Digitized by the Internet Archive in 2010 with funding from NCSU Libraries http://www.archive.org/details/practicaldraughtOOarme THE PRACTICAL DRAUGHTSMAN'S BOOK OF INDUSTRIAL DESIGN: FORMING A COMPLETE COURSE OP ccjmnintl, (foghumitg, anfo ^rtjritcctoal glraluhig. TRANSLATED FROM THE FRENCH OP M. AKMENGAUD, AlNE, PROFESSOR OF DESIGN IN THE CONSERVATOIRE DES ARTS ET METIERS, PARIS, AND MM. ARMENGAUD, JEUNE, AND AMOUEOUX, CIVIL ENGINEERS. REWRITTEN AND ARRANGED, TOH ADDITIONAL MATTER AND PLATES, SELECTIONS FROM AND EXAMPLES OF THE MOST USEFUL AND GENERALLY EMPLOYED MECHANISM OF THE DAY. WILLIAM JOHNSON, Assoc. Inst., C.E., EDITOR OF "THE PRACTICAL MECHANIC'S JOURNAL." "The Philosopheb may very justly be delighted with the extent of his views, and the AimFiczE with the readiness of his hands : bat let the one remember, that without Mechanical performances, refined speculation is an empty dream; and the other, that without Theoeetical reasoning, dexterity is little more than a brute instinct." — Johnson. " The weakness of Accident is strong, where the strength of Design is weak."— Tapper. LONDON: LONGMAN, BROWN, GREEN, AND LONGMANS. 1853. 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 conspi- cuously 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 the 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 powers in compelling him to observe — as if the objects he delineates were really before his eyes. Besides, he always does that the best, which he best under- stands ; 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 pencillings — the jotted memoranda of a contem- plative brain, prying into the corners of contrivance — often form the nucleus of a splendid invention. An idea thus pre- served at the moment of its birth, may become of incalculable value, when rescued from the desultory train of fancy, and treated as the sober offspring of reason. In nice gradations, it receives the refining touches of leisure— becoming, first, a finished sketch, — then a drawing by the practised hand — so that many minds may find easy access to it, for their joint counsellings to improvement — until it finally emerges from the workshop, as a practical triumph of mechanical invention — an illustrious example of a happy combination 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 It PRE] tarnish that tnnscript of their minds which might make tlirir thonghts immortal. I only in mitigating this evil, it will not entirely fail in il I it will at least add ■ I the [adds of Intelligence, and fom a few more approach lot Perfection — i luut not lo«t an hoar whereof there ii a It A written thought at midnight will redeem the livelong day." . of Industrial 1' ngn is really as indispensably neoeesary a-* the ordinary rudiments of leamii • form an • --.-utial baton in the education of young persons for whatever profession or employment they may intend I the great business of th.ir lives ; for without ■ knowledge of Drawing, no scientific work, whether relating to Mechanics, Agriculture, or Manufactures, can 1"- advantageously studied. This is now beginning to acknowledgment, and the routi y in all varieties of educational establishments are being benefited by the introduction of the art. The special mission of the /'>■■ ■'<■<' Draughtsman's Book of Industrial Design may almost be gathered 1 fide-page. It is intended to tarnish gradually developed lessons in Geometrical Drawing, applied directly to the various branches of the Industrial Arts: comprehending Linear Design proper; Isometrical Perspective, or th< of Projections; the Drawing of Toothed W heels and Eccentrics ; with Shadowing and Colouring ; Oblique Projections; and tin Btudy of parallel and exact Perspective ; each division being accompanied by Bpecial applications to the litecture, 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 student- in elementary form has been carefully considered ; and after « very gi ometrical problem, a practical example of its application lias been added, to facilitate its comprehension and increase its value. The work is comprised within nine division-, approprated to the different branches of Industrial Design. The first, which ooncenm Linear Drawing only, treats particularly of straight lines — of circles — and their application to the delineation of Mouldings, Ceilin . Floors, 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 <.<( constructing plain curves, such as are of frequent occurrence in the arts, and in mechanical combinations ellipse, the oval, the parabola, and the volute; and certain figures, accurately shaded, to represent reliefs, exemplifying casca when these Curves are employed. The second division illustrates the geometrical representation of objects, or the study of projections. This forms of all descriptive geometry, practically considered. It Bhows that a Bingle figure is insufficient for the deter- mination of all the outlines and dimensions of a given subject; hut that two projections, and one or more sections, are alu ry lor the due interpretation of interna] tonus. The third division points out the conventional colours and tints f.r the expression of the sectional details of ol 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 \ atially valuable curves, as Helices and different kinds of Spirals and Serpentines, with the intersection of surfaces and their development, and work-hop applications to Pipes, ' i id Cocks. This Btudy IS obviously of importance in many professions] and clearly so to Ironplatc- and Boiler-makers, Tinmen and Coppersmiths. The fifth division is di of carves relating to the teeth of Spur Wheel-. Screws and Recks, and the details of the construction of their patterns. The latter branch is of peculiar importance here, inasmuch as it PREFACE. v 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, and effecting 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 principal 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 parts 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 columns, 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 appro- priate 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 are so essentially necessary at every stage fn 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 comprehending her laws, and knowing those forces in relative value and measure." All art is the true application of knowledge to a practical end. We have outlived the times of random construction, and the mere heaping together of natural substances. We must now design carefully and delineate accurately before we proceed to execute — and the quick pencil of the ready draughtsman is a proud pos- session for our purpose. Let the youthful student think on this; and whether in the workshop of the Engineer, the studio of the Architect, or the factory of the Manufacturer, let him remember that, to spare the blighting of his fondest hopes, and the marring of his fairest prospects — to achieve, indeed, his higher aspirations, and verify his loftier thoughts, which point to eminence- — he must give his days and nights, his business and his leisure, to the study of En&ustrtal Besian. abbreviations AND CONVENTIONAL signs. In order to simplify the language or expression of arithmetical and geometrical operations, the following conventional signs are used: — -ign + signifies plus or more, and ^o or more terms to indicate addition. Exaui'i i : 4 4- S, is 4 plus 3, that is, 4 added to 3, or 7. ■n — signifies minus or lest, and indicates subtraction. 4 — 3, is 4 minus 3, that is, 8 taken from 4, or 1. The sign X signifies multiplied by, and, placed between two terms, indicates multiplication. 5 x 3, is 5 multiplied by 3, or 15. i quantities are expressed by letters, the sign may be suppressed. Thus we write, indifferently— a x b, or ab. The sign : or (as it is more commonly used) -r, signifies divided by, and, placed between two quantities, indicates division. Ex. j 18 12 : 4, or 12 -r 4, or T , is 12 divided by 1 The sign = signifies equals or equal to, and is placed between two expressions to indicate their equality. l.\. : + 2 = 8, meaning, that C plus 2 is equal to 8. The union of these signs, : :: : indicates geometrical proportion. 1 ; \ . : 2 : 3 : : 4 : C, meaning, that 2 is to 3 as 4 is to G. The 6ign V 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, V, indicates the degree of the root Thus — #21 = 3, expresses that the cube root of 27 is equal to 3. The signs L. and y indicate ntpi iiiv.lv, smaller Uian and greater than. 3 Z. 4, = 3 smaller than 4 ; and, reciprocally, 4 "7 3, = 4 greater than 3. Fig. signifies figure; and pL, plate. rm'.NVII AND l.V.USH UNT.AK MEASUBE8 COMPARED. 1" I ...t.m.tr.* 10 Decimetres : MttN l" Dmattm ■Mm 1 Millimetre = 1 Cenlini.tn-s = 1 Decimetre = 1 M = 1 Decametre = I Had = 1 Kilometre = 1 Mjriamitre : : Kntflih. '0394 Inches. •SI 17 " 80.171 " 1-0936 Yards. l'988i Poles or Rods. 19-8844 49-7109 Furlongs. 62 1 89 Miles. 621380 " F.nftlfh. 11 Inches 3 Feet 6J Yanls 40 I'olca 8 Furlongs ) 17C0 Yards I I Inch = (25-400 Millim.tr,>. • nUmitrv*. 1 Foot = 3048 Decimetres 1 Yard = 9144 1 1-. Ii Of Rod = ■ n tth 1 Furlong = 2-012 Decametres 1 Mil. — 1-610 Hectometres. CONTENTS. Preface, .... Abbreviations and Conventional signs, CHAPTER I. LINEAR DRAWING, 7 Definitions and Problems : Plate I. Lines and surfaces, - - - - - ib. Applications. Designs for inlaid pavements, ceilings, and balconies : Plate II., 11 Sweeps, sections, and mouldings : Plate III., - - 13 Elementary Gothic forms and rosettes : Plate IV., - 14 Ovals, Ellipses, Parabolas, and Volutes : Plate V., - - 15 Rules and Practical Data. Lines and surfaces, - - - - - 1 9 CHAPTER II. 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. A, Projections of a right square-based prism, or rectangular paral- lelopiped: Fig. IS, Projections of a quadrangular pyramid : Fig. Ig, Projections of a right prism, par.ially nollowed, as Fig. ®, Projections of a right cylinder : Fig. II, - Projections of a right cone: Fig. f, - Projections of a sphere: Fig.©, - Of shadow lines, ------ Projections of grooved or fluted cylinders and ratchet-wheels : Plate VIII., The elements of architecture : Plate IX., Outline of the Tuscan order, - Rules and Practical Data. The measurement of solids, - CHAPTER III. ON COLOURING SECTIONS, WITH APPLICATIONS. Conventional colours, _ _ _ - - Composition or mixture of colours : Plate X., Continuation of the Study of Projections. Use of sections— details of machinery : Plate XL, - Simple applications — spindles, shafts, couplings, wooden pat- terns : Plate XII., - Method of constructing a wooden model or pattern of a coupling, Elementary applications — rails and chairs for railways : Plate XIII., Rules and Practical Data. Strength of materials, - Resistance to compression or crushing > roe, - - - Tensional resistance, _ - - - - Resistance to flexure, - Resistance to torsion, - - - - - Friction of surfaces in contact, - - - - CHAPTER IV. THE INTERSECTION AND DEVELOPMENT OF SURFACES, WITH APPLICATIONS, The Intersections of Cylinders and Cones : Plate XIV. Pipes and boilers, ------ Intersection of a cone with a sphere, - Developments, ------ Development of the cylinder, - - - - Development of the cone, - - - - - TnE Delineation and Development of Helices, Screws, and Serpentines : Plate XV. Helices, -------- Development of the helix, - - - - - Screws, ------- Internal screws, ------ Serpentines, ------- Application of the helix — the construction of a staircase: Plate XVI., ------ The intrncctloa of rarficM— a; . 6fl . - K - Ml -fair. Calculation of the dlroeiui . it,. iuImu of firegrate, -61 ueyt, ..... - i'A. Safety-valves, ...... . i i: v I ::, C3 . III. .ml XIX. Will, - - - - i'A. lit, - - - • I External ,., . . - ib. Extern. \l\., - - . • ..'. I \ . - - Hi. ..f a rack and ph. i > Will., ib. GearnR of a wonn with a wonn-wh..| : Piga. 5 ami 6, Pi ttl Mill, 07 Crum xix. •• I, - i'A. IliQg internally: Fin. .'., - C8 I*r»' :,.:.!. I'i mi. XX. , 09 The Hi i im mam uro ( 1 i ■ , ■ X '. I. SjiUr-n h.-. 1 iall.ni-, - - . . - 70 1'attcm of the pinion, - - - - - i'A. Pattern of I) . . 71 Core moulds . . . . - i'A. (M. I'I (.11. U I - 7- l.ir ami eireutn' . - 71 Bjioaiof gwiogj - - - -75 Thkfamtaf t&ataath, - - - - TA. 76 f 111.' wen, - . - . - 77 ib. ro«, - . . . . - 78 i II Al I rra sn ax of i bed gear. Conical or bevil gajring, ----- Dalgnftr apaJrof barfl-wbaalilo gaari Pi mi XXII., XIII. - . \ I V. - ' ' - i uniform an. I | - - ■ ■ ;.lc uiachitx*. •v, - ■ Calcul .', - - ferm. . v. \ II. I A1;Y PRDJCIPL1 S "I SHADOWS, . C\ UM. ' SlIAl". Priam, ...... Pyramid, -..---. Truncated pyramid, ..... Qjrflnte, - - - - ■ prim, r, Shadow ca»t by a prl PrdTi in i - ..i Sll\|.I\..: I'i vri XXVII., Illumined .urfuec.% ... Surf. lttu shading, ..... ; by softened v. . .... atom i - .\ III. Under, ■«• cast by one cylinder U|»n another, - Sli. i l.ws of cones, ------ H of an invert) I - - Bhadon cad npon n • . Applications, ...... ibi i b; Pi mi xxix. ..... Shadows of surfaces of r. volut in, ■ ID I'm. ii. u I > v ■ v. Pnmpa, --....- ....... i.'-puuirM, ...... l.ifuiiL,' and for. in^ pojn| -. - The h ,...-. Il_v.lt different orifices, ------ - ... ■ nation of the diat b u •• of a tti i thi orili. .... ' I'llioll of the .Ii rinine the widlh of an .mine the .1. |,'li ,.f the outlet, <»uilet Kith aapoal - i ii M-n B vm. AlTl [CATION 01 SB tDOWS TO COOTHl D Ql IB: Pi « xx\ . Barll Aril i. mi. .v.. i Sum., Pi .. , , \\\|., - t'ylin.lii, ,1 -.,11. ,re tli. (angular threads: Kigs. 4 and 5, 100 ■A. I"7 i'A. h>. i ib. no •A III 111 i'A B IK CONTENTS. Triangular-threaded screw: Figs. 6, 6", 7, and 8, - Shadows upon a round-threaded screw ; Figs. 9 and 10, 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 XXXIII., - rAriE 118 119 a. 122 CHAPTER IX. THE CUTTING AND SHAPING OF MASONRY: Plate XXXIV., 123 The Marseilles arch, or arriire-voussure : Figs. 1 and 2, - ib. Rules and Practical Data. Hydraulic motors, - - - - - -126 Undershot water-wheels, with plane floats and a circular channel, - - - - - - ib. Width, ------- a. Diameter, - - - - - -127 Velocity, - - - - - - - ib. Numher and capacity of the buckets, - ti. Useful effect of the water-wheel, - ib. Overshot water-wheels, - '- - - - - 128 "Water-wheels, with radial floats, _ - - - 129 Water-wheels, with curved buckets, - 130 Turbines, - - - - - - - ib. Remarks on Machine Tools, - 131 CHAPTER X. THE STUDY OF MACHINERY AND SKETCHING. Various applications and combinations, - - - 133 The Sketching of Machinery: Plates XXXV. and XXXVI., ib. Drilling Machine, - - - - - - ib. Motive Machines. Water-wheels, - - - - - -135 Construction and setting up of water-wheels, - - ib. Delineation of water-wheels, - 13G Design for a water-wheel, - 137 Sketch of a water-wheel, .-■-'-.- ib. Overshot Water-Wheel: Fig. 12. - - - - ib. Delineating, sketching, and designing overshot water-wheels, 138 Water-Pusifs : Plate XXXVII. Geometrical delineation, - - - - ib. 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, - - _ - - - -«S. Connecting-rod and crank, - - - - - ib. Fly-wheel, ______ ib. Feed-pump, - - - - - - ib. Ball or rotating pendulum governor, - - - 114 Movements of the Distribution and Expansion Valves, ib. Lead and lap, - - _ - - - 145 Rules and Practical Data. Steam-engines: low pressure condensing engine without expan- sion 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, - - - 1 53 CHAPTER XI. OBLIQUE PROJECTIONS. Application of rules to the delineation of an oscillnting cylinder : Plate XLL, - - - - - - 154 CHAPTER XII. PARALLEL PERSPECTIVE. Principles and applications: Plate XLII., - - 155 CHAPTER XIII. TRUE PERSPECTIVE. Elementary principles : Plate XLIII., ... 158 First problem — the perspective of a hollow prism : Figs. 1 and 2, ib. Second problem — the perspective of a cylinder : Figs. 3 and 4, 159 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 : Figs. 5 and 6, 1G0 Fourth problem — the perspective of a bearing brass, placed with its axis vertical : Figs. 7 and 8, - - - ib. Fifth problem — the perspective of a stopcock with a spherical boss : Figs. 9 and 10, - - - - - ib. Sixth problem — the perspective of an object placed in any posi- tion with regard to the plane of the picture: Figs. 11 and 12, 161 Applications— flour-mill driven by belts: Plates XLIV. and XLV. Description of the mill, - - - - - ib. Representation of the mill in perspective, - 16." Notes of recent improvements in flour-mills, - - 164 Schiele's mill, --____ ib. Mullin's " ring millstone," - 166 Barnett's millstone, - 16(1 Hastie's arrangement for driving mills, - ib. Carrie's improvements in millstones, - ib. Rules and Practical Data. Work performed by various machines. Flour-mills, - - - - - -168 Saw-mills, - - - - - - 170 Veneer sawing machines, - - - - - 171 Circular saws, - - - - - - ib. CHAPTER XIV. EXAMPLES OF FINISHED DRAWINGS OF MACHINERY. Example Plate &, balance water-meter, - 172 Example Plate _., engineer's shaping machine, - - 1 74 Example Plate ©1 ©, II. express locomotive engine, - - 178 Example Plate IF, wood planing machine, - - - 180 Example Plate (£., washing machine for piece goods, - - 182 Example Plate _., power-loom, - ib. Example Plate fl, duplex steam boiler, - 183 Example Plate -I, diicct-acting marine engines, - - 184' CHAPTER XV. DRAWING INSTRUMENTS, - . - - - - 185 INDEX TO THE TABLES. rial f. Multiplier, foe regular poh-guas of from 8 to 12 sido>, - 19 Approximate ratios between circle* and ■quart*, M Comparison of Continental measures with French millimetre* and - I'l Surfaces and volumes of regular |>oMi^n, - ■ 30 Proportional measurement* of the various parts of tin- (modem) 1 1 order, - - - - - - - 83 Proportional measurements of the various parts of the Tuscan order, 8 1 . lappOTta, will sus- tain without bal ----- -12 v. - wtruh prisms and cylinders will sustain whan submUtad to a train, - - - - - - -44 Diameters of the journal* of wit'-rwh-.l and .tier >Ii:»f: - for I. work, ........ .jo aft Journal* i-.i1cul.itnl with reference to torsional strain, ->•-.-- 1*8 49 Ratios of friction for joarnall in bearing*, - . &, Preaann- - • 68 Amount of beat developed by one kilogramme of fuel, - . 69 Thlcknease»ofplali-i incvlin.lric.il I.. il.r«, - - - - ib. Dimcnalona of boilers and thickness of pi it. - for a pressure of Bra atmmphrm, ----- (52 Numbers of teeth, and diameters of spar gear, - - - 73 ritch and thickness of spur . of the *ur- ich sub- tendi it. are drawn by the ail of the ■ ■ i il. when rijlit linu ir inter- • : . other witbool being An acute angle • hi a right ingl ■ generally I • I by two 1 1. with the apex of nn anj' the pop- it off by the : divide an entire oirole and instruments called the B— her of degrees contained in any angle are a . which is to be found in ■ of mathemati . parts. In nn.. . manner that its dimn. ' when the measure of the a: ■ d by the other side of the angle. Thneth h it will alwav - i qucutly whatei t'"r the measuring arc niurt nl ■ 60 seconds (or 60"); or when the circle is divided into 400 degrees, each degree is inbdit . and each minute into The other protractor, fig. 5. of modem i: i ■ i ach half beii the hmer side into 180 degrees, bat externally I isaqnare. It is placed against a rule, B, made I ■ . of the - "ii the circle indicating the number of degrees contained in the aj will be seen that the ■ • one of 50*. ' [tight lines, which do not form right ai those they intersect, are said to be oblique, or inclim I other. The right lino. DC a to the vertical line, k i . er the horixontal fim ParaUi I rallel with each Other when they are an equal distance apart throughout their thl lini -. I K. \ B, and I kt, li- r . 1. are parallel. Triangle*. — The spa,, enclosed by three inten called a triangle; when the three side-, a- i>i, i r, and i d .1, the triangle is egwifrrtsmf; if ti as o n, and Q I. tiL:. '.'. are , qua], it i- il It* ; and it isecoi when the three sides are i::u ipial. as in fig. 6. The ti called rnrtiinjafar whitti any two of iis sides, as d i and i I form aright angle; and in this case the sid. subtending the right angle, is called t!.. An instrument constantly used in di i mmoaly called aii'jU ; it is in the shape of a rectangular triangle, and is construi ted .as fig. having one I icfa form the right angle at mast double thi I ther. og nt any auglo It i- plane when all the lilies 1 '• and the hu and Its outline i.- called it- \ /. hrjXa- the sides of which, ■ - Mvt.lr III* .. r, mvouou lur il.- ■ in ol division. BOOK OF INDUSTRIAL DESIGN. and 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 quadrilateral, having two sides equal, as A B and F n, fig. 14, and perpendicular to two other equal and parallel sides, as A F 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 quadrilateral, of which only two sides, as H I and m l, fig. 9, are parallel. Polygons are regular when all their 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. We 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 their frequent occurrence in practice, it is important that the student should be well acquainted. The first step, how- ever, to be taken, 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 will 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 formation of the actual outlines of the drawing, we have in the plates represented the former by dotted hues, 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 centres, and with a radius greater than half the line, describe the arcs which cross each other in 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 perpendiculars, I k and l m, dividing the line isto four equal parts, and we can thus divide any given right line into 2, 4, 8, 1G, &c, equal parts. This problem is of constant application in drawing. For instance, in order to obtain the principal lines, v x and Y z, which divide the sheet of paper into foiu- equal parts; with the points, r s t u, taken as near the edge of the paper as possible, as centres, we describe the arcs which intersect each other in P and Q ; and with these last as centres, describe also the arcs which cut each other in y, z. The right lines, v X and Y z, drawn through the points, P, Q, and y, z, respectively, are perpendicular to each other, and serve as guides in drawing on different parts of the paper, and are merely pen- cilled in, to be afterwards effaced. 2. To erect a perpendicular on any given point, as u, in the line c d, Jig. 1. — Mark off 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 line 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. IF". 4. To draw parallels to any given lines, as v x and, Y z. — For regularity's sake, it is well to construct a rectangle, such as rstu, 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 applying the rule tangentially to the two first, draw the line rs, and then in the same manner the line t u. The lines rt and su are also obtained in a similar manner. In general, however, such parallels are more quickly drawn by means of the T-square, which 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 rigid line, as A B, Jig. 3, into several equal piarts. — AVe have already shown how a line may be divided into 2 or 4 equal parts. We shall now give a simple method for dividing a line into any number of equal parts. From the point A, draw the line AC, making any convenient angle with AB; mark off on AC as many equal distances as it is wished to divide the line AB into; in the present instance seven. Join cb, and from the several points marked 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 making any angle with AB, asA J, may be employed in- stead of A c, with exactly the same results. This is a very useful problem, especially applicable to the formation of scales for 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 which they are derived. The object of the 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 necessary to state that the metre (= 39-371 English inches) is the unit of measurement, and is divided into 10 decimetres, 100 centimetres, and 1000 millimetres. If it is intended to execute the drawing to a scale of J or j- ; the metre is divided by 4 or 5, one of the divisions being the length of a metre on the reduced scale. A line of this length is drawn on the paper, and is divided into reduced decimetres, &c, just as the metre is itself. Fig. 7 is part of a scale for reducing a drawing to one-fifth. In this scale an extra division is placed to the left of zero, which is subdivided, to facilitate the obtainment of any re- quired measure. For example, if we want a length corresponding to 32 centimetres, we place one point of the compasses on the division marked 3 to the right of zero, and the other on the second 10 THE PRACTICAL DRA.UQHTSMAS'H to the left, and the length comprised betw. ire*. The iluiijonal scab. Whan m -ry minute measurement* are re- ii U obtained with a diagonal scale, mch as fig. 8. It U thus t ll ll ll ln il llni : - Having drawn a line nil It, a* in fig. 7, draw, parallel and equal to it, ten uther lines, an r.d.r.j wit " peroen- dicular* at tli. " one of the smaller l , 0,.. Lit ol ssjm, draw the diagonal, b i, and draw parallel* to it from th. tilUOtrs dtviail Krum Um divi-ion corrcspon i. .4. out by tha zero pacpan- dicular, and draw also i I— 8, 2—3, M It will bo evident, that as in I horizontal lines. the diagonal BXtaodi one division tO ill. 1. ft, it will inter- a.itc line, as the 1-t. U, Bd, *C, :>< tlie di-t ancc I tin of such division, in the same direct ion, so that the diagonal Una, ".!, will CUl the Ml line at n point •.','•„ of a division distant from /.'To. '1 Bg placed en the point /, and the other on the intersection of the same horizontal line with the perpendicular of the deoa the mea- B than will be 3 decimetres, 2 cen- timetre*, and A, or 5 millimetres — 325 millino' I 8. To dirndl « gi<-< n angle, at WOO, I ' aaglm. — With the apei.e, as a centre, describe the arc, It I, and with the two point- of intersection, II, I, as centres, describe the arcs cutting each other in J ; join J C, and the right line. J < , will divide the . i>. int.. two equal angle-, n c i and i i. These may be subdivided in tha same manner, as shown in the figure. An anglo may also be divided by means of either of the protractors, E>, X. BJH ii rirrlr, O II I) 11, fig. 4. — If it is raqubi .1 to draw the tangent through a given point, aa D, in the circle, a radius, C D, must be drawn meeting the point, and be pro- duced beyond it, say to K. Tlien, by the method already given, draw a line, F «!, perpendicular to B, cutting it in D, and it will be the tangent reqnta 1 If. however, it is required to draw tho tangent through B given point, as A, outside the circle, a straight line must be drawn joining the point. \. and the centre, r, of the circle. Al':. r bisO ting this line in the point, 0, with tliis point as a centre, daeeribe a circle pairing through a and 0, and tha given circle in b and n ; right linei Joining a B and A II will both be tangent- to tlie given circle, and the radii C B and C It will be perpendiculars to A n and a ii respectively. 10. 7 •.''■• a I i'r.7.-, ..r that irilh nhi' are, aim a, <; '■>. f draw* With any throe points, B, f, a, as centres, describe arcs of equal circle*, cutting each other, and through hmw light lin.-. I and I 0; .., the point of mtai action of thaaa two Unas, ia the required 11. To ifansiTu a afrafa through any three paint* not in a rigid Vat. rela can j>ass through tho same three points, and since any circle may bl '' is found and a point in the cir.-iin.f. r. nee given — this problem is solved in exactly the same manner a- the pn I] To iiarri)* a rirrlr in a girrn trinnglr, as A n 0, jig. C. — ■ i« said to be inscribed in a figure, when all the sides of the latter are tangent* to it. Bisect any two of tho angles 1 line*, as A o, n o, or C o ; and i Call perpendiculars to the side*, a* o E, u r, and o u. Tlicse per- pendiculars will be equal, and radii of being trc. 1 t /'. iliritls a triangle, as O K I, fig. 9, into tiro equal farts. — If the parts are not required to be simil.r ' i. , as o I, in the point, 0, with which, a* a centre, describe tho (emicircle, Q K l, of which Q I is the diamett r. This semicircle will be cut in t I K.bvth. :. k •.; mark off on o I a distance, (. : to o K, and draw the line, I. v. parallel to ll I. The triangh nil, ii l l. M. will be equal ft) each oth. r. | . qua] to h it the triangla, a B i. If the given trim it would al-u be divid. A into two equal parts by tha line, l h. 1 1. /'. dra oaaaa square, a n c n. Jig. 10. — After producingfrom dill, rent corner-any two aides which are at right angles to each other, as D a and D c, to ii and i.. with the centre. i>, and radius, D n, describe tho Juni'fUlt or quarter of a circle. I B l . and t h rou g h the point- of intersection, I and 1'.. with the i and pi.; draw parall. 1- 1.. i. i and p A re spec tiv ely, 01 tangent* to the quadrant, t li I ; the square, r I I', will be d.ul.le the area of tho given square. A li D ; and in the same manner a square, ii K I. n, may be drawn double the area of the square, F o B ii. It is evident that the diagonal of one square ia equal to one side of a square twice the size. 15. To (feennTVl " rircle half the size e/ a giren rirrlr, as A c n D, fig. 11. — Draw two diameters, A n and D, at right :ni_-l. - BO each other ; Join an extremity of each, as a, i'. by tha chord. \ . . Biaed this cliord by the perpendicular, B r. The radius of the required circle will he equal to B a. It follows that tha annular space shaded in the figure i- equal to the smaller circle within it. lg. To timriTm ia swm eanaiai as in jig. IS, aw aj tmda regular hexagon. — Draw any dianu for, c; r, and with O, as a centre, describe the arc, DOE, its radius being equal to that of the given circle ; join !■ r. I f. and I D, and D B r will be the triangle required. The side of a regular hexagon ia equal to tha radius of the circuni-i Tilling circle, and. thl r. fore, in ord( r to in- scribe it in a circle, all that is necessary is to mark off on the cir- cumferen. .■ the length of the radiu-. and. joining the point t enaction, a- km.h ■ >, tha resulting figure will be tha I required. To inaariba figures of ISorSf aides, ii is merely na- ccssary to divide or sul«li\i.le the arcs mhtandad by the sides obtained sa shovo, and to Join tha points of intersection. It is frequently necessary to draw very minute hexagon-, such . nut- and boll bl id-. Thi- i< done more qui. klv by means of the 60", H . which is placed against a rule, R, or the square, in ditl'erent positions, a- iudi. it. 1 u. IT. To foatlllNJ ii SOWarS ia a girrn rirrlr. mACB D, / Draw two diaiii n. perpendicular to one another, and join the point- of intersection with : I At n 1) will be the square r. quired. 18. Tothtrriltr a regu la r OOtOfiOt r the horizontal line.'. This pattern is generally produced in black and white marble, or in stone* of different colours, whereby the effect is distinctly ■ out 1'ilar hexagon-. With a radius, a o, equal to a side, a b, of the hexagon, I circle, in which inscribe the regular hexagon, A n ' D] r. hexagons will readily lie obtained by producing, in sides and diagonals of this one. In fig. IE, the hexagons are plain and shaded alternately, to show their arrangement ; but in practice they are generally all of one colour. 87. / t, combined in Drawl ■ D; also its diagonals, a * , BO; oonitrnct the smaller square, abed, concentric with the lir-t. On tlu- diagonal, n d, mark the equal fliirtannnn. oe, of, and through • and/draw paralleli to the diagonal, \ i : join the points f these with the smaller square) by the lines, kl, iii ii, which wiO give all the lines required to form the pattern, ; merely to be produced and repeated to the desired cx- ■ ry beautiful combinations may thus be formed in differ- ent kiiids of wood for furniture and panels. 28. Todnaoi ''./*. ©n/uf 7. — On a straight line, ap, mark off the length of a tide of thi twice; i ; iil.iter.il triangle, a nc ; draw the line, CD, perpendicular to ah; and draw a Band Br parallel to dc, and 1 tract the equilateral triangle, E nr, cut- . t bc, bo and B, and join a h. In this manner aro obtaiiK'd the losengea, a <; n d and i: a n o, and by oontinuing the lines and drawing parallels a! regnlar i the re- mainder of the pattern will be readily constructed— this being • I any desire 1 mill 1 — - — 1 f in the last-in. intioned fig g, we draw the longitudinal diagonal of each lozenge, we shall obtain the type of the |.] and 8. — The lozenge, aid. 1 .. , being given, its corners forming the <■ four of the small lozenges, draw the lines, IF, n. dil into four equal parts; next mark off the of the lozenges, as a i, a o, and join l and o. The centre lines of the pattern being tlm- obtained, the half-breadths, fg, fk, are marked on each aide of these, and the a ppr o pri ate par.ill. Is t,. then drawn. In extending the patten by repetition, tl spending to i and o will be readily obtained by drawing a series of parallel lines, as 1 1 and on. liy varying th the losengea an 1 the octagons, as ah I ns of each, a number of patterns may be produced of very vari.d appearance, although I ants. 32. To draw .i at m lxilii- h may accordingly be drawn, with given radii. 11 SO, <' and w •. b i, oono ntric, but with the \i mi ii" ;ui«l ■ ■ • '■ IIKHII OIIN BOOK OF INDUSTRIAL DESIGN. 13 circle, the diameter of which is equal to the width of the indented crosses, the sides of these being drawn 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 re- gular figures, lines, and circles. There is no limit to the multipli- cation 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 audi. — Let a b be a side of the square ; draw the diagonals cutting each other in 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 semicircles of the given radius, D a, which must be less than the distance, D b. This completes the figure, the symmetry of which may be verified by drawing circles of the radii. C G, C II, which should touch, the former the larger quadrants, and the latter the smaller semi- circles. If, instead of the smaller semicircles, larger ones had been drawn with the radius, d6, the outline would have formed a perfect sweep, being free from angles. This figure is often met with 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, AC, and a radius equal to a b, describe arcs cutting these lines, and through the points of intersection draw parallels to them, bo, CO, cutting each other in O, which will be the centre of the required arc. Draw perpen- diculars from it to the straight lines, ab, AC, meeting them in r> and E, which will be the points of contact of the required arc. Secondly, if a point of contact be given, as B, fig. 3, the lines being ab, a c, making any angle with each other, 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, will 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 stand out from, instead of being tangents to, the given straight lines. This problem meets with an application in drawing fig. [§, which represents a section of various descriptions of castings. 36. To draw a circle tangent to three given straight lines, which make any angles with each other, Jig. 4. — Bisect the angle of the lines, A B and A C, by the straight line, A E, and the angle formed by c D and c A, by the 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 consequently the centre of the requiied circle, which may be drawn with a radius, equal to a line from the point, o, perpendi- cular to any of the sides. This problem is necessary for the com- pletion of fig. US. 37. To draw the section of a stair rail, jig- (g. — This gives rise to the problems considered in figs. 5 and 6. First, let it be re- quired 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 O F, through 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. 5. Bisect the angle, b c d, by the straight line, c e ; with the centre, c, and the radius, c ii, equal to that of the given arc, o a, describe the arc, o g ; parallel to b c draw I n J, cutting e c in J. Join o J, the line, o J, cutting the arc, n 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, perpendicu- lar 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. First, 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 given arc ; join o D, and draw 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 arc, A E C, a b being the radius. Secondly, it is required to draw 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, 0, of the given arc, and with a radius, O E, equal to O C, plus the given radius, a, draw the arc E; and with the given point, A, as a centre, and with a radius equal to a, describe an arc cutting the former in E — e will be the centre of the required arc, and its point of contact with the given arc will be in C, on the line, E. It will be seen that in fig. ID, 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 diffi- cult to make them symmetrical without such a guide. This is an ornament frequently met with in machinery, and in articles of va- rious materials and uses. 39. To draw a wave curve, formed by arcs, equal and tangent to each other, and passing through given points, a, B, their radius being equal to half the distance, A B, Jigs. [I and 7. — Join A B, and draw the perpendicular, ef, bisecting it in c With the centres, A and c, and radius, A C, describe arcs cutting each other in G, and witli the centres, B and C, other two cutting each other in n ; G and n will be the centres of the required arcs, forming the curve or sweep, a c b. This curve is very common in architecture, and is styled the cyma recta. 40. To draw a similar curve to the preceding, but formed by arcs of a given radius, as a i, Jigs. (F and 11. — Divide the straight line into four equal parts by the perpendiculars, ef, g h, and c d ; then, with the centre, a, and given radius, a i, which must always be greater than the quarter of A b, describe the are II •llli: PRACTICAL DRAUGHTSMAN'S (iiliing CD in C; also with the centra, n, a similar arc cutting i. li in ll ; and 11 will •• arcs forming the re urve. Wh»! ■'• h ""I too small, the centres of the arcs will always bo in the linca, C D It will be seen tliat the arc", I *>"! » l-, cut the Straight line*, CD ami i. li. in two points respectively. If we take the t, we shall form a similar oarra to thl but, hut with the concavity anil convexity tran-p. -,■.!, and called lir.-t at a, ami the second at l>. - window frame — it is one well known to carpenters ami masons. The little instrument known as the " Cymametcr," affords a nt means of obtaining rough measurements of contours i'f various classes, as mouldings and baa roHafa. It is simply a light adjustable frame, ai ting as a species of holding socket for a mas* of parallel slips of wood or metal — a bundle of straight r example. Previous to applying this for taking an im- of measurement, the whole aggregation of pieces is dressed up on a flat surface, so that their ends form a perfect plane, III I the bristle! in a square cut bru.-h ; and : • pieces are held in close parallel contact, with just enough of stilT friction to keep them from slipping and foiling away. The ends of the pieces arc then applied well up to the moulding or surface whose cavities and projections are to be mca- ■ I the frame is thl n mewed up to retain the slips in the position thus assumed. The surface thus moulds its sectional i nt our upon the needle ends, as if the surface made up of these ends was of a plastic material, and a perfect impression is there- ad away on the instrument. The nicety of delineation is obviously bounded by the relative fineness of the measuring ends. ■1 1 . '/'o draw . and to pan through the point, r, is drawn with the centre, 0, the chord, nr. by a perpendicular which cuts tie- radius of the are. OD. This surra ha-, in Sg, -.to be repeated both on each side of the vertical line, n n, and of the horizontal 42. 7b dross ft i i /;./.• mitHiir, a* Jig. y. We bavC lore t0 draw ail arc passing through tWO points, a, B, bra being in a straight line, BC : this arc. moreoTor, requiring to join at i>, and form a sweep with another, Di I iraOel to b o. Joining n a,aperp< odicular il cm bi in o, which will be the centre of the first arc. and that of the SeOO&d may DOW be obtained, U in probli in B7| • i.a curve, I I SePtio, It may be drawn hy \arioiis methods. The following are two of the staples! aooording to the Bret, the curvo may be formed by arc- m fa Othl r. and tan- gents at * and I A and C draw the perpendiculars, c and A i:, and divide t' into three equal part*. With one di\isioti, t A, as a radius, describe the first arc, A a n ; make l equal to r a. Join I i by the perpendicular, OK, which cuts < in O. O will t' the other arc required. The line. II, passing •r. -. ii and r, will cut the arcs in the point of junc- tion, II. It i- in thi- manner that the mrve in fig H ia obtained. I by two arcs n into each other and passing through tin grran points. 1 I tin ir centres, I in the same horizontal tin parallel to two straight tinea, l l and BI -h the :i\.n pints. 'I'lir ujli A, draw the perpendicular, A I. 1. of inti r.-e, ti,,ii with i ii. i- t!ie centre of one arc, A p. Next draw I, bd, the per] li. which, will out CD fat O, the centre of the other arc, the radius being ODoro n. Thi- curve is more particularly met with in the construction of bases of the ' rinthian.an.il ll rs of architecture. With a view to accustom the Stud tton his designs to the rules adopted in practice in the more obvious app! .-.. and on the corres po nding outlines, the meaaurt n irioos parte, in millimetres, It must, however, ai the amna time bound that the \arioiis probl other lis I. the number Of applications of which the forms considered are susceptible, will give rise to a considerable variety of these. ELEMENTART GOTHIC FORMS AND BOSET1 I I.A I I IV. 44. Having ioIvi l the foregoing problems, the student may now attempt tli. ■ mOTB complex objOOta. Ilen.id not. li ... n r. ai ;- ' anticipate much difficulty, merely giving his chief attention to the a I iuatioii of the principal lines. which serve ai guides to the minor details of the dm It is in Gothic architecture that we meet with the more numer- ous application.- of outline- formi d by smoothly joint d cdrcli I and thii order in 1 Pig. 6 represents the upper portion of a window, composed of a I arcs, combined so as to form what are denominated rut/lid arches. The width or spin, v r.. being riven, and the I joining a c i i.. draw the bisecting perpendiculars, cutting \ B in These utter are the oentrea oi the sundrj oonoentrie arcs, which, severally Cutting each Othl r on the vertical, i I . form tin arch of the window. The small interior cuspids are drawn in the same manner, as indicated in the figure ; thl h< i. b, being given, also the span Interior arches are sometimes surmounted hy the ornament, M. termed an nil-dt- ' - -i-t iiiLT simply of concentric circle-. 45. Fig, l repreaenl • irmed by concentric circles, the outer interstices containb mailer circles, forming an interlaced fillet or ribbon, The radius, 10, otthecb taming the centres of all the small circle-, i- luppoaed to be given. Divide it Into a given number of equal parts, with the point- of division. 1, '.'. .. • . ntial to I in I ^/\ r II 4 \ rii i i-i 1 1 ' .i 1 1 . 1 I i \fii()iimti\ BOOK OP INDUSTRIAL DESIGN. lfi 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, A b. 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 quadrant of a Gothic rosette, distin- guished as radiating. It is formed by a series of cuspid arches and radiating mullions. In the figure are indicated the centre lines of the several arches and mullions, and in fig. 6 d , the capi- tal, connecting the mullion to the arch, is represented drawn to double the scale. With the given radii, A B, A c, A d, a e, de- scribe the different quadrants, and divide each into eight equal parts, thus obtaining the centres for the trefoil and quatrefoil ornaments in and between the different arches. We have drawn these ornaments to a larger scale, in figs. 6 a , 6 b , and 6 C , in which are indicated the several operations required. 47. Fig. 4 also represents a rosette, composed of cuspid arches and trefoil and quatrefoil 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 a is a portion of the details on a larger scale, from which it will be seen that the entire pattern is made up 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 archi- tecture 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 IV. are partially shaded, to in- dicate 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 his 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, a b and CD, fig. 1, being given, per- pendicular to each other; with the point of intersection, o, as a centre, first describe the circle, cade, half of which forms the 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 b bv the per- pendicular, g H, cutting c d in h. n will be the centre, and h c the radius of the arc, c J; and I, the point of intersection of HC with a b will be the centre, and I b the radius of the smaller 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. A more accurate and beautiful ove may be drawn 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 horizon f al 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 ove may be varied as required. 51. The oval differs from the ove in having the upper portion symmetrical with the lower ; and to draw it, it is only necessary to repeat the operations gone through in obtaining the curve, L b h, fig.l. 52. The ellipse is a figure which possesses the following pro- perty : — The sum of the distances from any point, a, fig. 2, in the 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 with reference both to the horizontal line or axis, D e, and to the vertical line, F G, bisecting the former in o, the centre of the ellipse. Lines, as B A, c A, B f, c f, &c, joining any point in the circumference with the foci, b and c, are called vectors, and any pair proceeding from one point are together equal to the longer axis, d e, which is called the transverse, f g being the conjugate axis. There are different methods of drawing this curve, which we will proceed to in- dicate. 53. First Method. — This is based on the definition given above, and requires that the two axes be given, as d e and f g, fig. 2. The foci, b and c, are first obtained by describing an arc, with the extremity, G or F, of the conjugate axis as a centre, and with a radius, F c, equal to half the transverse axis ; the arc will cut the latter in the points, Band c, the foci. If now we divide de unequally in H, and with the radii, d H, e h, and the foci as centres, we describe arcs severally cutting each other in I, j, k, a ; these four points will lie in the circumference. If, further, we again unequally divide D e, say in l, we can similarly obtain four other u THE PRACTICAL DBADOHT8MA1T8 rl umferenee, anil we can, in like manner, obtain any i.uml" ■ may If traced thn by ban J Tli' I which are sometimes required in con- illy drawn with a inpatses, the Ira- rule with a : gardener'i tllijue : To ; 1 ice a rod in each of the foci of the i '■■ which, ..111 form the rector* : and t! ■ drawn by carrying the tracer round, k. alwa*. -Take a strip of paper, liaving one edge ■!i. a '•. i qn >l to halfthe id tn half the conju- gate ai. • it the point, ". ot" the i a, end the other point, <-, on tl li tl»' strip be now canted to rotate, al | the two point*, a and c, on the reept Btive I point, b, will, in every position, indicate a point in the circumference which may be marked with a pencil, the thai obtained. ■',-. :t. It ii demonstrated, in thai branch of I which treat! of solids, as we shall see later on, that if a cone, or cylinder, be cut by a plane inclined to its axis, the It is on this property that i:- transverse and conjugate axee . \ r. and i d, catting each other in the centre, o, draw any line, a v.. equal in length to the conjugate aii-. i D, I K, as a diani the semicircle, B G A. Join l. u, and through any i ken at random, on F. a, ■i l. •_'. S, dec, draw paralleli to i b. Then, al each point of e a, erect perpeodicnl cutting ; icirela, and, at the corresponding divisions obtained on ind make them the oorree] ling perpendiculan on i: a. A li: i i ints thus obtained, that is, the extremities, i fthe lines, «ill form the required ellipse. On the I verse axis, l b, and with the . i b, the axis forming it ^ dia- • the diam ter, n t. equal to the i the smaller semicircle, bdi. Draw radii, cntting the in the points, i,j.l-.l, Ac, ami the •mailer in the points, i',j',l It I tot neci ury that ihould be »t aqauuo • apart, thongfa they i the plats for i . Through the latter to the I -. A ii. nod through the CD, tin points "I" int. iv ill be so many points in the required which may, accordingly, be traced through them. It from this, that, in order to draw an ellipse, it is sufficient i, and s point, r, in the circumfi n nee, which most always lb within per] liculan pw'nfl through tl ran aii*. 'I ; r draw al i iralM to a ii, and a line, rj. perpendicular to it ; with the centre, o, and radius, A, e<|ual half tin given tl the are, Cutting rj in/; join jo, and the Una, j o, will cut rj' in j' : oy"' will be equal to halfthe conjugate axis, C D. If the conjugate axis, C D, I . '■ it); the arc, however, in this case, having the smaller radius, on, and cutting rj' in;'; I _• the line till it cuts rj, which will be in j, and OJ will equal half the transverse axis, A n. It lias already been shown how to describe an ell . two axes are . . ihotl time back by Mr. Crane of Birmingham, for constructing an ellipse with tl passes. This method applies to all proportions, and pro . near an approximation to a true ellipse, as it is possible to obtain by means of four arcs of tan By a] i 1\ ing compasses to any true ellipse, it will lie n Certain parts of the curve appn. uli very near to ares of and that these parts arc about I it- two axes; and by m of an ellipse, the curve on each side of either axis is equal and similar; oonsequently, 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 be indefinitely mar in ellipse. Pour ribed from four differ! nt points bnl with only two different radii, are then n These four points may be all within the figure ; tl.- two greater circles may either be within or without, but the Of the two circles at the citrcm.' JOT axis must always In- within, ami, conseipn nlly. the wl can never be without the figure. ' ..rtioiis of the major and minor axes may vary infinitely, but tbi y can I Inn fore, any rule for describing ellipses most suit all pos- sible proportions, or it does not possess the necessary requfa Moreover, if any rule apply to one certain proportion ami not to another, it is evident thai the more the proportions differ from that one — win! ' or nJJSMsusaadb thl greater will be the difference of the result from the true one. From t lii- it • that if a rule applies DOt to all, it can only apply to MM 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. I ■ i. .iv major axis, and c D any minor axis; produce them both in either direction, say towards F and H, and make A F equal then join A, and through F draw F II parallel to C A. Bet off B t. a .'. and i K. equal to n r; join .i k, and bisect it in n. and at Bl ■. li.nlar. Cntting • D, or I Dpi tOOSl; J i.l. M, will be the centres of tho Through the points, j and i. draw y v u 1 1, i r. i y. each equal love; then v N and t t will be the radii of i j s. i ii. of the lees : the points oi will tin i •. P, v. and the figure drawn through a, n, e\ 0, B, •,'. p. r. will be the required ellipse. BOOK OF INDUSTRIAL DESIGN. 17 Several instruments have been invented for drawing ellipses, many of them very ingeniously contrived. The best known of these contrivances, are those of Farey, Wilson, and Hick — the last 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 arectangularbase 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 one end of the latter, a vertical pillar, B, is screwed down, for the purpose 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-piece, 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 21 inches by Is, to -is by J inch. In general, how- ever, these instruments do not appear to be sufficiently simple, or convenient, to be used with advantage in geometrical drawing. 57. Tangents to ellipses. — It is frequently necessary to deter- mine the position and inclination of a straight line 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 given apart from the ellipse ; and when a straight line is given, to which it is necessary that the tangent should be parallel. First, then, let the point, A, in the ellipse, fig. 2, be given ; draw the two vectors, c A, b a, and produce the latter to II ; bisect the angle, M a c, by the straight line, N P ; this line, n p, will be the tangent required ; that is, it will touch the curve in the point, A, and in that point alone. Secondly, let the point, h, be given, apart from the ellipse, fig. 3. Join l with 1, the nearest focus to it, and with l as a centre, and a radius equal to l 1, describe an arc, mi n. Next, with the more distant focus, n, 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 II, and the ellipse will be cut in the points v and x ; a straight line drawn through either of these points from the given point, l, will be a tangent to the ellipse. 58. Thirdly, let the straight line, Q R, fig. 2, be given, parallel to which it is required to draw a tangent to the ellipse. From the nearest focus, b, let fall 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, QR, the accuracy of which may be verified by observing whether it bisects the line, S B, which it should. 59. The oral of Jive centres, fig. 4.— As in previous cases, the transverse and conjugate axes are given, and we commence by obtaining a mean proportional between their halves ; for this 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-circle, 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 O, the semi-axes constituting its dimensions ; joining C A, let fall from the point, G, on the diagonal, C A, the per- pendicular, G H D — which, being prolonged, cuts the conjugate axis or its continuation in D. Having 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, 11 n, describe the arc, N a, cutting the former in a. The points, 11, a, on one side, and h', b, obtained in a similar manner on the other, together with the point, d, will be the five centres of the oval ; and straight lines, r n «, s 11' b, and P a d, q b d, passing through the respective centres, will 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 vaults ; an example of its use is given in fig. g. GO. The parabola, fig. 5, is an open curve, that is, one which does not return to any assumed starting point, to however great a length it may be extended ; and which, consequently, can never enclose a space. It is so constituted, that any point in it, D, is at an equal distance from a constant point, C, termed the focus, and in a perpendicular direction, from a straight line, A B, called the directrix. The straight line, F G, perpendicular to the directrix, A B, and passing through the focus, c, is the a.eis of the curve, which it divides into two symmetrical portions. The point, A, midway between F and c, is the apex of the curve. There are several methods of drawing this curve. 61. First method : — This is based on the definition just given, and requires that the foeus and directrix be known, as c, and a b. Take any points on the directrix, a b, as a, e, h, i, and through them draw parallels to the axis, F G, as also the straight lines, 1- Tlii: PRAf I I'M. DRAUGHTSMAN'S ■ «ith the focui. I)r»w | dicuUn ,! >c'n u,lt '' ,,u '}' cut the »hich may bo traced I • : ■ straight line* which were just drawn, cutting the parallels in different puint> of I If, then, it I draw a tangent lined simply by joii. | a] to c t\ ami bisecting tho angle, ii c 0, by the ' which will be the required hMgffllt Ii' tin- point .ij.^rt from the curve, tho procedure will be tho same, but the I Dng to H <" will not bo parallel to tho axis. We l,;ive lure given the axis, m;, the I ':■ 111 the point, /. let fall on the .. olicul.ir. / ii, and prolong ttkil to ., making o t equal to/... Iiivile Id into any number ..f equal parts, as in the point*, i, j, I; through which draw par. 11. K to the axis; divide il ■. into the same number of equal \ .... i j through tin -c draw lines radiating from the del tie v will interned the parallels in the points, m,n,o, which are so many point* in the curve. C4. If it is required to draw a line tangent to a given parabola, . | D line, J K. we h t fall a perpenilieiilar, L, on this last; thil perpendicular will cut the directrix in />. ami // n drawn parallel to tho axis will cut the curve in the point of con- tact, n. We find frequent applications of this curve in constructions and o a. -fount of the peculiar propi rties it ] which the Mil. lent will find .li U". d a* lie proceeds. :. tented in Age. 'J. I)', are an example of the application of this curve. They are call. 1 /'..- and are employed in philosophical rescare'e IB. The angles of ''. are equal to the i of the parallel [| foUowi from this pro- perty, that if in the focoa, 0, of one mirror, h /, the flam* of I lamp, or . ly be placed, and in the Fo the opposite mirror. i tinder, the latter will be i b the two fociinaybe.it a considerable dis- tance apart ; for all the rav - of OSloric Calling 00 the mirror, bf t •..1 fr.nn it in paralli 1 lint . ooU b I In i mirror, b'/\ and concentrated at it- focus, with B, the point of in: of the lines, a n, n 0> I'i 'it B l> in ■*, which will be a point in the .loin 1. 1, l: A, and bias I thl lim -. i: ' . i: A, b\ I pendieulara, a b, c BUB] 08. The square new* is the unit of surface measurement, just as the linear in. Ire is that of length. The square metre is sub- divided into the ■fNOr the fijiinrc a atsM '•:■• . and the BSWOn tniUini'tri-. Whilst the linear .1. .inn' Ire 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 number mul- tiplied into it.-. It. 01 m. x -1 m. = 001 squaro n. In the same manner the square i. ntituetre is the t. n-thousandth part of the square metre ; for i M il m. X 001 m. = 00001 sqn And the square millimetre is the millionth (.art of tho square in. tre j for, o-ooi m. x i m , tare metre*. It is in this way that a relation is at once dcti the units of linear and surface IIIIHHIIHIIIII i.t. Similarly in English measures, a square foot is the ninth part of a square yard ; for 1 foot x 1 foot = J yard x 3 yard = J square yard. A square inch is the 1-1 Ith part of a square foot, and the 1296th part ; for 1 inch X 1 inch = j^ foot X iS foot = , } t square foot, and 1 huh x 1 inch B .,'„ yard X j,\, yard = , ; ,'„,i square yard. This illustration places the simplicity and adaptability of the d. . nnal sy-t. in of m.-asur. -. in ItrOBg . 'iilrast with the complicity Of other methods. 69. Ml r\ mart of turfiir, m. The surface or area of a square, as well as of all rectangles and parall rpre*B*d by the product of tho base or length, and height or breadth measured BOOK OP 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, the 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 •9375 „ r25=' 75m - 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. Example. — The parallel sides of a trapezium being respectively 1-3 m., and 1-5 in., and the breadth '8 m., the area will be *±+l* x -8 = 112 sq.m. The area of a triangle is obtained by multiplying the base by half 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 115 23X-r- = 1-3225 sq. m. The area of a triangle being known, and one of the dimensions given — that is, the base or the perpendicular height — the other di- mension can be ascertained by dividing double the area by the given dimension. Thus, in the above example, the division of (1-3225 sq.m. x 2) bythe height 1-15 m. gives for quotient the base2 - 3m., and its division by the base 2'3 m. gives the height 1-15 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 given, 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 e, of the triangle, a b C, fig. 16, PI. I., being 3 m., the side b c, 4 m., the hypothenuse, a c, will be ac= vF+T 2 = V9+ 16 = V25 = 5m. 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 = V25 — 9 = Vl6 = 4m. The diagonal of a square is always equal to one of the sides mul- tiplied by V 2; therefore, as V 2 = 1-414 nearly, the diagonal is obtained by multiplying a side by 1-414. Example. — The side of a square being 6 metres, its diagonal = 6 X 1-414 = 8-484m. The sum of the squares of the four sides of a parallelogram are 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 the perpendicular distance from the centre to one of the sides 5-6 m., will have for area — 9-8X5X 5 ! 6 : 2 The area of an irregular polygon will be obtained by dividing it into triangles, rectangles, or trapeziums, and then adding together the areas of the various component figures. 137-2 sq.m. TABLE OF MULTIPLIERS FOR REGULAR POLYGONS OF FROM 3 TO 12 SIDES. Triangle, Square, Pentagon, ... Hexagon, Heptagon,... Octagon, Enneagon, .. Decagon, Undecagon,.. Duodecagon, 2000 1-414 1-238 1156 1-111 1-080 1-062 1-050 1-040 1037 B 1-730 1-412 1-174 radius. •867 ■765 •681 •616 •561 ■516 •579 •705 •852 side. 1-1C0 1-307 1-470 1-625 1-777 1-940 •433 1-000 1-720 2-598 3-634 4-828 6-182 7-694 9-365 11-196 Internal Angle. 60° 0' 90° 0' 108° 0' 120° 0' 128° 34'f 135° 0' 140° 0' 144° 0' 147° 16' T 4 T 150° 0' Apothegm Perpendicular. •2886751 ■5000000 •6881910 •8660254 10382607 1-2071069 1-3737387 1-5388418 1-7028436 1-8660254 By means of this table, we can easily solve many interesting problems connected with regular polygons, from the triangle up to the duodecagon. Such are the following : — First, The width of a polygon being given, to find the radius of the circumscribing circle.— When the number of sides is even, the width is understood as the perpendicular distance between two opposite and parallel sides ; when the number is uneven, it is twice the perpendicular distance from the centre to one side. Bute. — Multiply half the width of the polygon by the factor in column A, corresponding to the number of sides, and the product will be the required radius. Example. — Let 18-5 m. be the width of an octagon ; then, H^ X 1-08 = 9-99 m.; 2 or say 10 metres, the radius of the circumscribing circle. -" Till: PRACTICAL DRAUGHTS*! Second, The radius of a circle briny gicxn, to find the length of 0.' n.I. ■ /ii .'_. • ' ■ i .». -Multiply the radim by the tactor in column B, corre- / j -'«•.— The radius being 10m., the side of tn inscribed octagon will be — 10 x -765 = 7 65 m. Third, The tide of a polygon being giccn, to find the radiiu of tkt circumtcr —Multiply i! factor in column C, corro- Examj>le. — l. ofanoctago:. i ! y the factor in column D, i :>e side of an octagon being 765 m., the area will be— T 5 X 4-828 = 36-93 sq.ra. L 01 a I m Us. 71 1: • reumference of any circle be divided l.y its diame- -.t will be a Dumber which is called, At ratio c/ ■ r. This ratio is found to be (ap- prtudmal 31410, or -J that ii, the circumference equals 8-1416 tim.s the length of the diameter. It i- npi raic formulas, by the Greek resents the circumference of a circle, and 1) its diameter, tho following formula, • L416 X I), expresses the •! I the eiretimierenee. Th\is. If the diameter of a oiroU, or 1), = J 7 m., or tho radius, It = 1 •:!.'■ in.. :.!• r. not Hill be equal to — J, or 8-1416 x 1-86 X 2 = 8-483 m. Hm oirctunference of n circ la I H - .Ii urn tar, or radios, i. farad by dividing this etroarafere by B 1416 for thi former, lismi b r. D, of ■ ehri la, the circumference of wbi< . is— and the radius, K, is— 3 in-; B 481 1 35 m. The area of a circle iefou . j the circumfertmet fry half the radim. — Tins rule is expressed in the folluwiug formula: — The area of a circle = 2 r It x — = This term, r II : . i« merely the simplification of the formula. The nui: multiplier and divisor, mav be can- ,-id the product of It into R is expressed by It', or the square of the radius. It follows, then, that the area of a equal to the square of the radius multiplied by the circumference, or 81416. . ,'<•.— The radius of a circle being l-05m., the area will be— 3 1410 X 105 X It'.', - 3-46S5sq.m. The area of a circle being known, the radius is determined by dividing the area by 31410, and extracting the square root of the quotient. Kcumjilc. — The area of a circle being 3-4035 sq. m., the radius V Area = 3-1416 The area of a circle is derived from the diameter ; thus— rllXD tD 1 1— > « — ; 81416 then, since -r or — = -7854, •t 4 the formula resolves itself into Aim = -7854 XD 1 . That is to say. if in multiply the faction, 7 putrc of the diameter, the product will be the area. / ./ •!<■.— The area of a circle, the diameter of which mea- sures 21 m., is — •7854 X 21 X 21 = S-46S5aq.ro, It follows from this, that if the ar. . wn, tliat of en inscribed obeli ii obtainable, by multiplying l>v '7864; that i-, the area of a square is to tho area of the inscribed circle, as, 1416, or 1 : 7 TABU Of LFFKOXIal VTr. RATIOS Bl rWI in OIBI lis \M> (QUASI - tho side of a square or equal area. 1 1 . 8. Tho side of a snuarc x x The diameter of a circle X ' X X x of aotrcle x I fan x -seal ■707 1 •2261 1-4142 1-1280 tho side of tho inscribed square. ■ the ili o ; ibing circle. laaeribing circle. tho circii irclc. table aflbrdl a ready solution of tho following amongst other 1 1 i I be diameter of a circle being 125 m. or 125"/, (railli- leof • square of equal area is 3862- 1 In 775-/.. !. The circiin.t'-rcuce of I iirJ. being S00"/ M , the the Ins cribed square i- 860 x S86"V Third. The si.h of I square being 21586"/,„ the diameter of the circurascril I ill.' = 305-27-/.. BOOK OP INDUSTRIAL DESIGN. 21 The 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 follows, hence, that if the radius or diameter be doubled, the circumference will only be doubled, but the area will be quadrupled ; thus, a drawing reduced to one-half the length, and half the breadth, 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 or 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 degrees contained in the arc, and dividing by 3G0°. Example. — The circumference of a circle being 3-5m., an arc of 45° will be 3-5 X 45 300 = -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 ■ 108X8—344 ^ = -4133 m. 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 „„„, 5 = -0225 sq. m., the area of the sector. The arc of a segment is obtained by multiplying the width ; 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 multiplying 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, 2x^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 3 and, 48 x 2 X 2 x 18 60-7 = 576-0 Total, 636-7 sq. m. A still simpler method, is to obtain the area of the sector of which the segment is a part, and then subtract the area of the COMPARISON OF CONTINENTAL MEASURES, WITH FRENCH MILLIMETRES AND ENGLISH FEET. Austria, . Bavaria, . Belgium,.... Bremen, Brunswick, . Cracovia,.. Denmark,., Spain, Papal States. Frankfort, Hamburg,. Hanover,. Hesse,.... '"{ Designation of Measure. Vienna) Foot or Fuss r= 12 inches = 141 lines, (Bohemia) Foot, (Venice) Foot, Foot (Palmo), Foot (Architect's Measure), (Carlsruhe) Foot (new) = 10 inches 100 lines, (Munich) Foot = 12 inches = 144 lines, (Augsburg) Foot (Brussels) Ell or Aune = 1 metre,.. Foot, (Bremen) Foot = 12 inches = 144 lines, (Brunswick) Foot = 12 inches = 144 lines (Cracow) Foot, (Copenhagen) Foot, (Madrid) Foot (according to Lob- man), , Castilian Vara (according to Liscar), (Havanna) Vara = 3 Madrid feet, .. (Rome) Foot, Architect's Span = ^ foot, Ancient Foot, Foot, Foot = 3 spans = 12 inches =2 96 parts, (Hanover) Foot = 12 inches = 144 lines, (Darmstadt) Foot = 10 inches = 100 lines, 316-103 296-416 435-185 347-398 396-500 300-000 291-859 206-168 1,000000 285-588 289-197 285-362 356-421 313-821 282-655 835-906 847-965 297-896 223-422 294-246 284-610 286-490 291-995 300-000 1-037 •970 1-460 1-140 1-301 •984 •958 •972 3-281 •937 •949 •936 1 169 1-029 •965 •933 Holland Lubcck, Mecklenburg, . Modena, Ottoman Empire,. Parma, Designation of Measure. Poland, ... Portugal,.. Prussia,... Russia,.... Sardinia, .. Saxe, Sicilies, ... Sweden, . Switzerland,.... (Amsterdam) Foot = 3 spans : inches, (Rhine) Foot, (Lubeck) Foot, Foot, (MoJena) Foot, (Reg^io), (Constantinople) Grand pie Aims-length = 12 inches = 1728 atomi, (Varsovie) Foot = 12 inches = 144 lines, (Lisbon) Foot (Architect's Measure), Vara = 40 inches, (Berlin) Foot = 12 inches, (St. Petersburg) Russian Foot, .. ' Archine, (Cagliari) Span, (Weimar) Foot, Span = 12 inches (ounces = 60 111 in lit i), (Stockholm) Foot, (Bile and Lurich) Foot (Berne and Neufchatel) Foot = 1 Tuscany, Wurtemburg,. inch'- (Geneva) Foot, (Lausanne) Foot = 10 inches = 10 lines, (Lucerne and other Cantons) Foot,.. Foot , Foot = 10 inches = 100 lines, 283-056 313-854 291-002 291-002 523048 530-898 069-079 544-670 297-709 338-600 1,008-363 309-726 538-151 711-480 202-573 281-972 263-670 296-838 304-537 293-258 487-900 300-000 313-854 548-167 286-490 •928 1-030 •954 •954 1-716 1-742 2-195 1-787 •977 Mil 3-636 1-016 1-765 2-334 •664 •925 •865 •974 ■999 •962 1-600 •984 1-030 THE lKAi lit AI. DRAUGHTSMAN'S triangle constituting the difference between the sector and H .-•: ■ :.* To find the area of an annular (pace contained between two con- centric articlci, multiply the sum of the diaii difference, and by the fraction Example. — Let 100m. and GOm. be the respective dl then, (100 + 60) x (100 — 60) x -T8M = 6026'M »q. dl the area of the annular space. The area of a fragment of such annular space will bo found by multiplying its radial breadth by half tin- sum of the arcs, or, more com | tly, 1 y tin- arc which is a mean proportional to them. mi: asp ai;i \ Of an i;i i ii sr. 74. The rircumfi B1 nee of II Dal to that of a circle, of which the diameter is a mean proportional Dl twaen the two axes ; it will be obtained by multiplying such mean propor- tional by 3 MIC, the ratio between thvdiaiuetcrand circumference of a circle. ' \fU,—\M 10 m. and 6-4 m. be the length* of the respec- tive axis ; then, \ 10 x 6 4 x 3 1416 = 251328 m. The are* of the ellipse is obtained by multiply- of the two axes by '7854, thl B the diameter and the I ■ • I ■ 7 I = —50-2656 sq. m. ■ in the indus- trial arts, nu 1 partieiihiily in meclianies, as will Ik- seen further on. The aiamplai given will enaUatha student to understand the various operations, as Well as to solve other a: . problems. CHATTER II. THE BTUDY OF PEOJECTIl NS 75. To indicate all the dimensions of on object by pictorial de- lineation, it is necessary to represent it under ■even] different as- pects. These various views arc compr. h. nl. .1 under the general denomination of I ind usually consist of derations, plans, ■ • n, of the ttndy of projections, or i tiun on paper of the appear- ances of all bodies of many dimensions as viewed from different : It is customary to determine the projeotioni of a body on two principal planes, one of which i- distinguished as the I- tharai the serttoaij a, These two plans are also called ;, | tions or plans. Tliey are an- aaefa other, the horiaontal plan being the lower; the line ting than is called the bom lias, and is always parallel to one of the sides of the draw i- It is of great Importance t" have a thorough knowledgo of the ■ ■■■■ prim Iplai of d, ■scriptive geometry, iii ordi r to l"' able .;' forms, thi i ontonn of all kinds of objects; and wo shall now . ioh explanatory details as are necessary, commencing primarily with the pn of a point and of a line. BLSMENTAKY 1'KINfII'I.KS. Till: PROJECTIONS OP A POINT. I'LATK VI. It A n c D, figs. 1 and 1 , bo a horizontal plane tlio board on which the drawing i urfacc of a pavement Al ... le) \ i: , i be .1 plane, such as a wall at one side of the mi nt ; the straight line, which is I planes, is the base line. Finally, let O be any point in space, the representation of which it is d : If. from this point, ii. we suppose a perpendicular, o o, to be let fall on the horizontal 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 snppi pendioular, o<>', to be let fall on the vertical plana, A li i: r, the point of contact, o\ or foot of t li i - perpendicular, will be the vertical projection of the same point. These perpendiculars are reproduced in the vertical ami horiaontal plant >, 1 j i. parallel and equal to o o andoo. 77. It follows from this construction, that, when the two pro- of any point arc ci\in. the position in space of the point itself is deb rminable, it being oecoisarfly the point of intt of perpendit mlan arectad on the reapecth of the point. As in drawin:;. on! i |j , the sheet Of paper, and | to one and tl plane.it is customary to luppOM the Vertical plane. A B i: F, fig. 1, as forming a continuation of the fa riaontal plane, i H id, being turned on thi I {a, so as to coincide with it — just as a book, half open, is folly opened tlat on a table. \\ , thus obtain tin figure, Dt I I . fig, I . n | rt lentil the two plant - a paratt d bj the base Una, a n, and tin po.: nl the horizontal and vertical pro- jections of tin- given point. It will be remarked, that these points. In in one line, perpen- dicular to the bee line, a B j thil i' 1 I ' ause. in the t iinii li ■; down Of the previously \crtical plane, the line, n <• . b( comes a prolonga- tion of the line. no. It || neci ssary to observe, thai the linc,fio', the distance of the point from the horizontal plane, whilst -. ni the \ertical plane. In other words, if on o we erect a perpendicular to the plane, and I ' ,. 1 1 <£ 4 1 iff 4 ,„• 1 i 1 1 1 1 i i_ Y i />' "' "<' 1 1 I I 1 1 I 1 1 > ( •/' /'' ! i > Wmencvaud \i troujf BOOK OF INDUSTRIAL DESIGN. 23 measure the distance, no-, 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 with 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, Mm', oo, 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, in 0, is a mere point, m, because the line lies wholly in a perpendicular, M m, 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 horizontal and ver- tical projections of the given right line, II 0, are respectively the point, m, and the right line, m o . 80. If we suppose that the given straight line, m o, 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, d, will be the vertical, whilst the straight line, mo, will be the horizontal projection. In both the preceding cases, the projections lie in the same perpendicular line, m m, fig. 2", and 6 o, fig. 3*. 81. When the given straight line, M o, is parallel to both the horizontal and the vertical plane, as in figs. 4 and 4", its two pro- jections, m o and in 6 ', will be parallel to the base line, and they will each be equal to the given line. 82. When the given straight line, BI o, figs. 5 and 5", is parallel to the vertical plane, A b e f, only, the vertical projection, in o', will be parallel to the given line, whilst the horizontal 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 parallel to it, whilst its vertical projection will be parallel to the base line. 83. Finally, if the given straight line, 11 o, figs. 6 and 6", is inclined to both planes, the projections of it, mo, m'o', will both be inclined to the base line, A B. These projections are in all cases obtained by letting fall, from each extremity of the line, per- pendiculars to each plane. The projections of a straight line being given, its position in space is determined by erecting perpendiculars to the horizontal plane, from the extremities, m o, of the projected line, and making them equal to the verticals, nm and po'. The same result follows, if from the points, in, 6, in the vertical plane, we erect perpendiculars, respectively equal to the horizontal distances, mn and po 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 representations. 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 counterpart 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, mopq, equal and parallel to itself, whilst the vertical projection will be a straight line, po, parallel to the base line, A B. 86. Similarly, in fig. 8, the object being supposed to be parallel to the vertical plane, its projection in that plane will be the equal and parallel figure, m op q, whilst that in the horizontal plane will be the straight line, m o. 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", whilst its projection in the other plane will assume the form, mopq, 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", which will be easily understood, the same letters in various characters indicating corresponding points and perpendiculars. Nor does the obtain- ment of the projections of surfaces bounded by curved lines, as circles, require the consideration of other principles, as we shall proceed to show, in reference to figs. 10 and 11. 88. In the first of these, fig. 10, the circular disc, mopq, is sup- posed to be parallel to the vertical plane, abef, and its projec- tion on that plane will be a circle, m op' q, equal and parallel to itself, whilst its projection on the horizontal plane, A B c D, will be a straight line, qmo, equal to its diameter. If, on the other hand, the disc is parallel to the horizontal plane, as in fig. 11, its vertical projection will be the straight line, pom, whilst its horizontal pro- jection will be the circle, op m q. If the given circular disc be inclined to either plane, its projec- tion in that plane will be an ellipse ; and if it is inclined to both planes, both projections will be ellipses. This will be made evi- THE PRA< H( \l. DRA in of various points in the circum- ■ Alien constructing the pr> facilitatcA tlic process considerably ■. 11, iad IS. I of nil plane l found, il known how to obtain the \ 1 1 1 1 » » I. And, mi .inJ Ik t > tion of their | ilows the same rule*. PROMS AND OTHEB BOLDDS. PLA'Ii Before ("ntprin.- upon the principlee involved in the repri •entation of solids, the student should make himself acqnainted with the de scripti v e denominarioni adopted in science and art, with reference to such objects; and we here subjoin such as will I i i il riOKB.— A solid is an object baring three dimensions; that is, its extent oon . width, and height. A 1 also possesses magnitude, volume, or capacity. I nns of solids. The polyhedron is a solid, . and the tphere, ' y curved surfaces. Thoae are termed solids of re- i, which may bo defini I by tin- revolution of a ni i fixi 1 straight line, termed the axis. Thus, a ring, or aiiiui! Derated by the revolution of a circle about a straight line, lying in the plane of the circle, an 1 I the plane of revolution. lion, the 10M of which are parallelograms, and the ends equal and parallel polygons. A pri-m is termed right, when the lateral faces, or fate is, arc perpendicular to the ends; and it is regular, when the ends are regular polygons. A pri-m i- also called a when the ends are rectangles, or parallelograms; and when it ii i and square facets, it u termed represented in fig. A. Other i ides the cube, are distingnJ appropt ia, the tttrah and the . which are bounded externally, respectively, by four, eight, and twenty equilateral triangles; and the duodeeahedron, which is terminated by twelve regular pentagons. A pyramid i- i polyhedron, of which all thi lateral uniting In one point, the op t, and ba , which i- the base of the pyramid, as fig. ©. The prism an. I pyramid are triau-iilar, .|iiadr.iiu-ul.ir. pentagonal, hi oording as the ining the bases are triangles, • By the height of a pyramid is meant the length of a perpendi- • fall from the apex on the base ; the pyramid IsaryiU I whan this pa] lieular mat i- the oentre of thi pyramid, or the frustum of a pyramid, is ■ solid which ms I as a] yramid having the apex cut otr by a plane paralh I, or In lno A to the base. \ . Vm/.r i» a solid which may ! a straight line, revolving ubout, and at any given distann IT axis to which it ir paralll 1. A cylinder which . g alniut one of its sides as an axis, is said to be a right cylinder; such a one ia ■ ted in fig. 2. A . one of its sides as an axis. \ terminated short of the apex by a plane parallel, or inclined to the base. Thil '•■ cone is said to be right wh< n its base is a circle, and when a per- pi ndicular let fall from the apex passes through i \ ' • lution of a semicircle about its diami tl t as an axis, as fig. (§. ■ . ' ■ | lail'r.is of the sphere of which the sector forms a part. When tin revolution is exterior to thi generating sector, thi obtained will be annular or sonic. The 1 by the are. t. i '. i- the The rono b. spheric ore whan the axis of n vt lution i- 1 oi of the radii forming •■•r. A rtion, a* i n a comprised between two semii ircular planes inclim 1 I and meeting in a diameter, as I o, of the sphere. Th il port, surface of the sphere which forms the base of the ungula, i a 'jure. A ipl i- any part of a sphere cut off by a plane, and may be considered as a -olid of rev. ; i by the revolution of a plane segment shout its centre line. The plane surface is the bast of the segment. When thi through the centre of the sphere, two equal segments are obtained, termed hemisphi ret. A mental wiiiidim is a solid general. 1 by the revolution of a •merit, ii' It' K. fig. 7. about any diameter.