'INS BURY lECHN. iviETAL 1 late Work C. T. MILLIS. pRflNKLm Institute L'^R^^^ FMILflbELFHIfl lass ^ ..7 / Book E Accession 7.5.Z.6 Article V.— The Library shall be divided into two classes : the hrst comprising such works as, from their rarity or value, should not be lent out, all unbound periodicals, and such text books as ought to l)e tound in a library of reference except when required by Commlttet^s ot the Institute, or by members or holders of second class stock, who have obtained the sanction of the Committee. 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Article VIII. — No member or holder of second class stock, whose annual contribution for the current year shall be unpaid or ;^ ho is in arrears for fines, shall be entitled to the privileges of the Library or Reading Room. i i n Article IX.— If any member or holder of second class stock, shall refuse or neglect to comply with the foregoing rules, it shall be tlie duty of the Secretary to report him to the Committee on the Library. Article X.— Any Member or holder of second class stock, detected in mutilating the newspapers, pamphlets or books belonging to the Insti- tute shall be deprived of his right of membership, and the nauu' of the offender shall be made public. ffin^iyut^ ^tt'bnttal Mmml&. METAL-PLATE WORK : ITS PATTEKNS AND THEIE GEOMETEY. ALSO, NOTES ON METALS, AND EULES IN MENSURATION, FOE THE USE OF TIN, IRON, AND ZINC PLATE WORKERS, COPPERSMITHS, BOILER-MAKERS, AND OTHERS. I BY C. T. MILLIS, M. Inst. M.E., LECTURER ON METAL-PLATE WORK AND PRACTICAL GEOMETRY AT THE CITY AND GUILDS OF LONDON TECHNICAL COLLEGE, FINSBURY. E. & F. N. SPON, 125, STRAND, LONDON. NEW YOKK: 35, MUEEAY STREET. 1887. COA/S TS 6?o PREFACE. In tlie pages that follow, the setting- out of patterns for metal- plat;e workers is systematised (for the first time, so far as I am awaire), and by the system laid down, nearly all the patterns reqiaired by sheet-metal workers can be set out on one genieral geometrical principle. Thus treated, the subject becomes a comparatively easy one, and the workman learns how, on the given principle, to develop for himself the surface of any article he may have to make, to the saving, as against various methods practised hitherto, of both time and matterial. Of the methods heretofore taught in the workshop or othierwise, some have no application beyond the particular article with respect to which they are described, some are absiolutely impracticable. The methods that these pages explain are applicable always ; and have been proved by abmndant experience in my classes. The commencement of the book was a series of articles that were written by me for The Ironmonger. These and my cla)ss and lecture notes have greatly aided me in my effort to make the book comprehensive, as well as at the same time a welcome workshop companion. Not only does it contain IV PEEFACE. patterns whicli are essentially those of numerous articles of every-day manufacture, but a s;pecial feature is made of large work ; — and how it may be dealt with under ordinary work- shop conditions. Each of the problems is complete in itself; but although solved independently they follow each other in due order. It is my hope that the book will be an aid to students in engineering and geometry, as well as to those for whom it is particularly written ; that it will be a serviceable addition to the scanty and insufficient literature on the application of geometry to metal -plate work ; and that it will in some degree assist the cause of technical education. C. T. M. City and Guilds of London Technical College, FiNSBUEY, London, June 1887. CONTENTS. Book I. CHAPTER I. PAGE Claissification ,. .. .. .. •• •• •• ^ CHAPTER II. o Intieodtjctory Peoblems .. ■• •• •• •■ ^ Definitions, 3-5 ; Problems on Angles, Lines, Circles, Poly- gons, Ovals, Ellipses, and Oblongs, 6-20 ; Measurement of Angles, 20-3. CHAPTER III. Ar'ticles or Equal Taper or Inclination of Slant .. .. 24 Eight cone defined and described, with problems, 24-7. CHAPTER IV. Patterns for Eodnd Articles of Equal Taper or Inclina- tion OF Slant .. Development of right cone, and problems on same, 28-31 ; allowance for lap, seam, and wiring, 32-3 ; frustum (round equal-tapering bodies) defined and described, with pro- blems, 33-37 ; patterns, for round equal-tapering bodies (frusta) in one, two, or more pieces, for both small and large work, 37-43. vi CONTENTS. CHAPTER V. PAGE Equal-tapering Bodies and their Plans .. .. .. 46 Plans and elevations, 46-50 ; plans of equal-tapering bodies ; their characteristic features, 50-6 ; problems ou the plans of round, oblong, and oval equal-tapering bodies, 56-65. CHAPTEE VI. Patterns for Flat-faced Equal-tapeking Bodies ,. .. 66 Definition of right pyramid, and development, 66-7 ; pattern for right pyramid, 68-9 ; frustum of right pyramid defined and described, 69-71 ; patterns for frusta of right pyramids (hoppers, hoods, &c.) for both small and large work, 71-7 ; baking-pan pattern in one or more pieces, 77-83. CHAPTER VII. Patterns for Equal-tapering Bodies of Flat and Curved Surfaces combined ., ,. .. §4 Pattern for oblong body with flat sides and semicircular ends, 84-90; for oblong body with round corners, 90-6; for oval body, 96-104. The patterns for each of these bodies are given in one, hoo, and four pieces, as well as for both small and large work. Book II. CHAPTER I. Patterns for Bound Articles of Unequal Taper or Incli- nation op Slant Oblique cone, definition, description, and development, 105-111; oblique cone frustum (round unequal-tapering body) and oblique cylinder, 111-3 ; patterns for round unequal-tapering bodies (oblique cone frusta), also oblique cylinder— work small or large, 113-23. CONTENTS. vii CHAPTEE II. Unequal-tapeeing Bodies and theie Plans Plans of unequal-tapering bodies and their characteristic features, 121-9 ; problems on plans of oblong, oval, and other unequal-tapering bodies, 130-3 ; plans of Oxford hip-bath, Athenian hip-bath, sitz bath and oblong taper bath, 134-42 CHAPTER III. Patterns foe Flat-faced Uneqtjal-tapeeing Bodies .. 143 Oblique pyramids and their frusta, 143-5 ; pattern for oblique pyramid, 145-8 ; patterns for frusta of oblique pyramids (flat- faced unequal- tapering bodies) — work small or large, 148-54; pattern for a hood which is not a frustum of oblique pyramid^ 154-6. CHAPTER IV. Patteens foe Unequal-tapeeing Bodies of Flat and Cueved Suefaces combined ,. ,. ,. .. ,. .. 157 Pattern for equal-end bath (unequal-tapering body having flat sides and semicircular ends), 157-68 ; for oval bath (oval unequal-tapering body), 168-83 ; for tea-bottle top (unequal-tapering body having round top, and oblong bottom with semicircular ends), 183-91 ; for oval-canister top (unequal-tapering body having round top and oval bottom), 192-205; for unequal-tapering body having round top and oblong bottom with round corners, 205-16. (_Tlie;patterns for each of these bodies are given in one, two, or four pieces, as well as for both small and large work.) Pattern for Oxford hip-bath, 216-30 (two methods) ; for oblong taper bath, 230-6 (two methods). Vlll CONTENTS. Book III. PAGE Patterns foe Miscellaneous Articles .. .. .. .. 239 Elbow patterns, 239-41 ; X patterns various, and at any angle, 242-54 ; patterns for pipe joining two unequal or equal circular pipes not in line with one another, 255 ; Y patterns, where the pipes joined are equal or unequal, 255-63 ; pattern for tall-boy base (body having rect- angular base and round top), 264-6 ; for dripping-pan with well, 266 ; for compound bent surfaces, as vases, aquarium stands, mouldings, 267-9. Book IV. CHAPTER I. Metals; Allots; Solders; Soldering Fluxes 270 Characteristics ; properties ; specific gravity ; melting points, 270-5; iron and steel, 275-7; copper, 278; zinc, 279; lead, 279 ; alloys, 280-7 ; of copper and tin, 281 ; of copper and zinc, 282-4; of tin and lead, 284; solders, 284-7 ; soldering fluxes, 288. CHAPTEE II. Seams or Joints .. .. .. .. ,. .. 291 CHAPTEE III. Useful Kules in Mensuration ; Tables op Weights of Metals 294 Index ., .. ., .. .. .. .. ,. .. 301 METAL-PLATE WOEK. Book I. CHAPTEE I. Classification. (1.) Notwithstanding the introduction of machinery and the division of labour in the various branches of metal- plate work, there is as great a demand for good metal-plate workers as ever, if not indeed a greater demand than formerly, while the opportunities for training such men are becoming fewer. An important part of the technical edu- cation of those connected with sheet-metal work is a know- ledge of the setting-out of patterns. Such knowledge, requisite always by reason of the variety of shapes that a,re met with in articles made of sheet-metal, is nowadays especially needful ; in that the number of articles made of sheet metal, through the revival of art metal-work, the general advance of science, and the introduction of new designs (which in many cases have been very successful), in articles of domestic use, has considerably increased. It is with the setting-out of patterns that this volume princi- pally deals. To practical men, the advantages in saving of time and material, of having correct patterns to work from, are obvious. Whilst, however, the method of treat- ment here of the subject will be essentially practical, an B 2 METAL-PLATE WORK. amount of theory sufficient for a tliorongli compreliension of the rules given will be introduced, a knowledge of rules without principles being mere ' rule of thumb,' and not true technical education. (2.) Starting in the following pages with some introduc- tory problems and other matter, we shall proceed from these to the articles for which patterns are required by sheet-metal workers and which may be thus conveniently classed and subdivided : Class I. — Patterns for Ar- ^ tides of equal taper or inclination (pails, oval teapots, gravy strainers, &c.). Class IT. — Patterns for Ar- ticles of unequal taper or inclination (baths, hoppers, canister - tops, &c.). Sub- divisions. Sub- divisions. / a. Of round surfaces. &. Of plane or flat surfaces, c. Of curved and plane surface combined. a. Of round surfaces. &. Of plane or flat surfaces. ^ c. Of curved and plane surface combined. Class III. — Patterns for Miscellaneous Articles (elbows, and articles of compound bent surface, as vases, aquarium stands, mouldings, &c.). All these articles will be found dealt with in their several places. We shall conclude with a few technical details in respect of the metals that metal-plate workers mostly make use of. (3.) The setting out of patterns in sheet-metal work belongs to that department of solid geometry known as "Development of Surfaces," which maybe said to be the spreading or laying out without rupture the surfaces of solids in the plane or flat, the plane now being sheet metal. METAL-PLATE WOEK. 3 CHAPTER II. Intkoductoky Peoblems ; with Applications. Definitions. StraigM Line. — A straight line is the shortest distance between two points. Note. — If not otherwise stated, lines are always supposed to be straight. Angle. — An angle is the inclination of two lines, which meet, one to another. The lines A B, C B in Fig. 1 which are inclined to each other, and meet in B, are said to form an angle with one another. Ta express an angle, the letters which denote the two lines forming the angle are employed, the letter at the angular point being placed in the middle ; thus, in Fig. 1, we speak of the angle ABC. Fig. 1. Fig. 2. Fig. 3. Ferpendicular. Biglt Angles. — If a straight line, A B (Fig. 2), meets or stands on another straight line, C D, so that the adjacent angles (or angles on either side of A B) A B D, ABC, are equal, then the line A B is said to be perpendicular to, or at right angles with (' square with ') D C, and each of the angles is a right angle. 4 METAL-PLATE WOEK. Parallel Lines. — Parallel lines are lines which, if produced ever so far both ways, do not meet. Triangle. — A figure bounded by three lines is called a triangle. A triangle of which one of the angles is a right angle is called a right-angled triangle (Fig. 3) ; and the side which joins the two sides containing the right angle is called the hypothenuse (or hypotenuse). If all the sides of a triangle are equal, the triangle is Equilateral. If it has two sides equal, the triangle is Isosceles. If the sides are all unequal, the triangle is Scalene. Polygon.— A figure having more than four sides is called a polygon. Polygons are of two classes, regular and irregular. Irregular Polygons have their sides and angles unequal. Begular Polygons have all their sides and angles equal, and possess the property (an important one for us) that they can always be inscribed in circles ; in other words, a circle can always be drawn through the angular points of a regular polygon (Figs. 12 and 13). Special names are given to legular polygons, according to the number of sides they possess ; thus, a polygon of five sides is a pentagon; of six sides, a hexagon; of seven, a heptagon; of eight, an octagon; and so on. Quadrilaterals. — All figures bounded by four lines are Fig. ia. Fig. 46. called quadrilaterals. The mo^t important of these are the square and ohlong or rectangle. In a square (Fig. 4a) the sides are all equal and the angles all right angles, and con- sequently equal. An oblong or rectangle has all its angles right angles, but only its opposite sides are equal. (Fig. ih.) METAL-PLATE WORK. 5 Circle. — A circle is a figure bounded by a curved line such that all points in the line are at an equal distance from a certain point within the figure, which point is called the centre. The bounding line of a circle is called its circumference. A part only of the circumference, no matter how large or small, is called an arc. An arc containing a quarter of the circum- ference is a quadrant. An arc containing half the circumfer- ence is a semicircle. A line drawn from the centre to any point in the circumference is a radius (plural, radii). The lino joining the extremities of any arc is a chord. A chord that passes through the centre is a diameter. A line drawn from the centre of, and perpendicular to, any chord that is not a diameter of a circle, will pass through its centre. In practice a circle, or arc, is ' described ' from a chosen, or given, centre, and with a chosen, or given, radius. If two circles have a common centre, their circumferences are always the same distance apart. Fig. 5. In Fig. 5. 0 is the centre. A D (the curve) is an arc. A B (or B C) is a quadrant. A C B is a semicircle. O A (or 0 B, or B C) is a radius. A D (the straight line) is a chord. A C is a diameter. 6 METAL-PLATE WOEK. PEOBLEM I. To draw an angle equal to a given angle. Case I. — Where the ' given ' angle is given by a drawing. This problem, though simple, is often very useful in practice? especially for elbows, where the angle (technically called ' rake ' or ' bevil ') is marked on paper, and has to be copied. Fig. 6. Let ABC (Fig. 6) be the given angle. With B as centre and radius of any convenient length, describe an arc cutting B A, B C (which may be of any length, see Def.) in points A and C. Draw any line D E, and with D as centre and same radius as before, describe an arc cutting D E in E. With E as centre and the straight line distance from A to C as radius, describe an arc intersecting in F the arc just drawn. From D draw a line through F ; then the angle F D E will be equal to the given angle ABC. Case II. — Where the given angle is an angle in already existing fixed work. The angle to which an equal angle has to be drawn, may be an angle existing in already fixed work, fixed piping for instance ; or in brickwork, when, suppose, a cistern may have to be made to fit in an angle between two walls. In such cases a method often used in practice is to open a two- fold rule in the angle which is to be copied. The rule is then laid down on the working surface, whatever it may be (paper, board, &c.), on which the work of drawing an angle equal to the existing angle has to be carried out, and lines are drawn on that surface, along either the outer or inner edges of the METAL-PLATE WORK. 7 rule. The rule being then removed, the lines are produced ; meeting, they give the angle required. Case III.— Where the given angle is that of fixed work, and the method of Case II. is inapplicable. With existing fixed work, the method- of Case II. is not always practicable. A corner may be so filled that a rule cannot be applied. The method to be now employed is as follows. Draw lines on the fixed work, say piping, each way from the angle ; and on each line, from the angle, set off any the same distance, say 6 in., and measure the distance between the free ends of the 6 -in. lengths. That is, if A C, A B (Fig. 7) Fig. 7. represent the lines drawn on the piping, measure the distance between B and C. Now on the working surface on which the drawing is to be made, draw any line D E, 6 in. long ; and with D as centre and radius D E, describe an arc. Next, with E as centre, and the distance just measured between B and C as radius, describe an arc cutting the former arc in F. Join F D ; then the angle F D E will be equal to the angle of the piping. Note.— When points are 'joined,' it is always by straight lines. PROBLEM 11. To divide a line into any nimler of equal parts. Let A B (Fig. 8) be the given line. From one of its extre- mities, say A, draw a line A 3 at any angle to AB, and on it, from the angular point, mark off as many parts,— of any con- 8 METAL-PLATE WORK. venient length, but all equal to each other,— as A B is to be divided into. Say that A B is to be divided into three equal parts, and that the equal lengths marked off on A 3 are A to Fig. 8. 1, 1 to 2, and 2 to 3. Then join point 3 to the B extremity of A B, and through the other points of division, here 1 and 2, draw lines parallel to 3 B, cutting A B in C and D. Then A B is divided as required. PROBLEM III. To bisect (divide a line into two equal parts) a given line. Let A B (Fig. 9) be the given line. With A as centre, and any radius greater than half its length, describe an indefinite arc ; and with B as centre and same radius, describe an arc intersecting the former arc in points P and Q. Draw a line through P and Q ; this will bisect A B. Fig, 9. P Q Note.— It is quite as easy to bisect A B by Problem II. ; but the method shown gives, in P Q, not only a line bisecting A B, but a line perpendicular to A B. This must be particularly remembered. METAL-PLATE WORK. 9 PEOBLEM lY. To find the centre of a given circle. Let ABC (rig. 10) be tlie given circle. Take any three points A, B, C, in its circumference. J oin A B, B C ; then AB, BC, are chords {see Def.) of the circle ABC. Bisect A B, B C ; the point of intersection, 0, of the bisecting lines is the centre required. Fig. 10. > < PEOBLEM V. To describe a circle which shall pass through any three given points that are not in the same straight line. Let A, B, C (Fig. 10) be the three given points. Join A B, B C. Now the circle to be described will not be a circle through A, B, C, unless A B, B C, are chords of it. Let us therefore assume them such, and so treating them, find (by- Problem IV.) 0 the centre of that circle. With 0 as centre, and the distance from 0 to A as radius, describe a circle ; it will pass also through B and C, as required. PEOBLEM VI. Given an arc of a circle, to complete the circle of which it is a portion. Let A C (Fig, 10) be the given arc ; take any three points in it as A, B, C ; join A B, B C. Bisect A B, B C by lines 10 METAL-PLATE WORK. intersecting in O. With 0 as centre, and 0 to A or to any point in the arc, as radius the circle can be completed. PROBLEM VII. To find wJieiher a given curve is an arc of a circle. Choose any three points on the given curve, and by Problem V describe a circle passing through them. If the circle coincides with the given cnrve, the curve is an arc. PEOBLEM VIII. To bisect a given angle. Let ABC (Fig. 11) be the given angle. With B as centre and any convenient radius describe an arc cutting A B, B C in D and E. With D and E as centres and any convenient distance, greater than half the length of the arc Fig. 11. Z? DA D E as radius describe arcs intersecting in F. Join F to B ; then F B bisects the given angle. PROBLEM IX. In a given circle, to inscribe a regular polygon of any given number of sides. Divide (Problem II.) the diameter A C of the given circle (Fig. 12) into as many equal parts as the figure is to have METAL-PLATE WOEK. 11 sides, here say five. Witli A and C as centres, and C A as radius, describe arcs intersecting in P. Through P and the second point of division of the diameter draw a line P B Fig. 12. A C cutting the circumference in B ; join B A, then B A will be one side of the required figure. Mark off the length B A from A round the circumference until a marking off reaches B. Then, beginning at point A, join each point in the circumference to the next following ; this will complete the polygon. Note. — By this problem a circumference, and therefore also one-half of it (semicircle), one-third of it, one-fourth of it (quadi'ant), and so on, can be divided into any number of equal parts. PEOBLEM X. To describe any regular polygon, the length of one side heing given. Let AB (Fig. 13) be the given side of, say, a hexagon. With either end, here B, as centre and the length of the given side as radius, describe an arc. Produce AB to cut the arc in X. Divide the semicircle thus formed into as many equal parts (Problem IX., Note) as the figure is to have sides (six), and join B to the second division point of the semicircle counting from X. This line will be another side of the required polygon. Having now three points. A, B, 12 METAL-PLATE "WOBK. and the second division point from X, draw a circle through them (Problem V.), and, as a regular polygon can always be inscribed in a circle (see Def.), mark off the length B A round Fig. 13. A B X the circumference from A until at the last marking-off, the free extremity of the second side (the side found) of the polygon is reached, then, beginning at A, join each point in the circumference to the next following ; this will complete the polygon (hexagon). PEOBLEM XI. 2^0 find the length of the circumference of a circle, the diameter being given. Divide the given diameter A B (Fig. 14) into seven equal parts (Problem II.). Then three times A B, with C B, one of the seven parts of A B, added, that is with one-seventh of Fig. 14. A C B A B added, will be the required length of the circumference. The semicircle of the figure is superfluous, but may help to make the problem more clearly understood. METAL-PLATE WORK. 13 PKOBLEM XII. To draw an oval, its length and width being given. Draw two lines A B, C D (the axes of the oval), perpendi- cular to one another (Fig. 15), and intersecting in 0. Make Fig. 15. v /^\ c 3 0 A and 0 B each equal to half the length, and 0 C and 0 D each equal to half the width of the oval. From A mark off A E equal to C D the width of the oval, and divide E B into three equal parts. With 0 as centre and radius equal to two of the parts, as E 2, describe arcs cutting A B in points Q and Q'. With Q and Q' as centres and QQ' as radius describe arcs intersecting C D in points P and P'. Join P Q, P Q', P' Q and P' Q' ; in these lines produced the end and side curves must meet. With Q and Q' as centres and Q A as radius, describe the end curves, and with P and P' as centres and radius P D, describe the side curves ; this will complete the oval. Note. — Unless care is taken, it may be found that the end and side curves will not meet accurately, and even with care tliis may sometimes occur. It is best if great accuracy be required in the length, to draw the end curves first, and then draw side curves to meet them ; or, if the width is most important, to draw the side curves first. The centres (P aud P') for the side curves come inside or outside the curves, according as the oval is broad or narrow. This figure is sometimes erroneously called an ellipse. It is, however, a good approximation to one, and for most purposes where an elliptical article has to be made, is very convenient. 14 METAL-PLATE WOEK. PEOBLEM XIII. To draw an egg-shaped oval, having the length and width given. Make A B (Fig. 16) eq[ual to the length of the oval, and from A set off A 0 equal to half its width. Through 0 draw Fig. 16. an indefinite line Q Q' perpendicular to A B, and with 0 as centre and 0 A as radius describe the semicircle CAD. Join D B ; and from D draw D E perpendicular to Q Q' and equal to 0 D. Also from E draw E G parallel to Q Q' and METAL-PLATE WOKK. 15 intersecting D B in G, and from G draw G F parallel to D E and intersecting Q Q' in F. . From B set off B P equal to D F, and join P F. Bisect F P and through, the point of bisection draw a line cutting Q Q' in Q. Join Q P and produce it indefinitely, and with Q as centre and Q D as radius descrihe an arc meeting Q P produced in H. Make 0 Q' equal to 0 Q, and join Q' P and produce it indefinitely. With Q' as centre, and Q' C (equal to Q D) as radius, describe an arc meeting Q' P produced in H'. And with P as centre and P B as radius describe an arc to meet the arcs D H and C H' in H and H' ; and to complete the egg-shaped oval. PROBLEM XIV. T descrihe an ellipse. Before working this as a problem in geometry, let us draw an ellipse non-geometrically and get at some sort of a defini- tion. This done, we will solve the problem geometrically, and follow that with a second mechanical method of de- scribing the curve. METHOD I.— Mechanical. A. Irresjpedive of dimensions. — On a piece of cardboard or smooth-faced wood, mark off any two points F, F' (Fig. 17) and fix pins securely in those points. Then take a piece of thin string or silk, and tie the ends together so as to form a loop ; of such size as will pass quite easily over the pins. Next, place the point of a pencil in the string, and take up the slack so that the string, pushed close against the wood, shall form a triangle, as say, F D F', the pencil point being at D. Then, keeping the pencil upright, and always in the string, and the string taut, move the pencil along from left to right say, so that it shall make a continuous mark. Let us trace the course of the mark. Starting from D, the pencil, constrained always by the string, moves from D to P, 16 METAL-PLATE WORK. then on to B, P', C, P^, P^, A, P*, and D again, describing a curve which returns into itself ; this curve is an ellipse. Having drawn the ellipse, let us remove the string and pins, draw a line from F to F', and produce it both ways to terminate in the curve, as at B and A. Then A B is the major axis of the ellipse, and F, F' are its foci. The mid- point of A B is the centre of the ellipse. Any line through the centre and terminating both ways in the ellipse is a diameter. The major axis is the longest diameter, and is commonly called the length of the ellipse. The diameter through the centre at right angles to the major axis is the shortest diameter, or minor axis, or width of the ellipse. Eeferring to the Fig. : — A D P B C is an ellipse. F, F' are its foci (singular, focus). ^ A B is the major axis. C D is the minor axis. 0 is the centre. Fig. 17. D We notice with the string and pencil that when the pencil point reaches P, the triangle formed by the string is F P F' ; when it reaches P', the triangle is F P' F' ; when it reaches P^ the triangle is F P^ F' ; and when P^ is reached, it is F P3 F'. Looking at these triangles, it is obvious that METAL-PLATE WORK. 17 F F' is one side of each of them ; from which it follows, seeing that the loop of string is always of one length, that the sum of the other two sides of any of the triangles is equal to the sura of the other two sides of any other of them ; that is to say, F D added to D F' is equal to F P added to P F', is equal to F P' added to P' F', and so on. Which leads us to the following definition. Definition. Ellipse. — The ellipse is a closed curve (that is, a curve returning into itself), such that the sum of the distances of any point in the curve from certain two points (foci), inside the curve is always the same. B. Length and width given. — Knowing now what an ellipse is, we can work to dimensions. Those usually given are the length (major axis), and width (minor axis). Draw A B, C D (Fig. 17), the given axes, and with either ex- tremity, C or D, of the minor axis as centre, and half A B, the major axis as radius, describe an arc cutting A B in F and F'. Fix pins securely in F, F' and D (or C). Then, having tied a piece of thin string or silk firmly round the three pins, remove the pin at D (or C) ; put, in place of it, a pencil point in the string ; and proceed to mark out the ellipse as above explained. METHOD II. — Geometrical. — The Solution of the Problem. Length and Width given. Draw A B, C D (Fig. 17), the major and minor axes. With C or D as centre, and half the major axis, 0 B say, as radius, describe arcs cutting A B in F and F'. On A B, and between 0 and F', mark points — any number and anywhere, except that it is advisable to mark the points closer to each other as they approach F'. Let the points here be 1, 2, and 3, With F and F' as centres and A 2, B 2 as radii respectively, describe arcs intersecting in P ; with same centres and A 3, B3 as radii respectively, describe arcs intersecting in P'. With F' and F as centres and A3, B 3 as radii respectively, 0 18 METAL-PLATE WORK. describe arcs intersecting in P^. With same centres and A 2, B 2 as radii respectively, describe arcs intersecting in P*. Similarly obtain P^. We have thus nine points, D, P> B, P', C, P^, P^, A and P*, through which an even curve may be drawn which will be the ellipse required. A greater number of points through which to draw the ellipse may of course be obtained by taking more points between 0 and F', and proceeding as explained. METHOD III. — Mechanical. — Length and Width GIVEN. As it is not always possible to proceed as described at end of Method L, for pins cannot always be fixed in the material to be drawn upon, we now give a second mecha- nical method. Having drawn (Fig. 18&) A B, CD, the Fig. 18a. Fig. 18&. HE F given axes, then, on a strip of card or stiff paper X X (Fig. 18a), mark off from one end P, a distance P F equal to half the major axis (length), and a distance P E equal to half the minor axis (width). Place the strip on the axes in such a position that the point E is on the major axis, and the METAL-PLATE WOEK. 19 point F on the minor, and mark a point against the point P. Now shift X X to a position in which E is closer to B, and F closer to 0, and again mark a point against P. Proceed similarly to mark other points, and finally draw an even curve through all the points that have been obtained. The following prollems deed with shapes often required hy the metal-plate worher, and will give him an idea of how to adapt to his requirements the problems that precede. The explanation of the measurement of angles that concludes the chapter will further assist him in his work. PEOBLEM Xy. To draw an oblong with round corners. Draw two indefinite lines AB, CD (Fig. 19) perpen- dicular to one another and intersecting in 0. Make 0 A and 0 B each equal to half the given length ; and 0 C and 0 D each equal to half the given width. Through C and D c 2 20 METAL-PLATE WOEK. draw lines parallel to A B, and through A and B draw lines parallel to C D. We now have a rectangle or oblong, and require to round the corners, which are quadrants. Mark off from E along E D and E A equal distances E G and E F according to the size of corner required. With E and G as centres and E F or E G as radius, describe arcs intersecting in 0'. With 0' as centre and same radius describe the corner E G. The remaining corners can be drawn in similar manner. PEOBLEM XYI. To draw a figure having straight sides and semicircular ends (oblong with semicircular ends). Draw a line AB (Fig. 20) equal to the given length; make A 0 and B 0' each equal to half the given width. Fig. 20. n E 0 o 5 F G Through 0 and 0' draw indefinite lines perpendicular to A B ; with 0 and 0' as centres and O A as radius describe arcs cutting the perpendiculars through O and 0' in D E and GE. Join D E, GF; this will complete the figure required. ANGLES AND THEIE MEASUREMENT. The right angle B 0 C (Fig. 5) subtends the quadrant B C. If we divide that quadrant into 90 parts and call the parts METAL-PLATE WORK. 21 degrees, then a right angle suhtends or contains 90 degrees (written 90°), or as nsually expressed, is an angle of 90 degrees, the degree being the unit of measurement. If each division point of the quadrant is joined to 0, the right angle is divided into 90 angles, each of which subtends or is an angle of 1 degree. That is to say, an angle is measured by the number of degrees that it contains. Suppose the quadrant B A is divided as was B C, then BOA also is an angle of 90 degrees. If the division is continued round the semicircle ADC, this will contain 180 degrees, and the whole circumference has been divided into 360 degrees. As an angle of 90, which is a fourth part of 360 degrees, subtends a quadrant or fourth part of the circumference of the circle, so an angle of 60, which is a sixth part of 360 degrees, subtends a sixth part of the circumference, and similarly an angle of 30 degrees subtends a twelfth part, an angle of 45 an eighth part, and so on. And this angular measurement is quite independent of the dimensions of the circle; the quadrant always subtends a right angle ; the 60 degrees angle always subtends an arc of one sixth of the circum- ference ; and the like with other angles. From our defini- tion p. 5 we have it that a chord is the line joining the extremities of any arc. The chord of a sixth part of the circumference of any circle, we have now to add, is equal to the radius of that circle. This being the case, and as an angle of 60 degrees subtends the sixth part of the circumfer- ence of a circle, it follows that an angle of 60° subtends a chord equal to the radius. SCALE OF CHOEDS. Construction. — We have now the knowledge requisite for setting out a scale of chords, by which angles may be drawn and measured. On any line 0 B (Fig. 21) describe a semicircle 0 A B, and from its centre C draw C A perpendicular to 0 B. Divide 0 A into nine equal parts. Then, as 0 A, being a quadrant, 22 METAL-PLATE WOKK. contains 90°, eacli of tlie nine divisions will contain 10°. The points of division, from 0, of the quadrant, are marked 10, 20, 30, &c,, np to 90 at A. With 0 as centre, describe arcs from each of these division points, cutting the line 0 B. Note that the arc from point 60 cuts 0 B in C, the centre of the semi- circle ; the chord from 0 to 60 (not drawn in the Fig.), that is, the chord of one-sixth of the circumference of the circle whose centre is C, being equal to the radius of that circle. Draw a line 0 E parallel to 0 B, and from 0 let fall 0 0 perpendicular to OB. Also from each of the points where the arcs cut 0 B let fall perpendiculars to 0 B and number these consecutively to correspond with the numbers on the quadrant 0 A. The scale is now complete. How to use. It is used in this way. Suppose from a point A in any line A B (Fig. 22) we have to draw a line at an angle of 30° with it. Then with A as centre and the distance from 0 to 60 on 0 E (Fig. 21) as radius, describe an arc C D cutting AB in C. And with C as centre and the distance from 0 to 30 on 0 E (Fig. 21) (the angle to be drawn is to be of 30°) as radius describe an arc intersecting are C D in D. Join D A, then DAG will be the required angle of 3X)i°. Similarly with angles of other dimensions. In taking the lengths of arcs, we really take the length of their chords, and it is these lengths that (Fig. 21) we have METAL-PLATE WORK. 23 set oflf along 0 E. The angle (Fig. 21) 0 C F (the point F is the point 60) being an angle of 60° subtends a chord equal to the radius ; therefore in 0 to 60 we have the radius C 0. In the example (Fig. 22), the distance C D (0 to 30) is the chord of 30° ; and it is clear that we must set this off on an arc C D of a circle of the same size as that employed in the construc- tion of the scale, and this we do by making A C equal 0 to 60 on the same scale. When a scale of chords has been constructed as explained, the semicircle may be cut away, and we thus get a scale convenient for shop use in the form of a rule. 24 METAL-PLATE WOEK. CHAPTEE III. Patterns for Articles of Equal Taper or Inclination. (CLASS I.) (4.) It is necessary here at once to remark that ordinary workshop parlance speaks of slant,' — not as meaning an angle, but a length ; not as referring to the angle of inclination of a tapering body, but to the length of its slanting portion. It is in this sense that we shall use the word, and shall employ the word ' taper ' or the term ' inclination of slant ' when meaning an angle. (5.) In order that the rules for the setting out of patterns for articles of equal taper or inclination may be better under- stood and remembered, it is advisable to consider the principles on which the rules are based, as a knowledge of principles will often enable a workman himself to find rules for the setting out of patterns for odd work. The basis of the whole of the articles in this Class is the right cone. It is necessary, therefore, to define the right cone and explain some of its properties. Definition. (6.) Bight Cone. — A right cone is a solid figure generated or formed by the revolution of a right-angled triangle about one of the sides containing the right angle. The side about which the triangle revolves is the axis of the cone ; the other side containing the right angle being its radius. The point of the cone is its apex ; the circular end its hose. The hypotenuse of the triangle is the slant of the cone. From the method of formation of the right cone, it follows that the axis is perpendicular to the base. The height of the cone is the length of its axis. (7.) Referring to Fig. la, 0 B E represents a cone gene- METAL-PLATE WORK. 25 rated or formed by the revolution of the right-angled tri- angle 0 AB (Fig. 1&) about one of its sides containing the right angle, here the side 0 A. Similarly the cone 0 D F, Fig. 2a, is formed by the revolution of 0 C D (Fig. 26) about its side 0 0. As will be seen from the figs., 0 A, O C are respectively the axes of the cones 0 B E, 0 D F, as also their heights. Their bases are respectively BGEH, DKFL Fig. 26. and the radii of the bases are A B and C D. The slants of the cones are 0 B and 0 D, the apex in either being the point 0. Other lines will be seen in figs., namely, those repre- senting the revolving triangle in its motion of generating 26 METAL-PLATE WORK. the cone. The sides of these triangles that start from the apex and terminate in the base are all equal, it must be borne in mind ; and each of them is the slant of the cone. Likewise their sides that terminate in A are all equal, and each shows a radius of the base of the cone. How these particulars of the relations to one another of the several parts of the right cone apply in the setting-out of patterns will be seen in the problems that follow. PEOBLEM I. To find the height of a cone, the slant and diameter of the hose heing given. Draw any two lines 0 A, B A (Figs. 3 and 4) perpen- dicular to each other and intersecting in A. On either line Fig. 3. Fig. 4. 0 1 0 ■ Reight \ 1 1 A JJxxjdiuje of Base B A Koutucs of Base B mark off from A half the diameter of the base, in other words, the radius of the base, as A B. With B as centre, and radius equal to the slant, describe an arc cutting A 0 in 0. Then 0 A is the height of the cone. METAL-PLATE WOEK. 27 PEOBLEM II. To find the slant of a cone, the height and diameter of the base being given. Draw any two lines 0 A, B A (Figs. 3 and 4) perpendicular to each other and intersecting in A. On either line mark off from A half the diameter of the base (radius of the base), as A B, and make A 0 on the other line equal to the height of the cone ; join 0 B. Then 0 B is the required slant. 28 METAL-PLATE WORK. CHAPTER IV. , Patterns for Eound Articles op Equal Taper or Inclination op Slant. (Class I. Subdivision a.) (8.) If a cone has its inclined or slanting surface painted, say, white, and be rolled while wet on a plane so that every portion of the surface in succession touches the plane, then the figure formed on the plane by the wet paint (see "Fig. 6) Fig. 5. will be the pattern for the cone. As the cone rolls (the figure represents the cone as rolling), the portion of it touching the plane at any instant is a slant of the cone (see § 7). (9.) Examining the figure formed by the wet paint, we find it to be a sector of a circle, that is, the figure contained between two radii of a circle and the arc they cut off. The length of the arc here is clearly equal to the length of the circumference of the base of the cone, and the radius of the METAL-PLATE WORI^. 29 arc evidently equal to the slant of the cone. From this it is obvious that to draw the pattern for a cone, we require to know the slant of the cone (which will be the radius for the pattern), and the circumference of the base of the cone. PEOBLEM III. To draw the pattern for a cone, in one piece or in several pieces, the slant and diameter of the base being given. Pattern in one Piece.— With 0 A (Fig. 66) equal to the slant as radius, describe a long arc ACE. What has now Fig. 66 3 to be done is to mark off a length of this arc equal to the cir- cumference of the base of the cone. The best and quickest way for this is as follows. Draw a line F B (Fig. 6a) equal to the given diameter of the base, and bisect it in G ; then G B is a radius of the base. From G draw G E perpendicular 30 METAL-PLATE WOKK. to FB ; and with G as centre and radins GB describe from B an arc meeting GE in E. The arc B E is a quadrant (quarter) of the circumference of the base of the cone. Divide this quadrant into a number of equal parts, not too Fig. 6a. E many, say four, by points 1, 2, 3. From A (Fig. 6&) mark off along arc ACE four parts, each equal to one of the divisions of the quadrant, as from A to B. Take this length A B equal to the four parts, that is, equal to the quadrant, and from B set it off three times along the arc tovs^ards E as from B to C, C to D, D to E. Join E to 0 ; then 0 A C E 0 will be the pattern required. Note.— It must be noted that when this pattern is bent round to form the cone, the edges O A and O E will simply butt up against each other, for no allowance has been made for lap or seam. Let us call the junction of 0 A and O E the line of butting. Nor, further, has any allowance been made for vyiring of the edge ACE. These most essential matters will be referred to immediately. Pattern in more than one Piece. — If B be joined to 0, then the sector OAB will be the pattern for one-quarter of the cone. If C be joined to 0, then the sector 0 A C is the pattern for one-half of it. Similarly 0 A D will give METAL-PLATE WOEK. 31 three-quarters of the cone. A cone pattern can thus be made in one, two, three, or four pieces. If the cone is required to be made in three pieces, then instead of dividing, as above, a quadrant of the circumference of the base, divide one-third of it into parts, say five ; set off five of the parts along ACE from A, and join the last division point to the centre ; the sector so obtained will be the pattern for one-third of the cone. If required to be made in five pieces, divide a fifth of the circumference of the base into equal parts, and proceed as before. Similarly for any number of pieces that the pattern may be required in. PEOBLEM IV. To draw the pattern for a cone, the height and the diameter of the lase heing given. First find the slant 0 B (Fig. 7a) by Problem II. Then with A as centre and radius A B, describe B C a quadrant of Fig. 7a. Fig. 76. the circumference of the base, and proceed, as in Problem III., to draw the pattern Fig. 7b (the plain lines). 32 METAL-PLATE WOEK. ALLOWANCE FOE LAP, SEAM, WIEING, &o. (10.) It has already heen stated that the geometric pattern Fig. 6& has no allowance for seam, wiring, or edging. (For the present it is assumed that these terms are understood ; we shall come hack to them later on.) In the pattern Fig. 7b the dotted line 0' B' parallel to the edge 0 B shows ' lap ' for soldered seam. For a ' grooved ' seam not only must there he this allowance, hut there must he a similar allowance along the edge 0 D. These allowances, it must he distinctly remembered, are always extras to the geometric pattern ; that is to say, the junction line of OD and 0 B, or line of butting (see Note, Problem III.) is not interfered with. And here a word of warning is necessary. Suppose instead of marking off a parallel slip or lap for soldered seam, a slip D 0 D' going off to nothing at the centre 0, is marked off, and that then, for soldering up, there is actually used not this triangular slip, but a parallel one as D D' 0 0^, the result brought about will be that the work will solder up untrue; there will be, in fact, a ' rise ' at the base of the work. We can understand the result in this way. If the parallel slip D D' 0 0^ used for soldering were cut off, there would remain a pattern which is not the geometric pattern, but a nondescript approximation, having a line of butting other than the true line. And it being thus to an untrue pattern that the parallel slip for seam is added, the article made up from the untrue pattern must of course itself necessarily be untrue. In the fig. the dotted line parallel to the curve of the pattern shows an allowance for wiring. For a grooved seam there must be on the edge 0 D an addition 0 D D' 0^ similar to the addition on the edge 0 B, as above stated. (11.) In working from shop patterns for funnels, oil-bottle tops, and similar articles, workmen often find that if they take a good lap at the bottom, and almost nothing at the top of the seam, the pattern is true. And so it is, for these patterns have the triangular slip D O D' added. Whereas, if a parallel piece D 0=^ 0 D' is used for lap, the pattern is METAL-PLATE WORK. 33 untrue. WMch. again is the case, because, now, in addition to D 0 D', an extra triangular piece D 0^ 0 is used, and this extra is taken off the geometric pattern. Consequently, the line of butting is interfered with; that is to say, the two lines OB and 0 D, instead of meeting, overlap ; 0 B forming a junction with 0^ D instead of with 0 D ; with which 0 B must always form a junction, for the pattern to be true. In setting out patterns, to prevent error, the best rule to follow and adopt is, to first mark them out independent of any allowance for seams, or wiring, or edging, and to afterwards add on whatever allowances are intended or requisite. In future diagrams, allowances, where shown, will be mostly shown by dotted lines. Definition. (12.) Frustum.— If a right cone is cut by a plane parallel to Fig. 8«. Fig. 8h. its base, the part containing the apex is a complete cone, as 0 C G D L (Fig. 8a), and the part C A B D containing the D 34 METAL-PLATE WOKK. base A H B K is a frustum of the cone. In other words a frustum of a right cone is a solid having circular ends, and of equal taper or inclination of slant eveiywhere between the ends. Conversely a round equally tapering body having top and base parallel is a frustum of a right cone. (13.) Comparing such a solid with round articles of equal taper or inclination of slant, as pails, coffee-pots, gravy Figs. 9. strainers, and so on (Fig. 9), it will be seen that they are portions (frusta) of right cones. (14.) In speaking here of metal- plate articles as portions of cones, it must be remembered that all our patterns are of surfaces, seeing that we are dealing with metals in sheet ; and that these patterns when formed up are not solids, but merely simulate solids. It is, however, a con- venience, and leads to no confusion to entirely disregard the distinction ; the method of expression referred to is therefore adopted throughout these pages. (15.) By Fig. 86 is shown the relations of the cone 0 A B of Fig. 8a with its portions 0 C D (complete cone cut off), and C A B D (frustum). The portion 0 C D is a complete cone, as it is the solid that would be formed by the revolution of the right-angled triangle OFD (both figs.) around OF. The triangles 0 F G and 0 F C (Fig. 8a) represent the METAL-PLATE WOKK. 35 triangle 0 F D in progress of revolution. Tlie triangle 0 E B (both figs.) is the triangle of revolution of the uncut cone 0 A B (Fig. 8a) and 0 E H, 0 E A represent 0 E B in pro- gress of revolution. The height of the cone 0 A B heing 0 E (both figs.), the height of the cone 0 C D is 0 F (both figs.). The radius for the construction of pattern of the uncut cone 0 A B will be 0 B (both figs.) ; for the pattern of 0 C D, the cone cut off, the radius will be 0 D (both figs.). In F E, or D M, we have the height of the frustum. Just as (§ 8) the portion of the rolling cone touching the plane at any- instant is a slant of the cone, so the slant of a frustum is that portion of it, which, if it were set rolling on a plane, would at any instant touch the plane. D B is a slant of the frustum C A B D. The extremities of a slant of a frustum are ' corresponding points.' Other details of cone and frustum are shown in Fig. 86. Fig. 10. o (16.) It is obvious that, if the patterns for the cones 0 A B, 0 C D (Fig. 8a) be drawn (Fig. 10) from a common centre 0, the figure A C D B will be the pattern for the frustum D 2 36 METAL-PLATE WORK. ACDB (Fig. 8a). From which we see that in order to draw the pattern for the frustum of a cone, we must know the slant of the cone of which the frustum is a portion, that is, we must know the radius for the construction of the pattern of that cone, and also the slant (radius for pattern) of the cone cut off. PROBLEM V. Given the dimensions of the ends of a round equal-tapering body (frustum of right cone), and its upright height. To find the slant, or the height, of the cone of which it is a portion. Draw any two lines 0 A, AB (Fig. 11) at right angles to each other and intersecting in A. From A on either line, Fig. 11. say on B A, mark off A B equal to half the diameter of the larger of the given ends, and from A on the other line make A C equal to the given upright height. Draw a line C D METAL-PLATE WORK. 37 parallel to A B, or, whicli is the same thing, at right angles to A O, and make C D equal to half the diameter of the smaller end. Join B D, and produce it, meeting A 0 in 0. Then 0 A is the height of the cone of which the tapering body is a portion, and 0 B the slant. To draw the pattern for a frustum of a cone, the diameters of the ends of the frustum and its upright height being given. The Frustum. — Draw any two lines OA, BA (Fig. 12a) perpendicular to each other and meeting in A ; on one of the perpendiculars, say B A, make A B equal to half the longer diameter (radius), and on the other make A C equal to the given upright height. Draw a line 0 D perpendicular to A 0 and make C D equal to half the shorter diameter. Join B D, and produce it, meeting A 0 produced in 0. With A PKOBLEM VI. Fig. 12a. 0 jf Half Longer Diojneter METAL-PLATE WOEK. 39 centre, and -with radius OB (Fig. 12a) describe an arc ACE; also with same centre and radius, 0 D (Fig. 12a), describe an arc A' C E'. From any point in the outside curve, as A, draw a line through 0', and cutting the inner arc in A'. From A mark off successively parts equal to those into which the quadrant B E (Fig. 12a) is divided, and the same number of them, four, to B. And from B, along the outer curve, set off B C, C D, D E, each equal to A B. Join E 0', cutting the inner curve in E'. Then A A' E' E is the pattern required. Just as 0 B (Fig. 12a) is the slant of the cone that would be generated by the revolution of right-angled triangle O A B around OA, so D B is the slant of the frustum of which A A' E' E (Fig. 12h) is the pattern. In the pattern the slant D B appears as A A', B B', C C, &c. Parts of the Frustum — If B be joined to 0', the figure A A' B' B will be one-quarter of the pattern of the frustum ; and if C be joined to 0', the figure A A' C 'C will be pattern for one-half of it, and so on. The paragraph " Pattern in more than one Piece " in Problem III. should be re-read in connection with this " Parts of a Frustum." (17.) The problem next following is important, in that, in actual practice, the slant of a round equal-tapering body is very often given instead of its height, especially in cases where the taper or inclination of the slant is great ; as for instance in ceiling-shades. The only difference in the work- ing out of the problem from that of Problem VI. is that the radii required for the pattern cf the body are found from other data. Let us take the problem. PEOBLEM VII. To draw the jyattern for a round equal-tapering body (frustum of right cone), the diameter of the ends and the slant being given. To find the required radii, draw any two lines O A, B A (Fig. 13) perpendicular to one another, and meeting in A. 40 METAL-PLATE WOEK. On either line, as A B, make A B equal to half the longer of the given diameters and A C equal to half the shorter. From C draw C D perpendicular to A B. With B as centre and 0 \ Fig. 13. D F \ Half Shorter j< Ealf Longer HiaiTieter >j • J the given slant as radius, describe an arc cutting C D in E. Join B E and produce it to meet A 0 in 0. Then 0 B and O E are the required radii. By E E being draw^n parallel to A B, comparison may be made between this Fig. and Fig. 12a, and the diflFerence between Problems VI. and VII. clearly apprehended. To draw the pattern, proceed as in Problem VI. (18.) For large work and for round equal- tapering bodies which approximate to round bodies without any taper at all, the method of Problem VI. is often not available, for want of space to use the long radii that are necessary for the curves of the patterns. The next problem shows how to deal with such cases ; by it a workiDg=:0§ntr§ and long radii can be dispensed with. The method gives fairly good results. METAL-PLATE WORK. 41 PKOBLEM VIII. To draw, without long radii, the pattern for a round equal' tapering body (frustum of right cone), the diameters of the ends and the upright height being given. First draw one-quarter of tlie plan. (To do this, we fore- stall for convenience what is taught in the following chapter.) Fig. 14. Draw any two lines B 0, C 0 (Fig. 14) perpendicular to each other and meeting in 0. With 0 as centre and radius equal to half the longer diameter, describe an arc meeting the lines B 0, C 0 in B and 0. With 0 as centre and radius equal to 42 METAL-PLATE WOEK. half the shorter diameter describe an arc B' C This completes the one-quarter plan. Kow divide B C, the largest arc, into any nnmher of equal parts, say fonr ; and join the points of division to 0 by lines cutting B'C in 1', 2', 3'. Join 3'C, and through 3' draw 8'E perpendicular to 3'C, and equal to the given upright height. Join C E ; then C E may be taken as the true length of C 3'. Through C draw C D perpendicular to C 0 and equal to the upright height. Join 0 D ; then C D is the true length of C C. If it is inconvenient to find these irue lengths on the plan, it may be done apart from it, as by the triangles P and Q. To set out the pattern. Draw (Eig. 15) any line C C equal to C D (Fig. 14). With 0' and C as centres and radii respectively CE and C3 (Fig. 14) describe arcs intersecting in 3 (Fig. 15). With C and C as centres and radii respec- tively CE and C'3' (Fig. 14) describe arcs intersecting in •3' (Fig. 15). Then C and 3 are two points in the outer Fig. 15. curve of the pattern, and C 3' two points in the inner cut ve. To find points 2 and 2', proceed as just explained, and with the same radii, but 3' and 3 as centres instead of C and C. Similarly, to find points 1' and 1, and B' and B. A curved line drawn from C through 3, 2, and 1 to B will be the outer curve of one-quarter of the required pattern, and a curved METAL-PLATE WORK. 43 line from C through 3', 2', and 1' to B' its inner curve ; that is C C B' B is one-quarter of the pattern. Four times the quarter is of course the required pattern complete. Note. — In cases where this method will be most useful, the pattern is generally required so that the article can be made in two, three, four, or more pieces. If the pattern is required in three pieces, one-third of the plan must be drawn (see end of Problem III., p. 31) instead of a quarter, as in Fig. ]4; the remainder of the construction will then be as described above. (19.) It is often desirable in the case of large work to know fwhat the slant or height, whichever is not given, of a round equal-tapering body (frustum of right cone) will be, before starting or making the article. Here the following problems will be of service. PROBLEM IX. To find the slant of a round equal-tapering body (frustum of right cone), the diameters of the ends and the height being given. Mark off (Fig. 16) from a point 0 in any line OB the lengths of half the shorter and longer diameters, as 0 C, 0 B. Fig. 16. From C draw C D perpendicular to 0 B. Make C D equal to the given height, and join BD. Then BD is the slant required. 44 METAL-PLATE WOKK. PEOBLEM X. To find the height of a round equal-tapering body (frustum of right cone), the diameters of the ends and the slant being given. Mark off (Fig. 17) from a point O in any line 0 B tlie lengths of half the shorter and longer diameters, as in Problem IX., and from C draw C D perpendicular to 0 B. With B as centre and radius equal to the given slant, describe an arc cutting C D in E. Then C E is the height required. Essentially this problem has already been given, in the working of Problem VII. *0 C £ c DA PEOBLEM XI. Given the slant and the inclination of the slant of a round equal- tapering body ; to find its height. Let A B (Fig. 18) be the slant, and the angle that A B makes with 0 A the inclination of the slant. From B let fall B D perpendicular to A C. Then B D is the height required. (20.) In the workshop, the inclination of the slant of a tapering body is sometimes spoken of as the body being so many inches "out of flue." This will be explained in the following chapter. If the inclination of the slant METAL-PLATE WORK. 45 is given in these terms the problem is worked thus. From any point D in any line C A (Fig. 18) make D A equal in length to the number of inches the body is " out of flue," and draw D E perpendicular to C A. With A as centre and radius equal to the given slant, describe an arc intersecting D E in B. Then B D will be the height required. 46 METAL-PLATE WORK. CHAPTEE V. Equal-tapering Bodies of which Top and Base are Parallel, and their Plans. (21.) First let us understand what a plan is. Fig. 19 represents an object Z, made of tin, say, having six faces, Fig. 19. of which the A B C D and G H K L faces are parallel, as also the BDKH and CALG. The ABC D and CD KL faces are square. The A B 0 D face has, soldered flat on it centrally, a smaller square of tin ahcd with a central METAL-PLATE WORK. 47 circular hole in it. Now suppose wires (represented in the fig. by dotted lines), soldered perpendicularly to the A B C D face, at A, B, C, D, a, h, c, d, E, and F (the points E and F are points at the extremities of a diameter of the circular hole). Also suppose wires soldered at Gr and H parallel to the other wires, and that the free ends of all the wires are cut to such length that they will, each of them, butt up against a flat surface (plane), of glass say, X XX xi parallel to the A B C D face. Lastly suppose that all the points where the wires touch the glass are joined by lines corresponding to edges of Z (see the straight lines in the Fig. 20. E' L figure on the plane) ; also that E' and F' are joined, that the line joining them is bisected, and a circle described passing through E' and F'. Then the complete representa- tion obtained is a projection of Z. Instead of actually pro- jecting the points by wires, we may make the doing of it another supposition may, find, as if by wires, the required points, and draw the projection. The ABC D face being 48 METAL-PLATE WOEK. say 2 inches square, the flat piece, say 1 inch square, and the hole ^ inch diameter, and the back face GHKL, say 2i in by 2 in., then the projection that is upon the glass would he as shown in Fig. 20. The plane X X X X is here supposed vertical, and the projection G'C D' H' is therefore an elevation; if the plane were horizontal, the projection would be a plan, and we might regard A B C D as the top of the body, and G H K L as its base, or rice versa. We may define a plan then as the representation of a body obtained by projecting it on to a horizontal plane, by lines perpen- dicular to the plane. (22) The plane X XXX was supposed parallel to the A B C D face of Z ; the plan A' B' C D' of it is therefore of the same shape as A B C D, and in fact A B C D may be said to be its own plan. Similarly the G' H' D' C is the plan of the back-face G H KL and is of the same shape as that face. But the plan of the face A G H B to which the plane is not parallel is by no means the same shape as that face, for the long edges BH and AG of the face AGHB are, in plan, the short lines B' H' and A' G'. We need not, however, go farther into this, because in the case of the bodies that now • concern us, the horizontal plane on which any plan is drawn is always supposed to be parallel to the principal faces of the body, so that the plans of those faces are always of the same shape as the faces. In this paragraph the plane X X X X is supposed to be horizontal. (22a ) We are now in a position to explain the getting at the true length of C C in the fig. of Problem VIII., p. 42 ; or putting the matter generally, to explain the finding the true lengths of lines from their apparent lengths in their plans and elevations. Horizontal lines being excepted, there is, manifestly, for any line, however positioned in space, a vertical plane in which its elevation will appear as (if not a point) a vertical line. Let B E (Fig. 17, p. 44) be any line in the plane of the paper, and let C D be the vertical plane seen edgeways on which the elevation E C of B E is a vertical line. Then if 0 B be a horizontal plane seen edgeways METAL-PLATE WOEK. 49 passing through C, the line joining the B extremity of B E to the C extremity of its elevation will be the plan of B E. We get thus the figure E C B, a figure in one plane, the plane of the paper, a right-angled triangle in fact, of which the E C side is the elevation of B E, the C B side its plan, and the hypotenuse the line itself ; a figure, which, as combining a line, its plan, and its elevation, we have under no other conditions than when the elevation in question is a vertical line. In the plane passing through E C and B E, that is, in the plane in which these lines wholly lie, we have in the line that we get by joining C with B the plan, full length, of B E. In respect of this plan of B E, we are concerned with no other measurement, because, in a right-angled ./ triangle representing a line and its plan and elevation, no other measurement of the plan line can come in. Not so, however, with the elevation line of B E. Here other mea- surement of it than its length can and does come in, because that length varies according to the position of the vertical plane with regard to it ; the plan length is always the same. But to have in the three sides of E C B, the representation of B E, and its plan and elevation, it is evident that the plane which contains B E and its plan C B must also wholly contain the elevation E C ; that is, the plane must be perpen- dicular to the plane of the triangle. Now, no matter on what vertical plane the line B E is projected, although the length of the projection will vary, the vertical distance between its extremities, that is, its height, never varies. Hence, if, in any right-angled triangle, we have in the hypotenuse the representation of a line, in one of its sides the plan of the line, and in the other side, not necessarily the elevation that comes out vertical, but the height of any elevation of the line, it comes to the same thing as if in the latter side we had the actual elevation that is vertical. And hence, further, if we have given the plan-length of an unknown line, and the vertical distance between its extre- mities, we can, by drawing a line, say C B, equal to the given plan-length, then drawing from one of its extremities and at E 50 METAL-PLATE WORK. right angles to it, a line, say C E, equal to the given vertical distance, and finally joining the free extremities, as by B E, of these two lines, construct a right-angled triangle, the hypotenuse, BE, of which must be the true length of the unknown line ; for there is no other line than B E of which C B and C E can be, at one and the same time, plan and elevation. We have explained this true-length matter fully, because we have to make use of it abundantly in problems to come. (23.) Proceeding to the bodies we have to consider, we Fig. 21. a, b take first a frustum of a cone, Fig. 21a. To draw its plan, let us suppose the extremities of a diameter of its smaller face top (namely points A and F of the skeleton drawing Fig. 216) (neither drawing is to dimensions), to be projected, in the way just explained, on to a plane parallel to the face, then, also as there explained, we can draw the circle which is a projection of that face. Suppose the smaller circle of Fig. 22 to be that circle, and to be to dimensions. Projecting now, similarly, the extremities of a diameter of the larger face (base), namely the points C and D of the skeleton drawing, on to the same plane, we can get the projection of the larger face. Let the larger circle of Fig. 22 be that projection. The two circular projections will be concentric (having the METAL-PLATE WORK. 51 same centre) because the body Z is of equal taper, and they will, together, be the plan of Z, that is Fig. 22 is that plan. A. C and F D each show the slant, and B A and E F the height. B C and D E each show the distances between the plans of corresponding points. Fig. 22. (24.) Turn to the skeleton drawing of Z, Here A C shows a slant of the frustum (§ 15), A B its height (see D M, Fig. 8h), and A and C are 'corresponding points' (§ 15). Looking at C D E B as at the plan of the frustum, we have, in the point B, the plan of the point A. Joining B C, we get a right-angled triangle ABC; the slant A C is its hypotenuse, the height A B is one of the sides containing the right angle, and the other side containing the right angle, B C, is the distance between the plans of the corresponding points A and C, as also between plans of corresponding points of Z any- where. This distance is that of how much the body is ' out of flue ' (a workshop expression that was referred to at the end of the previous chapter), in other words, how much A 0 is out of parallel with A B. What points, in the plan of a frustum, are the plans of corresponding points is shown E 2 62 METAL-PLATE WOEK. by tlie fig., as the line joining the plans of corresponding points (the line joining B and C or that joining D and E, for instance) will always, if produced, pass through the centre of the circles that constitute the plan of the frustum ; the centre of the circles being the plan of the apex of the cone of which the frustum is a part. Which leads us to this ; that the distance, actually, between the plans of corresponding points in the plan of a frustum is equal to half the difference of the diameters of its two circles ; for, the difference between E B and D C is the sum of D E and B C, and D E and B C are equal ; in other words, either D E or B C is half the difference between E B and D C. Fig. 23a. (25.) Let us now consider another equal-tapering body which has top and base parallel, and we will suppose it to have flat parallel sides, flat ends, and round (quadrant) corners. Such a body is represented, except as to dimen- sions, in Z, Figs. 23a and h; Fig. 236 being a skeleton drawing of the body represented in Fig. 23 a. Extending our definition of ' slant ' to apply to such a body, a ' slant ' becomes the shortest line that can be drawn anywhere on the slanting surface ; and ' corresponding points ' become, in accordance, the extreme points of such line. Either of the lines F A, G B, E C, H L, or MO represent a slant of the body, and F and A are corresponding points ; as also are G and B, E and C, H and L, and M and 0. The height METAL-PLATE WORK. 53 of the body is represented by either of the lines F A', G B', E C, H L', or M O'. The plane for the plan being parallel to the M P Q E face (here the top) of Z, the plan of that face is of the same shape as the face. The round-cornered rect- angle A' F' G' D' B' 0' of Fig. 29 is the plan to dimensions. For the same reason the plan of the OATS face (here the base) is of the same shape as that face. The round-cornered rectangle A F G D B C of Fig. 29 is the plan to dimensions. How actually to draw these plans we shall deal with presently as a problem. The two circles constituting the plan of the frustum were concentric, that is, symmetrically disposed with respect to one another, because the frustum Fig. 236. was an equal-tapering body ; and the plans of top and base of the body we are now dealing with are symmetrical to each other for the same reason. The two plans (Fig. 29) together are the plan of the body Z. (26.) Looking at A B C D A' B' C D' of the skeleton draw- ing (Fig. 236) as at the plan of Z, we have, just as with the cone frustum, in the point A' the plan of F, in the point B' the plan of G, in the point C the plan of E, in the point L' the plan of H, and in the point 0' the plan of M. Further as in the case of the frustum, if we join any point in the plan of the base, as A, to the plan of its corresponding point A', then we have a right-angled triangle, F A A', of which the 54 METAL-PLATE WOKK. hypotenuse F A represents the slant of the body, F A', one of the Bides containing the right angle, its height, and A' A, the other side containing the right angle, the distance between the plans of the corresponding points F and A, which is also the distance between B and B', 0 and C, L and L', O and 0', and between plans of corresponding points of the body anywhere, the body being of equal taper. As with Fig. 216 what points, in the plan, are the plans of corresponding points is clear from the fig. Where the plan of the body consists of straight lines, the plans of corresponding points are always the extremities of lines joining these straight lines perpen- dicularly ; the extremities of A A', B B', C C, and L L', for instance. Where the plan of the body consists of arcs, the plans of corresponding points (compare with cone frustum) are the extremities of lines joining the arcs, and which, pro- duced, will pass through the centre from which the arcs are described ; the line O 0' for instance. To make all this quite plain, reference should again be made to Fig. 29 ; also to Fig. 28, which is the plan of an equal-tapering body with top and base parallel, and having flat sides, and semicircular ends. In Fig. 29, A A', B B', C C, D D', are lines joining the plan lines of the flat sides and ends perpendicularly, and the extremities of each of these lines are plans of correspond- ing points, that is to say, A and A' are plans of corresponding points, as are also B and B', C and C, and D and D'. Also F and F' are plans of corresponding points, being the extremities of the line F F' which is a line joining the ends of the arcs which are the plans of one of the quadrant corners of the body. Similarly G and G' are plans of corre- sponding points. In Fig. 28, F F', G G', D D', E E', are lines joining perpendicularly the plan lines of the flat sides of the body at their extremities where the semicircular ends begin ; and F and F', G and G', D and D', E and E' are plans of corresponding points. Also A and A' are plans of corre- sponding points, and B and B', seeing that the lines joining these points, produced, pass respectively through O and 0', the centres from which the semicircular ends are described. METAL-PLATE WOEK. 55 In the cone frustum, the actual distance between the plans of corresponding points was, we saw, equal to half the difference of the diameters of the two circles constituting its plan. Similarly with the body Z of Fig. 23a, and indeed with any equal-taperirig body of which the top and base are parallel, if we have the lengths of the top and base given, or their widths, the distance between the plans of corresponding points (number of inches ' out of flue ') is always equal to half the diflerence between the given lengths or widths. Thus, the distance between the plans of corresponding points of Z is equal to half the difference between A B and A' B' (Fig. 29) or between C D and C D'; and the distance be- tween the plans of corresponding points of the body of which Fig. 28 is the plan, is equal to half the difference between A B and A' B' of that fig., or between F D and F' D'. Summarising we have a. In the plans of equal-tapering bodies which have their tops and bases parallel, there is, all round, an equal distance between the plans of corresponding points of the tops and bases. 6. Conversely.— -li, in the plan of a tapering body with top and base parallel, there is an equal distance all round between the plans of corresponding points of the top and base, then the tapering body is an equal-tapering body, that is, has an equal inclination of slant all round. c. The plan of a round equal-tapering body having top and base parallel, consists of two concentric circles. The plan of a portion of a round equal-tapering body having top and base parallel, consists of two arcs having the same centre. The corners of the body Z (Fig. 23a) are portions (quarters) of a round equal-tapering body ; their plans are arcs (quadrants) of circles having the same centre. d. Conversely. — If the plan of a tapering body having top and base parallel, consists of two concentric circles, then the body is a frustum of a right cone. Also if the plan of a tapering body having top and base parallel, consists of two 56 METAL-PLATE WOEK. arcs having the same centre, then the body is a portion of a frustum of a right cone. The plan of each end of the tapering body represented in plan in Fig. 28 consists Of two arcs (semicircles) having the same centre ; the ends are portions (halves) of a frustum of a right cone. The plan of each corner of the tapering body Z (Fig. 23a) consists of two arcs (quadrants, Fig. 29) having the same centre ; the comers are portions (quarters) of a frustum of a right cone. The fig. annexed represents a quadrant corner in plan separately. We conclude the chapter with some problems. PROBLEM XII. Given the height and slant of an equal-tapering body with top and base parallel; to find the distance between the plana of corresponding points of the top and base (number of inches ' out of flue'). Let C A' (Fig. 25) be the given height. Draw A' B per- pendicular to A' C ; with C as centre and the given slant as radius, describe an arc cutting B A' in A. Then A A' is the distance required. Fig. 24. METAL-PLATE WOEK. 57 PEOBLEM XIII. Given the height of an equal-tapering body with top and base parallel, and the inclination of slant (number of inches * out of flue ') ; to find the distance between the plans of corresponding points of the top and base. Let OA' (Fig. 26) "be the given height. Through A' draw a line A' B perpendicular to C A' ; from any point, D, in A' B draw a line D E making with A' B an angle equal to that of the given inclination. From C draw C A parallel to E D and cutting A' B in A ; then A A' is the distance required. Fig. 26. C B D A A PEOBLEM XIV. To draw the plan of a round equal-tapering body with top and base parallel (frustum of right cone), the diameter of either end being given and the height and slant. Case I. — Given the height and slant and the diameter of the smaller end. On any line 0 B (Fig. 17) set oflF 0 C equal to half the given diameter, and from 0 draw 0 D perpendicular to 0 B. 58 METAL-PLATE WOEK. Mark off C E equal to the given height, and with E as centre and radius equal to the given slant, describe an arc inter- secting OB in B ; then C B will be the distance in .plan between corresponding points anywhere in the frustum ; that is to say (by c, p. 55) 0 C will be the radius for the plan of the smaller end of the frustum, and 0 B the radius for the plan of the larger end. Case II. — Given the height and slant and the diameter of the larger end. On any line 0 B (Fig. 27), set off 0 B equal to half the given diameter, and now work from B towards 0 instead of from 0 towards B ; thus. From B draw B C perpendicular to O B. Mark off B D equal to the given height, and with D as centre and radius equal to the given slant, describe an arc intersecting 0 B in E ; then B E will be the distance in Fig. 27. c D plan of corresponding points anywhere in the frustum ; that is to say (by c, p. 55) 0 B will be the radius for the plan of the larger end of the frustum, and O E the radius for the plan of the smaller end. METAL-PLATE WOKK. 59 PEOBLEM XV. To draw the plan of a round equal-tapering body with top and base parallel {frustum of rigid cone), the diameter of either end being given, and the number of inches ' out of flue ' (distance between plans of corresponding points). Case I.— Given the number of inches ' out of flue,' and the diameter of the smaller end. The radius for the smaller circle of the plan will be half the given diameter ; the radius for the larger circle of the plan will be this half diameter with the addition of the number of inches ' out of flue.' Case II. — Given the number of inches ' out of flue,' and the diameter of the larger end. The radius for the larger circle of the plan will be half the given diameter ; the radius for the smaller circle of the plan will be the half diameter less the number of inches ' out of flue.' (27.) It should be noted that with the dimensions given in this problem, we can draw plan only, we could not draw a pattern. To do that we must also have height given, for a plan of small height and considerable inclination of slant is also the plan of an -infinite number of other frusta (plural of frustum) of all sorts of heights and inclinations of slant. PEOBLEM XVI. To draw the plan of an oblong equal-tapering body with top and base parallel, and having flat (plane) sides and semicircular ends. Case I. — Where the length and width of the top are given, and the length of the bottom. Commencing with the plan of the top, we know from § 25 that it will be of the same shape as the top ; we have there- 60 METAL-PLATE WOEK. fore to draw tliat shape. On any line A B (Fig. 28) mark off A B equal to the given length of the top. From A set off AO, and from. B set off BO' each equal to half the given •width of the top. Through O and 0' draw lines perpen- dicular to A B ; and with 0 and 0' as centres and O A or O' B as radius describe arcs meeting the perpendiculars in D F and EG. As D F and E G pass through the centres O and 0' respectively they are diameters, and the arcs are semicircles; these diameters, moreover, are each equal to the given width. Join D E, F G, and the plan of the top is complete. Fig. 28. D ^ a] \ ^ 0 (P G' 1 B F Gr The plan of the base will be of the same shape as the base, and we will suppose it smaller than the top. What we have then to do is to draw a figure of the same shape as the base, and to so place it in position with the plan of the top that we shall have a complete plan of the body we are dealing with. By a, p. 55, we know that the distances between the plans of corresponding points of the top and base all round the full plan will be equal. We have therefore first to ascertain the distance between the plans of any two corresponding points. This by § 26 will in the present instance be equal to half the difference between the given METAL-PLATE WOKK. 61 lengths of the top and base. Set off this half-difference, as the base is smaller than the top, from. A to A'. Then with O and 0' as centres, and 0 A' as radius, describe the semi- circles D'A' F', E'B'G'. Join D'E', F'G', and we have the required plan of the body. Case II. — Where the length and width of the top are given, and the height and slant, or the height and the inclination of the slant (number of inches 'out of flue '). First draw the plan of the top as in Case I. Then if the height and slant are given, find by Problem XII. the distance between the plans of corresponding points. If the height and inclination of slant are given, find the distance by Problem XIII. If the inclination of the slant is given in the form of ' out of flue,' the number of inches ' out of flue ' is the required distance. Set off this distance from A to A' in the fig, of Case I., and complete the plan as in Case I. Case III. — Where the length and width of the base (bottom) are given, and the height and slant, or the height and the inclination of the slant. On any line A B (Fig. 28) mark off A' B' equal to the given length of the bottom. From A' set off A' 0 and from B' set off B' 0' each equal to half the given width of the bottom. Through 0 and 0' draw indefinite lines D F, E G perpendicular to A B ; and with 0 and 0' as centres, and O A' as radius describe the semicircles F' A' D', G' B' E', join D' E', F' G', and we have the plan of the bottom. Now by Problem XII. or Problem XIII., as may be required, find the distance between the plans of corresponding points, or take the number of inches ' out of flue,' if this is what is given. Set off this distance from A' to A. With 0 and 0' as centres and 0 A as radius describe semicircles meeting the perpendiculars through 0 and 0' in D and F and in E and G. Join D E, F G, and the plan of the body ie completed. 62 METAL-PLATE WOEK. PROBLEM XVII. To draw tJie plan of an oblong equal-tapering body with top and base parallel, and having flat sides, flat ends, and round (^quadrant) corners. Case L— Where the length and width of top and bottom (base) are given. Draw any two lines A B, C D (Fig. 29) perpendicular to each other and intersecting in 0. Make 0 A and 0 B ( ach equal to half the length of the top, which we will suppose Fig. 29. larger than the bottom, and 0 A' and 0 B' each equal to half the length of the bottom. Also make 0 C and 0 D each equal to half the width of the top, and 0 C and 0 D' each equal to half the width of the bottom. Through C, D, C, and D' draw lines parallel to A B, and through A, B, A', and B' draw lines parallel to C D and intei secting the lines parallel to A B. We have now two rectangles or oblongs, and we require to draw the round corners, which are quarters of circles. METAL-PLATE WORK. 63 From the intersectiBg point E along tlie sides of the rectangle mark equal distances E F and E G, according to the size of quadrant corners required. With F and G as centres and E F or E G as radius, describe arcs intersecting in 0' ; and with O' as centre and same radius, describe the arc F G, which will be a quadrant because if the points F and G be joined to 0' the angle F 0' G will be a right angle (p. 21). Draw F F' parallel to A B and G G' parallel to C D, and with 0' as centre and radius 0' F' describe the arc F' G', which also will be a quadrant. We have now the plan of one of the quadrant corners; the other corners can be drawn in like manner. (27a.) It is important to notice that the larger corner deter- mines the smaller one. In practice it is therefore often best to draw the smaller corner first, otherwise it may sometimes be found, after having drawn the larger corner, that it is not possible to draw the smaller curve sufficiently large, if at all. To draw the smaller corner first, mark off from the intersect- ing point E' equal lengths E' F', E' G', according to the size determined on for the corner. With F' and G' as centres and E' F' or E' G' as radius describe arcs intersecting in 0'. Then 0' will be the centre for the smaller corner. It will also be the centre for the larger corner, which may be described in similar manner to the smaller corner in the preceding paragraph. Case II. — Where the dimensions of the top are given and the height and slant, or the height and the inclination of the slant. Draw the plan of the top, A D B C. Find the distance between the plans of corresponding points of the top and base by Problem XII., or Problem XIII., according to what is given ; and set off this distance, as the base is smaller than the top, from A and B inwards towards 0 on the line A B, and from D and C inwards towards 0 on the line D G. Complete the plan by the aid of what has already been explained. ■ . % 64 METAL-PLATE WOEK. Case III.— Where the dimensions of the bottom are given, and the length and slant, or the height and the incli- nation of the slant. Draw the plan of the bottom, A' D' B' C',find the distance between the plans of corresponding points of top and bottom, set this off outwards from A', D', B', and C and complete the plan by aid of what has already been stated. PEOBLEM XVIII. To draw the plan of an oval equal-tapering hody with top and base parallel, the length and width of the top and bottom being given. Draw (Fig. 30) any two lines A B, C D intersecting each other at right angles in 0. Make 0 A and 0 B each equal Fig. 30. to half the given length of the larger oval (top or bottom, as may be), and 0 C and 0 D each equal to half its given width. A B and C D will be the axes of the oval. From A, on A B, METAL-PLATE WOEK. 65 mark off A E equal to C D the width of the oval, and divide E B into three equal parts. With 0 as centre and radius equal to two of these parts, as from B to 2, describe arcs cutting A B in Q and Q'. With Q and Q' as centres and Q Q' as radius describe arcs intersecting in P and P' ; and from P and P' draw lines of indefinite length through Q and Q'. With P and P' as centres and radius P D describe arcs (the side arcs), their extremities terminating in the lines drawn through Q and Q' ; and with Q and Q' as centres, and radius Q A, describe arcs (the end arcs) to meet the extremities of the side arcs. This completes the plan of the larger oval. To draw the plan of the smaller oval. Make 0 A' and 0 B' each equal to half the length of the smaller oval, and with Q and Q' as centres and QA' as radius describe the end curves, their extremities terminating, as do the outer end- curves, in the lines drawn through Q and Q' ; the point R is an extremity of one of the smaller curves. With P and P' as centres and radius PE, describe the side curves. The plan of the oval equal- tapering body is then complete ; of which either the larger or smaller ovals are plan of top and bottom according to the purpose the article may be required for. (28.) The plans of corresponding points in the plan of an oval equal-tapering body will be the extremities of any line joining the inner and outer curves anywhere, and that, pro- duced, will pass through the centre from which the curves where joined by the line are described. To draw the plan of an oval equal-tapering body with top and base parallel, other dimensions than the above may be given. For instance the top or bottom may be given, and either the height and slant, or the height and the inclination of the slant (number of inches out of flue). It will be a useful practice for the student to work out these cases for himself by the aid of the instruction that has been given. F 66 METAL-PLATE WORK. CHAPTER VI. Patterns for Articles of Equal Taper or Inclination OF Slant, and having Flat (Plane) Surfaces. (Class I. Subdivision &.) Definition. (29.) Pyramid. — A pyramid is a solid having a base of three or more sides and triangular faces meeting in a point above that base, each side of the figure forming the base being the base of one of the triangular faces, and the point in which they all meet being the apex. The shape of the base of a pyramid determines its name ; thus a pyramid with a triangular base is called a triangular pyramid; with a square base, a square pyramid; with a hexagon base, an hexagonal pyramid (Fig. 31) ; and so on. The centre of the base of a pyramid is the point in which perpendicular lines bisecting all its sides will intersect. If the apex of a pyramid is perpendicularly above the centre of its base, the pyramid is a right pyramid (Fig. 31 represents a right pyramid), in which case the base is a regular polygon and the triangular faces are all equal and all equally inclined. In a pyramid, the line joining the apex to the centre of the base is called the axis (the line V V, Fig. 31) of the pyramid. (30.) An important property that a right pyramid possesses is that it can be inscribed in a right cone. (31.) A pyramid is said to be inscribed in a cone when both the pyramid and the cone have a common apex, and the base of the pyramid is inscribed in the base of a cone ; in other words, when the angular points of the base of the pyramid are on the circumference of the base of the cone and the apex of cone and pyramid coincide. METAL-PLATE WOEK. 67 (32.) Fig. 31 shows a right pyramid inscribed in a right cone. The apex V is common to both pyramid and cone, and the A, B, C, &c., of the base of the pyramid are on the circumference of the base of the cone. Also the axis V V is common to both cone and pyramid. Further, the edges V A, V B, V C, &c., of the pyramid are lines on the surface of the cone, such lines or edges being each a slant of the cone, or in other words a radius of the pattern of the cone in Fig. 31. V which the pyramid is inscribed. It hence follows, that if the pattern of the cone in which a right pyramid is inscribed be set out with the lines of contact of cone and pyramid, as V A, V B, &c., on it, and the extremities of these lines be joined, we shall have the pattern for the pyramid. Thus, the drawing a pattern for a right pyramid resolves itself into first determining the cone which circumscribes the pyramid, and next drawing the pattern of that cone with the lines of contact of pyramid and cone upon it. F 2 68 METAL-PLATE WOEK. PEOBLEM XIX. To draw the pattern for an hexagonal right pyramid, its height and base being given. Draw (Fig. 32a) the plan A B C D E F of the base of the pyramid, which will be of the same shape as the base (see Chap, V.) ; the base in fact will be its own plan. Next draw any two lines 0 A, B A (Fig. 32&), perpendicular to each other and meeting in A ; make A B equal to the radius of the circumscribed circle (Fig. 32a), and A 0 equal to the given height of the pyramid. Join B 0 ; then B 0 is a slant of the Fig. 32a. Fig. S2b. cone in which the pyramid can be inscribed, that is to say, is a radius of the pattern of that cone. The line B 0 is also a line of contact of the cone, in which the pyramid can be inscribed, that is, is one of the edges of the pyramid. To draw the pattern (Fig. 32c). With any point 0' as centre and BO (Fig. 32i!>) as radius, describe an arc ADA, and in it take any point A. Join A 0', and from A mark off A B, B C, C D, D E, E F, and F A, corresponding to A B, B C, C D, D E, E F, and F A of the hexagon of Fig. 32a, and join the points B, C, D, E, F and A to 0'. Join A B, B 0, C D, D E, E F, and F A, by straight lines ; and the figure bounded METAL-PLATE WOEK. 69 by 0' A, the straight lines from A to A, and A 0', will be the pattern required. The lines B 0', C 0', &c., correspond to the edges of the pyramid, and show the lines on which to ' bend- up ' to get the faces of the pyramid, the lines 0' A and 0' A then butting together to form one edge. Similarly the pattern for a right pyramid of any number of faces can be drawn, the first step always being to draw the plan of the base of the pyramid ; the circle passing through the angular points of which will be the plan of the base of the cone in which the pyramid can be inscribed. Fig. 32c. D O Suppose, instead of the dimensions from which to draw the plan of the base of the pyramid, the actual plan be given. The centre from which to strike the circumscribing circle can then be found by the Definition § 29. Definition. (33.) Truncated pyramid. Frustum of pyramid. — If a pyramid be cut by a plane parallel to its base, the part containing the apex will be a complete pyramid, and the other part will be a tapering body, the top and base of which are of the same shape but unequal. This tapering body is 70 METAL-PLATE WOEK. called a truncated pyramid, or a frustum of a pyramid. The faces of a truncated pyramid which is a frustum of a right pyramid are all equally inclined. In Fig. 33 is shown such a frustum standing on a horizontal plane. (34.) Comparison should here be made between this defini- tion and that of a frustum of a cone (see § 12), which it closely follows ; also between Fig. 21 and Fig. 33. (35.) Articles of equal taper or inclination of slant and having flat (plane) surfaces and top and base parallel (hexagonal coffee-pots ; hoods ; &c.), are portions of right pyramids (truncated pyramids), or portions of truncated pyramids. (36.) Exactly as a pyramid can be inscribed in a cone, so a truncated pyramid can be inscribed in a frustum of a cone, and the edges of the truncated pyramid are lines on the surface of that frustum. The skeleton drawing, Fig. 33&, shows a right truncated pyramid inscribed in a cone frustum. It also represents the plan of the cone frustum, and that of the pyramid frustum, with the lines of projection (see Chapter v.), of the smaller end of the latter on to the horizontal plane. This inscribing in a cone gives an easy construction for setting out the pattern of a truncated pyramid ; which construction is, to first draw the pattern for the pyramid of Fig. 33. a b METAL-PLATE WORK. 71 wh.icli the truncated pyramid is a portion ; and then mark off on this pattern the pattern for the pyramid that is cut off. Here again comparison should be made with what has been stated about the pattern of a frustum of a cone (see § 16), and the resemblance noted. PEOBLEM XX. To draw the pattern for an equal-tapering body made up of flat surfaces (truncated right pyramid), the height, and top and bottom being given. Suppose the equal- tapering body to be hexagonal. To draw the required plan of the frustum. The plans of the top and bottom are respectively of the same shape as the top and bottom (§ 25). Draw (Fig. 34a) A B C D E F the larger hexagon (Problem X., Chap. II.) and its diagonals Fig. 34a. Fig. 346. A D, B E, C F, intersecting in Q. On any one of the sides of this hexagon mark off the length of a side of the smaller hexagon, as A G on A F, and through G draw G F' parallel to the diagonal A D, and cutting the diagonal F C in F'. With Q as centre and Q F' as radius describe a circle. The 72 METAL-PLATE WORK. points in which this cuts the diagonals of the larger hexagon will be the angular points of the smaller hexagon. Join each of these angular points, beginning at F', to the one next following, as F'E', E'D', &c. Then T'E'D'C'B'A' is the plan of the smaller hexagon, and, so far as needed for our pattern, the plan of the ' equal-tapering body made up of flat surfaces ' is complete. The lines A A', B B', C C, will be the plans (see and compare lines D E and B C of Fig. 33&) of the slanting edges of the frustum. Next draw (Fig. 34&) two lines 0 A, B A, perpendicular to each other, and meeting in A ; make A B equal to the radius of the circle circumscribing the larger hexagon of plan, and Fig. 34c. A C equal to the given height of the body. Through C draw C D perpendicular to A 0, and make C J) equal to the radius of the circle which passes through the angular points of the smaller hexagon (Fig. 34a). Join B D, and produce it to meet A 0 in 0. To draw the required pattern (Fig. 34a). Draw any line 0' A, and with 0' as centre and 0 B, 0 D (Fig. 346) as radii, describe arcs A D G, A' D' G' (Fig. 34c). Then take the METAL-PLATE WORK. 73 straight line length A B (Fig. 34a), and set it off as a chord from A (Fig. Sic) on the arc A D G. Do the same successively, from point B, with the straight line lengths (Fig. 34a) B C, C D, D E, E F, F G, the terminating point of each chord as set off, being the starting point for the next, the chord F G- (Fig. 34c) corresponding to the straight line length F A (Fig. 34a). Join (Fig. 34c) the points B, 0, D, E, F and G to 0' by lines cutting the arc A' D' G' in B', C, D', E', F', and G'. Join A B, B 0, C D, &c., and A' B', B' C, C D', &c., by straight lines ; then A D G G' D' A' is the pattern required. The frustum of pyramid is here hexagonal, but by this method the pattern for any regular pyramid cut parallel to its base can be drawn. The next problem will show methods for larger work. (37.) If 0'ADG(Fig. 34c), the pattern for the cone in which the frustum of pyramid is inscribed, be cut out of zinc or other metal, and small holes be punched at the points A, B, C, &c., and A', B', C, &c. ; and if the cone be then made up with the lines 0' A, 0' B', &c., marked on it inside, and wires be soldered from hole to hole successively to form the top and bottom of the truncated pyramid, then (1) the whole pyramid of which the truncated pyramid is a portion, (2) .the pyramid that is cut off, as well as (3) the truncated pyramid, will be clearly seen inscribed in the cone. The making such a model will amply repay any one who desires to be thoroughly conversant with the construction of articles of the kind now under consideration. PEOBLEM XXI. To draw, without using long radii, the pattern for an equal- tapering body made up of flat surfaces, the height and top and bottom being given. Again suppose the body to be hexagonal. Case I. — For ordinarily large work. Draw (Fig 35a) the plan as by last problem. Next join A B', and draw B' G perpendicular to A B' and equal to the given height of the body. Also draw B' H perpendicular to 74 METAL-PLATE WORK. B B' ; and equal to the given height. Join A G and B H ; then A G and B H are the true lengths of A B' and B B' respectively. Fig. 3oa. To draw the pattern. Draw (Fig. 356) B B' equal to B H (Fig. 35a). With B as centre and radius equal to BA METAL-PLATE WORK. 75 (Fig. 35a), and with B' as centre and radius equal to G A (Fig. 35a) describe arcs intersecting in C, right and left of BB'. With B as centre and the same GA as radius, and with B' as centre and radius B' A', describe arcs intersecting in C, right and left of B' B. Join B C, C C, C B' (Fig. 356) ; then BB'C'C is the pattern of the face BB'C'C in plan (Fig. 35a). The other faces C D C D', B C B' C, &c. (Fig. 35&) of the frustum are described in exactly similar manner, G A being the distance between diagonally opposite points of any face as well as of the face B B' C C. The triangles A B' G, BB'H, can be drawn apart from the plan, as shown at K and L. (38.) It should be observed that if the pattern is truly drawn, the top and bottom lines of each face will be parallel, as B C, B' C, of face B 0 B' C (Fig. 35 h) ; and that this gives an easy method of testing whether the pattern has been accurately drawn. Case II. — For very large work ; where it is inconvenient to draw the whole of the plan. Draw AB (Fig. 36a) equal in length to the end-line of one of the faces of the frustum at its larger end, and produce it both ways. With B as centre and radius B A describe a semicircle, which divide into as many equal parts as the frustum has faces (Problem IX.). Here it is hexagonal, and the points of division working from point 1, are 2, C, 4, &c. Through the second division point, here C, draw a line to B, then A B C is the angle made in plan by two faces of the frustum one with another, and A B, B C are two adjacent end-lines of the plan of its larger ends. Bisect the angle ABC (Problem VIII.) by B E ; and draw a line C C from C making the angle C C B equal to the angle CBE (Problem I.). On B C set off B D equal to the end-line of one of the faces of the frustum at its smaller end ; and draw D C parallel to B E, cutting C C in C. Through C draw C B' parallel to C B and meeting B E in B' ; and draw B' A' parallel to B A and equal to B D or C B'. Join A' A, and we 76 METAL-PLATE WOEK. have in A'ABCC the plan of two adjacent faces of the tapering body or truncated pyramid. Next from B' let fall B' G perpendicular to A B, and make G F equal to the given height of the frustum. Join FB', then F B' is the true length of B'G. Through B' draw B'H perpendicular to B' B and equal to the given height. Join H B, then H B is the true length of B B' one of the edges of the frustum. To draw the pattern. Draw (Fig. S6h) B B' equal to H B (Fig. 36a). With B and B' as centres and radii respectively B G and F B' (Fig. 36a) describe arcs intersecting in G (Fig. 86&). Join B G and produce it making B A equal to B A (Fig. 36a). Through B' draw B' A' parallel to B A and equal to B' A' (Fig. 36a), and join A A' ; then B A A' B' will be the pattern of one face of the frustum. The adjacent face B C C B' is drawn in similar manner. The fig. shows the pattern for two faces only of the equal-tapering body, because in cases where this method would have to be employed, two faces are probably the utmost that could be cut out in one piece. Sometimes each face would have to be cut out separately, or perhaps even one face would have to be in portions. Any point in B' A' (Fig. 36a) instead of B' can be chosen from which to let fall a perpendicular to B A, and the true length of B' G found as explained. The choice of position depends upon the means at hand for drawing large METAL-PLATE WOKK. 77 arcs, the radius of llie arc B' G(Fig. 366) increasing in length as the point G approaches nearer to A. If the pattern be truly drawn, B A will be perpendicular to B' G. It must not be forgotten that these methods are, both of them, quite independent of the number of the sides of the pyramid. Also it should be noted that BE does not of necessity pass through a division point, nor of necessity is C C parallel to A B. These are coincidences arising from the frustum being here hexagonal. PEOBLEM XXII. To draw the pattern for an ohlong or square equal-tapering body with top and base parallel, and having flat sides and ends. {The bottom is here taken as- part of the body, and the whole pattern is in one piece.) Note. — TMs problem will be solved in the problem next following. We adopt this course because the article there treated of is so important an example of the oblong equal-tapering bodies in question, that it is desirable to make that, the special problem, the primary one, and this, the general problem, secondary to it. Its solution will be found at the end of Case I. PEOBLEM XXIII. To draw the pattern for a bahing-pan. A baking-pan has not only to be water-tight, but also to stand heat ; hence when made in one piece the corners are seamless. Case I. — Where the length and width of the bottom, the width of the top, and the slant are given. Draw two lines XX, Y Y, intersecting at right angles in 0 (Fig. 37) ; make 0 A' and 0 B' each equal to half the length of the bottom, and 0 C and 0 D' each equal to half its width. Through C and D' draw lines parallel to Y Y ; also through A' and B' draw lines parallel to X X and inter- METAL-PLATE WOEK. 79 secting the lines drawn through C and D' ; we get hy this a rectangular figure, which is the shape of the bottom. Make A' A, B' B, C C, and D' D each equal to the given slant ; through B and A draw E F and G H parallel to X X ; through C and D draw S T and E P parallel to Y Y ; and make B E, B F, A H, and A G each equal to half the given width of the top. Join Q F and with Q, which is one of the angular points of the bottom, as centre and radius Q F describe an arc cutting PR in P. Join QP (the working can here be best followed in Fig. 38) ; bisect the angle F Q P by Q M (Problem VIII., Chap. II.) ; and draw a line Q L making with F Q an angle equal to the angle F Q M, The readiest way of doing this is by continuing the arc P F to L, then setting off F L equal to F M, and joining L Q. Now on B B' set off B N equal to the thickness of the wire to be used for wiring, and through N draw NF' parallel to E F (Fig. 37) and cutting Q L and Q F in V and F' ; make Q P' equal to Q F', and Q M' equal to Q V ; and join P' M' and M' F'. Repeat this construction for the other three corners and the pattern will be completed. This is not done in Fig. 37, for a reason that will appear presently. We shall quite under- stand the corners if we follow the letters of the Q corner in Fig. 38. These are B F F' M' P' P. The dotted lines drawn outside the pattern (Fig. 37) parallel to E F, P R, G H, and S T show the allowance for fold for wiring. It is important that the ends of this allowance for fold shall be drawn, as, for instance, P K (both figs.), perpendicular to their respec- tive edges. Now as to bending up to form the pan. When the end adjacent and the side adjacent to the corner Q come to be bent up on B' Q and D' Q so as to bring F Q and P Q into junction, it is evident that as F Q and P Q are equal we shall obtain a true comer. To bring F Q and P Q into junction it is likewise manifest that the pattern will have also to be bent on the lines F' Q, P' Q, and M' Q. This fold on each other of P'QM' and F'QM' is generally still further bent round against Q V. 80 METAL-PLATE WOEK, (39.) The truth of the pan when completed, and the ease ■with which its wiring can he carried out, depend entirely on the accuracy of the pattern at the corners. This must never be forgotten. In marking out a pattern, only one corner need be drawn, as the like to it can be cut out separately and used to mark the remaining corners by. The points E, S, and T, the fixing of which will aid in this marking out, can be readily found, thus : — For the E corner to come up true it is clear that D E will have to be equal to D P, from which we learn that D P is half the length of the top. Having then determined the point P we have simply to set off D E, C S, and C T each equal to D P. If the pattern Fig. 37 were completed with the corners as at E, S, and T, that would be the solution of Problem XXII., and the ends and sides being bent up, we should get an oblong equal-tapering body with top and bottom parallel and having flat sides and ends. Case II. — Where the length and width of the bottom, the length of the top, and the slant are given. The only difference between this case and the preceding is that D P (Fig. 87), half the length of the top, is known instead of B F the half- width of top. To find the half-width of top ; with Q as centre and Q P as radius describe an arc cutting B F in F. Then B F will be half the width of the top, just as (§ 39, previous case) we saw that D P was half the length of the top. The remainder of the construction is now as in Case I. (40.) It will be evident that, in this problem, choice of dimensions is not altogether arbitrary. The lines Q F and Q P the meeting of which forms the corner, must always be of the same length. This limits the choice ; for with the dimensions of bottom given, and the slant, and the width of the top, the length of the top cannot be fixed at pleasure, but must be such as will bring Q F and Q P into junction ; and vice versa if the length of the top is given. It is un- necessary to follow the limit with other data. METAL-PLATE WOEK. 81 Case III.— Where the length and width of the top, and the slant and the height are given. The data in this case and the next are the usual data when a pan has to be made to order. The difference between this case and the preceding is that the size of the bottom is not given, but has to be determined from the data. Excepting as to finding the dimensions of this, the case is the same as Cases I. and II. All that we have now to do is therefore to find the dimensions of the bottom, thus : — Draw O C (Fig. 39) equal to half the given length of the top, and through C draw C B perpendicular to 0 C and equal to the given height. With B as centre and radius equal to the given slant describe an arc cutting 0 C in B'. Then O B' is the required half-length of the bottom. Next draw 0 E (Fig. 40) equal to half the given width of the top, and through E draw E D perpendicular to O E and equal to the given height. With D as centre and radius equal to the given slant describe an arc cutting 0 E in D'. Then 0 D' is the required half- width of the bottom. Case IV.— Where the length and width of the top and the length and inclination of the slant are given. This is a modification of Case III. Let B B' (Fig. 39) be the length of the given slant, and the angle that B B' make with B' C be the inclination of the slant. Through B draw B C perpendicular to B' C. Then half the given length of the top less B' C will be the required half-length of the bottom; and similarly half the given width of the top less B' C will be the required half-width of the bottom. Fig. 39. Fig. 40. B G 82 METAL-PLATE WOEK. PEOBLEM XXIV. To draw the pattern for an equal-tapering body with top and base parallel, and having flat sides and ends {same as Problem XXIL), but with bottom, sides, and ends in separate pieces ; the length and width of the bottom, the width of the top, and the slant being given. To draw the pattern for the end. Draw B B' (Fig. 41) equal to the given slant, and through B and B' draw lines C D and C D' perpendicular to B B'. Make B C and B D each equal to half the given width of the top ; also make B' C and B' D' each equal to half the given width of the bottom. Join CC and DD'; then C'CDD' will be the end pattern required. Fig. 41. Fig. 42. End Pattern. Side Pattern. To draw the pattern for the side. Draw E E' (Fig. 42) equal to the given slant, and through E and E' draw F G and F' G' perpendicular to E E'. Make E' F' and E' G' each equal to half the given length of the bottom, and with F' and G' as centres and radius D'D (Fig. 41) describe arcs cutting F G in points F and G. Join F' F and G' G ; then F' F G G' will he the side pattern required. It should be noted that, as in the preceding problem, the width of the top determines the length of the top and vice versa, also that lines such as D D', G G' must be equal. METAL-PLATE WORK. 83 (41.) If the ends of tlie body are seamed (' knocked up ') on to the sides, as is usual, twice the allowance for lap shown in Fig. 41, must be added to the figure C C D' D. Similarly if the sides are seamed on to the ends, a like double allowance for lai> must be added to FF'G'G, instead of to the end pattern. 84 METAL-PLATE WOKK. CHAPTEE VIL Patterns for Equal-tapering Articles op Flat and Curved Surfaces combined. (Class I. Subdivision c.) From wHat has been stated about the plans of equal- tapering bodies, and from d, p. 55, it will be evident that the curved surfaces of the articles now to be dealt with, are portions of frusta of cones. PEOBLEM XXV. To draw the pattern for an equal-tapering body with top and bottom parallel, and having flat sides and equal semicircular ends {an ' equal-end ' pan, for instance), the dimensions of the top and bottom of the body and its height being given. Four cases will be treated of ; three in this problem and one in the problem following. Cj^se I. — Patterns when the body is to be made up of four pieces. We may suppose the article to be a pan. Having drawn (Fig. 43) the plan AECSA'D'C'B' by the method of Problem XVI. (as well as the plan, the lines of its construction are shown in the fig.), let us suppose the seams are to be at A, B, C, and D, where indeed they are usually placed. Then we shall require one pattern for the flat sides and another for the curved ends. To draw the pattern for the sides. Anywhere in A D (Fig. id) and perpendicular to it draw METAL-PL A.TE WORK. 85 F F', and make F H equal to the given height ; join F' H ; then F'H is the length of the slant of the article, and therefore the width of the pattern for the sides. Fw. 43. Draw A A' (Fig. 44) equal to F' H (Fig. 43). Through A and A' draw lines perpendicular to A A'. Make A D (Fig. 44) equal to A D (Fig. 43), and draw D D' parallel to Fig. 44. \U V Side Pattern. A A'. The rectangle ADD' A' will be the side pattern required. The fig. also shows extras for lap. To draw the pattern for the ends. Draw (Fig. 45a) two lines DA, OA, perpendicular to one another and meeting in A ; make D A equal to O D (Fig. 43), the larger of the radii of the semicircular ends ; and on A 0 set off A G equal to the given height. Draw G D' perpen- dicular to A G and equal to 0 D' (Fig. 43), the smaller of the radii of the semicircular ends ; join D D' and produce it, meeting AO in 0. Then with any point 0' (Fig. 456), as 86 METAL-PLATE WOEK. centre, and 0 D (Fig. 45a) as radius, describe an arc D C, and with tlie same centre and radius 0 D' (Fig. 45a) describe an arc D' C. Draw any line D 0', cutting the arcs in points D and D'. Divide D E (Fig. 43) into any number of equal parts, say three. From D (Fig. 45&) mark off these three Fig. 45a, Fig. 45&. 0' dimensions to E, make E C equal to D E, and join C 0' ; then D C C D' will be the end pattern required. If E be joined to 0', then D D' E' E will be half the pattern. The centre line E E' is very useful, because, in making up the article, the point E' must meet the line S E (axis) of Fig. 43, otherwise the body will be twisted in consequence of the bottom not being true with the ends. Case II. — Pattern when the body is to be made up of two pieces. Secondly, suppose the article can be made in two pieces (halves), with the seams at E and G (Fig. 43) ; the line E G (part only of it shown) being the bisecting line of the plan. It will be seen by inspection of the fig. that we require one pattern only, namely, a pattern that takes in one entire end of the article with two half-sides attached. METAL-PLATE WORK. 87 Draw the end pattern DD'C C (Fig. 46) in precisely the same manner that it is drawn in Fig. 45&. Through C and C draw indefinite lines C G C Gr', perpendicular to C C, and through D and D' draw indefinite lines D E, D' E' perpen- dicular to D D'. Make D E, D' E' each equal to D E (Fig. 43) Fig. 46. and join EE'; also make CGr, C'G', each equal to CG (Fig. 43), and join G G'. Then E E' G' G will be the pattern required. Case III. — Pattern when the body can be made of one piece. Thirdly, suppose the article can be made in one piece, with the seam at S (Fig. 43). Then evidently the pattern will be made up of the pattern of one entire end, side patterns attached to this, and half an end pattern attached to each side pattern. Draw DD'C'C (Fig. 47) the end pattern. Through C and C draw lines perpendicular to C C, and each equal to C B (Fig. 43) ; and join B B'. Produce B B', and with B as centre and 0 D (Fig. 45a), the larger of the radii for the end pattern, as radius, describe an arc cutting the produced line 88 METAL-PLATE WOEK. in 0. With 0 as centre and 0 B and 0 B' as radii respec- tively, draw arcs B S, B' S'. Make B S equal to D R (Fig. 47) ■■--10 and join S 0. Eepeating this construction for D D' A' A the other side pattern, and A A' S' S the remaining half-end pattern, will complete the pattern required. METAL-PLATE WORK. 89 PEOBLEM XXVI. To rfrat., without long radii, the pattern for an equal-tapering body with top and bottom parallel, and having flat sides and equal semicircular ends; the dimensions of the top and bottom of the body and its height being given. This problem is a fourth case of the preceding, and exceedingly useful where the work is so large that it is inconvenient to draw the whole of the plan, and to use long radii. => Draw half the plan (Fig. 48a). Divide D C into six or w .^'^vl^f ^""^ •'■^^^ ^' ^' 0' cutting U U in 1 , 2 , &c., and join D 1'. Draw D E perpendicular to FiQ, 48a. Fig. 486. D 1' and equal to the given height, and join E 1'. (The line 1 to 1' appears to, but does not, coincide with E 1'.) Then E 1' may be taken as the true length of D 1'. Next, producing as necessary, make D A equal to the given height. Joining D' to A gives the true length of D D'. To draw the end pattern. Draw Fig. (48&) D D' equal to D'A (Fig. 48a). With D and D' as centres, and radii respectively D 1 and E 1' (Fig. 48a) describe arcs intersecting 90 METAL-PLATE WOEK. in point 1 (Fig. 485). With D' and D as centres and radn respectively D' 1' and El' (Fig. 48a) describe arcs inter- secting in point 1' (Fig. 486). By nsing points 1 and 1 (Fig. 486) as centres, instead of D and D', and repeating the construction, the points 2 and 2' can be found. Next, using points 2 and 2' as centres and repeating the construction, find points 3 and 3', and so on for the remaining points necessary to complete the end pattern, which is completed by joining the various points, as 3 to 3' by a straight Ime ; D, 1, 2, and 3, by a line of regular curve ; and D', 1', 2 and 3', also by a line of regular curve. Only half the end pattern is shown in Fig. 486. The side pattern can be drawn as shown "in Fig. 44. PKOBLEM XXVII. To draw the pattern for an equal-tapering body with top and bottom parallel, and having flat sides and ends, and round corners {an oblong pan with round corners, for instance) ; the height and the dimensions of the top and bottom being given. Again four cases will be treated of; three in this problem, and one in the problem following. Case I.— Pattern when the body is to be made up of four pieces. The plan of the article, with lines of construction, drawn by the method of Problem XVII., is shown in Fig. 49. ^ We will suppose the seams are to be at P, S, Q, and E, that is, at the middle of the sides and ends. The pattern required is therefore one containing a round corner, with a half-end and a half-side pattern attached to it. The best course to take is to draw the pattern for the round corner first, which, as will be seen from the plan, is a quarter of a frustum of a cone. METAL-PLATE WORK. 91 Draw two lines (Fig. 50a) OA, OA, meeting perpendi- cularly to one another in A ; make A C equal to 0 A (Pig. 49) the radius of the larger arc of the plan of one of the corners, Fig. 49, and make A D equal to the given height of the body. Draw D E perpendicular to A D and equal to the radius of the smaller arc of the plan of a corner, and join C E and produce Fig. 50a. Fig. 506. it to meet A 0 in 0. With any point 0' (Fig. 60&) as centre and radius equal to 0 C (Fig. 50a), describe an arc A B, and with the same centre and radius equal to O E (Fig. 60a) describe an arc A' B'. Draw any line A 0' cutting the arcs 92 METAL-PLATE WOKK. in the points A and A', and make A B equal to A B (Fig. 49) by marking off the same number of equal parts along A B (Fig. 506) that we divide (arbitrarily) AB (Fig. 49) into. Join B O', cutting in B' the arc A' B'. Through B and B' draw B P, B' P' perpendicular to B B', make B P equal to B P (Fig. 49) and draw P P' perpendicular to B P. Through A and A' draw A R and A'' R' perpendicular to A A', make A R equal to A E (Fig. 49) and draw R R' perpendicular to A R. Then R R' P' P will be the pattern of one round corner, with a half-end and a half-side pattern attached left and right. Case II. — Pattern when the body is to be made up of two pieces. Now suppose the seams are to be at the middle of each end, at R and S (Fig. 49). The pattern required will then be of twice the amount shown in Fig. 50&. It will be found best to commence with the side pattern. Draw in the plan (Fig. 49), any line X X perpendicular to the Q side-line ; then X and X' will be plans of corre- sponding points (§ 26). Make XF equal to the given Fig. 51. height of the body and join X' F. Then X' F is the length of a slant of the body. Draw B B' (Fig. 51) equal to X' F (Fig. 49), and through B and B' draw B C and B' C perpen- dicular to B B'. Make B C equal to B C (Fig. 49) and draw METAL-PLATE WORK. 93 C C perpendicular to B C ; then B C C B' will be the pattern for the side. We have now to join on to this, at B B' and C C, the patterns for the round corners, which can be done thus. Produce B B' and C C, and make B 0' and C 0" each equal to 0 C (Fig. 50a) the radius of the larger arc of the corner pattern. With 0' and 0" as centres and O'B as radius, describe arcs B A and C D, and with the same centres and O'B' as radius, describe arcs B'A' and CD'. Make BA and C D equal each to B A in the plan (Fig. 49), and join AC, cutting the arc B' A' in A', and D 0" cutting the arc C D' in D'. Through A and A' draw A E and A' E' perpendicular to A A' ; make A E equal to A E (Fig. 49), and draw E E' per- pendicular to A E. Next through D and D' draw D S and D' S' perpendicular to D D' ; make D S equal to D S (Fig. 49), and draw S S' perpendicular to D S. Then E E' S' S will be the pattern required. It should be noted that .0' B' and 0" C should be each equal to 0 E (Fig. 50a), or the pattern will not be true. This gives a means of testing its accuracy. Case III. — Pattern when the body is to be made up of one piece. When the body is made in one piece it is usual to have the seam at S (Fig. 49). It is now best to commence with the end pattern. Draw A A' (Fig. 52) equal to X' F (Fig. 49), which, the body being equal-tapering, is the length of its slant any- where, and draw A H and A' H' perpendicular to A A'. Make A H equal to A H (Fig. 49) and draw H H' perpendi- cular to A H ; then A A' H' H will be the end pattern. Next produce A A', and make A 0' equal to C 0 (Fig. 50a). Then O' will be the centre for the arcs of the corner pattern, which, drawn by Case II. of this problem, can now be attached at A A'. Through B and B' draw B C and B' C perpendicular to B B' ; make B C equal to B C (Fig. 49), and draw C C perpendicular to B C. That completes the addition of the side pattern 94 METAL-PLATE WOEK. B B' C 0 to the corner pattern. Next produce C C, mark the necessary centre, and attach to C C the comer pattern CDD'C, just as ABB' A' was attached to A A'. Then through D and D' draw D S and D' S' perpendicular to D D' ; make D S equal to D S (Fig. 49) and draw S S' perpendicular Fig. 52. to D S ; this adds half an end-pattern to D D'. By a repeti- tion of the foregoing working on the H H' side of the end- pattern H H' A' A, the portion H H' S' S on the H H' side may be drawn, and the one-piece pattern S H A S S' A' H' S' of the body we are treating of completed. PROBLEM XXVIII. To ,A distance C 0. With 0 as centre and radu 00 and LTZ C B and C B: Make 0 B eqnal to C B (Fig. 64) 100 METAL-PLATE WOEK. and join B 0, cutting C B' in B'. Then C B B' C is a half-end pattern attached to C C. The half-end pattern E A A' E' is Fig. 57. added at E E' by repeating the construction just described. This completes A B B' A' the pattern required. Case III. — Pattern when the body is to be made up of one piece. We will put the seam at A (Fig. 54), the middle of one end. Draw C D D' C (Fig. 58), an end pattern in the way described in Case I. Produce C C and make C 0 equal to METAL-PLATE WOKK, 101 C 0 (Fig. 56a), the radius for a side pattern. With 0 as centre and radii 0 C and 0 C describe arcs C E, C E'. Make C E equal to CE (Fig. 54) and join E 0 cutting C'E' in E'. Make E 0' equal to B 0 (Fig. 55a) and with 0' E as radius 102 METAL-PLATE WORK. describe an arc E A. With same centre and O'E' as radius describe an arc E' A'. Make E A equal to E A (Fig. 54) and join AO', cutting E' A' in A'. The remainder D AA'D' of the pattern can be drawn by repeating the foregoing con- struction. The figure A C D A A' D' C A' will be the pattern required. PEOBLEM XXX. To draw, without long radii, the pattern for an oval equal- tapering body with top and bottom parallel, the height and the dimensions of the top and bottom being given. This is a problem which will be found very useful for large work, especially with the pattern for the sides, the radius for which is often of most inconvenient length. End Pattern. J)' To draw the ends' pattern. Draw (Fig. 59a) the plan C D D' C of the end of the body. Divide C D into four or more equal parts, and join, 1, 2, &c., METAL-PLATE WOEK. 103 to Q, by lines cutting CD' in 1', 2', &c. From C draw CD perpendicular to C C and equal to the given height. Join C' D, then C D is the true length of C C. Join C 1 ; draw V E perpendicular to it and equal to the given height ; and ioin C E. Then C E may he taken as the true length otL I. Draw C C (Fig. 59b) equal to C D (Fig. 59a). With C and C as centres and radii respectively equal to 01 and Lih (Fig. 59a) describe arcs intersecting in point 1. With O and C as centres and radii respectively equal to C' 1' and (Fig. 59a) describe arcs intersecting in point 1'. By using points 1 and 1' as centres and repeating the construction, points 2 and 2' can be found. Similarly find points 3 and 3 , and D and D'. Join D D', draw a regular curve from G through 1, 2, and 3, to D, and another ^^g^^^r curve from C through r, 2', and 3', ix. D'. The figure C C D' D is the pattern required. Fig. 60a. Side Pattern. To draw the sides' pattern. Draw (Fig. 60a) the plan E 0 C E' of the side of the body. Divide E C into four or more equal parts, and join 1, 2, &c„ to P, by lines cutting E' C in 1', 2', &c. From E draw E F 104 METAL-PLATE WOKK. perpendicular to E E' and equal to the given height. Join E' F ; then E' F is the true length of E E'. Next join E 1' ; draw I'D perpendicular to El' and equal to the given height, and join ED; then ED maybe taken as the true length of El'. Now draw (Fig. 606) EE' equal to E' F (Fig. 60a), and with E and E' as centres and radii respec- tively equal to E 1 and DE (Fig. 60a) describe arcs inter- secting in point 1. With E' and E as centres and radii respectively equal to E' 1' and D E (Fig. 60a) describe arcs intersecting m point 1'. Eepeating the construction with points 1 and 1' as centres, the points 2 and 2' can be found • and similarly the points 3 and 3' and the points C and c'. Join C C ; draw a regular curve from E through 1, 2, and 3, to C, and another from E' through 1', 2', and 3', to C The figure E E' C C will be the pattern required. METAL -PLATE WOEK. 105 Book II. CHAPTEE I. Patterns for Eound Articles of Unequal Taper or Inclination of Slant. (Class II. Subdivision a.) (42.) We stated at the commencement of the preceding division of our subject that it was advisable, in order that the rules for the setting out of patterns for articles having equal slant or taper should be better understood and remem- bered, to consider the principles on which the rules were based ; and the remark is equally true and of greater im- portance in respect of the rules for the setting out of patterns for articles having unequal slant or taper. We have shown that the basis of the rules for the setting out of patterns for equal tapering bodies is the right cone ; we purpose showing that the basis of the rules for the setting out of patterns for articles of unequal slant or taper is what is called the oblique cone. The consideration of the cone apart from its species, that is, apart from whether it is right or oblique, which becomes now necessary, immediately follows. Definition. (43.) Cone. — A cone is a solid of which one extremity, the base, is a circle, and the other extremity is a point, the apex. The line joining the apex and the centre of the base is the axis of the cone. (44.) Given a circle and a point in the line passing through the centre of the circle at right angles to its plane. If an indefinite straight line, passing always through the given point, move through the circumference of the given circle, 106 METAL-PLATE WOEK. there will be thereby generated between the point and the circle, a solid ; this solid is the right cone. (45.) If, all other conditions remaining the same, the given point is out of the line passing through the centre of the circle at right angles to its plane, the solid then generated will be the oblique cone. (4:6.) Figs. 1 and 2 represent oblique cones. The lines V A, V C, V F, and V G (either fig.) drawn from the apex V to the circumference of the base of the cone, are portions of the generating line at successive stages of its revolution. It is but a step from this and will be a convenience, to regard these lines, first, as each of them part of an independent generating line, and then as each of them a complete generating line. Shaded representations of various oblique cones will be found later on. (47.) Comparing now the right and oblique cones. A right cone may be said to be made up of an infinite number of equal generating lines, and an oblique cone of an infinite number of unequal generating lines. (48.) If a right cone is placed on a horizontal plane, the apex is vertically over the centre of the base. (49.) If when a cone is placed on a horizontal plane the apex is not vertically over the centre of the base, the cone is METAL-PLATE WOKE. 107 oblique. Hence all cones not right cones are oMique cones. Hence also a cone is oblique if its axis is not at right angles to every diameter of its base. (50.) In the right cone, any plane containing the axis is perpendicular to the base of the cone, and contains two generating lines (see Figs, la and 2a, p. 25). (51.) In the oblique cone, only one plane containing the axis is perpendicular to the base of the cone, and this plane contains its longest and shortest generating lines. In Fig. 1 or 2 the lines V A and Y G are respectively the longest and shortest generating lines, and the plane that contains these lines contains also V 0 the axis of the cone. (52.) The obliquity of a cone is measured by means of the angle that its axis makes with that radius of the base that terminates in the extremity of the shortest generating line. Thus the angle V 0 G gives the obliquity of either cone V A G. The angle V 0 G also gives the inclination of the axis. As the angle V A G is in the same plane as V 0 G and smaller than that angle, it will be seen that not only is V A the longest generating line, but it is also the line of greatest inclination on the cone. Similarly as the angle that VG makes with A G produced is greater than V 0 G, the line V G is not only the shortest generating line, but also the line of least inclination on the cone. (53.) The plane that contains the longest and shortest generating lines bisects the cone; consequently the generating lines of either half are, pair for pair, equal to one another. (54.) If the elevation (see § 21, p. 48) of an oblique cone be drawn on a plane parallel to the bisecting plane, the elevations of the longest and shortest generating lines will be of the same lengths respectively as those lines. Thus if the triangle V A G (either fig.) be regarded as the elevation of the cone represented, then V A will be the true length of the longest generating line of the cone, and V G of the shortest. In speaking in the pages that follow, of elevation with regard to the oblique cone, we shall always suppose it to be on a plane parallel to the bisecting plane. (55.) If the hypotenuse of a right-angled triangle represent 108 METAL-PLATE WORK. a generating line of any cone, right or ohlique, then one of the sides containing the right angle is equal in length to the plan of that line, and the other side is equal to the height (distance between the extremities) of any elevation of it. See, p. 25, the triangles 0 B A, 0 G A, 0 E A, 0 D C, 0 K C, 0 F C, and, p. 109, the triangles V A V, V B V, V C V, &c. See also § 22a. (56.) As the generating lines of the oblique cone vary in length, the setting out of patterns of articles whose basis is the oblique cone (that is to say, the development of the curved surfaces of such articles) differs from that apper- taining to articles in which the right cone is involved. The principles, however, are the same in both cases. In develop- ments of the right cone, if a number of its generating lines are laid out in one plane, then as they are all equal and have a point (the apex) in common, the curve joining their extremities is an arc of a circle of radius equal to a generating line ; while in developments of the oblique cone, as the generating lines, though they still have a point in common, vary in length, the individual lengths of a number of them must be found, as well as the distances apart of their extremities, before, by being laid out in one plane at their proper distances apart, a curve can be drawn through their extremities, and the required pattern ascertained. The problem which follows gives the working of this in full. PEOBLEM 1. To draw the pattern {develop the surface) of an oblique cone, the inclination of the axis (§ 52), its length and the diameter of the base of the cone being given. First (Fig. 3) draw V A G the elevation of the cone, and AdQ half the plan of its base ; thus. Draw any line X X, and at any point 0 in it, make the angle VOX equal to the given inclination of the axis (a line 0 V is omitted in the fig. to avoid confusion), and make 0 V equal to the given length of the axis. "With 0 as centre and half the METAL-PLATE WOEK. 109 given diameter of the base as radius describe a semicircle AdG cutting XX in A and G. Join AV, GV, then V A G is the elevation required, and A G is the half-plan of base. Next divide AdG into any convenient number of parts, equal or unequal. The division here is into six, and the parts are equal ; to make them so being an advantage. Now let fall V V perpendicular to XX; and join V to the division points h, c, d, e, f (to save multiplicity of lines this is only partially shown in the fig-)« The lines VA, V6, V c, &c., will be the lengths in plan of seven lines from apex to base of cone, that is, of seven generating lines. Now with V as centre and radii successively V &, V c, V c?, V e, and V/, describe arcs respectively cutting X X in B, C, D, E, and F. Join y B, V C, &c. ; then as V V is the height of any elevation of either of the seven generating lines (see § 22a, p. 48, and § 55, p. 107) and as V A, Y B, &c., are their plan lengths, we have in V A, V B, V C, &c., their true lengths. V G is not only so however, it is (§ 54) the shortest of all the generating lines of the cone. To draw the pattern of the cone with the seam to corre- 110 METAL-PLATE WORK. spond with V G the shortest generating line. Draw (Fig. 4) V A equal to V A (Fig. 3), and with V as centre and V B, V C, V D, V E, V F, and V G (Fig. 3) successively as radii describe arcs, respectively, hh, cc, dd, ee,ff, and g g. With A as centre and radius equal to A 6 (A 6 being the distance apart on the round of the cone at its base of the generating lines VA and V'B') (Fig. 3) describe an arc cutting the arc 6 h right and left of Y' A in B and B. With Fig. 4. A — 5^ f---^^ ■ — 7* 1 / ^ ' 1- XZ-^—^ — TVi ^ \ i ^ \ \ \ ^ ^ ' \ ^ \ ' \ ^ \ \ 1 \ / 1 ' / ' / / / 1 / ' ' ' 1 1 / \ \ \ ^ M 11/ / / / / / / / / / / / ,' / / ' / \ ^ \ \ \ \ \V II'// ' '// I''// II /// w i"/ these points B and B as centres and radius as before (the distances apart of the extremities of the generating lines at the base of the cone being all equal), describe arcs cutting the arc cc right and left of V A in 0 and 0. With same radius and the last-named points as centres describe arcs cutting dd right and left of V A in D and D. With D and D as centres and same radius describe arcs cutting e e right and left of V A in E and E, and with E and E as METAL-PLATE WOKK. Ill centres and same radius describe arcs cutting // right and left of y A in F and F. Similarly with same radius and F and F as centres find points G and G. Draw through A and the points B, C, D, E, F and G, right and left of V A an unbroken curved line. Also join G V right and left of V A, then G A G V will be the pattern required. The dotted lines V B, V C, V D, &c., are not necessary for the solution of the problem, and are only drawn to show the position of the seven generating lines on the developed surface. (57.) As the plane of the elevation V A G bisects the cone (§§ 53 and 54), it is clear that the seven generating lines found, . correspond with seven other generating lines on the other half of the cone; so that in finding them for one half of the cone, we find them for the other, as the halves necessarily develop alike. (58.) Bound articles of unequal slant or taper having their ends parallel are portions (frusta) of oblique cones. Definition. (59.) Frustum. — If an oblique cone be cut by a plane parallel to its base, then the part containirg the apex is still an oblique cone, as V A' C G' (Fig. 5), and the 112 METAL-PLATE "WORK. part A' A C G G' containiBg the base is a frustum of an oblique cone. A shaded representation of such frustum will be found in Fig. 15. This frustum, however, differs from that of the right cone (§ 12, p. 33), in that, though having circular ends, its sides are not of equal, but of un- equal slant. Conversely, a round unequally tapering body, having its top and base parallel, is a frustum of an oblique cone. A tapering piece of pipe A' A G G' (Fig. 6) joining two cylindrical pieces which are not in line with each other is a frustum of an oblique cone. (60.) Eeferring to Fig. 5, it is evident that if the pattern for the larger cone, V A G, and the pattern for the smaller cone, V A G', be drawn from a common centre V (Fig. 9), the figure G' G A G G' (Fig. 9) will be the pattern for the portion AGG'A (Fig. 5) of the cone V A G. The line C C shows a generating line of the frustum (&, p. 126). Fig. 76. Fig. la. (61.) A case that not unfrequently occurs needs mention here. Suppose the diameters of the top and base of a round METAL-PLATE WORK, 113 unequal-tapering tody of parallel ends are very nearly equal, then it is evident that the apex of the oblique cone of which the body is a portion will be a long distance off, and, if the diameters be equal, then the body becomes what is called an oblique cylinder, (a cylinder is a round body without any taper at all). Of course this is an extreme case, but it is quite an admissible one. For, if the diameters of the ends of the body differ by only ^^^o i^^ch, then, clearly, however short the body may be, we are dealing with a frustum of an oblique cone, although so nearly a cylinder that, for almost any purpose occurring in practice, it could be treated as a cylinder. Later on the advantages of looking upon the oblique cylinder as a special case of frustum of an oblique cone, and considering its generating lines as parallel,^ will be seen. Such a frustum is represented in A A' G' G in Figs. 7a and 7h, the line 0 0' being the axis of the frustum and C C and r F' generating lines. A construction easily dealing with its development will be given presently. PEOBLEM II. To draw the pattern of a round unequal-tapering hody with top and base parallel (frustum of an oblique cone, as in Fig. 6), the diameters of top and base, the height, and the inclination of the longest generating line being given. First (Fig. 8) draw the elevation and half the plan of the body, thus° Draw a line XX', and at any point A in it, make the angle X' A A' equal to the given inclination of the longest generating line. At a distance from X X' equal to the given height, draw a line A'G' parallel to XX'. From the point A' where A' G' cuts A A', make A' G' equal to the diameter of the top, also make AG equal to the diameter of the base ; and join G G'. Then A A' G' G is the elevation of the body. Now on A G describe a semicircle A d G, this will be half the plan of the base. Produce A A', G G', to intersect in V ; this point will be the elevation of 114 METAL-PLATE WORK. the apex of the cone of which the body is a portion. Through y draw V V perpendicular to X X' ; divide the semicircle into any convenient number of equal parts, here six, in the points h, c, d, e, and/; with V as centre and radii successively Yb,Yc,Yd, &c., describe arcs respectively cutting X X in B, C, D, &c. ; and join B, C, D, &c., to V by lines cutting A' G' m B', C, D', E', and F'. Then B B', C C, D D', &c, will be (see construction of last problem) the true lengths of various generating lines of the frustum (b, p. 126). Now to draw the pattern (Fig. 9) so that the seam shall correspond with GG'(Fig. 8) the shortest generating line Draw (Fig. 9) V A equal to V'A (Fig. 8) and with V (Fig. 9) as centre and radii successively equal to V B V 0 Y' B, Y' E. V F, and Y' G (Fig. 8), describe, respectively, arcs bh, cc,dd,ee, ff, and g g. With A as centre and radius equal A 6 (Fig. 8) (see preceding problem for the reason of this), describe arcs cutting the arc & 6 right and Igft of V'A in B and B. With these points, B and B as centres and radius as before, describe arcs cutting the arc c c right and left of V'A in C and C. With same radius and the last- METAL-PLATE WORK. 115 named points as centres, describe arcs cutting dd right and left of V A in D and D. With D and D as centres and same radius, describe arcs cutting ee right and left of V A in E and E, and with E and E as centres and same radius describe arcs cutting //in F and F. Similarly, with same radius and E and E as centres find points G and G. Join the points B, C, D, E, &c., right and left of V A to V. With V as centre and \' A' (Eig. 8) as radius describe an arc Fig. 9. cutting V A in A'. With same centre and V B' (Eig. 8) as radius describe an arc h' b' cutting V B right and left of V A in B'. With same centre and V C (Fig. 8) as radius describe an arc c' c' cutting V C right and left of V A in C Similarly, with the same centre and V D', V B', V E', and V G' (Fig. 8) successively as radii describe, respectively, arcs d'd', e'e\ff', and g' g', cutting V'D, V'E, V'F, andV'G, right and left of V'A respectively in D', E', F', and G'. Draw through A and the points B, C, D, E, F, and G, right 116 METAL-PLATE WOKK. and left of V A an unbroken curved line. Also draw through A| and the points B', C, D', E', F', and G', right and left of V A an unbroken curved line. Then G A G G' A' G' will be the pattern required. The small semicircle (Fig. 8) and perpendicular lines from its extremities are not needed in this problem, but are introduced in illustration of that next following. (62.) In the applications of the oblique cone, it is generally in the form next foUowing that the problem presents itself. PROBLEM III. To draw the pattern of a round unequal-tapering body with top and base parallel (frustum of oblique cone), its plan and the perpendicular distance between the top and base (the height of the body) being given. The working of this problem should be carefully noted tor the reason just above stated. Let aga'g' (Fig. 10) be the given plan of the body. The side lines of the plan are not drawn, but only the circles of Its top and base, as we do not make use of the side lines In fact, all that we really make use of is, as will be presently seen, the halves of the plan-circles of the top and base of the body, not however, placed anyhow, but in their proper positions relatwely to one another in plan. This is the all-essential point If the plan-circle a' g' (Fig. 10) of the top of the frustum were further removed than represented from the plan-circle a g of Its base, we should, by the end of our working, ^et at the pattern of some other oblique-cone frustum than that the pattern of which we require. Through 0 O' the centres of the circles draw an indefinite line aV, and draw any line X X parallel o 0 ^. Through a and g draw a A and g G HnTa I't d r"""' f ^^"^^ ^' ^' -definite hues a A and g' G" perpendicular to X X, and cutting it in METAL-PLATE WORK. 117 points A', G'. Make A' A" and G' G" each equal to the given height or perpendicular distance that the top and base are apart. Join A A" and G G", and produce these lines to inter- sect in v. The problem can now be completed by Problem II. Fig. 10. A' / / /g" X l4 G A' G 'cu d c", c ci", and d e" may be taken as the true lengths required. To diaw the pattern (Fig. 12) the seam to correspond with the shortest generating line. Draw A A' equal to A A" (Fig. 11) and with A and A' as centres and radii respectively A 6" and Ml' (Fig. 11) describe arcs intersecting in Next with V and A as centres and radii respectively B F and A h (Fig. 11) describe arcs intersecting in &. Then A, l, A', &', are points in the curves of the pattern. With h and h' as Fig. 12. A centres and radii respectively & c" and V c' (Fig. 11) describe arcs intersecting in c'. With c' and h as centres and radii respectively C F and 5 c (Fig. 11) describe arcs intersecting in c ; and with c and c' as centres and radii respectively c d" and c' d' (Fig. 11) describe arcs intersecting in d'. With d' and c as centres and radii respectively D F and eci (Fig. 11) describe arcs intersecting in d. Similarly find E' and E. Draw unbroken curved lines through A 6 c E and A' 6' c' d' E METAL-PLATE WOEK. 121 and join E E' ; that will give us half tlie pattern. By like procedure we find the other half of the pattern, that to the left of A A'. (66.) The lines h V, c c', &c., and the dotted lines A h', h c', &c., are drawn in Fig. 12 simply to show the position that the lines which correspond to them in Fig. 11 (hh', Ah', &c.) take upon the developed surface of the tapering body. It is evident that it is not a necessity to make distinct operations of the two halves of the pattern ; for as the points h', h, c', c, &c., are successively found, the points on the left of A A ' corresponding to them can he set off. PEOBLEM V. To draw the pattern of an oblique cylinder (inclined circular pipe for example), iJie length and inclination of the axis and the diameter being given. Draw (Fig. 13) any line A' G', and at the point A' in it make the angle G' A' A" equal to the given inclination of the axis. Make A' G' equal to the given diameter, and draw a line G' G" parallel to A' A". Make A' A" and G' G" each equal 122 METAL-PLATE WORK. to the length of the cylinder (the length of a cylinder is the length of its axis) and join A" G". Then A' A" G' G" is the elevation of the cylinder. Now on A' G' describe a semi- circle, and divide it into any number of equal parts, in the points 1, 2, 3, &c. ; through each point draw lines perpendi- cular to A' G', meeting it in points B', C, J)', &c., and through B', C, D', &c., draw lines parallel to A' A". Draw any line A G perpendicular to A' A" and G' G", cutting the lines B' B", C C", &c,, in points B, C, &c. Next make B b equal to B'l, Cc equal to C'2, Bd equal to D'3, Ee equal to E' 4, and F / equal to F' 5, and draw a curve from A through the points 6, c, d, &c., to G. It is necessary to remark that this curve is not a semicircle, but a semi-ellipse (half an ellipse). C 0 E F & F E J) C B A To draw the pattern (Fig. 14). Draw any line A A, and at about its centre draw any line G" G' perpendicular to METAL-PLATE WOKK. 123 it and cutting it in G. From G, right and left of it, on the line A A mark distances GF, F E, ED, D C, C B, and B A equal respectively to the distances Gf,fe, ed, d c, &c. (Fig. 13). Through the points F, E, D, &c., right and left of G, draw lines parallel to G" G'. Make G G', G G" equal to G G', GG" (Fig. 13) respectively. Similarly make F F', F F", E E', E E", D D', D D", &c., right and left of G' G" equal respectively to P F', F F", E E', E E", D D', D D" &c. (Fig. 13). Draw an unbroken curved line from G" through F", E", D", &c., right and left of G" and an unbroken curved line through F', E', D', &c., right and left of G'. The figure A"G"A"A'G'A' will be the pattern required. The two parts G"A"A'G' of the pattern are alike in every respect. 124 METAL-PLATE WOEK. CHAPTER II. Unequal-tapering Bodies, of which Top and Base are Parallel, and their Plans. (66.) Before going into problems showing liow to draw the patterns of unequal-tapering bodies with parallel ends, bodies which are {as the student will realise as he proceeds) partly or wholly portions of oblique cones, it will be neces- sary to enter into considerations in respect of the plans of frusta of such cones (see § 58, p. Ill), similar to those appertaining to the plans of frusta of right cones treated of in Chap. V., Book I. ; but to us of greater importance, because the constructions in problems for the setting out of patterns of bodies having unequal taper or inclination of slant are a little more difficult than those in problems for patterns of equal-tapering bodies. The chapter referred to may be now again read with advantage. As much use will be hereafter made of the terms Pro- portionate Arcs and Similar Arcs we now define them. We also extend our explanation of Corresponding Points. Definitions. (67.) Proportional Arcs : Similar Arcs. — Arcs are pro- portional when they are equal portions of the circumferences of the circles of which they are respectively parts; they are similar when they are contained between the same generating lines. Similar arcs are necessarily proportional. In Fig. 15 the arcs A D and A' D' are proportional because each is a quarter of the circumference of the circle to which it belongs. They are similar because the generating lines V A and V D contain them both. (68.) Corresponding Points. — Points on the same gene- rating line are corresponding points (compare § 24) ; thus, METAL-PLATE WORK. 125 the points A and A' are corresponding points, because they are on the same generating line V A ; also the points C and C on the generating line V C. The point A' on V A is the plan of A' on V A ; the point C on V C is the plan of C on V 0 ; and so on. (69.) From the figure, which shows a frustum of an oblique cone standing on a horizontal plane, it will be seen that the plan of a round unequal-tapering body (frustum of oblique cone) consists mainly of two circles C A D B, C A' J)' B', the plans of the ends of the body. In Fig. 16 is shown a complete plan of an oblique-cone frustum. With the con- necting lines of the two sides we are not concerned, but may simply mention that they are tangents (lines which touch but do not cut) to the circles. Further from Fig. 15 it will be seen that, completing the cone of which the tapering body is a portion. a. The plan of the axis (line joining the centres of the ends) of a frustum is a line joining the centres of the circles which are the plans of its ends ; thus, the line 0 0' is the plan of the axis 0 0" (see also 0 0', Fig. 16). 126 METAL-PLATE WOEK. Similarly, the plan of the axis of the complete cone, is the plan, produced, of the axis of its frustum ; thus, O V is the plan of 0 v. b. The plan, produced hoth ways, of the axis of a frustum contains the plans of the lines of greatest and least inclina- tion on the frustum (see § 52) ; that is to say, of the longest and shortest lines on it. Thus, 0 0', produced both ways, contains the plans of A A' and B B'. It is convenient to regard lines joining corresponding points of a frustum (cori'esponding points of a frustum are points on one and the same generating line of the complete cone) as generating lines of the frustum. (See in connection with this, § 46). Then lines A A', B B', for instance, may be spoken of as generating lines of the frustum represented. Similarly, the plans of the longest and shortest generating lines of a cone are contained in the plan, produced, of the axis of its frustum ; thus, the plans of V A and V B are contained in 0 0' produced, both ways. c. The line, produced, which joins the centres of the plans of the ends of a frustum, contains the plan of the apex of the cone ; thus, 0 0', produced, contains Y the plan of Y'. d. The line, produced, which joins the plans of corre- sponding points of a frustum (see definition, § 68) contains the plan of the apex of the complete cone, and, produced only as far as the plan of the apex is the plan of a generating line of the cone ; thus, C and C being corresponding points on the cone, the plan, C C, produced, of the joining C and C, contains Y ; and C Y is the plan of the generating line CY'. e. The plans produced of all generating lines of a frustum intersect the plan of its axis produced, in one point, and that point is the plan of the apex of the complete cone ; for example, the plans produced, of the generating lines CC and D D' of the frustum intersect 0 0' produced in Y. (70.) It follows from e that the plan of the apex of the complete cone of which a given frustum is a portion can easily be found if we have given the plans of the ends of the METAL-PLATE WOEK. 127 frustum and the plans of two corresponding points not in the line passing through the centres of the plans of its ends. This is a valuable fact for us, as it spares us elevation drawing which in many cases is very troublesome, and indeed, sometimes practically impossible, as, for instance, where an unequal-tapering body is frustum of an exceed- ingly high cone the axis of which is but little out of the perpendicular. This is a case in which although the apex cannot be found in elevation because of the great length of the necessary lines, it can readily be found in plan, because, in plan, the requisite lines are short. An oblique cone may of course not only be exceedingly long, but also very greatly out of the perpendicular. In this case it is impracticable anyhow to find the plan of the apex. Problem IV., just solved, meets both cases. It was by e that we there found the plan of the apex when accessible, that is, where the lines of the plan are not unduly long (see Fig. 11) by joining the plans of corresponding points c and c' (c and c' are corre- sponding points in that they are mid-points on the half- plans of the ends of the frustum, and therefore necessarily on one and the same generating line), and producing cc' to intersect 0 0', the line joining the centres of the plans of the ends, that is to intersect the plan produced of the axis of the frustum. (71.) Passing the foregoing under review, it will be seen that if we have two circles which are the plans of the ends of a round unequal-tapering body (frustum of an oblique cone) standing on a horizontal plane, and the circles are in their proper relative positions as part plan of the frustum, then the line produced, one or both ways as may be necessary, which joins the centres of the circles, contains :— The Plan of the Axis of the frustum (see a, p. 125). The Plan of the Axis of the cone of which the frustum is a part (see a, p. 125). The Plan of the Apex of the cone (see c, p. 126). The Plans of the Longest and Shoetest Generating lines of the frustum (see 6, p. 126). 128 METAL-PLATE WORK. The Plans of the Longest and Shortest Generating Lines of the cone, of which the frustum is a part (see b, p. 126). The Plans of the Lines of Greatest and Least Inclination of the frustum (see 6, p. 126). The Plans of the Lines of Greatest and Least Inclina- tion of the cone, of which the frustum is a part (see § 52, p. 107). And this is a matter of very great practical importance, as will be seen later on. (72.) As with circles under the conditions stated, so exactly with arcs which form the plans either of the ends or of portions of the ends, of an unequal-tapering body. (73.) Further, referring to c and d of p. 55 as to round equal-tapering bodies we are now in a position to deduce (see Fig. 15) the following as to round unequal -tapering bodies. /. The plan of a round unequal-tapering body with top and base parallel (frustum of oblique cone) consists essentially of two circles, not concentric, definitely situate relatively to one another. See Fig. 16. Fig. 16. D X ^ -^P ' 0 0' 1 \^ — C C Similarly the plan of a portion of such round unequal- tapering body (frustum of oblique cone) consists essentially of two arcs definitely situate relatively to one another, and METAL-PLATE WORK. 129 not concentric. See Fig. 17. 0 0' is the axis of the complete frustum. g. Conversely. — If two circles, not having the same centre, definitely situate, relatively to one another, form essentially the plan of a tapering body having parallel ends, that body is a round unequal-tapering body (frustum of oblique cone). In Fig. 16, if the two circles represent essentially the plan of a tapering body having parallel ends, then the body, of which the circles are the essential plan, is a round unequal- body (frustum of oblique cone). Similarly if two arcs definitely situate relatively to one another, and not having a common centre, form, the essential part of the plan either of a tapering body or of a portion of a tapering body having parallel ends, then that boily or por- tion is a portion of a round unequal-tapering body (frustum of oblique cone). In Fig. 17 if the arcs form the essential part of the plan, either of a tapering body or of a portion of a tapering body having parallel ends ; then the body or portion of body, of which that fig. is the plan, is a portion of a round unequal- tapering body (frustum of oblique cone). In the particular plan represented the arcs are similar; the points B and B', and A and A' are therefore corresponding points. 130 METAL-PLATE WOEK. We will now proceed to draw tlie plans of some unequal - tapering bodies, of which patterns will be presently set out. PEOBLEM YI. To draw the plan of an unequal-tapering hody with top and base parallel and having straight sides and semicircular ends (an " equal-end " bath with semicircular ends), from given dimensions of top and bottom. Draw A'B'C'D' (Fig. 18) the plan of the bottom by Problem XVI., p. 20. Bisect A'B' in 0, and through O draw C D perpendicular to A' B'. Make 0 A and 0 B each Fig, 18. A D D' \ 0 b\ ' c' / equal to half the length, and 0 C and 0 D each equal to half the width of the top. The plan of the top can now be drawn in the same manner as that of the bottom, completing the plan required. METAL-PLATE WOKK. 131 PEOBLEM VII. To draw the plan of an oval unequal-tapering body with top and base parallel (an oval bath), from given dimensions of top and bottom. Draw (Fig. 19) A'B'C'D' the plan of the bottom, the given length and width, by Problem XII., p. 13 ; and make Fig. 19. A A' D D' -"^o.^ 0 B' B C 0 A and 0 B each equal to half the length, and 0 0 and 0 D each equal to half the width of the top. The plan of the top can now be drawn in the same manner as that of the bottom ; this completes the plan required. K 2 132 METAL-PLATE WOEK. PEOBLEM VIII. To draw the plan of a tapering body with top and base parallel and having oblong bottom with semicircular ends and circular top (tea-bottle top), from given dimensions of top and bottom. Draw (Fig. 20) the plan of the oblong bottom by Problem XVI., p. 20, and with 0 the intersection of the axes of the oblong as centre and half the diameter of the top as radius, describe a circle. This completes the plan. Fig. 20. J. I / 0 j We here, for the first time, extend the use of the word 'axes' (see Problems XII. and XIV., pp. 13 and 15). It is convenient to do so, and the meaning is obvious. PEOBLEM IX. To draw the plan of a tapering body with top and base parallel, the top being circular and the bottom oval (oval canister-top), from given dimensions of top and bottom. Draw (Fig. 21) the plan of the oval bottom by Problem XII., p. 13, and with 0 the intersection of the axes as centre, METAL-PLATE WOEK. 133 and half the given diameter of the top as radius, describe a circle. This completes the plan. Fig. 21. / \ / \^ / \ / \ A 'f 1 B « \ / \ / 0 -p \ \ PEOBLEM X. To draw the plan of a tapering body with top and base parallel and having oblong base with round corners and circular top, from given dimensions of the top and bottom. Draw (Fig. 22) the plan of the oblong bottom by Problem XV., p. 19, and with 0 the intersection of the axes of the Fig. 22. A V oblong bottom as centre and half the diameter of the top as radius, ^describe a circle. This completes the plan. 134 METAL-PLATE WOEK. PEOBLEM XI. To draw the plan of an Oxford Mp-hath. Fig. 23 is a side elevation of tlie bath, drawn here only to make the problem clearer, not because it is necessary for the working. No method that involves the drawing of a full- size side elevation is practical, on account of the amount of space that would be required. The bottom of an Oxford hip-bath is an egg-shaped oval. The portion O X' of the top is parallel to the bottom A' B', and the whole X X' top, the portion 0 X E of the bath being removed, is also an egg-shaped oval. In speaking of the plan of the * bath,' we mean the plan of the XX' B' A' portion of it, as the plan of this portion is all that is necessary to enable us to get at the pattern of the bath. We will first suppose the following dimensions given : — The length and width of the bottom, and the length of the X X' top, the height of the bath in front, and the inclination of the slant at back. First draw (Fig. 25) the plan of the bottom A'D' B' C by Problem XIII., p. 14. To draw the horizontal projection of METAL-PLATE WOKK. 135 the X X' top, make (Fig. 24a) the angle A A'E equal to the given inclination of the slant at the back. Through A' draw A' H perpendicular to A A', and equal to the height of the bath in front ; through H draw H X parallel to A A' and cutting A' E in X ; and draw X A perpendicular to X H ; then A A' will be the distance, in plan, at the back, between the curve of the bottom and the curve of the XX' top (Fig. 23). Make A' A (Fig. 25) equal to A A' (Fig. 24a), and make AB equal to the length of the XX' top. With 0 as centre and 0 A as radius describe a semicircle ; the remainder of the oval of the X X' top can now be drawn as was that of the bottom. This completes, as stated above, all that is necessary of the plan of the bath to enable its pattern to be drawn. Fm. 24a, Fig. 24&. If the width of the X X' top is given, and not the inclina- tion of the bath at back, make 0 A (Fig. 25) equal to half that width, and proceed as before. The seam in an Oxford hip-bath, at the sides, is on the lines of which C C and D D' are the plans. If the length of the X X top (Fig. 23) is not given, it can be determined in the following manner : — Let the angle X B' B (Fig. 246) represent the inclioation of the slant of the front, and B' X its length. Through X 136 METAL-PLATE WOKK. draw X B perpendicular to B' B ; then B B' will be the dis- tance in plan, at the front, between the curve of the bottom and the curve of the X X' top (Fig. 23), and this distance, marked from B' to B (Fig. 25), together with the distance A' A at the back end of the plan of the bottom, fixes the length required. Fig. 25. Ij C If the lengths only of the slants of the bath at back and front are given and not their inclinations, the plan of the X X' top can be drawn as follows : Draw two lines X B, B'B (Fig. 246) perpendicular to one another and meeting in B; make BX equal to the given height of the bath in front, and with X as centre and radius equal to the given length of the slant at the front, describe an arc cutting B'B in B'. Make B'B (Fig.- 25) equal to B B'^ (Fig. 246), and B A equal to the given length of the X X' top ; this will give the distance A' A. Now make A A' (Fig. 24a) equal to A A' (Fig. 25), draw AX and AH per- pendicular to A A' and equal to the given height of the bath in front; join HX and draw A'E, through X, equal to the length of the slant at the back ; the remainder of the plan of the X X' top can then be drawn as already described. It will be useful to show here in this problem how to METAL-PLATE WORK. 137 complete the back portion already commenced in Fig 2ia of the side elevation of the bath. Make A' E equal to the slant (§ 4, p. 24) at back, which must of course be given, and make X 0 equal to half the given width of the X X' top ; join 0 E, and draw 0 0 perpendicular to A A' produced ; then A' E 0 0 is the elevation required. PROBLEM XII. To draw the plan of an Athenian hip-hath or of a sitz-hath. Fig. 26 is a side elevation of the bath, drawn for the reason mentioned in the preceding problem. Fig. 26. The bottom of an Athenian hip-bath or a sitz-bath is an ordinary oval. The portion X' F of the top is parallel to the bottom A' B', and the whole X X' top, the portion F X E of the bath being removed, is also an ordinary oval. Simi- larly as with the bath of the last problem ; we mean by plan of the ' bath,' the plan of the X X' B' A' portion of it ; no more being required for the drawing of the pattern of the bath. We will first suppose the given dimensions to be those of 138 METAL-PLATE WOEK. the bottom and the X X' top of the bath, also height of the bath in front. First draw A' D' B' C (Fig, 27) the plan of the bottom by Problem XII., p. 13. To draw the plan of the XX' top (Fig. 26) set off 0 A and 0 B each equal to half the given length of that top, and 0 C and 0 D each equal to half its given width. The plan of the X X' top can now be drawn Fig. 27. D as was that of the bottom. This completes, as stated above, all that is necessary of the plan of the bath to enable its pattern to be drawn. If the length of the X X' top (Fig. 26) is not given but the inclination of the slant at front and back, these inclinations being the same, the required length can be determined in the following manner : — ■ Make the angle AA'E (Fig. 28a) equal to the given inclination. Through A' draw A' H perpendicular to A A' and equal to the given height of the bath in front ; through H drawHX parallel to A A' and cutting A'E in X, and draw X A perpendicular to A A' ; then A A' will be the distance in plan, at back and front, between the curve of the METAL-PLATE WORK. 139 bottom and the curve of tlie X X' top. Make A A' (Fig. 27) and B B' each equal to A A' (Fig. 28a) ; then A B will be the length required. Fig. 28a. Fig. 286. E If the length of the XX' top of the bath (Fig. 26) is not given, nor the inclination of the slant at front and back, but only the length of the slant at front, the required length can be thus ascertained. Draw two lines XB, B' B (Fig. 286) perpendicular to one another and meeting in B ; make B X equal to the given height of the bath in front, and with X as centre, and radius equal to the length of the slant at the front, describe an arc cutting B B' in B'. Make A' A and B' B (Fig. 27) each equal to BB' (Fig. 28&), then AB is the length wanted. The remainder of the plan can be drawn as described above. By a little addition to Fig. 28a we get at the back portion of the side elevation of the bath. It will be useful to do this. Produce A'X and make A'E equal to the slant at back, which must of course be given. Then, on the plan (Fig. 27), E being the meeting point of the end and side curves of the oval A D B C, draw E F perpendicular to A B. Make X F (Fig. 28a) equal to AF (Fig. 27) ; join F E ; this completes the elevation required. 140 METAL-PLATE WOEK. PEOBLEM XIII. To draw the plan of an oblong taper hath, the size of the top and bottom, the height, and the slant at the head being given. To draw D E F C (Fig. 30) the plan of the top. Draw A B equal to the given length of the top, and through A and B draw lines perpendicular to A B. Make A E and A D each equal to half the width of the top at the head of the bath, and B F and B C each equal to half the width of the top at the toe ; and join E F and D 0. Next from E mark off along E F and E D equal distances, E G and E H, according to the size of the round corner required at the head. (It will be useful practice for the student to work this problem, com- mencing with the plan of the bottom, and its smaller corners, for the reason given in § 27a, p. 63). Through G and H draw Fig. 29a. Fig. 296. lines perpendicular to E F and E D respectively, intersecting in 0 ; and with 0 as centre and 0 G as radius describe an arc H G to form the corner. The round corners at D F C, &c., are drawn in like manner. To draw the plan of the bottom. Let the angle A" A' A (Fig. 29a) be the angle of the inclination of the slant at the head, and A' A" the length of the slant. Through A" draw METAL-PLATE WOKK. 141 A" A perpendicular to A A', then A A' will be tlie distance between the lines, in plan, of the top and bottom at the head. Make A A' (Fioj. 30) equal to A A' (Fig. 29a), and A' B' equal to the length of the bottom. Through A' and B' draw lines each perpendicular to A B ; make A' E' and A' jy each equal to half the width of the bottom at the head, and B' F' and B' C each equal to half the width of the bottom at the toe. Join E'F' and D'C. The round corner of the bottom at the head must be drawn in proportion to the round corner of the top at the head, and this is done in the following manner. Join EE' and produce it, to meet Fig. 30. A B in P, and join H P by a line cutting D' E' in H' ; make E' G' equal to E' H', and complete the corner from centre O' obtained as was the centre O. Draw the other corners in similar way, and this will complete the plan required. The D corner is like the E corner ; the corners also at F and 0 correspond. Similarly with the E' and D', and F' and C corners. If the length of the bath is given and the length of slant at (but not its inclination) head or toe, the distance A A' can be 142 METAL-PLATE WORK. foundby drawingtwo lines A" A, A' A (Fig. 29a) perpendicular to one another and meeting in A, and making A A" equal to the given height ; then, with A" as centre and A" A', the given length of the slant at the head, as radius, describe an arc cutting A A' in A'. Then A A' is the distance required. Similarly (Fig. 296) the distance B B' can be found. METAL-PLATE WORK. 143 CHAPTEE III. Patterns for Articles of Unequal Taper or Inclination OF Slant, and having Flat (Plane) Surfaces. (Class II. Subdivision h.) Articles of unequal taper or inclination of slant, and having plane or flat surfaces (hoppers, hoods, &c.), are frequently portions (frusta) of oblique pyramids, or parts of such frusta. Definitions. (74.) Oblique Pyramid : Frustum of Oblique Pyramid : — Oblique pyramids have not yet been defined. For our purpose it will be sufficient to define an oblique pyramid negatively, that is, as a pyramid which is not a right pyramid ; and when cut by a plane parallel to its base (that Fig. 31. is, when truncated), to define its frustum (§ 33, p. 69) as the frustum of a pyramid which is not a right pyramid. In the oblique pyramid the faces are not all equally inclined. Articles of which the faces are not all equally inclined are 14i METAL-PLATE WOEK. not necessarily portions of oblique pyramids. One sucli case will be given later on. The problems immediately following deal With articles of which the faces are not all equally inclined, but which are portions of oblique pyramids. (75.) Further, an oblique pyramid, when it has a base through the angular points of which a circle can be drawn, can be inscribed in an oblique cone like as a right pyramid in a right cone, and this property gives constructions for solving most of our oblique-pyramid problems, somewhat similar to those in Book I., Chapter VI., where the right pyramid is concerned. Fig. 31 represents an oblique hexagonal pyramid inscribed in an oblique cone. This fig, should be compared with Fig. 31, p. 67. The edges of the oblique pyramid are generating lines of the cone. Fig. 32. (76.) Also from Fig. 32 it will be seen that the plan of a frustum of an oblique hexagonal pyramid standing on a hori- zontal plane consists of two hexagons A B C D and A' B' C D' (the plans of the ends), whose similarly situated sides, A B and METAL-PLATE WORK. 145 A' B', B C and B' C, C D and C D' for instance, are parallel, and whose corresponding points (§ 68, p. 124) A, A' and B, B', for instance, are joined by lines A A', B B', which are the plans of the edges of the frustum. Just as in the case of the frustum of the oMique cone (see d and e, p. 126), if a line joining corresponding points in plan be produced, it will contain the plan of the apex of the complete pyramid of which the frustum is a portion ; and if another such line be produced to intersect the first line, the point of intersection will be that plan of apex. For example, the lines A A', B B' and C C produced meet in a point which is the plan of the apex of the pyramid of which the frustum ABDD"B"A" is a portion. (77.) From this it follows that if the plan of a tapering body with top and base parallel and having plane or flat surfaces be given, we can at once determine whether the tapering body is or not a frustum of an oblique pyramid by producing the plans of the edges. If these meet in one point, then the given plan is that of a frustum of an oblique pyramid. PROBLEM XIV. To draw the pattern of an oblique pyramid. Case 1. — Given the plan of the pyramid and its height. Let A B C D E F y (Fig. 33) be the plan of the pyramid (here a hexagonal pyramid), V being the apex, and 0 V the plan of the axis. Draw X X parallel to 0 Y, and through V draw V V perpendicular to X X, and cutting it in ; make V V equal to the given height of the pyramid. Next make va, vh, vc, vd, ve, and v f equal respectively to V A, V B, VC, VD, VE, and V F, the plans of the edges of the pyramid. Joining V'a, V'&, V'c, &c., will give the true lengths of these edges. To draw the pattern of the pyramid with the seam at the edge V A. Draw V A (Fig. 34) equal to V a (Fig. 83) ; with L 146 METAL-PLATE WORK. V as centre and V h (Fig. 33) as radius describe arc b, and with A as centre and A B (Fig. 33) as radius describe an arc inter- secting the arc h in B. The other points 0, D, E, F, A, are Fig. 33. METAL-PLATE WOEK. 147 found in similar manner. Thus, with V (Fig. 34) as centre and V'c, Yd, Y' e, Yf, and V'a (Mg. 33) successively as radii, describe arcs c, d, e, f, and a (Fig. 34). Next, with B (Fig. 34) as centre and B C (Fig. 33) as radius, describe an arc intersecting arc c in C ; with C D (Fig. 33) as radius and C (Fig. 34) as centre describe an arc cutting arc dinj); with D (Fig, 34) as centre and D E (Fig. 33) as radius describe an arc intersecting arc e in E ; and so on for points F and A. Join A B, B C, C D, DE, E F, F A and A V, and this will complete the pattern required. Fig. 34. Joining the points B, C, D, &c., to V, it will be seen that the pattern is made up of a number of triangles, each triangle being of the shape of a face of the pyramid, also that the construction of the pattern is very similar to the construction of that of an oblique cone. Should it be inconvenient to draw XX in the position shown in Fig. 33, the true lengths of the edges of the pyramid L 2 148 METAL-PLATE WOEK. can be found in the following manner. Draw X X quite apart from the plan of the pyramid, and from any point v in it draw v Y perpendicular to X X, and equal to the height of the pyramid, and proceed as just described. Case II.— Given the plan of the pyramid and the length of its axis. Draw X X (Fig. 33) parallel to 0 V, the plan of the axis ; through y draw V V perpendicular to XX, and through 0 draw 0 O' perpendicular to X X. With 0' as centre and the given length of the axis as radius describe an arc cutting V V in V ; then v V will be the height of the pyramid, and we now proceed as in Case 1. Or, draw Vo; perpendicular to 0 V, and with 0 as centre and radius equal to the length of the axis describe an arc cutting Yxinx; Y x will be the height of the pyramid. PEOBLEM XV. To draw the pattern of a frustum of an oblique pyramid. Case I.— Given the plan of the frustum and its height. Let A B C D D' A' B' C (Fig. 35) be the plan of the frustum (here of a square pyramid). Produce A A', BB', &c., the plans of the edges to meet in a point V ; this point is the plan of the apex of the pyramid of which the frustum is a part. Join 0, the centre of the square which is the plan of the large end of the frustum, to V. The line 0 V will pass through o', the centre of the plan of the small end ; 0 0' will be the plan of the axis of the frustum, and O V the plan of the axis of the pyramid of which the frustum is a portion. ^ Draw X X parallel to O V ; through V draw V V perpen- dicular to XX, and cutting it in v. Make a; equal to the given height of the frustum, and through x draw xx parallel to X X ; through 0 draw 0 Q perpendicular to X X and METAL-PLATE WORK. 149 meeting it in Q and through 0' draw 0' Q! perpendicular to X X and cutting xx in e'. Join Q e' and produce it to intersect v V in V. Next make va, vi, vc, v d equal to V A, V B, V C, V D respectively ; join a, b, c, and d to V by lines cutting a; £c in points a\ b', c', and d' ; a a', b b', &c., are the lengths of the edges of the frustum. Fig. 35. To draw the pattern with the seam at A A'. Draw V A (Fig. 36) equal to V a (Fig. 35) ; with V as centre and V 6 (Fig. 35) as radius describe an arc b, and with A as centre and A B (Fig. 35) as radius describe an arc intersecting arc 6 in B ; with V c (Fig. 35) as radius and V as centre describe arc c, and with B C (Fig. 35) as radius and B as centre describe an arc intersecting the arc c in C. Next with V d and V a (Fig. 35) as radii and V as centre describe arcs d and a ; with C as centre and radius C D (Fig. 35) describe an arc intersecting arc dinJ); and with D A (Fig. 35) as radius and D as centre describe an arc intersecting the arc a in A. J oin A, B, C, D, 150 METAL-PLATE WOEK. and A to V ; make A A', B B', C C, D D' respectively equal to a a', hh', cc', dd' (Fig. 35), and join AB, B C, CD, DA, A' B', B' C, C D', &c. Then A B C D A A' D' C B A' is the pattern required. Fig. 36. (78.) The dotted circles (Fig. 35) through the angular points of the plans of the ends show the plans of the ends of the frustum of the oblique cone which would envelop the frustum of the pyramid. From the similaritj'- of the con- struction above to that for the pattern of a frustum of an oblique cone, it will be evident that we have treated the edges of the frustum as generating lines (see h, p. 126) of the frustum of the oblique cone in which the frustum of the pyramid could be inscribed. Should it be inconvenient to draw X X in conjunction with the plan of the pyramid draw X X quite apart, and from any METAL-PLATE WOEK. 151 point V in it draw v Y' perpendicular to X X ; make v x equal to the height of the frustum and draw x x parallel to X X. Make va,vh,vc,vd equal to V A, V B, V C, V D (Fig. 35) respectively ; and make x a', x h', x c', x d' equal to V A', V B', V C, V D' (Fig. 35) respectively. Join a a', b h\ c c\ and d d' by lines produced to meet v V in V, and proceed as stated above. Case II. — Given the dimensions of the two ends of the frustum, the slant of one face and its inclination (the slant of the face of a frustum of a pyramid is a line meeting its end lines and perpendicular to them). Draw (Fig. 37) a line E E" equal to the given slant, make the angle E" E E' equal to the given inclination, and let fall E"E' perpendicular to EE'. Draw A B C D (Fig. 35), the plan of the large end of the frustum, and let B C be the plan of the bottom edge of the face whose slant is given. Bisect B C in E and draw E E' perpendicular to B C and equal to Fig. 37. E' E E' (Fig. 37). Through E' draw B' C parallel to B C ; make E' C and E' B' each equal to half the length of the top edge of the BC face, through C and B' draw CD' and B'A' parallel to C D and B A ; make C D' and B' A' each equal to B' C ; join D' A', also A A', B B', C C, and D D' ; this will complete the plan of the frustum. E' E" (Fig. 37) is the height of the frustum. The remainder of the construction is now the same as that of Case L 152 METAL-PLATE WOEK. For large work and where the ends of a frustum are of nearly the same size, it would be inconvenient to use long radii. For unequal-tapering bodies which are not portions of oblique pyramids, as in Problem XVII., the method now given, or a modification of it, must be used. PEOBLEM XVI. To draw, without long radii, the pattern for a frustum of an oblique pyramid. The plan of the frustum and its height being given. Let A B C D D' A' B' C (Fig. 38) be the plan of the frustum. From any point E in B C draw E E' perpendicular to B C and B' C, the plans of the bottom and top edges of the face B C B' C of the frustum. Draw E' E" perpendicular to E E' and Fig. 38. equal to the height (which either is given or can be found as in Case II. of last problem), and join E E", then E E" is the true length of a slant of the face BOB' C, of the frustum. Join D C, and find its true length (D C") by drawing C'C" METAL-PLATE WORK. 153 perpendicular to D C and equal to the height of frustum and joining D C". Xext join D' A and B' A ; through D' and B' draw lines D' A", B' B" perpendicular to D' A and B' A respectively, and make D' A" and B' B" each equal to the given height of the frustum ; join A A" and A B", then A A" and A B" are the true lengths of D' A and B' A respectively. To draw the pattern of the face BCB'C, draw EE' (Fig. 39) equal to EE" (Fig. 38), and through E and E' draw B C and B' C perpendicular to E E'. Make E C, E B, E' C, and E' B' equal to E C, E B, E' C, and E' B' (Fig. 38) respec- tively ; join 0 0' and B B' ; this completes the pattern of the face. The patterns of the other faces are found in the following manner : — With C (Fig. 39) and C as centres and Fig. 39. D C" and C D (Fig. 38) as radii respectively, describe arcs intersecting in D ; join C D, draw C D' parallel to C D and equal to CD' (Fig. 38); and join DD'. With D' and D (Fig. 39) as centres and A A" and DA (Fig. 88) as radii respectively describe arcs intersecting in A ; join D A, draw D' A' parallel to D A and equal to D' A' (Fig. 38), and join A 154 METAL-PLATE WOEK. to A'. Next, with B' and B as centres and A B" and B A (Fig. 38) respectively as radii, describe arcs intersecting in A ; join B A and draw B' A' parallel to B A and equal to B' A' (Fig. 38). Join A A', and this will complete the pattern required. PKOBLEM XVII. To draw the pattern for a hood. The plan of the hood is necessarily given, or else dimensions from which to draw it. Also the height of the hood, or the slant of one of its faces. The hood is here supposed to be a body of unequal taper with top and base parallel, but not a frustum of an oblique pyramid. Let A B C D A' B' 0' D' (Fig. 40) be the given plan of the Fig. 40. / D G S hood (a hood of three faces), A D being the ' wall line,' A B and D C perpendicular to A D and B C parallel to it, also let the length of F C", a slant of face B B' C C, be given. Draw C F perpendicular to B C and through C draw C C" perpendicular to C'F, and with F as centre and radius METAL-PLATE WORK. 155 equal to the given length, describe an arc cutting C C" in C". Join FC"; then CO" is the height of the hood, which we need. If the height of the hood is given instead of the length F C", make C C" equal to the height and join F C", which will be the true length of F C. Next, through C draw C E perpendicular to CD; draw C C" perpendicular to 0' E, make C C" equal to the height and join E C". Now produce C B' to meet A B in G ; draw B' B" perpendicular to B' G and equal to the height, and join G B". To draw the pattern of the hood. Draw F C (Fig. 41) equal to F C" (Fig. 40) ; through F and C draw B C and B' C, each perpendicular to F C ; make F B equal to F B (Fig. 40) ; make F C equal to F C (Fig. 40), and C',B' equal to C B' (Fig. 40). Join B B' and C C, then B B' C C will be the pattern of the face of which B B' C C (Fig. 40) is the plan. To draw the pattern of the face C D' D 0 (Fig. 40). With C and 0 (Fig. 41) as centres and EC" and CE Fig. 41. (Fig. 40) as radii respectively, describe arcs intersecting in E. Join C E and produce it, making C D equal to C D (Fig. 40), and through C draw C D' parallel to C D and equal to C D' (Fig. 40). Join D D', then C C D D' is the pattern of the face of which C C D D' (Fig. 40) is the plan. With B' and B as 156 METAL-PLATE WORK. centres and radii respectively equal to B" G and B G (Fig. 40), describe arcs intersecting in G. Join B G and produce it, making B A equal to B A (Fig, 40), and through B' draw B' A' parallel to B A and equal to B' A' (Fig. 40). Join A A' ; and the pattern for the hood is complete. METAL-PLATE WORK. 157 CHAPTER IV. Patterns for Unequal-tapering Articles of Flat and Curved Surface combined. Class II. (Subdivision c.) From what has been stated about the plans of unequal- tapering bodies and from g, p. 129, it will be evident that the curved surfaces of the articles now to be dealt with are portions of frusta of oblique cones. (79.) The advantages referred to in § 61 of looking upon the oblique cylinder as frustum of an oblique cone will be evident in this chapter. For there is to each of the problems a Case where the plan arcs of the curved portions of the body treated of have equal radii. To deal with these as problems exceptional to a general principle would be most inconvenient. As extreme cases, however, of the one principle that the curved portions of the bodies before us are portions of frusta of oblique cones, their solution presents no difficulty. It will be sufficient to take one such Case in connection with only one of the bodies. This nve shall do in Case IV. of the next problem. PEOBLEM XVIII. To draw the pattern for an unequal-tapering body with top and base parallel and having flat sides and semicircular ends (an ' equal-end ' bath, for instance), the dimensions of top and bottom of the body and its height being given. Five cases will be treated of ; four in this problem and one in the problem following. 158 METAL-PLATE WOEK. Case I. — Patterns when the body is to be made up of four pieces. Draw (Fig. 42) the plan of the body (see Problem VI., p. 130), preserving of its construction the centres 0, 0' and the points A, A' in which the plan lines of the sides and curves Fig. 42, E Tfl X I— H r X F A of the ends meet each other. Join A A', as shown (four places) in the fig. The ends A D A A' D' A' and A E A A' E' A' of the body (see g, p. 129) are portions of frusta of oblique cones. Let us suppose that the seams are to be at the four A corners where they are usually placed, and to correspond with the four lines A A'. Then we shall require one pattern for the flat sides, and another for the semicircular ends. To draw the end pattern. (80.) Draw A 6 D D' A' (Fig. 43) the A 6 D D' A' portion of Fig. 42 separately, thus. Draw any line X X and with any point 0 (to correspond with O, Fig. 42) in it as centre and 0 D (Fig. 42) as radius describe an arc (here a quadrant) D & A equal to the arc D 6 A (Fig. 42). Make D O' equal to D 0' (Fig. 42), and with 0' as centre and 0' D' (Fig. 42) as radius describe an arc (here a quadrant) D' A' equal to the arc D' A' (Fig. 44). Joining A A' completes the portion of Fig. 42 required. Now divide D A into any number of equal parts, here three, in the points h and c. From D' draw D' D" per- pendicular to XX and equal to the given height. Then D, D" are, in elevation, the corresponding points of which METAL-PLATE WOEK. 159 D, jy are the plans. Being corresponding points, they are in one and the same generating line (§ 68). Join D D" and produce it indefinitely, then somewhere in that line will lie the elevation of the apex of the cone of a portion of which Fig. 43 A6DD'A' is the plan. Now from 0' draw 0' 0" perpen- dicular to X X and equal to the given height. Then 0, O" are, in elevation, the centres of the ends of the frustum in the same plane that D, D" are represented in, that is in the plane of the paper ; 0, 0' being the plans of these centres. Join 0 0" and produce it indefinitely ; then in this line lies the elevation of the axis of the cone of a portion of which A 6 D D' A' is the plan, and necessarily therefore the elevation of the apex. That is, the intersection point P of these two lines is the elevation of the apex. Next, from P let fall P P' perpendicular to XX ; then P' will be the plan of the apex. Join D" 0". With P' as centre and P' c, P' h, and P' A sue- 160 METAL-PLATE WOEK. cessively as radii, describe arcs cutting X X in C, B, and A". Join these points to P by lines cutting 0" D" in C, B', and A'. Next draw a line P D (Fig. 44) equal to P D (Fig. 43), and with P as centre and P C, P B, and P A" (Fig. 43) successively as radii describe arcs c c, hh, and a a. With D as centre and radius equal D c (Fig. 43) describe arcs cutting arc c c in C and C right and left of P D. With same radius and these points C and C successively as centres describe arcs cutting arc 6 6 in B and B right and left of P D. With B and B Fig. 44. successively as centres and same radius describe arcs cutting arc a a in A and A right and left of PD. Join the points C, B, and A right and left of P D to P. With P as centre and P D" (Fig. 43) as radius describe an arc cutting P D in D". With same centre, and P C (Fig, 43) as radius, describe arc c' c' cutting lines P C right and left of P D in C and C With same centre, and P B' (Fig. 43) as radius, describe arc METAL-PLATE WOKK. 161 h' V cutting lines P B right and left of P D in B' and B'. Similarly find points A' and A'. Through the successive points A, B, C, D, C, B, A, draw an unbroken curved line. Also through the successive points A', B', C, D', C, B', A,, draw an unbroken curved line. Then ADAA'D'A' will be the required pattern for ends of the body. To draw the pattern for the sides. Through A' (Fig. 42) draw A' F perpendicular to A' A' make F G equal to the given height and join A' G. Then A' G is the slant of the body at the side. Next draw (Fig. 45) A A equal to A A (Fig. 42), and make A F equal Fig. 45. F A' A' to AF (Fig. 42) ; through F draw F A' perpendicular to A A and equal to A'G (Fig. 42), and through A' draw A' A' parallel to A A. Make A' A' equal to A' A' (Fig. 42). Join A A', A A', then A A' A' A is the pattern for the sides. Case II. — Pattern when the body is to be made up of two pieces. We will take it that the seams are to be at D D' and E E' (Fig. 42). It is evident that we want but one pattern, which shall include a side of the body and two half-ends. First draw as just explained A' A F A A' (Fig. 46) a side- pattern of the body. Produce one of the lines A A' of this pattern, and make A P' equal to A" P (Fig. 43). With P' as centre and P B, P C, and P D (Fig. 43) successively as radii describe arcs 6, c, and d, and with A as centre and A& (Fig. 43) as radius describe an arc cutting arc h in B. With same radius and B as centre describe an arc cutting arc -c in C ; similarly with C as centre and same radius find D. 162 METAL-PLATE WOKK. Join B P, C F, D P'. Now with P B' (Fig. 43) as radins and P' as centre describe an arc V cutting P B in B', and with P C, P D" (Fig. 43) successively as radii describe arcs c' and d cutting P' C and P' D in C and D'. Through the points Fig. 46. A, B, C, and D draw an unbroken curved line. Also through the points A', B', C, and D' draw an unbroken curved line. Then A D D' A' will be a half-end pattern attached to the right of the side pattern. Draw the other half-end pattern A E E' A' in the same manner ; then E F D D' A' A' E' will be the complete pattern required. Case III. — Pattern when the body is to be made up of one piece. In this case we will put the seam to correspond with D D' (Fig. 42). First draw A E A A' E' A' (Fig. 47) an end pattern of the body in the same manner that A D A A' D' A' (Fig. 44) was drawn. With A' and A (right of E E') as centres and A' G and A F (the small length A F) (Fig. 42) respectively as 164 METAL-PLATE WOEK. A' A' (Fig. 42). Join A A', tlie extremities of the lines j'ttst drawn, and produce it indefinitely ; and make A P' equal A" P (Fig. 43). Witli P' as centre and P B, P C, and P D (Fig. 43) successively as radii describe arcs h, c, and d, and with A (of A P') as centre and A b (Fig. 43) as radius describe an arc cutting arc h in B. With same radius and B as centre describe an arc cutting arc c in C, and similarly with C as centre and same radius find D. Join BP', CP', and DP'. Now with P' as centre and radii successively equal to P B', PC, and PD" (Fig. 43) describe arcs b', c', and d' cutting P' B, P' C, and P' D in B', C, and D'. Through the points A, B, C, and D draw an unbroken curved line. Also through the points A', B', C, and D' draw an unbroken curved line. We have now in D F A A' A' D' attached to the right of the end pattern we started with, a side pattern and a half-end pattern. By a repetition of the foregoing construction we can attach A ADD' A' A' to the left of the end pattern we started with. The figure D E D D' E' D' will be the complete pattern required. Case IV.— Where the plan arcs D A, D' A' (Fig. 42) have This is the extreme case above (§ 79, p. 157) referred to, where the cone becomes cylindrical. Problem V., p. 121, may equal radii. Fig. 48. X A' A b advantageously be compared with the work now given. The arcs (Fig. 48) D A and D' A' (here quadrants) being equal, METAL-PLATE WORKi 165 tlieir radii 0 D, 0' D' are also equal. Through D' and 0' draw D' D", and O' A" perpendicular to X X and each equal to the given height of the body. Join D D", A" D" ; divide the arc D A into any number of equal parts, here three, in the points h and c ; and through c, h, and A draw c C, 6 B, and A 0 each perpendicular to X X and cutting it in C, B, and 0 respectively. The arc D A being here a quadrant the point where the line from A perpendicular to XX cuts XX is necessarily 0, the centre whence the arc is drawn. Through C, B, and 0 draw C C", B B", and 0 A" parallel to D D" and cutting A" D" in points C", B", and A". Also through D" draw a line D"A' perpendicular to DD" and cutting the lines just drawn in C, B' and A'. Make c 2 equal to C c ; B' 1 equal to B h, and A' 0 equal to 0 A. From D" through 2, 1, to 0 draw an unbroken curved line. To draw the pattern. Draw D D" (Fi^. 49) equal to D D" (Fig, 48) and through D" draw an indefinite line A' A' perpendicular to DD", Mark off on A' A', right and left of D", D"C', C'B', and B A' respectively equal to the distances between D" and 2, 2 and 1, and 1 and 0 (Fig. 48) ; and through C, B', and A', right and left of D" draw indefinite lines each parallel to D D", Make C C right and left of D D" equal to C C (Fig. 48) ; and make B'B right and left of DD" equal to B'B (Fig. 48). Also make A' 0 right and left of D D" equal to A' 0 (Fig. 48). 166 METAL-PLATE WORK. Next make C'C" right and left of DD" equal to C'C" (Fig. 48). Similarly find points B", A" right and left of D D" by making B' B", A' A" respectively equal to B' B", and A' A" (Fig. 48). Through the points 0, B, 0, D, 0, B, O, draw an unbroken curved line. Also through the points A", B", C", D", C", B", A", draw an unbroken curved line. Then 0 D 0 A" D" A" will be the pattern required. PKOBLEM XIX. To draw, without long radii, the pattern for an unequal- tapering body witTi top and base parallel and having flat sides and equal semicircular ends [an ' equal-end ' bath, for instance). The dimensions of the top and bottom of the body and its height being given. This problem is a fifth case of the preceding, and is exceedingly useful where the work is so large that it is inconvenient to draw the whole of the plan, and to use long radii. To draw the pattern. Fig. 50. ^i— ^ \. A BC First draw half the plan (Fig. 50). It is evident that the drawing of the side pattern presents no difficulty, as long METAL-PLATE WORK. 167 radii are not involved. It can be drawn as in Case I. of preceding problem. Divide tke quadrants D A, D' A', each into the same number of equal parts, here three, in the points c, h, c', y ; join c c', h V. Through D' draw D' E perpendicular to D' D and equal to the given height of the body. From D' along D' D mark off D' A, D' B, and D' C respectively equal to A' A, V h, and c' c ; and join points D, C, B, and A to E, then E A, E B, E C, and E D, will be the true lengths of A' A, h' h, c' c, and D' D respectively. Next join c D', and draw B'd" perpendicular to D'c and equal to the given laeight. Join c d", then c d" may be taken as the true length of D'c. Similarly join he' and Afe', and through c' and V draw e' c" and 6' h" perpendicular to c' h and b' A respectively, and each equal to the given height. Jom. h c" and A h", then b c" and A h" may be tak«n as the true lengths of 6 e' and A V respectively. Now draw (Fig. 51) a line DD' equal to D E (Fig. 50) and with D' and D as centres and radii respectively equal to d" c Fig. 51. and D c (Fig. 50) describe arcs right and left of D D', inter- secting in c and c. With c, right of D, and D' as centres, and radii respectively equal to 0 E and D' c' (Fig. 50) describe arcs intersecting in c', right of D'. With c, left of D, and D' as centres, and radii as before, describe arcs intersecting in c', left of D'. With successively c' and c right and left of D D', 168 METAL-PLATE WOKK. as centres and radii respectively equal to c" 6 and c h (Fig. 50) describe arcs intersecting in b and h right and left of D D'. With successively h and c' right and left of D D' as centres and radii respectively equal to B E and c' V (Fig. 50) describe arcs intersecting in V, and 6' right and left of D D', and with suc- cessively V and h right and left of D D' as centres and radii respectively equal to A 6" and h A (Fig. 50) describe arcs in- tersecting in A and A right and left of D D'. Similarly with A E and h' A' (Fig. 50) as radii and centres respectively A and h' describe intersecting arcs'to find points A' and A', right and left of D' D. Through the points A, h, c, D, c, 6, A draw an unbroken curved line. Also through the points A', h\ c', T>', c', V, A', draw an unbroken curved line. Join A A', A A', right and left of D D', then A D A A' D' A' will be the pattern required. The lines c c', c D', h h', &c., are not needed in the working ; they are drawn here to aid the student by showing him how the pattern corresponds with the plan, line for line of same lettering (see also § 65, p. 121). PEOBLEM XX. To draw tJie pattern for an oval unequal-tapering body with top and base parallel (an oval bath, for instance). The height and dimensions of the top and bottom of the body being given. Four cases will be treated of, three in this problem, and one in the problem following (see also § 79, p. 157). Draw (Fig. 52) the plan of the body (see Problem VII., p. 131), preserving of its construction, the centres 0, 0',P,P' and the several points d and d' in which the side and end curves meet each other. Join d d', as shown (four places) in the fig. From the plan we know (see g, -p. 129) that dGd d'G'd', dBdd'B'd', the ends of the body are like portions of the frustum of an oblique cone; we also know that METAL-PLATE WOEK. 169 dAdd' A'd', dEdd' E' d', the sides of the body are like por- tions of the frustum of an oblique cone. In Plate I. (p. 181), is a representation of the oval unequal- FiG. 52. / CL ?' 1 ■6?' G X ye f 5^ tapering body for which patterns are required, also of two oblique cones {x and Z). The oblique cones show (except as to dimensions) to what portions of their surfaces the several portions of the surface of the oval body correspond. Thus the sides, A', of the body correspond to the portion A of cone sc, and the ends, B', B', correspond to the B portion cone Z, The correspondence will be more fully recognised as we proceed with the problem. The difference of obliquity between B' and B is seeming only, not real ; and arises simply from Z being turned round so that the whole of the dV>ddi! B' dl (Fig. 52) of the cone shall be seen. If the representation of Z showed its full obliquity, then the line on it from base to apex would be the right-hand side line of the cone, and only half of the B portion could be seen. METAL-PLATE WORK. 171 Case I. — ^Patterns when the body is to be made up of four pieces. It is clear tbat we require two patterns ; one for the two ends, and one for the two sides ; also that the seams should correspond with the four lines d d', where the portions of the respective frusta meet each other. To draw the pattern for the ends. Draw separately (Fig. 53) the G' Gfd d' portion of Fig. 52, thus. Draw any line X X, Fig. 53, and with any point O (corresponding to 0, Fig. 52 ) in it as centre and 0 G (Fig. 52) as radius, describe an arc G d equal to G of of Fig. 52. Make G 0' equal to G O' (Fig. 52), and with 0' as centre and O' G' (Fig. 52) as radius describe an arc G' d' equal to G' d' of Fig. 52. Joining d d' completes the portion of Fig. 52 required. Now divide the arc G d into any number of equal parts, here three, in the points / and e. At G' and 0' draw G' G", O' 0" perpendicular to X X, and each equal to the given height of the body. Join 0 0", G G" ; produce them to their intersection in S (§ 80, p. 158) ; and from S let fall S S' perpendicular to XX. Join 0"G". With S' as centre and 8'f, S'e, and S' d successively as radii, describe arcs cutting X X in F, E, and D. Join these points to S bylines cutting 0" G" in F', E', and D'. Next draw S G (Fig. 54) equal to S G (Fig. 53), and with S as centre and S F, S E, and S D (Fig. 53) successively as radii describe arcs ff, ee, and d d. With G as centre and radius equal to G/ (Fig. 53) describe arcs cutting arc // right and left of S G in F and F. With each of these points, F and F as centre and same radius describe arcs cutting arc e e right and left of S G in E and E. With same radius and each of the last-obtained points as centre describe arcs cutting d d right and left of S G in D and D. Join all the points right and left of SG to S. With S as centre and SG" (Fig. 63) as radius, describe an arc cutting S G in G'. With same centre and SF' (Fig. 53) as radius, describe an arc/'/' cutting the lines S F right and left of S G in F' and F'. With same centre and S E' (Fig. 53) as radius, describe an METAL-PLATE WOEK. 173 arc e' e' cutting the lines S E right and left of S G in E' and E'. Similarly by arc d' d' obtain points D' and D'. Through the points D, E, F, G, F, E, D, draw an unbroken curved line. Also through points D', E', G', E', E', D', draw an unbroken curved line. Then D G D D' G' D' will be the required pattern for the ends of the body. It is in fact the development of the B portion of cone Z of Plate I. To draw the pattern for the sides. Draw separately (Fig. 55) the A' Abdd' portion of Fig. 52, thus. Draw any line X X, Fig. 55, and with any point P Fig. 55. A a\^\: \d (corresponding to P, Fig. 52) in it as centre and P A (Fig. 52) as radius, describe an arc A d equal to A of Fig. 52. Make 174 METAL-PLATE WOEK. A P' equal to A P' (Fig. 52) and with P' as centre and P' A' (Fig. 52) as radius describe an arc A! d' equal to A! d' of Fig. 52. J oining d d' completes the portion of Fig. 52 required. Now divide the arc A d into any number of equal parts, here three, in the points h and c. At A' and P' draw A' A", P' P" perpendicular to X X, and each equal to the given height of the body. Join PP", A A", produce them to their inter- section in Q (§ 80, p. 158) ; and from Q let fall Q Q' perpen- dicular to X X. Draw a line through P" A". With Q' as centre and Q' h, Q' c, and Q' d successively as radii describe arcs cutting X X in B, C, and D. Join these points to Q by lines cutting that through P" A", in B', C, and D'. Next draw Q A (Fig. 56) equal to Q A (Fig. 55), and with Q as centre and Q B, Q C, and Q D (Fig. 55) successively as Fig. 56. radii describe arcs hb, cc, and d d. With A as centre and radius equal to A & (Fig. 55) describe arcs cutting arc 6 6 right and left of Q A in B and B. With each of these points METAL-PLATE WOEK. 175 B and B as centre and same radius describe arcs cutting arc c c right and left of Q A in C and C. With same radius and each of the last-named points as centre describe arcs cutting arc d d right and left of Q A in D and D. Join all the points right and left of QA to Q. With Q as centre and QA" (Fig. 55) as radius, describe an arc cutting Q A in A'. With same centre, and Q B' (Fig. 55) as radius, describe an arc b' h cutting the lines Q B right and left of Q A in B' and B'. With same centre and Q C (Fig. 55) as radius, describe arc c' c' cutting the lines Q C right and left of Q A in C and C. ^ Similarly by arc d' d' obtain points D' and D'. Through the points D, C, B, A, B, C, D, draw an unbroken curved line. Also through the points D', C, B', A', B', C, D', draw an un- broken curved line. Then D A D D' A' D' will be the required pattern for the sides of the body, and is in fact the develop- ment of the A portion of cone x of Plate I. Case II. — Pattern when the body is to be made up of two pieces. In this case the seams are usually made to correspond with B B' and G G' (Fig. 52). It is evident that only one pattern is now required, made up of a pattern for the side A' of the body (Plate I.) with right and left a half-end (B', B', Plate I.) pattern attached. Draw (Fig. 57) a side pattern D A D D' A' D' as described in Case I. Produce D Q and make D S equal to D S (Fig. 53). With S as centre and S E, S F, and S G (Fig. 53) successively as radii describe arcs e, f, and g, and with D as centre and d e (Fig. 53) as radius describe an arc cutting arc e in E. With same radius and E as centre describe an arc cutting arc / in F, and similarly with F as centre and same radius find G. Join E S, F S, G S. Now with S E' (Fig. 63) as radius and S as centre describe an arc e' cutting S E in E', and with S F' and S G" (Fig. 53) successively as radii describe arcs /' and g' cutting S F and S G in F' and G'. From points D to G draw an unbroken curved line. Also from points D' to G' draw an unbroken curved line. Draw the other half-end 176 Fig. 57. METAL-PLATE WORK. 177 pattern D B B' D' in the same manner ; then G A B B' A' G' will be the pattern required. Case III. — Pattern when the body is to be made up of one piece. We will put the seams at the middle of one end of the body, say, to correspond with B B' (Fig. 52). We now need an end pattern (the end dGdd' G' d' in plan), with side pattern attached right and left (dEdd' E' d', dAd d' A' d' in plan), and attached to each of these a half-end pattern {dBB' d', dBB' d' in plan). For want of space we do not give the pattern, but it is evident from what has just been stated, that the pattern will be double that shown in Fig. 57. It will be a useful exercise and should present no difficulties to the student, io himself draw the complete pattern, first drawing an end pattern (see Case I. and Fig. 54) and attaching right and left, a side pattern and a half-end pattern. PEOBLEM XXI. To draw, without long radii, the pattern for an oval unequal- tapering body, with top and base parallel (an oval hath, for instance). The height and the dimensions of the top and bottom of the body being given. This problem is a fourth case of the preceding, and will be found very useful for both the end and side patterns, the radii of which are often of a most inconvenient length. To draw the end pattern. First draw (Fig. 58) the plan of the end of the body ; that is thedGdd'G d' portion of Fig. 52. Divide the arcs G d, G' d' each into the same number of equal parts, here three, in the points /, e, f, e' ; join //', ee'. Through G draw GH perpendicular to G G' and equal to the given height of the body. From G along GG' mark off GF, GE, and GD re- spectively equal to //', e e', and d d' ; join G', F, E and D to H ; then G' H, F H, E H, and D H will be the true lengths of N 178 METAL-PLATE WORK. G G', //, e e', and dd' respectively. Next join /G' ; draw G' g" perpendicular to/G' and equal to the given height and join fg"; then fg" may be taken as the true length of G' /. Similarly join ef, d e\ and through / and e' draw /' /" and Fig, 58. e' e" perpendicular to/'e and e' respectively, and each equal to the given height, and join e/"and de" ; then e f" and de" may be taken as the true lengths of e f and d e' respectively. Next draw (Fig. 59) a line G G' equal to G' H (Fig. 58) and with G and G as centres and radii respectively equal to //and G/(Fig. 58) describe arcs right and left of GG', intersecting in / and /. With /, right of G, and G' as centres' and radii respectively equal to F H and G'f (Fig. 58) describe arcs intersecting in /', right of G'. With /, left of G, and G' as centres, and radii respectively as before, describe arcs intersecting in /, left of G'. With successively /' and /, right and left of G G, as centres and radii respectively equal to ef" and fe (Fig. 58) describe arcs intersecting in e and e. With successively e and /' right and left of G' G as centres and radii respectively equal to E H and /' e' (Fig. 68) describe METAL-PLATE WORK. 179 arcs intersecting in e' and e' ; and witli successively e' and e right and left of G G' as centres, and radii respectively equal to d e" and e d describe arcs intersecting in d and d. Also with successively d and e' as centres and D H and e' d!' re- spectively as radii describe arcs intersecting in dl and Fig. 59. END PATTERN. Through d, e, f, Gf, e, d, draw an unbroken curved line. Also through d', e', f, G', /', e', d', draw an unbroken curved line. Join d d', right and left of G G', then d Q d d' G' d' will be the end pattern required. The lines //', e e', G'/, fe, &c., are not needed in the work- ing, they are drawn for the reason stated in § 65, p. 121. To draw the side pattern. First draw (Fig. 60) the plan of the side of the body ; that is the dAdd' A'd' portion of Fig. 52. Divide the arcs A d, A' d' each into the same number of equal parts, here three, in the points h, c, h', c' ; join h h', c c'. Through A draw A E perpendicular to A A' and equal to the given height of the body. From A along A A' mark off A B, A C, and A D re- spectively equal to hh\ cc', and dd'. Join A', B, C and D to E ; then A' E, B E, C E, and D E will be the true lengths of A A', h b', c c', and d d' respectively. Next join h' A and draw b' b" perpendicular to b' A and equal to the given height. 180 METAL-PLATE WOEK. Join A b" ; then A h" may be taken as the true length Ah'. Similarly join c' b and d' c ; through c' and d' draw c' c", and d' d" perpendicular to c' b and d' c respectively, and each equal to the given height ; join b c", c d", then b c" and c d" may be taken as the true lengths of b c' and c d' respectively. Fig. 60. Next draw (Fig. 61) a line A A' equal to A'E (Fig. 60) and with A and A' as centres and radii respectively equal to A h" and A' b' (Fig. 60) describe arcs right and left of A A', intersecting in b' and b'. With b', right of A', and A as centres, and radii respectively equal to B E and A b (Fig. 60) describe arcs intersecting in b, right of A. With b', left of A', and A as centres, and radii respectively as before, describe arcs intersecting in b left of A. With successively h and b' right and left of A A' as centres, and radii respectively equal to h c" and b' c' (Fig. 60) describe arcs intersecting in c' and c'. With successively c' and b right and left of A A' as centres, and radii respectively equal to C E and b c (Fig. 60) describe arcs intersecting in c and c ; and with successively c and c right METAL-PLATE WOEK. 183 and left of A A' as centres, and radii respectively equal to c d" and c' d' (Fig. 60) describe arcs intersecting in d' and d'. Also with successively d' and c right and left of A A' as centres, and D E and c d respectively as radii describe arcs Fig. 6L m SIDE PATTERN. intersecting in d and d. Through d, c, h, A, b, c, d, draw an unbroken curved line. Also through d', c', h', A ', b', c', d', draw an unbroken curved line. Join d d\ right and left of A A' ; then dAdd' A' d' will be the side pattern required. The remark about lines //', e e', &c., in (end pattern) Fig. 59, applies to lines c c', b V, b c', &c., in the present pattern. PEOBLEM XXII. To draw the pattern for a tapering body with top and base parallel, and having circular top and oblong bottom with semicircular ends (tea-bottle top, for instance), the dimensions of the top and bottom of the body and its height being given. Four cases will be treated of ; three in this problem and one in the problem following (see also § 79, p. 157). Case I. — Pattern when the body is to be made up of four pieces. Draw (Fig. 62) the plan of the body (see Problem VIII., p. 132) preserving of its construction the centres 0, 0' ; the points b in plan of bottom where the extremities of the plan 184 METAL-PLATE WOEK. lines of the sides meet the extremities of the plan semicircles of the ends ; and the points b' in plan of top where the sides and ends meet in plan. Join b' b at the four corners. The ends bBbb'B' b', and & E 6 b' E' b' of the body (see g, p. 129), are portions of frusta of oblique cones. Making the body in four pieces it will be best that the seams shall correspond with the lines A b', B B', D b', and E E', then one pattern only, consisting of a half-end with a half-side pattern attached, will be required. Fig. 62. 4 ^ b B \ E 1 ^ ^ — — ^=^r C d b A b To draw the pattern. Draw separately E' E & fc' (Fig. 63), the YlY.dbV portion of rig. 62, thus. Draw any line X X and with any point O (to correspond with 0, Fig. 62) in it as centre and 0 E (Eig. 62) as radius describe an arc (here a quadrant) EcZ&, equal to E 6 of Fig. 62. Make E 0' equal to E 0' (Fig. 62), and with O' as centre and 0' E' (Fig. 62) as radius describe an arc (here a quadrant) E' V equal to E' V of Fig. 62. Joinino- b b' completes the portion of Fig. 62 required. Now divide E b into any number of equal parts, here three, in the points d and c. From E' and 0' draw E' E", 0' 0" perpendicular to X X and each equal to the given height of the body. Join E E", O 0" ; produce them to intersect in P (§ 80, p. 158) ; from P let fall P P' perpendicular to X X, and join E" 0". With P' METAL-PLATE WOEK. 185 as centre and P' d, P' c, and P' h successively as radii describe arcs cutting XX in D, C, and B. Join these points to P by lines cutting 0"E" in D', C, and B'. Fig. 63. Next draw (Fig. 64) a line P E equal to P E (Fig. 63), and with P as centre, and P D, P C, and P B (Fig. 63) successively as radii describe arcs d, c, and h. With E as centre and radius equal to E d (Fig. 63) describe an arc cutting arc d in D, With same radius and D as centre describe an arc cutting arc c in C, and with C as centre and same radius describe an arc cutting arc h in B. Join the points D, C and B to P. With P as centre and P E" (Fig. 63) as radius describe an arc e' cutting P E in E'. With same centre and P D' (Fig. 63) as radius describe arc d' cutting P D in D'. Similarly 186 METAL-PLATE WORK. with same centre and P C and P B' (Fig. 63) successively as radii find points C and B'. Through h' (Fig. 62) draw h' H perpendicular to h' D and equal to the given height, and join D H, then D H will be the true length of the line of which h' D is the plan, that is, will be the length of a slant of the body at the middle of the side, where one of the seams Fig. 64. will come. With B' and B (Fig. 64) as centres and radii respectively equal to D H and h A (Fig, 62) describe arcs intersecting in A. Through the points E, D, C, and B draw an unbroken curved line. Also through the points E', D', C, and B' draw an unbroken curved line. Join B A, A B' ; then E C A B' E' will be the pattern required. METAL-PLATE WOEK. 187 Case XL — Pattern when tlie body is to be made up of two pieces. We will suppose the seams are to correspond with the lines BB' and EE'. It is evident that here we need but one pattern only, which will combine a side of the body and two half-ends, in fact will be double that of Fig. 64. First draw B B' B' B (Fig. 65) a half-end pattern exactly as the half-end pattern E E' B' B in Fig. 64 is drawn, and with B' as centre and B'B as radius and B as centre and hb Fig. 65. (Fig. 62) as radius, describe arcs intersecting in F. Join B F, F B' ; and produce F B' indefinitely. Make F P' equal to P B (Fig. 63), and with P' as centre and P C, P D and P E (Fig. 63) successively as radii describe arcs c, d, and e. With F as centre and b c (Fig. 63) as radius describe an arc cutting arc c in C, and with C as centre and same radius describe an arc cutting arc d in D. With same radius and D as centre describe an arc cutting arc e in E. Join C P', D P', E P'. Now with P' as centre and P C (Fig. 63) 188 METAL-PLATE WOEK. as radius describe an arc c' cutting P' C in C, and with same centre and P D', P E" (Fig. 63) successively as radii describe arcs d' and e' respectively cutting P' D and P'E in D' and E'. Through F, C, D, and E draw an unbroken curved line. Also through B', C, D', and E' draw an unbroken curved line. Then B B F E E'B' B' will be the complete pattern required. Case III. — Pattern when the body is to be made up of one piece. In this case we will put the seam to correspond with B B' (Fig. 62). We now need an end pattern (the end 6 E 6' E' 6' in plan), with right and left a side pattern attached (6 A 6 h', hBhh' in plan), and joined to each of these, a half-end pattern (h h' B' B, hh'B'B in plan). First draw Fig. 63 ; then draw (Fig. 66) P E equal to PE (Fig. 63) and with P as centre and PD, PC and PB (Fig. 68) successively as radii describe arcs dd, cc, and h h. With E as centre and radius equal to E (Fig. 63) describe arcs cutting arc d d right and left of P E in D and D. With points, D, D, successively as centres and same radius describe arcs cutting arc c c right and left of P E in C and C ; and with same radius and the last found points as centres describe arcs cutting arc h h right and left of P E in B and B. Join all the points found to P. With P as centre and P E" (Fig. 63) as radius describe an arc cutting P E in E', With same centre and PD' (Fig. 63) as radius describe an arc d' d' cutting lines P D right and left of P E in D' and D'. With same centre and P C (Fig. 63) as radius describe an arc c' c' cutting lines PC right and left of PE in C and C. Similarly by arc h'h' find points B' and B'. Through B, C, D, E, D, C, B, draw an unbroken curved line. Also through B', C, D', E', D', C, B', draw an unbroken curved line. This gives us B E B B' E' B' a complete end pattern. Now with B' on the right-hand side of the end pattern as centre and B' B as radius, and B as centre and b b (Fig. 62) as radius describe arcs intersecting in F. Join B F, F B' ; produce F B' indefinitely, and to F B' attach the half-end pattern F E E' B' in precisely the same manner that METAL-PLATE WORK. 189 FEE'B' tlie half-end pattern in Fig. 66 is attached to the side pattern B F B'. By a repetition of the foregoing Fig. 66. E TV construction on the left of the end pattern B E B B' E' B' we can attach B B E E' B' and complete E E E E' E' E' the pattern required. PEOBLEM XXIII. To draw, without long radii, the pattern for a tapering body with top and base parallel, and having circular top and oblong bottom with semicircular ends. The dimensions of the top and bottom of the body and its height being given. This problem is a fourth case of the preceding, and is exceedingly useful where the work is so large that it is 190 METAL-PLATE WOEK. inconvenient to draw the whole of the plan, and to use long radii. To draw the pattern (with the body in four pieces, as in Case I. of preceding problem). (81.) Draw (Fig. 67) E' E 6 A 6' one quarter of the plan of the body. Divide the quadrants E h, E' b', each into the same number of equal parts, here three, in the points d, c, d', c' ; join d d', c c'. Through E' draw E' F perpendicular to E' E and equal to the given height of the body. From E' along E'E mark off E' D, E C, and E'B respectively equal to dd', c c', and h h' ; and join E F, D F, C F, and B F; then E f' D F, C F, and B F will be the true lengths of E E', d d', c c', and h 6' respectively. Next join d E', and draw E' e" perpen- dicular to it and equal to the given height, and join d e" ; then d e" may be taken as the true length of d E'. Also join c d' and b c' ; through d' and c' draw d' d" and c' c" perpendicular to c d' and b c' respectively, and each equal to the given height, and join c fZ" and h c" ; then c d" and b c" may be taken as the true lengths of d' c and c' b respectively. Through b' draw METAL-PLATE WORK. 191 6' h" perpendicular to h' A and equal to the given height, and join A h", then A h" will be the true length of h' A. Next draw (Fig. 68) E E' equal to E F (Eig. 67), and with E' and E as centres and radii respectively equal to d e" and E d (Fig. 67) describe arcs intersecting in d, and with d and E' as centre and radii respectively equal to DF and E' (Fig. 67) describe arcs intersecting in d'. With d' and d as centres and radii respectively equal to c d" and d c (Fig. 67) describe arcs intersecting in c, and with c and d' as centres Fig. 68. and radii respectively equal to C F and d' c' (Fig. 67) describe arcs intersecting in c'. With c' and c as centres and radii respectively equal to 6 c" and c b (Fig. 67) d scribe arcs intersecting in h. Similarly with 6 and c' as centres and radii respectively equal to B F and c' b' (Fig. 67) describe arcs intersecting in b'. With b' and b as centres and radii respectively equal to b" A and b A (Fig. 67) describe arcs intersecting in A. Through E, d, c, b, draw an unbroken curved line. Also through E', d', c', b', draw an unbroken curved line. Join b' A, b A[; then E c A 6' c' E' is the pattern required. (82.) The lines dd', cc', dE', &c., are drawn in Fig. 68 simply to show the position that the lines which correspond to them in Fig. 67 (dd', cc', dl&', &c.) take upon the developed surface of the tapering body. 192 METAL-PLATE WORK. PEOBLEM XXIV. To draw the 'pattern for a tapering body with top and base parallel, and having an oval bottom and circular top (ovaZ canister top, for instance). The height and dimensions of the top and bottom of the body being given. Again four cases will be treated of; three in this problem and one in the problem following (see also § 79, p. 157). Draw (Fig. 69) the plan of the body (see Problem IX. , p. 132), preserving of its construction the centres 0, 0', P, P, and the four points (^d) where the end and side curves of the plan of the bottom meet one another,ialso the points A'E' where the axis A E cuts the circular top. Join d A' (two places) and d E' (two places). From the plan we know (see g, p. 129) that dQdA'G'B',dBdA' B' E', the ends of the body, are like portions of the frustum of an oblique cone ; we also know that d Ad A', d E d E', the sides of the body, are like portions of the frustum of an oblique cone. METAL-PLATE WOEK. 193 (83.) It is evident that in this problem the arcs dGd, E'G'A' and dBd, E'B'A' are, neither pair, proportional (§ 67, p. 124). We have hitherto in Prohlems XVIII., XX., and XXII. been dealing with proportional arcs. The work- ing will therefore differ, though but slightly, from that of problems mentioned. In Plate II. (p. 203) is a representation of the tapering body for which patterns are required, also of two oblique cones (x and Z). The oblique cones show to what portions of their surfaces the several portions of the tapering body correspond. Thus the sides, B', of the body correspond to the B portion of cone Z, and the ends. A', correspond to the portion A of cone x. The correspondence will be more fully recognised as we proceed with the problem. Case I.— Pattern when the body is to be made up of four pieces. We will suppose the seams are to correspond with the plan lines G Q', B B', A A', E E', of ends and sides, as in Problem XXII. Then one pattern only, consisting of a half-end pattern, with, attached, a half-side pattern will be required. To draw the pattern. Draw (Fig. 70) separately G'G fd A' the G'GfdA' poi-tioii of Fig. 69, thus. Draw any line X X and with any point O in it (corresponding to 0, Fig. 69) as centre and OG (Fig. 69) as radius describe an arc Gd, equal to G d of Fig. 69. Make G 0' equal to GO' (Fig. 69), and with 0" as centre and O'G' (Fig. 69) as radius describe an arc G'A' equal to G' A' of Fig. 69. Joining d A' completes the portion of Fig. 69 required Now divide Gd into any number of equal parts, here three, in the points / and e. From G' and 0' draw G' G", 0' 0" perpendicular to X X and each equal to the given height of the body. Join G G", 0 0" ; produce them to inter.^ect in S (§ 80, p. 158) ; from S let fall S S' per- pendicular to X X, and join 0" G". Now join d to S', by a line cutting arc G'A' in d', then (§ 68, p. 124, and d, p. 126) 0 194 METAL-PLATE "WORK. d' and d are corresponding points and (§ 67, p. 124) the arcs G d, G' d' are proportional ; G G' (Z is the plan of a portion of an oblique-cone frustum, lying between the same generating lines, and d d' A! is plan of a portion of the same frustum outside the generating line 8' d. With S' as centre Fig. 70. and S'/, S'e, and S'd successively as radii describe arcs cutting X X in F, E, and D ; join these points to S by lines cutting 0"G" in F', E', and D'. With 8' as centre and radius the distance between S' and A' describe an arc A' A cutting X X in A ; from A draw A A" perpendicular to X X and cutting 0" G" in A", and join A" 8. Next draw any line X X (Fig. 71) and with any point P in it (corresponding to P, Fig. 69) as centre and radius PA (the radius of arc dKd, Fig. 69) describe an arc A d equal to Ac? of Fig. 69. Make A A' and AO' respectively equal to METAL-PLATE WORK. 195 A A' and A 0' (Fig. 69). Divide A d into any number of equal parts, here three, in the points h and c. From A and 0' draw A' A", 0' 0" perpendicular to X X and each equal to the given height of the body. Join A A", P 0" ; produce Fig. 71. them to intersect in Q (§ 80, p. 158) ; from Q let fall Q Q' per- pendicular to X X, and join 0" A". With Q' as centre and Q' 6, Q' c, and Q' d successively as radii describe arcs cutting XX in B, C, and D, and join these points to Q. Next draw (Fig. 72) a line S G equal to S G (Fig. 70) and with S as centre and S F, S E, and S D (Fig. 70) successively as radii describe arcs /, e, and d. With G as centre and radius equal to G/ (Fig. 70) describe an arc cutting arc / in F. With same radius and F as centre describe an arc cutting arc e in E, and with E as centre and same radius describe an arc cutting arc d in D. Join the points F, E, and D to S. With S as centre and S G" (Fig. 70) as radius describe an arc g' cutting S G in G'. With same centre and S F' (Fig. 70) 196 METAL-PLATE WORK. as radius describe arc /' cutting S F in F'. With same centre and S E' (Fig. 70) as radius describe an are e' cutting S E in E', and witb same centre and S D' (Fig. 70) as radius describe arc d' cutting S D in D'. With same centre and S A" (Fig. 70) as radius describe arc a', and with D' as centre and radius d' A' (Fig. 70) describe an arc intersecting arc a' in A'. Make D Q equal to D Q (Fig. 71) and with Q as centre and Q C, Fig. 72. S Q B, and Q A (Fig. 71) successively as radii describe arcs c, b, and a. With D as centre and d c (Fig. 71) as radii describe an arc cutting arc c in C, and with C as centre and same radius describe an arc cutting arc b in B. Similarly with same radius and B as centre find point A. Join A A'. Through the points G, F, E, D, C, B, A draw an unbroken curved line. Also through the points G', ¥', E', D', A' draw an un- broken curved line. Then GDAA'E'G' is the pattern required. METAL-PLATE WOEK. 197 Case II.— Pattern wken the body is to be made up of two pieces. We will suppose the seams are to correspond with tke lines GG' and B B'. It is evident that here we need but one pattern only, which will combine a side of the body and two half-ends, in fact will be double that of Fig. 72. First draw B D A'B' (Fig. 73) a half-end pattern exactly as the half-end pattern GD A'G' in Fig. 72 is drawn, and make DQ equal to D Q (Fig. 71). With Q as centre aiid Fig. 73. QC, QB, and Q A (Fig. 71) successively as radii describe arcs dd,cc,hh, and a a. With D as centre and radius equal to dc (Fig. 71) describe an arc cutting arc c c in C, and with same radius and C as centre describe arc cutting arc 6 6 in B. With B as centre and same radius describe an arc cutting arc a in A, and with same radius and A as centre describe an arc cutting arc 6 6 in B. Similarly with same radius and 198 METAL-PLATE WORK. points B and C, to the right of A, successively as centres find points C and D. Join D Q and produce it indefinitely ; make D S' equal to S D (Fig. 70). With S' as centre and S E, S F, and S a (Fig. 70) successively as radii describe arcs e, f, and g, and with D as centre and d e (Fig. 70) as radius describe an arc cutting arc e in E. With same radius and E as centre describe an arc cutting arc / in F. Similarly with F as centre and same radius find point G. Join the points E, F, and G to S'. With S' as centre and S D' (Fig. 70) describe arc d' cutting S' D in D'. With same centre and S E' (Fig. 70) as radius describe arc e' cutting S' E in E'. Similarly with same centre and S F' and S G" (Fig. 70) successively as radii find points F' and G'. Through the points D, C, B, A, B, C, D, E, F, G, draw an unbroken curved line. Also through points A', D', E', F', G' draw an unbroken curved Ime. Then B A G G' A B' is the complete pattern required. Case III.— Pattern when the body is to be made up of one piece. In this case we will put the seam to correspond with G G' (Fig. 69). We now need an end pattern (the enddBd A' B'E' in plan), with right and left a side pattern attached (cZ A A', d E d E' in plan), and joined to each of these a half- end pattern {d A' G' G, E' G' G in plan). First draw Figs. 70 and 71 ; then draw (Fig. 74) S G equal to S G (Fig. 70), and with S as centre and S F, S E, and S D (Fig. 70) successively as radii describe arcs//, ee, and dd. With G as centre and radius G/ describe arcs cutting arc// right and left of S G in F and F. With points F, F, succes- sively as centres and same radius describe arcs cuttino- arc e e right and left of S G in E and E, and with same^radius and the last found points as centres describe arcs cutting arc d d right and left of S G in D and D. Join all the points found to S. With S as centre and S G" (Fig. 70) as radius describe an arc cutting S G in G'. With same centre and S F' (Fig. 70) as radius describe an arc /'/' cutting lines SF right and left of SG in F' and F'. With same centre METAL-PLATE WOEK. 199 and S E' (Fig. 70) describe an arc e' e' cutting lines S E right and left of S G in E' and E', and with same centre and S D' (Fig. 70) as radius describe arc d! d' cutting lines S D right and left of S G in D' and D'. With S as centre and U' right and left of S Gr as centres and radii respectively equal to S A" and d' A' (Fig. 70) describe arcs intersecting in A' and E'. Through D, B, F, G, F, E, D, draw an unbroken curved line. Also through E', D', E', F', G', F', E', D', A', draw Fig. 74. an unbroken curved line, and join D E', D A'. This gives us D G D A' G' E' a complete end pattern. Now attach the side pattern D A D A' and the half-end pattern D G G' A' to the right and left of the complete end pattern we started with, in precisely the same manner that the side pattern D A D A' and half-end pattern D G G' A' in Fig. 73 is attached to D A', which corresponds to D A' in Fig. 74. This will complete G E G A G G' G' G' the pattern required. 200 METAL-PLATE WOEK. PEOBLEM XXV. To draw, without long radii, tie pattern for a tapering body with top and base parallel, and having an oval bottom and circular top. The height and dimensions of the top and base of the body being given. This problem is a fourth case of the preceding, and is exceedingly useful where the work is so large that it is inconvenient to draw the whole of the plan, and to use Ions radii. ° To draw the pattern (with the body in four pieces, as in Case I. of preceding problem). (84.) Draw (Fig. 75) E d & A A' d' E' one quarter of the plan of the body. J oin c (the point where the end and side curves Fig. 75. of the plan of the bottom meet) to A', the extremity of the quadrant E A'. We must now get at corresponding points in METAL-PLATE WORK. 201 the arcs E c, E' A'. To do this, as the arcs are not propor- tional (§ 67), we must find the plan of the apex of the oblique cone of a portion of which E c A' E' is the plan. It is in the finding of these points that our present working differs from the working of Problems XIX., XXL, XXIII., and XXVII., where the corresponding points d d', cc' (Figs. 50, 58, 60, and 82) are found. With radius 0 E produce arc E c inde- finitely, and through 0 draw 0 Q perpendicular to E'E and cutting E c produced in Q. Then E Q is a quadrant, and E Q, E' A' (each a quadrant) are proportional. In Q and A' therefore we have corresponding points (§ 68), as well as in E and E', which are points on the longest generating line (h, p. 126) ; and the intersection of 0 0' produced, of which E E' is part, and Q A' joined and produced will give us V, the required plan of the apex. Next divide E c into any number of equal parts, here two, in point d. Join dY, cY cutting E' A' in d' and c', respectively (the lines from d and c are not carried to V in the fig.), then d and d', c and c' are corresponding points. Next through E' draw E'E perpendicular to E'E and equal to the given height; from E' along E'E mark off E' D, E' C, respectively equal to d d' and c c', and join E F, DF, and CF; then EE, DF and CF will be the true lengths of E E', d d', and c c'. Join E d' and d c' ; through d' and c' draw d' d" and c' c" perpendicular to d' E and c' d re- spectively, and each equal to the given height, and join E d", d c" ; then E d" and d c" may be taken respectively as the true lengths of d' E and c' d. Now divide arc A c into any number of equal parts, here two, in the point h, and join h A' ; through A' draw three lines A' A" ; one perpendicular to A' c, the second perpendicular to A' b, and the third per- pendicular to A' A, and each equal to the given height, and join c A", A", and A A"; then cA", 6 A", and A A" may be taken as the true lengths of A' c, A' h, and A' A respectively. Next draw (Fig. 76) E E' equal to E F (Fig. 75), and with E and E' as centres and radii respectively equal to E d" and E' d' (Fig. 75) describe arcs intersecting in d', and with d' and E as centres and radii respectfully equal to D F and E d 202 METAL-PLATE WORK. (Fig. 75) describe arcs intersecting in d. With d and d' as centres and radii respectively equal to d c" and d' c' (Fig. 75) describe arcs intersecting in c', and with, c' and d as centres and radii respectively equal to C F and d c (Fig. 75) de- scribe arcs intersecting in c. With c and c' as centres and radii respectively equal to cA" and c' A' (Fig. 75) describe Pig, 76. arcs intersecting in A', and with A' and c as centres and radii h A" and c h (Fig. 75) describe arcs intersecting in h. Simi- larly with A' and 6 as centres and radii respectively equal to A A" and 6 A (Fig. 75) describe arcs intersecting; in A. Join A A'. Through E, d, c, b, A draw an unbroken curved line. Also through E', d', c\ A' draw an unbroken curved line. Then E c A A' E' is the pattern required. The lines d d', c c', E d', d c', &c., are not needed for the working, they are drawn for the reason stated in § 82, end of Problem XXIII. (85.) If V is inaccessible, corresponding points c, c', d, d' can thus be found. From the point E' along the arc E' A' set off an arc proportional to the arc E c in the following manner. Join Oc (line not shown in fig.) and through 0' draw O'c' (also not shown in fig.) parallel to O c and cutting arc E' A' in c' ; then arcs E' c' and E c will be pro- portional. (The student must particularly notice this METAL-PLATE WOEK. 205 method of drawing proportional arcs. It is outside the scope of the book to prove the method.) Now divide arcs E'c', E c each into the same number of equal parts, here two, in the points d and d' ; then d, d' and c, c' are corresponding pointsc PEOBLEM XXVI. To draw the pattern for a tapering body with top and base parallel, and having oblong bottom with round (quadrant) corners, and circular top. The dimensions of the top and base of the body and its height being given. Again four cases will be treated of, three in this problem, and one in the problem following (see also § 79, p. 167). Draw (Fig. 77) the plan of the body (see Problem X., p. 133) preserving of its construction the centres 0 0' and the Fig. 77. points E' B' where the flat sides and flat ends meet the circle of the top. Join B B', E E' each in four places. From the plan we know (see g, p. 129) that the round corners of the body are portions of frusta of oblique cones. 206 METAL-PLATE WOEK. (86.) Looking at the plan, we can at once see that what we have to deal with differs somewhat from what has as yet been before us. Hitherto a line passing through the centres of the plan arcs bisected the arcs, and the cone development was consequently identical each side of a central line. In Fig. 77, however, the line drawn through 0 0' does not bisect the plan arcs E B, E' B'. This affects the working but little, as will be seen. In Plate III. (p. 213) the tapering body is represented ; also an oblique cone Z, the A portion of which corresponds to the A' portion of the body, and the development of the former is the development of the latter. Case I. — Pattern when the body is to be made up of four pieces. We will suppose the seams to correspond with the plan lines C E', D E', F B', A B', of ends and sides, as in Problems XXII. and XXIV. just preceding. Then one pattern, com- prising a half-end, a complete corner, and a half-side, will be the pattern required. To draw the pattern. Draw separately (Fig. 78) an E E', B' B portion of Fig. 77, thus. Draw an indefinite line S' d (Fig. 78), and with any point 0 (corresponding to 0, Fig. 77) in it as centre and 0 B (Fig. 77) as radius describe an arc B E. Join O 0' (Fig. 77) and produce it cutting arc B E in ; make dB&ndd'E (Fig. 78) equal respectively to dB and d E (Fig. 77). Now (Fig. 78) make d O' equal to d O' (Fig. 77), and with 0' as centre and 0' B' (Fig. 77) as radius describe an arc B' E'. Make d' B' and d'E' equal respectively to d'B' and d'E' (Fig. 77). Joining E E', B B' completes the portion of Fig. 77 required. Now divide (Fig. 77) B E into any number of parts. It is convenient to take d as one of the division points, and to make d c equal to E ; leaving c B without further division, thus making the division of B E into three portions not all equal. In actual practice the dimensions of the work will suggest the number of parts expedient. Now (Fig. 78) make d c equal METAL-PLATE WOKK. 207 to dc (Fig. 77), then B E (Fig. 78) will be divided corre- spondingly to BE (Fig. 77). Draw XX parallel to S' d ; and at d and 0 draw dD, OQ perpendicular to 8' d; and meeting X X in D and Q ; also through d' and 0' draw d' D', Fig. 78. O' 0" perpendicular to S' d ; the line O' 0" cutting X X in Q'. Make Q' 0" equal to the given height of the body, and draw 0"D' parallel to XX. Join DD', QO"; produce them to intersect in S (§ 80, p. 158) ; and from S let fall a perpen- dicular to S' d, cutting S' d in S'. With S' as centre and S' E, 208 METAL-PLATE WORK. or S' c (which is equal to S E) and S' B successively as radii describe arcs cutting ^' d in e and /. Draw e C, /F perpen- dicular to X X and cutting it in C and F ; also join C S, F S, cutting 0"D' in C and F'. Next draw S D (Fig. 79) equal to S D (Fig. 78), and with S as centre and SO, SF (Fig. 78) successively as radii Fig. 79. T describe arcs c and 6. With D as centre and radius equal to d E (Fig. 78) describe arcs cutting arc c c in E and C. With C as centre and radius cB (Fig. 78) describe an arc METAL-PLATE WOEK. 209 cutting arc h in B. Join E, C, and B to S. Make S D' equal to S D' (Fig. 78) and with S as centre and S C (Fig. 78) as radius describe an arc c' c' cutting S E and S C in E' and C respec- tively. With same centre and S F' (Fig. 78) as radius describe an arc h' cutting S B in B'. Through E, D, C, and B draw an unbroken curved line. Also through E', D', C, and B' draw an unbroken curved line ; this will complete the pattern of the round corner. To attach the half-end and half-side patterns to E E' and B B' respectively, the true lengths of E' D and B' A (Fig. 77) must be found. Draw (Fig. 77) E' H perpen- dicular to E' D and equal to the given height of the body ; join D H, then D H is the true length of E' D. The lines B' E and B' A being equal, their true lengths are equal, we will therefore for convenience find the true length of B' A in that of B' F. Draw B' G perpendicular to B' F and equal to the given height, join F G, then F G is the true length required. Now with E' (Fig. 79) as centre and D H (Fig. 77) as radius, and E as centre and radius E D (Fig. 77) describe arcs intersecting in G. Join EG, G E' ; this attaches to E' E the half-end pattern. With B' (Fig. 79) as centre and F G (Fig. 77) as radius, and B as centre and radius B A (Fig. 77) describe arcs intersecting in A. Join B A, A B' ; this attaches to B' B the half-side pattern. Then A B' E' G is the complete pattern required. Case II. — Pattern when the body is to be made up of two pieces. Here it will be best that the seams shall correspond with the lines A B', F B', that is with the middle of each side. The required pattern will then be double that of Case I. Draw (Fig. 80) EBB' E', the corner pattern, in exactly the same manner that E B B'E' (Fig. 79) is drawn. With E' as centre and E' E as radius, and E as centre and E E (Fig. 77) as radius describe arcs intersecting in F. Join E F, F E'. Produce F E' indefinitely and make F S' equal to E S. Using S' as centre, the round corner F G B' E' can be drawn as was E B B' E. With B' as centre and F G (Fig. 77) p 210 METAL-PLATE WOEK. as radius, and B as centre and B A (Fig. 77) as radius de- scribe arcs intersecting in A. Similarly witli B' and G as Fig. 80. centres and- same radii respectively describe arcs intersecting in F. Join B A, A B', G F, F B', then A D F F B' E' B' will be the pattern required. Case III. — Pattern when the body is to be made up of one piece. We will suppose the seam to correspond with C E' the middle of one end. Draw GF DB B' E' B' (Fig. 81) in the METAL-PLATE WOKK. 211 same manner that G F D B B' E' B' (Fig. 80) is drawn. With B' as centre and B' B as radius, and B as centre and B B (Fig. 77) as radius describe arcs intersecting in A. Join B A, A B' ; produce A B' indefinitely. Make A P equal to S F (Fig. 78). With P as centre and S C and S D succes- FiG. 81. P. sively as radii describe arcs h and I, and with A as centre and B c (Fig. 78) as radius describe an arc cutting arc h in H. With H as centre and c d (Fig. 78) as radius describe an arc cutting arc Z in L ; and with L as centre and same radius p 2 212 METAL-PLATE WOEK. describe an arc cutting arc Ji in M. Join H, L, and M to P. With P as centre and S C (Fig. 78) as radius describe arc h' cutting PH and PM in H' and M' respectively. With same centre and SD' (Fig. 78) as radius describe arc I cutting P L in L'. Through the points A, H, L, M draw an unbroken curved line. Also through the points B', H', L', M' draw an unbroken curved line. With M' as centre and D H (Fig. 77) as radius, and M as centre and E C (Fig. 77) as radius describe arcs intersecting in N. Join MN, NM'. Repeating the working to the left of B'G, the GOO'B' portion of the pattern can be drawn which completes the pattern required. PEOBLEM XXVII. To draw, without long radii, the pattern for a tapering hody with top and base parallel, and having ohlong bottom with round quadrant corners, and circular top. The dimensions of the top and base of the body and its height being given. This problem is a fourth case of the preceding, and is exceedingly useful where the work is so large that it is in- convenient to draw the whole of the plan, and to use long radii. To draw the pattern (with the body in four pieces as in Case I. of preceding problem). Draw (Fig. 82) E c & A 6' c' d', one quarter of the plan of the body. Divide the arc (quadrant) d b into any number of equal parts, here two, in the point c, and the arc d' V into the same number of equal parts in the point c' ; and join c c'. Through d' draw d' F perpendicular to d! E and equal to the given height of the body. From d' along d' E mark off d' B, equal to b b', and d' D equal to c c' and d d' (which two lines have happened to come in this particular fig, so nearly equal that we may take them as equal), and join B F, E F, and D F ; then B F and E F will be the true lengths of b V and 'Etd' respectively, and DF may be taken as the true METAL-PLATE WOBK. 215 length of both c c' and d d'. Next join c' d,h'c; draw c c , b- perpendicular to c' c?, b' c respectively, and each equal to the given height ; and join d c" and c h" ; then d c' , c 6 may Fig. 82. be taken as the true lengths of c' d and b'c respectively. Draw h'h" perpendicular to 6' A and equal to the given height, and join A&", then A 5" is the true length of 6 A. Next draw (Fig. 83) d d' equal to D F (Fig. 82), and with d and d' as centres and radii respectively equal to dc and d'c' (Fig. 82), describe arcs intersecting in c'. With c and d as centres, and radii respectively equal to DF and (ic (Fig. 82) describe arcs intersecting in c; and with c and c as centres and radii respectively equal c h" and c' h (Fig. 82) describe arcs intersecting in 5'; also with h' and c as centres and radii respectively equal to B F and cb (Fig. 82) describe arcs intersecting in h. With d' and d as centres and radii respectively equal to EF and (Fig. 82) describe arcs intersecting in E; and with h' and 6 as centres and radii respectively equal to &" A and &A (Fig. 82) describe arcs 216 METAL-PLATE WOEK. intersecting in A. Through d, c, h, draw an unbroken curved line. Also through d', c', V draw an unbroken curved line. Fig. 83. Join cZE, d'E. 5 A, and &'A; then EcA6'cZ' is the patteni required. The lines cc\hh',&o., are not needed for the working they are drawn for the reason stated in § 82, end of Problem XXIII. PEOBLEM XXVIII. To draw the pattern for an Oxford Mp-bath, the like dimemions to those for Problem XI. being given. It is only necessary to treat of two cases, one in this problem, and one in the problem following (see also § 79, p- 157). Draw (Fig. 84) the plan of the body (see Problem XI., p. 134) preserving of its construction the centres 0 F P Q'' Q • the points D, D' (two sets) in which the arcs,' in plan' of the back and sides meet each other; and the points h, g' (two sets) m which the plan arcs of the sides and front meet each other. Join h g' (two places) as shown in the fig. Examining the plan of the bath we see (d, p. 55) that the back of it. METAL-PLATE WOEK. 217 D A D D' A' D', is a portion of a right cone ; that the sides D D' ^ are {g, p. 129) each of them a portion of an oblique cone; and that the portion h g' g' h is also a portion of an Fig. 84. Q Q'. D' ^^^^ A 1 A' 0 f 1 ^ D' D oblique cone. Similarly as in Problem XXIV. (§ 83, p. 193) the arcs D ^, D' g' and B A, B' are, neither pair, propor- tional. In Plate IV. (p. 227) is a representation of an Oxford hip- bath, also of a right cone a;, and two oblique cones Y and Z. The cones show to what portions of their surfaces the several portions of the bath correspond. Thus the back, A', of the bath corresponds to the A portion of right cone oa ; the half- fronts, C, of the bath correspond each of them to the C portion of oblique cone Y, and the sides, B', of the bath correspond each of them to the B portion of oblique cone Z. Patterns when the body is to be made up of three pieces. We will put the seams to correspond with the lines D D' 218 METAL-PLATE WORK. (two places), and B B'. Clearly, only two patterns will be required, one for the back of the bath, and the other for a complete side and a half-front. To draw the pattern for the back. Draw E A A' 0' 0 (Fig. 85) the elevation of the back (see Fio. 85. ,D Problem XI. and Fig. 24a, p. 135) and produce A A', 0 0' to intersect in 0". "With 0 as centre and 0 A as radius describe a quadrant A D (corresponding with A D, Fig. 84), and divide it into any number of equal parts, here three, in the points h and c. Draw c C, & B perpendicular each of them to A O and cutting it in points B and C. Join 0" C, and produce it to cut O E in C ; and through C draw C C" parallel to A 0 and cutting 0' E in C". Join 0" B and produce it to cut 0 E in B' ; and through B' draw B' B" parallel to A 0 and cutting 0" E in B". Then A C" is the true length of C C, and A B" the true length of B B'. METAL-PLATE WOEK. 219 Next draw (Fig. 86) 0" A equal to 0" A (Fig. 85), and with 0" as centre and radius 0" A describe an arc DAD, and with same centre and 0" A' (Fig. 85) as radius describe an Fig. 86. I' arc D' A' D'. Mark off, right and left of A, AB, B C, and C D, each equal to A 6 (Fig. 85), one of the equal parts into which the quadrant A D is divided. Join D 0", C 0", B 0" right and left of AO", and produce 0"A, 0"B, 0"0 indefi- nitely. Make A E equal to A E (Fig. 85) ; B B" equal to AB" (Fig. 85); and CC" equal to AC" (Fig. 85); and through D, C", B", E, B", C", D, draw an unbroken curved line. Then D E D D' A' D' is the required pattern for back of bath. To draw the pattern for a side and a half-front. 220 METAL-PLATE WOEK. Draw separately D' D fJi g' (Fig. 87), the B'Bfhg' portion of Fig. 84, thus. Draw any line X X and with any point Q (to correspond with Q, Fig. 84) in it as centre and (same Fig. 87. fig.) Q D (the D on the further side of A B) as radius describe an arc D h equal to D of Fig. 84. Make D Q' equal to D Q' (Fig. 84), and with Q' as centre and Q' D' (the further D', Fig. 84) as radius describe an arc D' g' equal to D' g' of Fig. 84. METAL-PLATE WORK. 221 Joining h g' completes the portion of Fig. 84 required. Now from D' and Q' draw D' D", Q' Q" perpendicular to X X and each equal to the given height of the D BD portion of the bath. Join DD", Q Q" ; produce them to intersect in S (§ 80, p. 158) ; and from S let fall S S' perpendicular to X X. Join 's' g', and produce it to cut arc Dh in g, then g and g' will he corresponding points. Divide D g into any number of equal parts, here three, in the points e and/. J oin Q" D". With S' as centre and S' e, S'/, and S' g successively as radii describe arcs cutting XX in E, F, and G; join these points to S by lines cutting Q" D", produced, in E', F', and G'. With S' as centre and S' h as radius describe an arc cutting X X in H, and join H S. Next draw separately B'Bhg' (Fig. 88) one of the B'Bhg' portions of Fig. 84, thus. Draw any line XX and Fig. 88. with any point P (to correspond with P, Fig. 84) in it as centre and P B (Fig. 84) as radius describe an arc B h equal to Bhoi Fig. 84. Make BP' equal to BP' (Fig. 84), and with P' as centre and P' B' (Fig. 84) as radius describe an arc 222 METAL-PLATE WORK. B' equal to B' of Fig. 84. J oining h g' completes the por- tion of Fig. 84 required. Now from B' and P' draw B' B", P'P" perpendicular to XX and each equal to the given height of the D B D portion of the bath. Join B B", P P" ; produce them to intersect in E (§ 80, p. 158) ; and from E let fall E E' perpendicular to X X. Join E' \ cutting arc B'^' in y, then h and A' will be corresponding points. Divide B h into any number of equal parts, here two, in the pointy. Join P"B". With E' as centre and E'y and E'A successively as radii describe arcs cutting X X in J and H ; join these points to E by lines cutting P" B" in J' and H'. \ Next draw (Fig. 89) a line D S equal to D S (Fig. 87). Fig. 89. With S as centre and S B, S F, S G, and S H (Fig. 87) succes- sively as radii describe arcs e, /, g, and h. With D as centre METAL-PLATE WOEK. 223 and radius equal to D e (Fig. 87) describe an arc cutting arc e in E, and with same radius and E as centre describe an arc cutting arc / in F. With F as centre and same radius de- scribe an arc cutting arc g in G, and with G as centre and radius g h (Fig. 87) describe an arc cutting arc h in H. Join the points E, F, and G (not H) to S. Make S D' equal to S D" (Fig. 87) ; and make S E', S F', and S G' respectively equal to S E', S F', and S G' (Fig. 87). Through the points D, E, F, G, H draw an unbroken curved line. Also through points D', E', F', G' draw an unbroken curved line, and join H G'. This completes the side pattern, to which we have now to attach, at H G', a half-front pattern. With H and G' as centres and radii respectively equal to HH' and g' h' (Fig. 88) describe arcs intersecting in H'. Join HH'; pro- duce it indefinitely and make H E equal to H E (Fig. 88). With E as centre and E J, E B successively as radii describe arcs J and 6. With H as centre and hj (Fig. 88) as radius describe an arc cutting arc j in J, and with same radius and J as centre describe an arc cutting arc h in B. Join the points J and B to E, and make E J' and E B' respectively equal to E J' and E B" (Fig. 88). Through H, J, and B draw an unbroken curved line. Also through H', J', and B' draw an unbroken curved line. Then DFHBB'G'D' is the complete pattern required. PEOBLEM XXIX. To draw, without long radii, the pattern for an Oxford Mp-hath ; given dimensions as lefore. This problem is a second case of the preceding. Patterns when the body is to be made up of three pieces with seams as in the preceding problem. To draw the pattern for the back. Draw E A A' 0' 0 (Fig. 90) the elevation of the back as in Fig. 85, and produce 0' 0. With 0 as centre and 0 A as 224 METAL-PLATE WOEK. radius describe a quadrant A D (corresponding witli A D, Fig. 84), and divide it into any number of equal parts, here three, in the points 6 and c. Join & 0, c 0, and with 0 as centre and radius O'A' describe a quadrant D'A" (corre- sponding with D' A', Fig. 84), and cutting lines 0 c, 0 6 in c' and V respectively. Then D D' A" A will be the plan of that portion of the back of the bath of which 0' A' 0 A is the elevation. Through h and c draw 6 B and c C perpendicular to 0 A ; and through 6' and c' draw V B' and c' C perpendi- cular to O'A'. (Here part of &'B' happens to coincide with Fig. 90. part of c C). Join B' B, C 0, and produce them to meet 0 E in ^B" and C" respectively. Through C" draw C" c" parallel to 0 A, and through B" draw B" h" parallel to 0 A. Through D' draw D' D" perpendicular to D' D and equal to the given height of that portion of the bath, and join D D", then D D" will be the true length of D' D. Join D c' ; through c' draw c' F perpendicular to c' D and equal to the given height, and join D F, then D F may be taken as the true length of D c'. In drawing the pattern, we will first set out that for the O A 0' A' portion of the back, and then attach to it the pattern for the 0 E A portion. It is evident that the 0 A 0' A' METAL-PLATE WOEK. 225 portion is half a right cone frustum, and therefore its pattern can be drawn (see Problem YIII., p. 41), thus. Draw (Fig. 91) a line D D' (the line D D' left of E A') equal to D D" (Fig. 90). With D and D' as centres and radii respectively equal to D F and D' c' (Fig. 90) describe arcs intersecting in c'. With D' and D as centres and radii respectively equal to D F and D c (Fig. 90) describe arcs intersecting in c. To find points 6 and h' proceed as just explained and with the same radii, but e and c' as centres instead of D and D'. Fig. 91. J? D Similarly to find A and A', and the points h and &c., on the right-hand side of A A'. Through the points D', c', &', A', h', c\ D' draw an unbroken curved line. Also through D, c, 6, A, h, c, D draw an unbroken curved line, and join D D'. This completes the pattern for the 0 A 0' A' portion of back of bath, to which we have now to attach, at D A D, the pattern for the OEA portion. Join A' A; produce it indefinitely and make A E equal to A E (Fig. 90). Next join, right and left of A A', h' h, c' c, and produce them indefinitely ; make 6 6", right and left of A A', equal to A 6" (Fig. 90). Also make cc", right and left of A A', equal to A c" (Fig. 90), and Q 226 METAL -PLATE WOEK. through the points D, c", h", E, b", c", D draw an unhroken curved line; then DEDD'A'D' is the complete pattern required. To draw the side and half-front pattern. Draw separately (Fig. 92) the B 7* D D' g' B' portion of Fig. 84, that is, the plan, of a side and half-front of the bath. Join h g' (as was done in Fig. 84). We now have to get at corresponding points in the arcs D h, D' g', and also in the arcs h B, g' B'. We do this by the method given in § 85, p. 202, Fig. 92. Q- thus. Join Q' g' and through Q draw parallel to Q'^^' and cutting arc D A in ^ ; then the arcs D g and D' g' will be the proportional. Now divide arc D g into any number of equal parts, here two only to avoid confusion, in the point e, and divide the arc D' g' into two equal parts in the point e' ; then e, e' and g, g' are corresponding points. Next join P h and through P' draw P' h' parallel to P A and cutting arc B' g' in h'. Divide arcs B A and B'h' each into the same number of equal METAL-PLATE WOKK. 229 parts, here two, in the points / and /' respectively; then/,/ and A, h' are corresponding points. Join e e', g g\ ff, and Ji h'. Through B' draw B' A perpendicular to B' B and equal to the given height of that portion of the bath ; and from B' along B' B mark off B' F, B' H, B' G, and B' E respectively equal to //', Jih', gg', and ee'. Join B A, FA, HA, GA and E A; then BA, F A, H A, GA, and EA will be respectively the true lengths of B B', / f, h h', g g', and e e'. Join /' B, h'f, e' g, and D'e; through/' draw //" perpendicular to/'B, and equal to the given height, and join B/", then B/" may be taken as the true length of /' B. Through h', g\ e', and D' draw h' h", g' g", e' e", and D' D", perpendicular to h'f, g' h, e' g, and D' e respectively, and each equal to the given height ; also draw D' D" perpendicular to D D' and equal to the given height; and join f¥, hg" ge", eD", and DD"; then /A", hg", ge", eD", and DD" may be taken as the true lengths of h'f, g' h, e' g, D' e, and D' D respectively. Fig. 93. J) Next draw (Fig. 93) B B' equal to B A (Fig. 92), and with B and B' as centres and radii respectively equal to B/" and B' /' (Fig. 92) describe arcs intersecting in /, and with /' and B as centres and radii respectively equal to F A and B/ (Fig. 92) describe arcs intersecting in /. With / and /' as centres and radii respectively equal iof¥ smdf'h' (Fig. 92) 230 METAL-PLATE WORK. describe arcs intersecting in l\ and with ^' and / as centres and radii respectively equal to HA and fh (Fig. 92) describe arcs intersecting in li. With h and A' as centres and radii respectively equal to I f and A' (Fig. 92) describe arcs inter- secting m g\ and with g' and l as centres and radii respectively equal to G A and 7*^7 (Fig. 92) describe arcs intersecting in g. With g and g' as centres and radii respectively equal to i/e" and g' ^ (Fig. 92) describe arcs intersecting in e', ' and with e' and g as centres and radii respectively to E A and e (Fig. 92) describe arcs intersecting in e. With e and e' as centres and radii respectively equal to e D" and e' D' (Fig. 92) describe arcs intersecting in D', and similarly with D' and e as centres and radii respectively equal to DD" and eD (Fig. 92) describe arcs intersecting in D. Join D D'. Through B, /, g, e, D draw an unbroken curved line. Also throiTgh B', /',;*', gr', e', D' draw an unbroken curved line. Then B (/ D D' gr' B' is the pattern required. PEOBLEM XXX. To draw the pattern for an ohlong taper hath, the like dimensions to those for Problem XIII. being given. Again, it is only necessary to treat of two cases— one in this problem, and one in the problem following (see also § 79, p. 157). Draw (Fig. 94) the plan of the body (see Problem XIII., p. 140), preserving of its construction the centres O, 0'; and the points 6, b', a, a', s, s', h, h'. A, A', B, B', in which the straight lines and arcs meet each other. Join b, b', a, a', s, s' h,h', A A' (two places), and BB' (two places) as shown 'in the fig. Examining the plan we see (d, p. 55) that each round corner A A' B B' of the toe is the same portion of a right cone frustum ; and each of the round corners a a' V h, s s' h' h, of the head are the same portion (g, p. 129) of an oblique cone frustum. As we proceed it will be seen that METAL-PLATE WORK. 231 the construction of the pattern for the round corners of the head of bath is exactly the same as that for the round corners of the hody in Problem XXVI. (see also § 86, p. 206). In Plate V. (p. 237) is a representation of an oblong Fig. 94. taper bath, also of an oblique cone Z, tbe A portion of which corresponds to the A' portion of the body, and the development of the former is the development of the latter. Patterns when the body is to be made up of four pieces. We will put the seams to correspond with the lines G A', G h', D D', and C C. The patterns required will be three, 232 METAL-PLATE WORK. one for the head of the bath, one for the toe, and one for the sides. The pattern for the toe can be readily drawn by Problem XXVII., p. 90. Likewise the pattern for the sides. The pattern for the head is drawn as follows. Fig. 95. Draw separately (Fig. 95) a head-corner portion of Fig. 94, say a a! h' h, thus. Draw an indefinite line 8' d (Fig. 95), and with any point 0 (corresponding to O, Fig. 94) in it as centre and 0 a (Fig. 94) as radius describe an arc ha. Join 00' (Fig. 94) and produce it to cut arc METAL-PLATE WOEK. 233 ba in d; make d b and d a (Fig, 95) equal respectively to db and da (Fig. 94). Now (Fig. 95) make dO' equal to d 0' (Fig. 94), and with 0' as centre and 0' a' (Fig. 94) as radius describe an arc b' a'. Make d'b' and d' a' equal respectively to d'b' and d' a' (Fig. 94). Joining bb', a a' completes tlie portion of Fig. 94 required. Now divide (Fig. 94) 6 a into any number of parts. It is convenient to take d as one of the division points, and to make d c equal to da; leaving c b without further division, thus making the division of 6 a into three portions, not all equal. In actual practice the dimensions of the work will suggest the number of parts necessary. Here &c is left without further division in order to make clear the correspondence of this problem to Problem XXYI., p. 205. Now (Fig. 95) make d c equal to d c (Fig. 94), and then b a will be divided correspondingly to 6 a (Fig. 94). Draw X X parallel to S' d ; and at d and 0 draw d D, 0 Q perpendicular toS' d; and meeting X X in D and Q ; also through d' and 0' draw d' D', 0' O" perpendicular toS'd; the line O' 0" cutting X X in Q'. Make Q' O" equal to the given height of the bath, and draw 0"D' parallel to XX, and cutting d'B' in D'. Join DD', QO"; produce them to intersect in S (§ 80, p. 158) ; and from S let fall a perpendicular to S' d, cutting S'd' in S'. With S' as centre and S' a, or S' c (which is equal to S' a) and S' & successively as radii describe arcs cutting S' d in g and /. Draw g C, fB perpendicular to XX and cutting it in C and B ; and join C S, B S, cutting 0" D' in C and B'. Next draw S D (Fig. 96) equal to S D (Fig. 95) and with S as centre and SO, SB (Fig. 95) successively as radii describe arcs c and b. With D as centre and radius equal to da (Fig. 95) describe arcs cutting arc c in A and C. With C as centre and radius cb (Fig. 95) describe an arc cutting arc & in B. Join A,1 C, and B to S. Make S D' equal S D' (Fig. 95) and with S as centre and S C (Fig. 95) describe an arc (not shown in the fig.) cutting S A and S 0 in A and C respectively ; make S B' equal to S B' (Fig. 95). 234 METAL-PLATE WOEK. Througli A,D,C, and B draw an unbroken curved line. Also through A', D', C, and B' draw an unbroken curved line ; this will complete the pattern for a head-corner. To attach the patterns for the flat portions of the head to A A' Fig. 96. and B B' respectively ; draw through a' (Fig, 94) a line a' F perpendicular to a' and through V draw V Gr perpendicular to6'D'. Now draw (Fig. 94) a'P perpendicular to a'F and equal METAL-PLATE WOEK, 235 to the given height of the bath, and join F P, then F P is the true length of F a'. Next draw V E perpendicular to V G (&' R will of course coincide with the line V B') and equal to the given height ; join GE, then GE is the true length of G h\ Now with B' (Fig. 96) as centre and G E (Fig. 94) as radius, and B as centre and radius h G (Fig. 94), describe arcs intersecting in G. Join B G, B' G. With A' (Fig. 96) as centre and F P (Fig. 94) as radius, and A as centre and radius a F (Fig. 94) describe arcs intersecting in F. Join AF and produce it indefinitely, and make As equal to as (Fig. 94); through A' draw A' s' parallel to A s and equal to a! s' (Fig. 94) ; and join s s'. The pattern for the portion, seen in plan in Fig. 94, G 6 a s s' a' 6' of the head of the bath is now completed. It is needless to work out in detail the addition of the portion (Fig. 96) s7t G Ji'f's' of the pattern, by which we complete the head pattern GEGA'E'B'. The extra lines in this latter portion of the pattern appertain to the next problem. PEOBLEM XXXI. To draw, without long radii, tJie pattern for an oblong taper hath ; given dimensions as in Problem XXX. This problem is a second case of the preceding. Patterns when the body is to be made up of four pieces with seams as in preceding problem. Again, the patterns required will be three ; one for the head of the bath, one for the toe, and one for the sides. The latter pattern needs no description. The pattern for the toe can be readily drawn by Problem XXVIII., p. 94. The pattern for the head can be drawn as follows. Draw half the plan of the bath, as the lower half of Fig. 94, and divide the arcs s^,s'^', each into any number of equal parts, here two, in respectively the points / and /', and join//'. Draw (Fig. 94a) any two lines KS, KL perpen- dicular to each other, and make K L equal to the given 236 METAL-PLATE WORK. height of the bath. From K along K S mark off K H equal to Jih',KF equal to //', and K S equal toss'; and join L H, L F, L S ; then L H, L F, and L S are the true lengths of ^ ff, and s s' respectively. Next join (Fig. 94) /' h, s' f; draw //", s' s", perpendicular to /' Ji, s'f respectively,' and each equal to the given height ; and join hf\ fs" ; then h /', fs" may be taken respectively as the true lengths of fh and s'f. The true length of h' G may be found along h' B' as was that of 6 G in Problem XXX. along 6' B' ; it will of course be equal to GR, and we shall speak of it as GR. Next draw (see Fig. 96, left-hand portion) ss' equal to LS (Fig. 94a), and with s' and s as centres and radii respectively equal to fs" and sf (Fig. 94) describe arcs intersecting in /. With / and s' as centres and radii respectively equal to LF (Fig. 94a) and s'f (Fig. 94) describe arcs intersecting in/'; and with /' and / as centres and radii respectively equal to h f" and /A (Fig. 94) describe arcs intersecting in Ji ; also with h and /' as centres and radii respectively equal to LH (Fig. 94a) and /'A' (Fig. 94) describe arcs intersecting in h'. With Ji' and h as centres and radii respectively equal to G R and JiG (Fig. 94) describe arcs intersecting in G. Through draw an unbroken curved line. Also through s',f, h! draw an un- broken curved line ; and join A G, G h'. Then sJiGh'fs' is the pattern for the portion of the head of the bath repre- sented in plan in Fig. 94 by the same lettering. It is unne- cessary to pursue the pattern further. METAL-PLATE WORK. 239 Book III. CLASS III. Patterns for Miscellaneous Astioles. The ■book we now reach is made up of miscellaneous problems, all of which are of considerable practical importance, as well as are also typical cases. PROBLEM 1. To ■draw a pattern for the elbow formed by two equal circular pipes (cylinders of equal diameters) which meet at any angle. First draw (Fig. 1) a side elevation and a part-plan of the elbow, as follows. Draw any two lines G Gr', 0 G' at an angle to each other equal to the given angle of the elbow ; through any point G in GGr' draw GA perpendicular to G G' and equal to the diameter of the pipes ; and through A draw an indefinite line A A' parallel to G G'. Then from any point 0 in G' 0 draw. 0 N perpendicular to G' 0 and equal to the diameter of the pipe ; and through N draw an indefinite line N A parallel to G' 0 and meeting A A' in A'. J oin A' G' (the line A' G' produced always bisects the angle G G' 0) ; then 0 N A G is a side elevation of the elbow, and the line A' G' represents the joint or ' mitre ' of the pipes. On G A describe a semicircle Gd A (this will be the part- plan of the elbow) ; which divide into any number, here six, of equal parts in the points 6, c, d, e, f. From each of these points draw lines parallel to A A', namely, the lines b B', c C, d D', eE', and /F', cutting G A in the points B, C, D, E, and F respectively. METAL-PLATE WORK. 241 To draw the pattern, witli the longitudinal seams of the pipes to correspond with the lines G' G, G' 0. Draw (Fig. 2) an indefinite line Gr G, and, from a point G in it set off along it distances G F, F E, E D, D 0, C B, and B A, equal respec- tively to Gf,fe,ed,dc,cb, and 6 A of Fig. 1. For small work the chord distances round the semicircle give satis- factory results ; of large work we shall speak immediately. Through G, F, E, D, C, B, and A draw indefinite lines GG', FF', EE', DD', CC, BB', and A A' perpendicular to G G ; and make G G' equal to G G' (Fig. 1), F F' equal to F F' (Fig. 1), and E E', D D', C C, B B', and A A' equal respectively to EE', D D', CC, BB', and A A' (Fig. 1). From G' through F', E', D', C, B', and A' draw an unbroken curved line ; this completes one-half the pattern required. The half to the right of A A' can be drawn in similar manner, setting off along A G the same above distances, but in reverse order and starting from A. (87.) The line G G (Fig. 2) must of course be equal to the circumference of the pipe. Therefore if a somewhat close accuracy is required, as in large work, it will be found best to make G G equal to the circumference of the pipe (Problem XL, p. 12) ; then to divide GGinto twice as many equal parts as the semicircle AdG (Fig. 1) is divided into ; then through each division point to draw lines perpendicular to G G and proceed as above explained. (88.) By making a start in Fig. 2 with G G' for an outer line of the pattern? we ensure that the longitudinal seams of the pipes shall, as we desired, correspond with G G' and G' 0 of Fig. 1. If the seams are to correspond with the lines A' A and A' N of this fig., we should letter our commencing indefinite line, not G G, but A A, and should start with, for an outer, line of the pattern, a perpendicular A A' equal to A A' of Fig. 1. The pattern we should then get would be like to that obtained by dividing Fig. 2 into halves along A A', turning each portion half-way round, and making the lines G G' coincide. R 242 METAL-PLATE WOEK. PEOBLEM II. To draw the pattern for the ~^ --piece formed hy two equal or unequal circular pipes [cylinders of equal or unequal diameter^, which meet at right angles. First draw (Fig. 3) a side elevation and a part-plan of the two circular pieces of pipe, which we will suppose unequal, thus. Draw two indefinite lines Z d, and K J, intersecting each other at right angles in 0. Make OZ equal to the Fig. 3. diameter of the larger pipe, and through Z draw an indefinite M L parallel to K J. Make 0 A' and 0 H' each equal to half the diameter of the smaller pipe, and through A' and H' draw indefinite lines A' A and H' H each perpendicular to K J. In A' A take any point A, on H' H set off H' H equal METAL-PLATE WOEK. to A' A, and join A H cutting OdinD; then A' A H H' will represent, in elevation, a piece of the smaller pipe. Next in A'K take any point K, and through K draw KM per- pendicular to K J and meeting ML in M ; also in H' J take any point J, and through J draw JL perpendicular to K J and meeting ML in L; then MKJL will represent, in elevation, a piece of the larger pipe, and M K A' A H H' J L a side elevation (except the curve of junction) of the X-piece. With D as centre and radius D A, that is, half the diameter of the smaller pipe, describe a semicircle AdH; divide the quadrant Ad of it into any number of equal parts, here three, in the points 6 and c; and through & and c draw indefinite lines b B' and c U' parallel to A' A. Now on Z 0 describe a semicircle Z 3 0 (this will be a part-plan of the large pipe), and with O as centre and radius D A describe a quadrant H' E (this may be regarded as a part-plan of the smaller pipe) which divide, exactly as quadrant Ad was divided in the points F and G ; through F, Gr, and H' draw lines F 1, G 2, and H' 3 parallel to Z <^ and cutting the semi- circle Z 3 0 in points 1, 2, and 3. Through point 1 draw a line 1 B' parallel to K J and meeting h B' in B' ; through 2 draw 2 C parallel to KJ and meeting cC in C; and through 3 draw 3 D' parallel to K J and meeting d D' in D' From D' to A' through the points C and B' draw an unbroKen curved line, then A' C D' is the elevation of one-half of the curve of junction of the two pipes. In practice it is only necessary to draw the A'ODA (Fig. 3) portion of the elevation of the smaller pipe. The other half elevation H' 0 D H of it is drawn here simply to make the full side- elevation of the "l"-piece clearer. To get at the whole Y-piece, it is evident that we require two patterns, one for the smaller piece of pipe, up to its junction with the larger, and one for the larger with the hole in it that the smaller pipe fits to. To draw the pattern for the larger pipe, with the longi- tudinal seam to correspond with the line M L. First set out apart from the pipe itself the shape for the R 2 244 METAL -PLATE WORK. hole in it. Draw (Fig. 4) two indefinite lines Z 0' and A' A' intersecting at right angles in 0 ; from 0, on Z 0', right and left of A' A' set off distances 0 1', 1' 2', and 2' D' equal respec- tively to 0 1, 1 2, and 2 3 (Fig. 3), that is, to the actual distances on the round curve of the pipe at Z 0 that the lines IB', 2C', and 3D' are apart. Through points 1' and 2' Fig. 4. right and left of A' A' draw B' B' and G"C' perpendicular to Z 0'. Make 1' B', above and below Z 0', and right and left of A' A', equal to 1' B' (Fig. 3) ; and make 2' C, above and below Z 0', and right and left of A' A', equal to 2' C (Fig. 3) and through all the points as above found, namely D', C, B', A', B', C, D', C, B', &c., draw an unbroken curved line ; then D' A' D' A' D' will be the shape of the hole required. To complete the pattern for the M K J L (Fig. 3) piece of the larger pipe, make 0 Z and 0 0' each equal to half METAL-PLATE WOEK, 245 its circximference (Problem XI., p. 12) ; and throtigli Z and 0' draw indefinite lines ML perpendicular to ZO'. Make ZM of left-hand line ML, and O' M of right-hand line M L each equal to Z M (Fig. 3) ; similarly make Z L and O'L each equal to ZL (Fig. 3). Then MLLM will com- plete the pattern required. (89.) We have shown how to mark out by itself the hole the larger pipe, because in cases where the pipe is already made up, it is convenient to be able to mark out the shape of the hole apart from the pipe, on, say, a thin piece of sheet metal, which shape can then be cut out and used as a tem- plate ; being applied to the pipe and bent to it, and the shape of the hole marked on it from the template. Even when the pipe is not made up, it is useful when the pipe is large to be able to mark out the hole quite apart from the pipe itself. To draw the pattern for the smaller piece of pipe, the longitudinal seam to correspond with the line A' A (Fig. 3). Fig. 5. A £ C D C 3 A Draw (Fig. 5) an indefinite line A A. In it take any point D, and from D, right and left, set off distances D C, 246 METAL-PLATE WOEK. C B, B A, equal respectively to dc,c b, and b a (Fig. 3), tliat is, equal to one another ; and from the point D and each of the points 0, B, and A draw lines perpendicular to A A. Make DD' equal to DD' (Fig. 3), and the lines CC, BB', A A', right and left of D D', equal respectively to C C, B B', and A A' (Fig. 3). From either point A' to A' on the other side of D D' draw through B', 0', D', C, and B' an unbroken curved line ; then A A' A' A will be half the required pattern. The other half can be similarly drawn. If a somewhat close accuracy in the length of A A (Fig. 5) is required proceed as in Problem I. (§ 87, p. 241). See also § 88, p. 241. PEOBLEM III. To draw tJie pattern for the ' slanting "Y ' formed by two equal or unequal circular pipes {cylinders of equal or unequal diameter) which, meet otherwise than at right angles. We will suppose the pipes unequal. First draw (Fig. 6) a side elevation and a part-plan of the pipes, thus. Draw two indefinite lines K 0, and A" A' at that angle to one another that the pipes are to be, the angle 0 A" A', say. In A" A' take any point A' ; through it draw A' G' perpendi- cular to A" A' and equal to the diameter of the smaller pipe, and through G' draw a line parallel to A" A' and meeting K 0 in G" ; then A" A' G' G" will represent, in elevation, a piece of the smaller pipe. IsText in K 0 take any point O, and through it draw an indefinite line L G perpendicular to K 0. Make O L equal to the diameter of the larger pipe ; through L draw an indefinite line L M parallel to K O ; in L M take any point M, and through M draw M K perpendi- cular to M L. Then M K 0 L will represent, in elevation, a piece of the larger pipe ; and M K G" G' A' A" O L a side elevation (except the curve of junction) of the slanting "p. On A' G' describe a semicircle A' d' G' ; and divide it into any number of equal parts, here six, in the points METAL-PLATE WOEK. 247 &', c', d\ e'j and /' ; and through these division points draw indefinite lines h' B", c' C", d' D", e' E", and / F" parallel to G' G", and cutting A' G' in the points B', C, D', E', and Fig. 6. M Now from A' let fall a perpendicular to L G meeting L G in A, and through B', C and D' draw indefinite lines B' h, C c, and D' d perpendicular to and cutting L G in B, C, and D. 248 METAL-PLATE WOKK. Make B h equal to B' h', C c equal to C c', and D d equal to D' d' ; and from A through h and c to d draw an unbroken curved line. On L 0 describe the semicircle L 3 O, and through the points h, c, d, just found draw lines 61, c 2, and d 3 parallel to L G and cutting the semicircle L 3 0 in points 1, 2, and 3 ; then L N 3 D is a part-plan of the slanting Through point 1 draw a line 1 F" parallel to 0 K and meeting B' B" and F' F" in B" and F" respectively ; through 2 draw 2 E" parallel to 0 K and meeting C C" and E' E" in C" and E" respectively ; and through 3 draw 3 D" parallel to 0 K and meeting D' D" in D". From A" through the points B", C", D", E", and F" to G" draw an unbroken curved line, then this curve is the elevation of the curve of junction of the two pipes. In the straight line A"G" take any point 0', about midway between A" and G", and through it draw 0' Z per- pendicular to K O, and cutting lines 1 F", 2 E", and 3 D" in points 1', 2', and 3'. It is evident that for slanting "J* we require two patterns, one for the smaller piece of pipe up to its junction with the larger, and one for the larger with the hole in it that the smaller pipe fits to. To draw the pattern for the larger pipe, the longitudinal seam to correspond with the line M L. First to set out apart from the pipe itself the shape of the hole in it. Draw (Fig. 7) two indefinite lines Z 0 and G" A" intersecting at right angles in 0' ; and from O', on Z 0, right and left of G" A" set off distances O' 1', 1' 2', and 2' 3' equal respectively to 0 1, 1 2, and 2 3 (Fig. 6), that is, to the actual distances on the curve of the pipe at Z 0 that the lines B" F", C" E", and 3' D" are apart. Through points 1', 2', and 3', right and left of G" A" draw F" B", E" 0", and D" 8' per- pendicular to Z 0. Make 0' G" equal to O G" (Fig. 6), 0' A" equal to O' A" (Fig. 6). Also make V F", 2' E", 3' D", right and left of G" A" and above Z 0, equal respectively to 1' F", 2' E", 3' D" above Z 0' (Fig. 6) ; and 1' B", 2' C" right and left of G" A and below Z 0, equal respectively to 1' B", 2' 0", below Z 0' (Fig. 6). Through all the points as above METAL-PLATE WORK. 249 found, namely D", E", P", G", F", E", D", C", B", &c., draw an unbroken curved line ; then D" G" D" A" will be the shape of the hole required. Fig. 7. G° F' -» D' Z d- \ c 2' r 0' r 2' i 3- 0 c' £' A.' To complete the pattern for the M K 0 L (Fig. 6) piece of the larger pipe, make 0' Z and 0' 0 each equal to half its circumference (Problem XL, p. 12) ; and through Z and 0 draw indefinite lines M L perpendicular to Z O. Make Z M of left-hand line ML, and 0 M of right-hand line ML each equal to Z M (Fig. 6) ; similarly make Z L and 0 L each 250 METAL-PLATE WOKK. equal to Z L (Fig. 6). Then MLLM will complete the pattern required. In connection with this see § 89, p. 245. To draw the pattern for the smaller piece of pipe, the longitudinal seam to correspond with the line A" A' (Fig. 6). Draw (Fig. 8) an indefinite A' A'. In it take any point A', and from A' set off distances A' B', B' C, C D', D' E', E' F', and F' G' equal respectively to A' 6', h' c', c' d', d' e', e'f, Fig. 8. A' £' C D' B' F' G' and / G' (Fig. 6), that is, equal to one another. Through A'', B', C, D', E', F', and G' draw indefinite lines A' A", B' B", C C", D'D", E'E", F'F", and G' G" perpendicular to A' A' ;' and make A' A" equal to A' A" (Fig. 6), B' B" equal to B' B" (Fig. 6), C C" equal to C C" (Fig. 6), and D' D", E' E", F' F", and G' G" equal respectively to D' D", E' E", F' F", and G' G" (Fig. 6). From A" through B", C", D", E", METAL-PLATE WORK. 251 F", and G" draw an unbroken curved line; this completes one-half the pattern required. The half to the right of G' G-" can be drawn in similar manner, setting off along G' A' the same above distances, but in reverse order and starting from G'. With this see § 87, p. 241, and § 88, p. 241. PEOBLEM IV. To draw the pattern for the J formed by a funnel-shape piece of pipe and a circular piece, the former being ' square ' to the latter {part-cone joined 'square' to a cylinder); the diameter of the circular pipe, and the diameters of the ends of the funnel-shape pipe and its length being given. By ' being square' we mean that the axes of the pieces of pipe intersect and are at right angles. The given diameter of the smaller end of the funnel-shape pipe is the diameter in the direction of the length of the circular pipe and that coincides with its surface. First draw a side elevation and a part-plan of the "J", thus. Draw (Fig. 9) any two indefinite lines Zd and KJ, intersecting at right angles at 0. Make 0 Z equal to the diameter of the circular pipe; and through Z draw ML parallel to K J ; make OD equal to the length of the funnel- shape pipe and through D draw a line A H perpendicular to O d. Now make D A and D H each equal to half the given diameter of the larger end of the funnel, and 0 A' and 0 H' each equal to half the given diameter of its smaller end, which small end we will suppose is not let into, but fits against the circular pipe ; join A A' and H H', then A' A, H H' will be, in elevation, the main portion of the funnel- shape pipe. Next in A' K take any point K, and through K draw KM perpendicular to KJ and meeting ML in M; also in H' J take any point J, and through J draw JL perpendicular to K J and meeting M L in L ; then M K J L will represent, in elevation, a piece of the circular pipe : and MKA'AHH'JL a side elevation (except the curve of 252 METAL-PLATE WOEK. junction) of the "f . Produce A A' and H H' to intersect Z 0 in V. With D as centre and radius D A describe a semi- circle AdB. and divide the quadrant Ad of it into any number of equal parts, here three in the points h and c ; through & and c draw h B and c C each perpendicular to A H and cutting it in B and C, and join B V and C V. Now on Z 0 describe a semicircle Z 3 0 (this will be a part-plan of the circular pipe) cutting V H' in point 3. With 0 as centre and radius O H' describe a quadrant H' E (this may be regarded as a part-plan of the funnel-shape pipe) which METAL-PLATE WORK. 253 divide into the same number of equal parts that the quadrant A d is divided into, in the points / and g. Through / and g draw /F and g G each perpendicular to A' H' and cutting it in F and G ; join F V, G V, cutting the semicircle Z 3 0 in points 1 and 2 respectivelj. Through point 1 draw a line 1 B" parallel to K J, meeting A V in B'^' and cutting Z O and B V in 1' and B' respectively ; through 2 draw 2 0" parallel to K J, meeting A V in C", and cutting Z 0 and C V in 2' and C respectively ; and through d draw 3 D" parallel to K J, cutting Z O in Jy and meeting A V in J}". From D' through C and B' to A' draw an unbroken curved line, then A' C D' is the elevation of one-half the curve of junction of the two pipes. In practice it is only necessary to draw the 0 A' A D (Fig. 9) portion of the elevation of the smaller pipe. The other half-elevation 0 H' H D of it is drawn here simply to make the full side elevation of the f clearer. ^ It is evident that the 7" requires 2 patterns, one for the circular pipe with the hole in it that the funnel-shape pipe fits to, and one for the funnel-shape itself. To draw the pattern for the circular pipe; the longi-- tudinal seam to correspond with the line M L, Proceed in exactly similar manner as explained for the pattern of the corresponding pipe in Problem II., p. 242. To draw the pattern for the funnel-shape pipe ; the longi- tudinal seam to correspond with the line A A' (Pig. 9). With V (Fig. 10) as centre and V A (Fig. 9) as radius describe an arc A A, and from any point in it set off along the arc distances A B, BC, CD, DC, BC, and BA each equal to A 6 (one of the equal parts in which quadrant d A (Fig. 9) is divided). Join A V, B V, C V, D V, C V, B V, and A V ; and make the extreme lines A A' right and left of D V equal to A A' (Fig. 9), also the lines B B", right and left of D V, equal to A B" (Fig. 9), C C" right and left of D V, equal to A C" (Fig. 9) and D D" equal to A D" (Fig. 9). Through points A', B", C", D", C", B", A' draw an unbroken curved line, then A A' A' A will be one-half the pattern required. By continuing to the right, say, the arc A A, and setting off on 254 METAL-PLATE WORK. it the same above equal distances A B, B C, &c., and pro- ceeding in exactly similar manner the other half pattern can be drawn to complete the pattern required. METAL-PLATE WOKK. 255 PEOBLEM V. To draw the pattern for a tapering piece of pipe joining two circular pipes of unequal diameter and not in the same line with each other. A representation of such tapering piece of pipe joining two circular pipes will be found in Plate VI. (p. 259) ; also (except apex portion) of an oblique cone (Z) the frustum B'D'DEB of which (the line B'D' should be continued to the outside of the cone) corresponds to the tapering piece of pipe. ^ This tapering piece of pipe being an oblique cone frustum, its pattern can be drawn by Problem II., p. 113, Book II! In connection here see also §§ 58, 59, p. Ill, and Figs. 5 and 6, p. 111. ^ PEOBLEM VI. To draw the 'pattern for a piece of pipe joining two circular pipes of equal diameters, and not in the same line with each other. ^ This problem is a special case of the preceding, and the piece of pipe joining the two equal pipes is an oblique cylinder. Being so, we can draw its pattern by Problem V., p. 121, Book II. See also § 61, p. 112, and Figs. 7a and Ih p. 112. PEOBLEM VIL To draw the pattern for the ' Y 'formed by two tapering pieces of pipe uniting two equal circular pipes at the arms of the Y to a third larger piece of circular pipe at its stem, the Y heing both loays symmetrical. In Plate VI. (p. 259) is a representation of such ' Y • The frustum B' D' D E B of the oblique cone Z in same 256 METAL-PLATE WOKK. Plate (the line B' D' being continued to tlie outside lines of the cone) corresponds to either of the tapering pieces of pipe of the except that at their junction C D' a piece (B C D of Fig. 11. JBT ^ J cone Z) of each frustum is cut off. It is manifest that the base of either frustum, supposing the frustum completed, would coincide with the top of the stem of the Y- Let M A H J GP Q A"N (Fig. 11) represent the Y METAL-PLATE WORK. 257 elevation. As it is Loth ways symmetrical, G' G is the shortest generating line. Produce A' A" to A ; then A' A is the longest generating line. Also produce A A', G G', to their intersection in V. On A G describe a semicircle A cZ G, which divide into any number of equal parts, here six, in the points h, c, d, e, f. Produce A G indefinitely and from V let fall a perpendicular to the produced line, meeting it in v. With V as centre and V'/, V e, V d, V c, and V 6 respectively as radii, describe arcs meeting A G in F, E, D, C, and B. Join each of these points to V by lines cutting A' G' in F', E', D', C, and B'. From c and h draw c c' and I h' per- pendicular to A G and join c' and h' to V, by lines cutting A" 0 in c" and h". Through c" draw c" C" parallel to A G and cutting V C in C" ; also through h" draw h" B" parallel to A G and cutting V B in B". Now to draw the pattern (Fig. 12), so that the seam shall correspond with G' G (Fig. 11) the shortest generating line. Draw VA (Fig. 12) equal to VA (Fig. 11) and with V (Fig. 12) as centre and radii successively equal to VB, V C, VD, VE, VF, and YG (Fig. 11), describe, respectively, arcs hh, cc, d d, ee, ff, and g g. With A as centre and radius equal to A& (Fig. 11), describe arcs cutting the arc h h right and left of V A in B and B. With these points B and B as centres and radius as before, describe arcs cutting the arc c c right and left of V A in 0 and C. With same radius and the last-named points as centres, describe arcs cutting d d right and left of V A in D and D. With D and D as centres and same radius, describe arcs cutting e e right and left of V A in E and E, and with E and E as centres and same radius describe arcs cutting // in F and F. Similarly, with same radius and F and F as centres find points G and G. Join the points B, C, D, E, &c., right and left of V A to V. With V as centre and VA' (Fig. 11) as radius describe an arc cutting V A in A'. With same centre and V B' (Fig. 11) as radius describe an arc h'h' cutting VB right and left of VA in B'. With same centre and V C (Fig. 11) as radius describe an 258 METAL-PLATE WOKK. arc c' c' cutting V C riglit and left of V A in C. Simi- larly, with same centre and VD', VE', V F', and V G' (Fig. 11) successively as radii describe, respectively, arcs d'd', e'e',ff, and g' g', cutting VD, V E, VF, and V Q, right and left of V A respectively in D', E', F', and G'. Fig. 12. Draw through A', and the points B', C, D', E', F', and G', right and left of VA, an unbroken curved line. Also through the points G, F, E, D, right and left of V A draw unbroken lines. Now make A A" equal to A A" (Fig. 11) j METAL-PLATE WORK. 261 B B" right and left of A V equal to B B" (Fig. 11) ; and C C" right and left of A V equal to CO" (Fig. 11). Through A", and the points B", 0", and D, right and left of V A draw an unbroken curved line. Then G D A" D G G' A' G' will be the pattern for the pipe A"OGG' A' (Fig. 11) as well as also for the pipe A" 0 A K L. The patterns for the circular pipes need no explanation. (90.) In this problem, the Y l^eing symmetrical, the two tapering pieces of pipe are equal, and the mitre or joint line A"0, perpendicular to AG, bisects AG, and, produced, bisects also the angle LA" A'. If the Y ^ere unsym- metrical, the line of junction of the tapering pipes should still be made perpendicular to A G ; it will, however, neither bisect it nor the angle L A" A', and a distinct pattern for each tapering piece of pipe will be required. PEOBLEM VIII. To draw the pattern for the Y formed hy two cylindrical pieces of pipe of equal diameter uniting two further pieces at the arms of the Y to a third piece at its stem, the Y ^^ing both ways symmetrical. This problem is a special case of the preceding, the oblique cone frusta of that problem now becoming oblique cylinders. Let MA'HJG'POa'N (Fig. 13) represent the Y elevation. Produce A" a' to A'. On A' G' describe a semi- circle A' 3 G', which divide into any number of equal parts, here six, in the points 1, 2, 3, 4, and 5 ; through each point of division draw lines perpendicular to A' G', meeting it in the points B', C, D', E', and F', and through these points draw lines B' B", C C", D' D", &c., parallel to A' A". From any point A in A' A" draw A G perpendicular to A' A", meeting G' G" in G, and cutting the lines B' B", C C", &c., in the points B, 0, D, E, and F. . Next make B h equal to 262 METAL-PLATE WOEK. To draw the pattern (Fig. 14), so that the seam shall correspond with G' G" (Fig. 13). Draw any line G G, and at or about its centre draw any line A" a' perpendicular to it and cutting it in A. From A, right and left of it, on the line G G mark distances A B, B C, C D, D E, E F, and F G equal respectively to the distances Ah,h c, cd, d e, ef, and fG (Fig. 13). Through the points B, C, D, &c., right and left of A, draw lines parallel to A" a'. Make A a', A A" equal to A a', A A" (Fig. 13) respectively. Similarly make METAL-PLATE WOEK. 263 B h\ B B", C c', C C", D D', D D", &c., right and left of A" a' equal respectively to Bh', BB", 0 c', C C", DD', D D", &c. (Fig. 13). Draw an unbroken curved line from A" through B", Q", &G., right and left of A", to G". Also draw unbroken curved lines from D' to G', right and left of a', and an Fig. 14, unbroken curved line from D' through c', h', a', h', and c', to D'. Then G" A" G" G' D' a' D' G' will be the pattern re- quired for either of the portions of oblique cylinders a' A" G" G' D', or a' L K A' D'. The patterns for the circular pipes need no explanation. (91.) In this problem, as in the preceding, the Y heing symmetrical the two obliquely cylindrical pipes are equal, and the mitre or joint line a' D', perpendicular to A' G', bisects A' G', and, produced, bisects also the angle L a' A". If the Y were unsymmetrical, the line of junction of the pipes should still be made perpendicular to A' G' ; it will, however, neither bisect it nor the angle L a' A", and a distinct pattern for each obliquely cylindrical piece of pipe will be required. 264 METAL-PLATE WOEK. PEOBLEM IX. To draw the pattern for a body of which, the bottom is rectangular and the top circular (base of a tall-boy). First draw the plan of the body, thus. Draw (Fig. 15) A B C D the rectangle of the bottom of the body. The rectangle is here a square, but the working is the same of whatever nature the rectangle may be. Draw the diagonals of the rectangle, and with their intersection, which will be at 0, as centre, and radius equal to half the given diameter of the circular top, describe a circle. Through 0 draw 0 1 Fig. 15. perpendicular to BA and cutting the circle in 1'. Also through 0 draw E 5 perpendicular to A D, and cutting the circle in E' and 5'. (Neither 0 1 nor E 5 are fully drawn in the fig. in order to keep this clear of confusing lines.) Divide the arc from 1' to 5' into any number of equal parts, here four, in the points 2', 3', and 4'. Join A 2', A 3', A 4', and A 5'; then I'l, 2'A, 3' A, 4' A, 6' A, and 5' 5 are the METAL-PLATE WOEK. 265 plans of lines on the body, of -wliioh we need the true lengths, lengths which we find as follows. Draw (Fig. 16) any two lines A F, 5'F at right angles to each other and intersecting in F. From F on F A set off FA equal to the height of the body, and from F on F 5' set off F 1' equal to 1' 1 or 5' 5 (Fig. 15), F 2' equal to A 2' (Fig. 15), F3' equal to A3' (Fig. 15), F 4' equal to A 4' (Fig. 15) (the point 4' here in Fig. 16 happens to coincide with 2', because of A B C D (Fig. 15) being a square), and Fig. 16. 5' 2'3' FS' equal to A 5' (Fig. 15). Join the points 6', 2', 3', and 1' to A ; then A' 1' will be the true length of 1' 1 or 5' 5 (Fig. 15), A 2' will be the true length of A 2' or A 4' (Fig. 15), and A 3' and A 5' will be respectively the true lengths of A 3' and A 5' (Fig. 15). To draw the pattern (body in two pieces ; seams to cor- respond with E E' and 5'5). Draw (Fig. 17) any line A B and from point 1 at or about its centre draw a line 1 1' perpen- dicular to A B. Make the line 1 1' equal to A 1' {Fig. 16). Now make 1 A and 1 B equal each to 1 A or 1 B (Fig. 15), and with A and 1' as centres and radii respectively equal to A 2' (Fig. 16) and the arc 1' 2' (Fig. 15) describe arcs intersecting in 2'. With A and 2' as centres and radii 266 METAL-PLATE WORK, respectively equal to A 3' (Fig. 16) and 2' 3' (Fig. 15) describe arcs intersecting in 3'. With A and 3' as centres and radii respectivelj^ equal to A 2' (Fig. 16) and 3' 4' (Fig. 15) de- scribe arcs intersecting in 4'. And witb A and 4' as centres and radii respectively equal to A 5' (Fig. 16) and 4' 5' (Fig. 15) describe arcs intersecting in 5'. Through the points Fig. 17. 1', 2', 3', 4', and 5' draw an unbroken curved line. With A and 5' as centres and radii respectively equal to A 5 (Fig. 15) and A 1' (Fig. 16) describe arcs intersecting in point 5. Join A 5, 5 5'; then 1 A 5 5' 1' is half the required pattern. The other half can be similarly drawn, and E B A 5 5' 1' E' will be the complete pattern. PEOBLEM X. To draw the pattern for a dripping-pan with ' well.' The pattern for the bottom can be drawn by the method given in the preceding problem, the rim and corners being added by means of Problem XXIII,, p. 77. METAL-PLATE WOEK. 267 The problem that next follows is a typical case for all compound bent surfaces, such as vases, aquarium stands, mouldings, &c. PEOBLEM XI. To draw the pattern for a stand (aquarium stand, for instance'), the edge of which is a moulding. Let A B C D D' C B' A' (Fig. 18) be the plan of the stand Fig. 18. and E F G H the front elevation. Through D' draw D'f per- pendicular to A D (the line D'f will be a continuation of the 268 METAL-PLATE WORK. line C D') ; and on it draw /' c' d", the curve of the moulding, which divide into any numher of parts equal or unequal. The division here is into six equal parts, in the points a', h', c', d', and e', but the student will perceive as he advances that it may sometimes be advantageous that the division shall be into unequal parts. Through the points of division draw lines 1 to 1, 2 to 2, 3 to 3, 4 to 4, and 5 to 5, parallel to A D, the extremities of these lines terminating in A' A and D' D, the ' mitre ' lines of the plan. These ' mitre ' lines A' A and D' D bisect respectively the angles BAD, C D A ; in fact, the ' mitre ' lines of a moulding which is joined at any angle always bisect that angle. From the points 1, 2, 3, 4, and 5 on the line D' D draw lines parallel to D C and terminating in C C. To draw the pattern for the A' D' A D portion of the moulding. Draw (Fig. 19) any line KL, and from any point D' in it set off distances D' a, ah, he, &c., equal respectively to the distances d"a', a'h', V c', &c., round the curve d" c' f (Fig. 18), and through the points D', a, h, c, &c., draw lines perpendicular to KL. Make al equal to a 1 (Fig. 18), and & 2, c 3, 4, e 5, and /' D respectively equal to the distances &2, c 3, d4, e5, and /'D (Fig. 18); and through the points D' 1, 2, 3, 4, 5, and D draw an unbroken curved line. Now from D' set off D' A' equal to D' A' (Fig. 18), and through A' draw A' M parallel to KL. From the points in A' M where the lines through a, h, c, &c., cut A' M set off distances to the left of the line corresponding to the distances a 1, & 2, c 3, &c., to the right of the line KL, and through the points thus found draw an unbroken curved line. Then A' D' A D will be the pattern for the A' D' A D (Fig. 18) portion of the stand. To draw the pattern for the D' C D C (Fig. 18) portion of the stand. It will be at once seen from the plan that this differs only from the A' D' A D portion in that the distance D'C is less than the distance A'D'. And thus that if in Fig. 19 the lines A'M and LK are brought closer METAL-PLATE WORK. 269 together, so that A'D' is equal to D'C (Fig. 18), that the pattern so ohtained will be the pattern for the D' C D C piece of moulding. Fig. 19. A' D' L a- \f h c V d V e \5 A M f D K The student will notice that the elevation F G E H is not used in the working of the problem although here drawn ; that indeed it is unnecessary to draw an elevation. 270 METAL-PLATE WORK, Book IV. CHAPTER I. Metals and their Properties; Alloys; Solders; Soldering Fluxes. In the pages tliat follow, we deal almost exclusively with the metals that are used largely in plate or sheet ; our business being the setting-out of patterns on sheets or surfaces. Characteristic Features. (92.) Metals are natural elementary substances, as fa,r as is known. They are opaque (not transparent), reflect light from their polished surfaces, and have a characteristic lustre, known as the metallic lustre. With the exception of mercury, they are all solid at the ordinary temperature of the atmo- sphere. (93.) Silver, tin, lead, mercury, antimony, zinc, cadmium, and bismuth, have a whitish or greyish colour. Gold stands alone as a metal having a yellow colour ; copper is the only red metal. (94.) Metals differ widely in their behaviour under the influence of heat; some, as tin and lead, are fusible below red heat ; others, as copper, gold, and silver, fuse readily in ordinary furnaces ; nickel, iron, and manganese fuse with great difficulty ; platinum is practically infusible. Arsenic, cadmium, zinc, and mercury are volatile, that is, vaporise easily. An interesting example of volatility is that of zinc, which when at a bright red heat takes fire, burns with a greenish flame, and oxidises (unites with the oxygen gas METAL-PLATE WORK. 271 of the atmosphere), being thereby converted into a dense white flocculent substance, called formerly 'philosopher's wool.' (95.) The fracture of metals is often characteristic ; we get crystalline, granular, fibrous, silky, and other fractures. We have fractures in iron, depending upon its kind, of the first three forms. In copper we find the silky fracture. Properties; Specific Gravity; Melting-points. Metals have various properties. Some remarks on these, and other particulars respecting metals, now follow. (96.) Malleability. ~K property which is possessed by metals in varying degree is that of malleability, that is, of permitting extension of surface without rupture, by, for example, hammering, pressure, or rolling. Gold, which is capable of being hammered into leaves of extreme thinness, is the most malleable of all metals. Other metals, though malleable to a considerable degree, require to be annealed (heated red and allowed to cool down slowly) once or even several times during the operation of rolling out, or extending by the hammer as in raising and stretching. Copper is an example;' though, curiously enough, copper is equally malleable whether, after heating, it is allowed to cool slowly or is cooled suddenly by dipping while at red heat in cold water. ^ Zinc is in its most malleable condition at a tempera- ture a little above the boiling-point of water ; when less than half as hot again as this, it is brittle and unworkable. (97.) Of the theory of annealing nothing definite appears to be known; but it is supposed that on rolling out or hammering a piece of metal the particles or molecules of which the metal is composed, become strained and dis- arranged (' the grain closed '), and the metal is hardened ; and that on heating, the metal expands, and the strain being removed, the molecules rearrange themselves. This, however, does not explain many matters connected with 272 METAL-PLATE WORK. annealing ; for example, why one heated metal is hardened by being suddenly dipped in cold water, and another metal softened when treated in the same manner. The following, which illustrates the effect of ' hammer hardening ' on iron, may be of interest : — (98.) In 1854, at the meeting of the British Association in Liverpool, a paper on the crystallisation of iron under certain circumstances was read by Mr. Clay (Mersey Iron and Steel • Works), who stated that he selected a piece bf good, tongh, fibrous bar-iron, which he heated to a full red heat, and then hammered by light, rapid, tapping blows, until it was what is called ' black-cold.' After complete cooling he broke it, and found that the structure of the iron was entirely changed, and that, instead of bending nearly double without fracture, as it should have done, and breaking with a fine silky fibre when fracture did occur, an entire alteration had taken place, and the bar was rigid, brittle, and sonorous, incapable of bending in the slightest degree, and breaking with a glassy, crystallised appearance. By simply heating the bar to full red-heat again, the fibre was restored exactly as before. (99.) Tenacity. — The property in metals of resisting being torn asunder by a tensile or stretching force, is called tenacity. The tenacity of metals varies with their purity and mole- cular condition, as due to modes of treatment or preparation: for example, the tenacity of steel is much influenced by its ' temper,' and that of cast iron made by the cold-blast process is greater than when the process is that of the hot-blast. (100.) Ductility. — The property of being permanently lengthened by a tensile or stretching force, as in wire drawing, is called ductility. All the malleable metals are more or less ductile, though the most malleable metals are not necessarily the most ductile ; ductility being influenced more by tenacity than by malleability. The table shows how some of the metals, starting from gold (see ' Malleability,' above, § 96), rank under the headings given. METAL-PLATE WORK, 273 Malleability. Ductility. Tenacity. Gold Silver Gold Silver Steel Iron Copper Tia Platinum Platinum Iron Copper Aluminium Silver Gold Zinc Tin Lead Copper Platinum Lead Zinc Iron Nickel Zinc Tin Lead (101.) Conductivity. — Of all solids, metals are the best con- ductors of heat. The order of conductivity for a few impoi- tant metals, beginning with the best conductors, is — silver, copper, gold, tin, iron, lead, platinum, and bismuth. (102.) Welding. — An important property of some of the metals is that pieces can be ' welded ' together, that is, in- corporated with each other. Iron at a white heat is in a pasty condition and can be ' welded ' ; that is, if two white-hot and clean surfaces of iron be brought into contact and pressed or hammered together, they thus are ' welded,' that is, become part of one and the same mass. If lead and gold in a fine state of division be strongly pressed together at the ordinary temperature of the atmosphere, they will form one mass. (103.) Hardness, — The comparative hardness of metals is usually estimated by the force required to draw the metals into wires of equal diameter. In order of hardness we have — steel, iron, copper, silver, tin, antimony, and lead. (104.) Specific Gravity. — Some substances are, in their nature, more weighty bulk for bulk than others. Thus, a cubic inch of lead is heavier than a cubic inch of iron ; and a cubic inch of iron than a cubic inch of water. By their specific gravity the weights, relatively to each other, of substances, are known. The standard of comparison is an equal bulk of pure distilled water, and if the specific gravity of a body is, say, 2, this means that it is twice as heavy as the same bulk of water. T 274 METAL-PLATE WOEK. The specific gravity of platinum is 21 ; platinum therefore t)ulk for bulk is 21 times heavier than water. The specific gravity of antimony is 6-7, and a cubic foot of pure distilled water weighs very nearly 1000 ounces. Therefore a cubic foot of antimony weighs 6 • 7 times 1000 ounces, that is, 6700 ounces. Knowing the specific gravity of a substance, we can find the wei ght of any volume of it, by multiplying the given volume, in cubic inches, by its specific gravity, and by 62 • 4 the weight in lbs. of a cubic foot (1728 cubic inches) of water, and dividing by 1728. Thus the weight of 48 cubic inches of cast copper, the specific gravity of which is 8 • 6, is 48 X 8-6 X 62-4 . q , IS, that is to say, 14 • 9 lbs. 1728 As the relative weights of equal volumes of metals have often to be taken into consideration in using metals for constructive purposes, for example, in the covering of roofs, where weight is sometimes a matter of importance, a table of specific gravities follows. Table of Specific Gteavities and Melting-points. Metals. Antimony Bismuth Copper (cast) „ (-wrought) Steel Cadmium Iron (wrought) . Lead Tin .. .. ■ . Zinc Aluminium .. Nickel Platinum Specific Gravities. Melting-points. (Centigrade.) Authority for Melting-points. 6-7 432 Buillet. 9-8 268-3 Eeimsdyk. 8-6 8-8 1 1054 VioUe. ( 1300 7-8 to Buillet. 1400 8-6 320-7 Person. 7-8 j T 1 Buillet. 1600 11-36 326-2 Person. 7-29 232-7 Person. 7-19 433-3 Person. 2-67 1045 Violle. 8-30 1450 Pictel. 21-25 1775 Violle. METAL-PLATE WOKK. 275 The melting-points of the metals named are added to the table, as it is often useful to be able to refer to these. The degrees of heat are according to the Centigrade thermometer. This portion of the table is taken from ' Melting and Boiling Point Tables,' by Thos. Carnelley, 1885. We now proceed to notice more particularly the metals iron (including cast iron and steel), tin, zinc, and copper. Iron and Steel. (105.) Iron in a state of purity is comparatively little known ; the ores of it are various and abundant. In its commercial forms, as plate or sheet, bar, and cast iron, it is well known. As sheet it can be cut into patterns and bent into desired forms ; as bar it can be made hot and ' wrought,' that is, shaped by means of the hammer ; and when molten it can be run or cast into all sorts of shapes. Cast iron is brittle, crystalline in fracture (§ 95), and not workable by the hammer. In sheet and bar form, iron is malleable mostly fibrous in fracture, and capable of being welded (§ 102). The presence of impurities in bar iron, that is, the presence of substances not wanted in it at the time being, seriously affects its malleability. Thus the presence of phosphorus, or tin, renders it brittle when cold (' cold- short '), and the presence of sulphur makes it unworkable when hot (' hot-short '). Iron quickly rusts (oxidises, § 94) if exposed to damp air, as in the case of iron exposed to all weathers ; or to air and water, as with vessels in which barely sufficient water is left to cover the bottoms, the rusting (oxidation) being then much more rapid than when the vessels are kept full. Heated to redness and above, ' scale ' (oxide of iron) rapidly forms and interferes greatly with welding. It is impossible to enter here into any consideration of the processes by which iron is prepared from its ores. To two modern processes, however, we shall presently have par- ticularly to refer. (106.) The effects of the presence of several foreign sub- T 2 276 METAL-PLATE WOEK. stances in iron as impurities has jnst been alluded to, but the presence in it of ' carbon ' we have not spoken of. This is a substance which in its crystalline form is known as the diamond, and in its un crystalline form as charcoal. The presence of carbon in iron destroys its malleability, but at the same time gives to it properties various, so remarkable and useful to mankind, that to say, as a defect, of a piece of iron with carbon in it, that it is not malleable, is simply equivalent to saying when we have a piece of brass, that it is not a piece of copper. Quite the reverse of being ' matter in the wrong place,' carbon in iron furnishes a compound so valuable on its own account, so entirely of its own kinds (in the plural because its kinds are several), that, if there were other substances not metals, the compounding of which with a metal gave products at all resembling those of iron and carbon, then all such compounds would form a class of their own. The iron and carbon compound, however, stands incon- veniently alone. There we shall not leave it, but as aiding the full comprehension of it, notwithstanding that we define alloys (§ 115) as compounds of metals, shall consider it not as outside but as within this class of substances, as well as also shall speak of iron as being alloyed with it. We shall deal with it, however, under the present heading, treating compounds of actual metals later on. (107.) Iron is alloyed with carbon in proportions varying from say ^ to 5 per cent. When in the proportion of from 2 per cent, upwards, the compound is cast iron, that is, iron suitable for casting purposes ; in other proportions it is known as steel. In cast iron the metallic appearance is somewhat modified; in steel it is maintained. Originally steel was made by the addition of carbon to manufactured iron, and the word had then a fairly definite signification ; meaning a material of a high tensile strength ; that by being heated dull red and suddenly cooled could be made so hard that a file would not ' touch ' it, that is, would slide over it without marking it ; and that could have that hardness modified or ' tempered ' by further application of heat. But METAL-PLATE WOKK. 277 witli tlie introdTiction of the Bessemer process of steel making, and of tlie Siemens' process of making steel direct from the ores, processes by which any desired percentage of carbon can be given, the signification of the word has become en- larged, and now includes all alloys of iron and carbon between malleable iron and cast iron ; except that the term ' mild ' steel is sometimes applied to those alloys that approach in qualities to malleable iron. Steel plates are now produced equal in toughness, and it is said even excelling the best ' charcoal ' plates, and as they are much cheaper, the old charcoal -plate-making process is very generally giving way to the direct process. In practice, however, these plates are found to be more springy than good charcoal plates, and not so soft and easy to work. (108.) As iron is very liable to rust, surface protection is given to it by a coating of tin, or of an alloy of lead and tin, (lead predominating), or of zinc. Plates coated with tin are termed ' tin ' plates ; with lead and tin have the name of ' terne ' plates, and if coated with zinc ai'e said to be ' galvanised.' Terne plates are used for lining packing-cases, also for work to be japanned. Usual sizes of tin and terne plates are 14" X 10", 20" X U", 20" X 28", and they are made up to 40" X 28". (109.) Large iron sheets of various gauges coated with tin and having the same appearance as a ' tin ' plate are called Manchester plates, and sometimes tinned iron. But the latter term is more generally applied to sheets of iron which are coated with lead and tin, and are dull like terne plates. (110.) Iron coated with zinc is not so easily worked as when ungalvanised. In galvanising, the zinc ' alloys ' with the surface of the iron, and this has a tendency to make the iron brittle. Galvanised iron is useful for water tanks and for roofing purposes, as the zinc coating prevents rust better than a tin coating. For roofing, however, ' terne ' plates are largely used in America, and, kept well painted, are found to be very durable. Owing to the ease with which zinc is attacked by acids, galvanised iron is not suitable for vessels exposed to acids or acid vapours. 278 METAL-PLATE WOEK. Copper. (111.) This, the only red metal (§ 93), is malleable, tenaciotis, soft, ductile, sonorous, and an excellent conductor of heat. For this reason, and because of its durability, it is largely made use of for cooking utensils. It is found in numerous states of combination with other constituents, as well as ' native ' (uncombined). Its most important ore is copper pyrites. Copper melts at a dull white heat and becomes then covered with a black crust (oxide). It bums when at a bright white heat with a greenish flame. No attempt at explanation of its manufacture will here be made, as any description not lengthy would be simply a bewilder- ment. For the production of sheet copper it is first cast in the forms of slabs, which are rolled, and then annealed and re-rolled, this annealing and re-rolling being repeated until the copper sheet is brought down to the desired thickness. In working ordinary sheet copper, it is hammered to stiffen it, and ' close the grain,' Hard-rolled copper is, however, nowadays produced that does not l equire hammering. (112.) In the course of the manufacture of copper it under- goes a process termed '■poling ' to get rid of impurities. We mention this because we shall find (§ 129) a similar process gone through in preparing solders. The poling of copper consists in plunging the end of a pole of green wood, prefer- ably birch, beneath the surface of the molten metal, and stirring the mass with it. Violent ebullition takes place, large quantities of gases are liberated, and the copper is thoroughly agitated. It is doubtful if this poling process is fully understood, for, though it is quite obvious that there may be insufficient poling (' under poling''), it is not easy to explain ' overpoling' But overpoling, as a fact, is fully recognised in the manufacture of copper, and the metal is brittle both if the poling is too long continued or not long enough. If duly poled, the cast slab when set displays a comparatively level surface ; if underpoled a longitudinal furrow forms on the surface of the slab as it cools ; if over- METAL-PLATE WOKK. 279 poled, instead of a furrow, the surface exhibits a longitudinal ridge. Copper, duly poled, is known as ' best selected,' and as ' tough cake ' copper. Zmc. (113.) Of this metal, known also very commonly as ' spelter,' calamine is a very abundant ore ; another abundant ore is Uende. The metal is extracted from its ores by a process of distillation, the metal volatilising (§ 94) at a bright red heat, and the vapour, passing into tubes, con- denses, and is collected from the tubes in powder and in solid condition. If required pure, further process is neces- sary. This metal does not appear to have been known till the sixteenth century. Henkel, in 1741, was the first, at least in Europe, who succeeded in obtaining zinc from cala- mine. Zinc is hardened by rolling, and requires to be annealed at a low temperature to restore its malleability. Until the discovery of the malleability of zinc when a little hotter than boiling water, it was only used to alloy copper with, and sheet zinc was unknown. Zinc expands ^^^th by heating from the freezing to the boiling point of water. The zinc of commerce dissolves readily in hydrochloric and in sulphuric acid ; pure zinc only slowly. If zinc is exposed to the air, a film of dull grey oxide forms on the surface ; it suffers afterwards little further change. Zinc alloys with copper and tin, but not with lead ; it also alloys with iron, for which it is largely used as a coating ; iron so coated being known as ' galvanised' iron (§ 108). Lead. (114.) Another metal that is prepared in sheet is lead. This metal was known in the earliest ages of the world ; it is soft, flexible, and has but little tenacity. One of its principal ores is galena. Being a soft metal, it is worked (' hossed ') by the plumber into various shapes by means of special tools, which often saves the making of joints. As it is compara- 280 METAL-PLATE WOEK. tively indestruotiWe ixnder ordinary conditions, it is largely •used for roofing purposes and for water cisterns. It is also used for the lining of cisterns for strong acids, in which case the joints are not soldered in the ordinary way with plumber's solder, but made by a process termed ' autogenous soldering' or 'lead burning.' Lead prepared in sheet by casting is known as cast lead, but when prepared by the more modern method of casting a small slab of the metal and then rolling it to any desired thickness is called milled lead. Alloys, (115.) An alloy is a compound of two or more metals. Alloys retain the metallic appearance, and whilst closely approximating in properties to the metals compounded, often possess in addition valuable properties which do not exist in either of the constituent metals forming the alloy. An alloy of copper and zinc has a metallic appearance and working properties somewhat similar to those of the individual metals it is made up of, and so with an alloy of gold, or silver, and a small percentage of copper. But the latter alloys have the further property of hardness, making them suitable for coinage, for which gold, or silver, unalloyed, is too soft. Like to this addition of copper to gold or silver is the addition of antimony to lead and to tin, by which alloys are obtained harder though more brittle than either lead or tin by itself. The alloy of lead and antimony is used for printer's type, for which lead alone is too soft. (116.) Alloys are often more fusible than the individual metals of which they are composed. Thus while lead melts at 326° C, tin at 233° C, bismuth at 268° C, and cadmium at 321° C, an alloy of 8 parts bismuth, 4 tin, and 4 lead, forms what is technically known as a ' fusible ' alloy, meaning an alloy very readily fusible, the particular alloy stated melting indeed at 95° C, that is, below the boiling-point of water. The addition of a little cadmium to the above forms a still more ' fusible ' alloy, called Wood's alloy, which melts at about 65° C. METAL-PLATE WORK. 281 Copper and tin. (117.) We give here the names of some of the more im- portant alloys, with those of the metals of which they are made up. Bronze Bell-metal Gun-metal Speculum-metal Brass Dutch-metal ^ Muntz' metal \ ^oppor and zmc. Hard solders (see p. 287) . German silver Copper, zinc, and nickel. Britannia metal ... . . / f ^timony, copper, bismuth. zinc. Lead and tin. Pewter Soft solders (see p. 287) Type-metal ' Lead and antimony. Alloys op Copper and Tin. (118.) Alloys of copper and tin are known as bronze, gun- metal, speculum-metal, &c. Some of these alloys possess the property of becoming soft and malleable when cooled suddenly while red-hot, by dipping into cold water, but of being hard and brittle when cooled slowly. Name of Alloy. Composition per cent. Copper. Tin. 78-0 22-0 90-0 10-0 66-6 33-0 95-0 4'0 and 1 zinc. (119.) Speculum-metal is a very hard, brittle, steel-grey alloy, capable of receiving a very smooth and highly- polished surface. 282 METAL-PLATE WOEK. Alloys of Copper and Zinc. (120.) Brass. — Brass is the general name given to alloys of copper and zinc (ordinary brass consists of two of copper to one of zinc) ; by some writers also to alloys of copper and tin, now better known as bronze. Brass was known to the ancients, who prepared it from copper and calamine (§ 113), as they were unacquainted with the metal contained in calamine. We are said to have learned the fusing of copper with calamine from the Germans, and until a comparatively recent date brass, called calamine brass, was thus prepared in this country. Between 1780 and 1800, various patents were taken out for improving the manufacture of brass, by fusing copper and zinc direct instead of employing calamine. The calamine method, however, did not at once die out, as it was thought by some that calamine brass was better than that made direct. In the manufacture of brass, the copper is first melted, because of its high melting-point, and the zinc, warmed, is then let down by tongs into the crucible containing the molten copper, plunged under its surface, and held there till melted. The mass is then stirred with a hot brass or iron rod, so as to mix the metals, great care being taken not to introduce any cold or damp matter. A little sulphate of sodium ' salt-cake,' or carbonate of sodium ' soda-ash,' thrown into the crucible at the moment of pouring, assists in raising any impurities to the surface, which can then be skimmed off as the mass is poured. With proper management, the loss of zinc is not so great as might be expected, consi- dering the comparatively low temperature at which it volatilises, and the relatively high temperature necessary to melt the copper. (121.) Brass is harder than copper, and therefore stands wear better ; it is very malleable and ductile, may be rolled into thin sheets, shaped into vessels by ' spinning' (see § 124), stamping, or by the hammer, and may be drawn into fine wire. It is well adapted for casting, as it melts easily at a METAL-PLATE WOEK. 283 lower temperature than copper and is capable of receiving very delicate impressions from the mould. It is said to resist atmospheric influences better than copper, but when its surface is unprotected by lacquer, it rapidly tarnishes and becomes black. It has a pleasing colour, takes a high polish, and is cheaper than copper. (122.) The malleability of brass varies with its composition, and the heat at which it is worked. The malleability is also affected in a very decided degree by the presence of various foreign metals in its composition, even though these are present in but minute quantity. Brass intended for door- plate engraving is improved by the presence of a little tin ; or by the presence of a little lead if to be used in the lathe or for casting. Brass for wire-drawing, however, must not contain lead ; nor must brass intended for rolling contain antimony, which renders it brittle. Some kinds of brass are only malleable while cold, others only while hot, others are not malleable at all. A good example of the remarkable malleability of certain kinds is furnished by Dutch Metal, which contains a large proportion of copper, and which can be hammered into leaves of less than -g^y^o^th of an inch in thickness. Though extremely tenacious, brass loses its tenacity in course of time by molecular change, especially if subject to vibration or continued tension. It is therefore unfitted for chains or for the suspension of weights. Chandelier chains have been known to lose their tenacity, become brittle, and break; and fine brass wire, which is of course brass in a state of tension, will, in time, become quite brittle, merely hanging in a coil. (123.) Muntz' Metal is a variety of brass consisting of above three parts of copper to two of zinc, with abou.t one per cent, of lead. The alloy is yellow, and admits of being rolled at a red-heat. It is extensively applied for the sheathing of ships, as it is said to keep a cleaner surface than copper sheathing. Hard Solders. — These are treated further on. 284 METAL-PLATE WORK. The number of alloys of copper and zinc is considerable, and there is great confusion in respect of their names and composition. The table on opposite page, showing the pro- portions of some alloys of copper and zinc, is an extract from a table by Dr. Percy. (124.) Britannia Metal. — We follow on with this alloy although not an alloy of copper and zinc alone. Britannia metal is highly malleable and one of the best of the sub- stitutes for silver. It is composed of tin, antimony, copper, bismuth, and zinc, in various proportions, according to the purpose for which it is required. If the alloy is to be ' spun,' that is, worked into shape by specially formed tools whilst revolving in a lathe, a greater proportion of tin is used than when the alloy is only to be rolled. If it is to be cast, the proportion of tin is much less. Alloys of Tin and Lead. (125.) Alloys of tin and lead furnish us with our soft solders, and are therefore of great practical value. Under the heading ' solders ' further particulars of these alloys will be found. The melting points of the soft solders in the Table of Solders (p. 287) can be compared with the melting points (p. 274) of the individual metals of which the solders are composed, as illustrating what has already been stated about the fusing points of alloys often being below those of the metals forming them. (126.) Pewter. — The composition of pewter varies consider- ably. Common pewter consists of tin and lead alone ; the best contains also small percentages of antimony, copper, and bismuth ; varying indeed but little from Britannia metal. Solders. (127.) A solder is a metallic composition, by the fusion of which metals are united. The requirements of a good solder are twofold. (1) Its melting point must be below that of 286 METAL-PLATE WOEK. whatever metal is to be joined; (2) it must run easily when melted. It is comparatively easy (§ 116) to fulfil the first condition of a good solder. To fulfil the second requirement, one of the constituents of the solder must be either the same metal as that to be soldered, or a metal which will readily alloy with it, or which will readily coat its surface. (128.) Solders are 'soft' or 'hard,' according to the temperature at which they melt. Hard solders fuse at a red heat. Soft solders are those which can be applied with a ' soldering iron,' that is to say, a ' copper bit,' or ' plumber's iron,' or with a mouth blowpipe ; these solders melt below say 300° C. Hard solders have a much higher fusing point, and require either a forge or a blast blowpipe to apply them. Soldering with hard solders is termed ' brazing.' (129.) An important particular in the preparation of solders is that they should be well stirred before pouring, preferably with a piece of green wood (§ 112), and the surface of the molten metal exposed as little as possible to the air, so that ' dross ' (oxide) shall not form on the surface. A few knobs of charcoal on the molten metal will to a very great extent prevent the formation of dross. (130.) Examining the soft solders, we see that plumber's solder melts at 227° C, that is to say, at a lower melting point than the metal (lead pipe), for soldering which it is used. Further, it is largely composed of lead. It thus fulfils both the requirements of a good solder. Tinman's solder melts at 160° 0. It is used for soldering tin-plate, which, remember, is iron coated with tin. Tin melts at 233° C, a higher temperature than that of its solder, and tin is a constituent of the solder. Again the conditions of a good solder are fulfilled. Tinman's solder is also used for soft-soldering copper, because an alloy of lead and tin will readily coat copper, as also readily alloy with it. 288 METAL-PLATE WORK. Soldering Fluxes. (131.) Substances that 'flux' or aid the flow of metals when melting or melted are termed 'fluxes.' The general subject of fluxes is outside our province ; we are, however, specially interested in what we have designated ' soldering fluxes,' namely, those fluxes that facilitate the flow of the solders and of the metals of which they are composed. (132.) Essentially this 'fluxing' consists in the prevention of the formation of oxide (§ 94) to which metals are very prone when highly heated or molten. The black scale (§ 111) that forms on the surface of copper, for instance on copper ' bits,' when highly heated, is an oxide ; also the scale that falls ofi" red-hot iron when hammered (§ 105) ; and also the ' dross ' that forms on the surface of molten lead or molten solder (§ 129). (133.) The employment of charcoal (carbon) for the purpose of preventing the formation of dross we have already alluded to in speaking of the preparation of solders. Sometimes a layer of it is spread over the surface of the molten metal to keep it from contact with the air ; some- times a layer of grease. In their character of aiding the flow of metals, fluxes are further applied to the surface of the metals to be soldered, which they clean, as well as aiding the flow of the molten solder when that is applied. (134.) ' Spirits of salts ' (hydrochloric or muriatic acid) when ' killed ' is a most useful flux for soft solders. The ' killing ' is done by dissolving zinc in the acid till gas is no longer given off. As the gas is most offensive, the dissolu- tion of the zinc should be effected in the open air. This flux is not one to be used where rust would be serious ; though there is very little danger of this, if, after solder- ing, the joint is wiped with a clean damp rag, and further cleaned with whiting. (135.) Besin, or resin and oil is a good flux for almost any METAL-PLATE WOEK. 289 kind of soft soldering. The surface to be soldered must, how- ever, be well cleaned before applying the flux. (136.) ^Killed spirits' (chemically, chloride of zinc) is specially useful for tin-plate soldering, because it assists in cleaning the edges to be joined ; whereas if resin, or resin and oil, is used, the edge must, as stated, be cleaned pre- viously. (137.) Spirits of salts not killed is used for soldering zinc because it cleans the surface of the zinc ; it acts indeed as chloride of zinc, for this is what it becomes on the applica- tion to the zinc, in fact the cleaning is the result of this action. The killed spirits, however, answers equally well as the strong acid if the zinc is bright and clean, so far as the experience of the writer has gone. The ' raw ' (unkilled) spirits of salts is improved, as a flux for soldering zinc, by adding a small piece of soda to it. (138.) Powdered resin, or resin and oil, as a flux, possesses the great advantage over chloride of zinc, that there is no risk of rust afterwards. For this reason resin, or resin and oil, is much used in the manufacture of gas-meters. It is also used, or should be, for the bottoms and seams of oil bottles. The resin and oil flux can easily be wiped off joints imme- diately after soldering ; it is for this reason better than dry resin which has to be scraped off. Even this trouble, how- ever, can be got over if the hot copper bit is dipped in oil before application to the joint to be soldered. (139.) In ' tinning ' a copper bit, that is, coating its point with solder before using it in soldering (a piece of manipula- tion of much importance as regards the easy working of the bit), the best thing to use is a lump of sal-ammoniac. In a small hollow made in the sal-ammoniac, the point of the bit, after having been filed smooth and bright, should be well rubbed, while hot, along with some solder ; the point of the bit will then become coated with solder ('tinned'). For * tinning ' copper utensils, that is, coating them with tin, sal-ammoniac both in powder and lump is largely used. Sal-ammoniac water is also used for cleaning copper bits ; u 290 METAL-PLATE WORK. the hot bits being dipped into it prior to being used for soldering. Killed spirits, however, acts better. Sal-ammoniac and resin, mixed, is used as a flux for soldering ' sights ' on gun-barrels. (140.) As a flux for lead soldering, plumbers use tallow (' touch '). For pewter, Gallipoli oil is the ordinary flux. (141.) For hard soldering, the flux is horax. This flux is also made use of in steel welding. METAL-PLATE WORK. 291 CHAPTER II. Seams or Joints. We notice here and illustrate the more important seams or joints used in metal -plate work. The drawings are intended to aid in the intelligent comprehension of the formation of joints, and not as exact representations of them. Lap Seam.— In No. 1 is shown how metal plates are arranged for a lap seam which is to be soldered. Circular Lap Seam. — No. 11 shows how the edge of the bottom of a cylindrical article is bent up previous to soldering. It is evident that this seam is essentially No. 1 seam adapted to the fitting a bottom to a cylinder. Such bottom is called a ' snuffed on ' or ' slipped on ' bottom. Countersunk Lap Seam.—This is represented in No. 2. It will be seen that the edge of one of the plates is bent down, so that the edge of the plate to be joined to it may lie in the part bent down, and that the two plates when joined may present an unbroken surface. Bivetted Lap Seam.—This is shown in No. 8. The amount of lap should not be less than three times the diameter of the rivet. Folded Seam. — No. 3 shows how the edges of plates are prepared for folded seam. Circular Folded Seams.— Ylfith. a circular article the folded seam is sometimes in the form of No. 12, which shows a ' paned down ' bottom to a cylinder. This seam is essen- tially No. 3. Another form of circular folded seam is shown in No. 13. It is really No. 12 seam turned up, so as to lie close against the cylinder (see reference letter A in Nos. 12 and 13). u 2 292 METAL-PLATE WORK. A bottom thus fitted is called a ' knocked up ' bottom. Here again comparison with. No. 3 should be made. Double Folded Seam. — This is shown in No. 6, and needs no explanation. It is used with thick plates, where these when joined have to present to the eye an unbroken surface ; as in the hot-plates of large steam-closets. Grooved Seam. — This is represented in No. 4. It will be seen that the seam is the same as No. 3, but one plate is countersunk. In fact No. 3 shows the seam as prepared for METAL-PLATE WOEK. 293 countersinking (' grooving ') with a * groover.' Seam No. 6 is used where plates are too thick for grooving. Countersunh Grooved Seam. — This seam (No. 5) is used when an unhroken surface is required on the outside of an article, for example, in toilet-cans, railway-carriage warmers. It is prepared as No. 3 and then countersunk the reverse way to No. 4. Box Grooved Seam. — This seam, shown in No. 14, is used for joining plates in ' square work,' as for example where the ends and sides of a deed-box are joined together. It is essentially No. 3 seam. Zinc -roofing Joint. — The arrangement for this joint is seen in No. 7. This joint admits of the expansion and contraction of the zinc sheets. The edges of two sheets, the wood ' roll,' and the ' roll cap ' are shown. The zinc ' clip,' by which usually the sheets of zinc are held down, is not represented. Brazing Joints. — A brazing joint for thin metal is shown in Fig. 9. The edge of plate A is cut to form laps as represented, and these laps are arranged alternately over and under the edge of plate B. For thick metal the brazing joint is shown in Fig. 10. It is essentially the same thing as the ' dovetail ' joint of the carpenter. 294 METAL-PLATE WOKK. CHAPTER III. Useful Eules in Mensueation. To find the circumference of a circle, the diameter being given. Multiply the diameter by 3i In other words, multiply the diameter by 22 and divide by 7. Or, should closer accuracy be required, multiply the diameter by 3-1416. Example Z— The diameter of a circle is 8 inches ; to find the circumference. 8 X 22 = 176, which divided by 7 gives 25| inches, the circumference required. Or, 8x3- 1416 = 25-13 inches. To find the area of a circle, the diameter being given. Multiply one-quarter of the diameter or, which is the same thing, half the radius, by the circumference. Example 11. — The diameter of a circle is 8 inches ; to find its area. One-quarter of the diameter is 2 inches ; the circumference is 25j- inches. 2 X 25f = 50f square inches, the area required. To find the area of an ellipse, the axes being given. Multiply the axes together, and multiply the result by •7854. Example HI. — The major axis of an ellipse is 6 inches and the minor 4 inches ; to find its area. 6 X 4 = 24, which multiplied by -7854 gives 18*85 square inches nearly. METAL-PLATE WORK. 295 To find the area of a rectangle. Multiply the length hj the breadth. Example IV. — The length of a rectangle is 16 inches and the breadth 9 inches ; to find its area. 16 X 9 = 144 square inches, that is, one square foot, the area required. To find the volume of a circular, elliptical, rectangular, or other tanle, or vessel, of which the sides are perpendicular to the base. Multiply the area of the base by the height. If the answer is required in cubic inches, all the dimen- sions must be multiplied in inches. If in cubic feet, the dimensions must be in feet. (See Examples.) Example Va. — The height of a circular tank is 6 feet, and the diameter of the base 8 feet ; to find its volume. By Example II. the area of the base is 50f square feet, which multiplied by 6 feet gives 301f cubic feet, the volume required. Example Vh. — The height of an elliptical tank is 1 foot 6 inches, the base is 6 inches by 4 inches; to find its volume. By Example III. the area of the base is 18*85 square inches, which multiplied by 18 inches (that is to say, by the height in inches, as the answer is to be in cubic inches) gives 339 • 3 cubic inches, the volume required. Example Vc. — The height of a rectangular vessel is 2 feet 3 inches, the length 1 foot 4 inches, and the breadth 9 inches ; to find its volume. We will suppose the answer is required in cubic feet. This being so, it is in feet that the dimensions must be multiplied. Stated in feet, the height is 2| feet, the length 1^ feet, and the breadth | foot. By Example IV. the area of the base is 1^ feet multiplied by i foot, that is, is | x f = If = 1 square foot, which multiplied by 2^ feet gives 2^ cubic feet, the volume required. 296 METAL-PLATE WOEK. To find the volume of a right cone. Multiply tlie area of the base by the height and divide by 3. _ Example FZ— The height of a cone is 6 inches, and the diameter of the base 3 J inches ; to find the volume. 3— X 22 The circumference of the base is — = 11 inches. The area of the base is one-half of If inches (the radius) Xll=|Xll = '-g^ = 9f square inches. And the area 9f x 6 inches (the height) = 57|, which divided by 3 gives 19^ cubic inches, the volume required. To find the volume of a frustum of a right cone. From the volume of the complete cone of which the frustum is a part subtract the volume of the cone cut off. For example, the volume of the frustum C A B D (Fig. 8a, p. 33) is equal to the volume of the complete cone 0 A B lees the volume of 0 C D the cone cut off. The height of the complete cone and that of the cone cut off from it to form the frustum can be found by Problem V., p. 36. To find the volume of a sphere^ the diameter being given. Multiply the diameter of the sphere by the area of a circle of same diameter. Example Vll. — The diameter of a sphere is 8 inches ; to find its volume. The area of a circle of same diameter is 50|^ (see Example II.) ; which area multiplied by 8 gives 402|- cubic inches, the volume required. Given the volume of a vessel, any vessel^ to find the number of gallons, quarts, or pints that it will hold. If the volume is in cubic feet, as in Example Va, then, to bring it to gallons, multiply by 6^, there being in a cubic foot of water gallons about. If the volume is in cubic METAL-PLATE WOEK. 297 inclies, divide by 277. The number of cubic inches in a gallon of' water is 277^ nearly; but in ordinary calculationp, the quarter may be omitted. If the volume is required in quarts, multiply it, if in cubic feet, by 25 ; if in cubic inches, divide it by 69. The number of cubic inches in a quart of water is about 69^ ; in our examples here we have disregarded the fraction. If the volume is required in pints, multiply it, if in cubic feet, by 50 ; if in cubic inches, divide it by 35. The number of cubic inches in a pint of water is rather more than 34^ ; in our examples we have taken it as 35. Example VIIL — To find the number of gallons, quarts, or pints, contained in the tank of Example Ya, Gallons.— SOl^ X 6| = 1885f gallons. Quarts.— dOl^ x 25 = 7542f quarts. Pints.— SOl^ X 50 = 15085|. pints. Example IX. — To find the number of gallons, quarts, or pints, contained in the tank of Example V6. Gallons.— 339 • 3 277 = 1-22 gallons about. Quarts.— 339 • 3 69 = 4 • 92 quarts about. Pints.— B39 • 8 -r 35 = 9 • 7 pints about. Given the number of gallons, quarts, or pints, that a tank or other vessel contains, any vessel, of which the sides are perpendicular to the base, also the dimensions of the base, to find its height. Divide the number of gallons by 6^ ; this will give the volume of the required tank in cubic feet. If the quantity is given in quarts, then to ascertain the required volume, multiply by 69. If the quantity is given in pints, multiply by 35. To find the required height for the tank, divide the volume found as just shown, by the area of the base. 298 METAL-PLATE WORK. Comparative Weights and Gauges op Sheet Iron, Copper, and Zinc, and op Tin-Plate. Tin-Plates Approximate Weight per Square Foot. Zinc. Approximate Weight. p bo Kquivalei Birming Wire-Sa of Kquiva.l6iit Strength. 13 C3 Iron. Copper. Per Square Foot. Per Sheet 1 ft. by 3 ft. 8 ft. by 3 ft. — ■ lb. oz. lb. oz. lb. oz. lb. oz. lb. oz. 4 33 • 0 4f 6 4 5 31 0 5f 7 9 6 30 1 c 0 8 0 9i 0 6f 9 0 10 5 7 29 (1X1 0 9 0 101 0 71 10 6 11 10 8 Zo < and > 0 lOi 0 12 0 9 jj 12 13 5 1 T\ 1 I I> 0 J 9 27 X X 0 111 0 14 0 lOi 13 5 15 8 — 26 S D 0 0 13 0 13§ 10 OK / SDX 1 \ DX ( (1 XX XXX] 0 15 1 0 0 15 17 3 11 24 DXX \ 1 0 1 0 131 17 4 20 0 I bDXXX J 12 23 DXXX 1 2 1 5 0 15 19 12 22 11 13 22 DXXXX 1 4 1 8 1 1 22 4 25 7 14 21 DXXXXX 1 6| 1 10 1 2| 24 12 28 2 15 20 DXXXXXX 1 9 1 12 1 51 28 11 32 10 16 19 1 12 2 0 1 8| 32 10 37 2 17 18 1 IH 2 4 1 111 36 8 41 8 18 1 14| 40 7 19 17 2 3 2 8 2 31 45 0 20 16 2 8 2 14| 2 4| 49 8 21 15 2 13 3 4 2 8| 55 0 22 2 12| 59 0 23 3 1 64 8 METAL PLATE WOEK. Strength, Sizes, and Weights of Tin-Plates. 299 Sizes of Sheets in Inclies. X 10 X 20 X 10 X 20 X 10 .9 a « las ho:* 30 28 26 easy 25 easy 25 full 24 24 full Weights per Box and Number of Sheets. cwt. qrs. lb. 225 sheets weigh 1 u u 112 )) 1 0 0 112 »♦ >» 2 0 0 225 )> 1 1 A 1 u 112 >» » 1 1 0 112 »> 2 2 0 100 0 3 14 50 » 8 3 14 225 1 1 21 112 1 1 Zl 112 " >» L o 14 200 1 1 27 100 >» >» 1 1 27 225 »> >f 1 2 14 112 » >> 1 2 14 112 » Q 1 0 100 1 0 14 50 »> 1 0 14 200 » 1 2 20 100 » »» I 2 20 225 » >» 1 3 7 112 »> >» 1 3 7 112 )» »» 3 2 14 200 >» 1 3 13 100 >J 1 3 13 112 55 2 0 0 112 » J> 4 0 0 100 »J 1 1 7 50 »> )» 1 1 7 200 »» 5> 2 0 6 100 » J» 2 0 6 200 »» l> 2 0 27 100 5J 2 0 27 300 metaltPlate woek. Strength, Sizes, and Weights op Tin-Plates — continued. strength. Sizes of Sheets in Inches. .3 a ^ S Ml Ha3^ Weights per Box and Number of Sheets. IXXXXXX DXXX DXXXX DXXXXX DXXXXXX 14 28 28 17 34 25 17 34 25 34 25 34 25 X 20 X 10 X 20 X 12i X 12^ X 17 X 12J X 12^ X 17 X 12i\ X 17 ; X 12J\ X 17 / 23 easy 23 22 21 20 cwt. qrs. lb. 112 sheets weigh 2 0 21 112 » 4 1 14 100 1 2 0 50 » i> 1 2 0 100 >» )» 1 2 21 50 >i )> 1 2 21 50 » 1 3 14 50 » 2 0 7 ( 301 ) INDEX. The references are to pages, except those in brackets, which are to paragraphs. A. Allots, defined, 280 ; fusible, 280 ; names of, 281 ; of iron and carbon (steel), 276 ; copper and tin, 281 ; copper and zinc, 282 ; tin and lead, 284. Angle, defined, 3 ; to draw, equal to a given angle, 6 ; to bisect, 10. Angles, measurement of, 20. Annealing, 271. Apex, of cone, 24, 105 (43) ; of pyramid, 66 (29). Aquarium stand pattern, 267. Arc, defined, 5 ; to complete circle from, 9 ; to find if given curve is, 10. Arcs, proportionate and similar, 124. Area, of circle, 294; ellipse, 294 ; rectangle, 295. Articles, equal tapering and other, see Bodies. Athenian hip-bath, plan, 137. Asis, of cone, 24 (6), 105 (43) ; of ellipse, 16; of pyramid, 66 (29). B. Baking pan pattern, 77. Bath, Athenian hip, plan, 137. , ♦ equal-end,' equal taper, pattern, 84-90 ; four pieces, 84 ; two pieces, 86 ; one piece, 87 ; short-radius method, 89. , , unequal taper, plan, 130 ; pattern, 157-68 ; four pieces, 158 ; two pieces, 161 ; one piece, 162 ; when ends are cylindrical, 16i ; short-radius method, 166. , oblong taper, plan, -140; representation (plate) 237 ; plate explained, 231 ; pattern, 230-6 ; short-radius method, 235. , Oxford hip, see that heading. , oval, plan, 131; representation (plate), 181; plate explained, 169; pattern, 168-83 ; four pieces, 171 ; two pieces, 175 ; one piece, 177 ; short-radius method, 177. 302 INDEX. Bath, sitz, plan, 137. Bevel (angle), 6. Bisect a line, to, 8 ; an angle, 10. Bodies, of equal taper, see equal-tapering bodies ; of unequal taper, see unequal-tapering bodies ; of rectangular base and circular top, 264. Brass, 282. Brazing, 286 ; joints for, 293. Britannia metal, 284. Bronze, 281-2. c. CiUfiSTEB-TOP, oval, plan, 132; representation (plate), 203; plate ex- plained, 193 ; pattern, 192-205 ; four pieces, 193 ; two pieces, 197 ; one piece, 198 ; short-radius method, 200, Carbon, 276 ; and iron, 276. Cast-iron, 275-6. Centre, of circle, 5 ; of elhpse, 16. Chord, defined, 5. Chords, scale of, to draw, 21 ; how to use, 22. Circle, defined, 5 ; sector of, 28 (9) ; to find centre, 9 ; to describe, that shall pass through three points, 9 ; to complete from arc, 10 ; to find if given curve is arc of, 10 ; to inscribe polygon in, 10 ; to find length of circumference geometrically, 12 ; to find same arithmeticaUy, 294 ; area, 294. Circular pipes, meeting at any angle, pattern, 239. 5 inclined, extreme cases of oblique cone frusta, 112(61); pattern for, 121, 255; pattern for, in Y-piece, 261. vessel, volume, 295. Circumference of circle, defined, 5 ; to find length of, geometrically, 12 ; arithmetically, 294. Classification of patterns, 2. Colfee-pots, 34 ; hexagonal, 70 (35). Colour of metals, 270 (93). Compound bent surfaces, 267. Conductivity of metals, 273. Cone, defined, 105 (43); axis, radius, apex, base, 24 (6), 105 (43); elevations of generating lines, 108 (55), , right, defined, 24 (6), 105 (44) ; basis of patterns for articles of equal taper, 24 (5) ; compared with oblique cone, 106 (from 47) ; representation (plate), 227 ; development of, by paint, 28 ; generating lines, 106 (47) ; corresponding points of generating lines, see corre- sponding points ; to find height, 26 ; to find slant, 27 ; to find dimen- sions of, from frustum, 36 ; volume, 296. > > pattern, 29-31 ; one piece, 29 ; more than one piece, 30. INDEX. 803 Cone, right, frustum, (round equal-tapering body), defined, 33 ; representa- tion, 50 ; representations of round equal-tapering articles, 34 ; relations of, with complete cone, 34 (from 13); development, 34 (15); volume, 296. , , , plan, 50 (23) ; characteristic features, 55 (c, d) ; to draw, see equal-tapering bodies (plans, to draw). , , , pattern, 37-43, ends and height given, 37 ; ends and slant given, 39 ; pattern for parts of, 39 ; short-radius method, 41. , oblique, defined, 106 (45) ; basis of patterns for articles of unequal taper, 105 (42); compared with right cone, 106 (from 47); obliquity, how measured, 107 (52); representations (plates), 181, 203, 213, 227, 237; development of, 108 (56); generating lines, 106 (46-7) ; longest and shortest generating lines, 107 (from 51) ; lines of greatest and least inclination, 107 (52) ; height of elevations of generating lines, 108 (55) ; true lengths of elevations of generating lines, 107 (54). , , plan of axis, 125-6 (a); of generating lines, 126 (d,e); of largest and shortest generating lines, 126 (6) ; of apex, 126 (from c). , , pattern, 108. , , frustum (round unequal-tapering body), defined, 111 (58-9); representation, 125 (plate), 259 ; generating lines, 126 (6) ; genera- ting lines of, when circumscribing oblique pyramid frustum, 150 (78) ; corresponding points of generating lines, see corresponding points ; oblique cylinder an extreme case of, 112 (61). , , , plan, 125 (69), 128 (from 73) ; of axis, 126 (a) ; of lines of greatest and least inclination (longest and shortest generating lines), 126 (6) ; characteristic features, 127 (from 71). , , , pattern, 113-23; ends, height, and inclination of longest generating line given, 113 ; height and plan given, 116 ; short-radius method, apex accessible or inaccessible, 118 ; extreme case of (oblique cylinder), 121. Copper, 270, 271, 278; gauges and weights of, 298; and tin alloys, 281 ; and zinc alloys, 282 ; table of, 285. Corresponding points, of right cone frustum, 35, 51 ; of equal-tapering body, 52 (25) ; of oblique cone frustum, 124 (68), 126 (6). , , in plan, of right cone frustum, 51 (23, 24),54 ; of equal-tapering body, 52 (25), 53 (26) ; of oblique cone frustum, 124 (68), 129 (g) ; of oblique pyramid frustum, 145 (76) ; of oval equal-tapering body, 65 (28) ; distance between, is equal in plans of equal-tapering bodies, 55 (from ft) ; to find distance between, height and slant given, 56 ; height and inclination given, 57. Cubic dimensions, conversion into gallons, 297. Cylinder, defined, 112 (61). , oblique, defined, 112 (61) ; an extreme case of oblique cone frustum, 112 (61) ; pattern for, 121, 255 ; pattern for, in Y-piece, 261. , equal meeting at any angle, pattern, 239. 304 INDEX. D. Definitions, of straight line, angle, perpendicular, 3; parallel lines, triangle, hypotenuse, polygon, pentagon, hexagon, heptagon, octagon, quadrilateral, square, oblong, rectangle, 4 ; circle, circumference, arc, quadrant, semicircle, radius, chord, diameter, 5 ; ellipse, 17 ; ellipse focus, axis, diameter, centre, 16 ; cone, 105 (43) ; cone axis, radius, apex, base, 24 (6), 105 (43) ; right cone, 24 (6), 105 (44) ; right cone frustum, 33 ; oblique cone, 106 (45) ; oblique cone frustum, 111 (58—9) ; pyramid and axis, 66 ; pyramid frustum, 143 (74) ; proportional arcs, similar arcs, 124 ; corresponding points, see that heading. Degi-ee (angle) explained, 21. Diameter of circle, 5. of ellipse, 16. Dripping-pan pattern, with well, 266. Ductility of metals, 272 ; table of, 273. Dutch metal, 283. E. Edoutg, allowance for, 83. Egg-shaped oval, to draw, 14. Elbow pattern, any angle, 239. Elevation, explained, 48. Ellipse, defined, 17 ; focus, axes, diameter, centre, 16 ; to describe mechani- cally, 15, 18; geometrically, 17; area, 294. Elliptical vessel, volume, 295. Equal end bath, see Bath. Equal-tapering bodies, 24. Bound 28-45 ; essentially right cone frusta, 84 (13) ; to find slant or height of cone of which body is portion, 36 ; to find slant of body, ends and height given, 43 ; to find height, ends and slant given, 44 ; slant and inclination given, 44 ; see also Cone (right, frustum). Of flat surfaces, 66-83 ; essentially right pyramid frusta, see Pyramid (right, frustum). Of flat and curved surfaces, 84-96 ; curved surfaces, portions of right cone frusta, 55 (d), 84 ; see also Cone (right, frustum). , plans, Characteristic features of, of round bodies, 51 ; characteristic features, body oblong with round corners, 52 ; features of plans summarisi d, 55 ; how to find from plan if article is of equal taper, 55 (&). — , , to draw, Bound bodies (frusta of right cones). Either end, height, and slant- given, 57; either end and distance between corresponding points (' out of flue ') given, 59. INDEX. 305 Equal-tapering bodies, plans, to draw, Oval bodies, 64. Of flat and curved surfaces. Oblong bodies, 59, 62. See also Corresponding points. , patterns. Eound bodies (frusta of right cones), 28-45 ; see also Cone (rigbt, frustum, pattern). Oval bodies. Patterns, 96-104 ; in four pieces, 96 ; in two pieces, 99 ; in one piece, 100 ; short radius method, 102. . Bodies having flat surfaces. Ends of body and height given, 71 ; short radius method, 73 ; body oblong or square (pan) pattern in one piece with bottom, 77, 80 ; same with bottom, sides, and ends in separate pieces, 82 ; baking-pan pattern, bottom, width of top, and slant given, 77 ; bottom, length of top, and slant given, 80; top, slant, and height given, 81; top, slant, and inclination of slant given, 81. . Bodies of flat and curved surfaces combined. Body having flat sides and semi-circular ends, see Bath (equal end and equal-taper) ; flat sides and ends, and round corners (oblong or square with round corners), see Oblong pan. F. Flub, out of, 55, 61. Fluxes, 288. Focus of ellipse, 16. . n t- Frustum of cone, see Cone (right, frustum ; and oblique, frustum) ; ot pyramid, see Pyramid (right, frustum ; and oblique, frustum). Galvanised Ieon, 277. Gauges, tables, 298. Generation of cone, 24. Gravy strainers, frusta of right cones, 34 (13). Grooved seam, 32, 292. H. Hardness op Metals, 273 ; table, 274. Heptagon, 4. Hexagon, 4. „ . , , ^ • i / ■ n- Hexagonal pyramid, 66 (29); pattern of right, see Pyramid (right frustum, pattern) ; oblique, see Pyramid (oblique, frustum, pattern ; coffee-pots, 70 (35). 306 INDEX. Hip-bath, see Athenian hip-bath, Oxford hip-bath. Hoods, their relation to truncated pyramids, 70, 143 ; pattern, 154. Hoppers, 143 ; see also Hoods. Hypotenuse, 4. I. IxcLiNATiON OF Slant, defined, 24 (4), 55 (6), 61 (Case II.) ; articles of equal, see Equal-tapering bodies ; of unequal, see Unequal-tapering bodies. Introductory Problems, 3. Iron, 270, 275; fusion, 270; fracture, 271, 272; hammer hardening, 272; impurities in, 275 ; oxide, 275 ; alloys with carbon, 276 ; galvanised, 277 ; tinned, 277. , sheet, gauges and weights of, 298. J. Joints, 291. K. Killing Spirits of Salts, 288. Lap, allowance for, 32. , seam, 291. Lead, 270, 279. Line, to divide into equal parts, 7 ; into two parts, 8. Lines, straight, defined, 3; points joined by, 7 ; parallel defined, 4; true lengths of, 48. Lustre of metals, 270 M. Malleability of Metals, 271 ; table, 273 Manchester plates, 277. Melting points, 274. Mensuration, rules in, 294. Metals and their properties, 270, Miscellaneous patterns, 239. Mouldings, 267. Muntz's metal, 283. INDEX. 307 O. Oblique Cone, see Cone (oblique) ; cylinder, see Cylinder. Oblong defined, 4; with round corners, to draw, 19; with semicircular ends, to draw, 20 ; oblong equal-tapering body, see EquaUtapering bodies. (or square) pan with round corners, pattern, 90 ; in four pieces, 90 ; in two pieces, 92 ; in one piece, 93 ; short-radius method, 94. taper batb, see Bath (oblong taper). Octagon, 4. Opaqueness of metals, 270. Oval, to draw, 13 ; egg-shaped, to draw, 14. batb, see Bath. canister top, representation (plate 203) ; plate explained, 193 ; plan, 132; pattern, 192; in four pieces, 193; in two pieces, 197; in one piece, 198. equal-tapering bodies, see Equal-tapering bodies (plans, to draw ; and patterns) ; corresponding points (in plan). unequal-tapering bodies, see Unequal-tapering bodies, and Unequal tapering bodies (plans ; patterns). Oxford hip-bath, representation (plate), 227 ; plate explained, 217 ; plan, 134 ; pattern, 216 ; short-radius method, 223. Oxide, oxidising, 270, 275, 288. P. Pails, frusta of right cones, 34 (13). Pan, baking, see Baking-pan ; dripping, see Dripping-pan ; oblong with round corners, see Oblong pan. Parallel lines, 4. Patterns, setting out, is development of surfaces, 2 (3), 34 (14). Pentagon, 4. Pewter, 284. Perpendicular, 3. Pipes, meeting at any angle, 239 ; joining equal pipes not in line, pattern, 255 ; joining unequal pipes, representation (plate), 259 ; plate ex- plained, 255 ; patterns, 255 ; tapering, are frusta of oblique cones, 112 (59); see also Circular pipes. Plan, explained, 46 ; of equal-tapering bodies, see Equal- tapering bodies (plans); of imequal-tapering bodies, see Unequal-tapering bodies, (plans). Plates, 181, 203, 213, 227, 237, 259; descriptions respectively, 169, 193, 206, 217, 231, 255. Points, angular, 3 ; always joined by straight lines, 7 ; corresponding, see Corresponding points. 308 INDEX. Poling, 278. Polygon, defined, 4; to inscribe in circle, 10 ; regular, to describe, 11. Projection, explained, 47. Properties of metals, 271. Proportional arcs, 124. Pyramid, Defined, 66 (29); base, apex, axis, 66 (29); triangular, square, hexa- gonal, 66 (29). , right, Defined, 66 (29) ; can be inscribed in right cone, 66 (from 30) : model] of, 73 (37). ) , pattern, Hexagonal, 68 ; any number of faces, 69. ) , frustum. Defined, 69 (83), 70, (34); representation, 70; can be inscribed im right cone frustum, 70 (86) ; the basis of articles of equal-taper havings- flat surfaces, 70 (85); model, 73 (37); slant of face of, 151 (Case II.) > ) , pattern, Hexagonal, 71 ; any number of faces, 73 ; short radius method, 73. , oblique, Defined, 143 (74) ; can be inscribed in oblique cone, 144 (75). ? , pattern. Plan and height given, 145 ; plan and axis given, 148. ' » frustum (unequal-tapering body), Defined, 143 (74) ; the basis of numerous articles of unequal-taper having;- flat surfaces, 143 ; can be inscribed in oblique cone frustum, 150 (78) ;; slant of face of, 151 ; model, 73 (37). » , , plan What it consists of, 144 (76) ; corresponding points in, 145 (76); how to determine from plan of unequal- tapering body if body is oblique; pyramid frustum, 145 (77). » » • , pattern. Plan and height given (frustum, hexagonal), 148; ends and slant audi inclination of one face given, 151 ; short radius method, 152. Q. Quadrant, defined, 5; to divide, 11. Quadrilaterals, 4. R. Kadius, defined, 5 ; of cone, 24. Eake (angle), 6. Rectangle, defined, 4 ; area, 295. Rectangular base and circular top, pattern for article of, 264. , vessel, volume, 295. **IN'D£X.** 309 Right angle, 31 1*1!: .*.*.; 'I' I I.'I.ir .**. , cone, see Cone* Cri'^t); pym*nn A History of Electric Telegraphy, to the Year 1837. Chiefly compiled from Original Sources, and hitherto Unpublished Docu- ments, by J. J. Fahie, Mem. Soc. of Tel. Engineers, and of the Inter- national Society of Electricians, Paris. Crown 8vo, cloth, ^s. Spons' Information for Colonial Engineers. Edited by J. T. Hurst. Demy 8vo, sewed. No. I, Ceylon, By Abraham Deane, C.E. 2s. 6d. Contents : Introductory Remarks- Natural Productions- Architecture and Engineering - Todo- graphy. Trade, and Natural History-Principal Stations- Weights and Melsures, etc., e?c No. 2. Southern Africa, including the Cape Colony, Natal, and the Dutch Repubhcs. By Henry Hall, F.R.G.S., F.R.C I With Map. 3j. dd. 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" It is certainly an extremely rare thing for a reviewer to be called upon to notice a volume measuring but 2i in. by if in., yet these dimensions faithfully represent the size of the handy iTtde b" ok before us. ^ The volume-which contains 118 printed pages, besides a few blank pages for memoranda-is, in fact, a true pocket-book, adapted for being carried m the waist- ?oft pockeTand containing a far greater amount and variety °f '"f°--'XtTur h^^^^^^^^ would imagine could be compressed into so small a space Ihe little volume has Ijeen compiled lith considerable care and judgment, and we can cordially recommend it to our readers as a useful little pocket companion."— -£??^2«^^rz«^. A P radical Treatise on Natural and Artificial Concrete, its Varieties and Constructive Adaptations. By Henry Reid, Author of the ' Science and Art of the Manufacture of Portland Cement. New Edition, with 59 woodcuts and 'i^ plates, 8vo, cloth, 1 5 J. 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The French- Polisher s Manual. By a French- Polisher; containing Timber Staining, Washing, Matching, Improving, Pamting, Imitations, Directions for Staining, Sizing, Embodying, Smoothing, Spirit Varnishing, French-Polishing, Directions for Re- polishmg. Third edition, royal 32mo, sewed, 6d. Hops, their Cultivation, Commerce, and Uses in various Countries. By P. L. SiMMONDS. Crown 8vo, cloth, 4^. ed. A Practical Treatise on the Manu/actttre and Distri- bution of Coal Gas. By William Richards. Demy 4to, with numerous wood engravings and 29 plates, cloth, 28^. Synopsis of Contents : Introduction— History of Gas Lighting — Chemistry of Gas Manufacture, by Lewis Thompson Lsq M.R.C S.-Coal with Analyses, by J. Paterson, Lewis Thompson, and O. R. Hislop, Esqrs.— Retorts, Iron and Clay— Retort Setting-Hydraulic Main— Con- densers— Exhausters— Washers and Scrubbers — Purifiers — Purification — History of Gas Holder — Tanks Brick and Stone, Composite, Concrete, Cast-iron, Compound Annular Wrought-iron — Specifications — Gas Holders — Station Meter— Governor — Distribution- Mains— Gas Mathematics, or Formula: for the Distribution of Gas, by Lewis Thompson Esq — Services-Consumers' Meters-Regulators-Burners-Fittings-Photometer-Carburization of Gas— Air Gas and Water Gas— Composition of Coal Gas, by Lewis Thompson, Esq.— Analyses of Gas—Influence of Atmospheric Pressure and Temperature on Gas— Residual iToducts— Appendix— Description of Retort Settings, Buildings, etc., etc. Practical Geometry.^ Perspective^ and Engineering Drawing; a Course of Descriptive Geometiy adapted to the Require- ments of the Engineering Draughtsman, including the determination of cast shadows and Isometric Projection, each chapter being followed by numerous examples ; to which are added rules for Shading, Shade-linin<^T etc., together with practical instructions as to the Lining, Colouring' Printmg, and general treatment of Engineering Drawings, with a chapter on drawmg Instruments. By George S. Clarke, Capt. R.E. Second edition, zvith 21 plates. 2 vols., cloth, 10s. 6d. The Elements of Graphic Statics. By Professor Karl Von Ott, translated from the German by G. S. Clarke, Capt. R.E., Instructor in Mechanical Drawing, Royal Indian Engineering College. With 93 illustrations, crown 8vo, cloth, 5^. The Principles of Graphic Statics. By George Sydenham Clarke, Capt. Royal Engineers. With 112 illustrations 4to, cloth, I2J-. ()d. Dynamo-Electric Machinery : A Manual for Students of Electro-technics. By Silvanus P. Thompson, B.A., D.Sc, Professor of Expermiental Physics in University College, Bristol, etc., etc. Second edition, illustrated, 8vo, cloth, I2s. 6d. PUBLISHED BY E. & F. N. SPON. 9 The New Formula for Mean Velocity of Discharge of Rivers and Canals. By W. R. Kutter. Translated from articles in the ' Cultur-Ingenieur,' by Lowis D'A. Jackson, Assoc. Inst. C.E. 8vo, cloth, I2J-. td. 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Third edition, with 148 wood engravings, post 8vo, cloth, is. 6d. Contents : Chap. I. How Work is Measured by a Unit, both with and without reference to a Unit of Time— Chap. 2. The Work of Living Agents, the Influence of Friction, and introduces one of the most beautiful Laws of Motion— Chap. i. The principles expounded in the first and second chapters are applied to the Motion of Bodies— Chap. 4. The Transmission of Work by simple Machines— Chap. 5. Useful Propositions and Rules. The Practical Millwright and Engineers Ready Reckoner; or Tables for finding the diameter and power of cog-wheels, diameter, weight, and power of shafts, diameter and strength of bolts, etc. By Thomas Dixon. Fourth edition, i2mo, cloth, y. Breweries and Mattings : their Arrangement, Con- struction, Machmery, and Plant. By G. Scamell, F.R.I.B.A. Second edition, revised, enlarged, and partly rewritten. By F. CoLYER, M.I.C.E., M.I.M.E. With 20 plates, Svo, cloth, i8j. 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Contents : Machinery for Prospecting, Excavating, Hauling, and Hoisting— Ventilation— Pumpiing— Treatment of Mineral Products, including Gold and Silver, Copper, Tin, and Lead, Iron Coal, Sulphur, China Clay, Brick Earth, etc. Tables for Setting out Curves for Railways, Canals, Roads, etc., varying from a radius of five chains to three miles. By A. Kennedy and R. W. Hackwood. Illustrated, 32mo, cloth, 2s. 6d. The Science and Art of the Manufacture of Portland Cement, with observations on some of its constructive applications. With 66 illustrations. By Henry Reid, C.E., Author of 'A Practical Treatise on Concrete,' etc., etc. Svo, cloth, iBj. TIu Draughtsman s Handbook of Plan and Map Drawing; including instructions for the preparation of Engineering, Architectural, and Mechanical Drawings. With numerous illustrations in the text, and 33 plates (15 printed in colours). By G. G. Andrk, F.G.S., Assoc. Inst. C.E. 4to, cloth, pj. Contents: The Drawing Office and its Furnishings — Geometrical Problems — Lines, Dots, and their Combinations — Colours, Shading, Lettering, Bordering, and North Points — Scales — Plotting — Civil Engineers' and Surveyors' Plans — Map Drawing — Mechanical and Architectural Drawing — Copying and Reducing Trigonometrical Formulae, etc., etc. The B oiler-maker s andiron Ship-builder s Companion, comprising a series of original and carefully calculated tables, of the utmost utility to persons interested in the iron trades. By James Foden, author of ' Mechanical Tables,' etc. Second edition revised, with illustra- tions, crown Svo, cloth, 5^. Rock Blasting: a Practical Treatise on the means employed in Blasting Rocks for Industrial Purposes. By G. G. Andre, F.G.S. , Assoc. Inst. C.E. With 56 illustrations aitd 12 plates, Svo, clotli, los. bd. Painting and Painters Manual: a Book of Facts for Painters and those who Use or Deal in Paint Materials. By C. L,. Condit and J. Scheller. Illustrated, Svo, cloth, \os. 6d. PUBLISHED BY E. & F. N. SPON. n A Treatise on Ropemaking as practised in public and Private Rope-yards, with a Description of the Manufacture, Rules, Tables of Weights, etc., adapted to the Trade, Shipping, Mining, Railways, Builders etc. By R. Chapman, formeriy foreman to Messrs. Huddart and Co.! Limehouse, and late Master Ropemaker to H.M. Dockyard, Deptford. Second edition, i2mo, cloth, 3J-. Laxtons Builders and Contractors Tables ; for the use of Engineers, Architects, Surveyors, Builders, Land Agents, and others. Bricklayer, containing 22 tables, with nearly 30,000 calculations. 4to, cloth, 5^. Laxtons Builders and Contractors Tables. Ex- cavator, Earth, Land, Water, and Gas, containing 53 tables, with neariy 24,000 calculations. 4to, cloth, 55-. Sanitary Engineering: a Guide to the Construction of Works of Sewerage and House Drainage, with Tables for facilitating the calculations of the Engineer. By Baldwin Latham, C.E., M. 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A Treatise on a Practical Method of Designing Slide- Valve Gears by Simple Geo^netrical Construction, based upon the principles enunciated in Euclid's Elements, and comprising the various forms of Plain Slide-Valve and Expansion Gearing ; together with Stephenson s, Gooch's, and Allan's Link-Motions, as applied either to reversing or to variable expansion combinations. By Edward J. Cowling Welch, Memb. Inst. Mechanical Engineers. Crown 8vo, cloth, bs. Cleaning and Scouring : a Manual for Dyers, Laun- dresses, and for Domestic Use. By S. Christopher. i8mo, sewed, (yd. A Handbook of House Sanitation ; for the use of all persons seeking a Healthy Home. A reprint of those portions of Mr. Bailey-Denton's Lectures on Sanitary Engineering, given before the School of Military Engineering, which related to the "Dwelling, enlarged and revised by his Son, E. F. Bailey-Denton, C.E., B.A. With 140 illustrations, 8vo, cloth, %s. 6d. B 4 12 CATALOGUE OF SCIENTIFIC BOOKS A Glossary of Terms used in Coal Mining, By William Stukeley Gresley, Assoc. Mem. Inst. C.E., F.G.S., Member of the North of England Institute of Mining Engineers. Illustrated with numerous woodcuts and diagrams, crown 8vo, cloth, ^s. A Pocket-Book for Boiler Makers and Steam Users, comprising a variety of useful information for Employer and Workman, Government Inspectors, Board of Trade Surveyors, Engineers in cl iarye of Works and Slips, Foremen of Manufactories, and the general Steam- using Public. By Maurice John Sexton. Second edition, royal 32mo, roan, gilt edges, 5j. Electrolysis: a Practical Treatise on Nickeling-, Coppering, Gilding, Silvering, the Refining of Metals, and the treatment of Ores by means of Electricity. By HiPPOLYTE Fontaine, translated from the French by J. A. Berly, C.E., Assoc. S.T.E. With engravings. 8vo, cloth, 9^. A Practical Treatise on the Steam Engine, con- taining Plans and Arrangements of Details for Fixed Steam Engines, with Essays on the Principles involved in Design and Construction. By Arthur Rigg, Engineer, Member of the Society of Engineers arud of the Royal Institution of Great Britain. Demy 4to, copiously illustrated with woodcuts and g6 plates, in one Volume, half-bound morocco, 2I. 2s.; or cheaper edition, cloth, 25J. This work Is not, in any sense, an elementary treatise, or history of the steam engine, hut is intended to describe examples of Fixed Steam Engines without entering into the wide domain of locomotive or marine practice. To this end illustrations will be given of the most recent arrangements of Horizontal, Vertical, Beam, Pumping, Winding, Portable, Semi- portable, Corliss, Allen, Compound, and other similar Engines, by the most eminent Firms in Great Britain and America. The laws relating to the action and precautions to be observed in the construction of the various details, such as Cylinders, Pistons, Piston-rods, Connecting- rods, Cross-heads, Motion-blocks, Eccentrics, Simple, Expansion, Balanced, and Equilibrium Slide-valves, and Valve-gearing will be minutely dealt with. In this connection will be found articles upon the Velocity of Reciprocating Parts and the Mode of Applying the Indicator Heat and Expansion of Steam Governors, and the like. It is the writer's desire to draw illustrations from every possible source, and give only those rules that present practice deems correct. Barlow's Tables of Squares, Cubes, Square Roots, Cube Roots, Reciprocals of all Integer Numbers up to 10,000. Post 8vo, cloth, 6^. Camus (M.) Treatise on the Teeth of Wheels, demon- strating the best forms which can be given to them for the purposes of Machinery, such as Mill-work and Clock-work, and the art of finding their numbers. Translated from the French, with details of the present practice of Millwrights, Engine Makers, and other Machinists, by Isaac Hawkins. Third edition, 7uitk 18 plates, 8vo, cloth. 55, PUBLISHED BY E. & R N. SPON. 13 A Practical Treatise on the Science of Land and Engineering Surveying, Levelling, Estimating Quantities, etc., with a general description of the several Instruments required for Surveying, Levelling, Plotting, etc. By H. S. Merrett. Fourth edition, revised by G. W. UsiLL, Assoc. Mem. Inst. C.E. 41 plates, with illustrations and tables, royal 8vo, cloth, \2s. 6d. Principal Contents : Part I. Introduction and the Principles of Geometry. Part 2. Land Surveying; com- pri-;ing General Observations— The Chain— Offsets Surveying by the Cham only— Surveying Hilly Ground— To Survey an Estate or Parish by the Cham only— Surveymg \yith the Theodolite— Mining and Town Surveying— Railroad Surveying— Mappmg— Division and Laying out of Land— Observations on Enclosures— Plane Trigonometry. Part 3. Levelling— Simple and Compound Levelling— The Level Boole— Parliamentary Plan and Section- Levelling with a Theodolite— Gradients— Wooden Curves— To Layout a Railway Curve- Setting out Widths. Part 4. Calculating Quantities generally for Estimates— Cuttings and Embankments— Tunnels— Brickwork— Ironwork— Timber Measuring. Part 5. Description and Use of Instruments in Surveying and Plotting— The Improved Dumpy Level— Iroughton s Level — The Prismatic Compass — Proportional Compass— Box Sextant— Vernier— Fanta- graph— Merrett's Improved Quadrant— Improved Computation Scale— The Diagonal Scale- Straight Edge and Sector. Part 6. Logarithms of Numbers — Logarithmic Sines and Co-Sines, Tangents and Co-Tangents— Natural Sines and Co-Sines— Tables for Earthwork, for Setting out Curves, and for various Calculations, etc., etc., etc Saws: the History, Development, Action, Classifica- tion, and Comparison of Saws of all kinds. By ROBERT Grimshaw, With 220 illustrations, 4to, cloth, I2s. 6d. A Supplement to the above; containing additional practical matter, more especially relating to the forms of Saw Teeth for special material and conditions, and to the behaviour of Saws under particular conditions. With 120 illustrations, cloth, 9 J. A Guide for the Electric Testing of Telegraph Cables. By Capt. V. Hoskicer, Royal Danish Engineers. With illustrations, second edition, crown 8vo, cloth, i^. 6d. Laying and Repairing Electric Telegraph Cables. By Capt. V. Hoskicer, Royal Danish Engineers. Crown 8vo, cloth, 3^. dd. A Pocket-Book of Practical Rules for the Proportions of Modern Engines and Boilers for Land and Marine purposes. By N. P. Burgh. Seventh edition, royal 32mo, roan, 4^. 6^/. The Assay ers Manual: an Abridged Treatise on the Docimastic Examination of Ores and Furnace and other Artificial Products. By Bruno Kerl. Translated by W. T. Brannt. With 65 illustrations, Svo, cloth, 12^. dd. The Steam Engine considered as a LJeat Engine : a Treatise on the Theory of the Steam Engine, illustrated by Diagrams, Tables, and Examples from Practice. By Jas. H. Cotterill, M.A., F.R.S., Professor of Applied Tvlechanics in the Royal Naval College. Svo, cloth, \2s. 6d. 14 CATALOGUE OF SCIENTIFIC BOOKS. Electricity: its Theory, Sources, and Applicati'ons. By J. T. Sprague, M.S.T.E. Second edition, revised and enlarged!, with numerous illustrations, crown 8vo, cloth, \^s. The Practice of Hand Tttrning in Wood, Ivory, S.hcll, etc., with Instructions for Turning such Work in Metal as may be recquired in the Practice of Turning in Wood, Ivory, etc. ; also an Appendlix on Ornamental Turning. (A book for beginners.) By Francis Ca.mun. Third edition, with wood engravings, crown 8vo, cloth, 6^. Contents : On Lathes— Turning Tools— Turning Wood— Drilling— Screw Cutting— Miscelllaneous Apparatus and Processes — Turning Particular Forms — Staining — Polishing — Spinning Metals — Materials — Ornamental Turning, etc. Health and Comfort in House Buildijtg, or Ven.tila- tion 7vith Warm Air by Self-Acting Suction Power, with Review cof the mode of Calculating the Draught in Hot- Air Flues, and with some :actual Experiments. By J. Drysdale, M.D., and J. W. Hayward, M.D. Second edition, with Supplement, with plates, demy 8vo, cloth, ^s. i6d. Treatise on Watchwork, Past and Present. By the Rev. H. L. Nelthropp, M.A., F.S.A. With 32 illustrations, crown 8vo, cloth, 6^. dd. Contents : Definitions of Words and Terms used in Watchwork — Tools — Time — Historical Sum- mary — On Calculations of the Numbers for Wheels and Pinions; their Proportional Sizes, Trains, etc. — Of Dial Wheels, or Motion Work — Length of Time of Going without Winding up — The Verge— The Horizontal — The Duplex — The Lever — The Chronometer — Repeating Watches — Keyless Watches — The Pendulum, or Spiral Spring — Compensation — Jewelliing of Pivot Holes — Clerkenwell — Fallacies of the Trade — Incapacity of \Vorkmen— How to Chuose and Use a Watch, etc. Notes in Mechanical Engineering. Compiled prin- cipally for the use of the Students attending the Classes on this subj^ect at the City of London College. By Henry Adams, Mem. Inst. M.E., Mem. Inst. C.E., Mem. Soc. of Engineers. Crown 8vo, cloth, 2r. 6^/. Algebra Self-Taught. By W. P. Higgs, M.A., D.Sc, LL.D., Assoc. Inst. C.E., Author of 'A Handbook of the Differ- ential Calculus,' etc. Second edition, crown 8vo, cloth, 2s. 6d. Contents : Symbols and the Signs of Operation — The Equation and the Unknown Quantity — Positive and Negative Quantities — Multiplication — Involution — Exponents — Negative Expo- nents—Roots, and the Use of Exponents as Logarithms — Logarithms — Tables of Loga rithms and Proportionate Parts — Transformation of System of Logarithms — Common Uses of Common Logarithms — Compound Multiplication and the Binomial Theorem — Division, Fractions, and Ratio — Continued Propjortion — The Series and the Summation of the Srcels between Paris and London, via Calais and Dover. By J. B. Berllier, C.E. Small folio, sewed, bd. List of Tests [Reagents), arranged in alphabeitical order, according to the names of the originators. Designed esptecially for the convenient reference of Chemists, Pharmacists, and Scieintists. By Hans M. Wilder. Crown 8vo, cloth, 4?. bd. Ten Years' Experience in Works of Intermiittrnt Downward Filtration. By J. Bailey Denton, Mem. Inst. C.E. Second edition, with additions. Royal 8vo, sewed, 4^, A Treatise on the Manufacture of Soap and Candles, Lubricants and Glycerin. By W. Lant Carpenter, B.A., B.Sc. (late of Messrs. C. Thomas and Brothers, Bristol). With illustrations. Crown 8vo, cloth, los. 6d. PUBLISHED BY E. & F. N. SPON. 17 The Stability of Ships explained simply, and calculated by a ne%v Graphic method. By J. C. Spence, M.I.N.A. 4to, sewed, 3J. (>d. Steam Making., or Boiler Practice. By Charles A. Smith, C.E. 8vo, cloth, \os. 6d. Contents : I. The Nature of Heat and the Properties of Steam— 2. Combustion. — 3. Externally Fired Stationary Boilers — 4. Internally Fired Stationary Boilers — 5. Internally Fired Portable Locomotive and Marine Boilers — 6. Design, Construction, and Strength of Boilers — 7. Pro- portions of Heating Surface, Economic Evaporation, Explosions — 8. Miscellaneous Boilers, Choice of Boiler Fittings and Appurtenances. The Fireman s Guide ; a Handbook on the Care of Boilers. By Teknoi.og, foreningen T. I. Stockholm. Translated from the third edition, and revised by Karl P. Dahlstrom, M.E. Second edition, Fcap. 8vo, cloth, 2s. A Treatise on Modern Steam Engines and Boilers, including Land Locomotive, and Marine Engines and Boilers, for the use of Students. By Frederick Colyer, M. Inst. C.E., Mem. Inst. M.E. With 36 plates. 4to, cloth, 25J. Contents : I. Introduction — 2. Original Engines — 3. Boilers — 4. High-Pressure Beam Engines — 5. Cornish Beam Engines — 5. Horizontal Engines — 7. Oscillating Kngines — 8. Vertical High- Pressure Engines — 9. Special Engines — 10. Portable Engines — 11. Locomotive Engines — 12. Marine Engines. Steam Engine Management ; sl Treatise on the Working and Management of Steam Boilers. By F. COLYER, M. Inst. C.E., Mem. Inst. M.E. i8mo, cloth, 2s. Land Surveying on the Meridian and Perpendicular System. By William Penman, C.E. 8vo, cloth, %s. 6d. The Topographer, his Instrtiments and Methods, designed for the use of Students, Amateur Photographers, Surveyoi-s, Engineers, and all persons interested in the location and construction of works based upon Topography. Illustrated with numerous plates, maps, and engravings. By LEWIS M. Haupt, A.M. 8vo, cloth, iSj. A Text-Book of Tanning, embracing the Preparation of all kinds of Leather. By Harry R. Proctor, F.C.S., of Low Lights Tanneries. With illustrations. Crown 8vo, cloth, loj. bd. In super-royal 8vo, 1168 pp., with 2400 illustrations, in 3 Divisions, cloth, pricce 13^.6^. each ; or i vol., cloth, tI. ; or half-morocco, 2/. 8^. A SUPPLEMENT TO SPONS' DICTIONARY OF ENGINEERING. Edited by ERNEST SPON, Memb. Soc. Engineers, Abacus, Counters, Speed Indicators, and Slide Rule. Agricultural Implements and Machinery. Air Compressors. Animal Charcoal Ma- chinery. Antimony. Axles and Axle-boxes, Barn Machinery, Belts and Belting. Blasting. Boilers, Brakes. Brick Machinery, Bridges. Cages for Mines, Calculus, Differential and Integral. Canals. Carpentry, Cast Iron, Cement, Concrete, Limes, and Mortar. Chimney Shafts. Coal Cleansing and Washing. Coal Mining. Coal Cutting Machines. Coke Ovens. Copper. Docks. Drainage. Dredging Machinery. Dynamo - Electric and Magneto-Electric Ma- chines. Dynamometers. Electrical Engineering, Telegraphy, Electric Lighting and its prac- ticaldetailsTelephones Engines, Varieties of. Explosives. Fans. Founding, Moulding and the practical work of the Foundry, Gas, Manufacture of. Hammers, Steam and other Power. Heat. Horse Power, Hydraulics. Hydro-geology. Indicators. Iron, Lifts, Hoists, and Eleva- tors. Lighthouses, Butoys, and Beacons. Machine Tools. Materials of C^onstruc- tion. Meters. Ores, Machinery and Processes emp^loyed to Dress. Piers. Pile Driving. Pneumatic Tramsmis- sion. Pumps. Pyrometers. Road Locomotives. Rock Drills. Rolling Stock. Sanitary Engineering. Shafting. Steel. Steam Navvy. Stone Machiner)y. Tramways. Well Sinking. London : E. & F. N. SPON, 125, Strand. New York : 35, Murray Street. NOW COMPLETE. With nearly 1500 illustrations, in super-royal 8vo, in 5 Divisions, cloth. Divisions 1 to 4, 13^. ^d. each ; Division 5, 17J. 6^/. ; or 2 vols., cloth, £,'3^ lOj. SPONS' ENCYCLOPEDIA OF THE INDUSTRIAL ARTS, MANUFACTURES, AND COMMERCIAL PRODUCTS. Edited by C. G. WARNFORD LOCK, F.L.S. mportant of the subjects treated of, are the Among the more i follow^ing : — Acids, 207 pp. 220 figs. Alcohol, 23 pp. 16 figs. Alcoholic Liquors, 13 pp. Alkalies, 89 pp. 78 figs. Alloys. Alum. Asphalt. Assaying. Beverages, 89 pp. 29 figs. Blacks. Bleaching Powder, 15 pp. Bleaching, 5 1 pp. 48 figs. Candles, 18 pp. 9 figs. Carbon Bisulphide. Celluloid, 9 pp. Cements. Clay. Coal-tar Products, 44 pp. 14 figs. Cocoa, 8 pp. Coffee, 32 pp. 13 figs. Cork, 8 pp. 17 figs. Cotton Manufactures, 62 PP- 57 figs- Drugs, 38 pp. Dyeing an_d Calico Printing, 28 pp. 9 figs. 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Salt, 31 pp. 23 figs. Silk, 8 pp. Silk Manufactures, 9 pp. II figs. Skins, 5 pp. Small Wares, 4 pp. Soap and Glycerine, 39 pp. 45 figs. Spices, 16 pp. Sponge, 5 pp. Starch, 9 pp. 10 figs. Sugar, ISS pp. 134 figs. Sulphur. Tannin, 18 pp. Tea, 12 pp. Timber, 13 pp. Varnish, 15 pp. Vinegar, 5 pp. Wax, 5 pp. Wool, 2 pp. Woollen Manufactures, 58 pp. 39 figs. London : E. & F. N. SPON, 125, Strand. New York : 35, Murray Street. Crown 8vo, cloth, with illustrations, 5j. WORKSHOP RFXEIPTS, FIRST SERIES. By ERNEST SPON. Bookbinding. Bronzes and Bronzing. Candles. Cement. Cleaning. Colourwashing. Concretes. Dipping Acids. Drawing Office Details. Drying Oils. Dynamite. Electro - Metallurgy — (Cleaning, Dipping, Scratch-brushing, Bat- teries, Baths, and Deposits of every description). Enamels. Engraving on Wood, Copper, Gold, Silver, Steel, and Stone. Etching and Aqua Tint. 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Varnishes, Manufaicture and Use of. Veneering. Washing. Waterproofing. Welding. Besides Receipts relating to the lesser Technological matters and prociesses, such as the manufacture and use of Stencil Plates, Blacking, Crayons, Paste, Putty, Wax, Size, Alloys, Catgut, Tunbridge Ware, Picture Frame and Architectural Mouldings, Compos, Cameos, and others too numero'us to mention. London : E. & F. N. SPON", 125, Strand. New York: 35, Murray Street. Crown 8vo, cloth, 485 pages, with illustrations, 5j. WORKSHOP RECEIPTS, SECOND SERIES. By ROBERT HALDANE. Synopsis of Contents. Acidimetry and Alkali- metry. Albumen. Alcohol . Alkaloids. Baking-powders. Bitters. Bleaching. Boiler Incrustations, Cements and Lutes. Cleansing. Confectionery. Copying. Disinfectants. Dyeing, Staining, and Colouring. Essences. Extracts. Fireproofing. Gelatine, Glue, and Size. Glycerine. Gut. Hydrogen peroxide. Ink. Iodine. Iodoform. Isinglass. Ivory substitutes. Leather. Luminous bodies. Magnesia. Matches. Paper. Parchment. Perchloiic acid. Potassium oxalate. Preserving. Pigments, Paint, and Painting : embracing the preparation of Pigments, including alumina lakes, blacks (animal, bone, Frankfort, ivoiy, lamp, sight, soot), blues (antimony, Antwerp, cobalt, cseruleum, Egyptian, manganate, Paris, Peligot, Prussian, smalt, ultramarine), browns (bistre, hinau, sepia, sienna, umber, Vandyke), greens (baryta, Brighton, Brunswick, chrome, cobalt, Douglas, emerald, manganese, mitis, mountain, Prussian, sap, Scheele's, Schweinfurth, titanium, verdigris, zinc), reds (Brazilwood lake, carminated lake, carmine, Cassius purple, cobalt pink, cochineal lake, colco- thar, Indian red, madder lake, red chalk, red lead, vermilion), whites (alum, baryta, Chinese, lead sulphate, white lead — by American, Dutch, French, German, Kremnitz, and Pattinson processes, precautions in making, and composition of commercial samples — whiting, "Wilkinson's white, zinc white), yellows (chrome, gamboge, Naples, orpiment, realgar, yellow lakes) ; Paint (vehicles, testing oils, driers, grinding, storing, applying, priming, drying, filling, coats, brushes, surface, water-colours, removing smell, discoloration ; miscellaneous paints — cement paint for carton-pierre, copper paint, gold paint, iron paint, lime paints, silicated paints, steatite paint, transparent paints, tungsten ]5aints, window paint, zinc paints) ; Painting (general instructions, proportions of ingredients, measuring paint work ; carriage painting — priming paint, best putty, finishing colour, cause of cracking, mixing the paints, oils, driers, and colours, varnishing, importance of washing vehicles, re-varnishing, how to dry pauit ; woodwork painting). London : E. &; F. N. SPON, 126, Strand. New York: 35, Murray Street. JTJST PUBLISHED. Crown 8vo, cloth, 480 pages, with 183 illustrations, 5 J. WORKSHOP RECEIPTS, THIRD SERIES. By C. G. WARNFORD LOCK. Uniform with the First and Second Series. Synopsis of Contents. Alloys. Indium. Rubidium. Aluminium. Tridium. Ruthenium. Antimony. Iron and Steel. Selenium. Barium. Lacquers and Lacquering. Silver. Beryllium. Lanthanum. Slag. Bismuth. Lead. Sodium, Cadmium. Lithium. Strontium. Caesium, Lubricants. Tantalum. Calcium. Magnesium. Terbium. Cerium. Manganese. Thallium. Chromium, Mercury. Thorium. Cobalt. Mica. Tin. Copper. Molybdenum. Titanium. Didymium. Nickel. Tungsten. Electrics. Niobium. Uranium. Enamels and Glazes. Osmium. Vanadium. Erbium. Palladium. Yttrium. Gallium. Platinum. Zinc. Glass. Potassium. Zirconium. Gold. Rhodium. London : E. & F. N. SPON, 125, Strand. New York: 35, Murray Street. WORKSHOP RECEIPTS, FOURTH SERIES, DEVOTED MAINLY TO HANDICRAFTS & MECHANICAL SUBJECTS. By C. G. WARNFORD LOCK. 250 Illustrations, with Complete Index, and a General Index to the Four Series, 5s. Waterproofing — rubber goods, cuprammonium processes, miscellaneous preparations. Packing and Storing articles of delicate odour or colour, of a deliquescent character, liable to ignition, apt to suffer from insects or damp, or easily broken. Embalming and Preserving anatomical specimens. Leather Polishes. Cooling Air and Water, producing low temperatures, making ice, cooling syrups and solutions, and separating salts from liquors by refrigeration. Pumps and Siphons, embracing every useful contrivance for raising and supplying water on a moderate scale, and moving corrosive, tenacious, and other liquids. Desiccating — air- and water-ovens, and other appliances for drying natural and artificial products. Distilling — water, tinctures, extracts, pharmaceutical preparations, essences, perfumes, and alcoholic liquids. Emulsifying as required by pharmacists and photographers. Evaporating — saline and other solutions, and liquids demanding special precautions. Filtering — water, and solutions of various kinds. Percolating and Macerating. Electrotyping. Stereotyping by both plaster and paper processes. Bookbinding in all its details. Straw Plaiting and the fabrication of baskets, matting, etc. Musical Instruments — the preservation, tuning, and repair of pianos, harmoniums, musical boxes, etc. Clock and Watch Mending — adapted for intelligent amateurs. Photography — recent development in rapid processes, handy apparatus, numerous recipes for sensitizing and developing solutions, and applica- tions to modern illustrative purposes. London : E. & F. N. SPON, 125, Strand. New York : 35, Murray Street. aUST PUBLISHED. In demy 8vo, cloth, 600 pages, and 1420 Illustrations, 6s. SPONS' MECHANICS' OWN BOOK ; A MANUAL FOR HANDICRAFTSMEN AND AMATEURS. Contents. Mechanical Drawing— Casting and Founding in Iron, Brass, Bronize, and other Alloys— Forging and Finishing Iron— Sheetmetal Worlkimg —Soldering, Brazing, and Burning— Carpentry and Joinery, embraiciing descriptions of some 400 Woods, over 200 Illustrations of Tools amd their uses. Explanations (with Diagrams) of 116 joints and hinges,, amd Details of Construction of Workshop appliances, rough furniituire. Garden and Yard Erections, and House Building— Cabinet-Malkimg and Veneering — Carving and Fretcutting — Upholstery— Paimtimg, Graining, and Marbhng — Staining Furniture, Woods, Floors, amd Fittings— Gilding, dead and bright, on various grounds— Pohsihimg Marble, Metals, and Wood— Varnishing— Mechanical movennenits, illustrating contrivances for transmitting motion— Turning in Woood and Metals— Masonry, embracing Stonework, Brickwork, Terrac:ottta, and Concrete— Roofing with Thatch, Tiles, Slates, Felt, Zinc, Sec- Glazing with and without putty, and lead glazing— Plastering amd Whitewashing— Paper-hanging— Gas-fitting— Bell-hanging, ordimaary and electric Systems — Lighting — Warming — Ventilating — Roacds, Pavements, and Bridges — Hedges, Ditches, and Drains — Waiter Supply and Sanitation— Hints on House Construction suited to mew countries. London: E. & F. N. SPON, 125, Strand. New York : 35, Murray Street.