' 1 » ]l t INI A ! A 1 ■ tt-'-'H 111 1 1 in 1 1! 1! ijij|-:!iiH|li' ■ jr * r 1 ' wJB i 1 i': fflji'i j Jl piiji •''jjp'jfll" J &fe i(J Mary Anni> duett (Decorati\/ecAr? Qoueffiotu STIRLING AND FRANC1NE CLARK, ART INSTITUTE L1BRART Digitized by the Internet Archive in 2012 with funding from Sterling and Francine Clark Art Institute Library http://archive.org/details/practicaldraught1896arme THE PRACTICAL DRAUGHTSMAN'S BOOK OF INDUSTRIAL DESIGN, AND MACHINIST'S AND ENGINEER'S DRAWING COMPANION: POEMING A COMPLETE COURSE OF fPtt|amaI, (fepurrag, aitfr §aT|itcdTOl §rairag + T — aa rce, - ib. Tensional resistance, - - - - - 43 Resistance to flexure, - - - - - 44 Resistance to torsion, - - - - 46 Friction of surfaces in contact, - - - 49 CHAPTER IV. THE INTERSECTION AND DEVELOPMENT OF SURFACES, WITH APPLICATIONS, 49 The Intersections of Cylinders and Cones : Plate XIV. Pipes and boilers, - - - - - 50 Intersection of a cone with a sphere, - ib. Developments, - - - - - - ib. Development of the cylinder, - - - - 51 Development of the cone, - - - - - ib. The DEr,rNEATioN and Development of Heuces, Screws, and Serpentines : Plate XV. Development of the helix, - - - - - Screws, ------- Internal screws, ------ Serpentines, - Application of the helix — the construction of a staircase: Plate XVI., ..---- CONTENTS. The intersection of surfaces — applications to stopcocks: Plate XVII., -.-._. 56 Rules and Practical Data. Steam, Unity of heat, Heating surface, Calculation of the dimensiu Dimensions of firegrate, Chimneys, - Safety-valves, CHAPTER V. THE STUDY AND CONSTRUCTION OF TOOTHED GEAR, Involute, cycloid, and epicycloid : Plates XVIII. and XIX. Involute: Fig. 1, Plate XVIII., - - - - Cycloid: Fig. 2, Plate XVIII., - - - - External epicycloid : Fig. 1, Plate XIX., - - - External epicycloid, described by a circle rolling about a fixed circle inside it : Fig. 3, Plate XIX., - Internal epicycloid: Fig. 2, Plate XIX., - Delineation of a rack and pinion in gear: Fig. 4, Plate XVIII., Gearing of a worm with a worm-wheel : Figs. 5 and 6, Plate XVIII., _-.--. Cylindrical or Spur Gearing : Plate XIX. External delineation of two spur-wheels in gear: Fig. 4, . Delineation of a couple of wheels gearing internally : Fig. 5, - Practical delineation of a couple of spur-wheels : Plate XX., The Delineation and Construction of Wooden Patterns for Toothed Wheels : Plate XXI. Spur-wheel patterns, ----- Pattern of the pinion, - Pattern of the wooden-toothed spur-wheel, - Core moulds, -_'____ Rules and Practical Data. Toothed gearing, ------ Angular and circumferential velocity of wheels, Dimensions of gearing, - Thickness of the teeth, - Pitch of the teeth, ------ Dimensions of the web, - Number and dimensions of the arms, - Wooden patterns, ------ CHAPTER VI. CONTINUATION OF THE STUDY OF TOOTHED GEAR. Conical or bevil gearing, - - - _ _ Design for a pair of bevil- wheels in gear : Plate XXII., Construction of wooden patterns for a pair of bevil-wheels : Plate XXIII., ------ Involute and Helical Teeth : Plate XXIV. Delineation of a couple of spur-wheels, with involute teeth : Figs. 1 and 2, - Helical gearing : Figs. 4 and 5, - - - - Contrivances for Obtaining Differential Movejiexts. The delineation of eccentrics and cams : Plate XXV., Circular eccentric, ------ Heart-shaped cam : Fig. 1, Cam for producing a uniform and intermittent movement: Figs. 2 and 3, - Triangular cam : Figs. 4and5, Involute cam : Figs. 6 and 7, - - - - Cam to produce intermittent and dissimilar movements: Figs. 8 and 9, Rules and Practical Data. Mechanical work or effect, - The simple machines, Centre of gravity, - On estimating the power of prime movers, Calculation for the brake, The fall of bodies, - - - Momentum, - Central forces, - CHAPTER VII. ELEMENTARY PRINCIPLES OF SHADOWS, Shadows of Prisms, Pyramids, and Cylinders : Plate XXVI. Prism, ___--- • Pyramid, ------- Truncated pyramid, - Cylinder, ------- Shadow cast by one cylinder on another, - Shadow cast by a cylinder on a prism, ... Shadow cast by one prism on another, - Shadow cast by a prism on a cylinder, - - - Principles of Shading: Plate XXVII., - Illumined surfaces, ------ Surfaces in the shade, - Flat-tinted shading, Shading by softened washes, -'.--.- Continuation of the Study of Shadows: Plate XXVIII. Shadow cast upon the interior of a cylinder, Shadow cast by one cylinder upon another, - - - Shadows of cones, ------ Shadow of an inverted cone, - Shadow cast upon the interior of a hollow cone, Applications, ------ Tuscan Order: Plate XXIX. Shadow of the torus, - Shadow cast by a straight line upon a torus, or quarter round, Shadows of surfaces of revolution, - - - - Rules and Practical Data. Pumps, ------- Hydrostatic principles, - Forcing-pumps, ------ Lifting and forcing pumps, - - - - - The hydrostatic press, - - - - - Hydrostatical calculations and data — discharge of water through different orifices, ------ Guaging of a water-course of uniform section and fall, Velocity at the bottom of water-courses, - Calculation of the discharge of water through rectangular orifices of narrow edges, - - - - - Calculation of the discharge of water through overshot outlets, To determine the width of an overshot outlet, To determine the depth of the outlet, - Outlet with a spout or duct, - - - 100 ib. ib. 101 102 103 ib. ib. 104 105 ib. ib. 107 108 ib. 110 ib. Ill 114 ib. 11G CHAPTER VIII. APPLICATION OF SHADOWS TO TOOTHED GEAR: Plate XXX. Spur-wheels: Figs. 1 and 2, - *'&• Bevil-wheels: Figs. 3 and 4, - 117 Application of Shadows to Screws : Plate XXXI., - - 118 Cylindrical square-threaded screw: Figs. 1, 2, 2 a , and 3, - ib. Screw with several rectangular threads : Figs. 4 and 5, - ib. CONTENTS. Triangular-threaded screw: Figs. G, 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., - PAGE 118 119 ib. 120 ib. 122 CHAPTER IX. THE CUTTING AND SHAPING OF MASONRY: Plate XXXIV., 123 The Marseilles arch, or amere-vousswe : Figs. 1 and 2, - ib. Rules and Practical Data. Hydraulic motors, - - - - - -126 Undershot water-wheels, with plane floats and a circular channel, - - - - - ib. Width, - - - - - - - ib. Diameter, - - - - - -127 Velocity, - - - - - - - ib. Number and capacity of the buckets, -■-'■..- ib. 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, - 136 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-Pumps : 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, - - - - - - - ib. Connecting-rod and crank, - - - - - ib. Fly-wheel, - - - - - - ib. Feed-pump, ---___ ;j. Ball or rotating pendulum governor, - 144 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, ---___ 14Q PAOE Diameter of piston, __._._.- 147 Velocities, - - - - - -148 Steam-pipes and passages, - - - - ib. Air-pump and condenser, - - - - ib. Cold-water and feed-pumps, - - - - 149 High pressure expansive engines, - - - - ib. Medium pressure condensing and expansive steam-engine, - 151 Conical pendulum, or centrifugal governor, - - _ 153 CHAPTER XI. OBLIQUE PROJECTIONS. Application of rules to the delineation of an oscillating cylinder : Plate XLI., - 154 CHAPTER XII. PARALLEL PERSPECTIVE. Principles and applications : Plate XLII., - - 155 CHAPTER XIII. TRUE PERSPECTIVE. Elementary principles : Plate XLIIL, - First problem — the perspective of a hollow prism : Figs. 1 and 2, Second problem — the perspective of a cylinder : Figs. 3 and 4, 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, Fourth problem — the perspective of a bearing brass, placed with its axis vertical : Figs. 7 and 8, - Fifth problem — the perspective of a stopcock with a spherical boss: Figs. 9 and 10, _____ Sixth problem — the perspective of an object placed in any posi- tion with regard to the plane of the picture : Figs. 11 and 12, Applications — flour-mill driven by belts : Plates XLIV. and XLV. Description of the mill, - Representation of the mill in perspective, - - _ Notes of recent improvements in flour-mills, Schiele's mill, --____ Mullin's " ring millstone," - Barnett's millstone, _____ Hastie's arrangement for driving mills, - Currie's improvements in millstones, - Rules and Practical Data. Work performed by various machines. Flour-mills, ______ Saw-mills, -_---_ Veneer sawing^ machines, - - - - _ Circular saws, --____ 158 ib. 159 163 164 ib. 165 16G ib. ib. 168 170 171 CHAPTER XIV. EXAMPLES OF FINISHED DRAWINGS OF MACHINERY. Example Plate .., balance water-meter, - 172 Example Plate _5, engineer's shaping machine, - - 174 Example Plate ► -'#'- 'f f I ► k- A ^ A Fio-,2. Kin' 3, A 2 1 J . - 1 i i i i J — -- "* — - . .. J_ L_ _ r' 1 < -iiflk ..iikii^^A,, A E t = 2JO Fiq\ 4 \ I --+» J Pelilcolin sculp FW E -¥; • I' Mr |<\ Fie 6 \ i.Tueno-aud r. < ■ . - 1 1 \mouroux Practical Draughtsman i„, \ KtQ" I I' liii-olin si-ulp Fie 'ie 4 'I..' I Fig- l». X ' ! e.B. I'-imiH,! I,v ivfa.» ie-.E. 40* '^7^///////A ¥////, m Hat« 3. Fig F. L ,,. _ < r: ; v Fi&27 = 3, expresses that the cube root of 27 is equal to 3. The signs A and -7 indicate respectively, smaller than and greater than. Ex. : 3 £. 4, = 3 smaller than 4 ; and, reciprocally, 4 V 3, = 4 greater than 3. Fig. signifies figure ; and pi., plate. FEENCH AND ENGLISH LINEAE MEASUEES COMPARED. 10 Millimetres 10 Centimetres 10 Decimetres 10 Metres 10 Decametres 10 Hectometres 10 Kilometres 1 Millimetre. = 1 Centimetres 3= 1 Decimetre = 1 Metre = 1 Decametre = 1 Hectometre = 1 Kilometre =: 1 Myriametre = { English. •0394 Inches. •3937 " 3-9371 " 3-2809 Feet. 1-0936 Yards. 1-9884 Poles or Rods. 19-8844 " 49-7109 Furlongs. 6-2139 Miles. 62-1386 " 1 Inch = j-25-400 Millimetres. 1 2.540 Centimetres. 12 Inches = lFoot = 3-048 Decimetres. 3 Feet = 1 Yard = 9-144 " 5£ Yards = 1 Pole or Rod = 5-029 Metres. 40 Poles = 1 Furlong = 2-012 Decametres. 8 Fnrlongs 1760 Yards } = IMile = 1-G10 Hectometres THE PRACTICAL DRAUGHTSMAN'S BOOK OF INDUSTRIAL DESIGN. CHAPTER I. LINEAR DRAWING. In Drawing, as applied to Mechanics and Architecture, and to the Industrial Arts in general, it is necessary to consider not only the mere representation of objects, but also the relative principles of action of their several parts. The principles and methods concerned in that division of the art which is termed linear drawing, and which is the foundation of all drawing, whether industrial or artistic, are, for the most part, derived from elementary geometry. This branch of drawing has for its object the accurate delineation of surfaces and the con- struction of figures, obtainable by the studied combinations of lines ; and, with a view to render it easier, and at the same time more attractive and intelligible to the student, the present work has been arranged to treat successively of definitions, principles, and problems, and of the various applications of which these are capable. Many treatises on linear drawing already exist, but all these, considered apart from their several objects, seem to fail in the due development of the subject, and do not manifest that general ad- vancement and increased precision in details which are called for at the present day. It has therefore been deemed necessary to begin with these rudimentary exercises, and such exemplifications have been selected as, with their varieties, are most frequently met with in practice. Many of the methods of construction will be necessarily such as are already known ; but they will be limited to those which are absolutely indispensable to the due development of the principles and their applications. DEFINITIONS. OP LINES AND SUEFACES. PLATE I. In Geometry, space is described in the terms of its three dimen- sions—length, breadth or thickness, and height or depth. The combination of two of these dimensions represents surface, and one dimension takes the form of a line. Lines. — There are several kinds of lines used in drawing — straight or rigid lines, curved lines, and irregular or broken lines. Eight lines are vertical, horizontal, or inclined. Curved lines are circular, elliptic, parabolic, &c. Surfaces. — Surfaces, which are always bounded by lines, are plane, concave, or convex. A surface is plane when a straight-edge is in contact in every point, in whatever position it is applied to it. If the surface is hollow so that the straight-edge only touches at each extremity, it is called concave ; and if it swells out so that the straight-edge only touches in one point, it is called convex. Vertical lines. — By a vertical line is meant one in the position which is assumed by a thread freely suspended from its upper ex- tremity, and having a weight attached at the other ; such is the line A b represented in fig. Ih. This line is always straight, and the shortest that can be drawn between its extreme points. Plumb-line. — The instrument indicated in fig. Ih. is called a plumb-line. It is much employed in building and the erection of machinery, as a guide to the construction of vertical lines and surfaces. Horizontal line. — When a liquid is at rest in an open vessel, its upper surface forms a horizontal plane, and all lines drawn upon such surface are called horizontal lines. Levels. — It is on this principle that what are called fluid levels are constructed. One description of fluid level consists of two upright glass tubes, connected by a pipe communicating with the bottom of each. When the instrument is partly filled with water, the water will stand at the same height in both tubes, and thereby indicate the true level. Another form, and one more generally used, denominated a spirit level — spirit being usually employed — consists of a glass tube (fig. [§) enclosed in a metal case, a, attached by two supports, b, to a plate, c. The tube is almost filled with a liquid, and the bubble of air, d, which remains, is always exactly in the centre of the tube when any surface, c D, on which the instrument is placed, is perfectly level. Masons, carpenters, joiners, and other mechanics, are in the habit of using the instrument represented in fig. ©, consisting simply of a plumb line attached to the point of junction of the two inclined side pieces, ab, be, of equal length, and connected near their free ends by the cross-piece, A B, which has a mark at its THE PRACTICAL DRAUGHTSMAN'S centre. When the plumb line coincides with this mark, the object, C D, on which the instrument is placed, is exactly horizontal. Perpendiculars. — If the vertical line, A E, fig. 1, be placed on the horizontal line, c D, the two lines will be perpendicular to, and form right angles with, each other. If now we suppose these lines to be turned round on the point of intersection as a centre, always preserving the same relative position, they will in every position be perpendicular to, and at right angles with, each other. Thus the line, I o, fig. 5, is at right angles to the line, e f, although neither of them is horizontal or vertical. Broken lines. — It is usual to call those lines broken, which con- sist of a series of right lines lying in different directions — such as the lines B, A, E, H, P, N, fig. 14. Circular lines — Circumference. — The continuous line, EFGH, fig. 5, drawn with one of the points of a pair of compasses — of which the other is fixed — is called the circumference : it is evi- dently equally distant at all points from the fixed centre, o. Radius. — The extent of opening of the compasses, or the dis- tance between the two points, 0, F, is called a radius, and conse- quently all lines, as O E, F, O G, drawn from the centre to the circumference are equal radii. Diameter. — Any right line, L H, passing through the centre O, and limited each way by the circumference, is a diameter. The diameter is therefore double the length of the radius. Circle. — The space contained within the circumference is a plane surface, and is called a circle : any part of the circumference, E I F, or F L G, is called an arc. Chords. — Right lines, e f, f G, connecting the extremities of arcs, are chords ; these lines extended beyond the circumference become secants. Tangent. — A right line, AB, fig. 4, which touches the circumfer- ence in a single point, is a tangent. Tangents are always at right angles to the radius which meets them at the point of contact, b. Sector. — Any portion, as boec, fig. 4, of the surface of a circle, comprised within two radii and the arc which connects their outer extremities, is called a sector. Segment. — A segment is any portion, as E F I, fig. 5, of the sur- face of a circle, comprised within an arc and the chord which sub- tends it. Right, continuous, and broken lines, are drawn by the aid of the square and angle ; circular lines are delineated with compasses. Angles. — We have already seen that, when right lines are per- pendicular to each other, they form right angles at their inter- sections : when, however, they cross each other without being perpendicular, they form acute or obtuse angles. An acute angle is one which is less than a right angle, as f c d, fig. 2 ; and an ob- tuse angle is greater than a right angle, as G C D. By angle is generally understood the extent of opening of two intersecting lines, the point of intersection being called the apex. An angle is rectilinear when formed by two right lines, mixtilinear when formed by a right and a curved line, and curvilinear when formed by two curved lines. Measurement of angles. — If, with the apex of an angle as the centre, we describe an arc, the angle may be measured by the por- tion of the arc cut off by the lines forming the angle, with reference to the whole circle ; and it is customary to divide an entire circle into 360 or 400* equal parts, called degrees, and instruments called protractors, and represented in figs. E), H, are constructed, whereby the number of degrees contained in any angle are ascertain- able. The first, fig. ®, which is to be found in almost every set of mathematical instruments, being that most in use, consists of a semicircle divided into 180 or 200 parts. In making use of it, its centre, b, must be placed on the apex of the angle in such a manner that its diameter coincides with one side, a b, of the angle, when the measure of the angle will be indicated by the division intersected by the other side of the angle. Thus the angle, abc, is one of 50 degrees (abbreviated 50°), and it will always have this measure, whatever be the length of radius of the arc, and conse- quently whatever be the length of the sides, for the measuring arc must always be the same fraction of the entire circumference. The degree is divided into 60 minutes, and the minute (or 1') into 60 seconds (or 60") ; or when the circle is divided into 400 degrees, each degree is subdivided intd 100 minutes, and each minute into 100 seconds, and so on. The other protractor, fig. H, of modern invention, possesses the advantage of not requiring access to the apex of the angle. It consists of a complete circle, each half being divided ou the inner side into 180 degrees, but externally the instrument is square. It is placed against a rule, e, made to coincide with one side, ce, of the angle— the other side, d c, crosses two opposite divisions on the circle indicating the number of degrees contained in the angle. It will be seen that the angle, d c e, is one of 50°. Oblique lines. — Right lines, which do not form right angles with those they intersect, are said to be oblique, or inclined to each other. The right lines, gc and fc, fig. 2, are oblique, as referred to the vertical line, k c, or the horizontal line, c J. Parallel lines. — Two right lines are said to be parallel with each other when they are an equal distance apart throughout their length ; the lines, I K, A B, and L m, fig. 1, are parallel. Triangles.— The space enclosed by three intersecting lines is called a triangle; when the three sides, as d e, e f, and f d, fig. 12, are equal, the triangle is equilateral; if two sides only, as GH, and gi, fig. 9, are equal, it is isosceles ; and it is scalene, or irregular, when the three sides are unequal, as in fig. 6. The triangle is called rectangular when any two of its sides, as D L and L K, fig. 10, form a right angle; and in this case the side, as dk, opposite to, or subtending the right angle, is called the hypothenuse. An instrument constantly used in drawing is the set-square, more commonly called angle ; it is in the shape of a rectangular triangle, and is constructed of various proportions ; having an angle of 45°, as fig. (S, of 60° as fig. H, or as fig. 0, having one of the sides which form the right angle at least double the length of the other. Polygon. — A space enclosed by several lines lying at any angle to each other is & polygon. It is plane when all the lines lie in one and the same plane ; and its outline is called its perimeter. A polygon is triangular, quadrangular, pentagonal, hexagonal, hepla- gonal, octagonal, &c, according as it has 3, 4, 5, 6, 7, or 8 sides. A square is a quadrilateral, the sides of which, as A b, b c, c d, * As another step towards a decimal notation, it was proposed, in 1790, to divide the circle into 4U0 parts. The suggestion was again revived in 1840, and actually adopted by several distinguished individuals. The facility afforded to calculators by the many submultiples possessed by the number 360, however, accounts for the still very general use of the ancient system of division. BOOK OP 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 redeliueated 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 lines, and the latter by full continuous lines. 1. To erect a perpendicular on the centre of a given right line, as c d, fig. 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 icto four equal parts, and we can thus divide any given right line into 2, 4, 8, 16, &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 four 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 T 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 ii, in the line C d, fig. 1. — Mark off on the line, on each side of the point, two equal distances, as c n and n 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 n, will be the line required. 3. To let fall a perpendicular from a point, as h, apart from the right line, c d. — With the point l, as a centre, describe an arc which cuts the line, CD, in G and d, and with these points as centres, describe two other arcs cutting each other in ji, 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 taugentially to the two first, draw the line R s, and then in the same manner the line t u. The lines R T 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 right line, as A B, fig. 3, into severed equal parts. — We have already shown how a line may be divided into 2 or 4 equal parts. We shall now giye 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, asi 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 i or \ ; 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 DRAUGHTSMAN'S division to the left, and the length comprised between these points will be 3 decimetres, 2 centimetres, = 32 centimetres. The diagonal scale. — When very minute measurements are re- quired, greater precision is obtained with a diagonal scale, such as fig. 8. It is thus constructed : — Having drawn a line and divided it, as in fig. 7, draw, parallel and equal to-it, ten other lines, as c, d, e, f, &c, at equal distances apart, crossing these with perpen- diculars at the decimetre divisions. From one of the smaller divisions to the left of zero, draw the diagonal, b i, and draw parallels to it from the remaining centimetre divisions, 1', 2', 3', &c. From the division corresponding to 1 decimetre, draw a diagonal to the point on the extreme parallel, ii, cut by the zero perpen- dicular, and draw also the parallel diagonals, 1 — 2, 2 — 3, and 3 — 4. It will be evident, that as in the space of the ten horizontal lines, the diagonal extends one division to the left, it will intersect each intermediate line, as the 1st, 2d, 3d, &c, at the distance of 1,2,3, &c, tenths of such division, in the same direction, so that the diagonal line, 2', will cut the 5th line at a point 2^ of a division distant from zero. Thus, one point of the compasses being placed on the point I, and the other on the intersection of the same horizontal line with the perpendicular of the decimetre division 3, the measure comprised between them will be 3 decimetres, 2 cen- timetres, and -j^y, or 5 millimetres = 325 millimetres. 8. To divide a given angle, asPCD, Jig. 2, into two equal angles. — With the apex, c, as a centre, describe the arc, h i, and with the two points of intersection, h, i, as centres, describe the arcs cutting each other in j ; join J c, and the right line, J c, will divide the angle, f c d, into two equal angles, H C J and J c I. These may be subdivided in the same manner, as shown in the figure. An angle may also be divided by means of either of the protractors, IS), I. 9. To draw a tangent to a given circle, o b d h, jig. 4. — If it is required to draw the tangent through a given point, as D, in the circle, a radius, C d, must be drawn meeting the point, and be pro- duced beyond it, say to e. Then, by the method already given, draw a line, F G, perpendicular to c E, cutting it in D, and it will be the tangent required. If, however, it is required to draw the tangent through a given point, as A, outside the circle, a straight line must be drawn joining the point, A, and the centre, c, of the circle. After bisecting this line in the point, o, with this point as a centre, describe a circle passing through A and c, and cutting the given circle in b and h ; right lines joining A B and A h will both be tangents to the given circle, and the radii c B and c n will be perpendiculars to A b and A H respectively. 10. To find the centre of a given circle, or that with which a given arc, as E F G, Jig. 5, is drawn. — With any three points, E, F, G, as centres, describe arcs of equal circles, cutting each other, and through the points of intersection draw right lines, I o and l o ; 0, the point of intersection of these two lines, is the required centre. 11. To describe a circle through any three points not in a right line. — Since only one circle can pass through the same three points, and since any circle may be described when the centre is found and a point in the circumference given — this problem is solved in exactly the same manner as the preceding. 12 To inscribe a circle in a given triangle, as A b C, Jig. 6. — A circle is said to be inscribed in a figure, when all the sides of the latter are tangents to it. Bisect any two of the angles by right lines, as A o, b o, or c o ; and from the point of intersection, o, let fall perpendiculars to the sides, as o E, o F, and o G. These per- pendiculars will be equal, and radii of the required circle, o being the centre. 13. To divide a triangle, asGEl, Jig. 9, into two equal parts. — If the parts are not required to be similar, bisect one side, as G I, in the point, o, with which, as a centre, describe the semicircle, G K I, of which G I is the diameter. This semicircle will be cut in the point, k, by the perpendicular, K ; mark off on G I a distance, G L, equal to G K, and draw the line, L M, parallel to h i. The triangle, G L M, and the trapezium, hum, will be equal to each other, and each equal to half the triangle, ghi, If the given triangle were ghi, it would also be divided into two equal parts by the line, l m. 14. To draw a square double the size of a given square, A B c D, Jig. 10. — After producingfrom different corners any two sides which are at right angles to each other, as d A and D c, to H and l, with the centre, D, and radius, d b, describe the quadrant or quarter of a circle, F b e, and through the points of intersection, F and e, with the lines, d A and d l ; draw parallels to D l and d a respectively, or tangents to the quadrant, fee; the square, F G E D, will be double the area of the given square, A b c d ; and in the same manner a square, eklb, may be drawn double the area of the square, F G E D. It is evident that the diagonal of one square is equal to one side of a square twice the size. 15. To describe a circle ludfthe sizeofa given circle, as A C B D, Jig. 11. — Draw two diameters, A B and c D, at right angles to each other ; join an extremity of each, as A, c, by the chord, A c. Bisect this chord by the perpendicular, e f. The radius of the required circle will be equal to e g. It follows that the annular space shaded in the figure is equal to the smaller circle within it. 16. To inscribe in given circles, as in Jig. 12, an equilateral triangle and a regular hexagon. — Draw any diameter, G F, and with G, as a centre, describe the arc, doe, its radius being equal to that of the given circle ; join D e, e f, and E D, and D E F will be the triangle required. The side of a regular hexagon is equal to the radius of the circumscribing circle, and, therefore, in order to in- scribe it in a circle, all that is necessary is to mark off on the cir- cumference the length of the radius, and, joining the points of in- tersection, as K i l H M J, the resulting figure will be the hexagon required. To inscribe figures of 12 or 24 sides, it is merely ne- cessary to divide or subdivide the arcs subtended by the sides obtained as above, and to join the points of intersection. It is frequently necessary to draw very minute hexagons, such as screw- nuts and bolt-heads. This is done more quickly by means of the angle of 60°, frO , which is placed against a rule, K, or the square, in different positions, as indicated in fig. 12. 17. To inscribe a square in a given circle, osACBD, Jig. 13. — Draw two diameters, as A B, c D, perpendicular to one another, and join the points of intersection with the circle, and A c B d will be the square required. 18. To describe a regular octagon about a circle having a given radius, as OF,, Jig. 13. — Having, as in the last case, drawn two diameters, as ef, gh, draw other two, I J, kl, bisecting the angles formed by the former; through the eight points of intersec- BOOK OF INDUSTRIAL DESIGN. 11 tion with the circle draw the tangents, e, k, g, j, f, l, h, i — these tan- gents will cut each other and form the regular octagon required. This figure may also be drawn by means of the square, and angle of 45°, ©. 19. To construct a regular octagon of which one side is given, as A B, jig. 14. — Draw the perpendicular, d, bisecting A b ; draw a f parallel to o d, produce A b to C, and bisect the angle, C A f, by the line, e A, making e a equal to A B. Draw the line, o G, perpendi- cular to, and bisecting e a. o g will cut the vertical, o D, in 0, which will be the centre of the circle circumscribing the required octagon. This may, therefore, at once be drawn by simply mark- ing off arcs, as eh, hf, &c, equal to A b, and joining the points, E, h, F, &c. By dividing and subdividing the arcs thus obtained we can draw regular figures of 16 or 32 sides. The octagon is a figure of frequent application, as for drawing bosses, bearing brasses, &c. 2i\ To construct a regular pentagon in a given circle, osABCDF, alsoadecagon in a given circle, as eem, Jig. 15.,— The pentagon is thus obtained ; draw the diameters, A I, e j, perpendicular to each other ; bisecting o E in k, with K as a centre, and k a as radius, de- scribe the arc, A L ; the chord, A L, will be equal to a side of the pen- tagon, which may accordingly be drawn by making the chords which form its sides, as A E, f d, d c, c b, and b a, equal to A l. By bisecting these arcs, the sides of a decagon may be at once ob- tained. A decagon may also be constructed thus : — Draw two radii perpendicular to each other, as m and o R ; next, the tan- gents, n si and N K. Describe a circle having N M for its dia- meter ; join E, and p the centre of this circle, the line, E P, cutting the circle in a ; e a is the length of a side of the decagon, and ap- plying it to the circle, as e b, &c, the required figure will be ob- tained. The distance, E a or E c, is a mean proportional between an entire radius, as e n, and the difference, c N, between it and the radius. A mean proportional between two lines is one having such relation to them that the square, of which it is the one side, is equal to the rectangle, of which the other two are the dimensions. 21. To construct a rectangle of which the sides shall be mean pro- portionals between a given line, as AC, Jig. 16, and one a third or two- thirds of it. — A c, the given line, will be the diagonal of the required rectangle ; with it as a diameter describe the circle, A b c d. Divide A c into three equal parts in the points, to, n, and from these points draw the perpendiculars, m d and n b ; the lines which join the points of intersection of these lines with the circle, as A b, a d, C b, c d, will form the required rectangle, the side of which, c D, is a mean proportional between c m and c A, or — Cto:CD::CD:CA; that is to say, the square of which c D is a side, is equal to a rec- tangle of which c A is the length, and c mthe height, because CDxCD = CmXCA* In like manner, A d is a mean proportional between c A and to A. This problem often occurs in practice, in measuring timber. Thus the rectangle inscribed in the circle, fig. 16, which may be con- sidered as representing the section of a tree, is the form of the beam of the greatest strength which can be obtained from the tree. * See the notes and rules given at the end of this chapter. APPLICATIONS. DESIGNS FOR INLAID PAVEMENTS, CEILINGS, AND BALCONIES. PLATE II. The problems just considered are capable of a great variety of applications, and in Plate II. will be found a collection of some of those more frequently met with in mechanical and architectural constructions and erections. In order, however, that the student may perfectly understand the different operations, we would re- commend him to draw the various designs on a much larger scale than that we have adopted, and to which we are necessarily limited by space. The figures distinguished by numbers, and showing the method of forming the outlines, are drawn to a larger scale than the figures distinguished by letters, and repre- senting the complete designs. 22. To draw a pavement consisting of equal squares, figs. Ihand 1. — Taking the length, a b, equal to half the diagonal of the required squares, mark it off a number of times on a horizontal line, as from A to b, b to C, &c. At A erect the perpendicular i h, and draw parallels to it, as d e, g f, &c, through the several points of division. On the perpendicular, I h, mark off a number of distances equal to A b, and draw parallels to A b, through the points of division, as h G, I f, &c. A series of small squares will thus be formed, and the larger ones are obtained simply by draw- ing the diagonals to these, as shown. 23. To draw a pavement composed of squares and interlaced rec- tangles, jigs. H5 and 2. — Let the side, as c d, of the square be given, and describe the circle, l m q b, the radius of which is equal to half the given side. With the same centre, o, describe also the larger circle, K n p i, the radius of which is equal to half the side of the square, plus the breadth of the rectangle ab. Draw the diameters, Ac, eb, perpendicular to each other ; draw tangents through the points, A, d, c, e, forming the square, jheg; draw the diagonals j f, g h, cutting the two circles in the points, I, b, k, L, m, n, P, Q, through which draw parallels to the dia- gonals. It will be perceived that the lines, A E, e c, c d, and d a, are exactly in the centre of the rectangles, and consequently serve to verify their correctness. The operation just described is re- peated, as far as it is wished to extend the pattern or design, many of the lines being obtained by simply prolonging those already drawn. In inking this in, the student must be very careful not to cross the lines. This design, though analogous to the first, is somewhat different in appearance, and is applicable to the construction of trellis-work, and other devices. 24. To draw a Grecian border or frieze, jigs'. © and 3. — On two straight lines, as A B, a c, perpendicular to each other, mark off, as often as necessary, a distance, a i, representing the width, ef of the ribbon forming the pattern. Through all the points of division, draw parallels to A b, a c — thus forming a series of small squares, guided by which the pattern may be at once inked in, equal distances being maintained between the sets of lines, as in fig. (g. This ornament is frequently met with in architecture, being used for ceilings, cornices, railings, and balconies ; also, in cabinet work and machinery for borders, and for wood and iron gratings. 25. To draw a pavement composed of squares ami regular octa- 12 THE PRACTICAL DRAUGHTSMAN'S gons,figs. © andL — With a radius, eo, equal to half the width, E F, of the octagon, describe a circle, E G F h, and, as was shown in reference to fig. 13, Plate I., draw the octagon circumscribing it — the square, A B c D, being first obtained, and its diagonals, A c, b d, drawn cutting the circle in the points, I, j, k, l, tangents being then drawn through these points. The octagon may also be formed by marking off from each corner of the square, a, b, c, d, a distance equal to A o, or half its diagonal — and thereby will be obtained the points of junction of the sides of the octagon. The pattern is extended simply by repeating the above operation, the squares being formed by the sides of four contiguous octagons, which are inclined at an angle of 45° to the horizontal lines. This pattern is generally produced in black and white marble, or in stones of different colours, whereby the effect is distinctly brought out. 26. To draw a pavement composed of regular hexagons, figs. S and 5. — With a radius, a o, equal to a side, a b, of the hexagon, describe a circle, in which inscribe the regular hexagon, abcdep. The remaining hexagons will readily be obtained by producing, in different directions, the sides and diagonals of this one. In fig. H, the hexagons are plain and shaded alternately, to show their arrangement ; but in practice they are generally all of one colour. 27. To draw a pavement composed of trapeziums, combined in squares, figs. IF and 6. — Draw the square, a«cd; also its diagonals, A C, b d ; construct the smaller square, a b c d, concentric with the first. On the diagonal, B d, mark the equal distances, o e, of, and through e and/ draw parallels to the diagonal, a C ; join the points of intersection of these with the smaller squares by the lines, hi, m n, which will give all the lines required to form the pattern, requiring merely to be produced and repeated to the desired ex- tent. Very beautiful combinations may thus be formed in differ- ent kinds of wood for furniture and panels. 28. To draw a panel design composed of lozenges, figs. © and!. — On a straight line, ab, mark off the length of a side of the lozenge twice ; construct the equilateral triangle, A B c ; draw the line, CD, perpendicular to AB; and draw a e and b p parallel to d c, and e p parallel to A b. Construct the equilateral triangle, edp, cut- ting the triangle, A b C, in G and h, and join G H. In this manner are obtained the lozenges, aohd and E G H c, and by continuing the lines and drawing parallels at regular distances apart, the re- mainder of the pattern will be readily constructed — this being repeated to any desired extent. 29. To draw a panel pattern composed of isosceles triangles, figs, n and 12. — If in the last-mentioned fig. @, we draw the longitudinal diagonal of each lozenge, we shall obtain the type of the pattern L. We will, however, suppose that the base, a b, of the triangle is given, instead of the side of the lozenge. Mark off this length twice on the line, ab, and construct the equilateral triangle, A CD, just as in the preceding case; also the second similar triangle, D e p, thus obtaining the points G and H. Join a h, g b, e h, and GF, &c, and each point of intersection, as I, l, &c, will be the apex of three of the isosceles triangles. The pattern, [L, is pro- duced by giving these triangles various tints. The patterns we have so far given are a few of the common arrangements of various regular polygons. An endless variety of patterns may be produced by combining these different figures, and these are of great use in many arts, particularly for cabinet inlaid mosaic work, as well as for pavements and other orna- mental constructions. 30. To draw an open-work casting, consisting of lozenges and rosettes, figs. C=Q andS. — The lozenge, aocrf, being given, the points, a, b, c, d, being each the centre of a rosette, draw and indefinitely produce the diagonals, ac,b d, which must always be perpendicular to each other. Through the points, a, b, c, d, draw parallels to these diagonals, also an indefinite number of such parallels at equal distances apart. The intersections of these lines will be the cen- tres of rosettes and lozenges alternately, and the former may accordingly be drawn, consisting merely of circles with given radii. The centres of the rosettes are joined by straight lines, and to right and left of these, at the given distances, fg, fh, parallels to them are drawn, thereby producing the concentric lozenges completing the pattern. 31. To draw a pattern for a ceiling, composed of small squares or lozenges, and irregular but symmetrical octagons, figs. Q and 9. — The rectangle, A b c d, being given, its corners forming the centres of four of the small lozenges, draw the lines, e f, g h, dividing the rectangle into four equal parts ; next mark off the semi-diagonals of the lozenges, as A I, A 0, and join I and o. The centre lines of the pattern being thus obtained, the half-breadths, fg, fh, are marked on each side of these, and the appropriate parallels to them drawn. In extending the pattern by repetition, the points corre- sponding to i and o will be readily obtained by drawing a series of parallel lines, as 1 1 and o o. By varying the proportions between the lozenges and the octagons, as also those between the different dimensions of each, a number of patterns may be produced of very varied appearance, although formed of these simple elements. 32. To draw a stone balustrade of an open-work pattern, com- posed of circular and straight ribbons interlaced, figs. J) and 10. — Construct the rectangle, A b c d, its corners being the centres of some of the required circles, which may accordingly be drawn, with given radii, as a 6, c d ; after bisecting A b in E, and drawing the vertical E G, make E P equal to E a, and with f as a centre, draw the circle having the radius, Fg, equal to Ab, drawing also the equal circles at c, b, e, &c. Draw verticals, such as g h, tangents to each of the circles, which will complete the lines required for the part of the pattern, J), to the left. The rosettes to the right are formed by concentric circles of given radii, as E e, e/. The duplex, fig. JJ, may be supposed to represent the pattern on the opposite sides of a stone balustrade. AVhere straight lines are run into parts of circles, the student must be careful to make them join well, as the beauty of the drawing depends greatly on this point. It is better to ink in the circles first, as it is practically easier to draw a straight line up to a circle than to draw a circle to suit a straight line. 33. To draw a pattern for an embossed plate or casting, composed of regular figures combined in squares, figs. K and 11. — Two squares being given, as abcd and fchi, concentric, but with the dia- gonals of one parallel to the sides of the other, draw first the square, abed, and next the inner and concentric one, efg h. The sides of the latter being cut by the diagonals, A c and b d, in the points, i,j,k,l, through these draw parallels to the sides of the square, abcd, and finally, with the centre, 0, describe a small 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. Ih and 1. — Let A B be a side of the square ; draw the diagonals cutting each other in the point, C, through which draw parallels, be, c f, to the sides ; with the comers of the square as centres, and with a given radius, a a, describe the four quadrants, and with the points, D, F, E, describe the small semicircles of the given radius, v> 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 H, 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, d b, 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 tioo 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, ab, 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, A b, ac, meeting them in d 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 A B, Ac, 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, wo 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. U, which represents a section of various descriptions of castings. 36. To draw a circle tangent to three given straight lines, which moJce any angles with each other, fig. 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 required 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. E. 37. To draw the section of a stair rail, fig. (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 n, 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. 0. — 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, o, of the given arc, and with a radius, o e, equal to o c, plus the given radius, a, draw the are 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, oe. It will be seen that in fig. ®, these problems arise in drawing either side of the object. The two sides are precisely the same, but reversed, and the outline of each is equidistant from the centre line, which should always be pencilled in when drawing similar figures, it being 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 radiusbeing equal to half the distance, a B, 7717s. [1 and 7. — Join a b, and draw the perpendicular, e f, bisecting it in c. With the centres, a and c, and radius, A c, describe arcs cutting each other in g, and with the centres, b and c, other two cutting each other in n ; g and 11 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, figs. IF and 11. — Divide the straight line into four equal parts by the perpendiculars, E F, 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 arc 14 THE PRACTICAL DRAUGHTSMAN'S cutting C d in C ; also with the centre, b, a similar arc cutting G h in h ; c and H will be the centres of the arcs forming the re- quired curve. Whatever be the given radius, provided it is not too small, the centres of the arcs will always be in the lines, c d and G H. It will be seen that the arcs, c I and H L, cut the straight lines, c D and G H, in two points respectively. If we take the second points, k l, as centres, we shall form a similar curve to the last, but with the concavity and convexity transposed, and called the cyma reversa. The two will be found in fig. IF, the first at a, and the second at b. This figure represents the section of a door, or window frame — it is one well known to carpenters and masons. ■ The little instrument known as the " Cymameter," affords a convenient means of obtaining rough measurements of contours of various classes, as mouldings and bas-reliefs. It is simply a light adjustable frame, acting as a species of holding socket for a mass of parallel slips of wood or metal — a bundle of straight wire.s, for example. Previous to applying this for taking an im- pression of measurement, the whole aggregation of pieces is dressed up on a flat surface, so that their ends form a perfect plane, like the ends of the bristles in a square cut brush ; and these component pieces are held in close parallel contact, with just enough of stiff friction to keep them from slipping and falling away. The ends of the pieces are then applied well up to the moulding or surface whose cavities and projections are to be mea- sured, and the frame is then screwed up to retain the slips in the position thus assumed. The surface thus moulds its sectional contour upon the needle ends, as if the surface made up of these ends was of a plastic material, and a perfect impression is there- fore carried away on the instrument. The nicety of delineation is obviously bounded by the relative fineness of the measuring ends. 41. To draw a baluster of a duplex contour, fins, (g and 8. — It is here necessary to drawan arctangent to, or sweepinginto two known arcs, a i and c D, and having its centre in a given horizontal, e i. Extend e i to h, making i h equal to g d, the radius of the arc, c D. Join G n, bisecting g h by a perpendicular ; this will cut e H in the point, e, which is the centre of the required arc — ei being its radius. A line joining e G cuts CDinc, the point of contact of the two arcs. The arc, d f, which is required to be a tangent to c D, and to pass through the point, f, is drawn with the centre, o, obtained by bisecting the chord, df, by a perpendicular which cuts the radius of the arc, CD. This curve has, in fig. @, to be repeated both on each side of the vertical line, m n, and of the horizontal line, fg. 42. To draw the section of a baluster of simple outline, as fig. C=j]. —We have here to draw an arc passing through two points, a, b, fig. 12, its centre being in a straight line, b c ; this arc, moreover, requiring to join at D, and form a sweep with another, de, having its centre in a line, fd, parallel to B c. Joining b a, a perpendicular bisecting b a, will cut B c in o, which will be the centre of the first arc, and that of the second may now be obtained, as in problem 37, fig. 6. 43. The base of the baluster, fig. C=3, is in the form of a curve, termed a scotia. It may be drawn by various methods. The following are two of the simplest — according to the first, the curve may be formed by arcs sweeping into each other, and tan- gents at a and c to two given parallels, A b, C d, fig. 13. Through A and c draw the perpendiculars, c and A E, and divide the latter into three equal parts. With one division, FA, as a radius, describe the first arc, A g h ; make C I equal to F a, join I F, and bisect i F by the perpendicular, o k, which cuts c o in o. will be the centre of the other arc required. The line, O H, passing through the centres, o and F, will cut the arcs in the point of junc- tion, h. It is in this manner that the curve in fig. $Q is obtained. The second method is to form the curve by two arcs sweeping into each other and passing through the given points, A b, fig. 14, their centres, however, being in the same horizontal line, c D, parallel to two straight lines, e f and B c, passing through the given points. Through A, draw the perpendicular, A I. I, its point of intersection with c d, is the centre of one arc, ad. Next draw the chord, B d, the perpendicular bisecting which, will cut CD in o, the centre of the other arc, the radius being od or ob. This curve is more particularly met with in the construction of bases of the Ionic, Corinthian, and Composite orders of architecture. With a view to accustom the student to proportion his designs to the rules adopted in practice in the more obvious applications, we have indicated on each of the figs. £\, H5, (g, &c, and on the corresponding outlines, the measurements of the various parts, in millimetres. It must, however, at the same time be understood, that the various problems are equally capable of solution with other data ; and, indeed, the number of applications of which the forms considered are susceptible, will give rise to a considerable variety of these. ELEMENTARY GOTHIC FORMS AND ROSETTES. PLATE IV. 44. Having solved the foregoing problems, the student may now attempt the delineation of more complex objects. He need not, however, as yet, anticipate much difficulty, merely giving his chief attention to the accurate determination of the principal lines, which serve as guides to the minor details of the drawing. It is in Gothic architecture that we meet with the more numer- ous applications of outlines formed by smoothly joined circles and straight lines, and we give a few examples of this order in Plate IV. Fig. 5 represents the upper portion of a window, composed of a series of arcs, combined so as to form what are denominated cuspid arches. The width or span, A b, being given, and the apex, c ; joining A c, c b, draw the bisecting perpendiculars, cutting A b in D and e. These latter are the centres of the sundry concentric arcs, which, severally cutting each other on the vertical, c f, form the arch of the window. The small interior cuspids are drawn in the same manner, as indicated in the figure ; the horizontal, G H, being given, also the span and apexes. These interior arches are sometimes surmounted by the ornament, M, termed an adl-de- bmuf, consisting simply of concentric circles. 45. Fig. 1 represents a rosette, formed by concentric circles, the outer interstices containing a series of smaller circles, forming an interlaced fillet or ribbon. The radius, a o, of the circle, con- taining the centres of all the small circles, is supposed to be given. Divide it into a given number of equal parts. With the points of division, 1, 2, 3, &c, as centres, describe the circles tangential to BOOK OP INDUSTRIAL DESIGN. 15 each other, forming the fillet, making the radii of the alternate ones in any proportion to each other. Then, with the centre, o, describe concentric circles, tangential to the larger of the fillet circles of the radius, 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 a , 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 exeixises will, at the same time, qualify him for the representation of a vast number of forms met with in machinery and architecture. OVALS, ELLIPSES, PARABOLAS, VOLUTES, &c. PLATE V. 50. The ove is an ornament of the shape of an egg, and is formed of arcs of circles. It is frequently employed in architecture, and is thus drawn: — The axes, ab and 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 r equal to b e, the difference between the radii, o c, o b. Bisect F b by the per- pendicular, G H, cutting CD inn. 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, n K, described with the centre, l, and radius, l d, equal to H c, form the lower portion of the required figure ; the lines, on, 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. Ik. 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. 9 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, lbh, fig. 1. 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, de, 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, b and c, the foci. If now we divide D e unequally in H, and with the radii, d ii, 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 16 THE PRACTICAL DRAUGHTSMAN'S points in the circumference, and we can, in like manner, obtain any number of points, when the ellipse may be traced through them by hand. The large ellipses which are sometimes required in con- structions, are generally drawn with a trammel instead of compasses, the trammel being a rigid rule with adjustable points. — The gardener's ellipse : To obtain this, place a rod in each of the foci of the required ellipse ; round these place an endless cord, which, when stretched by a tracer, will form the vectors ; and the ellipse will be drawn by carrying the tracer round, keeping the cord always stretched. 54. Second Meilwd. — Take a strip of paper, having one edge straight, as d b, and on this edge mark a length, a b, equal to half the transverse axis, and another length, b c, equal to half the conju- gate axis. Place the strip of paper so that the point, a, of the longer measurement, lies on the conjugate, f g, and the other point, c, on the transverse axis, D e. If the strip be now caused to rotate, always keeping the two points, a and c, on the respective axes — the other point, b, will, in every position, indicate a point in the circumference which may be marked with a pencil, the ellipse being afterwards traced through the points thus obtained. 55. Third Method, Jig. 3. — It is demonstrated, in that branch of geometry which treats of solids, as we shall see later on, that if a cone, or cylinder, be cut by a plane inclined to its axis, the resulting section will be an ellipse. It is on this property that the present method is based. The transverse and conjugate axes being given, as A b and c D, cutting each other in the centre, O, draw any line, A e, equal in length to the conjugate axis, c d, and on A e, as a diameter, describe the semicircle, EGA. Join E b, and through any number of points, taken at random, on e A, as 1, 2, 3, &c, draw parallels to E B. Then, at each point of division, on e A, erect perpendiculars, la, 2b, 3c, &c, cutting the semicircle, and, at the corresponding divisions obtained on A b, erect perpendiculars, as l'a', 2'b', 3 V, &c, and make them equal to the corresponding perpendiculars on E A. A line traced through the various points thus obtained, that is, the extremities, a',b',c', &c, of the lines, will form the required ellipse. 56. Fourth Method. — On the transverse axis, A b, and with the centre, o, describe the semicircle, A p b, the axis forming its dia- meter; and with the diameter, h i, equal to the conjugate axis, describe the smaller semicircle, hbi. Draw radii, cutting the two semicircles, the larger in the points, i, j, h, I, &c, and the smaller in the points, i', j', h', I', &c. It is not necessary that the radii should be at equiangular distances apart, though they are drawn so in the plate for regularity's sake. Through the latter points draw parallels to the transverse axis, A b, and through the former, parallels to the conjugate axis, c D, the points of intersection of these lines, as q, r, s, t, &c, will be so many points in the required ellipse, which may, accordingly, be traced through them. It follows from this, that, in order to draw an ellipse, it is sufficient to know either of the axes, and a point in the circumference. Let the axis, a b, be given, and a point, r, hi the circumference, which must always lie within perpendiculars passing through the extre- mities of the given axis. Through r draw a line, rj', parallel to A b, and a line, rj, perpendicular to it; with the centre, 0, and radius, o A, equal half the given axis, describe the arc, cutting rj'mj ; join j o, and the line, / o, will cut rj' in j' : oj' will be equal to half the conjugate axis, c D. If the conjugate axis, C d, be given, proceed as before ; the arc, however, in this case, having the smaller radius, o d, and cutting rj' inj'; then join oj', producing the line till it cuts rj, which will be in j, and oj will equal half the transverse axis, A B. It has already been shown how to describe an ellipse, when the two axes are given. "We may here give a method invented a short time back by Mr. Crane of Birmingham, for constructing an ellipse with the com- passes. This method applies to all proportions, and produces as near an approximation to a true ellipse, as it is possible to obtain by means of four arcs of circles. By applying compasses to any true ellipse, it will be seen that certain parts of the curve approach very near to arcs of circles, and that these parts are about the vertices of its two axes ; and by the nature of an ellipse, the curve on each side of either axis is equal and similar; consequently, if arcs of circles be drawn through all the vertices, meeting one another in four points, the opposite arcs being equal and similar, the resulting figure will be indefinitely near an ellipse. Four circles, described from four different points, but with only two different radii, are then required. These four points may be all within the figure ; the centres of the two greater circles may either be within or without, but the centres of the two circles at the extremities of the major axis must always be within, and, consequently, the whole four points can never be without the figure. Again, the proportions of the major and minor axes may vary infinitely, but they can never be equal ; therefore, any rule for describing ellipses must suit all pos- sible proportions, or it does not possess the necessary requirements. Moreover, if any rule apply to one certain proportion and not to another, it is evident that the more the proportions differ from that one — whether crescendo or diminuendo — the greater will be the difference of the result from the true one. From this it follows, that if a rule applies not to all, it can only apply to one propor- tion ; and also, that if it apply to a certain proportion and not to another, it can only be correct in that one case. Let ab be any major axis, and c d any minor axis ; produce them both in either direction, say towards p and n, and make A r equal to C G ; then join C A, and through p draw P H parallel to c A. c X. ? ''-A, °r~. .-'■* 1 ..-"I"-, C / ,-^v-> v j V J Set off b I, A J. and c k, equal to h c; join J K, and bisect it in E, and at it erect a perpendicular, cutting c d, or c D produced, at M ; then make G E equal to G m ; j, E, i, m, will be the centres of the four circles required. Through the points, J and I, draw M N, M o, E P, E Q, each equal to M c ; then M N and E P will be the radii of the greater circles, and j n, I o, of the less : the points of contact will therefore be at N, o, P, q, and the figure drawn through a, n, C, o, e, q, d, P, will be the required ellipse. Hate Practical Dmuilitsman. I'.o i - //' 1 / \ / ; Ai'iiicno.uiil i' i- i-i \ 01 ! ate 6. /'f ' 1 1 1 1 *l_^ '" , c-. lio- 8 [•10- n" /' r i 1 ! 1 ! i rnr g. 1 r / 1 hn' 4-' • i i i i ! 1 1 1 1 l i Y 1 i u _:l .>-- FW m: ,'i! ■. Practical Draughtsman Plate 8. \ J ■ i n . ■ 1 1 ■ j ■ . 1 1 1 ■ 1 I 'i A II I t . 1 1 v. 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 arectangular base plate, A, having sharp countersunk points on its lower surface, to hold the instrument steady, and cut out to leave a sufficient area of the paper uncovered for the traverse of the pen. It is adjusted in position by four index lines, setting out the trans- verse and conjugate axes of the intended ellipse — these lines being cut on the inner edges of the base. Near 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, a, 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 hue 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 14, to t 7 ? by k 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 M ; 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, t, be given, apart from the ellipse, fig. 3. Join l with I, the nearest focus to it, and with l as a centre, and a radius equal to L I, describe an arc, Ml 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 H and N. Join m n and n h, and the ellipse will be cut in the points v and x ; a straight fine 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 r 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 oval of five 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, 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 l) — 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, ali; and having also made A N equal to the mean pro- portional, o L, with the centre, h, and radius, h n, describe the arc, n a, cutting the former in a. The points, h, 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 h a, s n' b, and pod, 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. 60. 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 axis 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 focus 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, 18 THE PRACTICAL DRAUGHTSMAN'S A C, E C, H C, I C, joining them with the focus. Draw perpen- diculars bisecting these latter lines, and produce them until they cut the corresponding parallels, and the points of intersection, b, c, D, e, will be in the required curve, which may be traced through them. 62. The straight lines which were just drawn, cutting the parallels in different points of the curve, are tangents to the curve at the several points. If, then, it is required to draw a tangent through a given point, c, it is obtained simply by joining c c, making H c equal to e C, and bisecting the angle, H c C, by the straight line, c d, which will be the required tangent. If the point given be apart from the curve, the procedure will be the same, but the line corresponding to h c will not be parallel to the axis. 63. Second method : — We have here given the axis, a G, the apex, a, and any point, I, in the curve. From the point, I, let fall on the axis the perpendicular, I G, and prolong this to e, making G e equal to I G. Divide I G into any number of equal parts, as in the points, i, j, k, through which draw parallels to the axis ; divide also the axis, o g, into the same number of equal parts, as in the points, /, g, h ; through these draw lines radiating from the point, e, and they will intersect the parallels in the points, m, n, o, which are so many points in the curve. 64. If it is required to draw a line tangent to a given parabola, and parallel to a given line, J k, we let fall a perpendicular, c l, on this last ; this perpendicular will cut the directrix in p, and p n drawn parallel to the axis will cut the curve in the point of con- tact, n. We find frequent applications of this curve in constructions and machinery, on account of the peculiar properties it possesses, which the student will find discussed as he proceeds. 65. The objects represented in figs. ©, W, are an example of the application of this curve. They are called Parabolic Mirrors, and are employed in philosophical researches. The angles of incidence of the vectors, ab,ac,ad, are equal to the angles of reflection of the parallels, b b', c c, d d'. It follows from this pro • perty, that if in the focus, a, of one mirror, bf, the flame of a lamp, or some incandescent body be placed, and in the focus, a', of the opposite mirror, b'f, a piece of charcoal or tinder, the latter will be ignited, though the two foci may be at a considerable dis- tance apart; for all the rays of caloric falling on the mirror, bf, are reflected from it in parallel lines, and are again collected by the other mirror, b'f', and concentrated at its focus, a'. 66. To draw an Ionic volute, fig. 6. — The vertical, A o, being given, and being the length from the summit to the centre of the volute, divide it into nine equal parts, and with the centre, o, and a radius equal to one of these parts, describe the circle, abed which forms what is termed the eye of the volute. In this circle (represented on a larger scale in fig. 7) inscribe a square, its dia- gonals being vertical and horizontal ; through the centre, o, draw the lines, 1 — 3, and 4 — 2, parallel to the sides, and divide the half of each into three equal parts. With the point, 1, as a centre> and the radius, 1 A (fig. 6), draw the arc, A e, extending to the horizontal line, 1 e, which passes through the point, 2. With this latter point as a centre, and a radius equal to 2 e, draw the next arc, extending to the vertical line, 2f, which passes through the point, 3, the next centre. The points, 4, 5, 6, &c, form the sub- sequent centres ; the arcs in all cases joining each other on a line passing through their respective centres. The internal curve is drawn in the same way; the points, 1', 2', 3', &c, fig. 7 bis , being the centres of the component arcs. The first arc is drawn with a radius, 1' a', a ninth less than 1 A, and the others are consequently proportionately reduced, as manifest in fig. 6. The application of the volute will be found in fig. |J. 67. To draw a curve tangentially joining two straight lines, A B and e C,fig. 8, the points A and c being tlte points of junction. — Join A c, and bisecting A C in D, join D with B, the point of intersection of the lines, A b, b c. Bisect b d in e', whiGh will be a point in the curve. Join E c, e a, and bisect the lines, E c, E A, by the per- pendiculars, ab, cd; make ef and e'f equal to a fourth part of E D ; /and /' will be other two points in the curve. Proceed in the same way to obtain the points, gh and g K, or more if desir- able, and then trace the curve through these several points. This method is generally adopted by engineers and constructors, and will be met with in railways, bridges, and embankments, and wherever it is necessary to connect two straight lines by as regular and per- fect a curve as possible. It is also particularly applicable where the scale is large. RULES AND PRACTICAL DATA. LINES AND SUEPACES. 68. The square metre is the unit of surface measurement, just as the linear metre is that of length. The square metre is sub- divided into the square decimetre, the square centimetre, and the square millimetre. Whilst the linear decimetre is a tenth part of the metre, the square decimetre is the hundredth part of the square metre. In fact, since the square is the product of a number mul- tiplied into itself, 0-1 m. x -1 m. = O01 square metres. In the same manner the square centimetre is the ten-thousandth part of the square metre ; for 0-01 m. X 0-01 m. = 0-0001 square metres. And the square millimetre is the millionth part of the square metre ; for, 0-001 m. X 0-001 m. = 0-000001 square metres. It is in tliis way that a relation is at once determined between the units of linear and surface measurement. 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 144th part of a square foot, and the 1296th part; for 1 inch x 1 inch = jV foot X -fV wot = T Jj square foot, and 1 inch X 1 inch — jfe yard X -fa yard = tsW square yard. This illustration places the simplicity and adaptability of the decimal system of measures, in strong contrast with the complexity of other methods. 69. Measurement of surfaces. — The surface or area of a square, as well as of all rectangles and parallelograms, is expressed by the product of the base or length, and height or breadth measured BOOK OP INDUSTRIAL DESIGN. 10 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 „, ■ = -75m. 1-25 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 1-3 + 1-5 Q l1t) x -8 = 1-12 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 1-15 2-3 X — = 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) by the height 1-15 m. gives for quotient the base 2-3 m., 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 b, of the triangle, ABC, fig. 16, PI. I., being 3 m., the side b c, 4 m., the hypothenuse, A c, will be AC= V3 2 + 4 a = 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 — 5-6 9-8 X 5 X _ 2 137-2 sq. m. 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. TABLE OF MULTIPLIERS FOE REGULAR POLYGONS OF FROM 3 TO 12 SIDES. Triangle Square, Pentagon,... Hexagon,.... Heptagon,... Octagon, Enneagon, ... Decagon, Undecagon,.. Duodecagon, Multipliers, Sides. A b c 3 2-000 1-730 •579 4 1-414 1-412 •705 5 1-238 1-174 •852 6 1-156 radius. side. 7 1-111 •867 1-160 8 1-080 •765 1-307 9 1-062 •681 1-470 10 1-050 •616 1-625 11 1-040 •561 1-777 12 1-037 •516 1-940 •433 1-000 1-720 2-598 3-634 4-828 6-182 7-694 9-365 11-196 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 1-0382607 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. Ride. — 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, ]^1 X 1-08 = 9-99 m.: 2 or say 10 metres, the radius of the circumscribing circle. 20 THE PRACTICAL DRAUGHTSMAN'S Second, The radius of a circle heing given, to find the length of the side of an inscribed polygon. Rule. — Multiply the radius by the factor in column B, corre- sponding to the number of sides of the required polygon. Example. — The radius being 10 m., the side of an inscribed octagon will be — 10 x -765 = 7-65 m. Third, Tlie side of a polygon being given, to find the radius of the circumscribing circle. Rule. — Multiply the side by the factor in column C, corre- sponding to the number of sides. Example. — Let 7-65 m. be the side of an octagon ; then 7-65 X 1-307 = 10 m., nearly. Fourth, The side of a polygon being given, to find the area. Rule. — Multiply the given side by the factor in column D, corresponding to the number of sides. Example. — The side of an octagon being 7-65 m., the area will be— 7-65 X 4-828 = 36-93 sq.m. THE CIRCUMFERENCE AND AREA OP A CIRCLE. 72. If the circumference of any circle be divided by its diame- ter, the quotient will be a number which is called, the ratio of the circumference to the diameter. This ratio is found to be (ap- proximately) — 3-1416, or 22 : 7 ; that is, the circumference equals 3-1416 times the length of the diameter. It is expressed, in algebraic formulas, by the Greek letter