 times that of the water at its exit "rom the duct. The width of these wheels is to be calculated in the same manner as that for the preceding ones ; as to the diameter, it may be re- duced in proportion to the fall, but it should never be less than threo times the height of the latter. The depth of the curved buckets, or the width of the shrouding in the direction of the radii of the wheel, should be equal to mie- fourlli of the fall augmented by the height of orifice open. For falls of less than 1-2 m., the height of the orifice, or the depth of the outflowing water, should be from -2 to -22 m. ; it may be reduced to -18 or -16 m. for falls of from 1-2 to 1-5 m. The buckets or floats are in the form of a cylindrical curve, being a single circular arc, tangential to the radius at the inner part, and making an angle of about 24° or 25° with the stream of water flowing towards the inside of the crown of the wheel. The space between two consecutive buckets is measured at the outer circum- ference by an angle of 25°, and their thickness is 24 to 28 hun- dredths when made of wrought-iron plates, and 32 to 35 when of wood. The bottom of the channel should have a fall or inclina- tion of about -jLth or y'jth — that is to say, equal to that of the hypothenuse of a triangle, the base of which is 12 or 15, and the height 1 metre. TURBINES. 390. Among the varieties of turbines which receive the action of the water throughout their whole circumference, may be distin- guished those which discharge the water at their outer circumfer- ences, and those which allow it to escape behind. The useful effect given out by these wheels varies from 55 to 65 per cent, of the absolute force of the stream of water. For these descriptions of wheel, the discharge of water is cal- culated in accordance with the rules and tables already cited. For the first kind, termed centrifugal turbines, the internal diameter is determined by multiplying the fourth or fifth of the velocity due to the total fall by 785-4 ; then dividing the quantity of water to be discharged by the result obtained, and finally extracting the square root of the quotient. Example. — Let us suppose that the fall is 2-2 m., and the dis- charge of water 800 litres per second. It will be gathered from the table on page 111, that the velocity due to the height, 2-2 m. — 6-57 m. 1-314; We have then, 6-57 4 - 1-642, and ^'f^ - 5 and further. D=> / ^^° V 785-4 X 1-642 " or, ■ ./ r> 800 787 for the internal diameter of the cylindrical tank above the turbine. Add 4 or 5 centimetres for the Internal diameter of the latter, which gives •82 to -91 m. The external diameter should be equal to the internal diameter multiplied by 1-25 or 1-45, and is, therefore, 1-025 to 1-189 m.; BOOK OF INDUSTRIAL DESIGN. 131 or, 1-137 to 1-319 m. When the height of the fall and the discharge of water are ■\-ariable, the diameters should be calculated for the extreme cases, so that the most advantageous proportion may be adopted — that is, the one which will give the best result throughout the greater part of the year. If the variation is very considerable, there should be two or more (urbines employed, some calculated for the lowest discharges, others for the mean, and others again for the maximum discharges. The height of the buckets — that is to say, the vertical distance between the two discs which form their top and bottom — is gene- rally about a fifth, or a fourth at most, of the radius of the interior of the wheel. Thus, in the case before us, the diameter being -787 or -874, the radius is -3985 or -437, and, consequently, the height of the buckets should be -1 to 11 m. The buckets being cylindrical in form, their entrance is normal to the conducting channels which direct the water against them, and for these low discharges of water should make angles of 68° to 70° with the internal circumference of the wheel — that is to say, the conducting channels should make angles of 20° to 22° with the circumference. When the discharges are large, this angle may be increased to 30° or 45° ; thus, for a discharge of 600 to 700 litres per second, it is considered that the angle should be about 30°. In order to obtain the maximum useful effect, the velocity of the wheel should be equal to about -7 times that of the water ; in prat> tice, one-tenth may be added to this ratio, or one-fifth to one-sixth, without materially diminishing the useful effect. The space between each bucket, taken at the internal circum- ference, should be nearly equal to the distance between the top and bottom discs of the turbine; it should, however, never exceed 18 to 20 centimetres. The internal and external distances between the buckets are necessarily in the ratio of the internal and external diameters of the wheel. In the following table we give the principal dimensions, data, and results of several descriptions of tui-bines, constructed, within the last few years, by MM. Fourneyron, Fontaine, and Andre KcEchlin. These results have been selected under circumstances where the best useful effects were given out ; — FABLE OF DIMENSIONS AND PRAUTICAL RESULTS OF VARIOUS KINDS OF TURBINES. Data and Rpsults. Tiiial fiill, Discliarf^e per second Kxtttrnal diami.'ter, I-)<'plh of shrouding, IIiMKlit of outlet, NiiiiiliLT of l>uckels, NijinlM:r of director curves i>luijiljiT of revolution.H per minute, Usr-ful effect Ratio of useful effect to absolute force, ■ Names of the Turbines and c Constructors. f their Nfoussay Mulbach Bouchet Viidenay Turbine. Turbine. Turbine. Turbine. Fourneyron. Fourn«yron. Fontaine. Fonlolni'. 7-Ofi m. .■i-45 m. 100 m. r40 m. 52T lit. 2.500 lit. 218 lit. 1400 lit. ■850 m. 1-9 m. 1 33 m. 1-940 m. •110 m. 335 m. •23 ni. •071 m. ■270 m. •04 ni. 32 58 B4 24 24 32 185 55 30 35 II. v. 90 11. i>. 2 II. P. 18 11. P 7U°.'o 70O/o Tl-Zo Tl"/, Data and Results. Total fall, Discharge per second, Exteroal diameter, , Width of buckets, Number of buckets, , Area of outlets, Area of escape outlet below the wheel Number of revolutions per minute,. . . , Useful effect, Ratio of useful effect to absolute force, Jonval Turbines, constructed byM Andre Kcechlin, Mulhonse 2-720 m. 684 lit. 1 •800 m. •410 m. 16 •290 sq. m. ■450 sq. m. 90 to 158 1 13 H. P. 2-77 m. 470 lit. •800 m. .100 m. 18 •220 sq. m. •45 sq. m. 90 to 168 15 H. p. 72°/„ r70 m 355 lit. •810 m. •120 m. 18 0706 sq. m. ■2977 sq. ni. 90 6 H. P. 72°/„ REMARKS OX MACHINE TOOLS. VELOCITY OF THE TOOL, OR OPERATING PIECE, IN MACHINES INTENDED TO WORK IN WOOD AND METAL. 391. The principal machine tools, employed in machine shops, are — 1. The simple lathe, the self-acting lathe, and the wheel-cutting lathe, with adjustable table. 2. Boring machines of various dimensions, and radial drUiing machines. 3. Horizontal and vertical shaping machines. 4. Planing machines with a fixed tool, or with a moveable one, so as to work both ways. 5. Mortising or slotting machines, having a vertical tool with a revolving table below. 6. Machines for finishing nuts and screws. 7. Machines for cutting screws and bolts. 8. Dividing engines, for dividing and cutting toothed-wheela of all dimensions. 9. Straight and curved shears, for shearing plates. 10. Punching and, riveting machines. 11. Steam and other hammers. 12. Straight and circular saws. The velocity of the cutting tools, in tluse niacliincs, varie;. according to the nature of llie material, and tin- quality of work desired. In general, for soft cast-iron, it is convenient to give a velocity of seven to eight ceiitiinetres per second to ihn tool, in such ma- chines as latlius, and planing and slotting machines. This velocity should be reduced, at least, to four or five centimetres in shaping, drilling, and screwing machines. W'Irii the cast-iron is liani, the velocity is considerably diminished. For wrought-iron, the velocity may be advantagoously increased one half, because the tool is kci)t well lubricated with oil, or with soap and water ; thus, in turiiiug or planing, the velocity may bo raised to eleven or twelve cent iiiu^'l res : and iu shaping and screw- ing, to about six ceiitiiiiL'tres per second. For copper, bra.ss, and other analogous iiiotiils, with which llu' tool (hies not become heated whilst working, the velocity may be very much greater; and for wood, its only limit.s are tiioso dotor- minod by the si/c tif the tool, and by the pnwors of Uio nmciiino. Witii regard to the pressure and rate of ndvanco of tho tool jht revolution, or per stroke, it necessHrily varies uccoi-ding to tho dimensions of the macliine itaell', and ulsi. iiccordinjf to tlio dogn<o y 132 THE PRACTICAL DRAUGHTSMAN'S of finish vvhich is to be given to the surface ; we evidently cannot give as much pressure to the tool upon a small lathe as to that upon a large one — to a small drill, as to a powerful shaping roachme. This variation extends, for the different metals, from a tenth of a millimetre, in some cases, to as much as two millimetres in others. Amongst other things, the following table shows tho rotative velocity to be given to the tool — when it revolves, and ths work is fixed; or to the latter when it revolves, and the cutting tool does not ;^in lathes, and shaping and drilling machines. TABLE OF VELOCITY AND PRESSURE OF MACHIIfE TOOLS OR CUTTERS. Turning. Drilling an d Shaping. Diameter Number Work performed per hour Number Work performed per hour in of revolutions with of revi. lutions wi th centimetres. per minute. i ""/m of pressure. per minute. i "Im of pressure. Cast Wrought Cast Wroug-ht Cast Wrought Cast Wrought Iron, Iron. Iron. Iron. Iron. Iron. Iron. Iron. cent. cent. cent. cent. 1 152-9 229-4 458-5 687-8 76-4 114-6 229-2 343-9 2 76-4 114-6 229-2 343-9 38-2 57-3 114-6 171-9 3 50-9 76-4 152-8 229-2 25-5 38-2 76-4 114-6 4 38-2 57-3 114-6 171-9 191 28-7 57-3 85-9 6 30-6 45-8 91-7 137-5 15-3 22-9 45-8 68-7 6 25-5 38-2 76-4 114-6 12-7 19-1 38-2 57-3 8 19-1 28-7 57-3 85-9 9-5 14-3 28-6 42-9 10 15-3 22-9 45-8 68-7 7-6 11-5 22-9 34-3 12 12-7 19-1 38-2 67-3 6-4 9-5 19-1 28-6 15 10-2 15-3 30-5 45-8 5-1 7-6 15-2 22-9 With 1 "/m of pressure. Withln'lm of pressure. 20 7-6 11-5 45-8 68-7 3-8 5-7 22-9 34-3 25 61 9-2 36-6 55-0 3-0 4-6 18-3 27-4 30 5-1 7-6 30-5 45-8 2-5 3-8 15-2 22-9 35 4-4 6-5 26-1 30-0 2-2 3-3 13-0 19-6 40 3-8 5-7 22-9 34-3 1-9 2-9 11-4 17-1 45 3-4 5-1 20-3 30-5 1-7 2-5 10-1 15-2 60 31 4-6 18-3 27-4 1-5 2-3 91 13-7 55 2-7 4-2 16-2 24-9 1-4 2-1 8-2 12-6 60 2-5 3-8 15-2 22-9 1-3 1-9 7-6 11-4 65 2-3 3-5 14-1 21-1 1-2 1-8 7-0 10-5 70 2-2 3-3 13-0 19-6 11 1-6 6-5 9-7 75 2-0 3-0 12-1 18-3 1-0 1-5 6-0 9-0 80 1-9 2-9 11-4 17-1 •9 1-4 5-7 8-5 90 1-7 2-5 10-1 15-2 •8 1-3 5-0 7-6 100 1-5 2-3 9-1 13-7 •8 11 4-6 6-8 110 1-4 2-1 8-2 12-6 •7 1-0 4-1 6-2 120 1-3 1-9 7-6 11-4 •6 •9 3-7 5-7 130 1-2 1-8 7-0 10-5 •6 -9 3-4 5-2 140 1-1 1-6 6-5 9-7 •5 •8 3-2 4-8 150 1-0 1-5 6-0 9-0 •5 •8 3-0 4-5 175 •9 1-3 5-1 7-8 •4 •6 2-6 3-9 200 •8 11 4-5 6-8 •4 •6 2-2 3-4 225 •7 1-0 4-0 6-0 •3 •5 1-9 3-0 250 •6 •9 3-6 5-4 •3 •4 1-8 2-7 275 •5 •8 3-3 4-9 •3 •4 1-6 2-4 300 •5 •7 3-0 4-5 ■2 •4 1-5 2-2 350 •4 •6 2-5 3-9 •2 •3 1-2 1-9 400 •3 •5 2-2 3-4 •2 •3 1-1 1-6 This table will serve as a guide in designing machine tools for !ne various combinations of movements, the application of which may be called for according to the nature and dimensions of the work to be submitted to their action. Thus, a lathe which is only intended to turn articles of from four to twenty or thirty centi- metres in diameter, should have a considerable rotative velocity, whilst one that is to be cliiefly applied to turning and shaping bulky pieces, or such as measure from one to two metres in dia- meter, should, on the contrary, be actuated by a combination of very slow, but, at the same time, very powerful movements. BOOK OF INDUSTRIAL DESIGN. 133 CHAPTER X. THE STUDY OP MACHINERY AND SKETCHING. VAEIOUS APPLICATIONS AND COMBINATIONS. 392. Hitherto we have had to occupy ourselves with industrial drawing, as regards only the geometrical delineatfon of the princi- pal elements of machinery and architecture. This preliminary study being of great importance, we have thought it well to dwell more particularly upon it, since also it is the very basis of all designing, with a view to actual construction, comprehending not only the mere outline of objects, but also the proportions between the vaii- ous pai'ts, as dependent upon the functions which each is required to perform. Machines are, indeed, but well calculated and thoughtfully ar- ranged combinations of these elements, and afford innumerable applications of the rules and instructions laid down in reference to them. The study, therefore, of machines in their complete state, naturally suggests itself as the next step to be taken. 393. Machines may, in general, be classified under three catego- ries — machine tools, productive or manufacturing machinery, and prime movers. By machine tools are meant those by the instrumentality of which we work upon raw materials, as wood, metal, stone ; lathes, wheel-cutting machines ; drilling, boring, and shaping machines ; mortising, slotting, planing, and grooving machines ; riveting ma- chines; shears, saws, hammers — are of this class. The movements )f these machines should be so combined, that the tool or cutting instrument — that is, that part which attacks the material — should move wdth a velocity properly proportioned to the nature of the work. In the few notes accompanying our text will be found some experimental deductions, which may serve as guides for adjusting the movements in designing and constructing machinery of this description. Amongst productive or manufacturing machinery, are comprised spinning, weaving, and printing machines; pumps, presses, corn, and oil mills ; and, finally, prime movers consist of those worked by animals ; windmills, \\'ater-wheels, turbines, and steam-engines. For the study of complete machines, we have selected from each of these categories those possessing most interest and generality — as a drilling machine, an instrument so very useful and so much employed in machine-shops and railway works; a pump for raising water, serving for domestic purposes as well as for important manu- facturing establishments ; two examples of water-wheels, showing various arrangements and forms of floats or buckets ; a high pres- sure expansive steam-engine, with geometrical diagrams, deterniin- ing the relative positions of the principal pieces in various ciicuin- stances ; and, finally, a set of belt-driven flour mills, coiislruclcd on a system recently adopted. Before proceeding to the description of these machines, it will bo necessary to habituate the student to draw from the rc^iliiy, for up to the present time ho will have done notiiing but cdiiy the various graj)liic uxainiilcs to this or that .scale. Tiie opc^ialicm in question consibts in drawing with the hand, the elevation, |)l!ui, sections, and details of a machine, preserving, as much as possible, the forms and proportions of each part; and then taking the actual measurement of each part, and laying it down in figures in its par- ticular position upon the drawing : this duplex operation of sketch- ing and measuring constitutes the study of the rough draughting of machinery. THE SKETCHING OF MACHINERY. PLATES XXXV, AND XXXVI. 394. Before commencing the sketch or rough draught of a machine, it is absolutely necessary to look carefully into its organi- zation, the action of the various working parts, the motion of the intermediate mechanical connections, and finally, its object ai-d results. The object of this preliminary examination is to give the draughtsman a good general idea of the more important parts — those which he will have to render most prominent and detailed when he comes to make a complete drawing of the whole ; such drawing comprising a series of combined views, together with sepa- rate diagrams of such details as may not be apparent in the former, or require to be drawn to a different scale to render them intelligi- ble. In fact, this must be done in such a manner, that, with iho aid of the sketch, a perfect representation of the machine may be got up, which, if necessary, may serve La the construction of otlier similar machines. DRILLING MACHINE. PLATE XXXV. 395. In order to give an exact idea of the manner of sketching machinery, we take a simple machine as an example ; this we suiv pose to be represented in perspective* in fig. ^A, this view being instead of the machine itself This machine is for drilling metals : it consists of a verticil c;isl- iron column, a, which forms part of the building or workshop. This column is hollow, and rests by an enlarged base upon a stono plinth, B, imbedded in the ground, and at its upper end it supports the beam, c. Upon one side of this culunui is cast the vertical face, p. which is planed to receive three cast-irou brackets, e, f, g, attached to it by bolts. To the opposite side, d', of tiie sjinie column, is in liki^ niaiuier attaciied the bracket, ii, whici), with the middle one, K, on the other side, serves to carry tlio hori/.ontiil spindU>, i. This s|)indle caiiies on one side the cone-pulley, j, over which pnsxs the ll^i^ill^-lH■ll, k, ;iiuI oh the other extremity it Hjis kr\r<\ iipoii il tlir Ih'\ ilpiiiioii, I., which gears with ii larger liiNil-wlucl, M. This hist is allachcil to the verlical sliart, N, • In n sulim-qiu'iit I'liaplor, wo sliuU ixiilalii tlic in'iu'nil iiiliu-'i>Ir» of i>:iriUlol «n4 131 THE PRACTICAI. DRAUGHTSMAN'S \vlii'-li is, in fact, the drill-holder, and is moveable in the bracket- bearings, F and G. This shaft receives a duplex movement, that of continuous rotation, which is more or less rapid, according as the belt, K, is on the less or greater diameter of the cone-pulley ; and tlie other vertical and rectilinear, due to the action of the screw, o, which works in an internal screw in the end of the bracket, e. This screw carries at its upper end a spur-wheel, p, gearing with a tsmall pinion, q, the shaft, k, of which is prolonged downwards, and terminates in a small hand-wheel, s. The object to be drilled is held between a pair of jaws, a, a', set In grooves upon the table, t, and capable of adjustment back and forward by means of the screw, J, the head of which has a sliding jandle. The table, x, is made in two pieces, so as to form a collar about the column, a, and it is fixed at any convenient height upon this column by means of the pressure screw, c ; the exact distance of the table from the drill, d, according to the thickness of the piece of metal to be drilled, is settled by means of the vertical rack, u, n-hieh is fitted to the front of the column, and into which gears a [>inion on the shaft, e, carrying the handle, /, at its extremity. The rotation of this handle and the pinion necessarily causes the ascent or descent of the table, t. The diillmg machine, then, fulfils the following conditions : on the one hand, the drill, d, is woi'ked at a greater or less speed of rotation, whilst it descends vertically with a very slow motion, which latter varies, of com-se, with the nature of the material acted upon ; and, on the other hand, the table which carries the object to be drilled is capable of being set at the most convenient height, according to the forms and dimensions of the objects, whilst it may rIso be set eccentrically, when necessary, by turning it to the re- quired extent round the column. 396. After having thus taken note of the construction and action of each individual piece and element of the machine, the draughts- man may proceed to make his sketch. He should commence by drawing a rough general view, indicating, in mere outline, the rela- tive positions of the various pieces. For example, in fig. 1 will be seen the geometrical elevation '^i the column, a, with the positions of the brackets and the table, which are merely in outline. It is advisable, even in this rough draught, as well as in the finished drawing, with or without the assistance of a rule, to draw the centre lines for guides ; thus, after drawing the first centre line, g h, of the column, a, draw upon each side of it the portions forming its contour ; then draw parallel to it the centre line, ij, of the drill-stock, n ; then the hori- zontal, k 1, which represents the centre line of the bevil-wheel, l, and the driving-pulleys, j, and likewse the straight lines, m n, o p, q r, which are the centre lines of the brackets, e, f, and G ; finally, draw the lines, s t and u v, of the table, t, and of the bracket, h ; and liliewise the extreme lines, x y, w z, of the bottom and top of the column. At this stage it is neces- sary to lay down the measurements upon the sketch. The column being fitted with the principal parts of the drilling appa- ratus, so that no clear space can be found upon it for the height to be measured close to it, a plumb-line is suspended from the point, z, on the beam, c, which rests upon the column, imd this line is measured either by a foot-rule, or by a mea^ Buring-tape. and the measurement in feet, or metres, and frac- tions thereof, may bo written upon the centre line, g h, of the column. The draughtsman must next measure tlie diameters at the base and summit of the column, as well as those of the various mould- ings. These diameters may be measured with callipers, which open to such an extent that they can be applied to the place in question, the amount of opening being then measured upon the rule, and written down upon the corresponding place in the sketch; or the diameters may be obtained by applying a cord, or a very flexible rule, to the circumference. This latter method is always employed for cylinders of large diameter, when it is not possible to obtain the measurement from either base. In this case, to obtain the actual diameter, it is necessary, as has been seen (72). to divide the circumference found, by 3-1416. To obtain the distance of the line, i j, from g h, the centre line of the column, place the extremity of the rule at i', against the surface of the column, and let it lie across the centre, i, of the spur-wheel, p, or screw, o ; the measurement read off the rule will be that of i' i, to which must be added the radius, i' t°, of the column. If the centre, ?, were not approachable with the rule, we should have to take the internal distance between the surface of the column and that of the screw, and then add the respective radii of the screw and column. When these distances are greater than the length of the measuring-rule, a rod or tape must be ' employed. When, indeed, the draughtsman has attained a reasonable amount of skill, he may take the measurement, i i^, directly, by applying the rule upon the surface of the column opposite to its centre, and also opposite to the axis of the screw, in such a manner that the rule shall be tangential to the column, when the space between the two points will be the measurement sought. It is further necessary to quote upon the sketch, the vertica distances between the different horizontal lines, m n, o p, q r, s t. The measurements indicated upon fig. 1 will show how all these are obtained. The preceding operations will allow of the finished drawing being commenced, by laying off the relative position of the main parts which go to compose the machine to be sketched. We have next to sketch and measure al) the minor details of each separate piec(i of the machine. To this effect, and to avoid confusion, it is neces- sary to treat each of these pieces as detached, and to draw different views of them, upon which the dimensions of every part may be properly indicated. Figs. 2 and 3 represent, in elevation and plan, the detail of tlio principal bracket, f, which supports the shafts, i and n, with tlie bt'vil-wheels, l and m. Even these views are not sufficient to represent thoroughly all the dimensions of this bracket; thus it is necessary to draw a section such as that made at the line, 1 — 2, and projected in fig. 4, so as to show the exact form of the feathers of the bracket ; it is likewise necessary to make a side view (fig. 5) of the bearing, b', which holds up the shaft, n, to the bracket, and also a vertical section (fig. 6) made at the line, 3 — 4, to show the brasses which embrace the journal of the shaft, I. These details should always, if possible, be drawn to a larger scale, so as to indicate the adjustments clearly, and to BOOK OF INDUSTRIAL DESIGN. 135 give room for the measurements ; and it may be observed, that, for a draughtsman who has not much practical knowledge of machine- ry details, it will be necessary to take down or separate various parts, such as the cap and the upper brass. With regard to wheel- work, it will be sufficient to give the section of the web and boss, as indicated in figs. 2 and 7, and a section, as fig. 8, of one of the arms when the wheel has any, and then the numbers of teeth and Arms must be counted and set down. When all the parts of any detail are thus sketched out in elevation, plan, or section, the draughtsman must take the measurements of each, and set them down in their appropriate positions upon the sketches, as indicated in the figures ; being mindful to see that the principal measurements coincide with those laid ofl" in the complete general view already commenced. The measurements of the diameter of the pitch-circle, and of the width of the teeth, will be sufficient, in addition to what has already been directed to be done in reference to wheel-work, the proper ratios being maintained between those in gear with each other. As many parts of machinery require to be in proportion to each other, a knowledge of such relations will enable the draughtsman to dispense with a great deal of tedious measuring and sketching, as in the case just alluded to, of wheels working together. The remaining parts of the machine are to be detailed in the same manner. Thus, figs. 9, 10, and 11, represent a vertical sec- tion, a plan, and a side view of a portion of the table, t, with its solding-jaws, and its elevating pinion and shaft. Fig. 12 is a ver- tical section of the lower extremity of the drill-stock, or spindle, n, with the drill, d, in elevation. Fig. 13 is a section of the cone- pulley, J. Figs. 14 and 15 show, in vertical and horizontal section, the manner of jointing the screw, o, into the upper end of the spindle, n. Finally, figs. 16 and 17 give a complete detail of the mechanism for elevating the table, t, as well as that for fixing or adjusting it at any required height. 397. On all the preceding details, we have quoted the measure- ments of the different parts exactly as they should be upon an actual machine. These measurements are expressed in millimetres, as in former examples, this measuring unit being adopted because its minute scale renders fractions unnecessary. We have also slightly shaded various parts, as is generally done where the com- plication and variety of forms would otherwise lead to confusion and error. Besides, in this manner, a few touches of the pencil show at once whether this or that portion is round or square, and, in many instances, the labour of drawing additional views will thereby bo dispensed with. In order to facilitate the proceedings of beginners in sketching, we would recommend them to delineate the main centre lines with the aid of a rule, and the circles with compasses, though the dimen- sions of the latter need not bo exact. This will give the sketch a much neater appearance, and render the various objccta or details more regular. It is with this view that sketchers frequently employ cross-ruled paper, with horizontal and veitical lines 0(|uaily spaced. That poition of IMate XXXV., upon which are sketehetl figs. 9, 10, and 11, is of this- description. It will 1)0 understood, that, if th(^ lines ruled upon Iho paper arc at equal distances apart, corresponding to one or more units of Iho scale to which the sketches are being m-ide, these may be drawn in correct proportions at once, in which case it will be unnecessary to write on the various measurements. The example which we have given as an introduction to the study of sketching machines, will have s^omewhat familiarized the student with his operations even now. The applications contained in the subsequent examples will suffice to complete this study, which is one of gi-eat importance to the draughtsman and construct- ive engineer. MOTIVE MACHINES. WATER-WHEELS. PLATE XXKVI. 398. The water-wheel, represented in fig. 1, has plane floats, and works, through a portion of its circumference, in a concentric cir- cular channel. It receives the water from over a sluice-gate, a little below Its centre, and is of the undershot description. The wheel is composed of several parallel shroudings, a, in which are fitted the radial wooden bearers, b, carrying the floats, C. When the shroudings are of cast-iron, as is supposed in the preseut example, they are cast in one piece with the arms, d, and centr;d boss, E, and are firmly secured by keys, a, upon the shaft, f, also of cast-iron. The head of the channel, g, which embraces the lower part of the wheel, is constructed with a piece, h, in cast-iron, called the neck-piece, which is fitted upon the cross timber, i, and let into the two lateral walls. Agamst this neck-piece works the wooden sluice, J, above which overflows a certain depth of water, falling, in succession, upon each float of the wheel as it comes round, causing it to turn in the direction of the arrow. The rotatory movement of this water-wheel is taken off by the cast-iron spur- wheel, K, mounted upon the end of the shaft, f, and gearing with the cast-iron pinion, l, the shaft of which communicates with the machinery to be set in motion. In giving this example, our object has been to examine this motor, not only with reference to its accurate delineation, but also with a view to sketching similar wheels, as well as to constructing and setting them up, with their channel and sluice gear. THE CONSTRUCTION AND SETTING UP OF THE WATER-WHEEI.. 399. The channel, G,is built up of hewn stones, the lateral joints of which converge towards the centre, o, of the wheel, imd they are imbedded upon a foundation of ordinary stone-work. All li.i- masonry is put together with mortar, made with hydraulic limo, tiie joints being finished with Roman cement. In some localities the channel is of bricks or freestone, and sometimes oven of wootl. The apparent concave surfaco of the channel should be perleclly cylindrical, and concentric with the exteruiil circunil'criMice of llio wlieil. Also, before placing the latter in iis proper posili'"), llii.t surlace should be finished, and rendered quite suioolh and true, which may be done with the assistance of a tomponiry xhal^, o. with the actual shiil't of liie wheel, in the followinfj manner: — The shaft, k (IKi)- ''^ adjusted to the exact height at wliich it i.- to bo afterwards, and it is made capable of rotation m appiv \?,r, THE PRACTICAL DRAUGHTSMAN'S piiiite bearings, adjusted upon iron plates, let into and firmly bolted to the lateral walls. Upon this shaft are fitted the shroudings, a, each connected to its boss by eight arms. To the outside of these arms are then temporarily attached two radial pieces of wood, having a cross piece attached to them, the outer edge of which is made true and parallel to the shaft, and coincident with the external edges of tlie complete wliu(il. It will be evident that, if the shaft is now made to revolve, this frame upon it will describe a cylindrical surface, which is precisely that which the channel should possess; it will serve, therefore, as an accurate guide in giving the channel its appropriate contour. The lower part of the channel is continued on in a straight line, commencing at the vertical, o b, and in a direction, b c, slightly inclined to a short distance away from the wheel, to facilitate the escape of the water. The cross timber, i, which smmounts the masonry of the chan- nel, and which receives the neck-piece, h, is also rendered concave internally, like the channel, so as to allow the sluice-gate to be brought closer up to the wheel. The neck-piece, h, which forms the crest of the channel, is more frequently constructed of cast-iron than of either wood or stone, as that material does not require to be so thick, for resisting the pressure of the water. The top of the neck-piece is at a distance below the upper water-level, corre- sponding to the greatest depth of water which it is proposed to ad- mit to the wheel at any time. This depth of water varies very considerably, according to the quantity of water to be discharged, and the width which it is wished to give the wheel. Behind the neck-piece, a cavity, m, is formed in the masonry, which is intended to receive the sluice, J, when lowered, and is of a sufficient size to allow of its being cleaned out, so that it may not become choked op with sediment. The small raised portion of masonry behind this, again, serves to arrest floating bodies, as trees, &c., independ- ently of a grating placed further behind, and preventing their re- acting, and injuring the wheel. The sluice consists of two strong oaken planks, having grooved and tongued joints, and being made thicker at the middle than at the extremities, where the wheel is of a greater width than 1| m&tre. The amount of inclination of this sluice, J, is determined by drawing a perpendicular to the extremity of the radius, o f, drawn near the middle, or, perhaps, two-thirds of the depth of the overflowing body of water. The sluice is moveable in grooves, in two wooden side-posts, n, entirely imbedded in the lateral walls. At the upper parts of these are iron bearing-pieces, to receive two straight cast-iron racks, o, which rise above the cross timber, p, attached to the two side-posts, n. These racks rest, on one side, upon the friction-pulleys, h, which also guide them, and, on the other side, they gear \vith the pinions, g, keyed upon one horizon- tal axis. This latter is at one end prolonged, to receive the worm- wheel, Q, actuated by the worm, e, which may be worked at pleasure from above — a winch-handle, or hand-wheel, being fixed upon the upper extremity of its vertical spindle for this purpose. This arrangement permits of the regulation of the position of the ■'luice, and, consequently, of the depth of the overflowing water, with the greatest nicety, as well as of the total shutting off" of the water from the wheel. The shrouding, a, of the wheel, being of cast-iron, the weight has been reduced, by making panels in it, as at h, i, shown in the elevation, fig. 1, and section, fig. 2, made at the circular line, 1 — 2. It is also cast with mortises, to receive the tenons, or ends, of the carrier-pieces, b, to which the floats are bolted. When the wheel has counter-floats, as represented at the lower part of figure 1, which is only the case when the discharge, and consequently the depth, of water at the sluice-gate is very small, the carrier-pieces are very short, and do not project far upon the inner side of the shrouding. But when the wheel is without counter-floats, which is the case when the discharge, and conse- quently the depth, of water at the sluice is considerable, the floats, _ c, and theu- carrier-pieces, b, are prolonged to a considerable dis- I tance inside the shroudings, as has been supposed to be the case in the upper part of fig. 1. In both cases, the tenons, or ends, of the carrier-pieces, always lie in the direction of radii from the centre of the wheel, and they are retained by iron-keys, y, upon the inside M of the shroudings. Sometimes, in order to facilitate the adjust- ^ ment of the carrier-pieces upon the shroudings, in place of fitting them into closed mortises, they are received into slightly dovetailed recesses, formed upon the side, as shown in figs. 3 and 4, being retained in position by wedges, j. When this last arrangement is adopted, it is unnecessary to cut holes in the carrier-pieces for the reception of the keys. When the shroudings are of wood, they must necessarily be composed of several pieces, which are fitted together with mortise and tenon joints, as shown in figs. 5 and 6 ; and to consolidate the joint, iron straps, k, are added, secured at one side by bolts, and at the other by keys, or tightening screws, bj' means of which the perfect union of the component pieces can at all times be obtained, should they begin to get loose. In this system, the carrier-pieces are adjusted with tenons, keyed on the inside of the shrouding, as indicated in figs. 7 and 8, and the oaken arms are joined to the shrouding with tenons, being further secured by iron straps, as shown in figs. 9 and 10. The floats, c, of the wheel, are formed of oaken boards, and are attached to the carrier-pieces, b, by means of the bolts, 1. The counter-floats, s, extend from the inner ends of the floats, c, to the bottom pieces, s', and are nailed down upon the small ti'iangular pieces, m. The open spaces left between the ends of the floats and the bottom pieces serve for the escape of the air. When the floats are lengthened, they ai-e, of course, formed of several boards, joined edge to edge. DELINEATION OF THE WATER-WHEEL. 400. The explanations just given will have enabled the student to comprehend the details and peculiarities of construction of the wheel, channel, and sluice apparatus. He should now proceed to delineate these various objects in the following manner : — Place the centre, o, of the wheel, at the intersection of two lines which form a right angle ; and with this centre, describe a first circle, with a radius equal to that of the wheel and channel. Divide this circle into as many equal parts as there are to b& floats. The number of the floats should always be divisible by that of the arms of the shrouding, so as not to be restricted as to space m fittino- in the carrier-pieces. Through each point of division draw BOOK OF INDUSTRIAL DESIGN. 137 lines passing through the centre, and representing the sides of the carrier-pieces upon which each float is placed. Two circles must next be described, expressing the depth of the shrouding. Then the complete outline of one of the carrier-pieces must be drawn, with the dimensions quoted on the figure ; and the key and bolts may also be indicated upon it. Afterwards, to complete the drawing, it will be sufficient to describe a series of circles, passing through the bolts, the ends of the floats and canier-pieces of the key, and of the counter-float. With regard to the floats, and to the arms of the shrouding, as well as to the spur-gear for trans- mitting the motion, the student may refer back to the diagrams and explanations already given concerning similar objects. The same remark applies to the lifting apparatus of the sluice-gate, which is also composed of gearing already treated of in the course of the studies. DESIGN FOR A WATER-WHEEL. 401. If it is in contemplation to make a design for the con- struction of a water-wheel, analogous, we shall suppose, to the one above described, it is simply necessary to ascertain the height of fall, and the amount of discharge per second, of the water at our disposal, and to refer to the calculations and practical rules which accompany our text, to be able to determine, on the one hand, the diameter and width of the wheel, and, on the other, the depth and interstices of the floats, and their number. By referring back, also, to the tables and notes relating to the resistance of materials (Chapter III.), we shall be able to complete the remaining dimen- sions for the shaft and its journals, the shrouding and its arms. The study of water-wheels of this description will be much sim- plified, if we consider that certain dimensions, such as the thickness of the floats, the section of the carrier-pieces and shrouding, and the diameter of the bolts, as well as the details of the sluice appa- ratus, do not sensibly vary ; and for them the draughtsman may refer entuely to those indicated upon the drawing, which are themselves examples of actual construction. SKETCH OF A WATER-WHEEL. 402. The sketch of a water-wheel, already constructed and set up, is a very simple matter ; for the apparatus consists of a repeti- tion of various pieces, and it is sufficient to obtain the measure- ments of one only of each kind. Thus, after having measured the diameter and extreme width of the wheel wilh the aid of a long rule or tape, and counted the number of buckets or floats, of the shroudings, and of the arms, we have merely to take Iho sketch of a single float, with its carrier-piece and accompaniments, then to make a section of one of the shroudings, another of one of the arms, and, finally, a third of the boss and shaft. Th(^ details given in figs. 2 to 10 show the various jiarts of vvliicli llic sketches have to bo made, as detached, togcllicr with Iho I'.orrcNponding measurements. Fig. 20 is a transverse section of one of the arms, d, of cast-iron, taken near the boss. The sketching of the sluice apparatus consists in making ii Rcction of the side-posts, with their cap-piece, and of tiie sluice itnelf ; then a detailed view of one of the racks, with its pinion and fVictioii-pulley, and of the worm-wheel and worm. As to the Hinoiitit of inclioiition of the sluice and side-posts, it has already been seen that it is determined by a perpendicular to the radius, entering near the middle of the depth of water at the outlet, at the circumference of the wheel. It may, however, be found by means of a plumb-line, let fall from one of the edges of the cap- piece down to the level of the water, by measuring the horizontal distance, r s, of the plumb-line, from one of the sides of the side- post, and then the vertical height, r t. By applying a rule against the side-post, and down to the neck-piece, h. we can always obtain the actual distance of the top of the latter, either from the pro- longation of the horizontal, r s, "r from the caji-piece, p, of the sluice. To obtain the horizontal distance, r s, with exactitude, it should generally be taken at a given distance above the level of the water, and chalked upon one of the side-walls of ihe channel ; it is also advisable to make use of a spirit-level. (Plate I.) In order to take an accurate sketch of the neck-piece and the channel, it is almost always necessary to stop the water behind by means of a dam, so that the parts requiring to be examined may be dry and open. The sluice must also be taken away, as well as a few of the floats of the wheel. We may remark, that this labour may be avoided, when it is known that the height and thickness of the neck-piece are nearly always equal to those indi- cated in fig. 11; and as to the arrangement of the masonry or brickwork, of which the channel may be constructed, it will bo recollected that all the lateral joints are pointed towards the centre of the wheel. OVERSHOT WATER-WHEEL. Figure 12. construction of the wheel, and its sluice apparatus. 403. Overshot water-wheels, with buckets, receive the water from a duct placed immediately above them, and allow it to escape from as low a part only as possible. They are constructed of wood, or of cast-iron. In the first case, which is the simi>ler and more economical, the shaft, the arms, and the shroudings are of oak. The lower part of the wheel, represented in the drawing, fig. 12, is of this description. The buckets and the inner-rim are likewise of oak, or of iron plates. As this wheel is of small dia- meter, its shaft, F, has only six sides ; and consequently, each shrouding, a, of the wheel has only six arms, D, which are recessed into, and bolted ui)an, a central cast-iron frame, e, which is itself keyed upon the shaft. The transverse section, (ig. 13, shows fiio manner in which the arms arc attached to this I'ranu'. The wouitoti shroudings, a, are generally coniposeil of two rings, jjlaccd one on the other in such a manner that the joints of each are opposite lo solid ])ortions ol" the other, to "break bond," and obvi.-ite the teu- dency to wari). A portion of the shrouiiing is represented as de- tached in ligs. 14 and 15. The-so rings are held together by sitcws, r, or bv nails or pegs ; and at their junction with the anus, a couple of bolts are passed throiigh all, as indicated in the transverse si>c- tion, fig. 16. The buckets, c, are either let into grooves of sniai. depth, upon the inrur l':ue of the slirouding, as seen at c',in figs. 14 and 1.5, or they are retained by bracket-pieces, c: and, nddt^l ti> fliiii. strong feiision-nnls, li, hold the w hole together, being .s»vurwl lo fiS THE PRACTICAL DRAUGHTSMAN'S the shroudingrs, a, on either side of tlie buckets. These tension- rods are fixed, when the inner rim, s, or bottom of the buckets, has l)een nailed or screwed to the inner edges of the shroudings. The shroudings are further strengthened externally by a circular iron strap, G, similar to the felloe of an ordinary wheel, and covering up the joints of the duplex shrouding. Sometimes the buckets are partly of wood., and partly of iron plate, to give them greater strength. The edges, indeed, should always be defended with metal, as they arc most apt to wear soon. The lower portion of the di-awing, fig. 12, shows three dilibrent ways of constructing these buckets. When the wheel is of cast-iron, if it is not of a very givat di.-i- raeter, but of the size represented in the upper part of fig. 12, the arms and the boss, e', may be cast in one piece \\ itii the shroua. ings, a'. Where the diameter is considerable, these consist of several pieces bolted together. The bottom piece, or inside rim, s', and the buckets, c', are of iron plates, of about Jth inch in thick- ness. To secure the Mter, a series of feathers, figs. 12, 17, and 19, are cast upon the inner faces of the shroudings, to which they are fixed by screw-bolts, I In the width of the wheel, the buckets and the bottom rim are riveted together, as at t, or are fixed together by small screw-bolts, i', figs. 17 and 18. The advantage of making the buckets of iron plates, consists in the being able to give them a curved form, which enlarges their capacity, and allows of a more favourable introduction of the water; whilst the w"ooden buckets necessarily consist of two rectilinear portions, one of which is directed towards the centre of the wheel, whilst the other is inclined. The water is conducted by the wooden channel, ."vi, to the top of the W'heel, and its outflow is regulated by a sluice, j, moving in side gi-ooves, and worked by means of a couple of vertical racks, o, and pinions, d, the shaft of which last carries the winch-handle, q. The tW'O vertical sides, n, of the channel, are prolonged beyond the actual summit of the wheel, and their distance asunder should be a little less than that of the tw'o shroudings of the wheel, with the twofold object of better directing the water into the buckets, and of avoiding the splashing and loss of water by allowing the air to escape laterally. The depth of the outflow of water depends on the distance of the lower edge of the sluice above the bottom of the channel, and should always be less than the smallest distance existing between tw^o consecutive buckets. The pressure of the w-ater upon the buckets, produces the rotation of the wheel in the direction of the arrow, and this motion is given off by an internally- toothed wheel, attached to the outside of one of the shroudincrs. ''^his wheel, which in the drawing is simply indicated by its pitch circle, k, goars with the pinion, l, mounted on the extremity of the shaft, which communicates with the machinery in the interior of the factory or workshop. DELINEATING, SKETCHING, AKD DESIGNIXG 0\"ERSHOT -WATER- ■WrSEELS. 404. The delineation of the principal parts of an overshot Ducket water-wheel, is effected in the same w^ay as that of the undershot wheel with floats, the only difference being, in fact in the receptacles for the water. It has been seen, that when these buckets arc of wood, they are composed of two boards, one of w hich lies in the direction of a radius of the wheel, the other being inclined according to the direction of the water, and make an angle of 15 or 30 degrees, as the case maybe, with the tangent, to the outer circumference of the wheel, drawn through its extremity, as will be seen by the angle, a 6 c, in fig. 17. When the bucket is made of iron plates, the same angle is adopted near the outer edge, although the whole contour is a continuous curve, which may be made up of two or three arcs of circles, as show"n in figs. 12, 17, and 18. In sketching this wheel, the directions given in the preceding case n?ay be likewise followed here, by counting the number of buckets, and taking an accurate sketch of one of them, together with the accompanying measurements. We must also measure the internal and external diameters of the sliroudings ; then the least space existing between two consecutive buckets ," also the depth from b to d, fig. 17. Finally, if it is required to obtain the exact form or curvature of the bucket, it will be necessary to take one down, and to make a pattern of it, by applying a sheet of paper against one edge, and pencilling out the shape, as is done for the forms of wheel-teeth, or other curves, which are difficult to measure. As to the sketch of the other parts of the wheel, such as the boss, the arms, and also the sluice apparatus, no peculiarity or difficulty can present itself which need detain us here. The drawing, moreover, indicates all the figures and measurements which are necessary. In designing an overshot water-wheel, it is necessary to know the height of fall, and the daily discharge of the water. With regard to these particulars, we must simply refer to our accomna- nying Rules and Practical Data. WATER-PUMPS. PLATE XXXYII. GEOMETRICAL DELINEATION. 405. We have already indicated, in preceding notes and calcul* tions, the various classes of pumps, with their proper dimensions, in proportion to the quantities of water to be flimished by them. We now propose to enter upon more detailed and complete expla- nations, with regard to their construction, action, and performance. For this purpose we have selected, by preference, a combined lifting and forcing pump, the discharge of which is ahnost continu- ous, although its construction is analogous to what is termed a single-acting pump. Figure 1, on Plate XXXVU., represents a vertical sectioii. taken through the axis of this pump. It consists of a cast-iron cylinder, a, turned out for the greater portion of its length, and resting upon a feathered base, b, cast in one piece with the suction or lift-pipe, c, below. This base is bolted down either to stout timbers, d, or to a stonew'ork foundation. It encloses the valve- seat, E, which consists of a rectangular frame, divided by a central partition, a, and having the sides formed so as to present two in clined edges, upon which the brass clack, f, rests when shut. The pipe, c, terminates below" in a flange, by means of which the suction-pipe is attached, extending down to the water to be ele- vated. Towards the upper part of the pump cylinder, a, is cast BOOK OF INDUSTRIAL DESIGN. 139 a curved outlet, g, likewise terminating in flanges, to which the discharge-pipe is secured. The piston, or bucket, of this pump is composed of a brass ring, or short cylinder, h, upon the outer cir- cumference of which is formed a groove, b, (fig. 2,) to receive a packing-ring, c, which fits, air-tight, to the inside of the pump cylinder. The bucket, h, has also a central partition, d, to the top of which are jointed the two clacks, i, which rest upon inclined seats, formed by the elevated sides, e, of the bucket. This is further cast with a bridle,/, perforated in the middle, to receive the screw-bolt, g, which secures it to the stout hollow piston-rod, j. This rod, which, in the generality of pumps, is made of but small diameter, like the upper part, k, of the one represented in the plate, is, in the present instance, of a sectional area, equal to half that of the pump .•ylinder. It follows from this, as will be more particularly ex- ,)!;dned further on, that the water is discharged during both the up m J down stroke of the piston. The chicks, F, have projections, h, cast upon them, which pre- vent their opening too far, and falling over against the sides of the casing, b, so as not to shut again when required to do so. The clacks, i, in the bucket, have similar projections, i, for a like purpose, these projections striking against the top of the bridle,/, when the clacks open. It will have been observed, that the seats of these valves are inclined at an angle of 45°, with the view of facilitating their opening movement, and diminishing the concus- sive action of their own weight. The edges of the valve-seats are generally defended with a strip of leather, to facilitate their tight closing. Figure 2 represents, detached and in elevation, the bucket, h, with its clacks, i. Fig. 3 is a horizontal section of the bucket, taken at the line, 1 — 2. Figs. 4 and 5 give the details of the valve-seat, E, in elevation and plan, the clacks being removed. To prevent the entrance of air to the pump cylinder, it is closed at the top by a cast-iron cover, l, which is fitted with a stuffing- box for the passage of the piston-rod ; the packing is compressed by the gland, m, similar in general form to that represented in Plate XI. (81.) ACTION OF THE PUMP. 406. The upper extremity of the piston-rod, k, carries a cross- head, I, (fig. 6,) and is there jointed to the lower extremity of a connecting-rod, n, which is itself jointed to the pin of a crank, o ; this latter is mounted on the end of a horizontal shaft, p, actuated by a continuous rotatory movement. This movement is trans- formed by means of the crank and connecting-rod into an alternate rectilinear motion — that is, into the up-and-down strokes of Ihe pump bucket — this last being forced to move in a straight line, the cross-head, I, sliding in vertical guide-grooves, to maintain the piston-rod, k, in the same line with it. It follows, from this disposition of parts, that when the crank, 0, is in the position, r — o, fig. 6, the piston will be at the bottom of its stroke, that is, at ii' ; consequently, during (he tinu^ tlu^ crank turns, the piston must ris(^ tending to leave a vacuum below it, because the spac(! I)etween the clacks, f, and its under side increases, as well as the vohimi^ of air that may be Ihercin en- closed. Consequently, the pressure of this air upon the clacks is diminished, whilst that upon the surface of the water remains the same, and causes the water to rise up the suction-pipe, and, raising the claclcs, f, to enter the pump cylinder, filling it up nearly to the under side of the piston ; or if the apparatus is in a perfectly air-tight condition, it will rise quite up to the piston. When the crank has reached the position, p — 12, — that is, when it shall have described a semi-revolution, — the piston itself will likewise be at the highest point of its stroke, and, in this position, all the space left behind it in the body of the pump will be filled with water; if now the crank, continuing its rotation, makes a second semi-revolution, the piston will descend, and, pressing upon the water below it, will cause the clacks, f, to shut. Now, as the water is incompressible, it must find an e.xit, or else prevent the descent of the piston ; and it therefore raises the bucket-clacks, i, thus opening up tor itself a passage through the piston, h, above which it then lodges. But as the piston-rod, J, is of a large diameter, and therefore occupies a considerable space in the pump cylinder, a part of the water must necessarily escape through the outlet, g, in such a manner that, when the piston shall have reached the bottom of its stroke, there will not remain in the pump cylinder more than half the quantity of water which was contained in it when the piston was at the top of its stroke. Such is the effect produced by the first turn of the crank, which corresponds to a double stroke of the piston — that is, an ascent and a descent. At the second turn, when the piston again rises, it sucks up, as it were, anew, a volume of water about equal to the length of cylinder through which it passes, because the suction-clacks, f. which were shut, now open again, and the bucket-clacks, i, whiuli were open during the descent, are now shut by the upward move- ment of the piston. During this stroke, all the water which previously remained above the piston, finds itself forced to pass off through the pipe, g, so that, with this arrangement of piston and rod, or plunger, of large diameter, it follows that, at each up-stroke of the piston, the quan- tity of water which rises into the pump is equal to the length of cylinder through which the piston passes, the half of which quantity rises in the discharge-pipe during the descent, and the o.hcr half during the subsequent ascent of the piston, and the jet is conse- quently rendered almost continuous and uniform. When, or the contrary, the piston-rod is made very small in diameter, as in ordinary pumps (fig. 6), the discharge of the water only takes place dui'ing the ascent of the piston, and it is conse- quently intermittent. In a pump, as in all other machines in wli'eli an alternate ree- tilinear is derived from a continuous riitatory motion, by means of a crank and connecting-rod, the spaces passed through in a straight line by the i)iston do not correspond to the angular spaces described by the crank-pin ; in fact, it will be soeu I'roiu the diagram, lig. (i, that if Ihe crank-pin is sujjposed to desoribi' a series of etpial arcs, beginning from the point, 0, the corn>spond- ing distances, 0' 1', 1' 2', 2' 3', jiassed iIuouliIi bv the piston will not he unil'orm ; very small al llie roinmeiieemeiil o\' tlio slroko, (hey will gradually increa.-e towards llu' middle, alter im-ssiu^; whii-li they will similarly decrease whilst the piston approaches 111,, olher end ol its stroke. The slU'COSsivO posltior)s of iho 140 THE PRACTICAL DRAUGHTSMAN'S piston may bo obtained by describing with eacli of tlio points, 1, 2, 3, 4, upon the circumference traced by the crank-pin as cen- tres, and with a radius equal to the length of the connecting-rod, a series of arcs or circles cutting the vertical, passing through the centre, p, in the points, 0^ l^ 2', which indicate upon this line the various po-itions of the point of attachment, Z, of the connecting- rod to the piston-rod: these points are then repeated at 0', 1', 2', on the same line, at distances from the points, 0^ 1", 2^ equal to the length of the piston-rod, measured from the point, I, to the bottom of the piston. It will be easily understood, that, in consequence of this irre- gularity In the motion of the piston, the force and volume of tlio jet of water will vary throughout the whole stroke. We have endeavoured to show the nature of this variation in the diagiam, fig. 7, which represents the comparative volumes of the jet of water at successive periods for a single-acting pmnp, such as the one in fig. 6. This diagram is constructed by laying off upon any line, x y. as many equal parts as we have taken in divisions on the circle de- scribed by the crank-pin; then through each of these points, as 1, -J, 3, 4, drawing perpendiculars to x y. As during the ascent of ilie piston from to 12 (figs. 6 and 7) there is no discharge, as the piston only sucks up the water, there is nothing to indicate upon these first divisions ; as soon, however, as the crank-pin passes the highest point, and the piston begins to descend, it will pro- duce the jet of water ; it is considered then, that when it has passed through the first rectilinear space, 12' to 11', the quantity of water forced out by it may be represented by its base multiplied by the height, 11' — 12'. It is this distance which is set off from 13 to a, upon the perpendicular drawn through the point, 13; in the same manner, when the piston descends from 11' to 10', it is also taken as represented by its base multiplied by the height, 11' — 10', which last is therefore set off from the point, 14 to b. It ^fill be seen from this, that, in proceeding with the diagram, it is simply necessary to set off upon each of the perpendiculars drawn thi-ough the points of division, 15, 16, 17, the successive distances passed through by the piston during its descent, so as to represent intelligibly the actual volumes of water discharged for each f)ori:ion of the stroke, since these volumes are proportional to the distances passed through by the piston, the section of the cylinder remaining constant. If, through the various points, a, b, c, d, fig. 7, obtained in this manner, we trace a curve, we shall obtain the outline of a surface which we have distinguished by a flat shade, and which will give a good idea of the amounts of water discharged in correspond-'nce with any position of the crank. On continuing the rotation of the crank, the piston next ascends and sucks up the water, consequently the jet of water is interrupted during this up- stroke, but recommences on the down-stroke due to the sub- sequent part of the revolution; the quantity of water then discharged is indicated in fig. 7, by a curve equal to the first, and on which the same points are distinguished by the same letters. To avoid this irregularity in the discharge, pumping apparatus is sometimes constructed with two, or with three, distinct cylinders, in which the disposition of the pistons is such, that the points of attachment to the several crank-pins divide the circle described by them into two or three equal parts. Figure 8 represents a geometrical diagram of the performance of a two-cylinder pump ; it is evident that the product of each of the pistons is alternately the same, since one descends whilst the other rises ; it is thus that one of the pistons, having produced a jet corresponding to, and expressed by, the curve, a' b' d, the other one immediately afterwards produces a jet, expressed by the curve, abed; so that this diagram only differs from fig. 7, in that the unoccupied intervals, from to 12, and 24 to 12, in the latter, are in the former filled up by an equal figure, covered by an equal flat shade. This diagram of the performance of a two-cylinder pump may also be considered as representing that of the pimap, fig. 1, which, because of its trunk piston-rod, acts as a double-acting pump, as already explained. Fig. 9 represents the diagram of the performance of a three- cylinder pump, of which the pistons, h, h', h^ represented, for convenience' sake, as in the same cylinder, occupy the positions corresponding to those of the three crank-pins, 0, 0', 0", as placed at the angles of an equilateral triangle, inscribed in the circle described by them with the centre, p. In consequence of this disposition, there are at one time two pistons ascending and one descending, and at another time, on the contrary, only one ascending and two desi^ending. It is easy to represent the com- bined performance of these punips in a diagram, by using diffeieut colours, or different dep.lis of shade, for the performance of ci.vli, as dependent upon the siiccf^sive pcisitions, 1, 2, i^, 4, taken up by U'lcir successive crank-pins. By this means all contusion v\iii uo avoided, and it will be necessary to find the positions, n^ n', k", of the attachment of the connecting-rod to the piston-rod upon the vertical line passing through the centre, p, only as for one cylinder, as the distances wDl be the same for all, being merely placed at different parts of the diagram. In the diagram, fig. 10, we have laid down the performance of each of the three pumps, supposing them all to be of the same diameter, and taking care, when two pumps are discharging to- gether, to add together their performance ; thus, for example, when one of the pistons elevates a quantity of water, corresponding to the perpendicular, 13 a, that which is also discharging at the same time furnishes a quantity expressed by the distance, a a' ; conse- quently, the total volume of the discharge at this instant is repre- sented by the total height, 13a' ; when, on the other hand, only one of the three pumps discharges, whilst the pistons of the other two are ascending, as in fig. 9, the volume discharged is represent(;d by a single length of perpendicular, such as 18/. Now, it will be observed, that it is precisely at the moment when only one pump is discharging that it gives out its maximum performance ; from which it follows, that the jet of water is continuous, and almost uniform throughout its duration, as will be very evident from a consideration of the diagram, fig. 10, the outline of which is detei- mined by perpendiculars, or ordinates, reaching nearly to the straight line, m n, throughout. To compare the combined effect of a three-cylinder pump with that of two or of three double-acting pumps, we have, in figs. 8 and 11, repeated the corresponding diagranas for the -wo last arranirf BOOK OF INDUSTRIAL DESIGN. 141 ments ; and it \vill be remarked, that although, with cylinders of an equal sectional area, we necessarily obtain a much larger dis- charge, yet the regularity of volume is not so great as in the previous example. STEAM MOTORS. HIGH-PRESSURE EXPANSIVE STEAM-ENGINE. Plates XXXVIIL, XXXIX., and XL. 407. When the steam generated in a boiler is led into a vase or cylinder which is hermetically closed, it acts with its entire expansive foi-ce upon the sides and ends of the cylinder, so that, if this en- closes a diaphragm, or piston, capable of moving through the cylin- dei- in an air-tight manner, the force of the steam, in seeking to enlaige its volume, will make the piston ino\c. It is in this way that a mechanical effect is derived from the expansive action of the steam, and it is on the same principle that the generality of steam- engines are constructed. Thus, in most apparatus to which this name is given, the action of the steam is caused to exert itself alternately on the upper and under surface of the piston, enclosed in the cylinder, thereby caus- ing it to make a rectilinear back and forward movement or stroke. (187.) Steam-engines are said to be low or high pressure engines, according as the tension of the steam is only of about 1 atmosphere on the one hand, or of 2, 3, and upwards, on the other. Low- pressure engines are generally condensing engines, and high-pres- sure ones non-condensing ; so that the terms, low pressure or con- densing, high pressure or non-condensing, are used indiscriminately, although, in modern engineering practice, what are called high- pressure condensing engines are extensively employed. When the steam is made to act alternately above and below the piston, the engine is said to be double-acting ; and of this description are most of those employed at the present day; but if the steam acts only on one side of the piston, as is the case in many mine- pumping engines, the engine is called a single-acting one. Low-pressure engines are generally also condensing engines; that is to say, that after the steam has exerted its expansive action upon the piston, and is on its way out of the cylinder, it passes into a chamber immersed in cold water, and termed a condenser, whore it is condensed or reduced to the state of water. This condensa- tion produces a partial vacuum in the cylinder, and consequently considerably diminishes the resistsince to the movement of tlio piston. In high-pressure engines, the steam which has produced its effect upon the piston escapes directly to the atmosphere, so that the pis- ton has always to overcome a resistance equal to one atmosphere, or about 15 lbs. per square inch, acting in a direction opposite to its motion. Steam-engines are further distinguished as expansive and non- exjiaiisivo ; of Iho hitter descripfion arc those wherein the steam enters the cylinder throughout the entire stroke of the piston ; so that the pressure is uniform, since the volume of steam of a given pressure which enters is always equal to the space passed through by the piston. In expansive engines, on the contraiy, the steam is only allowed to enter the cylinder during a portion of the stroke ; so that the expansive power of the steam is called into action during the remainder of the movement. The machine detailed in Plates XXXVIIL, XXXIX., and XL., is a high-pressure engine, with a variable expansion valve. Fig. 1, Plate XXXVIIL, represents an external elevation or front view of the machine, the frame of which consists of a hollow column, with lateral openings. Fig. 2 is a horizontal section, taken at the height of the line, 1—2. Fig. 3 is an elevation of a fragment of the lower part of the column. Fig. 4 is another horizontal section, taken at the line, 3 — 4; and fig. 6 is an elevation of the capital of the column. Figs. 6 and 7 are diagrams, relating to the movemeiit of the governor, with its balls. Fig. 8, Plate XXXIX., represents a vertical section, taken through the axes of the column and the steam cylinder, at the plane, 5 — 6, parallel to that of the fly-wheel. Fig. 9 is another vertical section, at right angles to the preceding figure. And, finally, fig. 10 is a horizontal section, taken at the broken line, 7 — 8—9—10. This machine consists of a cast-u-on cylinder, a, truly ooied out, and enclosing the piston, b. On one side of the cylinder are i ast the passages, a, b, by which the steam enters alternately above :.nd below the piston. These passages are successively covered over by a cup or valve, d, the details of which are given in figs. 28 to 31, Plate XL.; and the valve is itself contained in the cast-iron chamber, e, called the valve casing, and communicating witn a second chamber, F, called the expansion-valie casing ; it is imo this latter chamber that the steam is first conducted by the pipe, G, from the boiler. The communication between the two vah u ca!-ings is intercepted for short periods during the action of tlu machine, by the expansion valve, h, detiiiled in figs. 38 to 41' Plate XL. The vertical rod, i, of the piston, b, is attached at its upper ex- tremity to a short cross pin, e\ which connects it to the wrouglit- iron connecting-rod, j, hung on the pin, /, of the crank, k ; this is adjusted and keyed upon the extremity of the horizontal shaft, i., which carries on one side the fly-wheel, M, and on the other the eccentrics, n, o, p. The first of these eccentrics is intended to ac- tuate the distributing valve, d, the rod, 5-, of which is connected to it by the intermediate adjustable rod, n'. The second works tlio exi)ansion valve, h, by means of the rods, 0' and h ; and, tinally. Iiu« third eccentric, i", gives an altern;ite movement to the i>i8ton or plunger, q, of the feed-pump, u. The steam cylinder is bolti'd in a tirm and .solid manner, by its uj)per flanges, to the top of the hollow cast-iron plinlii or (h-- dostal, s, on which also rests, and is bolted, tJie column, r. 'l'lu> pedestal is square ; and, at the corners of its base, lugs are cast, by means of which it is lirmly bolted down to 11 solid stone foniulaliori. The column, T, is cast liollnu, ami wilh four large lateral open- ings diametrically opposite to each other, llnlr obitvt W'inji t« 142 THE PRACTICAL DRAUGHTSMAN'S diminish the weight of the column, and to afford the necessary passages for the introduction of the various pieces when being put together, or when talcen down. This column also serves as a f'lame for the entire machine, and above the capital is placed a cast- iron pillow-block, u, furnished with bearing brasses to receive the principal journal of the first motion shaft, as well as the supporting brackets, k, k', of the spindle, /, of the ball governor. To its inner side are also bolted the two supports, i, of the parallel motion, and guide, y, of the valve-rod, g. ACTION OF THE MACHINE. 408. Before proceeding further, we shall give some idea of the general action of the machine. As already mentioned, the steam is generated in a boiler, such, for example, as that represented in Plate XIV. (189), and is conducted by the steam-pipe, g, into liie first chamber, f ; when the valve, h, in this chamber uncovers tne orifice, or port, d, the steam finds its way into the valve-casing, E, whence it passes either to the upper or to the lower end of the cylinder, accordingly as the valve, d, uncovers one or other of the two ports or passages, a, b. Now, when the piston is, for example, at the top of its stroke, the passage, a, is almost fully open, whilst the channel, b, is in communication with the exit orifice, c, from which the two pipes, e*, conduct it to the atmo- sphere. If, on the introduction of the steam to the cylinder, it has a pressure of, say four atmospheres, it follows that it will act upon the piston with all this force to cause it to descend ; since, how- ever, the lower part of the cylinder is at this time in communica- ■,ion with the external atmosphere, there is a resistance equal to jne atmosphere opposed to its movement, therefore the actual effective pressure acting on the top of the piston will be equal to three atmospheres. It is the same when the piston reascends ; the valve uncovers the port of the passage, b, to allow the steam to enter the lower end of the cylinder, whilst the port, a, is put in communication with the exit orifice, c, by the cup of the valve, to give an outlet for the steam which has just acted on the upper side of the piston during the down-stroke. It is to be remarked, that if the introduction of the steam takes place during the entire up-and-down stroke of the piston, which might be the case if the steam-pipe, g, communicated directly with the valve-casing, e, and if the valve, n, kept one of the ports unco- vered throughout the entire stroke, the pressure of the steam would remain constant ; in such case, it would be said that the machine was a high-pressure non-expansive engine — that is to say, that it vvorked with a full allowance of steam. In the machine, however, which at present occupies our atten- tion, the steam first introduces itself into the casing, f, the valve, H, of which, at each stroke, closes the passage, d, communicating vvith the second casing, e, before the piston reaches either end of (ts stroke. It follows, that the steam contained in the cylinder at the time of closing the passage, d, must augment in volume or expand, whilst its pressure will consequently decrease during the remaining advance of the piston: the engine is then said to be working expansively ; and in this case a quantity of steam is » upended for each stroke, equal only to a third, half, or two- tiiirds of the capacity of the cylinder, according as the intro- duction of the steam is intercepted at one-third, one-half, or two- thirds of the stroke ; it is the ratio between the quantity of full steam-pressure introduced, and the entire capacity of the cylin- der, which expresses the degree of expansion at which the engine works. PARALLEL MOTION. 409. The rectilinear alternate movement of the piston is trans- formed into a continuous circular motion on the first motion shaft, L, by the intervention of the connecting-rod, j, and crank, ic; but with this arrangement there is naturally a lateral strain upon the top of the piston-rod, i, and in order that its movement may be perfectly rectilinear and vertical, it is jointed to a system of articu- lated levers, forming what is termed a parallel motion. This mechanism is composed of two wrought iron rods, v (figs 1, 4, and 8), which oscillate on the fixed centres, i, and are articu- lated at their opposite extremities to the levers, x, near their middle, by means of the pin, n. The levers, x, are also of wrought-iron. and are jointed at one end to the cross pin, e", fig. 9, of the piston- rod end ; and at the other to the rod, y, attached to a cross spindle, 0, and oscillating in bearings, in a couple of cast-u-on brackets, z, bolted to the lower part of the frame. The head of this last-mentioned oscillating rod is detailed sepa- rately, in figs. 21 and 22, Plate XL. It has brasses, to embrace the journal of the spindle, p*, by which it is connected to the ends of the levers, x. The combination of this mechanism is such, that the point of attachment, e, constantly moves in a straight line throughout the entire stroke. It may be designed on geometrical principles, as indicated in the diagrams, figs. 8 and 11. To this effect we have supposed, that after having drawn the horizontal line, e" p, and the vertical, e e', distances, e e* and e e^ are set off on the latter, equal to the half stroke of the piston, or to the radius of the crank; theni with the points, e e^ describe an arc, with a radius, e p, equal to the length of the lever, x, which is taken at pleasure, but should never be less than the stroke of the piston. If we next lay off this distance from e" to p\ the space, p p', will express the amount of oscillation of the rod, y, the centre of oscillation, o, of which we place below, on the vertical line, drawn at an equal distance from and between the two points, p, p^. We next fix the point, n, of attachment of the rods, v, to the lever, x. This point, n, during the movement of the parallel motion, necessarily describes a circu- lar arc, of which it is requisite to find the centre. In investigating this problem, it is to be observed that, whatever may be the posi- tion of the lever, the point, n, is always at an equal distance from the extremity, p, or the other one, e. If, then, we in succession draw the lines, p e, p' e', /^^ e", p^ e^ indicating the different posi- tions of the lever, corresponding to those, e, e\ e', e', of the piston, rod end, we shall, on each of these lines, obtain the several posi. tions, n, n', ?^^ ?i', by laying off on them either of the distances, p n or e n. We can then very easily find the centre of the arc passing through these points. (10.) » Fig. 10 represents the diagram of an analogous parallel motion, but one in which the rods, v, are so disposed, that their point of attachment is exactly in the middle of the levers, x ; ar d in this BOOK OF INDUSTRIAL DESIGN. 143 case, their axis of oscillation lies in a plane passing through the vertical axis, e e^. DETAILS OF CONSTRUCTION. ' STEAM CYLINDER. 410. The cylinder is cast in one piece with its bottom cover and lateral steam passages. As it should be bored with great care, so as !o be perfectly cylindrical in the interior, a central opening is made in the bottom for the passage of the spindle of the boring tool; this opening, however, is afterwards closed by the small cover, a', cemented at its junction surfaces, and bolted down to the bottom of the cylinder. The upper end of the cylinder is closed by a cast-iron cover, a', which is formed into a stuffing-box in the centre, to embrace the piston-rod, which works steam-tight through it. The packing is compressed or forced down for this purpose by a gland (140), bolted to the stuffing-box, and hollowed out at the top to receive the lubricating oil. The valve-face, on the out- side of the cylinder, and on which the valve works, is planed very carefully, so as to be a true plane throughout. The same is done with the valve-casing at the flanges, where it is fitted to the valve- face. PISTON. The piston (figs. 8, 9, 19, and 20) is composed of two cast-iron plates, which have an annular space between them for the recep- tion of two concentric cast or wrought-iron or brass packing-rings, c'. These rings are cut through at one side, and are placed one within the other in such a manner, that the breaks in each are diametrically opposite to each other; their thickness gradually diminishes on each side towards the break, and they are hammered on the inside in a cold state, which renders them elastic, giving them a constant tendency to open. Since the diameter of the outer ring is equal to that of the cylinder when the two edges are brought together, the elasticity of the iimer ring, combining with that of the outer one, tending constantly to enlarge them, it follows that there must be a perfect coincidence between the outside of the ring and the inside of the cylinder throughout the whole extent of the latter. Thus the contact of the piston with the sides of the cylinder only fakes place through the packing- ring, and not by the plates, which are of a slightly less diam';ter. To prevent the passage of the steam through the break in the outer packing-ring, a rectangular opening is made in the two edges of the ring, and in this is placed a small tongue-piece, ffl^ screwed to the inner ring, this piece serving to close or break the joint without preventing the play of the rings. The principal plate of the piston is fixed to the piston-rod by means of a key (fig.' 9)- The pjston-rod is consequently of incre;ised diameter at its lower end. The upper end of the piston-rod is likewise fixed in a socket, l', (figs. 9 and 13,) which terniinalcs in two vertical brnnchos to receive the middle of the spindle, e'^ whicli is held down by means of a key. CONNECTING-ROD AND CKANK. 411. Tno connecting-rod, J, (figs. 8, 9, 14, and 1.5,) terminates at its lower end in a fork, by means of which it is jointed to tlio spindle, e^ brasses being fitted in either side, and secured by bridle- pieces passing under them and keyed above. The fork is jointed to the spindle, e', on each side of the piston-rod head, sufficient space, however, being left between them for the levers, x. The head of the connecting-rod (figs. 15 and 16) is likewise fitted with brasses to embrace the pin, /, of the crank ; these brasses are tightened up by means of the pressure screw,/'. The crank, k, like the connecting-rod, j, is of wrought-iron, being adjusted on the end of the first motion shaft, and secured to it by a key. This crank is very often made of cast-iron in sta- tionary engines, but in marine and locomotive engines it is gene- rally forged, so as to be better suited for resisting severe strains and shocks. The first motion shaft, l, is likewise either of cast or ^vrought- iron. In the notes, we have already given u.bles and rules for determining the respective dimensions of this detail. It is not only supported by the brasses of the pillow-block, u, but also by those of a similar one, fixed, we shall suppose, upon the wall which divides the engine-house from the workshop or factory. It should always be larger in diameter where it receives the fly- wheel, M. FLY-WHEEL. 412. The fly-wheel is of cast-iron — of a single piece in the pre- sent example, because its diameter is only 3-5 metres. When of larger dimensions, the rim and the arms are cast in separate pieces, and then bolted together. For wheels of from 5 to 8 metres in diameter, the rim is made in several pieces, and the arms are also cast separate from the boss, and all the parts are then bolted together. The arms are sometimes made of wrought-iron of small dimensions, with the view of reducing the weight near the centre, without reducing the effect of the wheel. FEED-PUMP. 413. This pump serves to force into the boiler a certain qu.inlity of water, to replace that which is converted into steam and expend- ed in actuating the engine. It is a simple force-pump, consisting of a cylinder, R, in which works the solid piston or plungei, g. The piston is not in contact with the sides of the pump cylinder, and the latter consequently only requires to be turned out at its upper part, where it is formed into a stuffing-box and guide for the plunger, being necessarily air-tight. On one side of the pump is cast a short pipe, lO which is atl..>.h- cd the valve-box, R-, generally made of brass. To the lower part of this is secured the suction-pipe, t', communicating witn ii cistern of water, and having a stopcock, s', uiion it, like the one rcprc^ sented in detail in Plato XVH. To one siiU> of the valvo-bo\ is likewise fitted the discharge-pipe, carrying a similar stopcock, s*; this last pipe is generally [tassed through the pipe which carries off the waste steam, so that the water may take up some of the heat of this steam before oiiteriii!.;- Ilie boiler. It will bo seen, from tigs. !» and 23, that tliis viilvo-box contaitia two valves, a', s'; the lower one of which, s', is the suction viilvo. and the iip|Hr one, n\ is the discharge-valve. The latter is much larger in diiiniel.r Ili.in \\\v former, so that its sent may hv wide enom'h for the low.r \:ilve to hr passed through it. The up|HM 144 THi; PRACTICAL DRAUGHTSMAN'S i^nd of the valve-box is closed by a cover, which is fiiinly held down by the screw, r', and iron bridle, q'. Both valves are made conical at the seat, as in fig. 24, so as to tit more easily. The under part of the valve is cylindrical, so as to guide it ; but it is cut away at the sides, to allow of the passage of the water when it rises. It is from the appearance this gives that these valves are called lantern lalves. The pump cylinder is further furnished with a safety-valve, s', of which fig. 25 is a detailed view. The object of this is to permit the air to escape, when it accumulates within to such an extent as to destroy the action of the pump. This valve is horizontal, and is kept in its place by the bell-crank lever, u, upon the horizontal arm of which is suspended a weight, sufficient to counterbalance the internal pressure. (195.) Tlie top of the plunger is surmounted by a small rod, /, adjust- able in the socket which terminates the long wrought-iron rod, p', tigs. 9 and 18, the upper extremity of which is formed into a collai', embracing the circular eccentric, p. (142.) The action of this pump is analogous to that of the pumps of which we have already given a description. Thus, when the machine is working, and the stopcocks, s' and s^ are open, the water rises from the cistern by the pipe, r', the valve, s', opening, to give it passage into the body of the pump, into which it flows as long as the piston ascends. When, however, the piston descends, the water is driven back, and, closing the lower valve, s', necessarily opens the upper one, s^ and proceeds along the discharge-pipe to the boOer. The quantity of feed-water is regulated by means of the stopcocks, and may be entirely shut off by closing them ; but then, in such case, as the eccentric, p, with its rod, p', will continue to move, it will be necessary to loosen the plunger, q, which is done by unscre-\ving the thumb-screw, v, by which the piston-rod is attached to the eccentric rod, in such a manner that the socket, &', fig. 18, at the end of the eccentric rod, p', will sin:ply slide up and down the rod, without moving it. BALL OR ROTATING PENDULTTM GOVERNOR. 414. The object of this piece of mechanism is the regulation of the velocity of the machine, in proportion to the resistances to be overcome ; and, accordingly, to this effect it opens or shuts a valve, c\ placed in the steam-pipe, g, and called a throttle-valve. Just as a freer or narrower passage is left for the steam by the opening or closing of this throttle-valve, which is contained in an especial box, to facUitate adjustment, and is actuated by a rod, passing through a stuffing-box at the side — so is the quantity of steam which finds its way to the cylinder more or less ; and, similarly, the consequent acceleration or retardation of the motion of the piston, as well as of the first motion shaft in connection with it. It is composed, as seen in fig. 1, of a vertical spindle, Z, stepped, at its lower extremity, in the end of a small bracket support, k', and is held higher up by a second bracket, k. To its upper end are jointed two symmetrical side rods, m', each terminating in cast- iron or brass spheres, o'. These side rods are also connected by means of the intermediate links, Z', to the wrought-iron or copper socket or ring, i', moveable upon the main spindle. A rotatory motion being given to the vertical spindle, and the Vails being carried round with it, will have a constant tendency to fly off from the vertical hne, by reason of the centrifugal force due to the rotation (262) ; as long as the rotative velocity remains the same, the balls wnW tend to occupy the mean position indicated upon the drawing, and corresponding to the normal velocity ; that is to say, the velocity to which the apparatus is regulated. When this velocity is exceeded — in consequence, as we may suppose, for example, of some parts of the machinery being put out of gear — the balls will fly asunder, and occupy the extreme position, o-, indicated upon fig. 6. In this position of the balls, the socket, i', will be lifted up. Now this socket is embraced, at its circular groove, by the prongs of the forked lever, /, fig. 9, which is con- nected to the vertical rod, h, and this, by a suite of levers and bell- cranks, g', g', g■^ g*, and g^, communicates with the throttle-valve, c', drawn with its box, a^ in figs. 26 and 27. It follows, from the combination of these connections, that, as the socket rises, the valve will be shut. If, on the contrary, the velocity should be reduced below the proper point, owing to an increased resistance, the balls will approach each other, and assume the position given in fig. 7. The socket, i', will descend, and, consequently, the throttle-valve will become more open, so as to allow a greater quantity of steam to enter the valve-casing, and thence pass into the cylinder. The extreme positions of the governor arms, beyond which they cannot go, are determined by the guides, m^ fixed upon the spindle, Z. The motion of this spindle is derived from the first motion shaft L, by means of the grooved pulley, ^^ fixed upon the intermediate spindle, r^ placed close to the capital of the column, t, and by the bevil-wheels, r^ receiving their motion from the pulley, r*, so that a constant ratio is maintained between the rate of the machine and that of the governor. The geometrical diagram, figs. 6 and 7, will sufficiently explain the respective positions of each of the pieces of the pendulum, and will show how the rising of the socket upon the spindle is caused by, and is in proportion to, the flying asunder of the balls, accord- ing to the number of revolutions of the spindle, and the length of the suspending arms. MOVEMENTS OF THE DISTRIBUTION AND EXPAN- SION VALVES. DISTRIBUTION VALVE. 415. We have seen that the valve, D, represented in different positions in figs. 28 and 31, and in horizontal section, fig. 32, is attached, by its rod, g, to the vertical rod, n', which is joined to the rod, n^ of the circular eccentric, n, figs. 33 and 34. When, as w-as customary until lately, the centre of the eccentric lies in a radius perpendicular to the direction of the crank, the movements of the steam piston and valve are diflTerent to each other — that is to say, when the crank passes from the left horizontal to the right horizontal position, the piston makes a corresponding rectilinear movement ; tlie eccentric, however, passes from the lower extremity of the vertical line, drawn through the centre of the first motion shaft, to the upper extremity, or rice rersd,and consequently gives the valve a rectilinear movement quite different to that of the pis- ton, in such a manner that, when the latter is at the middle of ita stroke, the valve, on the other hand, is at the end, and the .steam- BOOK OF INDUSTRIAL DESIGN. 145 ports are consequently fully open, to give the steam the freest passage into the cylinder. Whilst the piston is accomplishing its stroke in one direction, '-he valve moves up or down, and returns again to its central po- sit' on, the part which it covered being opened and again shut; when, however, the crank makes two fourths of a revolution in different directions, the piston rises and falls half a stroke each vav, whilst the valve makes a single rectilinear movement in one arection. Finally, for each of these movements, whilst the velocities of the piston are increasing from the commencement towards the middle of its stroke, those of the valve are decreasing, and reciprocally. It therefore follows, that the maximum space passed through by the piston, for a given portion of a revolution of the crank, corresponds to the minimum passed through by the valve. LEAD AND LAP OF THE VALVE. 416. Of late years, engineers have recognised the advantage of inclining the radius of the eccentric, with regard to the radius of the crank, instead of placing them perpendicular to one another, in such a manner that, at the dead points — that is, the extreme high and low positions of the piston — the valve shall already have passed the middle of its stroke to a slight extent ; it is this advance of the valve which is termed the lead. The effect of giving this lead to the valve, is to facilitate the introduction of the steam into the cylinder at the commencement of the piston's stroke, and at the same time to allow a freer exit to the v/aste steam on the other side of the piston ; a greater uni- formity of motion is in consequence obtained, whUst les.*^ force is lost. In order to avoid as much as possible the back pressure due to the slow exit of the waste steam, it is likewise customary, in addi- tion to the lead, to give the valve more or less lap ; that is to say. to make the width of that part of the valve which covers the ports, a, b, fig. 28, sensibly greater than that of the ports themselves. In explanation of the effects due to the lead and lap of the valve, we have, in fig. 35, given a geometrical diagram, indicating the relative positions of the crank, the piston, the eccentric, and of the valve. Let o o represent the radius of the crank ; with this distance as a radius, and with the centre, o, describe a semicircle, which divide into a certain number of equal parts. From each of the points of division, let fall perpendiculars upon the diameter, o c. The points of contact, 1, 2, 3, 4, &c., represent upon this diameter, considered as the stroke of the piston, the respective positions of the piston, corresponding to those, 2", 3', 4', &c., of the crank pin. It is un- necessary to take into account the length of the connecting-rod, which connects the latter to the piston, because, in the present caao, Uie connecting-rod is supposed to be of an indefinite length, and to remain constantly parallel to itself, so that it cannot modify the results. With the centre, o, likewise describe a circle with a radius, I) a', e(jual to that of the eccentric, n. Wo have assumed the point, a', to be the position the centre of the eccentric siiould have at the moment when the piston i« at the end of its stroke — that is to say, at o ; the distance of this point, a', from the vertical m n, expresses the lead of the valve, and consequently the angle m o a', is called the angle of lead. The position of the point, a' may likewise be obtained, after the following data are decided on— namely, the height of the ports, a, b, fig. 28, the width, r s, of the flange of the valve, which is equal to the height of opening, t r which properly expresses the amount of lead given to the port augmented by twice the lap, together with the amount of the intro- duction of the steam to the cylinder, and the amount of opening, s' i', expressing the lead given to the escaping steam, and which is always greater than the former, so that the exit passages may be in communication as long as possible. The diameter of the eccentric, n, is equal to the height of the port, augmciuted by the width, r s, of the flange of the valve, and the difference which exists between the two amounts of lead, s' t' and r / ; it is, then, with the half of this as radius that the circle, a' b' c' d', must be described ; and we then obtain the point, a', by setting off from the centre, o, to the right of the vertical, m n, a. distance equal to the lead of introduction, r t, augmented by the lap. Starting from this point, a', we then divide this circle into as many equal parts as we previously divided the one mto, described by the crank pin, and then through each of the points of division we draw perpendiculars to the vertical, m n. We further draw the straight line, a' g', parallel to m n, when the distance of the several points of division from this line ^\'ill indicate the successive positions of the valve in relation to those of the piston. Thus, after having drawn the horizontals, r u, through the extreme point, r, of the valve, at the moment when the piston is at the extremity of its stroke, make I'' — 1' equal to b' b^, and the point, 1', indicates how far the valve has descended during the time the piston has traversed the space, o 1, whilst the crank has described the first arc, o 1'. In like manner, set off the distances, c' c', d' d\ &c., which correspond to the third and fifth divisions, reckoning from the horizontal line, ru, from i' to 3', and from h^ to 5', on the verticals corresponding to the third and fifth positions of the piston, and consequently the positions, 3^ and 6^ of the crank. It will then be seen that the valve con- tinues to descend until the moment the centre of the eccentric reaches the point,/', upon the horizontal line, of, corresponding to the sixth position, and the valve then wholly uncovers the port, a, as shown in fig. 29. During the continued revolution of the eccentric, on passing this point tiie distances of the points of division from the line, a g', diminish, and the valve reiiscends, in such a manner as that, when the centre attains the point, p — that is to say, when the crank shall have performed a semi-revolution, and the piston have arrived at 18, at the other end of its stroke — the valve will occupy the position indicated in tig. 30. This figure shows that it uncovers tlio lower port, b, for tiie introduc- tion of the fresh steam, and the upper one, a, for the escape of the used steam. If the respective positions, U', 7', 8', 9', &c., of the valve, be determined tliroughoul I lie .ntire stroke, by sotting oir upon the verticals, 6, 7, 8, S), \c., the distances of tlio points of division of tiio eccentric from the straight line, u' i.-', lus already explained, a curve will br t'oriued, ;is at u 3' G' 9' 18', which is a species of ellipse. This diaeTani lias the advantage of bringing into a single \ie\\ the relative positions el' the crank, piston, wcon- 1 '.s THE PRACTICAL DRAUGHTSMAN'S trie, and valve, and facilitatea the determination of the position of the valve, corresponding to any position of the piston. Thus, to obtain the position of the valve to correspond to that, y, of the steam-piston, it is sufficient to draw the vertical, y x', which will cut the curve in the point, v'. The distance, i;' x', of this point, from the horizontal, t u', passing through the upper edge of the introduction port, «, shows how much of this is un- covered by the valve. It will be seen, also, that the curve is cut by the horizontal, t u', in the point, y\ which indicates the moment at which the valve closes the port. In this position the piston will only as yet have reached the point, xf, of its stroke ; and it has, consequently, to traverse the distance, y'' 18, before it can receive any more steam from the boiler, which shows that, with a valve which has lead and lap, we actually work the steam expansively to a slight extent. In the case before us, the steam is cut off at four- fifths of the stroke. It will be understood that, if the machine continues its action, the piston will retrace its stroke, the centre of the eccentric which had reached j) will continue to ascend, and the valve will shortly attain the position indicated in fig. 31, this taking place as soon as the centre of the eccentric reaches the point, z. In this position, the ports, a, h, are completely open — the first to the exit aperture, the other to the introduction of the steam, whilst the valve is at the highest point of its stroke, as will also be found by continuing the curve, u 9' 18', of which the prolongation, 18' 24' 30', is exactly symmetrical with regard to the inclined line, u 18'. On the same diagram, tig. 35, we have delineated a second elliptic curve, 0" 9" 18", equal and parallel to the first, and which indicates the respective positions of the point, s', of the lower flange of the valve, in relation to the port, h, so as to have, at first sight, the respective positions of this second flange. This outline is evidently obtained by setting oflT the constant distance, r s', of the valve, fig. 28, upon the verti- cals, drawn through 1, 2, 3, 4, &c. It may be remarked, that the distance between the ports, a and h, is arbitrary. It is, however, advisable to reduce it as much as possible, in order to diminish the surface of the valve, and, conse- quently, the pressure of the steam acting on the back of it In all cases, it is necessary that the height of the exit port, c, should be greater than that of the introduction ports, by a quantity at least equal to the difference which exists between the lap and the lead, l! s' and t r. EXPAMSION VALVE. 417. The action of the expansion eccentric, o, is analogous to rhat of the ordinary valve eccentric, except that the position of its centre is not regulated in the same manner. We may observe, in the first place, that this eccentric is not immovably fixed upon the main shaft, l, as is the case with the preceding one. It is only attached to the adjustable collar-piece. F^ figs. 36 and 37, by screws, v?. This arrangement allows of its throw being increased or diminished ; that is, of its centre being placed further from or nearer to that of the shaft, according to the icngth of stroke which it is wished to give it. To this end, its central opening is oblong in shape, and the holes for the securing screws are oblong likewise. If the centre of this eccentric happens to be in the same diree- tion as the crank, the expansion valve, h — the rod, hfoi which is guided by the socket-bracket, 7i*, attached to the pedestal, s, and drawn more detailed in fig. 43 — is wholly open when the piston is at the end of its stroke ; but we have supposed, as indicated in fio-. 35, that the centre of this eccentric is in the point, a*, upon the circle described with the centre, o, and radius, l a*, of the eccentric, and that the valve does not therefore wholly uncover the entrance port, d, at this moment, so that the time of closing it may be later than would otherwise be the case. As in the preceding case, we divide this circle into equal parts, starting from the point, a*; through the point, aS draw a vertical line, and then set off on the various verticals, 1, 2, 3, 4, &c., the distances of the points of division from this line, measuring these from the horizontal passing through the upper edge, r", of the valve, H ; we thus obtain a second elliptic curve, u m' n' p', the inside of which is flat — tinted with a slightly stronger shade than 1 J the ellipse corresponding to the distribution valve, so as to rendei • ' the diagram more distinct. This curve cuts the horizontal line drawn through the upper edge of the port, d, in the point, n, whi<;h indicates at what time the valve, H, closes the entrance port, fig. 39. It wUl be seen that this point corresponds to the position, b', of the steam-piston, thereby signifying that the cut-off takes place when the piston has performed no more than a fourth of its stroke. Con- tinuing the movement, it will be observed that the valve, H, rises higher and higher, so that it begins to uncover the entrance port a little before the piston reaches the end of its stroke ; but it is evi- dent that the steam cannot find its way into the cylinder at this point, for the distribution valve is in its turn closed, as soon as the position, y y', is passed ; no inconvenience, therefore, will be caused by the fact of the valve, h, being open before reaching the end of its stroke, as indicated in figs. 38 and 40, and as shown also in the diagram, fig. 35. By varying the radius of the eccentric, o, and the position of its centre relatively with the radius of the crank, it will be easily un- derstood, that *Fithin certain limits we can alter the time when the valve, H, opens and closes the entrance port, and are consequently enabled to vary the degree of expansion. Figures 41 and 42 show that the rod of the valve is attached to it by a T joint, which leaves the valve sufficiently free for the steam to press it constantly against the planed valve face ; and a similar adjustment is adopted v\'ith the distribution valve. The general explanations which we have ^ven in the preceding pages, with reference to the construction and action of this engine, evidently apply to other systems, which merely differ in some of the arrangements and forms of the component pieces. Moreover, in our notes, the student will find the rules and tables concerned in the calculations and designs of these engines. RULES AND PRACTICAL DATA. STEAM-ENGINES. LOW-PRESSURE CONDENSING ENGINE, WITHOUT EXPANSION VALVE. 418. In those engines which are called low-pressure engines, the steam is produced at a temperature very little over that of boiling water, or 100° centigrade (212° Fahrenheit) — it is, in fact, BOOK OF INDUSTRIAL DESIGN. 147 generally 106° cent. — in which case the tension of the steam will sustain a column of mercury of 90 centimetres in height ; that is to say, 14 centimetres above that due to atmospheric pressure. It is, consequently, equal to a pressure of ri7 atmospheres, or I'ii kilog. per square centimetre. It is for this pressure that what are gene- rally known as Watt's engines, without cut-off valves, are calculated; and the one we have been examining is regulated upon this datum. There is, however, a great difference between the pressure of the steam in the boiler, and that to which the effective power of the machine is due. It is evident that a part of the pressure will be absorbed by the back pressure due to an imperfect vacuum, as well as by the friction of the piston, and other moving parts, and the leakage and condensation in the steam passages. So that, taking into consideration these various causes of loss, the effective force may be estimated at '5 kilog. only, per square centimetre, in the majority of engines, whilst it may reach, perhaps, -65 kilog. in the most efficient. The rule for calculating the power of low-pressure steam-engines consists in — Multiplying the mean effective pressure of the steam upon the piston by the area of the latter, expressed in square centimetres, and the pro- duct by the velocity in metres per second. The result of this calculation will be the useful effect of the engine in kilogrammetres. To obtain the horses power, this result must be divided by 75. Thus, the diameter of the cylinder of a low-pressure non-expan- sive steam-engine being -856, and its section 5755 square centi- metres, if the effective pressure upon the piston is -63 kilog. per square centimetre, and the velocity 1-1076 — We have •63 X 5755 X 1-1076 = 4015-67 k. m. Whence — 4015-67 -^ 75 = 53-54 H. P. But the effective pressure upon the piston is not always -63 kilog. per square centimetre ; it is more frequently below than above this amount. It varies not only according to the power of the machine, but also according to the state of repair. Thus, some- times the effective pressure will not be more than -45 kUog. in small engines, whilst in large, powerful ones, it may at times reach •65 kilog. Single-acting engines, such as are employed in mines, are of the same dimensions as double-acting ones, but of only half the power. Thus, the cylinder of a low-pressure steam-engine, of 50 horses power, and only single-acting — that is to say, receiving the action of the steam during the descent only of the piston — is exactly the same as in a machine of 100 horses power, in which the steam acts alternately on both sides of the piston. In the following table, which applies to this kind of steam-engine, we have given the diameters and velocities of the steam-piston from 1 to 200 horses power. TABLE OF DIAMETERS, AREAS, AND VELOCITIES OF PISTONS, IN LOW-PRESSURE DOUBLE-ACTING STEAM-ENGINES, WITH THE QUANTITIES OF STEAM EXPENDED PER HORSE POWER. Horses Diameter Area of Piston. Length of Number of revolutions. Velocity of piston per second. Velocity of piston per minute. Effective pressure on the piston per square centimetre. Weight of steam ejpend- power. of piston. Total. Per stroke. ed per horse power per hour hor se power. cent. sq. m. s q. cent. m. per 1'. III. ni. kilo?. kiloj. 1 •15 •018 181 •62 50 -85 51 -49 38-81 2 -21 •036 178 •61 42 •86 62 -49 38^77 4 -30 -068 171 •76 34 -90 54 •49 38^-7 6 •35 ■098 163 •91 31 -94 57 •49 38-72 8 ■40 •128 160 r07 27 -96 68 •49 38-72 10 •45 •169 159 r22 24 •98 69 •49 38-64 12 •49 •189 157 V22 24 •98 59 •49 38-64 16 •55 •240 150 1-31 22 1-01 60 •50 37-80 20 •61 •292 146 1-62 20 1-02 61 •51 37-38 24 -66 •346 144 1^69 18 1-02 61 •62 3688 30 -73 •414 137 r83 17 1-04 62 53 36-04 40 •83 •535 134 1-99 16 1-06 64 63 36-70 50 •91 •658 132 2-13 15 1-07 64 64 36-32 60 TOO •779 130 2^28 14 1-07 64 64 34-94 70 ro7 •903 129 2^44 13 1-06 63 •65 3436 80 M4 1-032 129 2-44 13 1-06 63 •56 3431 90 1-21 1-138 126 2-59 12 1-04 62 •57 •58 3301 3297 100 1-27 1-264 126 2-59 12 1-04 ()2 120 1-39 i-512 126 2-74 11 1-00 60 •69 3 1 -92 160 r60 2-005 125 3-00 10 1-00 60 •60 3 1 •t)7 200 r78 2-480 124 3-00 10 1-00 (iO •(il 31-47 DIAMETER OF THE PISTON. By means of the above table, wo can, in a very simple inamuM-, Aetoriiiino the diaiiietor and velocity of the piston of a iow-picsMiiro ilouhlc-acting steam-ongino, supposing tlio nUmu to ho of tlio pn^ssuro of 1-17 iilmosiihcrfs in tlu« hoilor, i-orrespoiiding U» a coluiuii of iiicivinv "t!Hl .•.■iiliiii.ln'N in li.'iH:ht. 148 THE PRACTICAL DRAUGHTSMAN'S Rule. — It is sufficient to obtain from tlie table the area of piston per horse power, and to multiply it by the number of horses power of the engine to be constructed, which will then determine the cor- responding area of piston. Example. — What should be the diameter of the piston of a low- pressm-e double-acting steam-engine of 25 horses power ? In the fourth column of our table, it will be seen that the area to be given to the piston should be 144 square centimetres per horse power for 24 to 26 horses, with a velocity of 1-02 m. per second. We have, therefore, 144 x 25 = 3600 sq. cent, for the total area of the piston. Whence — ■V'3600 X -7854 = 67-7 cent. Thus, the diameter of the piston must be -677 m. VELOCITIES. The velocities per second, and per minute, given in the seventh and eighth columns of the table, are what are generally adopted as the regular working rates in establishments and manufactories where steam-engines are employed, whatever may be the number of revolutions of the crank, or strokes of the piston, per minute, for this number varies according to the length of stroke which it is wished to give to tne piston. Thus in stationary engines, the stroke of the piston is generally longer ; and, therefore, fewer strokes are made per minute than in marine engines, since in these latter the engineer seeks, as much as possible, to reduce the height of the machinery ; and the stroke is, consequently, much shorter for the same amount of power. The length of stroke of the piston is regulated at pleasure by the constructor, according to what he may find most advantageous in the transmission of the power to the machinery ; and he calcu- lates so that the crank may make a few revolutions per minute more or less, without occasioning any very sensible difference in the velocity of the piston, with regard to the velocities laid down in the table. If, notwithstanding, it is wished to construct an engine to work with a velocity somewhat less, or somewhat greater, than that given in the table, it will evidently be necessary to take this difference into consideration, and to augment or diminish the area of the piston in proportion, so as always to obtain the required power. The proper amount of alteration may be determined by a very smaple operation. Example. — Let it be proposed to construct our example engine of the effective power of 25 horses, wdth a velocity of piston of 1 metre per second, in place of 1'02 m. It will be sufficient to calculate the following inverse propor- tion : — 1 : 1-02 : : 144 sq. c. : x. Whence — X = 144 X 1-02 = 146-8 sq. c, the area, per horse power, to be given to the piston. Consequently, 146'8 X 25 = 3675 sq. centimetres for the total area; and V3675 "=" '7854 = 68-4 cent., for the diameter of the piston. As complemental to this table, we have given the expenditure of steam corresponding to the different powers, as well as the de ductions from this of the expenditure per horse power per houi. It will be observed from the last column, which gives the expendi- ture of steam that it is considerably more for engines of small force than for more powerful ones — the reason of which is self-evi- dent. Thus, for an engine of 12 horses power, the expenditure of steam is 38-64 kilog. per horse power per hour ; whilst for an engine of 100 horses power, the e.xpenditure only reaches 32-97 for a like power in the same time. The expenditures or weights of the steam have been calculat^a from the following formula : — W = AxSx2ox2Nx60. A representing the area per horse power ; S, the stroke of the piston ; w, the weight of a cubic metre of steam at the pressure employed; N, the number of revolutions. We need not here take into consideration the loss of steam re- sulting from leakage and condensation in the steam pipes and pa-ssages, which is generally estimated at one-tenth of the whole expenditure, as this item should e\-idently enter into the calcula- tions respecting the boiler. STEAM-PIPES AND PASSAGES. The section of the pipe which conveys the steam to the cylindei, as well as that of the introduction ports and passages, should be equal to a twentieth of the area of the piston. Whence it follows, that the diameter of the steam-pipe should be one-fifth of that of the cylinder. We must, however, remark, that the greater the velocity of the engine, the gi-eater should be the sectional area of the steam-pipes and passages. It is because of this that, in locomotive engines, this section is sometimes made a tenth or a ninth of that of the cylinder, and at the same time the pressure of the steam is much greater, being generally equal to 5 or 6 atmospheres, and sometimes more, in such engines. AIR-PUMP AND CONDENSER. The stroke of the air-pump piston is equal to half that of the steam-piston ; and as it gives the same number of strokes, but does not discharge in ascending, it can only raise a quantity of air and water equal to its own cubic contents, at each double stroke. Now, the sectional area of the pump is -2827 sq. m. ; and the length of stroke, -923 m. Its capacity is, therefore, -261 cubic metres; and as twice the cubic contents of the steam-cylinder is 2-125 cubic m., it follows that the pump discharges only a little more than an eighth of the volume sent out by the steam cylinder. This capacity is quite sufficient for the effective action of the engine. The sectional area of the condenser is the same as that of the pump, and its length is about 1 metre ; so that its capacity is, at least, as great. As the quantity of water to be injected into the condenser varies according to the temperature of the injection water, it will be well lo know how to regulate it. To this end, the following rule will answer : — BOOK OF INDUSTRIAL DESIGN. 149 Rule. — Take the excess of the temperature of the steam over that of the injected water, and, after adding 550 to it, multiply it by the weight of steam to be condensed, and divide the product by the differ- ence of temperature between the discharged and the injected water. The quotient will be the weight of cold water to be injected. Thus, let w represent the weight of the steam to be condensed; t, its temperature ; W, the weight of the cold water to be injected into the condenser ; i', its temperature ; and T, that of the water discharged : — We have w(o50 + t-T) T — t' If we make w = 26-16, t' — 12° cent., T = 38°, and i = 105°, we shall have „, 26-16 (550 + 105° — 38°) 38° — 12° Whence, W = 621 kilog. or litres, for the expenditure per minute of cold water in the condenser. That is to say, the quantity of water to be injected into the con- denser should, in this case, be about 24 times the weight of the steam expended. If the discharged water were of the temperature of 55°, the cold water remaining at 12° — We should then have _ 26-16(550 + 105° — 55°) 55° — 12° Whence — W = 365 kUog. or litres. That is to say, that in the la-st case the water injected would not be more than 14 times the steam expended. But it is to be remarked, that in this case the force of the steam in the condenser, at a temperature of 55°, is equal to a column of mercury of 12-75 centimetres in height; whUst, in the first case, it would only be equal to a column of 5-5 cent. There is, therefore, an advantage in employing sufficient injection-water to produce the lower of the two temperatures. From the preceding results, we may deduce what follows : — First, That the stroke of the air-pump piston, in low-pressure double-acting steam-engines, is ordinai-Uy equal to half the stroke of the steam-piston. Second, That the diameter of the air-pump piston is equal to about two thirds of the diameter of the steam piston ; and, conse- quently, its area is about half that of the latter. Third, That the effective displacement of the air-pump piston — that is, the cubic contents of the cylinder generated by the disc of the piston — is equal to an eighth, or at least a ninth, of the contents of the cylinder generated by a double stroke of the steam-piston. Fourth, That the capacity of the condenser is at least equal to that of the air-pump. Fifth, That the sectional area of the passjigc communicating between the condenser and air-pump is equal to one-fourth the area of its piston. Sixth, That the quantity of cold water to be injected into the condenser varies according to its temperature, and to the tcnijxni- ture of the water discharged. Seventh, Tha' this quantity is equal to 24 times the weight of steam expended by the cylinder, where the mean temperature of the cold water is 12°, and that of the water of condensation 38", which are generally what exist in low-pressure double-acting engines. COLD-WATER AND FEED PUMi-s. The capacity of the cold-water pump should be the 24th or 18th of that of the .steam cylinder. The capacity of the feed or hot- water pump should be the 230th or 240th, at least, of that of the steam cylinder. HIGH-PRESSURE EXPANSIVE ENGINES. Let the following dimensions be given for an engine analogous to that which we have just described: — Diameter of the cylinder, = -275 m. Stroke of the piston, = -680 m. Area of the piston, = -0594 square m. Number of double strokes per minute, = "40 Let us suppose, in the first place, that when the steam reaches the cylinder, its pressure is equal to 5 atmospheres, and that it is cut off during three-fourths of the stroke ; that is to say, that the cylinder only receives the steam during the first quarter of the stroke. This pressure of 5 atmospheres is equal to 5 x r033 = 5-165 kilog. per square centimetre. Consequently, the total pressure exerted upon the surface of the piston is — 5-165 X 594 sq. cent. = 3068 kilog. And as with this pressure the piston passes through a space equal to one-fourth of its stroke, or •680 -=- 4 = -170 m., it is capable, theoretically speaking, of transmitting an amount of force expressed by 3068 X -17 = 521-56 kilogrammStres. Next, dividing the length, -51 m., or the remaining three-fourths of the stroke, into an even number of equal parts — as four, for example — each of these parts will be equal to 51 , - —- = -12/0 m. 4 Now we know that, according to Mariotte's law, the successive volumes of a given quantity of any gas are in the inverse ratio of their tension or pressure, provided the gas is in the same condition throughout. This principle may be reg-arded a.s quiie true in steam- engines, because the expansion is never carried very far, and as tlio steam passes through the cylinder with groat rapidity, and is con- tinually being renewed, alter a certain time and wiieii the i-yliiuiiT has become warm, its teniperaturo is very little below that of Ilio steam itself, and the latter suffers no appreciable clinngo in y>ass\\\<; through it. Putting P for the pressure, 3068 kilog., as found lor the first (juarter of the stroke, we may state the relation.s of tlio vdlunus iuul pressures in the following manner: tlmt is, at Hio points, 1, 2, 3, l, 5, of the stroke, or lor the sum'ssivo ep*c«», -170 ni., -295 m., -i'lb ni., 5525 m., -080 in. 150 THE PRACTICAL DRAUGHTSMAN'S The corresponding pressures will be — P -1700 -170 - 1700 -170 3068 A:, -2975 ' -425 ' -5525 ' -680 ' or finally, 3068 k, 1764 k, 1227 k, 944 k, 767 k. Next, according to Simpson's method, we have The sunt of the extreme pressures, = 3068 + 767 = 3835 Twice the pressures at the odd intervals, = 2X 1227 = 2454 Four times the pressures of the even intervals, =4(1764+ 944)= 10632 Total, 17121 Taking the third of this quantity, and multiplying it by -1275, we shall have the work given out during the cut-off. Thus — 17121 X -1275 g = 727-64 k. m. Adding to this 521-56 k. m., the work given out before the cut- off, we shall have the total of the work given out by the steam during the entire stroke of the piston — = 1249-2 k. m. Deducting now from this the effect of the atmospheric pressure, which resists the motion of the piston throughout the stroke, and which is equal to 1-033 k. X 594 sq. c. x -68 m. = 417-25 k. m., there remains for the effective force of the piston — 1249-2 — 417-25 = 832 k. m., nearly, for each stroke ; and as the piston gives 40 double or 80 single strokes per minute, the effective force per minute becomes 832 X 80 = 56560 k. m. ; that is, 56660 kilogrammes, raised one metre high. The effective povver of this, as well as of most other expansive steam-engines, will be obtained in a much more simple and less tedious manner, by taking advantage of the following table : — TABLE OF THE FORCE, EST KILOGRAMMETRES, GIVEN OUT WITH VARIOUS DEGREES OF EXPANSION BY A CUBIC ilETRE OF STEAM AT VARIOUS PRESSURES. ■ Force 3;iven out, corresponding- with tlie pressure of Volume when expanded. 1 ^ 2 2^ 3 4 5 6 atmosph. atmosph. atmosph. atmosph. atmosph. atmosph. atmosph. atmosph. Mibic metres. k. ra. k. m. k. m. k. m. k. m. k. m. k. m. k. m. 1-00 10333 15500 20666 25833 31000 41333 51666 62000 1-25 12639 18968 26278 31597 37917 50566 63196 75834 1-50 14523 21784 29046 36257 43668 68092 72615 87138 1-75 16116 24174 32232 40290 48348 64464 80580 86696 2-00 17496 26244 34992 43740 62488 69984 87480 104976 2-25 18713 28069 37426 46782 66139 74852 93565 112278 2-50 19802 29703 39604 49605 59406 79208 99010 118812 2-75 20787 31180 41574 61967 62361 83148 103936 124722 3-00 21686 32629 43372 54216 65058 86744 108430 130116 3-25 22613 ' 33769 45026 56282 67539 90052 112666 135078 3-50 23279 34918 46658 58197 69837 93116 116395 139674 3-75 23992 35988 47984 59980 71976 95968 119960 143952 4-00 24658 36987 49316 61645 73974 98632 123290 147948 4-25 25285 37927 60570 63212 76855 101140 126425 151710 4-50 26875 38812 51750 64687 77625 103500 129375 165250 4-75 26434 39651 62868 66085 79302 105736 132170 158604 6-00 26964 40446 63928 67410 80892 107866 134820 161784 5-25 27467 41200 64934 68667 82401 109868 137335 164802 6-60 27949 41923 55898 69872 83847 111796 139745 167694 6-76 28408 42612 66816 710-20 85224 113632 142040 170448 6-00 28848 43272 67696 72120 86544 116392 144240 173088 6-25 29270 43905 58540 73175 87810 117080 146350 175620 6-50 29675 44512 69350 74187 89025 118700 148375 178060 6-75 30065 46097 60130 76162 90195 120260 150326 180390 7-00 30441 45661 60882 76102 91323 121764 152205 182646 7-25 ■ 30804 46206 61608 77010 92412 123216 164020 183224 7-50 31164 46731 62308 77886 93462 124616 165770 186924 7-75 31494 47239 62986 78732 94479 125972 167465 188958 8-00 31820 47730 63640 79550 96460 127280 159100 190920 8-25 32139 48208 64278 80347 96417 128556 160695 192835 8-50 32447 48670 64894 81117 97341 129788 162235 194682 8-75 32747 49120 66494 81867 98241 130988 163735 196482 9-00 33038 49557 66076 82595 99114 132162 165190 198228 9-25 33321 49981 66642 83302 99963 133284 166605 199926 9-60 33597 50395 67194 83992 100791 134388 167985 201582 9-75 33866 50797 67730 84662 101595 136460 169325 203190 10-00 34127 61190 68254 85317 102381 136508 170635 204762 BOOK OF INDUSTRIAL DESIGN. 151 According to this table, if we have to calculate the force acting upon the piston in this engine, in the same circumstances, we must, in the first place, ascertain the original volume of the steam intro- duced into the cylinder during the first quarter of the stroke of the piston. This volume is equal to •0594 X -17 — -010098 cubic metres. Now it will be seen from the table, that the force given out when a cubic metre of steam, of a pressure of 5 atmospheres, expands to four times its original volume, is equal to 123290 k. m. Consequently, that corresponding to a volume of -010098 cubic metres will be — 123290 X -010098 = 1245 k. m., And deducting from this the atmospheric pressure, which resists the motion of the piston, we have 1245 — 417 = 828 k. m., a quantity which differs very little from that obtained by the more tedious calculation. Thus, the calculation for determining the effective power of a steam-engine, of which we know the diameter and stroke of the piston, the pressure of the steam, and the amount of cut-off, reduces itself to the following rule : — Rule. — Multiply the area of tlie piston by the portion of the length of the stroke, during which the steam acts with full pressure, and you will determine the volume of steam expended. Multiply this volume by the amount of kilogrammetres in the table, corresponding to the pressure of the steam and to the final volume, and then deduct froin the product the amount, in kilogrammetres, of the atmospheric pressure opposed to the piston during the entire stroke, and the result will be ^he theoretic amount of force, in kilogrammetres, given out by the steam during a single stroke of the piston. A MEDIUM-PRESSURE CONDENSING AND EXPANSIVE STEAM- ENGINE. Let the following data be assumed : — The diameter of the steam-cylinder = -330 m. The stroke of the piston = -650 m. The diameter of the air-pump = -180 m. The stroke of its piston = -325 m. The diameter of the feed-pump . . . . = -035 m. The stroke of its plunger = -235 ra. It follows, from these dimensions, that we shall have — • The area of the steam-piston = 855-30 sq. cent. The area of the air-pump piston . . . = 254-47 " The area of the feed-pump = 9.62 " And for the displacement, or volumes of the cylinders generated by the pistons — That of the steam cylinder . . . . = 65-594 cubic dccim. That of the air-pump = 8-270 " That of the feed-pump — -226 " Wo shall suppose that, when tho engine is in rogiii.ir working ••iiiidition, tho pressure of tho sicam is 3J atmoHpiuM-es ; and we iiiiihI ascertain what is tho actual loivo given out, sM|)poHing the HtcuMi to be cut off during three-fourlhs of the stroke of the piston. That is to say, that the steam is admitted into the cylinder only during a quarter of the stroke, which corresponds to -1625 m. Since the sectional area of the cylinder is -0885 m., the volume of steam expended during a fourth of the stroke will be equal to -0885 X -1625 = -0139 cubic metres; or, 13-9 cubic decimetres. Now, according to the table of the amounts of force given out by the steam at various pressures, it will be found that the force due to a cubic metre of steam, of an initial pressure of 3^ atmospheres, when allowed to expand to four times its volume, is equal to 86303 kilogrammetres. As the table does not give the actual amount for 3i atmospheres, it may be taken by adding together that for 21 and 1 atmospheres. Thus — 61645 + 24658 = 86303 k. m. We have, therefore, in the present case — -0139 X 86303 = 1199-6 k. m., as the force due to a single stroke of the piston. From this quantity, however, we must deduct the back pressure due to the imperfect vacuum in the condenser. This back pressure is, in the generality of cases, equal to about -27 kilog. per square centimetre, when the temperature of the water of condensation is about 65" cent. Allowing this to be the case in the present example, we shall have to deduct from the preceding result the action of this back pressure upon the whole surface of the piston, and during the entire stroke. This is -27 X -0885 X -65 x 150-1 k. m., We have, consequently, 1199-6 — 150-1 — 1049-5 k. m., for the actual force given out by the piston during a single stroke ; and if this engine works at the rate of 42 revolutions per minute, which supposes the velocity of the piston to be -9 m. per second, we shall find that the mechanical effect per minute will be equal to 1049-5 X 84 = 981588 k. m ; or, 881598 -f- 4500 = 19-59 horses power. It is well known, however, that this amount is far from being all transmitted by the first-motion shaft, for a portion is absorbed in overcoming the friction of the various moving parts of the engine, and there are also other causes of loss. If we reckon that the force which is really utilised is not more than four-tenths of that theoretically due to the steam, in which case we must suppose that six-tenths are completely lost, we shall have for the effective force transmitted to the first-motion shall— 19-69 X -4 = 7-84 horses power; or almost 8 horses power, of 75 kilogrammetres each. If it is desired to know the quantity of fuel consiuuod per hour in producing this mechanical efVoct, we may remark, that a cuhiv metro of steam, at a pressure equal to 3i atmospheres, weighs 1-8518 kilog. ; and at a pressure of 4 atmospheres, it weighs 2-0'291 kilog. Now, although wo have supposed tho pressure in the cyliudor to be 3'f atmospheres, wo, novertlieless, allow tiiat it will bo consid»»- rably more in tho boiler, to compeusnto for tho leakjigo iu tho valve- casing, passages, and valves. Taking 1 atnios|ihores as the iirossiiro hi tho boiler, it will Ik< foiMul that the weight of stonm expended for each .singlo stroko of the piston is — 152 THE PRACTICAL DRAUGHTSMAN'S •0139 X 2-091 = -0291 kilog. ; and per hour — •0291 X 84 X 60 = 146-204 kilog. From which it follows, upon the hypothesis that one kiiog. of 2oal generates 6 kilog. of steam, that the quantity of fuel consum- ed will be 146-64 -4- 6 = 24-44 kilog. per hour. And since the power obtained is 7^84 horses power — We have 24^44 -f- 7^84 = 3^1 kilog. for the quantity of coal consumed, per horse power, per hour- To complete the rules here given, we add the two following tables, relating to the principal dimensions given to steam-engines of different kinds : — TABLE OF PROPORTIONS OF DOUBLE-ACTING STEAM-ENGINES, CONDENSING AND NONCONDENSING, AND WITH OR WITHOUT CUT OFF, THE STEAM BEING TAKEN AT A PRESSURE OF 4 ATMOSPHERES IN THE CONDENSING, AND AT 5 ATMOSPHERES IN THE OTHER ENGINES. Condensing en^nes, Noncondensing expansive en- Noncon Jensing Stroke of Velocity of piston Number of revolutions cutting off at one-fourth of the stroke. gines, cutting oiF at one-fourth. nonexpansive engines. Horses power. piston. per second. per minute. Diameter Area of piston Weight of Diameter Area of piston Diameter Weight of of the per steam per horse of the per of steam per horse piston. horse power. power per hour. piston. horse power. piston. power per hour. cent. cent. cent. sq. cent. kil..g. cent. sq. cent. cent. kilog. 1 40 70 52-5 16 189 24-90 14 148 10 50-76 2 50 75 45-0 20 160 22-62 19 135 14 49-56 4 60 80 40-0 27 148 22-38 25 124 18 46-98 6 70 85 36-4 32 138 22-08 31 123 21 45-30 8 80 90 33-7 36 127 21-54 33 106 23 42-00 10 90 95 31-7 39 119 21-36 36 100 25 41-.34 12 100 100 30-0 42 112 21-18 38 92 26 40-86 16 110 105 28-6 46 104 20-58 42 87 29 39-96 20 120 110 27-5 49 94 20-28 45 81 31 38-82 25 130 115 26-5 54 92 19-80 49 76 34 38-52 30 140 120 25-7 57 86 19-32 52 72 36 37-56 35 150 125 25-0 59 77 18-54 55 68 38 37-38 40 160 130 24-3 62 75 18-06 57 64 39 36-60 50 170 135 23-8 67 70 17-28 62 60 43 36-18 60 180 140 23-3 72 68 17-22 66 58 46 35-76 75 190 145 22-9 78 67 17-16 72 54 50 35-04 100 200 150 22-5 85 57 16-62 84 66 56 34-08 TABLE OF PROPORTIONS OF MEDIUM PRESSURE CONDENSING AND EXPANSIVE STEAM-ENGINFS, WITH TWO CYLINDERS ON WOOLFS SYSTEM; PRESSURE, 4 ATMOSPHERES. Diameter of cylinders in centimetres. Area of pistons in square centimetres. Stroke of pist ons in metres. Horse Revolutions power. d. D Per horse power. i. S. per minute. a. A. a. A. 4 16 27 201 572 50 143 •67 •90 30-0 6 19 35 283 962 47 160 67 •90 30-0 8 21 38 346 1134 43 141 75 1-00 30-0 10 23 42 415 1385 41 138 75 1-00 30-0 12 25 46 491 1662 40 138 82 1-10 27-3 16 28 52 616 2124 38 133 90 1-20 27-5 20 30 54 707 2290 35 114 97 1-30 25^4 24 32 59 804 2734 33 113 97 1-30 25-4 30 34 63 908 3117 30 103 20 1-60 21-6 36 37 67 1075 3526 29 98 20 1-60 21-6 40 37 67 1075 3526 26 88 27 1-70 22-1 45 39 71 1194 3959 26 87 27 1-70 22-1 50 41 75 1320 4418 26 88 35 1-80 20-8 60 45 82 1590 5281 26 88 35 1-80 20-8 70 48 87 1809 5945 25 84 50 2-00 19-5 80 51 93 2043 6793 25 84 50 2-00 19-5 90 54 99 2290 7698 25 85 57 2-10 18-6 100 57 104 2552 8495 25 85 57 2-10 18-6 110 60 109 2827 9331 25 84 57 2-10 18-6 120 62 114 3019 10207 25 85 57 2-10 18-6 130 65 118 3318 10936 25 84 1-67 2-10 18-6 BOOK OF INDUSTRIAL DESIGN. 1.-53 CONICAL PENDULUM, OK CENTRIFUGAL GOVERNOR. The centrifugal ball-governor is compared, in physics, to a pimple pendulum, the length of which is equal to the distance of the point of suspension from the horizontal plane passing through the centres of the balls ; and the duration of an entire revolution of the ball-governor is equal to that of a complete oscillation of the pendulum. The formula for determining the vertical height or the distance of the point of suspension above the plane of the balls is, conse- quently, the same as that employed to find the width of a pendu- 'um, of which we know the number of oscillations. It may be reduced to the following rule : — Rule. — Divide the constant number, 89,478, by (he square of the number of revolutions per minute. The quotient will give the height in centimetres. Example. — What is the vertical height or distance of the point of attachment, from the horizontal plane passing through the centres of the balls of a governor, revolving at the rate of 40 turns per minute ? We have 40' = 1600, and 89478 -f- 1600 = 56 centimetres, for the height sought. With this rule, it will be easy for us to calculate the heights of conical pendulums, from the velocity of 25 revolutions per minute, to that of 67 ; and within these will be found the rates of combi- nations more generally met with in practice. We have given them in the following table, adding a column, which gives the difference in height for each revolution. And as the angle which the arms of the governor make with the spindle is generally one of 30°, when the balls are in a state of repose, or are going at their minimum velocity, we have given, in the fifth column of the table, the lengths of these arms, from their point of suspension to the centres of the balls, assuming the angle of 30°, and making them to correspond with the number of revolutions given in the first column. In calculating the lengths of the arms, we have employed the following practical rule : — Rule. — Divide the constant number, 103,320, hy the square of the number of revolutions per minute, and the quotient will be the length in centimetres. Example. — Assuming the angle to be 30°, what should be the length of the arms of a conical pendulum, making 37 revolutions per minute? Wo have 37' = 1369. 10-3320 Then— 1369 75-46 ceutim^tres, for the length of the arms of the pendulum, or the diameter of the circle described by the balls. It is evident, that if, on the otiicr hand, the length of the arms, with this angle of 30°, is known, the number of revolutions which the balls make in a minute, will bo found bij dividing the number, 103,320, by the length of the arms expressed in centimetres, ami then extracting the square root of the quotient. The weight of the balls, according to the resistance they have to encounter, is as important to detcnnino as the length of the suspending-arms, in order that the governing action of the pendu- lum may be suflSciently powerful and quick. It often happens, in badly designed engines, that the governor produces no effect, be- cause the length of the suspending-arms is not proportionate to the velocity, or because the weight of the balls is not proportionate to the resistance to be overcome. We have considered that it would be a great convenience to engineers and artisans to possess a table, sho\^ing at sight the ve- locities and corresponding lengths, for the conical pendulums, or ball-governors, generally employed in steam-engines, so as to enable them to determine with certainty the exact proportions to be given them, in relation to their spindles and driving-gear. When these points are determined, the weights of the balls may be easily adjusted. TABLE RELATIVE TO THE DIMENSIONS OF THE ARMS AND TO THE VELOCITIES OF THE BALLS OF THE CONICAL PENDULUM OR CENTRIFUGAL GOVERNOR. Number of Square of the Velocities. Length of Difference of Length of Armt Revolutions per Minute. Pendulum in Centimetres. Length for one Revolution. with an Augle of SU". Cent. Mill. Cent. 25 625 143-1 108 16 26 676 132-4 96 153 27 729 122-7 86 142 28 784 114-1 77 132 29 841 106-4 70 123 SO 900 99-4 63 116 31 961 93-1 67 107 32 1024 87-3 52 101 83 1089 82-1 48 95 34 1156 77-4 44 89 35 1225 73-0 40 84 36 1-296 69-0 37 80 37 1369 65-3 34 75 38 1444 61-9 31 71 39 1521 58-8 29 68 40 1600 55-9 27 64 41 1681 53-2 25 61 42 1764 50-7 23 68 43 1849 48-4 22 56 44 1936 46-2 20 63 45 2025 44-2 19 51 46 2116 42-3 18 49 47 2-209 40-5 17 47 48 2304 38-8 16 45 49 2401 37-3 16 43 50 2500 35-8 14 41 51 2601 34-4 13 40 52 2704 33-1 12 38 63 2809 81-8 12 37 54 2916 30-7 11 88 55 3025 29-6 10 84 56 3136 28-5 10 S5 57 3249 27-5 9 83 58 3364 26-6 9 8] 59 3481 25-7 8 80 60 8600 24-8 8 29 61 3721 •24-8 8 28 62 3844 23-3 7 27 63 3969 22-6 1 20 64 4096 21-9 7 26 65 4225 21-2 6 24 66 4356 •20-5 6 24 67 4489 1 9 • 9 6 28 68 4624 19-;i 23 Note.— With an nnglo of 80", tlio oontrifugivl force is tho 8*me for nil loiitrlliH of pondiiliim. Tills l;il)lf may also Ix' ooiisulloil in llu> oiiso of singlo-nrmod pondu- linns, wlnchiii.MHc'ii-^ioiuilly .inployod, instciid of oontrifuicnl govor- inns. 154 THE PRACTICAL DRAUGHTSMAN'S CHAPTER XI. OBLIQUE PEOJECTIONS. APPLICATION OF RULES TO THE DELINEATION OF AN OSCILLATING STEAM CTLINDEK. PLATE XLI. 419. In geometrical drawing, the planes of projection on which the objects are represented, are chosen, when possible, so as to be [larallel to the faces of such objects ; from which it follows, that these are expressed in their exact shapes and dimensions. It is often, however, that the position of certain parts of the machine or apparatus to be drawn, are inclined in regard to the other parts, so that all the surfaces cannot be parallel to the geometrical planes. The projections of the inclined parts are oblique, and, consequently, are seen as foreshortened. The general method employed in projections is evidently appli- cable to the delineation of oblique projections. It is, however, necessary first to represent the objects as if parallel to the plane of the drawing, so as to obtain the exact proportions and dimen- sions, such views being auxiliary to the production of the oblique representations. 420. Thus it is proposed to represent a hexagonally-based prism or a six-sided nut, the edges of which are inclined to both the hori- zontal and vertical plane. We first of all represent this nut, in fig. 1, as placed with its base parallel to an auxiliary horizontal plane, represented by the line, L T, fig. 3. This gives the regular hexagon, abed ef. If we were to make the vertical projection of this prism on a vertical plane, parallel to one of the faces, or to a d, we should, in this second auxiliary plane, have the projection of the edges, abed. The straight line, l' t', fig. 3, indicates the line of intersection of these two auxiliary planes, when placed in their actual position with regard to the nut ; and it is, therefore, the base line of the two projections. This line forms, we shall suppose, the angle, l o l', with the base line of the actual drawing in hand, which angle, likewise, expresses the amount of inclination of the top and bottom of the prism, with the actual horizontal plane ; whilst the angle, y o o^, formed by the perpendiculars, drawn to each of the lines through the point, o, expresses the amount of inclination of the edges and axis of the prism with regard to the vertical plane. After this, it is merely necessary, in order to obtain the points, a', b', c', d', to set otf to the right and left of the point, o, on the line, l' t', the distances, a o or d o, and b g or e g, derived from fig. 1. Drawing peipendiculars to the line, l' t', through each of the points, a\ b\ c", d\ and limiting them by the lines, a' d' and a' d', fig. 2, parallel to the former, we obtain the entire vertical projection of the prism, as upon the auxiliary plane, parallel to one of the faces, as e b. When one of the bases of the nut is rounded, or terminated by a spherical portion, which is generally the case, as already seen (186), its con- tour is limited by circular arcs, expressing the intersection of each race "'ith the sphere. We can then, by means of the two projections, figs. 1 and 2, obtain the oblique projection, fig. 4, upon the vertical plane, l t ; fig. 1 giving the widths, the distances of each of the points from the axial line, a d, which passes through the centre, o, and fig. 2-, defining the vertical heights or distances of the various points above the horizontal plane. To this end, through any of the points, as c, for example, ex- pressing the horizontal projection of the edge, c" c', erect a vertical line, and through the corresponding points, c° c', fig. 2, draw a couple of horizontal lines, cutting the vertical in c" and c"'. The same operation is performed with regard to the points, b, a, d, &c., which are projected in b'", a'", d", d'", fig. 4. The whole matter consists, therefore, in drawing vertical lines through each of the points in fig. 1, and horizontal lines through the corresponding points in fig. 2. The intersections of these lines give the projec- tions of the extremities of each of the edges in the oblique view, fig. 4. If it is wished to obtain the projections of the circular outlines with minute exactness, it will be necessary to determine, at least, three points in each arc ; and as we have the extremities already, we only require now to find the middle of each. It is the same for the circle representing the central opening of the nut. Its oblique projection is necessarily an ellipse, the proportions of which are obtained by the projection of the two diameters perpendicular to one another, one of which, m n, is parallel to the vertical plane, and does not alter in magnitude ; consequently, giving the transverse axis of the ellipse, whilst the other is inclined and foreshortened, and gives the conjugate axis. 421. In general, the oblique projection of any circle is always an ellipse, the transverse axis of which is equal to the actual diameter of the circle, whilst the conjugate axis is variable, according to the inclination or angle which the plane of the circle makes with one of the planes of projection. The application of this principle wall be seen in figs. 5, 6, and 7. The two first of these figures repre- sent the horizontal and vertical projections made upon the auxiliary planes of a portion of the cylindrical rod. A, of the piston, b, work- ing in the oscillating steam-cylinder, c ; and the last, fig. 7, is the oblique projection of this part of the piston-rod upon the vertical plane, corresponding to that of the drawing. It will be remarked, that the upper part of the fragment of the rod being limited by a plane, k I, perpendicular to its axis, is pro- jected as an ellipse, the transverse axis, p q, of which is equal to k I, whilst its conjugate axis, I' k', is equal to the projection of this line, k I, on fig. 7. The cylindrical fillets, r s, t u, &c., of this rod, are projected obliquely, as similar ellipses, of which portions only are apparent. For the torus, or ring, which is comprised between these two fillets, the oblique projection is a curve, which results from the intersection of an elliptical cylinder, the generatrices of which are horizontal, and tangent to the external surface of the torus. If, therefore, we wish to determine this curve with great precision, we must use the very same method adopted in determining the shadow proper of the external surface of the torus (323). In BOOK OF INDUSTRIAL DESIGN. 155 practice, however, when the drawing is on but a small scale, we may content ourselves with determining the principal points in the eurve, by projecting first the point, v, situated upon the middle of tne diameter, y y', of the torus, and drawing through it the line, ii' t)', equal to the diameter ; and, secondly, di'awlng the horizontal lines touching the external contour of the torus in the points, z, z', fig. 6, over to z% z^ upon the axial line, I' o', fig. 7 ; then di-aw an ellipse with these two lines, d' d' and z^ z^ for the transverse and conjugate axes respectively. The key, d, which passes through the rod, a, being rectangular in section, is projected in fig. 7, by a couple of rectangles, as indicated by the dotted projection lines. 422. Proceeding upon these principles, we can make oblique projections, in a very simple manner, of various objects, more or less complicated in form, when we have already the projections of these objects upon auxiliary planes, making any known angle with the actual plane of the drawing. Thus, figs. 10 and 13 are the oblique projections of an oscillating steam-cylinder, the first repre- senting the cylinder in external elevation, whilst the second is a section made through the axis of the cylinder. It is easy to see that these projections have been obtained in the same manner as those already given in figs. 4 and 7 ; that is to say, the external projection, fig. 10, is derived from the two right pro- jections, figs. 8 and 9 — one made upon an auxiliary vertical plane, parallel to the axis of the piston-rod, and perpendicular to the axial lines of the trunnions, and the other upon a horizontal plane, parallel to the cylinder ends, and, consequently, perpendicular to its axis. All the diiferent parts of this cylinder are, in fig. 10, project- ed by straight lines and ellipses, accordingly as they are rectilinear cr circular in contour. It is the same with the section, fig. 13, ind the horizontal projection, fig. 14, which are derived from the two right projections, figs. 11 and 12, made upon auxiliary planes ; one vertical, and passing through the axis of the cylinder, and through the valve-casing, whilst the other is perpendicular to this axis, and passes through the line, 1 — 2, fig. 11. The dotted work- ing lines, indicated upon the various figures, show sufficiently clearly the various constructions necessary to obtain these oblique projections. We have, moreover, applied numbers to the diflerent parts projected, and more particularly to the axes or centre lines, which show at sight what parts correspond with each other upon the diiferent projections. 423. These drawings represent the cylinder of a steam-engine. different from that which we have already described. The pre- sent one is called an oscillating steam-engine, because, instead of the cylinder being vertical and immovable, it oscillates during the motion of its piston, b, upon the two trunnions, e, carried in suita- ble bearings in the engine-framing. This arrangement of oscillat- ing cyhnder has the advantage of dispensmg ^\-ith the parallel motion, and of attaching the rod, a, of the piston, directly to the crank-pin, to which its motion is transmitted, without the intervention of any connecting-rod. In the head, h, of the rod, there is, consequently, formed a bearing, which embraces the crank-pin. The bottom of the cylinder is cast in the same piece with it, but it has a small central opening, for the passage of the spindle of the boring tool, by means of which the interior of the cylinder is turned smooth and true. This opening is closed by a cast-iron cap, f, bolted to the bottom of the cylinder. Against a planed face, upon one side of the cylinder, is fitted the valve-casing, g, which receives the steam direct from the boiler, and has within it the valve, h, which has an alternate rectilinear movement, at the same time oscillating along with the cylinder. During this movement, the valve alternately uncovers the ports, a, b, fig. 11, which conduct the steam to the top and bottom of the cylinder. A blade spring, I, attached to the inside of the valve-casing, at the back of the valve, constantly keeps the latter well up against the planed valve face. The steam coming from the boiler introduces itself into the cas- ing through the passage, c, fig. 12, which communicates with ono of the trunnions, e, and the escape of the steam, when it has acted upon the piston, is effected through the e.xit channel, d, which com- municates with the other trunnion. The piston, b, is composed of a cast-iron body, on the outer sur- face of which is cut out a groove, to receive the hempen packing, i, partly covered by an elastic metal ring, h, coinciding exactly with the inside of the cylinder. Oscillating cylinder-engines have always been admired for their simplicity and beautiful action ; but it is only of late years, and now that such superior workmanship is attainable, that such en- gines have been constructed of considerable size. The aptness of this arrangement for engines of the largest size has lately been demonstrated by Penn, in the case of the Great Britain, and other large vessels. CHAPTER XII. PARALLEL PERSPECTIVE PKINCIPLES AND AITLICATIONS. PLATE XI,1T. 424. Wo give the name of parallel perspective to (ho represen- tation of objects by oblique projections, which diirer from the preceding, in so far that the visual rays, which we have hilluilo supposed to bo always iiorpeiidicular to the geometrical i)lMncs, form, oil the contrary, a ccirlain angle with tluwc pl;iims, r<'iii;iin- ing, however, coiistnnlly parallel to owh other ; from which it follows, that all the strMiirht lines, which •.no panillol in the object, niainl.'iiii their p;ii;illclisin in tin- iMcIure, according to this syslciu of |u'rsi)octivc. .Mthougii, in genonil, it is imin:itcri:il wiiat ln» ani'lo of inclination is, it is lU'Vcrllieh'ss prol'crable, in ri'jfuiai 156 THE PRACTICAL DRAUGHTSMAN'S drawing, to adopt some particular angle as a matter of convention, which will have the advantage of giving the entire dimensions of the object in a single projection or view. l^et A B and a' b', figs. 1 and 2, be the two projections of a straight line, to which we wish to make the visual rays all parallel ; the vertical projection, a b, of this straight line, forms an angle, CAT, with the ground line, l t, which angle is, we shall suppose, equal to 30°, and its horizontal projection, a' b', is such, that the distance of the point, a', from the point, a, where it touches the horizontal plane, is equal to twice the length, a b, of its vertical projection, the pouit, b, being that at which it touches the vertical plane. We shall proceed to show, by means of the various figures in Plate XT.TT., that, in taking the above straight lines as directrices for the visual rays, a single projection will be quite sufficient to express all the dimensions of any object. Instead of making the directrices of the different objects, represented in this plate, to coin- cide with the actual projection of the straight lines which we have just indicated, we have, by preference, chosen the mere setting of these same lines round at an angle of 30°. Thus, the lines, c' d", k' i', fig. 3, are the straight lines perpendicular to the planes of projection, set round to the angle in question ; whUst, on the contrary, the straight line, y z, represents the projection of these lines properly parallel to the horizontal projection, a' b', in fig. 2. 425. The finished view, fig. z^, is the representation in parallel, or, as it is sometimes called, ya/se perspective, of a prism, e, with a square base, resting upon a plinth, f, also prism-shaped and square. In the first place, we suppose this prism to be represented in the horizontal projection, fig. 3, by two concentric squares, a' d' ef and h' i' k I; and in the vertical projection, fig. 4, by the rectangles, abed and g li ij. These projections are made upon the suppo- sition, hitherto acted upon, that the visual rays are perpendicular to the geometrical planes. If now, on the contrary, the visual rays make with the planes of projection an angle equal to that of the given straight line, figs. 1 and 2, each of the faces parallel to the vertical plane continues to be parallel to this plane, and is represented by a figure equal to itself, whilst all the faces perpendicular to the two planes are in projecting rendered oblique, in such a manner, that the lines hori- zontally projected become parallel to a' b', fig. 2 ; and those verti- cally projected, to a b, fig. 1. Consequently, if through the points of the projecting angles, a, h, i,j, fig. 4, are drawn straight lines, parallel to a b, they will express the directions of all the edges perpendicular to the vertical plane. Since then, as we have already stated, the length of the projection, a b, fig. 1, is equal to one-half the perpendicular, a a', fig. 2, if from the points, a, /i, i, j, and on each side of them, we measure oflf, upon the oblique lines just drawn, the distances, a a' and a/", li h^ and h P, i i?, and i Ir, &c., respectively equal to half the lengths, a' m and h' n, &c., we shall, in fig. 4, have the perspective representation of the various straight lines perpendicular to the vertical plane ; and as all the other edges are parallel to this same plane, such lines as are actually vertical are represented as vertical, whilst all lines parallel to the base line remain horizontal. Thus, the edges, a b, h g, d c, ij, being vertical, are in the parallel perspective repre- sented bv the verticals, a' b", h^ g^. i' ;", d^ a' ; and likewise the edges, a d, b c,h. i, &.C., which are parallel to both planes of pro jection, are rendered by the straight lines, a" d?, b' c', K' -P, &c., parallel to the base line. It \\i\\ be easOy seen, that by adopting the angle we have indi cated for the direction of the visual, or more correctly termed, representative ray, that the one single view in parallel perspective is suflicient to make known all the dimensions of the object : for, on the one hand, we have the exact widths and heights of those faces which are parallel to the plane of the projection, as if the perspective view did not differ from an ordinary geometrical pro- jection, in which the representative rays are supposed to be per- pendicular to the plane ; and, on the other hand, the oblique lines representing all the edges actually perpendicular to the vertical plane, and which are exactly equal to half the actual lengths of the latter. We may here observe, that the base of the prism being square, the sides, li I and i' k, are equal to the sides, h' i' or I k. Conse- quently, in order to construct the perspective or oblique projection, fig. 4, the plan, fig. 3, is not needed, since it would have answered the purpose equally well to have made the lines, d'^ e' or i^ /f", equal to the half of a d or h i. 426. The shaded view, fig. [B, represents a frustum of a regular pyramid, g, resting upon an octagonal base, h, the horizontal pro- jection of which is indicated in full, sharp lines, in fig. 5, and tin vertical projection in dotted lines, in fig. 6. According to the principle thus laid down, the perspective view is obtained, in the first place, by dra\nng all the lines which are perpendicular and parallel to the base line, fig. 5, and passing through the opposite angles of each of the octagons, representing the upper and lower bases of the pyramidal frustum, and of the plinth. Of these lines, all, a' d'.f e', h' i', &c., which are parallel to the base line, as well as the sider, p' q[, t' u, x! x', which are likewise parallel to the former, remain horizontal in the perspective ^iew, fig. 6, whilst, on the other hand, all the straight lines, such as y r', i' y', v' z', as well as the sides, a' f, h' I', i' k!, which are per- pendicular to the base line, become inclined at an angle of 30° from this line, as in fig. 6, or, in other words, parallel to the straight line, A B, fig. 1. If now, through the points, a, g, p, q, &c., and the points, li, o, i, of the two bases of the pyramidal frustum, we draw parallels to the straight line in question, and then mark off from each of those points, aud on each side of them, the distances, a a', g g'jp p\ h ^ &c., respectively, equal to the semi-lengths of the corresponding sti-aight lines, mf, g' n, &.C., of fig. 5, we shall have the parallel- perspective representation of all these lines ; and consequently, by joining the extreme points of each of them, we shall also have all the lines representing the contours of the two bases ; and further, by joining the angles of these bases, we define the lateral faces, and complete the view, fig. 6. 427. The pai-allel-perspective representation of a cylindrical object, of which the axis is perpendicular to the vertical plane, as in the finished example, fig. ©, may be determined without the assistance of the horizontal projection : that is, when the length of the cylinder is known, as well as that of any other pai-t which may be perpendicular to the vertical plane. i Let ab c dgfe, fig. 7, be the vertical projection of this object, BOOK OF INDUSTRIAL DESIGN. 157 the perspective of its base, abed, will be parallel to a b. The circles which have their centres at o, being parallel to the vertical plane, are represented in perspective by two circles equal to them- selves ; and their position is obtained by drawing through the point, 0, the straight line, o' o% parallel to a b, fig. I, and marking off a distance, lying equally on both sides of the point, o, equal to half the length of the cylinder, measured in the direction of the axis, perpendicular to the vertical plane. Then with the points, o\ 0°, as centres, describe the circles with the equal radii, o'/' and o' i', straight lines,/'/'' and t' i", drawn tangential to the circles, and parallel to the axis, o' o^ express in perspective the genera- trices of the two cylinders forming the contour of the object. The cylindrical pieces which join the cylinder to the base are determined in the same manner by means of the line, n ri\ drawn through the centre, n, of the circle, d g, parallel to o' o", and by the distances, n w', n n', together equal to half the actual length of these cylindrical surfaces. The base is drawn as in the preceding example. 428. The example, fig. E), represents a cone resting upon a cylindrical base, both cone and base having the same axis perpen- dicular to the horizontal plane. This cone and cylinder are pro- jected on the plan, fig. 8, in sharp lines, and in the elevation, fig. 9, in dotted lines. The circles, fig. 8, representing the bases of the cone and cylin- der, are to be divided into a certain number of equal parts ; and through the points of division, 1, 2, 3, &c., perpendiculars are drawn to the ground line, and are prolonged as far as the horizontal line, a' o', which is the vertical projection of the two bases. Through the points, a', b', c', o', are drawn straight lines parallel to a b, fig. 1 ; and on each of these are set off the distances, a' 2', b' 3', c' 4', o' 5', &c., respectively equal to half the lengths of the perpendicu- lars, 2 ffl, 3 6, 4 c, 5 0. &c., which oi)eration gives the points, 2', 3', 4', 6', &e., through which an ellipse must be traced, to represent the perspective of the base of the cylinder. In the same manner we obtain the points through whicli passes the ellipse, representing the perspective of the base of thS cone. The heights of the cone and its base remain precisely what they really are, in consequence of their common axis being parallel to the vertical plane ; but this is not the case with their bases, which, being horizontal, are projected obliquely, in the form of the ellipses wo have just drawn. The apex of the cone, at the upper extre- mity of the vertical axis, does not change, and for the generatrices, or sides of the cone, it is simply necessary to draw through the apex the straight lines, o^ m and ff' n, tangents to the ellipse repre- senting the base of the cone, whilst the generatrices of the cylinder are tangents to the two ellipses representing its upper and lower bases. 429. The example, fig. (E, is the parallel-perspective reprcsenta^ tion of a metal sphere or knob, attached to a polygonal base by a circular gorge, forming altogether an ornamental iicad for a screw Figs. 10 and 11 are the horizontal and vertical prujictions of this piece. Wo must, in the first place, remark that tiio s])li('ro, tlio radius of which is o a, may be determined in its (lerspoctivo represeiitji- tion in several ways. First, by imagining the hori/ontal sections, rt h, c d, ef, which give in plan the circles, with the ladii, «' o, c' o', and e' o', and the perspectives of each of these circles may be obtained by operations similar to the preceding, v/hich will give a series of ellipses, to be circumscribed by another ellipse, tangen- tial to them all. Second, by drawing the planes, g h, ij, parallel to the vertical plane, and which are projected in perspective as circles, with the radii, I" g, i m, the centres being upon the oblique line, n n', parallel to the line, a b, fig. 1, and passing through the centre, o, of the sphere. The external curve, drawn tangential to all these circles, will be elliptical, as in the preceding method, and represent the perspective of the sphere. Thirdly, by at first drawing through the centre, o, an oblique line, n n' ; then a per- pendicular, e e", passing through the same point. Then set out from the centre, o, and on each side, the distances, o e, o e", equal to the radius, o a, of the sphere, which gives the conjugate axis of the ellipse. To obtain the transverse axis, it will be necessary to draw tangents to the great circle of the sphere, parallel to the oblique ray of projection, a b, fig. 1, as brought into the vertical plane. This line is brought into the vertical plane, as at a' b. in the following manner : — At the point, a, a perpendicular to a b is erected, and the distance, a" a, is made equal to a a', when a" b is joined. These tangents touch the sphere in the points,/, /', which may be obtained directly by drawing through the centre, o, the line,//, perpendicular to a" b. These tangents, further, meet the line, n n', in the points, n, n' ; and the distance between these points is, con- sequently, the transverse axis of the ellipse, which represents the perspective of the sphere, and which may be drawn according to any of the known methods. This last method is evidently the shortest and simplest of the three for obtaining the perspective of the sphere, but it is confined in its application ; for any other surface of revolution, as, for exam- ple, the gorge, which unites the sphere to the hexagonal base, cannot be defined in this way. In cases where the axis of the surface of revolution is vertical, as in this example, it will be neces- sary to adopt the first general method, which consists in taking horizontal sections. When, on the other hand, the axis of the surface of revolution is horizontal, we must employ the second process, which consists in taking sections parallel to the vertical plane, or perpendicular to the axis. The perspective of the thread of the screw, of which the sphere is the ornamental head, is do- terminable in a manner analogous to that of a circle. It is sufli- cient, in fact, first to draw the two geometrical projections, figs. 8 and 12, of one or two convolutions of the thread, and to find the perspectives of the very points wiiich iiavo served for tiic construc- tion of the helices. Thu.s, for example, wo put llio circle,/;)!/, fig. 8, into perspective, as at I* i>* q*, fig. 13, rotaining tor this jmr- pose the same points, /, /;, </, &c., which wore oinployod in dolino- ating the screw, fig. 12. 'J'hrough llicso points, l\p\q*, &c., draw viutical lines; and upon (Ikiii .set oil" the distances, V P, /' /', yi' ;<', &c. Then through the points, l\ f, &c., draw liio curve, whicli will he the p.i siuctivc of Ihc outi'r holix of Iho scrow-tliroad. liy giiiiiu' llniHinh Ihi' sniuc o|irra(ion for tlio iiinor circle, r s I, li.,'. 10, wo shall ohlain ihc sinilhiily |>c|-Niicclivo oullilio of llu> inilor helix. An I'Miiiiiiialion nt lig. 13 will furlhcr show iIimI iho hoiglil.>* on (lie vertical linos are precisely Uio siuiio for Ik li lulicos, for th«<> l68 THE PRACTICAL DRAUGHTSMAN'S are taken upon radii common to the two circles, l" p" and rst, con- sidered as bases. We have deemed it unnecessary to enter further into the deve- lopment of this species of perspective, of which we have already given a general application in the boring-machine, represented at fig. 1, Plate XXXV., an example in which are collected almost all the various forms which present themselves for delineation amongst mechanical elements and machinery. In that example, as well as in the figures in Plate XLIL, which we have just been studying, we have supposed the representative or visual rays to be in all cases parallel to each other, and to be inclined at an angle of 30° with the ground line in the vertical pro- jection, in such a manner that a single view serves to express all that two, or even three, geometrical projections can do, showing not only the external contours of the objects, but also whatevei- may be upon their surface. It will be easy to comprehend the utility of this system of giv- ing, at a single view, a general and precise idea of the actual relieve appearance of any object. It is a manner of representation often more intelligible to the generality of people than a series of geo- metrical projections, and in many cases it will greatly simplify the process of sketching buildings or machinery. CHAPTER XIII. TRUE PERSPECTIVE PRINCIPLES AND APPLICATIONS. — ELEMENTARY PRINCIPLES. PLATE XLIII. 430. Perspective, properly so called — ^but here defined as true or exact perspective — differs from parallel or false perspective, in its being founded upon the actual manner in which vision takes place ; that is, that instead of being parallel to each other, the visual rays converge to a point. An object is said to be drawn in perspective when, on viewing the drawing from a particular point, it presents the same appearance to the organs of vision as the object repre- sented itself does when similarly viewed. The visual rays, or impressions, travel from the object in straight lines, converging to a point at the eye, and forming a cone of rays. Let us suppose this cone to be intercepted by a trans- parent plane, or diaphragm, of any form — then, noting the points where the rays from the various parts of the object pierce this diaphragm, let us paint upon it the outline, and complete the picture with colours, which to the eye shall have the exact appear- ance of those of the object itself, modified, as they may be, by distance, position, form, and by their being in the light or shade. After doing this, we may remove the object, but the picture upon the diaphragm will make the same impression upon the eye that the object would itself. It is to the representation of objects in this exact and natural manner, that the art of drawing in perspec- tive is devoted. The fixed point, to which the rays converge, is called the point of sight ; and in diagrams explanatory of perspective, it is shown, together vsath the converging rays, as projected into the plane of the picture. To determine the perspective delineation of an object mathematically, we must have given us the horizontal and vortical projections of the object ; also those of the point of sight, and the position of the plane of the picture ; and the general problem of perspective then reduces itself to conceiving the visual rays as passing from the various points of the object to the point of sight, ana to find the intersection of these rays with the plane of the picture. FIRST PROBLEM. THE PERSPECTIVE OF A HOLLOW PRISM. 431. Let A and a', figs. 1 and 2, be the horizontal and vertical projections of the prism which we wish to delineate in perspective, the point of sight being projected in v and v', and the plane of the picture, in t and t', being supposed to be perpendicular to the planes of projection, and vertical, as is generally the case. Through the point, v, in the horizontal projection, draw visual rays from each of the points, a, b, c, d, appertaining to the external contour of the prism. The intersection of these points with the plane of the picture determines the points, b^, a", n\ cP, which give, upon the horizontal projection, t o, of the latter, the perspective of the points, a, b, c, d. In like manner, through the point, v', in the vertical projection, draw the visual rays, a! v', b' v',/v', e v', intersecting the plane of the picture in the points, a", b",f", and e" ; these last being, conse- quently, the vertical projections of the perspective of the points, a', b', e,f. As, because of the position given to the plane of the picture, the perspective of the object is not visible, since all the points are situated in the vertical line, t t', the plane being represented on edge, we must imagine this plane as turned over upon the vertical plane of the drawing; whilst we must suppose the line, t o, the horizontal projection of the picture-plane, as turned about the point, 0, as a centre, through a right angle, bringing it to coincide with the ground line, l m. When this is done, the points, a", i^ c^ d'', will describe arcs of circles, and, finally, coincide with the points, a\ b*, c\ d\ on the ground line. Next, upon these last erect perpendiculars, to meet the horizontals, drawn through the points, a", b",f", e" ; the points of intersection of these cross lines will be the perspectives, a*, b*, c*, d*, of the corners, a, b, c, d, ot BOOK OF INDUSTRIAL DESIGN. 159 (he top of the prism. We have, likewise, the points, e^/^ g, h, for the perspectives of the corners of the bottom of the prism, which is parallel to the top; consequently, by joining all these points together in couples, as indicated in fig. A, we obtain the entire perspective of the external outline of the prism. As the prism is hollow, we shall see in the perspective view the outline, i' m! v! o', corresponding to the edges of the part hollowed out. The point of sight, of which v and v' are the geometrical projec- tions, is projected upon the picture-plane in the points, d, d° ; and when the picture-plane is turned over, the point will be found at d", which is the position of the point of sight upon the perspective drawing. Tt must be observed, that in this example the lines, a* Z»S a* d^, and 6' c*, which express the perspective of the corresponding lines, ah,a d, and h c, are the intersections with the plane of the picture, of planes passing through these lines, and through the point of sight. Now, since the intersection of two planes is always a straight line, the following conclusions may be drawn ; that, 432. First, The perspective of a straight line upon a plane is a straight line. It may also be remarked, that the verticals, such as b* e\c*k,d*g, are the perspectives of the vertical edges, projected in the points, b, c, d; whence we deduce that, 433. Secondly, The perspectives of vertical lines are verticals, when the plane of the picture is itself vertical. It will further be seen, that the horizontals, I* c*, d* a*, ^ h,f^ g, of the perspective view, correspond to the straight lines projected horizontally in 6 c and d c, which are parallel to the picture ; whence it may be gathered, that, 434. Thirdly, The perspective of any straight horizontal line, parallel to the picture, is itself horizontal. Further, it follows from the two preceding principles, that all lines parallel to the plane of the picture are represented, in per- spective, by lines parallel to themselves. Finally, the straight lines, a* b*, d* c*, ^ f, and h g, which all converge to the same point, v", the projection of the point of sight upon the plane of the picture, correspond to the edges, ab, dc, ef, which are horizontal, but perpendicular to the plane of the picture; whence it follows, that, 435. Fourthly, The perspectives of lines which are horizontal, but perpendicular to the plane of the picture, are straight lines, which converge to the point of sight, and are consequently foreshortened. It will be seen, from figs. 1 and 2, that the whole width of the perspective representation, fig. A, is comprised between the points, i' and d^, which lie on the outermost visual rays, drawn in the horizontal projection ; and that its height is limited by the two points, a" and e", which correspond to the extreme visual rays in the vertical projection. The angle formed by the extreme visual rays is termed the optical angle. In the present example, this angle, U' V d'', in the horizontal projection, dilfers from the angle, a" v' e", in the vertical projection. The positions of the object and point of sight being given, the ditncnsions of the per.spoctivo rci)r(!scnt;iti()n vary acconling to the position of the plane of the picture. It will thus bo seen, on teforring to fig. 1, that if this plane bo removed from t x' to < t', loarer to the object, the limits to the porspuctivo roprusentatlon by the extreme visual rays will be enlarged ; whilst, on the con- trary, if we remove the plane of the picture to the position, f i\ nearer to the point of sight, the limits will be sensibly narrowed. Again, if, in place of moving the plane of the picture, the point of sight is removed further off, or brought nearer to, the size of the perspective outline will thereby be augmented or diminished. It may therefore be concluded, that, 436. Fifthly, The dimensions in the perspective representation do not wholly depend, either on the actual size of the object, or on the distance from which it is observed, but also on the relative distances of the point of sight and of the object from the plane of the picture. Thus the sides, d a and c b, fig. 1, are actually equal, but the former is furtiier from the plane of the picture than the latter; so that whilst this is represented by the space, c* b*, fig. A, that is limited to the much smaller space, d'^ a*, in the perspective view. SECOND PROBLEM. the perspective of a cxlinder. Figures 3 and 4. 437. To obtain the perspective outline of a vertical cylindei, sucn as the one projected horizontally in b, fig. 4, and vertically in li , fig. 3, we proceed, as in the preceding example, to draw througn the point of sight, v' v, a series of visual rays, extending to the various points, a, b, c, d, e, taken on the upper end of the cylinder, by preference at equal distances apart. These lines intersect the .plane, t t', of the picture, in the points, d'', c', a', g', &<;., in the horizontal projection, and in the points, a", c", d". Sic, in the verti- cal projection. By bringing the plane, t t', of the picture, into that of the diagram before us, or what is equivalent to it, by finding the points, g\ a\ &c., by means of arcs, drawn with tlie centre, o, on wliich the plane is supposed to turn, and drawing the horizontal lines through the corresponding points in the vertical projection of the picture-plane, we obtain the points, c*, d*, a*, g*, Slc., which are points in an elliptic curve representing the top of the cylinder in perspective, which is visible, in consequence of the point of sight being above it. The same points, a, b, c, d, of the horizontal projection, give, in combination with their vertical projections, g^',/', c", &c., the per- spectives, d°, e°,/', g•^ of the bottom of tlio cylinder, of which, obviously, only a part is visible. The two vertical generatrices, d* d'',g* g'', being drawn tangential to the upper and lower ellipses, complete tho perspective outlino of the cylinder, fig. S. As this cylinder is hollow, nn operation similar to tho preceding will 1)0 called for in delineating the upper visible edge of Uie hol- lowod-out i)()rti()n. It in\ist 1)0 observed that, in taking an oven nmnbor of divisions, at eiiu:il ili.stain< - ai)iirt, upon tiie horizontal projection of tho cylinder, and setting Ilieni off from tho dinniofor, c g, immllel to tho plane of the picture, wo have always ii couplo of points situated upon tho same perpendicular to the jilane, and of which the [ht- speetives are, conseiiuently, situated on the siune stniight lini- drawn through tho projection, i', of the point of sijjht Thui 160 THE PRACTICAL DRAUGHTSMAN'S occurs with the points, b, d, the perspectives, b*, d*, of which, are situated upon the line, v' d\ so that we have a means of verifying the preceding construction. THIRD PROBLEM. the perspective of a regular solid, when the point of sight is situated in a plane passing through its axis, and perpendicular to the plane of the picture. Figures 5 and 6. 438. Let V and v' be the projections of the point of sight, situ- ated in the vertical plane, v o, passing through the axis, o o', of the solid, c c', and perpendicular to the plane, t t', of the picture. It will be seen at once, that in the perspective view, fig. ©, this point must be projected on the vertical line, v" v\ representing the axis of the object, and in relation to which all the lateral edges of the object are synametrical. Such are the sides, a b and c d, which are perpendicular to the plane of the picture, and which are repre- sented in perspective by the Imes, a* b* and d^ c*, both directed to the point of sight, v". It is the same with the edges, /^, h i, of which the perspectives, /* g*, Ti* i*, likewise converge to the point of sight, i;'. As for the vertical edges, they retain their vertical position in the perspective, and the horizontal lines, a d,l m, nk, c b, parallel to the plane of the picture, are rendered in the perspective view by parallels, such as a* d*, I* m*, n* k*, c* b*. It may be gathered from the solution of this problem, that when- ever the point of sight is in a plane, passing through the axis of a regular solid, and perpendicular to the plane of the picture, the perspective representation will be symmetrical with reference to the centre line, and that it is, therefore, quite sufficient to go through the constructive operations for one side only of the figure. FOURTH PROBLEM. the perspective of a bearing-brass, placed wath its axis vertical. Figures 7 and 8. 439. The pomt of sight being situated, as in the preceding example, in a vertical plane, passing through the axis of the object, and perpendicular to the picture-plane, the perspective vnil like- wise be symmetrical in reference to the centre line, v" v" ; and it is unnecessary to notice this peculiarity further. The inside of the brass being cylindrical, and being terminated by horizontal semi- circular bases, the perspective of these bases will be rendered by a couple of regular semi-ellipses, of which it will be sufficient to determine the axes. The transverse axis, a* c*, is equal to the perspective of the straight line, a c, which is horizontal and parallel to the picture-plane ; and the semi-conjugate axis, b* d', is equal to Ine perspective of the line, b d or b' a', which is also horizontal, but is perpendicular to the plane of the picture, and is conse- quentlv foreshortened, whilst it coincides with a line passing througn the point of sight, v". It will be remarked, that the transverse axis of the ellipse, corresponding to the upper end of the semi-cylinder, is equal to a* c*, but that the conjugate axis is foreshortened to a greater extent than that, b* d*, of the lower one, in consequence of being at a less distance below the point of sight. The effect of the perspective, in this case, is well rendered in fig. E). FIFTH PROBLEM. THE PERSPECTIVE OF A STOPCOCK, WITH A SPHERICAL BOSS. FiGDKES 9 AND 10. 440. In can-ying out the general principle, it will be conceived that, in obtaining the perspective representation of a sphere, we should draw through the point of sight a series of rays, tangential to the outer surface ; but in order to ascertain the points of con- tact of these tangents, it will be necessary to imagine a series of planes as passing through the sphere, producing circular sections, and then to find the perspective of each of these circles. These being found, a curve, drawn to circumscribe them, will be the perspective of the sphere. In the example, figs. 9 and 10, the point of sight being still chosen as before — that is, as situated in a plane passing through the centre of the sphere, and perpendicular to the picture-plane — the perspective of the sphere wUl, on this account, simply be an ellipse, having for conjugate axis the base, a? ¥, of the optical angle, a' v b, in the horizontal projection ; and for transverse axis, the base, c" d\ of the angle, c" v' d", in the vertical projection ; because the right cone, formed by the series of visual rays tan- gential to, and enveloping the sphere, is obliquely intersected by the plane of the picture. If, however, the point of sight were situated upon a horizontal line, passing through the centre, o, of the sphere, the perspective of the latter would evidently bo a circle. The spherical part of the stopcock is traversed by a horizontal opening, for the reception of the key, and the edge, e /, of this opening, being situated in a plane parallel to the picture-plane, ■will, in the perspective, fig. [g, be rendered by a circle, of which the diameter is e'f\ As to the cylindrical flanges on either side of the stopcock, for forming its junction with a line of piping, the semicircle, a g b,is represented by a portion of an ellipse, having the horizontal lioe, a* b*, corresponding to a b; and the vertical, g' &, being the per- spective of ^ o" . The circle, alib g, the horizontal projection of the surface of the uppermost flange, is represented in the perspec- tive view by a perfect ellipse, as is also the upper visible edge oi the inner tubular portion. But this is the same as the case in fig. 3, and the ellipses may be formed in a similar manner. The perspective representation of a circle, however situated, otherwise than parallel with regard to the plane of the picture, is always a perfect ellipse ; but the transverse axis of the eOipse does not coincide with, or is not the representation of, any diameter of the circle ; for it is evident, that the more distant half of the circle must be more foreshortened in the perspective, and must occupy a less space than the anterior half, whilst the ellipse is equally divided by its transverse axis. BOOK OF INDUSTRIAL DESIGN. IP.l When the point of sight is in a line perpendicular to the circle, the rays from the latter will form a right cone ; and if the plane of the picture is not parallel to the circle, the section determined by it will be an ellipse, as is well known. Again, if the circle is not perpendicular to the central visual ray, the cone of rays will be nlliptical, and the sections of such cone will be ellipses, of various proportions, that parallel to the circle, however, being a circle. The transverse axis of the perspective ellipse is the perspective of that chord in the original circle, which is subtended by the arc, between the points of contact of the extreme visual rays, as pro- jected iu the plane of the circle. SIXTH PROBLEM. the perspective of an object placed in ant position with kegard to the plane of the picture. Figures 11 axd 12. 441. In each of the preceding problems, we have supposed one or other of the surfaces of the objects to be parallel or perpendicular to the plane of the picture ; but it may happen that all the sides of the object may form some angle with this plane. It is this case which we propose examining in figs. 11 and 12. Let a b c d he the horizontal projection of a square, of which the sides are inclined to the plane, t t', of the picture, and of which it is proposed to determine the perspective. The point of sight being projected in v and v', if we employ the method adopted in figs. 1 and 2, we shall find the points, a^ b^, c', d% to be the horizontal projections, and a", b", c", d", the vertical projections of the corners of the square ; and when we have brought the plane of the picture, T t', into the plane of the present diagram, as before, we shall find the actual positions of these points to be at a*, b*, c*, d*. If we join these points, we shall have a quadrilateral figure, of which the two opposite sides, a* b*, c* d\ converge to the same point, /, whilst the other two sides converge to the point, /'. These two points are termed vanishing points. They are determined geometrically, by drawing fhrough V, the horizontal projection of the point of sight, the itraight lines, v t and v t', parallel to the sides, a b and b c, of the given square, and prolonging these lines until they cut the line, T t', representing the plane of the picture. Having drawn through v' the horizontal, v' v', termed the horizontal lino or vanisliins -plane, set off the distance, v t, from v" to /, and the distance, 1) t', from v" to /', and / and /' will be the required vanishing points. It follows from the preceding, that When the straight lines which are inclined to the plane of the Yiclure are parallel to each other, their perspectives will converge in one point, situated on the horizontal line, and termed the vanishing voint. When several faces or sides, situated in difterent planes, are parallel to each other, their perspectives all converge to the same vanishing point, which allows of a great simplification of tho operations. Tims, fhe edges of the horizontal faces, h' i' and h" i", of tho <|iia(lrangular prism, being respectively parallel to the .sides of tho square, abed, are represented in perspective by the straight lines, i'e\ i* e*, converging to the first vanishing point,/, and the straight lines, i" g', i* g*, converging to the second point,/'. The cone, f f', which is traversed laterally by the prism, has its apex projected horizontally in the point, s, fig. 12, and vertically in the point, s', which, with its axis, appertains to fig. 11. The per- spective of the point, s s', on the plane of the picture, is found, in the usual way, to be at s in the horizontal projection, and at s' in the vertical projection ; and when the plane of the picture is brought into the plane of the diagram, these points are represented by the points, s" and s", upon the same vertical, s' o', which is, consequently, the perspective of the axis, o s', of the cone, and the point, s", is the