THE NEW WARN SHEET METAL WORKER. THE SHEET METAL WORKER'S INSTRUCTOR, FOR ZINC, SHEET IRON, COPPER, AND TIN PLATE WORKERS, AND OTHERS : CONTAINING A SELECTION OF GEOMETRICAL PROBLEMS, ALSO PRACTICAIi AND SIMPLE RULES FOR DESCRIBING THE VARIOUS PATTERNS REQUIRED IN THE DIFFERENT BRANCHES OF THE ABOVE TRADES. BY REUBEN HENRY WARN, PRACTICAL TIN PLATE WORKER. TO WHICH IS ADDED AN APPENDIX CONTAINING INSTRUCTIONS FOR BOILER MAKING, MENSURATION OF SURFACES AND SOLIDS, RULES FOR CALCULATING THE WEIGHTS OF DIFFERENT FIGURES OP IRON AND STEEL, TABLES OP THE WEIGHTS OP IRON, STEEL, ETC. ILLUSTRATED BY THIRTY-TWO PLATES AND NINETY-SEVEN ADDITIONAL ENGRAVINGS. A NEW AND ENLARGED EDITION TO WHICH ARE ADDED SHEET METAT. WORK PROCESSES, INCLUDING TOOLS, JOINTS, SOLDERS, FLUXES, ETC. ; A9 WELL AS GEOMETRY APPLIED TO SHEET METAL WORK. PHILADELPHIA : HENRY CAREY BAIRD & CO., INDUSTRIAL PUBLISHERS, BOOKSELLERS AND IMPORTERS, 810 W^i-NUT^ Street. :!>i9oa.*' v JS IP6 THE GETTY CENTER DBRARY PUBLISHER'S PKEFACE TO THE SIXTH EDITION, REVISED. Warn's Sheet Metal Worker's Instructor having been well tried and well proved, still enjoying great popularity and being in active demand, and a new edition, being called for, the publishers, with a view to increasing its useful- ness have caused to be added to it considerable matter of great practical value. This matter comprises, Sheet Metal Work Processes, such as Sheet Metal Working, Tools, and Manipulation, Joints, Galvanized Iron, Soldering and Solders, both Hard and Soft, Fluxes, Modes of applying Heat, Brazing, and Geometry as applied to Sheet Metal Work ; this latter being illustrated by thirty-one engravings. The book has of course been supplied with a ,good Table of Contents, and a full Index, thus (iii) 18920 iv PREFACE TO THE SIXTH EDITION, REVISED. rendering any subject prompt and easy of refer- ence. In its present form it is believed that it will find new popularity and new uses, and that its future sale will even rival that of the past, great as that has been. Philadelphia, February 22, 1906. , INTRODUCTIOK This Work is intended as a book for private study for artisans in the various branches of the Sheet Metal Trade. The first four plates contain a selection of Problems on Practical Greometry, which embrace principles in the above trades, and which it is hoped will serve as an introduction to the accompanying diagrams, enabling the student more readily to work out the several figures. Indeed, Geometry is of great service to working me- chanics, both when called upon to work from drawn plans, and when required (as is frequently the case) either to alter some device, or to form an original pat- tern ; nor is it less serviceable when instructing appren- tices in their respective arts. In selecting the subjects contained in the plates, the aim has been to place them in such easy stages of ad- vancement, that the figures may be easily worked, and so that, the earlier diagrams may assist the student both in comprehending and in working out those which follow. The student is recommended not merely to read the letterpress, but to take a pair of compasses, with shifting leg for pencil, a T and set squares, a square board, and some paper, and work out every figure in the book, for it is only in this way that he can hope to obtain a proper knowledge of the various diagrams herein con- tained. By working out carefully each figure, the mind will embrace the principles contained therein, and the figures themselves will be thereby better fastened on the memory ; and further, the student by this means will derive increasing pleasure as he proceeds from figure to (V) vi INTRODUCTION. figure. Some students will find it beneficial to work the figures tlirough in this manner several times. If any reader should think the description of the diagrams too diffuse, or even commonplace, the answer must be that the Author desires the work to be useful, not only to the most cultured of his fellow-workmen, but also to apprentices, and to such adults as 'may have had only a very limited education : and further, as the words used in one trade are often almost intelligible to others, such language has been used throughout as will, perhaps, be on the whole most readily understood. While the Author disclaims entire originality for the whole of this course of instruction, yet there are many portions thereof which he has never seen elsewhere, and believes to be original. And as, in the few years the Author has turned his attention to the subject, he has found considerable pleasure and benefit therefrom, he hopes that his fellow- workmen who use this book will derive equal or even greater pleasure and benefit from its study ; and if so he will feel great satisfaction in the effort herein be- stowed. R. H. WARN. CONTENTS. A SELECTION OF GEOMETRICAL PROBLEMS. PAGK. Plate I — To draw a straight line parallel to a given line, and at a given distance from it . . . . . .13 To bisect a given straight line. . . . . .13 To divide a given line into any number of equal parts . I t To draw a straight line equal to the circumference of n given circle . . . . . . . .11 To construct a plain scale for drawing a small plan pro- portionate to a larger one . . . . . .14 To find the centre of a circle or the radius of a curve . 15 To draw a circle through any three given points . .1-5 To erect a perpendicular . . . . . .16 Plate IT. — To strike a segment by a triangular guide, the chord and height being given . . . . .17 The chord and height of a segment of a circle of large radius being given to find the curve without recourse to the centre . . . . . . . .17 Having an arc of a circle given, to raise perpendiculars from any point or points without finding the centre . 18 To draw a tangent to a circle or portion of a circle with- out hiiving recourse to the centre . . . .18 Upon a given straight line to describe any regular polygon 19 Upon a given side to draw a regular pentagon . .19 Plate III. — Upon a given side to draw a hexagon . 20 To draw a heptagon ....... lil How to describe an octagon in a given square . . 21 To draw a straight line equal to any given part of a circle 21 To find the stretch-out of a semicircle . . . .22 Definition of polygon . . . . . . .22 Definition of parallelogram ...... 23 (Vii) Vlll CONTENTS. PAGE. Plate IV. — To strike an ellipse or oval with the compasses, the length or major axis being given . . . .25 To describe an ellipse -within a given square, or when the major and minor axes are given . . . . .24 Method of drawing an egg-shaped oval . . . .25 Graphic illustration of what an ellipse really is . . 25 Another method of drawing an ellipse . . . .25 To draw a parabola ....... 25 To draw an ellipse with a trammel . . . .26 Plate V. — To strike a pattern for a round, tapering, or flue article, or a frustum of a cone ..... 28 The plan of a round flue body to be cut in three pieces . 29 Two or three methods of describing ovals . . .30 Plate VI. — The pattern of an article where the sides are straight and the ends semicircular . . . .32 The pattern of an oblong tapering pan in two parts or sections ......... 33 Plate VII. — The pattern for a tapering oblong article in one piece, such as a flue oblong candlestick . . .35 The pattern for a tapering oval article in four pieces or sections ......... 37 Plate VIII. — To describe a tapering oval body in one piece . 38 A method of drawing an ellipse or oval with a string and pencil ......... 40 Platr IX. — The pattern of an egg-shaped oval tapering body 41 Another method of describing an oval tapering body . 42 Plate X. — To describe a tapering oval body where the taper- ing is not equal on all sides . . . . .44 Plate XI. — The pattern for a square tapering article or pyra- mid 48 The pattern for a tapering octagon body in one piece . 49 The pattern for a diamond-shaped tapering body, in one piece 50 The pattern of a square funnel where one side is straight or upright ......... 50 Plate XII. — The pattern for a square or rectangular tapering top or tray, with sides and bottom in one piece . . 62 The pattern for a hexagon mould or tray, having the bot- tom and sides in one piece ...... 52 The pattern of an irregular octagon pan or tray, with the sides or bottom in one piece . . . . .53 The pattern of an oblong pan, with round corners, but struck from different centres, and tapering more at the ends than sides ........ 54 CONTENTS. ix PAGE. Plate XTII. — A pattern for a tapering top, the base being straight at the sides, and with circular ends, the bole in the top to be circular and parallel with the base (Similar to a tea bottle top) 56 Plate XIV. — The pattern for a tapering article, oval at the base and round at the top (Such as an oval canister top, having a round hole for the leck and cover) ....... 60 Plate XV —A pattern for the tapering sides of a tray, having various curves ........ 64 Plate XVI. — The pattern for an oblong tapering bath . . 66 Plate XVII. — The pattern of an elbow at right angles, in a round pipe ........ 70 A pattern of an elbow in a round pipe at any angle re- quired (in this case an obtuse angle) . . . .71 The pattern of a tapering piece of pipe to join two up- right cylinders to form a double elbow . . .71 Plate XVIII. — The pattern for a T- piece, or to join two cylin- ders at right angles ....... 73 The pattern of two cylinders for joining at an oblique angle for slanting direction . . . . .74 Plate XIX. — The pattern for a lobster-back cowl . . 77 The pattern for a round pipe, to form a semicircle for connection to other pipes ...... 78 Plate XX. — The pattern for a cone and cylinder to intersect or meet at right angles with their axes . . .80 Plate XXI. — A cylindric section through any given angle . 84 To draw an ogee arch ....... 85 Another method .... .... 85 To find the covering of an ogee dome, the plan of which is hexagonal ........ 85 Plate XXII. — The pattern for a rectangular base and bottom in one piece, where the flue or curve is equal on all sides (Such as may be used as a base for either an aquarium or a ferii-case) ........ 86 The pattern for a hexagon base . . . . .87 Plate XXIII. — The pattern for a vase, octagon shape . . 88 Plate XXIV. — The pattern for a vase having twelve sides (duodecagon) 90 Plate XXV. — The pattern for a cone with an elliptic base . 92 The pattern of an oblique cone, or the frustum of a cone cut parallel with the base ...... 93 B X CONTENTS. Plate XXVI. — The pattern for a round-end bath tapering more at the ends than at the sides . . . .95 Plate XXVII. — The pattern for a hip bath . . .96 Plate XXVIII. — The pattern for a travelling sitz bath . . 98 Plate XXIX. — The pattern for a globe formed of twelve pieces joined together ...... 101 The pattern for a triangular pedestal or pyramid, with all three sides alike (an equilateral triangle) . . 101 To obtain the radius required for striking the pattern of a slightly tapering article, without the necessity of pro- ducing lines to meet ....... 103 Plate XXX. — The patterns for the sides of an irregular octa- gon pan ......... 104 Plate XXXI. — The pattern for a cover and neck of an irregu- lar octagon article, such as a tureen . . . .3 06 Plate XXXII. — The pattern for the top of a jack screen . 110 SHEET METAL WORK PROCESSES. I. SHEET METAL WORKING. Cutting out the pattern and tools used for this purpose . 11. S Hammers and mallets; Hatchet-stake; Taper-stake; (Jreas- . ing tool; Seam-set; Holliper or Oliver . . . .114 Shaping a vessel, such as a shallow tray . . . .115 The beak-iron; The crease and its uses. .... 116 Joints; Various methods of making joints; Butt and mitre joints; Lap joint . . . . . . . .117 Folded angle joint; Riveted joint . . . . . .118 Straight joints; Cramp joint. . ..... 119 Lap joint without solder; Patent strip overlap; Roll joint; Hollow crease; Joints united by screw bolts or rivets . 120 Zinc; Mode of forming gutters of zinc . . . . .121 Soldering zinc . . . . . . . . . 122 Galvanized iron and its uses for cornices and string courses. 12:-! Observations on galvanized iron circular work by Mr. 0. A. Vaile; Best plan for bringing mouldings to the required shape ..... ..... 124 The tools required 125 Mode of putting the work together 126 CONTENTS. xi PAGE n. SOLDERING. Hard solders and soft solders 127 Composition of the various solders, the fluxes suitable for each, and the manner of applj'ing the heat; Composi- tion of the soft solder mostly used . . . . .128 Mode of burning together for brass, iron and lead. . . 129 Alloys and their melting-points; Fluxes; Modes of applying heat 130 Brazing; Copper soldering bit or soldering iron and its use . 131 Tinning the copper bit ........ 132 Soldering coarse work ........ 133 Soldering copper works; Making a wiped joint; The blow- pipe and its use ........ 134 Use of the hot-air blast; Heat required for hard soldering . 135 The brazier's hearth; Preparation of the edges for hard soldering; Mode of brazing ...... 136 Brazing iron. ......... 137 The blow-pipe and its use in hard soldering .... 138 Use of gas in conjunction with the blow-pipe . . . 139 III. GEOMETRY AS APPLIED TO SHEET METAL WORKING. Utility of a knowledge of practical geometry . . . 140 The geometrical process mainly called in by the sheet-metal worker; Development of a hexagonal-sided tin or sheet iron box. ......... 141 Development of the cylinder. ...... 142 To make a mitre joint at right angles in a half-round rain- water gutter or trough ....... 144 Tools required to draw the various diagrams . . , 146 Terms or definitions used in geometry; The point; Lines; Definition of a line ....... 147 Straight line or right line; Curve or curved line; Simple, compound and concentric curves ..... 148 Parallels or parallel lines; Horizontal line; Vertical line; Perpendicular line; Angles; Definition of an angle . 149 Right angle; Acute angle; Obtuse angle; Triangles . .150 Equilateral triangle; Isosceles triangle; Scalene triangle; Right-angled triangle ....... 151 Obtuse-angled triangle; Acute-angled triangle; Vertex and altitude of a triangle . . . . . . . 152 Quadrilateral figures; The square; The rectangle or oblong; The rhombus 153 The rhomboid; The trapezoid; The trapezium; The polygon. 154 Irregular polygons; The circle ...... 155 Circumference of a circle; The radius; Diameter; Chord; Segment; Sector; Tangent 156 XII CONTENTS. PAGE To bisect any given straight line ...... 157 To draw a line parallel to a given line; To find the centre of any given circle or arc of any circle .... 158 Mode of measuring angles by instruments; The protractor and its use . . . . . . . . .159 To draw an angle of 60° geometrically; To draw an angle of 30° geometrically; To bisect any given angle . . 161 To trisect a right angle . . . . . . .162 To inscribe in a given circle any regular polygon . .163 Development of regular solids or polyhedrons . . . 164 To obtain the " stretch-out " of the solid; The tetrahedron or four-sided figure . 165 The cube or hexahedron or six-sided figure, and its develop- ment 166 The octahedron or eight-sided figure ..... 167 Dodecahedron or twelve-sided figure ..... 168 Icosahedron and its development ; Figures from which regular polyhedrons can be formed .... 169 Mode of obtaining regularly symmetrical solids whose sides are formed of two similar faces; Mode of getting at the figure produced by cutting a cylinder in a diagonal or slanting direction . . . . . . . .170 Method of finding the precise form of an ellipse . . .171 The cone and its development . . . . . .173 To find the development of the covering surface; To develop the surface or find the " stretch-out " .... 175 To find the section which a cone transected or cut at any particular angle will present . . . . . .177 The sphere and its development ...... 180 Methods of arriving at the shape of the segmental portions of the covering of a sphere . . . . . .181 To develop the covering of a hemisphere . . . .182 APPENDIX. RULES FOR BOILER-MAKING, ETC. Template making ........ 185 To build a tubular boiler of 36 inches, inside diameter of two plates in the circle ...... 186 To find the breadth of the template for a given dome at any particular place, by calculation ..... 190 To find the template for a given short egg-end spherical . 193 To find the template for a given long egg-end parabola; the breadth and length of the template at any particular place 195 To find the templates for a given barrel spheroid; the breadths and lengths at any particular place by calcu- lation .......... 198 CONTENTS. xill PAGE MENSURATION OF SURFACES. Four-sided figures . ....... 202 Triangles .......... 203 Trapezoids and trapeziums ....... 20(3 The circle .......... 207 Ellipse or oval , . . . . . . , .210 Mensuration of solids . . , , . , . .211 For superficies ......... 212 For the solidity ......... 212 Cj'linders and prisms . . . . . . , .213 Cones and pyramids . . . , . . . .215 Spheres and segments of spheres . . . . . .218 APPROXIMATE RULES FOR FINDING THE WEIGHT OF DIFFERENT FIGURES OF WROUGHT IRON AND STEEL. For round iron . . . . . , , , .221 For square iron . . . . . . . , .221 For square, angled T, convex, or any figure of beam iron . 221 For square cast steel . . . . , , . .221 For round cast steel ........ 222 Table I. Weight of wrought iron and steel .... 224 II. Weight of boiler plates ...... 224 III. Weight of malleable round iron .... 225 IV. Weight of malleable square iron .... 226 V. Weight of malleable flat iron .... 227 VI. Weight of a superficial square foot of plate or sheet iron 231 VII. For ascertaining the weight of malleable iron pipes. 231 Rule 231 VIII. Weight of ordinary angled iron .... 232 TABLES OF CIRCUMFERENCES AND AREAS OF CIRCLES 235 SIZES OF TINWARE IN FORM OF FRUSTUM OF A CONE. Pans ............ 244 Dish Kettles and Pails ....... 244 Coffee Pots .......... 244 Wash Bowls ......... 244 Dippers '245 Measures .......... 245 Druggists' and Liquor Dealers' Measures .... 245 Table of Weights, &c., of American Lap-Welded Iron Boiler Flues 246 Table of Effects upon Bodies by Heat 246 Index 247 TEE SHEET METAL WORKER'S INSTRUCTOR. PLATE I. Fig. 1. — To draw a straight line parallel tc a given line, and at a given distance from it. Let AB be the given straight line, and the line AC to represent the distance between the parallels. Then with A as a centre, with the radius AC describe the arcs- (or curves) C and D. Draw the line CD so as to touch these curves, and CD will be parallel to AB as required. Fig. 2. — To bisect a given straight line; that is, to divide it into two equal parts at right angles. Let AB be the given line. From any part, say o o, with radius' greater than half the length o o, describe curves cutting each other in CD. Then a straight line drawn through the- points of intersection^ will bisect the line AB. 14: Fig. 3. — To divide a given line into any number of equal parts (in this case seven). Let AB be the given line which is to be divided into seven equal parts. From the point A draw another line (not being particu- lar as to what angle with AB), and with any convenient opening of the compasses set off seven equal parts, as 1, 2, 3, 4, 5, 6, 7. Join the points 1 and B, and draw parallel lines from 6, 5, 4, 3, 2, 1, to cut the line AB, which will be divided into seven equal parts as re- quired. YiG. 4. — To draw a straight line equal to the circumference of a given circle. Let ADBC be the given circle. Draw the diameter AB, and from its centre o draw the perpendicular CD. Draw a diagonal line AC ; set the compasses in C, and with a radius at any distance beyond its centre a describe the arc E ; now with the compass in A draw an- other arc intersecting at E, and draw the line o E, then three times the diameter, with the distance a h added, will be a close approxima- tion to the length of the circumference. Fig. 5 is the same, but showing only the section required. YiG. 6. —To construct a plain scale, for 15 drawing a small plan proportionate to a larger one. Draw a line AB, say 12 inches long, and mark off the inches as from 1 to 12 ; then draw BC, say 2 inches long. Draw a line from 12 to C, and draw parallel lines from the points 11, 10, 9, 8, etc., to cut the line BC. By using the distances on the line BC, as 1, 2, 3, 4, etc., as inches, you would get a scale of 2 inches to the foot. Fig. 7. — To find the centre of a circle or the radius of a curve. From the point B as a centre, with radius greater than half the distance to the other points, draw a portion of a circle, as e d g and from A and C as centres, with the same radii, draw curves to intersect or cut the part of a circle first drawn e d and g Ti. From these points of intersection draw lines e d and g li until they meet at o, which will be the centre of the curve required. Fig. 8. — To draw a circle through any three given points (provided they are not in a direct line). Let ABC be the three given points; join AB and BC ; bisect AB and BC, and produce the bisecting lines until they cut each other in the point D, then D will be equi-distant from 16 each of the three points, and the centre of the circle required. Fig. 9. — From the point B on the line AB to erect a perpendicular.^ Above the given line AB take any point t/, and with the radius d B draw a portion of a circle ABC. Draw a line from points A d to meet the circle in C. Draw the line BC, which will be perpendicular to AB. NOTES. ^ A parallel line is one that runs in the same direc- tion as another line, but always keeps at the same dis- tance from it. ^ Arc is part of a circle. The word " curve" will be used frequently in the commencement instead of arc 5 but it should be remembered that a curve is not always part of a circle. ^ A circle is a figure bounded by a curve equally dis- tant in every part from its centre. A straight line across the figure through the centre is called its diameter, half this line is the radius, it being the length from the centre of a circle to its outer line or circumference. * Where one line or curve crosses or cuts through another, is called the point of intersection. ^ Perpendicular means square with another line. To say that a line is perpendicular does not necessarily- mean that it is upright, but at right angles or perpendi- cular to the line on which it is drawn. "When two lines are perpendicular to each other they form a right angle- A- line formed by a cord having a weight at its end ie really an upright line or a vertical line. 17 PLATE II. Fig. 1. — To strike a segment (or part of a circle) by a triangular guide, the chord' and height being given. Let AB be the chord of the segment and DC the height (or versed sine); join BC and cut CE parallel to AB, and make it equal BC, fix a pin in B and another in C, and with the triangle ECB describe the curve CB, then re- move the pin B to A, and by guiding the sides of the triangle against AC, strike the other part of the curve ACB. Fig, 2. — The chord and height of a segment of a circle of large radius being given, to find the curve without having recourse to the cen- tre, which is supposed to be unattainable. Let AC be the chord line and DB the height, through B draw EF parallel to AC, join AB and BC, draw AE at right angles to AB, and CF at right angles to BC, divide AD and EB into any number of equal parts (say 6), join the corresponding numbers 1 1, 2 2, 3 3, &c. Also divide AG into the same number of equal parts, and from each division draw lines to B, and the points of intersection will be points in thn curves. 18 Fig. 3. — Having an arc of a circle given, to^ raise perpendiculars from any point or points without finding the centre. Let AB be the given curve or arc, and A 1 2 3 4 5 the points from which perpendiculars are to be erected, in the space 5B make the point 6 equal to 4 5, from 4 and 6 as centres, with a radii greater than half the distance be- tween them describe arcs intersecting each other at 7, a line drawn at the point of inter- section at 7 to 5 gives one of the perpendiculars required, the other points as far as 11 will be found in the same manner. If a perpendicu- lar is to be raised at A, the extremity of the curve, a method somewhat different must be employed; suppose the perpendicular 1 11 to be erected, from 1 with the radius 1 A describe the curve A 11, and from A with the same distance describe 12 1, make o 12 equal oil and join A 12, will give the perpendicular wanted. Note.— The A should be exactly under the line 12, it has been drawn too far in error. YiG. 4.— To draw a tangent^ to a circle or portion of a circle without having recourse to the centre. Let A be the point from which the tangent is to be drawn, take any other point in the circle AC, join AC and bisect the curve AC at /'-it'fi/tn i,/: fA. 19 /, then from A as centre with a radius A/, the chord of half the curve, describe the curve e/D, making /D equal e/, then through the points AD draw the line DAB, which will be the tangent required. Fig. 5. — Upon a given straight line to de- scribe any regular polygon (in this case a pen- tagon). Produce ah indefinitely,^ from h as centre with a radius h a, describe the semicircle a c 5, which divide into as many equal parts as there are to be sides in the polygon, which in the present example is five, through the second division from 5 draw the line &c, which will form another side, bisect these sides as shown at fg h e, the point of intersection at o is the centre of the circle of which abc are points in the circumference, then by producing the dotted lines 6 1 to c, and 6 2 to c/, will divide the cir- cle into the number of parts required. Fig. 6. — Upon a given side to draw a regu- lar pentagon. Let AB be the given side, from its extremity B erect a perpendicular B / equal to half AB^ join Af and produce it till fb be equal to B /, from A and B as centres with a radius equal to B b, draw arcs intersecting at E, which will .20 be the centre of a circle containing five divisions equal to AB. NOTES. * Chord, a line cutting off any part of a circle. The part of a circle thus cut off or divided by a chord is called a segment. ^ Tangent, a line perpendicular to a radius, that is to say, a line required to be drawn from a curve or a circle without any perceptible point where it joins the curve ; the line should be at right angles with the centre that the curve was struck by, the line will then be tangent. ^ To produce a line, or draw a line indefinitely, is to carry it further or make longer in the same direction ; its required length is sometimes not known untiT intersected by another line. PLATE III. Fig. 1. — Upon a given side to draw a hexa- gon. Suppose AB to be the given side, from the extremities A and B draw curves intersecting at G ; from G with a radius GA describe the circle ABCDEF, which will contain six divi- sions equal to AB. Fig. 2 is drawn precisely the same. By drawing the lines ABC to the centre D, 21 the vertical and horizontal projections of a cube are given, with the plan and elevation in one view. Fig. 3 is a heptagon drawn on the same principle as the pentagon, Fig. 5, Plate IL, and which will give a sufficient explanation how this or any other polygon having a given number of equal sides is drawn. Fig. 4 shows how to describe an octagon in a given square. Let ABCD be the given square. Draw the diagonals ACDB, then from the angular points ABC and D, with a radius equal to AE, de- scribe curves cutting the sides of the square in 1234567 8, then by joining these points the polygon will be complete. Fig. 5.— To draw a straight line equal to any given part of a circle. Let AB or CD be the given arcs. From A ivith radius AB, and vice versa, describe arcs intersecting at G. Draw EF parallel to AB, then from G draw lines through A and B cut- ting at E and F, then EF in the length of the curve from A to B. Again, draw lines from G, through C and D, cutting at 1 and 2 will give the stretch-out from C to D. On the same principle the 22 stretch-out may be found from any other point in the semicircle. Fig. 6. — To find the stretch-out of a semi- circle by another method. Let ACB be the semicircle. Make AE equal AF, and draw EB, tlien draw BD at right angles to EB, and draw CD parallel to AB ; CD is the length of the quadrant CB, and twice CD the length of the semicircle ACB. Definition of Polygons. All figures having more than four sides are called polygons, and are distinguished by names, denoting the number of their sides, thus : — A Polygon of five sides is called a Pentagon. " six " Hexagon. " seven " Heptagon. " eight " Octagon. " nine " Nonagon. " ten " Decagon. " eleven " Undecagon. " twelve " Duodecagon. When all the sides of a polygon are equal and all its angles equal, it is called " regular." When they are not equal, the polygon is called " irregular." 23 Definition of four-sided figures, or Parallelograms. A parallelogram is a four-sided figure whose opposite sides are parallel to each other. When the four sides are equal and the four angles are right angles the figure is called a square, as shown by Fig. 4 (Plate III) ABCD Diagonals are lines crossing to opposite angles, as AC and DB. When one pair of sides is of a different length to the other, but the sides remain parallel to each other in op- posite pairs, the angles being right angles, the figure is called a rectangle or parallelogram, such as the four right lines within which an oval is described (Fig. 2, Plate TV). When the four sides are equal and the opposite sides parallel to each other, but the angles not right angles, the figure is called a rhombus or lozenge,, a figure frequently known as a diamond shape (see Fig. Plate II}. An angle is an opening formed by any two lines meet- ing at a point." If this opening be greater than that formed by a line meeting perpendicularly, it is called an obtuse angle. If the opening be less than one formed by a perpendicular line, it is called an acute angle. Triangles. — In a right-angle triangle, one of the an- gles is a right angle or square, as in Fig. 9, Plate I. A triangle having all three sides and angles alike, is called an equilateral triangle. PLATE IV. Fig. 1. — To strike an ellipse (or oval) with the compasses, the length or major axis being given. 24 Divide the given length 1 5 into five equal parts, then with 2 as the centre and a radius 2 4, and vice versa, describe curves intersecting at A and B ; then from the points A and B draw lines through 2 and 4 indefinitely, with 2 as centre and radius 2 1, describe the curve CD, and from 4 the curve EF ; now with centre A and radius AD, describe the curve DF, and from B the curve CE, which completes the ellipse. Fig. 2. — To describe an elhpse within a given square, or when the major and minor axes are given. Draw the major axis AB, and minor axis C D, make the diagonal BD, take the distance AE on the major axis, and transfer from B to 1 on the diagonal BD, also the distance ED to the point 3 ; take half 1 3 in the point 2 as centre, and any distance towards B greater than its half, and vice versa ; from B describe arcs intersecting at 4 and 5, through these points draw a line until it cuts the minor axis at C, make equal Eo, and ED equal EC, from C and D draw lines through o and g indefi- nitely, then with centre g and radius gA, de- scribe the curve 6A8, and from centre o, 5B7 ; then from centre C and radius CD, draw the curve 8D5, and centre D the curve 6C7, which will complete the ellipse. 25 Fig. 3 shows a method of drawing an egg- shaped oval. Draw the line AB, and bisect it in C; with centre C draw circle AoBd, and draw the diagonals Aoh and Bob, then with A as cen- tre, and radins AB, draw the curve B6, and with centre B the curve A b ; now with o for centre, and radius o b, describe the curve b b^ which makes the oval required. Fig. 4. — If two semicircles are described as shown in this figure, and both semicircles divided into the same number of equal parts, and if through the points of division of the larger semicircle lines are drawn perpendicular to AC, and through the corresponding points in the smaller one parallel to AC, the points of intersection will be points in the elliptic curve, giving a graphic illustration of what an ellipse really is. Fig. 5 is another method of drawing an ellipse by intersecting lines, so simple in construction as to need no further explanation. Fig. 6. — To draw a parabola by the inter- section of lines, its axis, height, and base or ordinate being given. Let AC be the base, and DE the axis, and E its vertex; produce the axis to B and make EB equal DE, join AB, CB, and divide them 26. into the same number of equal parts, join the divisions by the lines 1 1, 2 2, &c., and their intersections will produce the curve required. Fig. 7. — To draw an ellipse with the trammel. The trammel is an instrument consisting of n right-angled cross A BCD grooved on one side, and a tracer E with three movable studs 1,2,B, two of which slide in the grooves just mentioned, the other at B is provided with a pencil to trace the curve of the ellipse. For the application suppose AC to be the major axis, and BD the minor; lay the cross of the trammel on these lines ; then adjust the sliders of the tracer so that 1 B may be equal to o C, and 2 B equal o D ; then by sliding the tracer in the grooves of the cross, the pencil at B will describe the ellipse. Fig. 8. — This is precisely the same principle of drawing the ellipse as Fig. 7, and is inserted because the trammel (which is perhaps prefer- able to any other method of drawing this curve) is not always at hand ; and this is a trammel easily constructed and answers every purpose. Take A to o as the major axis, and D to o as the minor, on which another straight-edge is to be fastened and extended as shown ; put a bradawl or nail through at 1 and 2, and apply the pencil at B, then by sliding the tracer 27 round, keeping the bradawls against the axes of the ellipse, one quarter of the curve will be described; now move the tracer to another quarter, and describe it in the same manner, and continue in like manner until the ellipse is completed. NOTES. Fig. 3 (Plate lY.) represents an oval, which is egg- like (from ova, an egg) ; but according to the custom of many trades Figs. 1 and 2, and Fig. 8 in Plate V. are commonly accepted as ovals, although strictly they are ellipses, or methods of describing an ellipse with the compasses by means of arcs of circles, practically good and useful. Figs. 4, 5, 7, 8 show various methods employed to obtain perfect ellipses under the following definitions. An ellipse is a figure bounded by a curve having no cen- tre but two foci, from which it is generated. It owes its form to the section of a right cone oblique to its axis. In Plate VIII. is shown elliptical fig. described by a piece of string and pencil, which forms no part of a circle. The words major and minor axes are terms used to describe the length and width of the ellipse or the dia- meters ; they are also described sometimes as the trans- verse and conjugate axes. 28 PLATE V. To strike a pattern for a Round Tapering or Flue Article (or a Frustum of a Cone), Fig. 1 represents the diameter of both top and bottom, and Fig. 2 from G to F the upright height. Divide the circle with lines, as AB and CD, at right angles, then draw a line as ab in Fig. 3, and take upright depth required, as from F to G, mark off from a to d, and draw the lines a c and d e at right angles with a 5, take the radius of the larger circle EB with the compasses, and mark off the distance from a to c, take also the radius of the small circle E to q, and mark it off from d to e, then draw a line from the points c e to cut the line a d 5, with b as centre and radius b e, strike the curve e h, open the compass to c, still using b as cen- tre, and strike the curve cioq /. The circle Fig. 1 is divided into quarters ; take one of them and divide it into any convenient number of equal parts as D to B, from c (Fig. 3) measure off a corresponding number of dis- tances to i, the curve c to i shows one quarter of the pattern required, by adding on a like distance, as from i to o, represents half the 29 patter^, and the distance cioqf the whole pattern required in one piece, draw a line from / to the centre h, and the required lap to be added on as shown. To describe the plan of a Round Flue body to be cut in Three Pieces. Fig. 4 represents the top and bottom of the body; this is to be divided into three parts, which is done in a very simple manner. The radius by which a circle is struck will measure six times the distance on the circumference, as shown in Plate III, Figs. 1 and 2, so that by drawing a line from every alternate point to the centre will divide the circle into three, as BCD. The pattern for this body is shown by Fig. 6, draw ohe right angles ; take the perpendicular height FE (Fig. 5), and mark it off from h to d (Fig. 6). Draw line clf at right angles with h o. Take the radius of the outer circle at AB, and mark it off from h to e, and the radius AG to be marked off from d to /. Draw a line from the points e f to cut the line h o. Take o as centre, radius o /, to strike the curve fn. Open the compass to e, still using o as centre, and strike the curve e m (Fig. 6). Divide the part of circle DB into any convenient number of equal parts, and measure 3 ■60 off a corresponding quantity of equal parts in Fig. 6 from e to m. Draw a line from m to the centre o, which gives the pattern of one third of the body required, the perpendicular height of which will be equal to EF. Fig. 7 gives the Ilue or slanting height of the same articles ; the only difference between this and Fig. 6 is, that the radius is taken from d and 5, instead of / and e. Fig. 8 represents an oval (this is recom- mended for general practical use). Let AB be the given length, and hB the width. From B set off equal to hT> the given width. Divide g A into three equal parts. Set off two of these parts on each side of E, as s and t. From s and t, as centres, with radius s t, de- scribe curves or arcs, cutting each other in it w. From u and w draw hues through s and t, and produce them as onlm. Take s as centre, ^A as radius ; draw the curve o A I and t as centre ; draw the curve m B n. Then with u and w as centres, radius w ?, strike the curves Ihm and o D n, which will complete the oval required. Fig. 9 shows a kind of oval very frequently used in the manufacture of various articles ; a method of getting this shape is required so as 31 to cut a flue or tapering body (which will be shown in a future example). Take AB the given length, set the compass to nearly half the required width. From A and B mark off the points o o, and strike semi- circles a Ah and c B c?. Take any distance on this curve, as from A to 5, or further if required, and mark off a corresponding distance from B to c and d. Produce lines from h o and c o until they meet as at D. Then with radius D h strike the re- mainder of the curves h c and a d, which will .give the oval required. Gone and Frustum of a Cone. ' A cone is a solid the base of which is a circle, but which tapers to a point from the base upwards. Jf a cone be cut horizontally, that is, parallel to the base, all such sections will be circles. Fig. 2 (Plate Y.)— Take tFu for the base, us and ■is form a cone : the point s being cut off by the line Gr, the Fig. from F to G only being required, is called a section or frustum of a cone, the point or apex being -cut off. ii2 PLATE VI. To strike the pattern of an article where the Sides are Straight and the Ends Semi- circular. Fig. 1 shows the size and shape of the re- quired article at top and bottom. Fig. 2 is the upright height. Having drawn figs. 1 and 2, showing the plan and elevation, proceed with fig. 3; draw a h and ac at right angles, take the depth ED (fig. 2), and mark it off from a to d, draw d e parallel to a c, take the radius AB, and mark off the distance on the line ac, and the radius AG, marking off the point from d to e, then draw a line from the points c and e to cut the line a h, Avhich will give the slanting height and the radius. This pattern is to be made in halves joined at- each end as at B. To strike the pattern shown in fig. 4, make the straight part for the side fh equal DE (fig. 1). Extend the lines/??? and h n indefinitely, take the distance from c to ^ (fig. 3), and mark off the points from / to g and h to i, then with radius h c taking g and i as centres, strike the curves fo and with radius h e (fig. 3), still using g and i as centres. 33 strike the curves m r and ns. Divide the curve from D to B (fig. 1) into any convenient number of parts with the compasses, and mark a similar number of parts on the curve fo (fig. 4) ; make hp equal /o, and draw lines from p to i and o to ^ the centres, which will give one half the pattern required. To strike the pattern of an Oblong Tapering Pan in two parts or sections. Fig. 5. — Draw lines EC and FD, the dis- tance apart required for the straight part of the ends. Draw HJ and G ^, the distance apart required for the straight part of the sides. Take the points a hcd for centres, and then draw curves, or the corners, as at GC and e /, &c., at each of the corners, then draw straight lines to meet the curves as at HG and CD, and fg, and so on, which will give the size of the article both top and bottom ; the line AB shows where the two halves meet for jointing together in fig. 6. Draw ah andac at right angles, take the required depth from a to d, draw d e parallel to a h, then take the radius from a to G, mark off the point from a to b, also the radius of the small curve from a to e, mark oft' from d toe (fig. 6), draw the line from h to e 34 to cut the line a h ; from 6 to e will give the slanting depth. Fig. 7 is the development of the pattern. Make the straight part from i to I equal GH^ the depth from i n and / o to equal h e (fig. 6)^ take the distance h c (fig. 6), and mark off" from i to h and Z to m (fig. 7), take k and m as cen- tres, with radius c h strike the curves i q and I p, divide the length of the curve from C to G, and dot off the same distance from I to p, make i q equal l-p^ draw the lines p m and q h ; with radius c e, still using m and k as centres, strike the curves o s and n r. Draw lines p u and s t at right angles with p m, making p u equal to CB (fig. 5), draw u t parallel to p s, finish the opposite end in like manner, adding the lap in both instances as required. Where wiring or edging is required, add on accordingly. Fig. 8 shows two circles divided into four equal parts, ACBD, equal to the four corners of fig. 5, with a little calculation the pattern may be obtained without going through the process of again constructing fig. 5. To illus- trate this, take an article, say 10 inches from A to B (fig. 5), and 7 inches from H to y, the corner EH to be the section of a 4 inch circle^ as from A to C (fig. 8). The diameter AB being 4 inches, subtract 4 inches from 7 inches. 35 would leave 3 inches straight at end, as from F to E. Again, subtracting the 4 inch circle from 1 0 inches the given length, will leave 6 inches straight in the sides, as from H to G, then drawing line.s n o and i I, the required depth as previously described, draw the lo and iti^ the perpendiculars 7 inches apart ; then draw- ing the corners as previously described, adding on an inch and a half, as from p to u and q to N, at right angles with j9 m and q will give the required pattern. PLATE VII. To strike the pattern for a Tapering Oblong article in one piece, such as a flue oblong candlestick. Fig. 1 is the size, top and bottom, and fig. 2 upright height. Take the perpendicular height a h (fig. 2), and mark it off from b to d (fig. 3). Take the radius for the corners aC (fig. 1), and mark it off from b to c (fig. 3), also the radius a e, mark off from d to e, drawing a line from ce to cut the line ba, which gives the slanting height and the radii required for 36 striking the corners. Draw the lines e f and a ■c (fig. 4) the same distance apart as e to c in fig. 3. Draw the perpendiculars a e and c / {fig. 4) equally distant as AC, making the straight part of the side required. With radius a G (fig. 3), using h and d (fig. 4) as centres, strike the curves ap and c (/, and with radius ae (fig. 3), still using the same centres in fig. 4, strike the curves eq and fh. Take the length of the curve DH (fig. 1), and dot off the same distance from c to ^ (fig. 4), make ap equal to c g, draw lines from p and g to the centres h and cZ, draw p r and 5 at right an- gles with p h. Take the distance from E to G (fig. 1), and make the same distance from p to r and q to s (fig. 4). Draw r z parallel with p h, from 7- mark off point z, the same length as p to h, then using z as centre strike the curves rt and su, making the curve rt equal pa, draw line from t to centre z, draw tw and ux at right angles to tz, taking the distance from B to Jv (fig. 1\ mark off" the same distance from t to IV and u to x, draw lo x parallel with t u, and proceed in the same manner with the other end ; adding on the lap, as shown, will make the pattern complete in one piece, being joined together at k i. O c 37 To strike the pattern for a Tapering Oval article in Four pieces or sections. Fig. 5. — Draw the smaller oval (as ex- plained ill fig. 8, Plate V.) first the size re- quired for the bottom, then from the same centre, which in this instance is ahcd, de- scribe the outer oval as much larger as re- quired for the top of the article, drawing the diameters AB and CD at right angles. Fig. 6 shows the perpendicular height. Draw AB and DE (fig. 7) parallel, the same distance apart as a 5 (fig. 6), being the upright depth, make AC at right angles with AB and DE, take the radius which the end section of the oval is struck by, and mark it oft' on fig. 7, i.e., take the distance a B oxag (fig. 5), and set it off from A to r (fig. 7), and the radius am or an, and mark off the same distance from D to s. Draw a line from r s to cut the perpendicular AC at then with t for centre and ts as radius, strike the curve su- open the compasses from t to r as radius, and draw the curve r v. Take the length of the curve from k to i (fig. 5), marking off a convenient number of parts, then taking a like distance with a corresponding number of parts, as from r to V (fig. 7) ; now draw the line from v to '6S the centre t, which will give the pattern of the end section. For the pattern of the side take the radius dk or dg (fig. 5), mark off an equal distance from A w (fig. 7), and the radius d h (fig. 5), mark ofi" from D x (fig. 7) ; draw a line from points w x_ to cut the perpendicular at o ; with o as centre, radius occ, strike the curve xy^ open the compass from o to lo, strike the curve w 2, take the length of the curve It g (fig. 5), and take a corresponding length of curve from w to z (fig. 7), then draw a line from z to cen- tre o, which by adding the usual laps will complete the side. PLATE VIII. To describe a Tapering Oval body in one piece. Draw the two ovals (fig. 1) as previously ex- plained, and proceed with fig. 3 (as in Plate VII. fig. 7). Draw AB and AC at right angles, and the required depth from A to D ; draw D E parallel to AB; from centre a (fig. 1) take the radius a m, that the curve BC is struck by, 39 and mark off the distance on fig. 3 from A to e, also the radius of the smaller curve as a n • mark off from D to / (fig. 3) ; draw e f to cut the perpendicular line AC at ^. Take the radius of the curve of the side from 5 to A or h to B, mark off the distance from A to ^ (fig. 3), then take the radius from 5 to E (fig. 1), being the radius by which the curve EF is struck, and mark it off on the line DE to h (fig. 3), dravi^ a line from the points g h, to cut the perpendicular line AC at o, and this will give the radii for describing the pattern, the development of which will be found in fig. 4. To commence fig. 4, draw the line a b, set the compasses from i to e (fig. 3), and on the line a b (fig. 4), taking c for centre, strike the curve e a d, make the length of the curve ead the same as BmC (fig. 1). Draw lines from d and e through the centre c, and extend them indefinitely as / and g. Take tlie distance if (fig. 3) for radius, still using c (fig. 4) as cen- tre, strike the curve h i. Now take the dis- tance o to ^ (fig. 3), with the compasses, and from e (fig. 4) mark off the point 7i on the line e g, likewise mark off a like distance d to m on the line df, using ?i and m as centres, strike the curves d o and ej^9, with radius equal to o A (fig. 3), still using m and n as centres, strike the curves i r, and h q. 4U Divide off the length of the curve C to D (fig. 1), and take the same distance from d to o (fig. 4), draw line from o to the centre m, make the distance from e to ^ the same as from d to o, draw line from p to centre mark off the points t and s, on the lines p n and o m, equal to the distance from a to c, using t and s as centres, radius strike the curves p u and ov, then with radius ^5- strike the curves qw and r take the length of the curve from a to cZ, mark off like distances from o to and p to if, draw lines from u to centre t and from v to s, which will complete the pattern. A method of drawing an Ellipse or Oval with a string and pencil. Make the given diameters AB and CD at right angles to each other at their centre E. Take the distance from E to A, then using C as centre, draw an arc to cut the diameter AB in 00 (these two points 00, are called the foci of the ellipse). Place a pin at each point where the curve cuts the line AB, as at 00, and another at C, pass a string round the three pins, and tie it securely, thus forming a trian- gle with the string, as 00 C. Take out the pin at C and substitute the point of a pencil, which may be drawn along, moving with the 41 string, and the point will thus trace a perfect ellipse. PLATE IX. • To describe the pattern of an Egg-shaped Oval Tapering body. Fig. 1. Draw AB and CD at right angles, and from E, with radius EH, draw a circle cutting the line CD at F ; from G and H draw lines through F, and produce them indefinitely, and GH and F will be the centres to strike the remainder of the figure (as shown in Plate IV. fig. 3), then from the same centres draw the larger oval as much larger as the flue re- quires. Fig. 2. Draw AB and ED the required depth, and BC at right angles, mark off" B « and D e equal to EC and EJ (fig. 1 ) . Draw line from ae to cut the perpendicular line at h. Take HA, and mark off* from B to C on the line BA. Take HG and mark off" from D to on the line DE, and draw Q d to g. Take the radii FM and FK and mark off from B to e and D to /, and draw ef to i. Fig. 3. Draw line / o with radii h a and h e 42 <%. 2) ; using o (fig. 3) as centre, strike the curves klm and rp. Take the length of curve from A to C (fig. 1), and dot olf a like distance from I to m and I to k (fig. 3). Draw lines from m and h through the centre o, and produce them indefinitely, take radius ^ to C (fig. 2), and from k mark off q, and from m mark off r, take q and r as centres, radius r m, and draw curves m s and ^ t ; make m s and k t the same length as AN and BM, draw lines from t to centre q ; and from s to r, with radius g d, draw curves ^ u and r Take the radius i e, and from t and s mark off m and x for cen- tres, and strike the curves t y and s z : make sz same length as ND, and draw lines from z to the centre x\ and from y to w, with radius ^/, describe the curves from u and which will complete the pattern. These figures heretofore described are re- <;om mended to be well studied before reading the ensuing ones. Another method of describing an Oval Tapering body. Fig. 4 shows the oval the size required for the bottom. Draw the diameters AB and CD at right angles, and describe the oval as already •explained. 43 Fig. 5 shows the upright height and flue required. To strike the pattern G, which is the end section, d being the centre from which the curve m7i is struck, draw a line de right angles with AB. Produce or extend the curve m n to cut the line de at e, draw the line e g at right angles Avith d e, make e g the upright height as « 5 in fig. 5. Draw ^ A at right angles with ^ e ; c 5 in fig. 5 showing the flue required, mark off a like distance from g to A. Draw line from points he io cut the line AB at i, taking i for centre on the line A B, with the radius ih draw the curve A/ and with radius i e draw the curve e h. Measure off the length of curve m n from e to and draw line from i h to /. Proceed with the side section H in the same manner. Extend the line CD indefinitely, D being the centre by which the curve nl h struck, draw a line D o at right angles with C D ; extend the curve 7i / as dotted, until it cuts the line Do; draw line op at right angles with D o, make op the upright height as from a ioh (fig. 5). Draw p r at right angles with p o, make p r the required flue as h c (fig. 5), draw line from ro to cut the extended line C D at s ; with s for centre, and radius s r draw the curve r/, and with radius so strike the curve o u. Measure the length from m to 44 taking a like distance from o to u; now draw line from su to t, which completes the pattern. This pattern will, after being well studied, be found an excellent introduction to Plate X ; it is a different method from that described at Plate VII. figs. 5 to 7, and in other diagrams, but t]iie result will on practice be found pre- cisely the same. In these and the foregoing figures, the tapering must be equal on all sides. PLATE X. To describe a Tapering Oval body, where the tapering is not equal on all sides. (In this case more tapering at the ends than at the sides. ) Fig. 1 shows the diameters of two distinct ovals, each one being described by a separate set of centres. To proceed with the larger or outer oval, which (as well as the smaller or inner one) is constructed in the same manner as described in fig. 8, Plate V., take the re- quired diameters, as AB and CD, the centres being EF and GH. The smaller oval will have to be constructed in the centre of the 45 larger one, according to the given length and width required for the bottom of the article, the centres by which this oval is struck being ah and cd: fig. 2, from a to 5, shows the up- right height. Fig. 3 shows a pattern of the side, which is obtained in the following manner: H being the centre by which the curve KL (fig. 1) is struck, draw line HE, at right angles with the perpendicular line GH, and extend the curve KL as dotted, to meet the line HR. Draw the line RS perpendicular with the line HR, mark off the depth from R to S, equal to the upright height, as a 5 (fig. 2). Draw ST at right angles with RS, take the distance at D between the two ovals on the line CD (the width between them being the flue of the sides), mark off the same distance from S to T, draw a line from the points TR, and extend it, as shown by the dotted line, to cut the extended perpendicular line CD at u ; with radius u R, taking for the centre any part of the perpen- dicular line (there not being space enough on the plate to show the centre), and strike the curve mim : c being the centre by which the curve fg (the side of the smaller, oval) is struck, draw line cl at right angles with the perpendicular line CD, extending the curve / g with the same radius to meet the line c J 4 46 Draw line I m at right angles with c /, again taking the distance a b (fig. 2) from I to m (fig. 1). Draw a line m n at right angles with I m. Take the distance again between the two ovals at the side C, and mark off the distance on the line 7n n, draw a line from the points n and / to cnt the perpendicular line CD as at o. Take the distance from I to or from B, to T, and measure off a like distance from i to g (fig. 3), being the slanting height of the body at the centre of the side. Take the distance from o to /, and from g mark off the point ; with w as centre, radius o I, strike the curve h g /.-. Extend the line chg (fig. 1), which shows the division of the smaller oval, to k on the curve of the larger one, the point L being the right sectional line of the larger oval, while the ex- tended line c & to would be the sectional line of the inner one ; let the distance be equally divided as at i, draw line from i to h, being the centre by which the end of the smaller oval is struck, take the length of the curve from i to C, measuring off a like distance from i to 7n on each side of the perpendicular line, and draw lines from m to the point w, being the centre by which the bottom curve gJiJc is struck; this being done, will complete the pattern for the side. Fig. 4 shows a pattern of the end, which is 47 obtained as follows: E (fig. 1) being the cen- tre by which tlie end of the large oval JAK is struck, draw line from E to N at right angles with the diameter AB, extend the curve JAK, to cut the line EN at N. Draw line from N to O at right angles with NE, make NO equal to a h (fig. 2) the upright height. Draw OP at right angles with ON, take the distance be- tween the two ovals at the end on the line AB, being the flue of the end, and mark off a cor- responding distance from O to P. Draw line from PN to cut the diameter AB at Q. With radius QN, and x (fig. 4) as centre, describe the curve e hf. Take a (being the centre by Avhich the end of the smaller oval, as fe, is struck) and draw ap at right angles with the centre AB, extending the curve fe to cut the line a p at J), draw q at right angles with a p, again taking the upright height as at a 5 (fig. 2), and mark off" a like distance from p to q. Draw q r at right angles with q p, take the dis- tance from A to the end of the smaller oval, and mark off the same from q to r. Draw a line from the points rp to cut the centre line at s, take the distance from r to p, or from P to N, which will give the slanting depth of the centre of the end, mark off the distance from h to a (fig. 4), take radius from s to p, (fig. 1) and from a (fig. 4) mark off the point 48 w. With w as centre, radius sp, strike the curve cad, take the length of the curve from B to i (fig. 1) and mark off a corresponding distance on fig. 4 from b to /, and h to e, draw lines from / and e to the centre w, being the centre by which the curve of the bottom is struck, which will complete the pattern for the end. PLATE XI. To strike the pattern for a Square Tapering article (or Pyramid). Fig. 1 represents the size or. projection, and fig. 2 the upright height or elevation. Draw the diagonals, and take distance from the centre a to h (fig. 1), and mark olf the same in fig. 2 from g to (/. Also take the distance (in fig. 1 ) from a to I or and mark off" in fig. 2 from 7i to e. Draw a line from points d e to cut the perpendicular line at /. Then draw (in fig. 3) the perpendicular line df, and take the radius fd (in fig. 2), and in fig. 3 describe the circle 7i d k, and with radius fe in fig. 2, still using / as centre in fig. 3, draw the smaller ' circle e, take the length of one side from c to b 49 {fig, 1 ) , and mark off the same four times on the larger circle (fig. 3), as hg d i k. Draw lines from these points to the centr'^ /; join these points by lines, as hg^ gd, Sec, and from the points on the smaller circle in the same man- ner, which will complete the pattern. To strike the pattern for a Tapering Octagon body, in one piece. Fig. 4 represents the size of top and bottom (the method of striking this figure is given in Plate III. fig. 3 and 4), fig. 5 from g to /, being the upright height required, take the distance from the centre a to one of the extreme points as c (fig. 4), and from / mark off" the same distance at h (fig. 5), and the distance a to e, mark off from g to i, draw the line h i to cut the perpendicular line fg at h. With radius kh (fig. 5) draw portion of circle as he h (fig. 6), and with radius, k i, still using k as centre strike the curve ied (fig. 6). Take the distance he (fig. 4), and mark off" eight times the same distance on the larger curve (fig. 6), as 5 c 7i, &c. Draw lines from all these points to the centre /.•. Draw straight lines from these points as from h to c, and c to h, and likewise from the intersecting points of the smaller curve, which will complete the pattern. 6U To strike the pattern for a Diamond-shaped Tapering body, in one piece. Fig. 7 shows the size and shape required. Fig. 8 from i to /, the upright height. Carry the length of a c and a e (fig. 7), on fig. 8, from f to g and from / to 7i, also the distance from a to b and a to d, from i to I and i to k, and draw through g and I a line to cut the perpen- dicular at nt, also through h and k. draw a line which will cut the perpendicular at the same point m. With the length of m g, m 7i, m?, and mk (fig. 8) as radii, describe the curves (/, 7i, Z, Z;, in fig. 9, from the centre 7?^ and draw the line g carry the length e c (fig. 7), from g to r and (fig. 9), also from n to o and from r to h, and draw lines from r and h to the centre m, likewise from n and o. Con- necting these points by straight lines b r, r g, d s, si., &c., will complete the pattern. To describe the pattern of a Square Funnel, where one side is straight or upright. Fig. 10, abed shows the projection for the top, e fg li the hole or bottom of the funnel. Fig. 11, from i to shows the elevation (or upright height). Draw lines from the points \ f and eg to cut each other on the centre hne. 51 as at o. Carry the distance o h and o a (fig. 10) to fig. 11 from i to m and i to also the length of and o e, from to n and to o. iJraw the line mn to cut the perpendicular line at p. Also the line through the points I and o, which will cut the perpendicular line at the same point if the distances are taken correctly. Take the distance from pm,pl,p n, and p o in fig. 1 1 as radii, and describe the curves ml no from the centre p> (fig. 12). Take the length from b to c (fig. 10), and mark off" the same from m to c (fig. 12), and draw the lines from 7n and c to the centre p, also take the distance b to a, and mark off" the same from m to g and from c to d, and the distance a to d (fig. 10) mark oft" from d to a (fig. 12). Draw lines from these points to the centre p, and connect these points with straight lines, as ad and de., &c., and also from the correspond- ing points on the smaller curves n and o, will complete the pattern required. Note. — This will be found a very useful method for striking a square or rectangular tapering top or sides. Whether the tapering be proportionate or not, by draw- ing lines as 6/ and eg from the angles (which show the position of the top and bottom of the article required) to cut the centre line wherever the point o may come, by taking it as a working centre, one-half or a section of the pattern may be developed. 52 PLATE XII. To describe the pattern for a Square or Rectangular Tapering Top or Tray, with sides and bottom, in one piece. Fig. 1 shows the upright height and one half of the plan. Draw in fig. 2 the horizon- tal line h d and the perpendicular line op. Draw the rectangle efg h the same size as ef g li in fig. 1. Take the length a h (fig. 1), and mark off corresponding distances from e to li to cZ, and o to p (fig. 2), and draw through the points h p and d the lines (at right angles) hq, st, and d r ; and carry the length i I to hq and to dr; also the length of u I from p s and t. Then draw the lines qf, sf, t g, and rg, which will complete one half of the pattern. To describe the pattern for a Hexagon Mould or Tray, having the bottom and sides in one piece. Fig. 3 shows the elevation and half an hexa- gon for the plan. To obtain a development of the pattern, draw (in fig. 4) the perpendicu- lar h c, and draw the half hexagon efg h i, the same size as efghi (fig. 3). 53 Divide the lines It g and g f into equal parts, and draw the lines a k and a m ; then carry the length oi ah (fig. 3) from I to k (fig. 4). Draw through k the line n o parallel with h g. Take the length k I (fig. 3), and mark off the same from k to n and o, and draw the lines li ngo: kg no is the sixth part of the pattern. Proceed in the same manner to draw the re- mainder, one half of the pattern (as well as the plan) only being shown here. To describe the pattern of an Irregular Octa- gon Pan or Tray, with the sides or bottom in one piece. Figs. 5 and 6 show the required projection and elevation, having drawn which, proceed with the development of the pattern in fig. 7. Draw the half octagon uafh h and v to the same dimensions as the corresponding letters in fig. 5. Draw the horizontal line tto and the perpendicular line oc. Divide the sides a f and h h into equal parts, and draw the lines o r and op, then carry the length of the line a c (fig. 6) — being the slanting height of the larger sides — from q to c, from v to i, and from V to vj (fig. 7). Draw from t and tv lines per- pendicular, and through the point c draw the line e g parallel with t w. 54 Take the distance from t to c (fig. 5), mark off the same in fig. 7 from t to c, w to d, c to e, and c to g, and draw the lines ca, e/, ,y7i, and Then in fig. 6 draw the perpendicular line X r, and from fig. 5 take the projection of the small side the distance s r, and carry the same from r to s (fig. 6), and draw line s x, the length of which should now be transferred from s to r (fig. 7). Draw yz parallel to af. Take the distance r e (fig. 5), and mark off the same from r to y and z in fig. 7, draw lines ya and zf. For the other side j) proceed in like manner, which on being done will com plete half the pattern. To strike the pattern of an Oblong Pan, with Round Corners, but struck from different centres, and tapering more at the ends than the sides. To construct the plan fig. 8, first draw the larger rectangle and the diameter lines, also the diagonals, and from the diagonals draw the four lines showing the width and length for the bottom. Draw the quadrants (or quarter circles) for the corners, as gf, from the centre c any size required, and from the points g and /draw lines to the centre a, which will give 55 a proportionate size for the corner of the bot- tom, as shown in the curve de^ struck from h as centre. Having drawn the plan, proceed now with fig. 9. To obtain the radii required draw a h and ad dit right angles, from a to c, take the upright depth required, and draw ce parallel with a d, then carry the lengths of a g, a /, and a k (fig. 8), to a d^ a 7c, and a I fig. 9), also the distances a cZ, a e, and a I (fig. 8), to c e, c /, and eg (fig. 9), and draw lines from the points de, k /, and I g, to cut the perpendicular a h at h (all cutting at one point), then take the lengths from c to g and h to d (being the radii of the corners), and carry them from a to and a to n, and draw the lines mp and 7i o parallel with To proceed with the pattern drawn in fig. 10, the perpendicular line a take the lengths of h cZ, h k, and bl in fig. 9 as radii, and de- scribe from point a as centre the curves c d, e /, and gw (fig; 10), also take the radii be and bg, and from the same centre a (fig. 10) strike the curves i k and u o. Then carry the length mg (fig. 8) on the curve cd (fig. 10), and draw the lines c a, d a, and c d. From the points d, i, and Z?, draw perpendicular lines as e jJ, dq, i r, and k s. Take the length p m ''fig. 9) and carry the same from c to p and d to q 56 (fig. 10;; take also the length on, and mark oft' the same from i to r and k to s. From i) and q as centres, radius p m, strike the curves c t and dx to meet the curve e f \ draw the lines tp and X also the lines t a and x a ; then from 7' and s as centres, radius on, describe the curves i I and k m. Now take the dis- tance from / to k (fig. 8), and mark oft" the same distance from / to on the curve also the same distance from x to and draw lines from V to a and from w to a. Draw the lines from t to V and from x to id, also / u and m o, which develops one half of the pattern. PLATE XIII. To describe a Pattern for a Tapering Top, the base being straight at the sides^ and with circular ends, the hole in the top to be cir- cular, and parallel with the base. {Similar to a Tea Bottle Top). Fig. 1 shows the plan and elevation required. Draw the lines a d and c h, the required width of the top, and draw a c and & at right an- gles with them, and through the centres by 57 which the circular ends are struck. Then draw the diagonal Hues ah and cc?, which will give the centre o, and draw the diameters AB and CD at right angles through the centre o. Take the distance from E to F, being the up- right height, and mark off a like distance on the line H to G (fig. 2), draw the lines a h and cd at right angles with the line HG. Take the length of the diagonal line a b (fig. 1), and make it equal to aUb (fig. 2), take the dis- tances from the centre o to t, and o to e (being the diameter of the circle), and mark off cor- responding distances on fig. 2 from G to c and G to d. Draw lines from points b d and a c to cut each other at g, with radius g a or g with i (fig. 3) as centre, strike the curve ABC (which will give a boundary line to describe the pattern on) . The cuTve of the end (fig. 1) being a semi- circle, extend the line 5 c to g, which will be at right angles with a c, take the distance EF (being the upright height) and mark off a like distance from c to g, draw g h at right angles with c g, take the distance from k to I, being the taper of the end, and mark off' a like dis- tance from g to the; point h, draw line from h c to cut the line AB at i; with radius ic, taking d as centre, strike the; curve a 5 c (fig. 4)^ now take the length of the curve a A c (fig. 1), and 58 mark off a corresponding distance from ah to c (fig. 4), draw the chord from a to c (fig. 4). In fig. 3 draw a Une from the centre i to B on the curve ABC, and take the length of the chord a c in fig. 4, marking off an equal dis- tance from a to c (fig. 3) on the curve ABC. Take the distance from a to d (fig. 4), and from a (fig. 3) mark off the point d on the line ■{ B ; with d as centre (with the same radius as the curve a hc is struck by) strike the curve ahc (fig. 3), take the distance cb (fig. 1), which is the straight part of the side, and mark off on fig. 3 from a to e and c to / on the cir- cle ABC, draw lines from eac and / to the centre take the distances from B to c or B to «, marking off the same distance from e to A and / to C, draw lines from the centre i to A and i to C, and produce them as at k and I : with radius id draw the circle as dotted g d Ji, take g and h as centres with radius- d b, and strike the curves from / to k and e to I, draw lines a e and c / from the highest part of the curves, which will make them tangent, and the base of the pattern will be finished. To get the curve for the hole in the top, bisect the lines a e and c f (fig. 3) through the centre and produce them indefinitely as n and 77i, from r on the circle (fig. 1) draw a line rj) at right angles with CD, take the up- 59 right depth as EF, and mark a like distance from r to p. Draw p m at right angles with r take the distance from k I [ which shows the slant of the end), and mark off a corresponding distance from p to m, take the distance from r to D (being the slant of the side) and mark off a corresponding distance from p to n on the line pm, draw line from points m r to cut the centre line AB at s, also draw line from points 71 r to cut the centre line at t, take the dis- tances from m to r or h to c, and mark off like distances from b to r, I to £c, and k to v (fig. 3), being the slanting depth of the end of the pattern. Take the distance from s to r (fig. 1), and from r (fig. 8) mark ofi' the point o, and from v the point s, and from x the point f ; using ^ o and 5 as centres strike the curves xw, vu, ^nd J) rq, with the radius S7' (fig. 1). Now take radius t r (fig. 1), using m and n as centres, strike the curves iv q and p u, which will complete the pattern required. 60 PLATE XIV. To describe the pattern for a Tapering articlet oval at the base and round at the top, (Such as an oval Canister Top, having a round hole for the neck and cover). Fig. 1 represents the plan and elevation of the top required. Take AB and CD, being the given diameters. Draw a diagonal line from the points a and h, being the sectional points of the curves ; make the line a b (tig. 2) equal to ab (fig. 1). Now take the distance from Y to E (fig. 1), and mark oft" a like dis- tance from F to E (fig. 2), draw c d parallel to ab, make cd (fig. 2) equal in length to the diameter of the circle from c to d (fig. 1). Draw b d and a c to cut the perpendicular line at with radius gb from centre i strike the circle ABc (fig. 4), or boundary line; v being the centre from which the curve a Cf is struck, draw line v id at right angles with CD, and ex- tend the curve « C/ as dotted to cut the line vw. Draw w£c at right angles with vio, take the upright height EF and mark ofi" from w to £c, draw x y at right angles with id take the distance from D to r, being the flue of the 61 side, and mark off a like distance from x to y; draw line from y and w to cut the perpendicu- lar GD at z. Draw a perpendicular line as A B (fig. 3) ; taking B as centre, with radius z w (fig. 1) strike the curve fAg (fig. 3). Take the length of the curve from C to / (fig. 1), and like distances from A to g and A to / (fig. 3). Draw lines from / and ^ to the centre B, and draw line fg, which will give one section of the base of the pattern. The curve of the end of the base being struck from g (fig. 1) as centre, draw line from g at right angles with AB, and extend the curve e a to cut the line produced from g to i. Draw line i k at right angles with i g, making i k equal the upright height, as EF. Draw ke at- right angles with i k, make k to e equal the fine of the end, as from A to g, or the distance from the curve of the oval to the circle on the line AB. Draw line from points e and { to cut the centre line AB at n, now with radius ni from B (fig. 3) as centre, strike the curve c d, take the length of the end curve from e to a (fig. 1), and mark off a like distance as from c to d (fig. 3), draw chord line cd, which will g^ve the end section of the base. Take the distance from c to d (fig. 3), and mark off an equal distance on the circle ABC (fig. 4) as from c to d, and draw lines from the points c and d to the centre i, bisect these hnes by the perpendicular 13 i, take the dis- tance from either B to d (fig. 3) or n to i (fig. 1) as radius, and from c or d (fig. 4) mark oft' point e on the line B i, with e as centre, strike the curve c d. Take the distance from f io g ■ (fig. 3), and mark oft* a like distance on the circle or boundary line from c to g (fig. 4), and d to /. Draw lines from g to the centre i, and irom / to i bisect the distances from g to c as at h, and from d to / at h, extend these Hnes as k and /. Take the distance from z to as radius (fig. 1), and from d (fig. 4) mark the point I on the line h I, and from g mark off the point k. Using I and A; alternately as cen- tres, strike the curves df and eg. Take the distance from d to B, and mark off from g to A and / to c, and draw Hnes from the cen- tre i to A and c. Again using n i (fig. 1) as radius, strike the curves f n from a centre on the line n also from a centre on the line w i strike the curve g which will complete the i;urve for the base. Now to descril)e the curve for the circular hole in the top: the line CD being drawn through the centre by which the circle is struck, from point r draw r;y'at right angles with the diameter line CT), take the distance 63 EF, being the upright height required, and mark off the same distance from r to y, draw the line y to u at right angles with r y^ take the distance from D to r, and mark off from y to being the slant of the side. Draw line from points t r to cut the diameter line AB at then take the distance from the oval to the circle, as from A to being the slant of the end, and mark off a like distance from y to u, draw line from a and ?• to o on the line AB, take the distance from u to r or from e to i, which should be the same, being the slanting depth of the end, and mark off like distances from n to o (fig. 4), from m to and from B to e (the outer curve). Take the distance o r (fig. 1) as radius, making 6- (fig. 4) as centre on the line B e, and strike the curve vew through the point e\ with the same radius strike the curve o x, with u as centre on the line c i ; also with t as centre on the line m i, strike the curve p y ; with radius pr (fig. 1) strike the curves y r (fig. 4) and w q:, with centres found on the lines h I- and h /, which will complete the pattern required. 64 PLATE XV. To strike a pattern for the Tapering Sides of a Tray having various curves. Fig. 1 shows the plan and elevation of the article, for which a pattern for the tapering sides is required. Having drawn the plan, it is required to show the points or centres from which the various curves are struck, as shown here by nm^ hli^ and I. The tapering being equal on all sides, the curves for the bottom and top are struck from the same centre, that is, the curves % e and g c are both struck from one centre, viz., h. To prepare for the development of the pat- tern construct fig. 2, making the distance from A to 5 the required upright height (fig. 1), and take the radius by which the curves a g and de are struck, that is tlie distance from h to a and h to and mark off the same on fig. 2 from A to c and from h to d ; draw the line from points c and d to cut the perpendicular at e, also the distance h e or h /, and mark off the same from A to i (fig. 2), and the distance from li to t mark off from h to t (fig. 2). Draw a line from points i and t to k. (The radius 65 m r, and m v, in this case, being the same as from h to i and h to t, do not require to be transferred to fig. 2.) To commence describing the pattern take e c (fig. 2) as radius, and from b as centre de- scribe the curve a c (fig. 3), and take the length of the curve from a to c (fig. 1) and mark off a corresponding distance from a to c (fig. 3), and draw lines from a and c to the centre b. Now take the radius from e to d (fig. 2), and again using b as centre (fig. 3) strike the curve from e as far as the line a b. Take the dis- tance from a k) o in fig. 1, and mark oft' the same from c to i (fig 3), likewise the distance from c to s (fig. 1) ; and take a like distance from a to s (fig. 3), and draw lines through the l)oints thus received from the points where the curve e intersects the lines a b and c b, and pro- duce them indefinitely as s g and i h. Note. — In further describing the pattern the letter d will be used, it ought to have been placed on the line a h, as e on the line c b. Take the radius hi (fig. 2), and from the curve e (fig. 3) mark oft' the point k on the line e /, also from d the point g on the line ds;^ using k as centre strike the curve e x, and from g as centre strike the curve d f. Again from k and g as centres, and radius k t (fig. 2), strike the curves a u and i w, take the length of curve t)6 from d to r (fig. 1) and mark off a correspond- ing distance from e to x (fig. 3) and draw the line X k ; the distance v in fig. 1 will show what is required to be added on the curve from V to ic in fig. 3 ; and draw the line x w, which will give one end of the pattern to meet for joining at /• u {-^g. I). Now take the length of the curve from e to i (fig. 1) and mark oft' a corresponding distance from d to / (fig. 3). Draw line from /' to the point AB describe arcs intersecting at e/C, which are the centres of the four arcs composing the arch. To find the covering of an Ogee Dome, the plan of which is Hexagonal. Fig. 4.— Let ABCDEF be the plan, and H IJ the elevation. Divide HJ into any number of equal parts, as 1 2 3 4 5 6, and through these points draw perpendiculars to FG; through the points in FG draw lines parallel to FE (the side of the hexagon) to EG, bisect EF in a, and draw a G, which is the seat of one side of the dome. Now to find the de- velopment of one section, set off the lines 1 2 3 4 5 6 K the same distance apart as 1 2 3 4 5 6 on the elevation from H to J. Then take 86 a E or a F on the plan, and transfer it to 1 o each side of 1 on the pattern ; now take & 6 on the plan, and transfer it from 2 to Z> on each side ; then c 7 on the plan transfer to 3 m, and so on to K, and through the points oh 711, etc., trace the curve as shown, and it will form the covering for one side of the dome. All the sides being equal, of course, the pattern of one side is all that is required. PLATE XXII. To describe the pattern for a Rectangular Base and Bottom in one piece, where the flue or curve is equal on all sides. Such as may be used as a base for either an Aquarium or a Fern-case. Draw fig. 1, which represents Tialf the pro- jection ; fig. 2 shows the elevation and profile of the base. Next, in fig. 3, draw the rectan- gle Ckta, same size as ACDB in fig. 1. Take the upright distance from a to h, and divide the curve into any number of equal parts, as c def, etc, and mark off corresponding dis- tances on the perpendicular line from a to i, also from a to ^ on the line BD, likewise from C to A, and draw parallel lines from these 87 points, and the distances from h to «, m to c, and 71 to d etc., will show the required dis- tances, as from h to 7, c to m, and d to to be taken on each side of the centre line, fig. 3. Then by taking the distance from B to D in fig. 1, and marking off the same from a to ^, fig. 2, and drawing the perpendicular th, the required length of the lines will be obtained, as from h to c d w, e x, etc., in fig 3. A curve drawn from the points thus obtained will give one half the required pattern. To describe the pattern for a Hexagon Base. Draw fig. 4, the half hexagon so placed that half of a side, as q will be perpendicular and at right angles with the base. Draw a per- pendicular line, o A, and lines from B and q to the centre. Now in fig. 5 (the elevation) divide the curve into a convenient number of parts, and draw the horizontal lines, and also the perpendiculars, and extend them to cut the line drawn from q, to the centre at qrs t, &c. Now draw the perpendicular a n in fig. 6, and carry on this line all the distances of the straight lines and angles, likewise the points of division on the curve of the eleva- tion (fig. 5), as nmlki, &c., to the points marked by corresponding letters in fig. 6, and 88 through the points thus received draw parallel lines at right angles Avith the perpendicular a n, and on each side mark off the points az, b y, c X, and d w, the same distance as the corre- sponding letters in fig. 4. By connecting these points by curves and right lines (according to the plan in fig. 5) the required pattern will be obtained. PLATE XXIII. To describe the pattern for a Vase, Octagon Shape. Draw the profile (fig. 1) the required design, and the half octagon (fig. 2) the size corre- sponding with the extreme points in fig. 1 (as described in fig. 4, Plate 3) and to be placed so that one half of the side, as from A to E, may be perpendicular with the base, and draw section lines from points A, B, C, and D, to the centre G, now divide the curve in the plan into any convenient number of parts, as abed. &c., and from these points draw horizontal lines across the plan, also draw perpendiculars and extend them to cut the line AG. Now mark ofi" the same distances as shown by cor- responding letters in fig. 3, and from these Fuji. 89 points draw parallel lines mm, nn^oo^ etc., take the length of the line (produced from a) from c to m (fig. 2) and mark oif the same from a to m on each side of the perpendicular line in fi.g. 3, also the distance from h to 7h (being the line produced from 5, fig. 1), and mark off the same from b to n (fig. 3), also let the distance from A to o (fig. 2) be carried from c to o (fig. 3). Now take the distance from d to 2> (tig- 2) and carry the same from d to p (fig. 3), and so on ; tracing all the dis- tances between the lines EG and AG, until all the points in the development of the pattern are obtained, and draw a curve from these points, which will complete the pattern. To make the pattern look more complete, and to give a prospective view of the article, the course of the curves N, F, R, may be ob- tained in the following manner (although not necessary for the development of the pattern) : From the points ahcdef, etc., on the line A G (fig. 2) draw lines parallel to AB, to cut the section line BG, and again produce them to cut the line CG parallel with BC. Now if a vertical line was raised from B and C in the projection (fig, 2) it would cut the horizontal line c in the elevation at N and N ; also by following the line from / in the elevation to / M/ in the projection, and raising perpendicu- 90 lars as before, they will give the points FF in the elevation; likewise by following the per- pendicular from /, produced at KR in fig. 2, perpendiculars raised from these points will give the points RE in fig. 1. By tracing all the points in like manner the course of the curves NFR, etc., will be found. And it will be ob- served that these curves will show the same width as the pattern (fig. 3) all the way through, while the length of the pattern will correspond with the length of the curve of the side, as at a & c e/, etc. PLATE XXIV. To describe the pattern for a Vase having twelve sides (Duodecagon). Draw the profile fig. 1 (the two outer curves only), also draw the perpendicular line CD through the centre, and draw the horizontal line AB (fig. 2), on which construct half the plan of the projection as in Plate XXIII., and draw the sectional lines from points FGHIK and L, to the centre E, having half of a side from F to A perpendicular, and at right angles with the base. Now divide the outer curve in 91 the plan into any convenient number of parts, as r^, 5, c, d, e,/, &c,, and from these points draw horizontal lines across the plan, also draw per- pendiculars, and extend them to cut the line F E, observing the points from which they are produced, take the distances between the lines AE and FE, and transfer them to the lines marked by corresponding letters in fig. 3. Now connect the points FG, GH, and HI, etc. (fig. 2), and draw lines parallel from all the points produced on the line FE ; observing that where there are straight parts in the ele- vation, as from m to ri, and from t to the same distance be taken, as marked by the cor- responding letters in the development of the pattern (fig. 3), and the two lines as t and u will be connected by lines at right angles (as they are both the same length). Now carry parallel lines from all the points on the line FE to all the other sectional lines in the projection, as from F to G, G to H, and H to I, etc., and by raising perpendiculars from these points, or by marking off points perpendicular to them on the corresponding horizontal lines in fig. 1, the course of all the curves may be obtained, showing all the joints and angles, or giving a prospective view of the Vase. For example, by following the per- pendicular line drawn from e (fig. 1), to e on 92 the line FE (fig. 2), and all the other sectiona] lines marked by e, perpendiculars raised from these points, on the lines GHI and K, will give the direction of the various curves on the horizontal line ee (fig. 1), as oooo. PLATE XXV. To describe the pattern for a Cone with an Elliptic Base. In fig. 1, let AB represent the major dia- meter, and DD the minor, and E (the centre of the base) to F (the apex) represent the vertical height. Now draw half of the ellipse, as from A to B, and divide it into a convenient number of equal parts, as 1, 2, 3, 4, 5, 6 ; and from these points, using E as centre, describe arcs to cut the base line AB. Now taking F for centre, radius FB, draw the curve BA in fig. 2, and so from all the points from B to D, describe the curves in fig. 2 to A. With the same compass set as the divisions 1 2 3 4 5 6 in fig. 1 were obtained by, take twice that number in fig. 2, but in measuring ofi" the dis- tances with the compasses in fig. 2, commenc- ing on the outer curve, from each point step 93 into the next one, as 1 2 3 4 5 6, and then re- treating back in like manner to A ; lines 4ra\vn from those points to the centre F, and a curve drawn from the points so obtained, will com- plete the pattern ; the length of the lines drawn from FA, Fl, F2, F3, &c. (fig 2) will be equal to the length FA, F5, ¥ c, ¥ d, &c. (fig-i)- To describe the pattern of an Oblique Cone, or the Frustum of a Cone cut parallel with the Base. ' . The vertical position of the two diameters is shown by the two circles in fig. 3. Now take the upright height from F to h, and draw FG and h a parallel to AB, the diameter line (being drawn across the two centres by which the cir- cles are struck), and draw AH perpendicular to AB ; also draw perpendiculars from E ,to a and from B to G, and draw a line from points G and a to cut the line AH. Now FH repre- sents the vertical height of the cone, and FG the base, and the line h a shows the frustum or section required, the apex or point being cut off parallel with the base. Divide half the plan into equal parts, as 1, 2, 3, 4, 5, and draw lines from these points to A. Now using A as centre, draw arcs from these points, as 1 D, 2 C, etc., to cut the dia- 94 meter AB, and draw perpendiculars from these points to the base hne FG. Next using the point H as centre (fig. 4), describe arcs from Ghfedc and F to A, also from b i etc. Now with the same compass set as the plan is divided by, as 1 2 3 4 5 B, mark off the same distances in fig. 4 from G to 5 4 3 2 1 and A, stepping from the outer curve into the second and third, and so on, and draw lines from these points to the centre H. By drawing curves from these points of intersection as A 1 2 3 4 5 G, and also from wvutsr and a, one half fit the development will be obtained. Fig. 5 is a further illustration of the same principle. The two circles struck from centres on the line AB, show the vertical position of the section of the cone required, and from D to F the elevation. Draw perpendicular lines from C to H and from B to E, being the dia- meter of the base, also carry perpendiculars showing the diameter of the top of the cone to J and G, join HJ and EG, and produce them to meet at the point H, and draw a perpen- dicular line from H to cut the line AB at o, which will be used as a working centre, as the point A is in fig. 3 ; likewise the development will be obtained in the same manner. 95 PLATE XXVI. To describe the pattern for a Round-end Bath, tapering more at the ends than at the sides. It will readily be seen that the pattern re- quired for the end of this bath is a section of the oblique cone in the last plate. The semi- circles CBD and EFG being struck from N and M as centres, extend these lines to N and M on the line AB (fig. 2), and draw two cir- cles corresponding with the semicircles in fig. 1 ; one quarter of the circumference is all that is required to be divided here, as from B to C ; the lines d f and gh being drawn the same distance apart as from t to the upright height. The vertical line from m to o will be obtained as explained in Plate 25, figs. 4 and 5. It will be observed that lines drawn from 12 3 and C to o, the working centre, will also divide the same section of the smaller circle into a like number of equal parts (a line drawn from C to o being perpendicular with the cen- tre jN",. also cuts the perpendicular from M, the centre of the smaller circle). Now using o as centre describe the arcs from C 3 2 1 to cut fctie diameter AB» and draw the perpendicu- 96 lars to meet the line df\ as at 4 3 2 1/; from the points thus obtained strike the various curves in tig. 3, as previously explained in Plate 25, figs. 2 and 4 ; and from the point m draw line //, and with the same compass set as the section BC (fig. 2) is divided by, mark off the same distances from / to 1 2 3 4 (fig. 3) from the outer to the inner curve, also from / to 10. Now draw lines from these points to the centre m, and draw a curve from the points of intersection, as from 4 3 2 1/, etc., also from the points rsuv, etc., which will give the development of the pattern for the end, so far as shown by the semicircles in the plan CBD and EFG. PLATE XXVII. To describe the pattern for a Hip Bath. Fig. 1 shows the plan, also the position of the bottom, which may be drawn to any given dimensions. Let the lines QE, and ST repre- sent the perpendicular height, as from M to N (fig. 2), both drawn parallel to the diameter line AB (fig. 1). Draw perpendicular lines from the extreme 97 points B and A in the plan to /rand g on the hne QE,, which will represent the top of the Bath; also perpendiculars from a to /i, and from n to i, on the line Sl\ showing the posi- tion and length of the boJttoni. The smaller oval need not necessarily be drawn, only the points a and n marked off, showing the posi- tion of the bottom or the required slant of the toe and back, as the shape of the bottom will be found in the development to come in pro- portion. Draw lines from the points g h and k i, and produce them to meet at id (fig. 3) ; now from w draw a perpendicular line to cut the dia- meter AB at X in the plan, which is to be used as a working centre ; next let one-half the plan from A to B be divided into any number of parts, as 1, 2, 3, 4, 5, 6, 7, and from these points, using x as centre, describe arcs cutting the diameter line AB, as at t s r, etc., and draw perpendiculars to cut the line QE,, as marked by figures corresponding with those on the curve. Now, by using w as centre, and describ- ing the various portions of circles, as shown, from 1, 2, 3, 4, 5, 6, 7 and likewise from the points from y to i, and taking the same distances from Ic to A and from A to m (on the arcs k, 7, 6, 5, etc.) as the divisions in the plan from B to A (in fig. 1); and by drawing 98 lines from 7, 6, 5, 4, etc., leading towards the centre w, all the points of intersection will be obtained, from which the curves may be drawn (by free hand) to give the development of the pattern in one piece. To obtain the shape of the back, draw the curve def, in fig. 2, as required, and mark off points h and c, perpendicular to 1 and 2 in fig. 1, and draw he and c /parallel with Ad; now transfer the lengths of A.d, he, and c/, from A to c?, 1 to 6, and 2 to /, in fig. 3, which will give the course of the curve of the back to correspond with the plan of the same in fig. 2. PLATE XXVin. To describe the Pattern for a Travelling Sitz Bath. Fig. 1 represents the plan of the top and bottom required, and fig. 2 the elevation. (It will be seen that the tapering is as much in the front as at the back, but the back being much higher than the front, the tapering is not equal in proportion to the depth ; this, however, may be governed according to dimen- sions required.) Let the horizontal line gd 99 (fig. 3) be drawn to represent the required length of the bottom, as shown by the dotted lines brought down from the plan (fig. 2), and let the Hne AB represent the slanting length" of the top, as shown by its meeting the per- pendiculars drawn from the top of the plan (fig. 2) ; now draw lines A g and B and produce them to meet at the apex E, and draw a vertical line from E to cut the line AB (fig. 1) at F, which will be used as a working cen- tre. Let one-half the ellipse be divided into any number of parts, as 1, 2, 3, 4, 5, 6, 7, A, and using F as centre, describe the arcs cutting the diameter line AB, as at /, e, c?, g, b, a; and from these points draw perpendiculars, as shown by the dotted lines. It will now be observed that the radius from F to A and from F to 7 will be (in this case) the same as from F to 6, by which the arc from 6 to a is drawn, which will prevent separate arcs being de- scribed from points 7 and A, as shown from all the other points ; therefore a perpendicular from point 6 must be drawn to cut the line AB (fig. 3) as at i, and from this point draw a line to s, parallel with gd (to cut the perpendicular from a), and from point s draw a line to B, which will give the various heights (where in- tersected by the perpendiculars drawn from h 100 cdef) of the Bath froiii B to the point, 6- (fig. I). Now, from point 7, drtiw a perpendicular to cut the line AB tfig. 8) at //, and a perpen- dicular from A (fig. 1) to A (fig. 3) ; now, from y and A draw lines parallel to g d, to cut the perpendicular dotted line from a (fig. 1), .as at o and z, which will give the two remain- ing points which were deficient while in the process of describing the arcs ; and from all the points, as z, o, s, t, v, iv, etc., describe arcs indefinitely ; now draw lines from all these points to the apex E, and where the line g d is cut, a§ at d, h, g, f, etc., will be the points from which another set of arcs are required to be drawn. Take the same compass set as half of the plan (fig. 1) is divided by, and measun? of the same number of distances from B 1 2 3* etc., in the development (fig. 3), stepping from the first arc into the second, third, and so on ; and draw lines from all those points to the apex E, which will give the points of intersec- tion on the smaller set of arcs. Now by draw- ing the curves from these points of intersection from m to n, and from the points A to 7 6 5, etc., to B, will give one-half of the pattern re- quired. 101 PLATE XXIX. To describe the pattern for a Globe, formed oi twelve pieces joined together. In fig. 1 describe a circle the required size, and draw the perpendicular a w ; now divide the half circle into any number of parts, as a b cd, etc., and draw the horizontals, as eq.fx, g y, etc. ; next draw in fig. 2 half a circle, and divide it into six equal parts, having half a side from e to h perpendicular with the base ; now in fig. 3 draw line a i, and mark ofi* the points a bcde, etc. the same distance apart as the cor- responding letters in fig. 1, also draw perpen- diculars in fig. 1, from points hg/e, etc. to cut the line k o (fig. 2), and let the distances from /; to t, c to d to r, and so on, be trans- ferred to corresponding points in fig. 3, which will give the course of the curve to complete the pattern. To describe the pattern for a Triangular Pedestal or Pyramid, with all three sides alike (an equilateral triangle). To draw the projection for this figure, let the point x, and the centre point o, come in a 102 horizontal line, also at right angles with the hne uo^ which represents the real centre of the article, not the centre of either of the sides. Now in order that the curves may be equal on all sides, divide the circle into three equal parts, from x to g and A, and draw lines to the centre o. Now draw the required shape for the side in fig. 4, from ahcdef, etc., and let this side be divided into any number of parts, and draw horizontal lines across the profile, likewise perpendiculars to intersect the lines g o and ^ o in fig. 5, and from these points draw lines parallel to h x and g x. Now by raising perpendiculars from all the points on the line o X, as shown from x and r, the direction of the curve will be obtained, showing the point or angle of the article, which at first may appear to have more curve, but by referring to the plan (fig. 5) it will be seen that the one gives a view of the side, and the other that of a point or angle. Now to develop the pattern (fig. 6) will require but little further explana- tion ; the distances between the lines ahc etc., are transferred from the corresponding letters in fig. 4, and the lengths from h to g, c to etc. are equal to those similarly marked in fig. 5, which will give the course of the curves to complete the pattern. 103 To obtain the radius required for striking the pattern of a slightly Tapering article, without the necessity of producing lines to meet. Let the two circles in fig. 7 represent the diameters at the top and bottom, as e f and g hf then the distance from a to h will show the flue on all sides. Take the distance from a to h with the compasses, and measure off or find how many times that distance there is between a and the centre o on the diameter line : in this case 9. Now let the upright depth, as from c to d, be multiplied by 9, and the result will be the radius required, or the length of string or wire to strike the curve with. To show an example. Suppose the diame- ter of the larger circle is 18 inches, and that of the smaller one 16 inches, the distance from a to h would be 1 inch, and from a to o would be 9 times as much as from a to h. Now sup- pose the upright depth from to c be 2 feet. Therefore, 9 times 2 being 18, a radius of 18 feet would strike the required curve for the pattern. , 104 PLATE XXX. To describe the patterns for the sides of an Irregular Octagon Pan. Figs. 1 and 2 show the plan and elevation of the article required, having the flue or curve equal on all sides, as shown by the distance from G to I and from i/ to these being alike ; therefore all that is needed to be described is to divide the curve (fig. 1) into equal parts, as a, 1, 2, 3, h, and take the same from a, 1, 2, 3, b on the perpendicular lines in figs. 3, 4, and 5 ; and the widths of these patterns will be obtained from the plan (fig. 2), as previ- ously described in Plates XXII. and XXIII. Figs. 6 and 7 show also the plan and eleva- tion of an irregular octagon article, where the curve will not be alike on all sides, but pro- portionate, and its angles or section lines all leading to the centre. Draw the elevation (fig. 6) ; also draw half the projection (fig. 7) AGFEDC, as required, and draw the sectional lines, from G, F", and E, D, to the centre B. Divide the curve from c to f into equal parts, as 1, 2, 3, 4, 5, and draw horizontal lines across the plan from JFTcUeSO 105 these points ; also perpendiculars to cut the line DB, and carry the lines from D e to E ^ and F g parallel with UE and EE. Now to obtain the pattern for the end, draw the perpendicular (fig. 10), atid mark off the same distances as 1, 2, 3, 4, 5 (fig. 6), and take the distances from CD to p e on each side, which will give the widths of the pattern. Next draw the line e n (fig. 7) at right angles with g e ; and as this line e jo is the same length as from p to C, draw the perpen- dicular G n (fig. 9), and draw the parallel lines 1, 2, 3, 4, 71, the same distance apart as in tig. 10. Now mark off the distance from 7i, to E and D, the same as from 7i to E and D (fig. 7) ; likewise transfer all the distances on each side of the line 7i e in the same order on fig. 9, as shown ; and lines drawn from the points thus obtained will give the pattern for the small side. Now the projection of the side from w to s being much less than that of the end from p to C, proceed as follows : — take the distances (fig. 7 ) from B, the centre, to u, and s, and mark off the same (fig. 6) from the per- pendicular a h, to g, h, i, ?, and m, and draw a curve from these points, which shows the fall of the side before mentioned; now draw the perpendicular (fig. 8} , and draw the 1G6 parallel lines g, h, i, k, /, m, the same distance apart as the corresponding letters in the eleva- tion (these will not be equal distances apart, as in figs. 9 and 10), and make them the same length as the lines 6', i, u, w (fig. 7) ; this being done, will give the course of the curve required to complete the pattern. [The few points of variation between this and the plates heretofore referred to are re- commended to bo well studied ; they will be found of great assistance in studving Plate XXXI.] PLATE XXXL To describe the pattern for a Cover and Neck of an Irregular Octagon Article, such as a Tureen. Fig. 1 represents the elevation and the re- quired curves, and fig. 2 shows one half the plan. Tc obtain the pattern for the cover, first -draw the half octagon zxuhivn (fig. 3) the i5ame size aszxabidn (fig. 2), and draw through the centre the perpendicular line b L Draw the line from x to / at right angles to 107 X u ; also from v draw the line v at right angles with v i ; then take the length of the line xw (fig. 1), and carry the same from zto y and n to m (tig. 3), and draw the perpendicu- lars y w and m I ; then take the length oi yw (fig. 2) , and mark oft' the same from y to w and m to I (fig. 3), and draw the lines w x and / V, Now, in the projection (fig. 2), draw a line X f from x, at right angles with x a, and carry the length of x/from y to /(fig. 1), and raise the perpendicular y x, and draw line from X to /. Now take the length of x /, and mark off the same from x to / and v to k (fig. 3), and draw the line w g through the point /, parallel to X u\ take the distances from f to g and /to w (fig. 2\ and mark off the same from f to g and w (fig. 3), and draw lines as £c and g u. Now take the distances from o to h and o to /i, in the projection, and mark oft' the same from o to 5 and i to li, in the elevation, and draw the line h h. Now take the length of h h (fig. 1], and mark off the same from h to h (fig. 3), and through the point h draw g n parallel to u i ; take the length oi h g (fig. 2) and mark off" the same from h to g and n (fig, 3), and draw lines g u and 4, which completes half the pattern for the top. Now to describe the patterns for the sides or neck. The octagon being irregular, and the 108 angles leading towards the centre, the several sides will take different curves ; therefore each section will have a little variation. To obtain a pattern for the end of which B A represents one half, the process is simply to divide the curve from a, c, etc. to rj and w (fig. 1), and draw the horizontal lines from these points, and also draw perpendiculars from the same points, and extend them to cut the lines AO and BO, in the projection (fig. 2). On the perpendicular line (fig. 5) take a corresponding number of distances, and draw the horizontal lines, as a, ^, c, etc., and on each side of these points mark oft' the points e e,ff, g g, etc., the same length as AB, h f\ c g, d h, etc. (fig. 2). A curve drawn from the points so obtained will give the pattern for the end. Note. — The lines in fig. 2, as A B, 6 /, c g, etc., are not all that would be obtained from the points in the curve (fig. 1), but they will sufficiently illustrate the principle. Now, to obtain the pattern for the side from B to C, draw the lines from / to p, from g to q, and from h to r, etc., parallel to BC, and draw a line from lo to E, at right angles with If) f or BC, precisely the same as the line x f was drawn at right angles with x a. Now draw the perpendicular line (fig. 6), also the 109 horizontals, as from a b, c, d, etc., the same distance apart as in fig. 5 ; and take the dis- tances from E to B and C, and mark off the same from a to e and n (fig. 6), also the dis- tances from k to f and j9, and mark off from b to f and j9, and so on until all the points are obtained, which will give the direction for the curve to be drawn to give the pattern for the small side. Now draw lines from j9, 5, i\ etc., parallel to CD (fig. 2), and take the distances from o t?, o 0 o and o A, and transfer the same from A to a, g tob, f to c, and e to c?, and so on, to i h ; and from the points thus obtained draw the curve from li and / to a. Take the distances between the horizontals (previously drawn from the curve at the extreme end), and on the perpendicular (fig. 4) mark off corresponding distances, and draw the hori- zontal lines, and take the distances from v to C, u to p, t to and s to ?% etc. (fig. 2), and transfer the same on each side of the perpen- dicular line (fig. 4) , from a to b to c to etc. A curve drawn from these points wQl complete the necessary patternSo 8 110 PLATE XXXII. To describe the pattern for the Top of a Jack Screen. [It will be seen that in striking this pattern, sufficient allowance must be made for hollow- ing in addition to the leading points obtained. This must be left to the judgment of the work- man^ as there are no known rules to describe it.] Fig. 5 represents the article of which the development of the top is required. Fig. 1 shows the elevation or the shape of the front of the screen, and fig. 2 gives the shape of the top or projection, the top being intended to be made in three pieces. Draw in the projection the shape of the hole for the jack to work through, as shown by m q v., and draw lines from n and r to the centre />, from any part of the outer curve that will make the back and side-pieces look propor- tionate or convenient for material. Now di- vide one side of the elevation into any num- ber of equal parts, and draw perpendiculars to cut the line w v in the projection, as marked by corresponding figures ; and from these Ill points describe arcs cutting the lines r p and u p. Now the height of the top, as far as the back piece, will come, is shown by the arc drawn from 6 intersecting it at s; therefore, from B (fig. 3) mark off 1, 2, 3, 4, 5, 6, the same distance apart as the same figures in fig. 1, and draw the horizontal lines from B to 6. Now the curve a eh m fig. 1 represents the opening in the top for the doorway, and the curve i (fig. 2) gives its course in the pro- jection, shown by the point g coming between the second and third arc, as the point h (fig. 1 ) comes between the second and third per- ])endicukr ; draw FGH at right angles, and take the distance from D to e (fig. 1), and mark ofi" the same from G to F, also the dis- tance from g to the outer curve in the projec- tion, and mark off" from G to H, drawing the line FH ; now let difterence between the dis- tances from FG to FH to be added on from B to A (fig. 3), and draw the line a c ; draw the curve a h c about one-fourth wider than a h in fig. 1, as it will draw in much closer by hollowing. Draw line from B to 6 (being the height of the back piece as previously stated) and extend it to cut the perpendicular at E ; now with radius EB describe the curve for the bottom of the pattern from points a and c, and 112 take the distance from / to u (fig. 2) and an allowance for the seam, and mark off from a to D, and c to F, and draw lines from points so obtained to 6 ; now draw the dotted curves for the hollowing as shown according to judg- ment. To obtain the pattern for the side draw the line r v at right angles with r s, and also the line s ^ in the same manner, now draw c b a (fig. 4) at right angles, and by taking the dis- tance from r to s in fig. 2, and by measuring off the same distance on the line AB (fig. 1) from B it will just reach the perpendicular 7 ; now take the distance from B to 7 on the curve, and mark off the same from 6 to 7 on the perpendicular in fig. 4, and draw c a at right angles with c b. Take the distance from s to t (fig. 2) and mark off the same from c to a (fig. 4) also take the distance from r to v (fig. 2) and mark off the same from ^ to a (at the base, fig. 4) and draw line from points a a and extend it ; now take the distance n to t (fig. 2) and let the same be added on from a to e (fig. 4) and draw line c e, this will give the main points and size required for the pat- tern of the side ; curves for hollowing and wiring to be added on as shown by dotted lines. SHEET-METAL WOEK PROCESSES. I. SHEET-METAL WORKING. The patterns and the manner of laying them out havmg been given in the foregoing pages, a cursory glance at the processes of sheet-metal work will now be taken. When the form of the pattern has been marked out on the sheet metal, the next busi- ness is to cut it out : this is generally effected with a large pair of shears, either screwed up in the vice or with their shank dropped into a hole in the bench, and worked by hand (Fig. 1). In some instances, however, the cold chisel and ham- mer are em- ployed, the work being either laid on the anvil di- rect or on a cutting-plate ; in others, being screwed up in the jaws of the vice and cut off by the hammer and chisel, the latter being kept in contact with the upper surface of the Tice-jaw . as a guide. Sometimes, the thick (113) 114 plates employed for boilers are screwed up in very long vices with a screw at each end and cut off by the chisel. There are also sUtting- plates in large works. The hammers (Fig. 2) are alike at both ends Fig. 2. ^ rule, sometimes with large faces either flat or convex. The faces or panes are always kept very bright, in order that they may impart some of their polish, to the work, a process which is termed '• planishing." Wooden hammers or mallets are often used to prevent stretching the sheet metal. The anvils are of very varied shapes, and generally placed in a hole in the work-bench. The smaller ones are p,^^ 3 usually called " stakes," and go down r ji to half an inch square. Fig. 3 is the " hatchet-stake," and is much usc^d for turning over edges, etc.; this var- ies from 2 inches to 10 inches wide. Fig. 4 is a " taper-stake," also much used. Fig. 5 is the " creasing tool," which is used for making small beads, tubes, etc. Fig. 6 is the seam-set," used for closing the seams prepared at the hatchet stake. Fig. 7 is the "-Holliper" or "Oliver:" it consists of two jointed arms, in which various kinds of top- 115 and-bottom tools, swages, etc., can be fixed, and the metal being fig. 4. placed between the dies, and the top forcibly struck with a hammer, the piece of tin, etc., is at once stamped out exactly to the contour of the dies. The sides of the vessel (which is a shallow tray) represented in Fig. 8, if the metal be thin, Fig. 5. Fig. 7. Fig, 8. would be bent to the required angles by laying the metal horizontally on the hatchet-stake, with each angle line exactly over the edge of the same, and blows would be given with the mallet, or with the hammer for more accurate angles, so as to indent the metal with the edge 116 of the stake ; it would then be bent down by the fingers, unless the edges were very narrow, as for the seam, when the mallet would alone be used. Thicker metal is more commonly hent over the square edge of the anvil, a square set-up hammer being held upon its upper surface, and sometimes the work is pinched fast in the vice, and is bent over with the blows of a flat-ended punch or set, applied close to the angle, and then hammered down square with the hammer. Very stout metal is seldom bent, but cut and the angles riveted. Thin metal is bent to curves by holding one edge and placing the other edge on the beak- iron, around which the sheet is (Figs. 9 and 10) Fig. 9. Fig. 10. curled by the mallet. The crease (Fig. 5) is frequently used for making seams or edging. A strip of sheet metal is laid in the appropriate groove, and an iron wire is driven down upon it by the mallet. The wire, of course, bends the strips when driven down ; the edges are then folded down upon the wire by the mallet, 117 and it is then finished by a punch or top tool (Fig-. 11) matching the groove in the crease. Joints. Let us now glance at the various methods of making joints at angles of sheet metal, as at Fig. 12. A and b are for the thinnest metals, such as tin, which requires a film of soft solder on one or the other side. Sheet lead Fm. ii. is similarly joined, and both are usually soldered from within. C and D are the and re joints, used for thicker metals, with hard solders. Sometimes d is dove-tailed together, the edges being ^ filed to correspond coarsely ; B sometimes they are partly Q riveted before being sold- D ered from within. These joints are very weak when jj united with soft solder. E is the hip joint, the Fig. 12. metal being creased over the hatchet-stake. Tin plate requires an external I layer of solder ; spelter solder runs through the K crack and need not project. F is folded by means of the hatchet-stake. 118 the two are then hammered together, but re- quire a film of solder to prevent their sHding asunder. G is the folded angle joint, used for fireproof deed-boxes and other strong work, in which solder would be inadmissible. It is common in tin and chopper work, but less so in iron and zinc, which do not bend so readily. H is a riveted, joint, which is very commonly used in stroni? iron Fig. 13. ^ L _ plate and copper M . work, as in boilers, etc. Generally a rivet is inserted at each end, then the other holes are punched through the two thicknesses on a block of leatl. The head of the rivet is put within, N ^ % '•^ — -vT^ 0 R the metal is flnt- tened around it by . placing the small hole of a riveting ^ 'J^^-lSL ~ set over the pin of V ^ , ^ the rivet, and giv- ing a blow ; the rivet is then clenched, and is finished to cir- cular form by the concave hollow in the rivet. , ing set. In I K one plate is punched with a long mortise, the other being formed into tenons, which are inserted and riveted, k, however, has tenons with transverse keys, which can be taken out and the plate released. Let us now see to the straight joints. L (Fig. 13) is the lap joint, employed with solder for tin plates, sheet lead, etc., and for tubes bent of these materials. M is the butt joint, used for plates and small tubes of the various metals. When united by hard solders or brazed, such joints are moder- ately strong, but with soft solders the joints are very weak, from the limited superficies of the adhering surface. N is the cramp joint. The edges are thinned by the hammer, the one is left plain, the other is notched obliquely with shears for one-eighth of an inch deep ; each alternate cramp is bent up, the other down, for the insertion of the plain edge ; they are then hammered together and brazed; after which they may be made nearly flat by the hammer, and quite so by the file. The cramp joint is used for thin work requiring strength, and amongst numerous others for the parts of musical instruments. Sometimes the lap joint (l) is feather-edged. 120 This improves it, but it is still inferior to the cramp joint in strength. o is the lap joint, without solder, for tin, copper, iron, etc. It is set down flat with a seam set, and is used for smoke-pipes and numerous works not required to be steam and water tight. P is used for zinc works and others. It saves the double bend of the preceding. It is some- times called the "patent strip overlap." Q is the roll joint, used for lead roofs. R is a hollow crease, used till recently for vessels and chambers for making sulphuric acid. The metal is scraped perfectly clean, filled with lead heated nearly to redness, and the whole united by burning with an iron also heated to redness. Solder which contained tin would be attacked by the acid. Now super- seded by autogenous soldering. S T, joints united by screw-bolts or rivets, for iron and copper boilers, etc. u, united with rivets, in ordinary manner of uniting the plates of marine boilers and other work requiring to be flush externally. v is a similar case, used of late years for con- structing the largest iron steam-ships, etc. The ribs of the vessel are made of "J" iron, varying from about 4 inches to 8 inches wide, which is bent to the curves by the employment of very 121 large surface-plates cast full of holes, upon which the wood model of the rib is laid down, and a chalk mark is made around its edge. Dogs or pins are wedged at short intervals in all these holes, which intersect the course ; the rib, heated to redness in a reverberating furn- ace, is wedged fast at one end and bent around the pins by sets and sledge hammers, and as it yields to the curve each pin is secured by wedges until the whole is completed. Zinc. Our illustrations of this metal principally refer to junctions of external rainwater gutters or troughs. In large towns the gutters and pipes are usually of tin or galvanized iron ; but occa- sionally tin is employed for this purpose. Gutters are very easily formed of zinc. The slip of the desired width being cut off the roll with shears or knife, is gently hammered to the correct curvature over a mould of wood made to order by the carpenter, something like in sec- tion A, Fig. 14, which is screwed up in one or a couple of vices, or otherwise fixed firmly on the shop-bench. When this is done, the trough is turned right way up, and the " stays," which are formed of a small piece of zinc, rolled up round into a kind of close tube, are 122 soldered across from side to side of the top at intervals (a a, Fig. 14), to hold the trough together and brace it. Of course, the angles at which the guttering joins at any internal or external angles of the roof will be cut to shape before the zinc is curved, and it is in this case that plans of proper cutting out are useful. It must be remembered that zinc is a less pliable metal than lead or copper, or even than tinned Fig. 14. iron, and very springy. This last quahiication renders it difficult to get zinc to take and retain a new shape when worked cold. But if it be heated over the fire to nearly boiling-point (212° Fahr.) there will be no more trouble on this score. It is not so easy to solder as tin, and rosin is rather uncertain with it. The hydrochloric acid (generally called " muriatic acid ") acts better, and so does " Baker's 123 soldering fluid." The copper bit, well tinned, is the tool used. There are several gas blow- pipes or soldering-jets which act well with moderate care. The surface of the zinc at the joints should be clean and scraped bright. Do not use too much solder. Galvanized Iron. This is comparatively a recent material. Of course, ordinary thin sheet-iron has been in use almost from time immemorial, but its range was limited from its excessive tendency to rust, and it was chiefly for such purposes as stove-pipes, etc., that it was applicable. The discovery of coating it with zinc [i. e., " galvan- izing " it) has largely added to its utility. In this country there is a large industry for the production of galvanized iron cornices for architectural purposes. lu place of using cornices and string-courses of stone in the Ironts of brick houses, as formerly, we now prefer those of galvanized sheet-iron made in long lengths, and fixed to wooden blocks let into the brickwork, or to suitable rod-iron supports similarly fixed. Some of these corn- ices, when containing many members of mould- ings, especially if they are circular in plan, need much skill. In general the metal is bent over the hatchet-stake with mallet or hammer, 124 much as in making zinc guttering, assisting with swages where necessary. The following observations on " Circular Work " are by Mr. C. A. Vaile, late Superintendent of the Cornice Works at Richmond, Indiana.* "In making up circular mouldings, it is necessary to have the material sufficiently heavy to bear shrinking and stretching without breaking or becoming brittle. The best plan for bringing mouldings to the required shape is in the following manner : Take a piece of hard wood (oak) 4 inches by 4 inches and 12 inches long, make a profile of work intended, and on one end of this piece make a die of the desired shape ; to this must be fitted a plunger, allowing the thickness of iron to intervene. The die is shown in the following figures: Fig. 15 is the top; Fig. 16 is the sectional view of the plunger and die for a half-round mould. Fig. 15 is to be made in the same circle as work. Figs. 17 and 18 are the same, of a different moulding. Figs. 16 or 18 is to be placed in an oak block, as Fig. 19. The right-hand portion should be of sufficient length to answer for a seat to the operator. Fig. 20 is a mallet about 12 inches long. To make these dies, imagine the cap to be stamped * Galvanized-Iron Cornice-Worker's Manual. Philadelphia. Henry Carey Baird & Co. 125 from one piece, and get out the die and plunger accordingly. The tools required will be a saw, Fig. 16. brace, and J-inch bit, a straight chisel, two or three sizes of gouges, a straight rasp, and a Fig. 18. rasp curved at one end. When the iron is cut 126 to the required pattern, it is raised in these dies, shifting- the mould to and fro each time, it is forced into the die with a blow on the plunger from the mallet, until it is brought to the re- quired shape. A httle practice will soon demonstrate the utihtry of this method, and also its superiority over the hammering process. When work is to be put together, never place two raw edges together. On one of the members turn | of an inch edge, and lap the member on this and soak the solder in well, so as to firmly unite the pieces,, and on the top strip that is to be built in the wall turn a J-ineh edge, to stiffen and answer the purpose of straps to hold the cap in position. An edge of the same kind should also be turned on bottom strip, to extend over the frame ; and if the cap is to have a drop or corbel, let the inside of the drop or corbel extend back past the frame at least one inch, to secure the corbel to the frame, and the other side of corbel have a |-inch wedge to fit against the wall. " Should the work be for a building already up, the strip should have an edge sufficient to nail through into mortar joints. Good judg- ment is required in putting up work of this character to make it a success." II. SOLDERING. Soldering is the process of uniting the edges or surfaces of similar or dissimilar metals and alloys by partial fusion. In general, alloys or solders of various and greater degrees of fusi- bility than the metals to be joined or placed between them, and the solder, when fused, unites the three parts into a solid mass ; less frequently the surfaces, or edges, are simply melted together with an additional portion of the same metal. The solders are alloys of various kinds, and are broadly distinguished as hard-solders and soft-solders. The former only fuse at the red heat, and are consequently suitable alone to metals and alloys that will endure that temper- ature ; the soft-solders melt at very low d(>- grees of temperature, and may be used for nearly all the metals. The forms of soldered joints in the sheet metal have been already given at pages 117 ^nd 118. The following table exhibits most of the facts necessary to be known relating to the solders ( 127 ) 128 and their use. It contains the composition of the various solders, the fluxes suitable lor each, and the manner of applying- the heat. This is abridg-ed from Holtzapftel s " Mechanical Manipulations." " Soldering may be divided generally into two branches, viz., 'hard-soldering' and 'soft- soldering.' The first process may be used with all metals less fusible than the solders, the modes of treatment being nearly similar. The hard-solders used are generally spelter solders, the flux usually borax, A, and the mode of heating the naked Are, the muflle, or furnace, and the blow-pipe (a, />, g). Lami- nated gold is used for soldering platinum, cop- per for iron, gold lor gold alloys ; spelter solders, granulated, for iron, copper, brass, gun- metal, German silver, etc. Soft-soldering is applicable to most of the metals. The methods pursued are very various. The soft-soldt^r mostly used is composed of two parts of tin and one part of lead ; sometimes, from economical motives, much more lead is employed, and IJ of tin to 1 of lead is the most fusible of the group, unless bismuth is used. Tlie fluxes B to G, and the modes of heating a to are all used with the soft-solders. In the following- examples the metals to be soldered are placed first, then the number of the alloy to be used 129 as solder, next the capital letter signifying- the flux to be employed, and lastly the italic letter which indicates the mode by which the heat should be applied. (See p. 130.) " Iron, cast-iron, and steel, 8, B D ; if thick heated by «, />, or c, and also by g. " Tinned iron, 8, C, D,/. " Silver and gold are soldered with pure tin or with 8, E, «, ad pipe, which require to be very sound. These SLYo generally extremely clumsy in appearance, as by the aid of the hot iron and a piece of tick held in the left hand the plumber manages to plaster a great bulbous patch of solder round the point of jimction, which they term a ^' wiped " joint. The blow-pipe (mouth) is used to some de- 135 gree in soft-soldering, principally by the gas- fitter, who is generally remarkably expert in making joints in his composition pipes there- with. These are not made like the plumber's, by inserting one end of the pipe in the other and plastering a bulb of solder around the place, but by cutting off the pipes with a fine saw and filing them up square and smooth to butt together into a mitre or a T-joint. These joints have frequently to be made in very awk- ward and confined situations amongst joists under floors, etc., and are generally eftected by applying by some convenient means the heat from one side only, and forcing the flame thus obtained upon the joints with a blow- pipe. They generally use a rich tin solder, and employ a flux of oil and rosin in equal parts. The pewterers generally use the hot-air blast, by means of a peculiar cast-iron apparatus em- ployed only in their trade. They use fusible solder containing bismuth, and for flux a com- mon green olive oil termed Gallipoli oiL For hard-soldering an intense fire-heat is re- quired, similar to that obtained in the smith's forge. In fact, the ordinary blacksmith's forge is frequently used for brazing, although the process is injurious to the fuel as concerns its normal purpose. 136 The brazier's hearth, for extensive works, is generally a plate of iron about 4 feet by 3 feet, supported on four legs at its corners, and with a central opening about 2 feet by 1 foot and 6 inches deep for the fuel. The blast is gener- ally supplied by a fan, and the tuyere-irons have large apertures Fresh coal should never be used, but char- coal, or, failing that, coke or cinders. Lard in the fire is very prejudicial. In all cases of hard- soldering or brazing the meeting edges are to be scraped or filed clean (especially when the heat used will not reach the red degree). The work in copper, iron, brass, etc., having been prepared and the joints retained in position by binding with iron wire when needful, the granulated spelt er and powdered borax are mixed in a cup with a very little water, and spread along the joints by a slip of sheet metal or a small spoon. The work is now placed above the clear fire, first at a small distance to gradually evaporate the moisture and deprive the borax of its water of crystallization. During this process the flux boils up with a frothy look, and sometimes shifts the solder away. The heat is now in- creased, and when the metal assumes a faint red the borax melts like glass. As the metal gets deeper red the solder fuses also, generally 137 with a slight blue flame if it contains any zinc. Generally at this point the solder " flushes " or disappears in the work. Should it not do so, and appear refractory on the score of running into the joint, the work may be tapped with the tongs, in order to make it move. Care must, of course, be taken that the heat is not so much raised as to melt the work as well as the solder. If the w^ork be iron, there is, of course, little need of precaution. If it is iron which you wish to braze, you have to file the meeting surfaces bright ; make a little borax into a paste with water, and smear them over with this. Next tie th(^m together with some fine iron wire, just enough to prevent the pieces from coming apart. Then wind them round and round at the place of the joint with several coils of fine brass wire, rub- bing them over with the borax paste. This is then laid on the fire and the blast put on. Presently a small blue flame will be seen re- flecting over th(^ placfe. This is a sign that the brass wire is meriting and that the heat is dis- sipating the zinc constituents of the brass, and the brass having melted and run into the joint the job is done. It is only iron, however, to which you can apply so much heat. For brass and copper you must have a more fusible metal than 138 l^rass. This solder is called " spelter " (incor- rectly), and is composed of copper and zinc in equal parts. Indeed it is a very soft kind of brass, and liquefies at a much lower tempera- ture than would melt copper or ordinary brass. There are two varieties of spelter, hard and soft, both procurable at any metal ware- house. The borax (borate of soda) can be got at the same place, or at a druggist's. We have mentioned its quality of swelling up when heated, and that this swelling displaces the solder on the work. In order to obviate this it is not unusual to heat the borax previously, till this considerable swelling up has subsided and the water of cry stallization is driven off, when it can be pounded and kept in a stop- pered jar. The blow-pipe is largely used in hard-solder- ing and brazing, especially lor work in the [m^cious metals. The ordinary blow-pipe is a light conical brass tube, about 10 inches or 12 inches long, from J inch to J inch in diameter at the end Cor the mouth, and from tV inch to -nr inch at the aperture or jet. The small end is bent in a quadrant that the flame may be immediately under observation. Very usually it is fitted with a small liollow brass ball just below the 139 quadrant, to serve as a receptacle for the con- densed vapor from the lungs. This mstru- ment is generally used with a lamp of a wick from I inch to 1 incr hi diameter and pro- duces a flame of great heat, the object exposed to it being generally placed upon charcoal. Gas is frequently used in conjunction with the blow-pipe, and this is especially useful lor sheet brass, the work being held in place by wire ties if necessary, and either laid upon a tlat piece of pumice-stone or held in a pair of pliers. Til. GEOMETRY AS APPLIED TO SHEET-METAL WORKING. The utility of a tolerable knowledge of prac- tical g(>omctry to those engaged in the sheet- metal trades scarcely needs be insisted upon. It is next to impossible to strike difficult pat- terns by mere rule of thumb, and although in many workshops templates may be found for the great number of ordinary patterns, still, even then, occasions will certainly arise for the construction of others for special work. Be- sides, in the present days of technical instruc- tion and active competition no young man who desires to excel in his trade should be content without the best knowledge available about it. If he will take the trouble, however, to acquire a certain amount of geometrical information, he will be prepared for all emergencies. He will be enabled to work from the roughly- drawn outline sketches of a customer with the same unfailing certainty as if the job was one which he had executed hundreds of times in- stead of beijig, perhaps, quite new to him. And besides having the pleasant conscious- \U0) 141 ness of mastery of his work, the artisan will effect a considerable saving in time, material, and temper on many occasions. The subject appeals to every worker in sheet metal in a greater or less degree. In the manipulation of tin, sheet iron, zinc, copper, lead, and brass it is brought into practice. The geometrical process mainly called in by the sheet-metal worker is that known techni- cally as the " development of solids ;" in other words, the representation on a plane of the ex- terior surface of a cylinder, cone, prism, or many-sided figure. But, besides this, the manner in which such solid bodies are cut, and the " sections " thus arising and their intersec- tions are not less necessary to be studied, as will become apparent as we proceed. In many cases the sheet-metal worker's pattern or template for a certain job is simply a development of the geometrical form of the article. If it is one (as is usually the case) which requires to be soldered or brazed together, and there are two or three possible ways of cutting the pattern, the operator will select that whereby he may, as far as possible, reduce such joints. Thus, take a hexagonal- sided tin or sheet-iron box shown at A (Fig. 22). For the purpose of the artisan it may be developed into either the pattern shown at b 142 or that at c. The saving of time effected over cutting its sides and bottom into separate pieces is evident. Let us now, to render our purpose more plain, detail the process to be pursued in "developing" one of the simplest of the geometrical solids, namely, the cylinder. When the surface of a cylinder is developed a right-angled parallelogram (all the geometri- FiG. 22. cal terms will be explained as we proceed) is obtained, as at f G h i (Fig. 23), the height of which, G H, is equal to the length of the cylinder, which we will imagine in this case to be equal to the diameter of the cylinder, and the length, F G or H i, is equal to the length of the circumference of the circle, as b E J c. The development of this cylinder will indicate the principle upon which all problems of this 143 Fig. 23. 5l kind are based. Let it be required to have the surface of a half-cylinder, as b e c, devel- oped, the height, b G or f d, being equal to the radius, A C. Through A draw the diameter, B c, and extend it in- definitely, as to D ; from E draw parallel to € D a line E F, and from B a line at right angles cutting e f in G. Divide the semi-cir- cumference, B c, into any number of equal parts, as ten in the present case. From B on B D set olF these parts to D, from it draw D F at right angles to b D ; then F D, B G is the develop- ment, or " stretch-out," as it is frequently called, of the semi- cylinder, B E c, and if cut out and wrapped around the said half-cylinder would exactly cover it. If the entire cylinder, as at B e c J, needed to be developed, the " stretch-out " would be twice that of b d, f G. 05 S .>o. r* > "o" 1^ . , s 144 Again, let us suppose that it is required by the zinc worker to make a mitre-joint at right angles in a half-round rain-water gutter, or trough. He will proceed geometrically as fol- FiG. 24. A B Fig. 25. A lows : Let the semi-circle ABC (Fig. 24) rep- resent the sectional outline of the gutter. Draw the line a b, and draw the lines a f and 145 B E at right angles to a b, also draw the line D E parallel to a b. Make D f equal to A b, and draw the line f e. Divide the semi- circle into any number of equal parts (in the present case ten). Draw lines parallel to A f through these points in the semi-circle, as at 1 , 2, 3, 4, etc. Next draw (Fig. 25) A c equal in length to the semi-circle, A c b (Fig. 24). Draw the lines A b, c d (Fig. 25) at right angles to A c, and make A b (Fig. 25) equal to b e (Fig. 24), and c d in the former figure equal to A F in the latter. Set off on the line A c (Fig. 25) the same number of equal dis- tances as the semi-circle was divided into. Draw lines parallel to c D (Fig. 25) from each point of division, as 1, 2, 3, 4, etc., and make each of these of equal length to the line corre- spondingly numbered in Fig. 24. Finally trace the curved line b d (Fig. 25), through the extremities of these lines, and the required pattern of the mitre-joint will be obtained. As in many other cases, there is a certain amount of preliminary dry details to be mast- ered before the subject can be fairly ap- proached. It is just these preliminary, simple and apparently needless processes that often disgust the learner. He is apt to think that the special knowl- edge he desires to gain can be attained by a 146 " hop, skip and jump " over these — hey, presto ! — to the point which appears to him useful and practical. This is the greatest of mistakes. It is as if a child should hope to learn to read without first painfully acquiring the alphabet. There is no royal road to any knowledge, although care on the part of an instructor may help to smooth the roughness of the way, and this, in the present instance, we shall endeavor carefully to do. Let us now speak of the tools required, that is, the appliances to enable us to draw the var- ious diagrams. These are neither numerous nor costly. The following will be sufficient for the present : A drawing-board of seasoned pine (any board per- fectly square at its angles will do), a y-square, two set squares, a flat foot-rule with scales, a pair of compasses with moveable leg for pencil, a protractor, a drawing-pen, a pair of dividers, a couple of black-lead pencils (H and HB or F), and a dozen drawing-pins. The entire outfit need cost but a trifle. We will speak of the use of the instruments as we proceed. Stout cartridge paper is the best for the purpose. Before proceeding to teach our readers to construct the various flgures most usually required in the trades comprehended by the 147 title of this book, it is requisite that certain terms used in geometry should be explained, as without a good understanding of these our subsequent instructions will not be properly comprehended. We wish our readers to clearly understand that we do not profess in this les- son to teach them the science of geometry, or the art of practical geometry, but merely to illustrate so much of the latter as is applicable to certain special purposes. It would be infinitely to the advantage of every artisan concerned in these trades to make himself master of the rudiments of geometry. An elementary work on the subject can readily be gotten, and can be mastered in a month. But to proceed with our terms or definitions : A point simply marks position. Theoretically it is said to have no magnitude or size. Prac- tically the smallest point or dot that we can make has size, and therefore is really a surface^ and not a point. A mathematical point would be the centre of a dot of ink, etc., but for practical purposes the dot itself is spoken of as the point. Sometimes a point is represented by a dot with a small circle around it. Lines. A line has, theoretically, length or direction only, without breadth. We all know that the 148 finest line which we can produce by a pen or any tool has some breadth. This is not, therefore, the mathematical or ideal line, although we call it a " line " for the conveni- ence of practical purposes. A straight line or rigid line is the shortest distance from one point to another. In draw- ing, heavy lines are called " strong," and light ones " fine." Dotted lines are also used for various purposes. Those formed of different- sized dots (principally employed on plans) are termed " chain lines." At A (Fig. 26) the top line is "strong," the next "fine," the next " dotted," and the lower one " chain." To produce a line signifies to lengthen it at either end. A curve or curved line constantly changes its direction, and is, therefore, nowhere straight (c. Fig. 26). Curves are infinitely variable, and may be simple or compound. Curved lines may be parallel. When they are parts of different circles struck from the same centre they are termed concentric (c. Fig. 26). Fig. 26. A B a 149 Parallels or parallel lines are those which are everywhere the same distance apart, and which if produced or lengthened forever would never meet (see A and b, Fig. 26). A horizontal line is one perfectly level, a vertical line is one perfectly upright, having regard to the horizon, as, for example, the line of a plumb-bob. A perpendicular^ or perpen- dicular line, is one that is vertical or at right angles to some other line. It is not necessarily vertical in the strict sense, but may incline to or even be parallel with the horizon line. (The Fig. 27. horizon is the line where sea and sky appear to meet when one looks from the shore.) It is thus clear that while a vertical line is perpen- dicular to a horizontal one, a horizontal line is perpendicular to a vertical one. The line A b. Fig. 27, is perpendicular to the line c D. An oblique line is one neither vertical or hori- zontal, but slanting in regard to some other line, as E F and i K (Fig. 27.) Angles. An angle (from the Latin angulus, " a 150 corner ") is formed by the inclination of two lines until they meet in a point called the vertex of the angle. The magnitude or size of an angle does not depend upon the length of the lines forming it, but upon their inclination to each other. Thus in an angle of 45° (or any other number) the lines may be an inch in length or may be produced or lengthened to a foot or a yard without affecting the angle, which still remains one of 45°. A rigid angle is one formed by one straight Ime standing upon or being perpendicular to another. Thus the line A b (Fig. 27), being perpendicular to the line c D, both the adjacent angles are right angles and equal. This is the angle of 90°. An ac'ute angle is sharper or less than a right angle, as at e f g (Fig. 27). An obtuse angle is blunter or greater than a right angle, as at i k l (Fig. 27) . Triangles. Triangles are figures bounded by three straight sides, and having in consequence three angles. They are also termed trilateral (mean- ing " three-sided ") figures. There are six varieties of triangles, three named with refer- ence to the length of their sides, and three with regard to the sizes of their angles. 151 The first three are • The equilateral triangle, which has its sides equal (a, Fig. 28). The angles are also equal, and each contains 60°. The isosceles triangle (b, Fig. 28), which has two sides equal. These sides may be longer or Fig. 28. ABO shorter than the third side. The unequal side is always termed the hase^ in whatever position the triangle may be represented ; the angles at the base are equal to each other. A scalene triangle (c, Fig. 28) has all its sides and angles unequal. Fig. 29. The second division of angles embraces : The right-angled triangle (left-hand figure of Fig. 29). The side opposite the angle is called the hypotenuse^ the others being termed the hase Sind perpendicular, as shown. These 152 terms remain the same in whatever position this triangle is placed. The obtuse-angled triangle (central figure of Fig. 29) has one obtuse angle. The acute-angled triangle (right-hand figure of Fig. 29) is that which has three acute angles. Although we have specified six kinds of triangles it will become clear that, on a little consideration, one of the three latter kinds must also belong to one of the three former classes. In this manner a right-angled triangle must be either an isosceles or a scalene triangle, and an acute-angled triangle may be also an equilateral, isosceles, or scalene triangle. The highest angle of a triangle is termed its vertex (in the plural vertices)^ or apex (plural apices or apexes) , or vertical angle ; the lowest side is called the base. With the exception of the isosceles and the right-angled triangle (see page 151), the terms just given are applied to each angle that may be uppermost, or each side that may be lowest when the position of the triangle is altered. The altitude of a triangle is a straight line drawn from the apex to the base, as at a b (Fig. 29). _ ^ Any two sides of a triangle, if added together, are greater than the remaining side. It would 153 hence not be possible to form a triangle whose respective sides were, say, 4 inches, 6 inches, and 10 inches in length. The three angles of a triangle when added together always equal 180°, or the half of a circle. Quadrilateral Figures. Quadrilateral figures are those bounded by four straight sides. They are also called quad- rangles or four-angled figures. Their united angles always amount to 360°, or four right angles. If the opposite sides of a quadrilateral are parallel to each other it is termed a paral- lelogram. Fig. 30. The square (left-hand figure, Fig. 30) is a parallelogram of four equal sides and four equal angles. A line drawn across a parallelogram from opposite corners is called a diagonal. The rectangle or oblong (centre figure, Fig. 30) is a parallelogram, all of whose angles are equal, but only its opposite sides are equal. The rhombus (right-hand figure. Fig. 30) is a parallelogram with four equal sides, having 154 two obtuse angles opposite to each other, and two acute angles opposite to each other. The rJiomboid (left-hand figure, Fig. 31) is a parallelogram having only the two opposite «ides equal and also the opposite angles equal. The trapezoid (centre figure, Fig. 31), has two parallel sides only, but may have some of the sides or some of the angles equal to each other or not. The trapezium (right-hand figure. Fig. 31), has none of its sides parallel, but some of the sides and some of the angles may or may not be equal to each other, or all the sides and the angles may be unequal. A Fig. 31. trapezium, one of whose diagonals will divide it into a couple of unequal isosceles triangles (see A B, Fig. 31) is called a ti'apezion or kite. A polygon is a rectilinear or straight-lined figure, bounded by more than four straight lines. Polygons are sometimes called multi- lateral (or " many-sided ") figures. They may have any number of sides. A regular polygon has all its sides and angles 155 equal, and can always be so surrounded by a circle, that the circumference thereof shall pass through all the angles of the polygon. The forms shown at Fig. 32, are regular polygons, that on the left being a pentagon, that in the centre a hexagon, and that on the right a heptagon. An irregular polygon may have unequal sides and equal angles, or equal sides and unequal angles, or neither may be equal. Fig. 32. Polygons are named according to the num- ber of sides they possess. A polygon may have any number of sides, but for general purposes is seldom found with more than 12 sides. A Poly- gon having 5 sides is a Pentagon ; 6, Hexagon ; 7, Heptagon; 8, Octagon; 9, Nonagon ; 10, Decagon; 11. Un-decagon; 12, Do-decagon; 1 3, Tri-decagon ; 14,Tetra-decagon ; 15, Penta- decagon; 16, Hexa-decagon ; 17, Hepta- decagon; 18, Octa-decagon ; 19, Nona-decagon ; 20, Bis-decagon; 21, Un-bis-decagon, etc. Irregular polygons have the same names, but the word " irregular " is added. The circle is a plain figure bounded by one 156 continuous curved line called the circumference; every portion of which is equidistant from a point vs^hich is called the centre (see Fig. 33). The radius (plural radii) is a straight line drawn from the centre to any point in the cir- cumference ; the diameter is a straight line drawn through the centre and terminating at the circumference at each extremity. A diam- eter divides a circle into two equal portions, called semi-circles. The arc is a portion of the circumference of any circle. Fig. 33. A chord is any straight line drawn across a circle which does not pass through the centre ; a segment is a slice cut off from a circle by a chord ; a sector is a portion of a circle enclosed by an arc and two radii. When that portion is exactly the fourth part of a circle, it is also called a quadrant. A tangent is a straight line drawn outside of a circle, and which just touches the circum- ference in one point ; in other words, it does not cut off a portion of the circle. 157 Fig. 34. To bisect (divide equally) any given straight line, as A B (Fig. 34) : Take the compasses, and v^^ith the centre A, describe (that is, draw) the arc of a circle 2, 3. With the centre b and the same radius, de- scribe the arc 4, 5, cutting (or crossing the first arc at 6 and 7. Lastly, through the points 6 and 7 draw the right (or straight) line shown, and this will bisect the hne A B in the point 8, and be perpendicular to the line A b. From a given point, as c (Fig. 35), to draw Fig. 35. A C B a line perpendicular to A b : With c as centre, 158 and any radius, mark off the points 1 and 2 at equal distances from c. With 1 as centre, and any radius, describe the arc 3, 4 ; with 2 as centre and the same radius, cut this arc at 5. Join this point 5 and c by a right hne, and this hne will be perpendicular to A b and at right angles thereto. To draw a line parallel to a given line, A b, and at a given distance, equivalent to c d, (Fig. 36) : — From it, with the centres A and b respectively, and the distance c d as a radius, Fig. 36. describe the arcs 1 and 2. Draw the line 3, 4, resting upon these arcs at their highest point, and this line will be parallel to A b and at the required distance from it. To find the centre of any given circle or arc of any circle : Draw any two chords, as 1, 2 and 2, 3 (Fig. 37). Bisect each chord by a perpen- dicular (this can be accomphshed by the means indicated at page 157 for bisecting a right line), and produce these perpendiculars 4, 5 and 6, 7, until they intersect at A. The point 159 A thus found is the centre of the required circle. We have spoken before of angles (see page 149), and it may be well here to allude to the Fig. 37. manner of measuring them by instruments. The circumference of a complete circle con- tains 360° ; a semi-circle 180°, and a quadrant Fig. 38. (or quarter-circle) 90°. If, then, we take a semi-circle of thin brass and divide it into 180 equal parts wo form a protractor (Fig. 38) , or 160' instrument for measuring angles in drawings, etc. Let A B be the base line, from which, ascends a Hne c e. If we apply the lower straight edge of the instrument to the former line, and bring the small nick or mark in the centre of its straight side to c, wo shall find that the line EDO cuts the circumference of the protractor at 90°; e c b is, therefore, an angle of 90°, or a right angle. Similarly F c a is an angle of 45°, or half a right angle, and Fig. 40. G c B is an angle of 60° (the mark ° indicates a degree or degrees), and g c A is an angle of 120°. Sometimes the protractor has the form of a parallelogram, as at Fig. 39, but its use is 161 the same. A protractor of one of these forms is generally found in every box of instruments. To draw an angle of 60° geometrically : With centre b (Fig. 40), and any radius, de- scribe the arc 1, 2. With centre 1, and the same radius, describe the arc B 3. Draw the right line, A B, through the point found by the intersection of the arcs, and A b c is an angle of 60°. To draw an angle of 30° geometrically : Fig. 41. With centre b (Fig. 41) describe the arc 1, 2. With the centre 1, and the same radius, de- scribe the arc B 3. With centre 2, and the same radius, describe the arc 1, 4. Join B 4, and the angle A b c is an angle of 30°. To bisect (divide into two equal angles) any given angle, as A B c (Fig. 42) : With B as centre and any radius, describe the arc 1 ; with 1 as centre, and any radius, describe the arc 3 ; with the opposite point as centre, and the same radius, cut the arc 3 at 2. Join b 2, and the 162 line B 2 will bisect the angle A b c — that is to say, the angle A b 2 will be equal to the angle 2bc. Fig. 42. To trisect (divide into three equal angles) a right angle A b c (Fig. 43) : With centre b, and any radius, describe the arc 1,2; with the centres 1 and 2, and the same radius, describe Fig. 43. B C the arcs 3 and 4. Draw b 3 and b 4, and the right angle will be trisected or divided into three equal angles. 163 In a given circle to inscribe any regular polygon, say a pentagon. One method : First draw the diameter A 5 (Fig. 44) , and divide it into as many equal parts as it is required that the polygon should have sides (in the present instance five). With points A 5 as radius de- scribe arcs intersecting each other at 6. From 6 draw a line through point 2 to B. Join A B, which is one side of required polygon. Mark off distance A B from b to F, from F to D, and D to H, round circumference. Join b f, F d, D H, and H A, and these lines will all be equal, 164 and the figure will be the required regular polygon ; in this instance a pentagon. By this plan a regular polygon having any desired number of sides can be inscribed within a given circle. If, for instance, it was required to inscribe an octagon, the student would divide the diameter into eight equal parts, and then proceed as above ; but to obtain the first side of the polygon he would invariably draw a line from point 6, though the second division of the diameter, no matter how many sides the poly- gon was to have. The centre of a polygon coincides with the centre of the circumscribed circle. In any polygon having an even num- ber of sides a line drawn from one angle to the angle opposite (which would be a diagonal) must go through the centre. When there are odd sizes, a line drawn from any angle, through the centre, bisects the side opposite. The development of regular solids, or poly- hedrons (viz., " many sided " figures), which are bounded by planes, is very simple and easy. Indeed, in most instances the instincts of the operator could scarcely fail to guide him aright. Still, in order that our lessons may be tolerably complete, we think it is just as well to advert to the subject here. All solids having plane (or " flat ") surfaces must form " solid " angles where their faces 165 unite. And as three plane angles at least are required to form a solid angle, it follows that the most elementary and simple of the solids is a pyramid whose base is triangular, and whose sides are formed by three triangles, which unite in the angle at the apex, or top, of the pyramid. The " stretch-out " of this solid (h, Fig„ 45) is obtained by first describing the equilateral triangle, d f e, by the method previously ad- FiG. 45. verted to, and then erecting on the three sides or base lines the three triangles d A f, F b e, and DOE (Fig. 45), whose surfaces are in- clined when the development is closed up, so that the three triangles meet at the apex g. The solid just spoken of is the simplest of the five regular polyhedrons. It is termed, geometrically, a tetrahedron^ or " four-sided " figure. 166 The next most simple solid is the cube (a, Fig. 46). This is known by the geometri- cal name of a hexahedroii^ or " six-sided " Fig. 46. figure. The development (shown also at Fig. 46) needs no explanation. A square, c b e D, is first formed by any process, and the adja- 167 cent squares bgfe, mncb, eopd, and C D H I, added to its sides, the last side being completed by the addition of the square h i k l (Fig. 46). The octahedron^ or " eight-sided " figure (a, Fig. 47), is composed of eight equilateral Fig. 47. . triangles as shown. One face, c d e, having been constructed in the usual manner, the other seven sides are subsequently added, as 168 shown at Fig. 47. (One face has been omitted in engraving.) The next regular solid is the dodecahedron, or " twelve-sided " figure (Fig. 48). The faces Fig. 48. of this solid are composed of twelve regular pentagons (or " five-sided " figures, and it is Fig. 49. hence necessary to construct a pentagon accord- ing to any approved method, and then form others on its sides in the manner shown at Fig. 49. 169 The last regular polyhedron is the icosahe- drori (Fig. 50), which is bounded by twenty equilateral triangles. For obtaining the " stretch-out " these may be arranged as shown at Fig. 51. Fm. 50 All the preceding developments, if cut in card-board, scale-board, or thin metal, will, when their edges are brought together, assume the appearance of regular solids. Fig. 51. Although the equilateral triangle, the square, and the pentagon are the only figures from which can be formed regular polyhe- drons whose angles and sides are equal, yet by 170 cutting tho solid angles of the said polyhedrons in a regular manner, we can obtain regularly symmetrical solids whose sides are formed of two similar faces. Such is, for example, the polyhedron of eight sides obtained by cutting equally the angles of a tetrahedron. Of these eight faces four are hexagons (or " six-sided " figures), and four are equilateral triangles. In the same manner if we cut the solid angles of the cube regularly we obtain a polyhedron of fourteen sides, viz., six octagonal (or " eight- sided ") faces and eight triangular. The octahedron, similarly dealt with, gives also a polyhedron of fourteen faces — six square and eight octagonal. The dodecahedron, when thus cut, yields a solid of thirty-two sides, of which twelve are pentagons and twenty are hexagons. We have already given the method of get- ting the stretch-out of a cylinder from the cir- cumference, and now present another problem having to do with cylindrical bodies and exem- plifying the use of ordinates. The sections obtained by cutting a cylinder otherwise than longitudinally, or at right angles to its length, are of considerable importance in many works. Fig. 52 shows how to get at the figure produced by cutting a cylinder in a diagonal or slanting direction. If we cut a 171 cylinder at right angles to its length, or, in other words, parallel to its base, as at c E (Fig. 52), we get a circle ; but if we cut the cylinder Fig. 52. obliquely to its base, as at F g (Fig. 52), the section produced is an ellipse. In many cases a knowledge of the method of finding the pre- 172 cise form of the ellipse produced by such ob- lique cuttings of a cylinder is of considerable importance to the artisan, and this we proceed to describe. Let A B c E (Fig. 52) be a cylindrical pipe, or tube, or rod, which has to pass through some flat surface (as a roof, ceiling, iron-plate, etc.), F G, which lies obliquely to the base of the pipe, or tube, and let it, moreover, be desired to find the form of ellipse that will need to be made or perforated in such roof or plate, to allow it to pass through. Through H (Fig. 52) draw c e at right angles to c A and e b respectively. Divide the semi-circumference cahcdefghik^ into any number of equal parts (the more the better, as the ordinates will give a greater number of points through which to trace the curve of the ellipse). From the points thus obtained in the circumference draw lines parallel to c A or E B, as h i h etc. cutting the line c E in the points D H i K, etc., and produce them until they cut the diagonal line F G in I n jp r, etc. Next, from the latter points, and at right angles to F G, draw the lines I n o, pq^ rs, etc. Then from D meas- ure to the semi-circle, and set off this distance from I to m on the line / m. Next measure from H to the semi-circle and set the distance off from n to o on the line n o. In the same 173 manner transfer the other distances to p q, r s, etc. Repeat these operations upon the other side of the line f g. Finally, through the points thus obtained draw the ellipse by hand. Now let us treat of the cone and its develop- ment. A cone may be produced in any thin ma- FiG. 53. terial, as shown in Fig. 53. Let a be the circle of the base ; through B, its centre, draw a line b c ; make e c equal to the length of the sloping side ; from c with c e, describe the arc E D F ; take, in inches, or in parts of inches, the radius b d of the base, and multiply by 180°, and divide it by the number of parts there are in c e, the length of the slanting side. 174 The result is the angle with the sloping side^ as c E makes with the centre line c D. In the example there are two parts in the radius b EBt^l six in the length of side, which gives the Fig. 54. G angle,E C D, of 60°. From C, with a chord of 60°, describe the arc e f and set off 60° from the same scale of chords from E to D ; draw c D and 175 make c f equal to c E ; join c f. Then bend the outline E F till the edges c f E meet, the -edge E D f passing round the periphery of the circle A, the cone will be completed. To find the development, or the covering surface, of part of a cone (Fig. 54). Let A b c d 1)6 the portion of the cone to be covered ; the sides A B C D being produced to e to complete the cone. Divide the base A d into two equal parts in point f, and draw f E at right angles to A D. With radius F A from f describe a semi-circle, A 6 D. Divide this into any num- ber of equal parts, as twelve. From e as a centre, with e A as a radius, describe an arc A G, and with E B, another arc B H, and set oft' from A on the arc A G the same number of •^qual parts as A 6 d is divided into, the last ol these terminating at 12. From e, through each of these points, draw lines as in the draw- ing, and also from the points in A 6 D, ob- tained by drawing the ordinates, as 5 i parallel to 6 F E. Then the part A 12 H B is the, *' stretch-out " required, which, when cut out, will be found to cover the surface A B c D, part of the cone. This covering may be supposed to be made up of a number of boards, as shown by the crossed lined parts at 12, or a sheet of -metal. To develope the surface or find the " stretcj:^- 176 out " for part of a cone's surface as in Fig. 55 : Let a b c dhe the parts of the cone to be cov- ered, and the sides, a b, d c, produced to e to Fig. 55. complete the cone. Divide a d in the point / into two equal parts, and from /, with /a, de- scribe th(^ semi-circle a 6 d. Divide this by any number of equal parts, say twelve, and from these points, on a 6 c?, draw ordinates cutting the base line, a d, of the cone in the 177 points, as h /, etc., etc. From these points draw lines to the apex or vertex of the cone cutting the Hne 6 c in the points i k m, etc. From these, at right angles to c h, draw lines, as k I, i jp, m n, etc. From the point e, with radius e a, describe the arc a g, and from a set off towards g the same number of equal parts as the semi-circle, a 6 c^, is divided into, ter- minating in the point 12. From the points on the arc, a 12, draw lines to the point e. Then from 12 in a ^ measure to the point j9, making 12 p equal to c o, the first of the perpendicular lines drawn from the points on the line c h ; in like manner set off from the points 11, 10 and 9, on a g, the distances obtained from the line be; thus the distance 7 t is equal to m the distance 5 s equal to i h, and 4 r to ?, and so on. Then through the points thus obtained, jp t s r, draw a curve by hand, and the part a 12 p b will be the "stretch-out," which when cut out will cover the part of the cone, abed. The " stretch-out " may be considered as made up of a number of pieces, as 12/», 11 10 v. It is sometimes necessary to find the section Avhich a cone transected or cut at any particu- lar angle will present. For this purpose pro- ceed as follows : To find the section of a cone cut by a line oblique to its base (Fig. 56). Let a b c he the given cone, and d e the cutting 118 line. Divide the base line a h into two equaf parts at the point /, and draw / c perpendicu-^ Fig. 56. lar to the base line a h. Draw any number of lines parallel to the base a Z), as e g, h i, j h^. 179 and so on. From the points where these in- tersect the side d c of the cone, as g e^i etc., drop perpendicular lines, cutting a f in the points 1, 2, 3, 4, etc., and from the points n Z, jh, other perpendicular lines as in the diagram. From / as a centre, with / 1 as a radius, de- scribe the semi-circle 1,7; with / 3 as a radius, the semi-circle 3, 8 ; with / 4, the semi-circle 4, 9 ; with / 5, the semi-circle 6, 5, and with Fig. 57. / 10, 10, 11. Then from the point v, where the semi- circle 3, 8, cuts the line dropped from measure to 8, and set off this distance v 8 on a line h a', drawn at right angles to the cutting line e the distance h a' equal to v 8. Next from the point where the circle 4, 9 cuts the perpendicular 9 j, measure to 9, and set off 180 this distance from j to b' on the line J b\ at right angles to d e. In like manner set off the dis- tance, X 5, from I to c', and y 11 from n to d' ; a curve drawn by hand, or carried through the points d' d h' a', will give one-fourth of the ellipse, and the remainder of the ellipse will be found as described in connection with Fig. 52. The solid whose development we will next briefly consider is the sphere, Fig. 57. The sphere itself does not, perj^aps, enter very largely into the province of the sheet-metal worker, although it has occasionally to be con- structed ; but other solids (such, for instance, as the hemisphere, (Fig. 58) derived from it, are largely employed both in engineering and in architecture ; as, for ex- ample, in the former, the hemispherical ends of boilers, etc., and in the latter, cupolas, domes, pen- dentives, niches, etc. A sphere is a solid, the boundary of which is a curve, every point of which is situated at the same distance from the centre, the latter being the generating point of the sphere. It is not possible to develop a spheri- cal surface with accurate exactness, and we 181 must be satisfied with arriving at an approxi- mation, which, however, mostly answers all practical purposes. In order to obtain this it is usual to conceive of the outside or boundary surface of the sphere as divided into a number of parts, which form a series of polygonal sides of solids, the surfaces of these polygonal por- tions of the " stretch-out " terminating at com- mon points at the vertices of the sphere. Two methods of arriving at the shape of these segmental portions of the covering of a sphere are in use. The most common is to divide the surfaces into such parts as are indi- cated by the lines A, B, c, d (Fig. 57), which are usually termed "gores." If we look at an ordinary terrestrial globe or a map of the world we shall find that the meridian hues or lines of latitude, which are shown as equi- distant at the equator and meeting in points at the poles, divide the surface of the globe into a series of" gores." The other plan in use in the development of spherical surlac;es is to con- sider the sphere or hemisphere, or any segment of the sphere, as made up of a series of conical rings as at E, F, G, H, and G, H, i, K (Fig. 57), the " stretch-out " of which gives a series of curved slips. It may be observed that these latter lines correspond with those of longitude on a terres- 182 trial globe, and that this principle of develop- ment is the one adopted in the next example *>iven. To develop the covering of a hemisphere, as in Fig. 59, let a b he the hemisphere, and the part to be covered in depth equal to c d This is assumed to be the side of the portion of a Fig. 59. cone, of which c d / 14 is the elevation, and the sides produced to complete the cone. Draw the line c as to represent the base of the part of the cone, and divide it in the point e, and draw through e the line ^ e 7, at right angles from the base, a b, of the hemisphere. 1»3 From the point e, with c e as radius, describe the semi-circle c 7, 14. Divide this into any number of equal parts, as 14 in the drawing. From ^, the apex of the cone, as centre, with g g d Sis radii, describe the arcs, c //, d i. and set off in the arc c h the same number oi' equal parts as are in 14, 7 c towards h. Through the last of these, as 14, draw 11^, and through all the other parts similar lines converging to g ; j d is the covering of the part, c d f 14. The whole surface of the hemisphere may be covered by a series of such parts, the quadrant being divided into equal parts, to give an equal depth to the covering surfaces. APPENDIX. RULES FOR BOILER-MAKING, ETC. TEMPLATE MAKING. Remarks. — Boilers, when made of a cylindrical form, usually consist of a series of conic frustums inserted into each other, and riveted together; the height of each frustum being the breadth of plate of iron, and the inside diameter at large end equal to the given diameter of the boiler, that of the small end twice the thickness of plate less than the large end. This will be clearly understood on inspection of figure 60, ViG. 60. In preparing to make such a boiler, the Plater generally procures a template or pattern plate, the length of which is sSme known proportion of the Bircumference of the boiler, and breadth suitable to ( 185 ) 186 Fig. 61. the plates of iron of which the boiler is to be made; this template is usually a thin sheet of iron or frame of wood, made to the proper length and shape and pierced with holes for the rivets, so that it forms a complete pattern by which , the whole of the plates can be at once drawn and punched. The method of making- the template will be easily understood by the following — Example. — Let it he required to huild a tubular boiler of 36 in., inside diameter, to be formed of 2 plates in the circle, the breadth of the plates 26 inches, and the thickness of plates of an inch, the distance between centres of rivet holes about 2 inches. By the above example, it is clear the frustum of one of the cones composing the boiler will be according to figure 61, as ABGH. The first step will be to find the point or vertex of the cone E, of which the frustum is a part ; this will be found by the following proportion, which is derived from the sixth book of Euclid, viz.: BC : BD::BA: BE, or in words the thickness of the boiler plate is to the radius of the boiler, as the breadth of the plate is to the slant edge of the cone, which will be the radius with which the curve of the outside of the template must be drawn. The carlculation is given at length. 187 As • 1^ • • in. : slant edge of cone = 1497-6. 18 468 ■ 16 5) 7488 1497-6 inches. To 36 inches, the inside diameter of boiler, add ^% of an inch, the thickness of plate, equal to 36^*^ inches, the circumference of which is 114*0793 inches, and as it is proposed to have 2 plates in the circle, the length of each plate will be 57-0896 inches, on the outer curve line. And as the diameter of small end of the frustum is twice the thickness of the plate less than the larger, or 35| inches, to which add thickness of plate, equal to 36]^ inches, the circum- ference of which is 112-1158 inches; the length of plate on the inner curve line will be ^^y~^=z 66-0579 inches, the template therefore, will be as figure 62. Fig. 62. A B = 57 -0396 inches, C I) = 56 0579 inches,' A C'= 26 inches. 188 It will be perceived that as the radius with which the outside of template is to be drawn, is 1497*6 inches, it would be inconvenient to draw a circle with a radius of that length by the common mode; therefore some other method has to be adopted for finding the curve; the one usually taken is to calcu- late the versed line (versed sine) corresponding to the curve of the template, and then trace the curve by means of a thin lath bent round the three points, viz. : — the two extremities of the straight line and the versed line. The mode of calculating the versed line will be easily under- stood by the following process, which is given at length ; but let it be observed in calculating the versed line, it is only necessary to take the circumference of the given diameter, hence the circum- ference of 36 inches is equal to 113-0976 inches, this divided by 2 gives 56"5488 inches, for one of the plates. In the right angled triangle E B G, is given E B = 1497-6 in., BG-= 5-^-^1^ = 28-2744 inches. Then E B2— B G2 is= E ; hence E G^ can be found, and consequently the versed line. 189 28-2t44 inches. 1497-6 inches. 28-2744 1497-6 1130976 89856 1130976 104832 1979208 134784 565488 59904 2261952 14976 665488 — 2242805-76=EB3 799-44169536=B G« 2242805-76=EB2 799-44— BG'' 2242006-32 | 1497-33=EG. 1 24) 124 96 289) 2820 2601 2987) 21906 20909 29943) 99732 89829 299463) 990300 898389 91911 rem. And 1497-60 1497-33 •27 = GH, the versed line, equal to If -we now take the thin plate of whicli the tem- piate is to be made, and draw upon it a straight line, as AD=to 57 inches, whicli bisect by the line EH, 190 and set off GH=^^ths of an inch, the versed line; then by bending a thin rod or lath round the three Fig. 64. points, A H D, the curve of the outside of template may be drawn; the inside line may be drawn in a similar manner by marking off E B, E C, each equal 28 inches, half the length of the inside, and A B, D 0, and H I, equal to 26 inches, the breadth of the plate. To find the breadth of the template for a given dome at any particular place, by calculation. ■Suppose ABC, figure 65, to be the middle section 'of the dome, each plate to be a given portion of ^ part of the whole circumference. Bisect AB at B, and erect a perpendicular as EC ; divide the arc B C into a number of equal parts, and through the points of division, 3, 2, 1, 0, draw lines parallel to A B, and since the breadth of the plate is to be ^ of the whole circumference, it is evident that the breadth at points 3, 2, 1, 0, will also be | of the circumference at these points; you have there- 191 fore only to measure the diameter at those points, and proceed accordingly. Fig. 65. Thus, suppose A B, figure 65, to equal 6 feet di- ameter, you will find that 3 3, is by measurement, 5 feet 7f inches ; and 2 2, is by measurement 4 feet 192 7^ inches; and 1 1, is by measurement 2 feet llf inches; and 0 0, is by measurement 1 foot. Then the circumference at A B, figure 65, will equal 18 feet 10^ inches, divided by 8 gives 2 feet 4J inches for breadth of template at F H, figure 67. The circumference at 3 3, figure 65, will be 17 feet 8f inches, divided by 8 gives 2 feet 2y\ inches for breadth of template at 3 3, figure 67. The circumference at 2 2, figure 65, will be 14 feet 5| inches, divided by 8 gives 1 foot 9^ inches for breadth of template at 2 2, figure 67. The circumference at 1 1, figure 65, will be 9 feet 4|- inches, divided by 8 gives 1 foot 23^ inches for breadth of template at 1 1, figure 67. The circumference at 00, figure 65, will be 3 feet If inches, divided by 8 gives 4t5 inches for breadth of template at 0 0, figure 67. If you then take the thin plate of which the tem- plate is to be made, and draw upon it a straight line as F H, figure 67, and bisect it by a perpendicular I K, and set off on I K, equal distances, as 3, 2, 1, 0, same as on the dome, figure 65, and make the breadth of each point same as found by calculation, you will, upon tracing a line with a thin lath, through these points, have the shape of the template required, eight of which will make the whole dome. Take the fourth part of the circumference of base and it will be the radius for describing the arc line of rivet holes. N. B.— It will be seen by the plan, Fig. 66, that the plates are supposed to be jump jointed, therefore no allowance has been made for lap. Should a dome of the same description be lap jointed, an allowance must be made. 193 T'o find the template for a given Short Egg -end Spherical. Figure 68 is supposed to represent the egg-end, the plates for forming the same are to be over-lap jointed, the breadth of each plate to bear an equal portion of the whole circumference, saj \ part. Fig. 68. Fi«. 70. Fig. 69. The method as laid down in the figures will be very easily understood from the following observa- tions. Suppose the greatest diameter of egg-end to equal 3 feet 111 inches, this divided by 2, gives 1 foot llf inches for radius. The circumference equals 12 feet 4:f inches, divided by 6 gives 2 feet Of inch, breadth of template at the points A B, figure 70, centre to centre of rivet holes. The radius for describing the arc line for rivet holes will be found by dividing the circumference 194 of the base hy 4, which gives 3 feet l^^ inches for radius. In the first place draw a straight line as A B, figure- 68, and bisect A B at E, and erect a perpendicular line E C, then with the radius of 1 foot 11| inches, describe the semicircle ; divide the arc ah c d e into a number of equal parts, and through the points of division ahcde, draw lines parallel to AB until they meet at perpendicular E 0 — this part of the operation is now complete. The semicircle A B, figure 69, is the greatest breadth of template between centres of rivet holes ; bisect the line A B at C, by a perpendicular, divide the arc A D into the same number of equal parts as taken in the arc ahcde, figure68,and draw lines parallel to A B ; having done this, take the sheet iron plate for template, and draw a straight line as A B, and bisect it with a perpendicular, then set off on E F, equal distances, as ahcde, figure 68, and make the breadths at each point same as A B, 2 2, 3 3, 4 4, 5 5, figure 69, drawing the lines parallel to A B ; then bend a thin lath round the points, and through them trace the curve lines of both sides of the template; set the trammel points to 3 feet inches for radius, and fix one of its points at the points A and B, the other on the perpendicular line for a centre, then describe the arc line for rivet holes; having done this, shift the sliding point of trammel until it meets at the points 5 5, keeping the same centre as before, describe the arc line for rivet holes and add the required lap round the template and it will be the shape required. 195 To find the template for a given long Egg-end Para- bola ; the breadth and length of the template at any particular place. Let ahcdef, figure 71, represent the parabola, the plates to be either over-lap or jump jointed, the Fig. 73. Fig. 72. Fig. 71. breadth of each plate to be a given portion of the whole circumference, say \ part. The breadth of the template at each place to be fpund by calculation, as follows: — Diameter of aa=3 feet 6 inches; circumference of aa=10 feet 11| inches- 6=1 foot 9j g inches= breadth of template at points 1 1, figure 72. Diameter of &&= 3 feet 2 inches; circumference of Z>&=9 feet llf in. ^6=1 foot 7| inches= breadth of template at points 2 2. Diameter of cc=2 feet 8 inches; circumference of cc=8 feet 4| in. ^6 = 1 foot 4f inches= breadth of template at points 3 3. 196 Diameter of dd=l foot 11 inches; circumference of £^a?=6 feet \ inch -f- 6=1 foot= breadth of tem- plate at points 4 4. Diameter of e e=l foot ; circumference of e e=3 feet If inches-^ 6= 6:|- inches = breadth of template at points 5 5. Note — The points are the centres of the rivet holes for over-lap joints. Now take the sheet iron plate intended for the template, and draw upon it a straight line, as 11, and bisect it by a perpendicular A B, and set off on A B, equal distances as 1, 2, 3, 4, 5, same as on the parabola, ahcde, figure 71; make the breadth at each point same as found by calculation, then by bending a thin lath round the points and tracing a line through them, will give the shape of the tem- plate required. The radius for describing the arc line of template for rivet holes will be found by the following rule : — Rule — To the square of the ordinate (a g), add I of the square of the abscissa {g /), and the square root of the product will be the length of half the arc, (a fa), and the same length will be the radius required. 197 Example. inches. 21 ordinate a g. 21 21 42 441 square of ordinate a g. 1728 2169 ( 46-5 inches=3 feet 10^ inches 16 [for radius. 86) 569 516 925) 5300 4625 inches. 36=abscissa g f. 36 216 108 1296=square of abscissa g f. 4 3) 5184 n28 = product of f Now, having obtained the radius, set tte trammel points to 3 feet 10 1 inches, and place one of its points to line 1 1, and the other on the perpendicu- lar A for a centre, then describe the arc line for rivet holes ; having done this, contract the sliding point of trammel until it meets at line 5 5, then 9 198 mark the arc line for rivet holes and place the lap round the template if required, and it will be com- plete as to its figure. To find the Templates for a givtn Barrel Spheroid ; the breadths and lengths at any particular place, hy calculation. Let A B C D E F, figure 74, be one-half of the mid- dle frustum of tbe barrel ; the plates for forming the same are to be over-lap jointed, and the breadth Fig. 76. Fig. 74. Fig. 75. of each plate is to bear an equal portion of the whole circumference of each plate, saj ^'(5 pa^t. Thus, the diameter of A B to measure 7 feet 6 inches, the diameter of C D to measure 7 feet 2 199 inches, the diameter of E F to measure 6 feet 5 inches ; then the circumference of A B will equal 23 feet 6f inches, which, divided by 10, gives for breadth of template at 1 1 (figure 75), 2 feet 4:| inches, at that point, between centres of rivet holes. The circumference of CD will equal 22 feet 6 inches, divided by 10, gives for breadth of template at 2 2, 2 feet 3 inches at that point, between centres of rivet holes. The circumference of E F will equal 20 feet 1| inches, divided by 10, gives 2 feet O^^g inch for breadth of template at the point 3 3, between cen- tres of rivet holes. The other half of the frustum will measure the same dimensions as the above calculations. The length of the arc and the versed sine may be found as follows : — Suppose the cord line to measure 88 inches, this divided by 2 gives 44 inches, and the square of 44 inches equals 1936 inches. Find the versed sine by taking the difference be- tween the two diameters A B and E F, figure 74, and divide the remainder by 2 :— Thus AB = 7 feet 6 inches. EF = 6 " 5 " 2) 1 1 0 6| inches=versed siae. Then square the versed sine and add the product to the square of I the cord, and from their united sum take the square root, which gives the length of cord of half the arc A B, figure 77. 200 Examples. Fig. 77. 44 44 176 1Y6 1936=square ^ of cord 42-25 1978-25 ( 44-47=square root. 16 84 ) 378 6-5 336 6-5 884 ) 4225 325 3536 390 8887 ) 68900 42-25 square of 62209 versed sine. Now if we take the square root and multiply it by 8, and from the product subtract the whole cord line, and divide the remainder by 8, we shall have the length of the arc nearly. 44-47 8 Consequently the length of the template, 355-76 figure 75, between centres of the rivet 88 holes 3 and 4, will be 7 feet 5^ inches 3) 267-76 89-25 = 89^ inches. Having completed the dimensions of template for middle frustum we proceed to find the template for the spherical or Egg ends of the barrel. 201 Thus, the diameter of 1 1, figure 74, will' equal 6 feet 4^ inches; the diameter of 2 2, will equal 6 feet 4:| inches ; the diameter of 3 3, will equal 4 feet 9^ inches ; the diameter of 4 4, will equal 3 feet 7| inches; the diameter of 5 5, will equal 2 feet 2^ inches. Then the circumference of 1 1, ec^uals 19 feet 11| inches- 10 gives for breadth of template at 1 1, figure 76, 1 foot inches at that point, centre to centre of rivet holes. The circumference of 2 2, equals 17 feet 10 inches, -rlO gives for breadth of template at 2 2, figure 76, 1 foot 9 1 inches at that point, centre to centre of rivet holes. The circumference of 3 3, equals 14 feet 11 J- inches, -f-10 gives breadth of template at 3 3, figure 76, 1 foot 5 }- 1 inches at that point, centre to centre of rivet holes. The circumference of 4 4, equals 11 feet 3| inches, ~ 10 gives breadth of template at 4 4, figure 76, 1 foot I5 inch at that point, centre to centre of rivet holes. The circumference of 5 5, "equals 6 feet 10 inches, -^10 gives breadth of template at 5 5, figure 76, 8j% inches at that point, centre to centre of rivet holes. N.B. To mark out the template, follow the instruc- tions laid down for the dome, figure 66, page 191, and the template will be the shape required. MENSURATION OF SURFACES. FOUR-SIDED FIGURES. Fig. 78. Fig. 79. Fig. 80. Fig. 81. Square. Rectangle. Rhombus. Rhomboid. To find the area of a four-sided figure, whether it be a square, a rectangle, a rhombus, or a rhomboid. Rule— Multiply the length, A B, or C D, by the breadth or perpendicular height; the product will be the area. Examples. Required the superficial content of a plate of iron, measuring 2 feet 6 inches, on the side of square. Figure 78. 2-5=2 feet 6 inches. 2 5 125 50 6-25 = 6^ square feet. N.B. — For the weight of a square foot of plate iron, according to its thickness — see Table II, page 224. (202) 203 What is the number of supevficial square feet con- tained in a plate of iron measuring 6 feet 9 inches long, by 3 feet 3 inches wide ? Figure 79. 6 75=6 feet 9 inches. 3-25=3 feet 3 inches. 3375 1350 2025 21-9375=21 feet 11^ in., or nearly 22 square feet. TRIANGLES. Fig. 82. Fig. 83. Fig. 84. To find the area of a triangle, whether it be isos- celes, scalene, or right-angled. EuLE— Multiply the base AB, by the perpendic- ular C D, and half the product will be the area. Examples. Required the area of a triangle, whose base is 33 inches, and perpendicular 40 inches ? Figure 82. 32 40 2) 1280 640 square inches=area. How many square feet are there in a triangle, whose base is 8 feet 3 in., and perpendicular 7 feet 6 inches? Figure 83. 204 8-25 = 8 feet 3 inches. •7-5 = 7 feet 6 inches. 4125 6^5 2) 61 •875 30-935=30 sq. ft. 11 inches, nearly. Any two sides of a right-angled triangle being given, to find the third side. EuLE 1.— When the base A B, and perpendicular C B, are given, to find the hypothenuse, or longest side ; to the square of the base add the square of the perpendicular, the square root of the sum will give the longest side. Rule 2. — When the hypothenuse or longest side A 0, and one side are given, to find the other side ; from the square of the hypothenuse subtract the square of the given side, the square root of the re- mainder -will be the side required.' Examples. Given the base A B,=32 inches, the perpendicular B C, = 24 inches, required the length of the longest side A C. Figure 84. 24 32 24 32 96 64 48 96 576 1024 576 1600 (40 in. = longest side. 16 ' The trnth of these rules is evident from the 1st book of Euclid, 47 prop. 205 Given the base A B, 32 inches, and longest side 40 inches ; required the length of the perpendicular B G. Figure 84. 32 40 32 40 64 1600 96 1024 1024 5T6 (24 in. =perpendicular B C. 4 44) 176 176 Note — The diagonal line, or hypothenuse in a square is equal to the square root of twice the square of the side ; and the side of a square is equal to the square root of half the square of its diagonal. Examples. Suppose each side of a square to equal 24 inches ; find the diagonal. 24 in. =AB Fig. 78, page 202. 24 96 48 576 2 1152 (33 94 in. =the diagonal line from A to D. 9 68) 252 189 669) 6300 6021 6784) 27900 27136 764 rem. 206 The diagonal line of a square measures 34 inches ; fiud the side of square equal thereto. 34=A D, Fig. 78, page 202. 34 136 102 2)1156 5T8-(24- inches = side of square. 4 44) lis 176 2 rem. TRAPEZOIDS AND TRAPEZIUMS. Fig. 85. Fig. 86. To find the area of a Trapezoid. KuLE— Add together the two parallel sides A D and B C, multiply their sum by the distance between them— B B, and half the product is the area. To find the area of a Trapezium. jiuL^ — Divide the trapezium into two triangles by the diagonal line A C, and let two perpendiculars, 207 B F and D E, fall on the diagonal from the opposite angles; then multiply the sum of these perpendic- ulars by the length of the diagonal, and divide by 2, which will give the area. Examples. What is the area of a trapezoid whose parallel sides are 7 feet 6 in., and 9 feet 9 inches, the perpendicular distance 1 feet ? Figure 85. 7 -5 = 7 feet 6 inches. 9-75=9 feet 9 inches. 17-25 7 feet 2) 120-75 60-875=Ans. Required the area of a trapezium A B C D, the diago- nal A C 5 feet, D E 4 feet 2 inches, and B P 3 feet 4 inches. Figure 86. BF 4166 = 4 feet 2 inches. DE 3-333 = 3 " 4 7-499 5 feet = AC 2) 37-495 18 7475 = 181 square feet nearly. THE CIRCLE. To find the circumference of a circle when the diameter is given ; or the diameter when the circum- ference is given. Rule 1. — Multiply the diameter by 3-1416 and -the product which will be the circumference ; or 208 divide the circumference by 3'1416 and the quotient will be the diameter. Rule .2. — As 7 is to 22 so is the diameter to the circumference ; and as 22 is to 7 so is the circumfe- rence to the diameter. Examples. Required the circumference of a circle when the diameter is 3 feet. 31416 3 feet=diaineter. 9-4248=9 feet 5 inches. What is the diameter of a circle when the circumfe- rence is 9-4248 feet ? 3-1416 ) 9-4248 ( 3 feet diameter. 9-4248 What is the circumference of a circle when the diant- eter is 36 in. ? as 7 : 22 : : 36 22 72 12 7) 792 113^ inches. =circnm. Required the diameter of a circle when the circumfe- rence is 113^ in. as 22 : 7 : : 113^ inches. 7 22) 792( 36 in. =diameter. 66 132 132 209 To find the area of a circle. Rule 1, — Multiply the square of the diameter by •7854, or the square of the circumference by '07958 ; the product in either case will be the area. Rule 2. — Multiply the circumference by the diam- eter, and divide the product by 4. Examples. Required the area of a circle, the diameter being 3 feet. By rule 1. •t864 9=3^ feet. 7 0686=Ans. What is the area of a circle whe.n the diameter is 3 feet, and circumference 9-4248 feet ? By rule 2. 9 4248 circumference. 3 diameter. 4 ) 28'2U4 r0686 = Ans. The area of a circle given, to find the diameter or ? 6 FEET. o 1 I— 1 I— 1 1— ( I— 1 r-H I— 1 QOi— IOMH«D0500(M0qcD«Ci— liOCCi— (Of- I— li-Hr— 'I— ll— ll— l(N 1— ll— Ir— iCOG0C0C0C■^^- l— ii— li— li— ii— i(M ,— ir-ii— 1 4 FEET. qrs. lbs. oz. T-l 1— 1 — 1 1—1 lOir-t— CSOCNOliOOfOCOOi— ir-(iO-<*ao i"i 1— 1 1— 1 G oooooooooo ooooooo- poB saqonj ni qa,v\ JO aj^ao^ at ssau3iou[j, "i^-^l^ -nl^-ilrH >n'^ .-o|x t-j^ .-flx f'^-^lo-i t-|2Hr:|aoHa pav 83qaai ni q)p«ojg ^ ,— 1 ^ rH 1— 1 en cq (jq (ji cvj 3-1 G^i G^i CO CO 233 WEIGHT OF ORDINARY ANGLED IRON. 1— ( I— 1 o 1—1 CO CO GO CO rH I— 1 CM o 00 CM I— 1 CO GO o 00 CO \ 1 o CO o CO CM cq 1—1 CM 00 CM O r-H o CO CM (M CM CO CO CO CO CM CO CO CO CO >o CO CO lO CO rH CM CO CO CO CO o 1 — 1 GO r-H o CO o JC- CO CO CM 1— I 00 o 00 lO rH rH 00 iC 1—1 00 1—1 Ol Jr- 1—1 CO in CM (X) Os rH CM CO r-l CM CM o 00 CI 00 o 1 — 1 1—1 r-H 1—1 1—1 rH I— I 1—1 CM 1—1 1— 1 CM CM O 1—1 rH o r-H CM 00 o o O rH o O 1— 1 o GO o o O o 1— ( CM i— CM O CM CO CO CM o CM CM OC O CO 1—1 CM CO rH 00 rH CO CM CM lO CM (M O O o I— 1 O 1—1 o o I— 1 rH I— 1 rH o 1— < rH rH O o o o o CO CO I— I o CD o 1—1 o 1—1 GO lO CM CM J— O o CO GO CM o GO o CO 30 (M I— < 1—1 CO rH lO 1—1 1—1 CM lO 00 1—1 t— O CM rH lO 1— 1 CI CO CM CO Jr— CM CM o o o O O o o o O O o O O O o o O o o O o o o o o io|xt~^o|xn|Tjiic|xco|rto|xco|cci^!^w^ Hx Hx rH|00 c-(XeO|'*r-H ^ -H rH l>3|TtiHXlO|X"|-*rH in|XM|TCrH >o|xnHOiO^'^'^'^''^''>''^ rH|-* Wh* CO »0 <0 CO OF THE CIRCUMFERENCES OF CIRCLES, TO THE NEAREST FRACTION OF PRACTICAL MEASUREMENT, ALSO OF THE AREAS OP CIRCLES, IN INCHES AND DECIMAL PARTS, LIKEWISE IN FEET AND DECIMAL PARTS, TOGETHER WITH TABLES OF SIZES OP TINWARE, ETC. ETC. RULES THAT MAY RENDER THE FOLLOWING TABLES MORE GENERALLY USEFUL:— 1. Any of the areas in inches, multiplied by .04328, or the areas in feet multiplied by 6-232, the product is the number of imperial gallons at 1 foot in depth. 2. Any of the areas in feet, multiplied by .03704, the product equals the number of cubic yards at 1 foot in depth. (2.35) 236 CIECTJMFEEENCES AND Via. In inch. Circum. ia inch. Area In sq. inch. Side of = sq. Dia in inch. Cir. in ft. in. Area in sq. ir.ch. Area in Bq. ft. 1-16 1-8 3-16 1-4 5-16 3-8 7-16 -196 -392 -589 -785 -981 1-178 1-374 -0030 -0122 -0276 -0490 -0767 -1104 -1503 -0554 -1107 -1661 -2115 -2669 -3223 -3771 4 in. 4i H 45 41 4| H 42 1 01 1 05 1 15 1 li 1 25 1 21 1 21 1 3i 12- 566 13- 364 14- 186 15- 033 15- 904 16- 800 17- 720 18- 665 -0879 -0935 -0993 -1052 -1113 -1176 -1240 -1306 1-2 9-16 5-8 11-16 3-4 13-16 7-8 15-16 1-570 1-767 1- 963 2- 159 2-356 2-552 2-748 2-945 -1963 -2485 -3368 -3712 -4417 -5185 -6313 -6903 -4331 -4995 -5438 -6093 -6646 -7203 -7754 -8308 5 in. 5| 5i 55 51 5| H 1 35 1 45 1 41 1 4| 1 5^ 1 55 1 6 1 65 19- 635 20- 629 21- 647 22- 690 23- 758 24- 830 25- 967 27-108 -1374 -1444 -1515 -1588 -1663 -1739 -1817 -1897 1 in. H H n ■ n ii 31 3^^ 3| H 41 5| 5i 5| -7854 -9940 1-227 1-484 1- 767 2- 074 2-405 2-761 7 I &3-32 1 in. 1 3-16 1 5-16 1 7-16 1 9-16' 1 11-16 6 in. Gi H 65 61 H Gl 1 6^ 1 n 1 75 1 8 1 85 1 85 1 95 1 91 28- 274 29- 464 30- 679 31- 919 33-183 34471 35-784 37-122 -1979 -2062 -2147 -2234 -2322 -2412 -2504 -2598 2 in. 2i H n n 6i 6| 7 n 8| 9 3-141 3-546 3- 976 4- 430 4- 908 5- 412 5- 939 6- 491 H 2 in. 2 i 2 3-16 2 5-16 2 7-16 2 9-16 7 in. •7' '5 n 75 71 75 75 ' 8 1 10 1 105 1 lOJ 1 115 1 111 1 HI 2 05 2 Oi 38- 484 39- 871 41- 282 42- 718 44- 1 78 45- 663 47- 173 48- 707 -2693 -2791 -2889 -2993 -3392 -3196 -3299 -3409 3 in. 3| H 3^ 31 3| 3i 3| 95 9J lOi m 11 12i 1 7-068 7- 669 8- 295 8- 946 9- 621 10- 320 11- 044 11-793 21 2i 2| 3 in. 3| 35 3 7-16 8 in. 85 8i 85 81 8| H 2 15 2 11 2 15 2 2i 2 25 2 3 2 35 2 31 50- 265 51- 848 53-456 55- 088 56- 745 58-426 60- 132 61- 862 -3518 -3629 -3741 -3856 -3972 -4089 -4209 .4330 AREAS OF CmCLES. 23"; Dia. in C!r. in Area in Area in Cir. In Area in Area of Inch. ft. in. Bq. inch. 8q. ft. ft. iu. 8q. inch. 9 in. 2 63-617 -4453 14 in. 3 7| 153-938 1-0775 9i 2 41 65-396 -4577 14i 3 83 156-699 1-09G8 9i 2 5 6Y-200 -4704 iH 3 8i 159-485 1-1193 9S 2 5? 69-029 -4832 14S 3 9^ 162-295 1-1360 9^ 2 5| 70-882 -4961 14^ 3 9^ 165-130 1-1569 9| 2 H 72-759 -5093 14| 3 91 107-989 1 1749 H 2 6i 74-662 -5226 143 3 lOi 170-873 1-1961 9i 2 1 76-588 -5361 141 3 103 173-782 1-2164 10 in. 2 78-540 -5497 15 in. 3 lU 176-715 1-2370 2 'ii 80-515 -5636 15i 3 11^ 179-672 1-2577 m 2 8i 82-516 -5776 15i 3 111 182-654 1-2785 2 8h 84-540 -5917 15^ 4 Oi 185-661 1-2996 lOJ 2 8| 86-590 -6061 15^ 4 01 188-692 1-3208 I0| 2 9i 88-664 -6206 15i 4 1 191-748 1-3422 l.OJ 2 H 90-762 -6353 155 4 1^ 194-828 1-3637 10| 2 101 92-855 -6499 15| 4 1| 197-933 1-3855 11 in 2 10^ 95-033 -6652 16 in. 4 2i 201-062 1-4074 ni 2 10| 97-205 -6874 16J 4 2| 204-216 1-4295 Ui 2 111 99-402 -6958 16J 4 3 207-394 1-4517 Hi 2 115 101-623 -7143 163 4 3| 210-597 1-4741 111 3 0| 103-869 -7290 16i 4 33 213-825 1-4967 111 3 0^ 106-139 -7429 16| 4 41 217-077 1-5195 iij 3 0^ "a 108-434 -7590 163 4 ^ 220-353 1-5424 * ■'•8 3 1| 110-753 -7752 4 5 223 654 1-5655 12 in. 3 1| 113-097 -7916 17 in. 4 55 226-980 1--5888 121 3 2 115-466 -8082 Hi 4 53 230-330 1-6123 1.2i 3 2J 117-859 -8250 in 4 6i 233-705 1-6359 12^ 3 2| 120-276 -8419 17g 4 6i 237-104 1-6597 3 31 122-718 -8590 4 6| 240-528 1-683S 12| 3 ^8 125-185 -8762 171 4 n 243-977 1-7078 3 4 127-676 -8937 in 4 73 247-450 1-7321 12| 3 ^ 130-192 -9113 1 17J 4 81 250-947 .1-7566 13 in. 3 132-732 -9291 18 in. 4 8i 254-469 1-7812 13^ 3 5i 135-297 -9470 181 4 8J 258-016 1-8061 13i 3 5f 137-886 -9642 I8i 4 9i 261-587 1-8311 13^ 3 6 140-500 -9835 182 4 93 265-182 1-8562 13^ 3 6s 143-139 1-0019 18J 4 m 268-803 1-8816 13| 13| 3 145-802 1-0206 18| 4 lOi 272-447 1-9071 3 148-489 1-0294 18-i 4 101 276-117 1-9328 13J 3 n 151-201 1-0584 18i 4 Ui 279-811 X-95P& 238 CmCUMFEKENCES AND Area in Area in Dia. in Cir, in Area In Area la ft. sq. inch, | sq. ft. ft. ft. in. sq. inch.. •q. ft. 4 115 283-529 1-9847 2 0 6 3^ 452-390 3-1418 5 0 287-272 1-9941 2 O3 6 4^ 461-8641 3-2075 5 Ok 291-039' 2-0371 2 6 ^ 471-436 3-2731 5 0^ "8 294-831 2-0637 2 O3 6 481-106 3-3410 5 1? 298-648 2-0904 2 1 6 6^ 490-875 3-4081 5 ^ 8 302-489 2-1172 2 I3 6 n 500-741 3-4775 6 2 306-355 2 1443 2 6 510-706 3-5468 5 25 *'8 310-245 2-1716 2 U 6 81 520-769 3-6101 5 314-160 2-1990 2 2 6 9^ 530-930 3-6870 5 318-099 2-2265 2 2? 6 101 541-189 3-7583 5 322-063 2-2543' 2 25 6 IH 551-547 3-8302 5 i 326-051 2-2822 2 2? 7 0 562-002 3-9042 5 330-064 2-3103 2 3 7 OJ 572-556 3-9761 5 334-101 2-3386 2 3? 7 Is 583-208 4-0500 5 *^ 8 338-103 2-3670 2 3^ 7 2^ 593-958 4-1241 5 342-250 2-3956 2 3? 7 3^ 604-807 4-2000 5 51 345-361 2-4244 2 4 7 3| 615-753 4-2760 5 ^8 ^ a 330-497 2-4533 2 4? 7 43 626-798 4-3521 5 6} 354-657 2-4824 2 4J 7 51 637-941 4-4302 5 358-841 2-511 7 2 4i 7 6i 649-182 4-5083 5 7^ 363-051 2-5412 2 0 7 7 660-521 4-5861 5 7| 357-284 2-5708 o o.'f 7 7^ 671-958 4-6665 5 H 371-543 2-6007 2 5-' 7 85 683-494 4-7467 5 H 375-826 2-6306 2 54 7 9i 695-128 4-8274 5 380-133 2--6608 2 6 7 lOi 706-860 4-9081 5 384-465 2-6691 2 6i 7 11 718-690 4-9901 5 388-822 2-7016 2 7 II4 730-618 5-0731 5 393-203 2-7224 2 65 8 0} 742-644 5-1573 5 lOf 397-608 2-7632 2 7 8 754-769 5-2278 5 11 402-038 2-7980 2 8 21 766-992 5-3264 5 m 406-493 2-8054 2 n 8 2'-. 779-313 5-4112 5 410-972 2-8658 2 n 8 3i 791-732 5-4982 6 Oi 415-476 2-8903 2 8 8 41 804-249 5-5850 6 0^ 420-004 2-9100 2 8i 8 5; 816-865 5-6729 6 "8 1 424-557 2-9518 2 8^ 8 Gl 829-578 5-7601 6 429-135 2-9937 2 8 Gl 842-390 5-8491 6 13 433-737 3-0129 2 9 8 7", 855-3001 5-9398 6 2i 438-353 3-0261 2 9i 8 8i 868-338 6-0291 6 2t 443-014 3-0722 2 9i 8 9! 881-415 6-1201 6 3 447- 69C 0-1081 2 9J 8 10 894-61.9 6-2129 ABEAS OF CIRCLES. 239 Din. la Cir. ta Ar6& in Area in Ditt in Cir. in Area in Area ia ft. in. ft. in. sq. inch. sq. ft. ft. in. fi. in. sq. inch. sq. ft. o g 907-922 6-3051 3 g 11 61 1530-53 10-559 2 g 115 921-323 6-3981 3 8i 11 7 1537-86 10-679 2 Q 03 "5 934-822 6-4911 3 8J 11 72 1555-28 10-800 O Q i7 1 1 is 948-419 6-5863 3 8? 11 8^ "a 1572-81 10-922 2 1 1 9 962-115 6-6815 3 9 11 ^ 8 1590-43 11-044 2 1 1 1 975-908 6-7772 3 91 11 10| 1608-15 11-16'? 2 m 9 3J 989-800 6-8738 3 9,1 11 101 1625-97 11 291 2 Hi 9 H 1003-79 6-9701 3 H 11 lit 1643-89 11-415 Q Q 5 1017-87 7-0688 3 10 12 Oi 1661-90 11-534 3 q ^8 1032-06 7-1671 3 101 12 11 1680-02 11-666 o o ni "5 Q i7 05 1046-35 7-2664 3 10^ 12 2 1698-23 11-793 3 q •71 '2 1060-73 7-3662 3 lO? 12 ^8 1716-54 11-920 3 q 1075-21 7-4661 3 11 12 35 1734-94 12-048 3 1 1 q q 1089-79 7-5671 3 111 i J. 4 12 43 1753-45 12-176 3 \\ 9 9| 1104-46 7-6691 3 12 H 1772-05 12-305 3 9 10^ 1119-24 7-7791 3 115 12 6 1790-76 12-435 3 2 g 1 1 3 1134-12 7-8681 4 0 12 63 1809-56 12-566 3 9i 10 01 1 149-09 7-9791 4 01 12 71 1828-46 12-691 3 '),^- "t 10 OJ "5 1164-16 8-0846 4 Qi 12 83 1847-45 12-829 3 10 i? 1179-32 8-1891 4 04 12 93 1866-55 12-962 3 3 10 oi 1194-59 8-29511 4 1 12 95 "^8 1885-74 13-095 3 04 10 1209-95 8-4026 4 12 lOJ 1905-03 13-229 3 3^ 10 4 1225-42 8-5091 4 12 1924-42 13-364 3 31 10 4i 1240-98 8-6171 4 1? 13 01 1943-91 13-499 3 4 10 5t 1256-64 8-7269 4 2 13 1 1963-50 13-635 3 4i 10 61 1272-39 8-8361 4 21 13 n 1983-18 13-772 3 4J 10 n 1 288-25 8-9462 4 2i 13 n 2002-96 13-909 3 4i 10 8 1304-20 9-0561 4 H 13 35 2022-84 14-047 3 5 10 81 1320-25 9-1686 4 3 13 41 2042-82 14-186 3 5| 10 9^ 1336-40 9-2112 4 31 13 5 2062-90 14-325 3 5i 10 lot 1352-65 9-3936 4 3J 13 H 2083-07 14-465 3 5i 10 Hi 1369-00 9-5061 4 3i 13 6i 2103-35 14-606 3 6 10 Hi 1385-44 9-6212 4 4 13 ~i 2123-72 14-748 3 61 11 05 1401-98 9-7364 4 41 13 2144-19 14-890 3 6J 11 1418-62 9-8518 4 4J 13 2164-75 15-033 3 65 11 1435-36 9-9671 4 4J 13 2185-42 15-176 3 7 11 3 1452-20 10-084 4 5 13 m 2206-18 15-320 3 11 3| 1469-14 10-202 4 51 13 111 2227-05 15-465 3 11 4f 1486-17 10-320 4 b\ 14 0 2248-01 15-611 n 11 5S 1503-30 10-439 4 H 14 o\ 2269 06 15-757 240 CIRCUMFEKENCES AND Dia. in Cir. in Area in Area in Dia in Cir. in Area in Area m -t. in. ft. iu. aq. inch. sq. ft. ft. in. n. in. sq. inch. sq. 11. 4 6 14 I5 2290 22 15-904 5 4 16 9 3216-99 22-333 4 ei 14 25 2311-48 16-051; 5 4? 16 93 3242-17 22-51& i ei 14 3i 2332-83 16-200: 5 41 16 10| 3267-46 22-621 4 61 14 4 2354-28 16-349; 5 43 16 Hi 3292-83 22-866 4 1 14 43 2375-83 16-498 1 5 5 17 oi 3318-31 23-043 4 U 14 5i 2397-48 16-649, 5 5| 17 0| 3343-88 23-221 4 n 14 6J 241 9-22 16-800 5 5-J 17 13 3369-56 23-330 4 n 14 7s 2441-07 16-951 5 53 17 2h 3395-33 23-578 4 8 14 71 2463-01 17--104 5 6 17 3i 3421-20 23-758 4 H 14 85 2485-05 17-257 5 6? 17 4i 3447-16 23-938 4 14 9| 2507-19 1 7-41 1 5 65 17 4| 3473-23 24-119 4 14 10? 2529-42 17-565 5 6? 17 Ss 3499-39 24-301 4 9 14 11 2551-76 17-720 5 7 17 6h 3525-26 24-485 4 H 14 II5 2574-19 17-876 5 1-1 17 73 3552-01 24-666 4 15 Of 2596-72 18-033 5 17 8 3578-47 24-850 4 H 15 Is 2619-35 18-189 5 71 17 83 3605-03 25-034 4 10 15 2i 2G42-08 18-347: 5 8 17 91 3631-68 25-220 4 lOi 15 2| 2664-91 18-506i 5 8| 17 log 3658-44 25-405 4 15 3? 2687-83 18-665: 5 83 17 IH 3685-29 25-592 4 103 15 4^ 2710-85 18-825; 5 83 17 111 3712-24 25-779 4 11 15 51 273,3-97 18-9:5 5 9 18 03 3739-28 25-964 4 ii-i 15 61 2757-19 19-147 5 18 1^ 3766-43 26-155 4 lU 15 6| 2780-51 19-309 5 18 23 3793-67 26-344 4 113 15 2803-92 19-471 5 93 18 3i 3821-02 26-534 5 0 15 2827-44 19-635 5 10 18 3| 3848-46 26-725 5 Oi 15 9i 2851-05 19-798 5 10^ 18 4i 3875-99 26-916 5 01 15 10 2874-76 19-963 5 101 18 5^ 3903-63 27-108 5 03 15 103 2898-56 20-128 5 103 18 63 3931-36 27-301 5 1 16 HI 2922-47 20-294 5 11 18 7 3959-20 27-494 5 16 Of 2946-47 20-461 5 Hi 18 73 3987-13 27-688 5 U 16 n 2970-57 20 629 5 m 18 8| 4015-16 27-883 ■ 5 H 16 If 2994-77 20-797 5 113 18 95 4043-28 28-078 5 2 16 2:3 3919-07 20-965 6 0 18 lOi 4071-51 28-274 5 H 16 3i 3043-47 21-135 6 03 1"! 4099-83 28-471 5 2h 16 4i 3067-96 21-305 6 11 113 4128-25 28 663 5 H 16 5i 3092-56 21-476 6 03 19 0^ 4156-77 28-866 5 3 16 5| 3117-25 21-647i 6 1 19 13 4185-39 29-065 5 3i .13 6i 3142-04 21-819 6 u 19 21 4214-11 29-264 5 3^ 16 7^13166-92 2l-992i 6 11 19 2J 4242-92 29-466 5 3£ 16 8i 3191-91 22-166 6 ■13 19 31 4271-83 29-665 AREAS OF CIKCLB8. 241 Dis. in (Sr. In Area in Ar6a in Dia. in Ci r in Afda in AreA in ft. in. ft. in. &q. inch. eq. ft. ft in. ft ' in. sq. inch. sq. ft. 6 2 19 ^ 4300-85 29-867 6 8 20 m 5026-26 34-906 6 2? 19 4329-95 30-069 6 H 21 5058-02 35-125 6 2i 19 6 4359-16 30-271 6 81 21 5089-58 35-344 6 H 19 6? 4388-47 30-475 6 81 21 n 5121-24 35-564 € 3 19 •^1 441T-87 30-679 6 9 21 5153-00 35-784 6 3i 10 85 4447-37 30-884 6 9? 21 3k 5184-86 36-006 6 3J 19 9^ 4476-97 31-090 6 9^ 21 4 5216-82 36-2 2 T 6 3^ 19 9| 4506-67 31-296 6 9^ 21 5248-87 36-450 6 4 19 loi 4536-47 31-503 6 10 21 5281-02 36-674 6 4i 19 Hi 4566-36 31-710 6 101 21 6^ 5313-27 36-897 6 4i 20 Oi 4596-35 31-919 6 lOi 21 n 5345-62 37-122 6 4J 20 H 4626-44 32-1 14 6 103 21 n 5378-07 37-347 6 5 20 U 4656-63 32-337 6 11 21 8J 5410-62 37-573 6 5i 20 4686-92 32-548 6 ill 21 5443-26 37-700 6 20 3i 4717-30 32-759 6 Hi 21 5476-00 38-021 6 5i 20 4i 4747-79 32-970 6 Hi 21 11 5508-84 38-256 6 6 20 5 4778-37 33-183 6 6i 20 4809-05 33-396 6 6i 20 6i 4839-83 33-619 6 6J 20 75 4870-70 33-824 6 20 81 4901-68 34-039 € 20 81 4932-75 34-255 6 20 H 4963-92 34-471 H 20 lOi 4995-19 34-688 242 CIRCUMFERENCES AND Bin. \n CIrcum, in • Area in ft* Diam in Circum, in Area in ft* ft. aud in. ft, and in. ft. aud in. ft. and in. 7 0 21 111 38-4846 10 0 31 5 78-5400 1 22 3 39-4060 1 31 81 79-8540 2 22 6s 40-3388 2 31 ll? 81-1795 3 22 9i 41-2825 3 32 2? 82-5190 4 23 0? 42-2367 4 32 5| 83-862T 5 23 2i 43-2022 5 32 8| 85-2211 6 23 44-1787 6 32 111 66-590a 23 11 45-1656 7 33 2| 87-9697 8 24 i| 4i 46-1638 8 33 Gi 89-3668 9 24 47-1 730 9 33 n 90-7627 10 24 u 48-1926 lU 34 0? 92-1749 11 24 lOg 49-2236 11 34 3i 93-5986 8 0 25 U 50-2656 11 0 34 61 95-0334 1 25 4| 51-6178 1 34 9if 96-4783 2 25 52-3816 2 35 01 97-9347 3 25 11 53-4562 3 35 4i 99-4021 4 26 2^> 54-5412 4 35 U 100-8797 5 26 5i 55-6377 5 35 101 102-3689- 6 26 8J 56-7451 6 36 U 103-8601 V 26 in 5 7 -o 0^8 7 36 1 UD-O 8 27 H 58-99^0 8 36 n 106-9013 9 27 60-1321 9 36 101 108-4342 10 27 9 61-2826 10 37 n 109-9772 11 28 0| 62-4445 11 37 111-5319 9 0 28 3i 63-6174 12 0 37 8§ 113-0976 1 28 6? 64-8006 1 37 Hi 38 2t 114-6732 2 28 9J 65-9951 2 116-2607 3 29 0| 67-2007 3 38 H 117-8590 4 29 3J 68-4166 4 38 8| 119-4674 5 29 7 69-6440 5 39 0 121-0876 6 29 10^ 70-8823 6 39 H 122-7187 7 30 H 72-1309 7 39 6^ 124-3598 8 30 4i 7S-3910 8 39 9i 126-0127 9 30 7i 74-6620 9 40 ot 127-6765 10 30 111 75-9433 10 40 129-3504 11 31 n 77-2362 ) 11 40 131-0360- AREAS OF CIECLES. 243 !>!■. In Clrcnm. in Dwm in Circuni, In ft, and in. ft. and in. ft. and in. ft. and iu. 13 0 40 10 132-7326 L6 0 50 3| 1 41 U 134-4391 1 50 6i 2 41 4f 136-1574 2 50 9| 3 41 1h 137-8867 3 51 Oi 4 41 10| 139-6260 4 51 3i 5 42 li 141-3:71 5 51 6i 6 42 41 143-1391 6 51 10 >l 42 8 144-9111 7 52 U 8 42 lli 146-6949 8 52 4i 9 43 2i 148-4896 9 52 7| 10 43 51 150-2943 10 52 10^ 11 43 8^ 152-1109 11 53 1| 14 0 43 11-2 153-9484 17 0 53 4i 1 44 2| 155-7758 1 53 8 2 44 6 157-6250 2 53 11^ 3 44 9i 159-4852 3 54 21 4 45 OJ 161-3553 4 54 5| 5 45 31 163-2373 5 54 8J 6 45 6| 165-1303 6 54 lit 7 45 95 167-0331 i 00 ^ a 8 46 0| 4Q 4 168-9479 8 55 6 9 170-8735 9 55 91 10 46 1i 172-8091 10 56 OJ 11 46 111 174-7565 11 56 3i 15 0 47 U 176-7150 18 0 56 6J 1 47 4| 178-6832 1 56 9| 2 47 7| 180-6634 2 57 0| 3 47 10| 182-6545 3 57 4 4 48 21 184-6555 4 57 n 5 48 5i 186-6684 5 57 lOi 6 48 8i 188-6923 6 58 It 7 48 HI 190-7260 7 58 4i 8 49 2| 192-7716 8 58 7| 9 49 53 194-8282 9 58 105 10 49 8| 196-8946 10 59 2 11 50 0 198-9730 11 6S> 5i 201-0624 203-1615 205-2(26 207-3946 209-5264 211-6703 213-8251 215-9896 218-1662 220-3537 222-5510 224-7603 226-9806 229-2105 231-4625 233-7055 235-9682 238-2430 240-5287 242-8241 245-1316 247-4500 249-7781 252-1184 254- 256- 259- 261- 263- 266- 268- 271- 273- 276- 278' 281- 4696 8303 2033 •5872 ■9807 •38G4 ■8031 ■2293 •6678 ■1171 ■5761 •0472 344 SIZES OF TlN-AfAUTI. ■Sizes of Tin-ware in form of F' rui^trxim of a cone Diarn. of top. Dlam, of bot. Height. Size. 2 qt. 3 pt. 1 " Pie Dlam. of top. Dlam. of bot. Helgiit 20 qt. 16 " .14 " 10 " 6 " 19| in 18 15i 14J 12f 13 in Hi H 11 9 8 in H H 9 in 8} 6J- 9 6 in 4 Si in 2f If DISH KETTLES AND PAILJ. Size. DIam. of lop: Diam. of bot. Height. Size. Diam, of top. Diam, of bot. Height. 14 qt. 10 " 13 in Hi 9 in 1 9 in 8 6 qt. 2 " 9i- in 5J- in 4 6Jin 4 COFFEE POTS Siie. Dtsm. of top; Diam, of bot. Height. Size, of top; Dlam. of bot. Heigh., 1 gal. 4 in 1 in 8i in 3 qt. 3^ in 6 in H in WASH BOWLS. Size Diam. Diam. Height of top. of boU 11 in. 6^ in. 5 in 11 51 5 Small Wash Bowl 9i H 33 91 5i 33 SIZES OF TIN-WAEE. DIPPERS. Slie, DIsm, of top; Diam, of bot. Height. Size, Diam, of top; Diam; of bot. Height; i gal. 6i in 4 in 4 in 1 pt. 4i in 3f in 2| in UEASCREg. Diam, of top ; Clam, of bot. Height, Size. Diam, of top; Biam, of bot, Height, 1 gal. 1 qt. in 4 H H in 4 H in 8 53 1 pt. i " 2i in 2f H in 2^ 4iin H DBUGGISTS' AND LIQUOR DEALERS' MEASUBES, Size. Diam. Diam, Height. Size, Diam. Diam. Height of top. of bot. of top. of bot. 6 gal. 8 in 13^ in 12| in Tgal 3i in Gl in 6 in 3 " 1 Hi 1 qt. 2i H 2 " 6 lOf 1 pt. 2 4 1 " 34 8i n i " n u H 12 246 TABLES OP WEIGHT, ETC. American Lap Weled Iron Boiler Flues, Manufac- tured by the Reading Iron Company. Oitside W. G. Dinnieter. Nos, U in- 16 H 15 n 14 2 13 12 12 11 3 11 Weight per Fool, about, 1 lb. 1 MO H 2 ^ 2f 3i 3i Outside Diameler, 3i 4 5 6 7 W, G, 11 10 10 Weight per Foot, aboi.t 4 H n 10 13 TaUe of Effects upon Bodies by Heat. FAHRhNHEir 2754" 1983° 1850° 2160° 1900° 740° 594° 476° 421° 28^.° Tin 3 parts Bismuth 5 and Lead 2 melt . . - 212<* INDEX. Acute angle, 23, 150 angled triangle, 152 Alloys and their melting-points, 130 Altitude of a triangle, 152 Angle, 16, 23, 149, 150 to bisect a, 161, 162 to trisect a, 162 Angled iron, weight of, 232 Angles, 149, 150 measuring of, by instru- ments, 159-161 Anvils, 114 Apex of the triangle, 152 Arc, 16 of a circle, 156 Arch, ogee, to draw, 85 Area of circle, 209 of ellipse or oval, 211 of four-sided figures, 202 of trapezoids and trape- ziums, 206 of triangles, 203 Ascertaining weight of iron pipes, 231 Axes, 27 Barrel, spheroid, template for, 198 Base, 151 hexagonal, 87 Bath, hip, 96 oblong, tapering, 66 round end, 95 sitz, 98 Beam iron, 221 Bisect a line, 13, 157 Blow-pipe, 134, 135 Boiler making, rules for, 185 plates, weight of, 224 (2 Borax, 138 Brass, solder for, 129 Brazier's hearth, 136 Brazing, 131 Breadth of template, 190 Britannia metal, solder for, 129 Burning together, 129 Butt joint, 117 Candlestick, flue oblong, 35 Centre of circle, to find, 15, 158 Chain lines, 148 Chord, 20, 156 Circle, 16, 155, 156 area of, 137 to draw through three points, 15 to find centre, 15, 158 Circular work of galvanized iron, 124-126 Circumference of circle, 207 of ellipse, 210 straight line equal to, 14 Compasses, 146 Compound curve, 148 Concentric, 148 Cone, 31 and cylinder to intersect at right angles, 80 development of the, 173- 177 elliptic base, 92 oblique to its base, 79 transected, section pre- sented by a, 177-180 Cones, 215 Conjugate axes, 27 Convex surface of cylinders, 214 of frustrum of right cone, 217 :7) 248 INDEX. Convex surface of globe, 218 of right cone, 217 Copper, solder for, 129 soldering bit, 131 tinning of, 132 Copper works, soldering of, 134 Cover of a tureen, 106 Covering of an ogee dome, 85 Cramp joint, 119, 120 Creasing tool, 114 Cube, development of the, 166, 167 solidity of, 212 superficies of, 212 Curve, 16, 148 to find, chord and height of segment given, 17 to find radius, 15 Curves, drawing, 72 Cylinder, development of the, 142, 143 sections obtained by cut- ting a, 170 Cylinders, 213 convex surface of, 214 Cylindrical section through any given angle, 84 Definition of a parallelogram, 23 of polygons, 22 Diagonals, 23 Diagrams, tools required for drawing, 146 Diameter, 16 of circle, 156, 207 Diameters of ellipse, 27 Diamond-shaped, tapering body in one piece, 50 Divide line into number of parts, 14 Dividers, 146 Dodecahedron, 168 Dome, covering of an ogee, 85 Dotted lines, 148 Double elbow, 71 Drawing board, 146 pins, 146 Duodecagon vase, 90 I Egg-end parabola, template for, 195 -shaped oval tapering body, 41, 42 Elbow at right angles, 70 double, 74 in round pipe at any angle, 71 Ellipse, 27 arc of, 211 circumference of, 210 drawn with string and pencil, 40 form of, 172 to draw with the trammel, 26 to strike, 23, 25, 26 within a square, 24 Equilateral triangle, 151 Fine lines, 148 Flat iron, malleable, weight of, 227 Flue article, pattern for, 28 height of a body, 30 oblong candlestick, 35 Fluxes, 130 Folded angle joint, 118 Four-sided figures, 202 Frustrum of a cone, 31 pattern for, 28 of cone parallel with base, 93 solidity of, 215 of pyramids, solidity of, 216 Galvanized iron, 123-126 Gas, use of, in conjunction with the blow -pipe, 139 Geometry as applied to sheet metal working, 140-183 Gold, solder for, 129 Gutters of zinc, 121, 122 Hammers, 114 Hard soldering, 128 Hatchet stake, 114 Heat, modes of applying, 130 INDEX. 249 Hemisphere, development of the covering of a, 182, 183 Heptagon, to draw, 21 Hexagon mould in one piece, 62 on a given side, 20 Hexagonal base, 87 Hip bath, 96 _ Holliper or diver, 114 Hollow crease, 120 Horizon, definition of, 149 Horizontal line, 149 Hot air blast, 135 Hy pot hen use, 151 ICOSAHEDRON, 169 Indefinite line, 20 Intersection, point of, 16 Iron, angled, weight of, 232 solder for, 129 weight of, 221 Irregular octagon pan, 104 one piece, 53 polygon, 22, 155 Isosceles triangle, 151 Jack screen, top of, 110 Joints, 117-121 Kite, 154 Lap joint, 117 without solder, 120 Lead, solder for, 129 Line, definition of a, 147, 148 divide in number of parts, 14 indefinite, 20 straight, equal to part of circle, 21 to given line, 13 to bisect, 13 to draw a, parallel to a given line, 158 vertical, 16 Lines, 147-149 Lobster back cowl, 77 Lozenge, 23 Major axis, 27 Malleable flat iron, weight of, m ^ iron pipes, weight of, 231 round iron, weight of, 225 square iron, weight of, 226 Mallets, 114 Mensuration of solids, 211 of surface, 202 Metal, thin, bending to curves of, 116 _ Minor axis, 27 Mitre joint, 117 at right angles in a half-round gutter, 144, 145 Mouldings, shaping of, 124 Oblique cone, 93 line, 149 Oblong, 153 pan, round corners, 54 tapering bath, 68 pan in sections, 33 Obtuse angle, 23, 150 -angled triangle, 152 Octagon in a given square, 21 pan, 104 in one piece, 53 shape vase, 68 to inscribe a, in a circle, 164 Octahedron, 167, 168 Ogee arch, to draw, 85 dome, covering of, 85 Oliver or holliper, 114 Oval, 27, 30 area of, 211 canister top, 60 circumference of, 210 drawn with siring and pencil, 40 tapering, in four parts, 37 in one piece, 38 unequal, 44 to strike, 23, 25 Parabola, egg-cnJ, template for, 195 to draw, 25 ■250 INDEX. Parallel line, 16 Parallelogram, definition of, 23 solidity of, 213 superficies of, 212 Parallels, 149 Pattern for cone and cylinder to intersect at right angles, 80 . for cone, elliptic base, 92 for cover and neck of ir- regular octagon pan, 106 for diamond-shaped taper- ing body in one piece, 60 for hexagon mould, 52 base, 87 for hip bath, 96 for lobster-back cowl, 77 for oblong tapering batli , 66 for rectangular base and bottom in one piece, 86 for round end bath. 95 for round pipe to form a semi-circle for connec- tions, 78 for round, tapering, or flue article, 28 for square tray, 52 for tapering ailicle, oval at base and round at top, 60 for tapering oblong in one piece, 35 for tapering octagon in one piece, 49 for tapering oval in four sections, 37 for tapering sides of tray, having various curves, 64 for tapering top, 56 for top of jack-screen, 110 for T-piece to join two cylinders at right angles, 73 for traveling sitz bath, 98 for vase, octagon shape, 88 for vase, twelve sides, 90 -of article, straight sides and semi-circular ends, 32 Pattern of cone, oblique to base, 79 of egg-shaped oval taper- ing body, 41, 42 of elbow at right angles, 70 of elbow in round pipe at any angle, 71 of irregular octagon pan, 53 of oblique cone, 93 of oblong pan, round corn- ers, 54 of oblong tapering pan in sections, 33 of slightly tapering article, 103 of square funnel, 50 of square tapering article, 48 _ _ of tapering pipe to join two upright cylinders to form a double elbow, 71 of two cylinders for joining at oblique angles, 74 Patterns, cutting of, with cold chisel and hammer, 113 for sides of irregular octa- gon pans, 104 Pentagon, to describe, 19 Perpendicular, 149 meaning of, 16 to erect, 16 to raise arc of circle given, 18 Pewter, solder for, 1 29 Pipe, round, to form semi- circle, 78 Pipes, iron, weight of, 231 Plain scale, to construct, 14 Plan of round flue body to be cut in three, 29 Plate iron, weight of, 231 Point, definition of a, 147 Polygon, 154, 155 regular, to inscribe a, in a circle, 163, 164 to describe, 19 Polygons, definition of, 22 Polyhedrons, development of, 164 INDEX. 251 Prisms, 213 Protractor, 146 Pyramid, 48 Pyramids, 215 ■Quadrilateral figures, 153- 183 Eadius, 16 of circle, 156 of curve, to find, 15 to obtain for pattern of a slightly tapering article, 103 Hectangle, 23, 153 area of, 202 Rectangular base and bottom in one piece, 86 tapering tray, 52 Regular polygons, 22, 154, 155 lihomboid, 154 area of, 202 Rhombus, 23, 153, 154 area of, 202 Right angle, 16, 150 -angled triangle, 151 line, 148 Riveted joint, 118, 119 Roll joint, 120 Round end bath, 95 flue body, to cut in three, 29 iron, weight of, 221 pipe, with elbow at right angles, 70 tapering article, pattern, 28 Rules for boiler making, 185 for weight of iron bodies, 221 Scale, to construct a plain, 14 Scalene triangle, 151 Seam -set, 114 Sector of a circle, 156 Segment, 20 of a circle, 156 to strike, 17 Segments of spheres, 218 Semi-circular pipe, 79 Shears, 113 Sheet iron, weight of, 231 metal work processes, 113- 183 working, 113-126 geometry as ap- plied to, 140- 183 Silver, solder for, 129 Simple curve, 148 Sitz bath, 98 Slanting height of a body, 30 Soft soldering, 128 Soldering, 127-139 iron, 131 Solid content of globe, 218 spherical seg- ment, 220 to obtain the stretcli-out of a, 165 Solidity of cone or pyramid, 215 of cube, 212 of cvlinders and prisms, 214 of frustrum of cone, 215 of pyramid, 216 of parallelogram, 213 Solids, development of, 141 mensuration of, 211 regular, development of, 164 regularly symmetical,mode of obtaining, 170 Speculum metal, solder for, 129 Spelter, 138 Sphere, development of the, 180-182 Spheres, 218 Spherical egg-end, template for, 195 segment, solid content, 220 Spheroid, template for, 198 Square, 23, 146 area of, 202 funnel, one side straight, 50 iron, malleable, weight of, 226 weight of, 221 tapering article, 48 252 INDEX. Square tray, 62 Stakes, 114 Steel, weight of round cast, 222 weight of square cast, 221 Straight joints, 119 line, 148 line equal to circumference, 14 line equal to part of a circle, 21 line parallel to given line, 13 line, to bisect a, 157 sides and semi-circular ends, 32 Stretch-out of a semi-circle, to find,' 22 -out of a solid, how to ob- tain the, 165 Strong lines, 148 Superficies of cube, 212 of parallelogram, 212 Surfaces, mensuration of, 202 Tables of weights, 224, 225, 226, 227, 228 Tangent, 20, 15C to a circle, to draw, 18 Taper stake, 114 Tapering article, oval at base and round at top, 60 octagon in one piece, 49 oval in four sections, 37 oval in one piece, 38 oval, tapering unequal, 44 pan, oblong, 33 sides of tray having var- ious curves, 64 top, 56 Template making, 185 Tetrahedron, 165 Tin or sheet iron box, hexa- gonal, development of a, 141, 142 Tools for drawing diagrams, 146 Top of jack screen, 110 Trammel, ellipse to draw with, 26 Transverse axes, 27 Trapezion, 154 Trapezium, 154 area of, 206 Trapezoid, 154 area of, 206 Tray, shallow, 115, 116 Triangles, 23, 150-153 area of, 203 Tubular boiler, 186 Tureen, pattern for, 106 VAiiiE, C. A., observations by, on circular work, 124-126 Vase, octagon shape, 88 twelve sides, 90 Vertex of the angle, 150 Vertical angle of the triangle, 152 line, 16 Weight of a foot of plate or sheet iron, 231 of angled iron, 232 of boiler plates, 224 of malleable flat iron, 227 of malleable round iron, 225 of malleable square iron, 226 of round iron, 221 of square iron, 221 of steel cast, 221, 222 of wrought iron and steel, 224 Wiped joint, 134 Zinc, 121-123 solder for, 129 soldering of, 122, 123 (DJsJTJLlljOOrTTm OF practical and Scientific Boo^^ PUBLISHED BY Henry Carey Baird & Co. INDUSTRIAL PUBLISHERS, BOOKSELLERS AND IMPORTERS- 810 Walnut Street, Philadelphia. Any of the Books comprised in this Oatalogne will he sent hy mail, £fm if postage, to any address in the world, at the pnhlication prioesi A Descriptive Oatalogne, 90 pages, 8vo., will he sent free and free of postage to any one in any part of the world, who will fnrnish his addressi M~ Where not otherwise stated, all of the Books in this Catalogne are hatoA in muslin. •AMATEUR MECHANICS' WORKSHOP: A treatise containing plain and concise directions for tlie manipulft* tion of Wood and Metals, including Casting, Forging, Brazing, Soldering and Carpentry. By the author of the " Lathe and Itl Uses." Seventh edition. Illustrated. 8vo. . . . ;^2.5« ANDES.— Animal Pats and Oils: Their Practical Production, Purification and Uses; their Propettiea^ Falsification and Examination. 62 illustrations. 8vo. . ANDES. — Vegetable Fats and Oils : Their Practical Preparation, Purification and Employment; their Propeities, Adulteration and Examination. 94 illustrations. 8vo. ARLOT. — A Complete Guide for Coach Painters : Translated from the French o*" M. Arlot, Coach Painter, for eleven years Foreman of Pain.mg to M. Eherler, Coach Maker, Paris. By A. A. Fesquet, Chemist and Engineer. To which is added an Appendix, containing Informalioa '■esnecting the MateriaJi and the Practice of Coach and Car Painting w..d Varnishing in th« United States and Great Britain Tamo. . . . $1.3^ (I) HENRY CAREY BAIRD & CO.'S CATALOGUE. •VRMENGAUD, AMOROUX, AND JOHNSON.— The Practl cal Draughtsman's Book of Industrial Design, and Ma- chinist's and Engi^iieer's Drawing Companion : Farming a Complete Course of Mechanical Engineering and Archi tectural Drawing. From the French of M. Armengaud the elder, Prof, of Design in the Conservatoire of Arts and Industry, Paris, and MM. Armengaud the younger, and Amoroux, Civil Engineers. Re- written and arranged will) additional matter and plates, selections from and examples of the most useful and generally employed mechanism of the day. By William Johnson, Assoc. Inst. C. E. Illustrated by fifty folio steel plates, and fifty wood-cuts. A new edition, 410 , cloth ^600 ARMSTRONG. — The Construction and Management of Steam Boilers : By R. Armstrong, C. E. With an Appendix by Robert Mallet, C. E., F. R. S. Seventh Edition. Illustrateii. i vol. i2mo. .60 ARROWSMITH.— The Paper-Hanger's Companion: Comprising Tools, Pastes, Preparatory Work ; Selection and Hanging of Wall- Papers ; Distemper Painting and Cornice- ilnting ; Stencil Work; Replacing Sash-Cord and Broken Window Panes; and Useful Wrinkles and Receipts, By James Arrowsmith. A New, Thoroughly Revised, and Much Enlarged EdiUon. Illustrated by 25 engravings, 162 pages. (1905) .... ^i.oo \SHTON. — The Theory and Practice of the Art of Designing Fancy Cotton and Woollen Cloths from Sample : Giving full instruciions for reducing drafts, as well as the methods of spooling and making out harness for cross drafts and finding any re- quired reecf; with calculations and tables of yarn. By Frederic T. AsHTON, Designer, West Pittsfield, Mass. With fifty-two illustrations. One vol. folio'" *5-oo A.SKINSON —Perfumes and their Preparation: A Comprehensive Treatise on Perfumery, containing Complete Directions for Making Handkerchief Perfumes, Smelling-Salts, Sachets, Fumigating Pastils; Preparations for the Care of the Skin, the Mouth, the Hair; Cosmetics, Hair Dyes, and other Toilet Articles. By G. W. Askinson. Translated from the German by IsiDOR FuRST. Revised 'oy Charles Rice. 32 Illustrations. 8vo. ^3.00 BRONGNIART. — Coloring and Decoration of Ceramic Ware. 8vc. ^2.50 BAIRD. — The American Cotton Spinner, anc Manager's and Carder's Guide: A Practical Treatise on Cotton Spinning ; givmg the Dimensions and Speed of Machinery, Draught and Twist Calculations, etc. ; with notices of recent Improvements: together with Rules and Examples ror making changes in the sizes and numbers of Roving and Yarn. Compiled from the paper! ^f the late Robert H. Bairu. i2mo. 5) 50 ^ HENRY CAREY BAIRD & CO.'S CATALOGUE. 3 BAKER. — Long-Span Railway Bridges : Comprising Investigations of the Comparative Theoretical and Practical Advantages of the various Adopted or Proposed Type Systems of Construction ; with numerous Formul£e and Tables. By B. Baker. i2mo $i.oo BRANNT.— A Practical Treatise on Distillation and Rec tification of Alcohol : Comprising Raw Materials ; Production of Malt, Preparation of Mashes and of Yeast ; Fermentation ; Distillation and Rectification and Purification of Alcohol ; Preparation of Alcoholic Liquors, Liqueurs, Cordials, Bitters, Fruit Essences, Vinegar, etc. ; Examina- tion of Materials for the Preparation of Malt as well as of the Malt itself; Examination of Mashes before and after Fermentation ; Alco- holoraetry, with Numerous Comprehensive Tables ; and an Appendix on the Manufacture of Compressed Yeast and the Examination of Alcohol and Alcoholic Liquors for Fusel Oil and other Impurities. By William T. Brannt, Editor of " The Techno-Chemical Receipt Book." Second Edition. Entirely Rewritten. Illustrated by 105 engravings. 460 pages, 8vo. (Dec. ,1903) . . . $4.00 BAKR.— A Practical Treatise on the Combustion of Coal : Including descriptions of various mechanical devices for the Eco- nomic Generation of Heat by the Combustion of Fuel, whether solid, liquid or gaseous 8vo. ....... $2.50 B ARR. — A Practical Treatise on High Pressure Steam Boilers: Including Results of Recent Experimental Tests of Boiler Materials, together with a description of Approved Safety Apparatus, Steam Pumps, Injectors and Economizers in actual use. By VVm. M. Barr. 204 Illustrations. 8vo. ....... ^3.00 BAUERMAN. — A Treatise on the Metallurgy of Iron : Containing Outlines of the History of Iron Manufacture, Methods of Assay, and Analysis of Iron Ores, Processes of Manufacture of Iron and Steel, etc., etc. By H. Bauerman, F. G. S., Associate of the Royal School of Mines. Fifth Edition, Revised and Enlarged. Illustrated with numerous Wood Engravings from Drawings by J. B. 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Brannt. 41 illustrations. (1904.) ;^3.oo BILLINGS.— Tobacco : Its History, Variety, Culture, Manufacture, Commerce, and Various Modes of Use. By E. R. Billings. Illustrated by nearly 200 engravings. 8vo. ........ ^300 BIRD. — The American Practical Dyers" Companion: Comprising a Description of the Principal Dye- Stuffs and Chemicals used in Dyeing, their Natures and Uses ; Mordants and Hosv Made ; with the best American, English, French and German processes for Bleaching and Dyeing Silk, Wool, Cotton, Linen, Flannel, Felt. 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With an Appendix concerning American Marbles. i2mo., cloth $l.$0 BRANNT.— A Practical Treatise on Animal and Vegetabll Fats and Oils ; Comprising both Fixed and Volatile Oils, their Physical and Chem- ical Propesties and Uses, the Manner of Extracting and Refining them, and Practical Rules for Testing them ; as well as the Manufac- ture of .'\rtificial Butter and Lubricants, etc., with lists of American Patents relating to the Extraction, Rendering, Refining, Decomposing, and Bleaching of Fats and Oils. By William T. Brannt, Editor of the " Techno-Chemical Receipt Book." Second Edition, Revised and in a great pnrt Rewritten. Illustrated by 302 Engravings. In Two Volumes. 1304 pp. 8vo ^Sio.oo BRANNT. — A Practical Treatise on the Manufacture of Soap and Candles : Based upon the most Recent Experiences in the Practice and Science ; comprising the Chemistry, Raw Materials, Machinery, and Utensils and Various Processes of Manufacture, including a great variety of formulas. Edited chiefly from the German of Dr. C. Deite, A. Engelhardt, Dr. C. Schaedler and others; with additions and lists of American Patents relating to these subjects. By Wm. T. Brannt. Illustrated by 163 engravings. 677 pages. 8vo. . . ^lo.oo BRANNT —India Rubber, Gutta-Percha and Balata : Occurrence, Geographical Distribution, and Cultivation, Obtaining and Preparing the Raw Materials, Modes of Working and Utilizing them, Including Washing, Maceration, Mixing, Vulcanizing, Rubber and Gutta-Percha Compounds, Utilization of Waste, etc. By Will- iam T. Brannt. Illustrated. i2mo. (1900.) . . ^3-0° 6 HENRY CAREY BAIRD & CO.'S CATALOGUE. BRANNT— WAHL.— The Techno-Chemical Receipt Book: Containing several thousand Receipts covering tije latest, most im- portant, and most useful discoveries in Chemical Technology, and their Practical Application in the Arts and the Industries. Edited chiefly from the German of Drs. 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By Frederick Kick Imperial Regierungsrath, Professor of Mechanical Technology in the imperial German Polytechnic Institute, Prague. Translated from the second enlarged and revised edition with supplement by H. H. P. PoWLES, Assoc. Memb. Institution of Civil Engineers. Illustrated with 28 Platen, and 167 Wood-cuts. 367 pages. 8vo. . $10.00 iwINGZETT. — The History, Products, and Processes of the Alkali Trade : including the most Recent Improvements. By Charles Thomas v.vnzKTT. Consultinsj Chemist. With 23 illustrations. 8vo. ^^2.50 KIRK. — The Cupola Furnace : A Practical Treatise on the Construction and Management of Foundry Cupolas. By Edward Kirk, Practical Moulder and Melter, Con- sulting Expert in Melting. Illustrated by 78 engravings. Second Edition, revised and enlarged. 450 pages. 8vo. 1903. J^3 5° LANDRIN.— A Treatise on Steel: Comprising its Theory, Metallurgy, Properties, Practical Working, and Use. By M. H. C. Landrin, Jr. From the French, by A. A. 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