Class __jr§^^ Boole F55 GapiglitlJ", CflEffilOHI DEPOSm ESSENTIALS OF MECHANICAL DRAFTING ELEMENTS, PRINCIPLES, AND METHODS WITH SPECIFIC APPLICATIONS IN WORKING DRAWINGS OF FURNITURE, MACHINE, AND SHEET METAL CONSTRUCTION A MANUAL FOR STUDENTS ARRANGED FOR REFERENCE AND STUDY IN CONNEC- TION WITH COURSES IN MANUAL TRAINING, INDUSTRIAL, HIGH, AND TECHNICAL SCHOOLS BY LUDWIG Ij-RANK INSTRUCTOR IN DRAWING, HIGH SCHOOL OF COMMERCE, BOSTON, MASS. FORMERLY ASSISTANT TO THE DIRECTOR OF MANUAL ARTS, BOSTON, MASS. SUPERVISING DRAWING IN PREVOCATIONAL. INDUSTRIAL, AND HIGH SCHOOLS 1917 MILTON BRADLEY COMPANY SPRINGFIELD, MASSACHUSETTS Copyright 1917 BY MILTON BRADLEY COMPANY, Springfield, Mass. - / m 23 1917 ©C1A467139 PREFACE The purpose of this book is to provide the student with definite comprehensive text and illustrations comprising the theory and practice of mechanical drafting, which shall effective^ supplement and give emphasis to the work of the teacher, and at the same time afford complete freedom in the presentation and application of principles to meet different requirements, conditions, and individual needs. In view of this purpose, and for greater convenience of reference and connected study of related subject-matter, the text is presented in a progressive series of topically arranged articles with appropriate cross references. This arrangement is not intended, however, as a prescribed order of study to be rigidlj^ adhered to, nor is the content of the text intended to supersede necessary personal instruction or thoughtful study on the part of the student. The book is designed to be used as the teacher may determine in connection with his own course; to conserve the time and energy usually expended in repeti- tion; to secure a systematic study of such text and illustrations as relate to the oral presentation; and to enable the student to review any desired topic as individual need arises and to proceed with the minimum of dependence upon the teacher. It is beUeved that the content and arrangement will be found adaptable and adequate wherever mechanical drafting is taught; will assist the teacher in formu- lating specific courses; will stimulate the interest of the student by giving a greater appreciation of the utility and scope of the subject; and will prove an efficient aid in developing a working knowledge of the elements, principles, and methods of drafting as applied in general practice. The writer gratefully acknowledges his obligations to numerous engineers and draftsmen, and to the following directors and teachers of Boston, for many helpful suggestions: Mr. Arthur L. WiUiston, Director of Wentworth Institute; Mr. John C. Brodhead, Associate Director of Manual Arts; Mr. Edw. C. Emerson, Assistant Director of Manual Arts; Messrs. R. H. Knapp and E. H. Temple of the Mechanic Arts High School; Messrs. A. Roswall and E. M. Longfield of the Boys' Industrial School; Mr. A. L. Primeau, formerly of the South Boston Pre- vocational Center; and Mr. R. A. Day of the Hyde Park Co-operative Classes. Messrs. Temple, Primeau, and Day also gave valuable assistance in the prepara- tion of the drawings. LuDwiG Feank. Brookline, Januarv, 1917. CONTENTS CHAPTER I INTRODUCTION PAGE ART. 1. Nature and Uses of Mechanical Draft- ing 1 2. Geometric Terms, Definitions, etc. ... 2 3. General Instructions for Working Out Problems 11 CHAPTER II INSTRUMENTS, MATERIALS, AND THEIR USE. 4. List of Equipment 13 5. Care and Arrangement of Equipment 13 6. Drawing Board 14 7. T-Square 14 8. Triangles 15 9. Pencils and Writing Pens 16 10. Needle-point 17 11. Scales 17 12. Protractor 19 13. Compasses 19 14. Bow Compasses 20 15. Dividers 20 16. Bow Dividers 20 17. Ruling Pen 20 18. Curve Rulers 21 19. Erasers 22 CHAPTER III PENCILING AND FINISH RENDERING 20. Layout of the Sheet 23 21. Constructive Stage of the Drawing. . . 23 22. Finishing Stage of the Drawing 24 23. Shadow Lining 25 24. Line Shading 28 25. Lettering 29 CHAPTER IV GEOMETRIC CONSTRUCTION 26. Geometric and Practical Methods. ... 33 27. To bisect a straight Une or a circular arc 33 28. To bisect an angle 33 29. To construct an angle equal to a given angle 34 30. To divide a straight line into any number of equal parts 34 31. To divide a straight line into parts proportional to those of a given divided line 34 32. To layoff the length of a given circular curve upon a straight line 34 33 . To lay off the length of a given straight line upon an arc 34 34. To draw a perpendicular to a line, from or through a given point ... 34 35. To draw a Une at an angle of any given magnitude in a quadrant -with a given line 35 36. To draw a parallel to a given straight hne: (a) at a given distance; or (b) through a given point 36 37. To construct a triangle: having given (a) the sides; (b) a side and angles adjacent to it; or (c) a side, an adja- cent angle, and angle opposite the side 36 38. To construct an isosceles triangle when the base and vertex angle are given 36 39. To construct an equilateral triangle when the altitude is given 37 40. To circumscribe a circle about a given triangle 37 41. To inscribe a circle within a given triangle 37 42. To inscribe an equilateral triangle within a circle 37 43. To circumscribe an equilateral triangle aboutacircle 37 44. To construct a parallelogram when two sides and the included angle are given 37 45. To construct a square on a given side 37 46. To inscribe a square within a circle. . . 38 47. To construct a square on a given diagonal 38 48. To circumscribe a square about a circle 38 49. To inscribe a regular pentagon within a circle 38 50. To construct a regular hexagon on a given side 38 51. To inscribe a regular hexagon within a circle 38 52, 53, 54. 55. 56. 57. 58. 59. 60. PAGE To construct a regular hexagon on a given long diagonal 38 To inscribe a regular octagon within a circle ■ 39 To circumscribe a regular octagon about a circle '. 39 To construct any regular polygon. General methods: having given (a) a side; (b) the circumscribing circle; or (c) the inscribed circle 39 To construct a polygon similar to a given polygon, upon a given side. . . 40 To plot a figure similar to a given figure by means of a base line or center line, and offsets 40 To draw a tangent to a circle through a given point 41 To draw a tangent to two given circles 41 To draw a circular curve of given ART. PAGE radius tangent to a given circular curve and to a given straight line, or to two given circular curves 42 61. To draw a circular curve tangent to two given straight lines: having given (a) a point of tangency; or (b) the radius 42 62' To draw a circular curve tangent to three straight lines 42 63. To draw circular curves tangent to two given parallel straight lines and to each other, at a given point 43 64. To draw an ellipse when the axes are given: (a) By Focal Radii 43" (b) By Trammel Method 43 (c) By Revolution of a Circle 43 (d) By Parallelogram Method 43 (e) By Circular Arcs 44 CHAPTER V ORTHOGRAPHIC PROJECTION 65. General Principles 45 66. To draw the front, top, and side views of a rectangular object 49 67. Objects Having Surfaces Obhque to the Co-ordinate Planes 51 68. Objects Having Curved Surfaces .... 51 69. Projections upon Oblique Auxiliary Planes 52 70. Revolution of Surfaces 54 71. True Length and Position of Lines Oblique to the Co-ordinate Planes . . 55 72. Objects Obhque to the Co-ordinate Planes 56 73. Partial Views, and Use of Auxihary Views in Determining Required Views 58 74. Rules Governing the Position of Lines and Surfaces Relative to Any Two Perpendicular Planes of Projection. 61 CHAPTER VI PLANE SECTIONS 75. Principles and Methods 62 76. Objects Having Plane Surfaces 62 77. Objects Having Curved Surfaces . . 64 CHAPTER VII INTERSECTION OF SURFACES 78. Principles and Methods 66 79. Objects Having Plane Surfaces 67 80. Objects Having Curved Surfaces . CHAPTER VIII DEVELOPMENT OF SURFACES 81. Principles and Methods 71 82. Objects Having Plane Surfaces 72 83. Objects Having Cylindric or Conic Surfaces 73 84. Objects Having Double Curved or Warped Surfaces 77 85. Intersecting Surf aces 78 CHAPTER IX MECHANICAL PICTORIAL DRAWINGS 86. Character and Purpose of the Drawing 80 87. Isometric Projection 81 88. To draw the isometric of a rectangular object 81 89. To draw the isometric of an object involving non-isometric figures 83 90. Oblique Projection 86 91. Shade Lines, Shadow Lines, and Line Shading 87 CHAPTER X WORKING DRAWINGS 92. CharacterandPurposeof the Drawing 89 93. Tjfpesof Drawings 89 94. Position of Object and Arrangement of Views 92 95. Selection and Number of Views, etc. . . 92 96. Center Lines 94 97. Conventional Representations 94 98. Sectional Views 96 99. Broken Views 98 ART. PAGE Standard Sizes of Sheets and Scale of Drawings 99 101. Dimensioning 100 102. Lettering 105 103. Shadow Lining and Line Shading. . . 108 Sketching 108 Making Scale Drawings 109 Tracing and Blue-printing 115 Checking Drawings 120 100. 104. 105. 106. 107. 108. Reading Drawings 120 CHAPTER XI HELICAL CURVES, THREADED PARTS, AND SPRINGS 109. HeUces 122 110. Screw Threads 123 111. Pipe Threads 128 112. Bolts 128 113. Screws 131 114. Springs 132 SYMBOLS AND GENERAL ABBREVIATIONS II parallel; ||s parallels ± perpendicular; J_s perpendiculars Z. angle; Zs angles A triangle; As triangles G circle; Os circles C. L. center line; C. Ls. center lines diam. diameter; diams. diameters hor. horizontal; hors. horizontals pt. point; pts. points rad. radius St. straight vert. vertical; verts, verticals ESSENTIALS OF MECHANICAL DRAFTING CHAPTER I INTRODUCTION 1. Nature and Uses of Mechanical Drafting. Mechanical drafting or mechanical drawing is the art of making the conventional representations used by engineers, architects, and inventors in working out and recording the details of their constructive designs, and the means by which ideas of the exact form or shape, dimension, and arrangement of parts in objects of a structural charac- ter are universally expressed and made intelligible to others. Mechanical drafting enables constructive work of any kind to be carried on with accuracy and economy of time and material, and takes the place of lengthy verbal description which would fail to express with clearness and exactness the definite information required by the workman. It will be seen from Fig. 179 that certain general peculiarities of the form and structure of an object may be understood from an ordinary pictorial representa- tion, but that it cannot show the exact form, size, and relation of all the lines and surfaces; hence the necessity for mechanical drawings which show all hidden as well as visible parts of an object as they are and not as they would appear to the eye. Mechanical drafting is thus the graphic language of the constructive or mechanic arts, and ability to read or comprehend mechanical drawings is of as great importance to the workman, builder, and manufacturer as ability to make such representations is to the designer or draftsman; and a knowledge of general drafting principles is of value to almost all men irrespective of their vocations. Because of the exact nature of the facts which it is intended to record or convey the drawing is generallj^ executed with the aid of instruments. The mechanical character of the representation, together with its purpose and the usual means of execution, gives mechanical drafting its name. Machine drafting, architectural drafting, and engineering drafting are specific applications of mechanical drafting. A mechanical drawing properly dimensioned in figures and prepared as a guide in constructing the object is called a working drawing. 2 ESSENTIALS OF MECHANICAL DRAFTING 2. Geometric Terms, Definitions, etc. Geometry is the science which describes the definite figures (forms or shapes) upon which all objects however complex are based, and the principles and methods by which these figures may be measured and graphically constructed. Geometry is thus fundamental in mechanical drafting and in all the con- structive arts. The terms defined in this chapter are commonly involved in both. (a) General Definitions. A physical solid or material object occupies a certain portion of space and has shape, size, weight, color, etc. Geometry is concerned simply with the shape and size of the space which a physical solid occupies or is conceived to occupy; hence — a geometric solid is a limited portion of space. A soKd has dimensions or extent in three principal directions at right Zs to each other; namely, length, breadth (or width), and thickness {height, altitude, or depth). The boundaries of a solid are called surfaces. A surface has only two dimensions: length and breadth (or width). The boundaries of a surface are called lines. A line has only one dimension : length. The hmits or ends of a line are called points. A point has position but no dimension. Points, lines, and surfaces may be considered as apart from a solid, or as combined in any con- ceivable figure; also a line may be imagined as generated by a point, a surface by a line, and a solid by a surface, in motion. Similar figures are those having the same shape; equivalent figures those having the same size; and equal or congruent figures, those having the same shape and size. A figure that Ues wholly in one plane is a plane figure. (See Art. (h).) A figure whose lines are straight is rectilinear; one whose lines are curved is curvilinear. The axis of a figure is a st. line which passes through its center, and about which it is symmetrical or balanced. An axis of revolution is a st. line about which a figure is revolved. When two lines, two surfaces, or a line and a surface meet or cross they are said to intersect or cut each other and the pt. or line in which they intersect is their inter'section. A bisector is a pt., line, or plane which divides a figure into Fig. I two equal parts, that is, bisects it. To trisect is to divide into three equal parts; to quadrisect, into four equal parts. (b) Lines. A straight or right line has the same direction throughout. Fig. 1. A curved line or curve is one no part of which is straight. Fig. 2. A reversed curve is one whose direction of curvature changes. Fig. 3. A horizontal line is one that is level throughout. In draw- ing, the term is appUed to a st. line drawn from left to right, without slant. Fig. 1. A vertical line is one that is upright or plumb. In drawing, the term is applied to a st. fine drawn from bottom to top, with- out slant. Fig. 4. An oblique line is one that slants. Fig. 5. Two lines having the same relative direction are parallel to each other. They are the same distance apart throughout. Fig. 6. Fig 4 Fig. Fig. 6 INTRODUCTION Two St. lines which extend from the same pt. or which would intersect if extended, are said to be at an angle with each other (Fig. 7), and are perpendicular, or oblique, to each other according as the included Z is a right Z or an oblique Z. See Art. (c). Two curves, or a st. line and a curve, are tangent to each other when they touch in but one pt. and cannot intersect. The pt. is the point of tangency. Figs. 11(a), 12, 14. An ordinate or offset is the J_ distance of a pt. from a given st. hne, or plane, of reference. A-1, B-2, etc., Fig. 103. Co-ordinates are ordinates of the same pt., measured || to two, or three, mutually J. hnes, or planes, of reference. C-Cv, C-Ch, C-Qp, Fig. 118. \ Fig. 7 V. Z 1 [Fig. 8 ^~- Fig. 9 (c) Angles. These definitions refer to plane Zs only. An angle is the opening between two st. lines, called the sides of the angle, which extend from a pt. called the vertex. BAG, Fig. 7. An angle may be considered as generated by a st. Une revolved about one of its ends. The size of an angle depends upon the relative direction of the sides and not upon their length. When the sides extend in opposite directions, so as to lie in the same st. line, the Z is a straight angle. When the directions are such that the Zs formed by extending the sides beyond the vertex are equal, each Z is a right angle. A right Z is equal to half a st. Z. Fig. 8. An Z less than a right Z is an acute angle (Fig. 7) ; one greater than a right Z and less than a St. Z is an obtuse angle (Fig. 9); one greater than a st. Z and less than two st. Zs is a reflex angle. Note that two st. lines extending from a pt. always form two Zs, as Za and Zb, Fig. 9. Angles other than right and straight Zs a,ie oblique angles. Two Zs having the same vertex and a common side between them are adjacent angles. DCF and FCE, Fig. 10. When two st. lines intersect, the opposite or non-adjacent Zs are equal and are called vertical angles. GCD and FCE, also Zs DCF and ECG, Fig. 10. An angle is said to be measured by an arc, described from its vertex as center and included by its sides. The unit of measure is an angle whose arc is a degree (5-5^ of a circumference). (See Art. (d).) Thus a St. Z is one of 180°, a right Z one of 90°, and the whole angular space about a pt. in a plane equals 360°. See Fig. 10. Lines can be drawn in four directions from a given pt., at the same given Z with a given line; thus in Fig. 10, C-D, C-E, C-F, and C-G each make an Z of 30° with A-A. Each of these lines makes two Zs with the hor. A-A and two with the vert. B-B. Thus C-D makes Zs of 150° and 30° with A-A, and 60° and 120° with B-B. 4 ESSENTIALS OF MECHANICAL DRAFTING In speaking of the Zs formed by two lines, the lesser is the one usually named, and, unless other- wise stated, a line at 15°, 30°, 45°, etc., is understood to mean 15°, 30°, 45°, etc., with the hor. direction. (d) Curvilinear Figures. A circle* is a plane figure bounded by a curve called its circumference, all pts. of which are equidistant from a pt. within called the center. Fig. 11(a). Any part of a circumference is an arc. A St. line intersecting a circular curve in two pts. is a secant, F-G. A st. line joining two pts. in the curve is a chord, H-I, A-B. A chord through the center is a diameter. A straight Une from the center to the curve is a radius, C-A, C-D, C-E. A segment is a portion of a O boundetkby an arc and its'chord. A segment equal to half a O is a semicircle*. A sector is a portion of a O bounded by an arc and two radii. A sector equal to one-fourth of a O is a quadrant*. An Z formed by two chords from the same pt. is an inscribed angle. Fig. 11(b). An angle is inscribed in a segment when its sides join the ends of the arc. The Z included by two radii is a central angle. Fig. 13 Fig. 12 N-^-^b) c J Fig. 14 The circumference of a O is conceived to contain 360 equal parts called degrees (360°), each degree 60 equal parts called minutes (60'), and each minute 60 equal parts called seconds (60"). A St. tangent is X to a rad. at the pt. of tangency. J-K, Fig. 11(a). The pt. of tangency of two circular curves is in their line of centers. Fig. 12. Two circles or arcs having the same center are concentric. Fig. 13(a). Two circles not having the same center are eccentric when one is within the other. Fig. 13(b). An ellipse* is a plane figure bounded by a curve called its circumference, the sum of the distances of every pt. of which, from two pts. within, called the focuses or foci, is constant. Fig. 14. A pt. midway between the foci is called the center. A St. line joining any two pts. in the curve is a chord, A chord through the center is a diameter. A diam. containing the foci and center is the long or major axis. A diam. ± to the major axis is the short or minor axis. The major and minor axes are the longest and shortest diameters of the eUipse and bisect each other at the center. , A St. line from either focus to any pt. in the curve is a focal radius. The sum of the focal radii of any pt. is equal to the major axis. A st. tangent bisects the Z between one focal rad. and the other extended at the pt. of tangency. When one diam. is || to the tangents at the ends of another, the diams. are conjugate to each other. *The terms " circle," " semicircle," " quadrant, " and " ellipse " are also used to denote merely the curve. INTRODUCTION In general any side may be (e) Polygons. These definitions refer to plane polygons only. A polygon is a plane figure bounded by st. lines called its sides. The sum of the sides is the ■perimeler; the Z s formed by the sides are the angles; and the vertices of the / s, the vertices of the polygon. Figs. 15-30. A polygon is equilateral when all of its sides are equal (Figs. 15, 24, 27) ; equiangular when all its Zs are equal (Fig. 21); regular when both equilateral and equiangular (Figs. 15,22, 28-30) ; otherwise, it is irregular. The hase of a polygon is the side upon which it is supposed to rest, considered as the base. The altitude is the ± distance between the base, or base extended, and the farthermost vertex or side. A-B in Figs. 15, 17, 19, 23, 26. A diagonal is a st. line joining any two non-consecutive vertices. A-B, Fig. 21; A-B and A-C, Fig. 29. A polygon is inscribed in a O when all its sides are chords of the O, and circumscribed about a O when all its sides are tangents of the O. Also the O is circumscribed about the inscribed polygon and inscribed in the circumscribed polygon. Fig. 28. The center of a regular polygon is the center of the inscribed or circumscribed O . The rad. of the inscribed O is the apothem, and the rad. of the circimiscribed O the radius of the potygon. Fig. 28. In a regular polygon of an even number of sides the diameter of the inscribed O is often called the short diameter, and that of the circumscribed O the long diameter of the polygon. The sum of the Zs of any polygon is equal to two right Zs (180°), taken as manj^ times less two as the figure has sides. See Fig. 20. The Z included by the radii to the ends of any side of a regular polygon is called the angle at the center. It is equal to 360° divided by the number of sides. Fig. 30. A polygon of three sides is a triangle; of four, a quadrilateral; of 'five, a pentagon; of six, a hexagon; of seven, a heptagon; of eight, an octagon; of nine, a nonagon; often, a decagon. (f) Triangles. Triangles are classified according to relative length of sides; as equilateral, isosceles, and scalene. An equilateral triangle has aU sides equal; it is also equi- angular. Fig. 15. An isosceles triangle has two sides equal; two Zs are also equal. Fig. 16. The equal sides are usually called the sides, and the other side, the base. A scalene triangle has no two sides equal; its Zs are also imequal. Fig. 17. Triangles are classified according to kind of Zs; as right, obtuse, and acute. A A is a right triangle when one Z is a right Z. Fig. 18. The side opposite the right Z is called the hypotenuse, the others are usually called the sides. A A is an obtuse triangle when one Z is obtuse. Fig. 19. A A is an acute triangle when all Zs are acute. Fig. 20(a). The Z opposite the base of a A is called the vertex angle, and its vertex , the vertex of the triangle. Fig. 20 (g) Quadrilaterals. A parallelogram is a quadrilateral whose opposite sides are ||. Figs. 21-24. A rectangle is a right-angled parallelogram. Figs. 21, 22. A square is an equilateral rectangle. Fig. 22. A rectangle whose opposite sides only are equal is often called an oblong. Fig. 21. A rhomboid is an oblique-angled parallelogram. Figs. 23, 24. ESSENTIALS OF MECHANICAL DRAFTING Fig. 21 Fig. 25 Fig 30 A rhombus is an equilateral rhomboid. Fig. 24. The side 1 1 to the base of a parallelogram is called the upper base, the other is the lower base. A trapezoid is a quadrilateral which has two sides only ||. Fig. 25. The parallel sides are the upper and lower bases. A trapezium is a quadrilateral which has no two sides 1 1 . Fig. 26. (h) Surfaces. A plane surface or plane is a surface such that a st. line through any two pts. in it lies whoUy in the surface. A curved surface is one no part of which is plane. If a semicircular arc be revolved about its chord, it will generate a spheric surface. Fig. 46. A St. line which moves 1 1 to its first position and constantly touches a fixed curve not in the plane of the line, will generate a cylindric surface. Fig. 39. A moving st. line which constantly intersects a fixed curve and passes through a fixed pt. not in the plane of the curve will generate a conic surface. The fixed pt. is its vertex. Fig. 43. Curved surfaces generated by st. lines are single curved surfaces. The generating line in any of its positions is called an element. A warped surface is one of single curvature in which no two consecutive elements are 1 1 or inter- secting. A in Fig. 158. A double curved surface is one generated by a curve of which no two consecutive positions are II; as the surface of a sphere, elHpsoid, etc. Figs. 46, 47. A curved surface generated by the revolution of a line about an axis is called a surface of revolution; as a right circular cylindric or conic surface, etc. Figs. 39, 43, 46, 47. Each pt. of the generating line describes a O whose plane is _L to the axis. A plane surface is horizontal when it is level throughout; vertical when J. to a hor. plane; oblique when neither hor. nor vert. Two surfaces, or a line and a surface, are parallel when they are the same distance apart throughout. Two plane surfaces which extend from the same st. line or which would intersect if extended, are said to be at an angle with each other; and are perpendicular, or obliqiie, to each other according as the included Z is a right or an oblique dihedral. A dihedral or dihedral angle is the opening between two planes called the /aces of the angle, which intersect in a st. line called its edge. A dihedral is measured by the plane Z formed by two st. lines, one in each face and _L to the edge at the same pt. INTRODUCTION A dihedral is right, acule, or ohluse according as its measure is a right, acute, or obtuse plane Z . A St. line and a plane are ± to each other when the line is X to every st. line through its foot or pt. of intersection with the plane. The Z a line makes with a plane is the Z which it makes with a line in the plane passing through its foot and the foot of a ± from any other pt. in the line. A plane and a curved surface, or two curved surfaces, are tangent when they touch in but one pt. or in one line, and cannot intersect. The pt. or line is the point or line of tangency. A St. line or curve and a curved surface, or a curve and plane, are tangent when they touch in but one pt. and cannot intersect. (i) Solids. The base is the plane surface of the solid upon which it is supposed to rest. The altitude is the ± distance between the plane of the base and the farthermost vertex or part. A plinth is a prism, or c\'linder, whose altitude is its least dimension. Figs. 33, 42. A plane section is the figure formed by the intersection of a soUd with a plane passing through it. A solid of revolution is one which may be generated by a plane surface revolving about an axis. A solid bounded by plane surfaces is called a polyhedron. Figs. 31-38. The bounding surfaces are its faces; the intersection of its faces, the edges; and the vertices of the faces, the vertices of the polyhedron. A diagonal of a polyhedron is a st. line joining any two vertices not in the same face, as A-B, Fig. 31(a). ^KD^ Fig. 31 Fig. 32 Fig. 33 Fig 34 Fig. 35 (j) Prisms. A prism is a pol}'hedron bounded by two equal polygons called its bases, and by three or more parallelograms called its lateral faces. The intersections of its lateral faces are its lateral edges; the others are its base edges. Figs. 31, 32, 33, 35. Prisms are named from their bases; as triangular, square, etc. A prism whose base centers lie in a JL to its bases is a right prism. Figs. 31-33. All others are oblique. Fig. 35. A regular prism is a right prism whose bases are regular polygons. Its lateral faces are equal rectangles. A cube is a regular prism whose six faces are equal squares. Fig. 32. A right section of a prism is a section ± to its lateral edges. A truncated prism is the portion of a prism included between a base and a section oblique to the base. Fig. 34. (k) Pyhajiids. a pyramid is a polyhedron bounded by a pol}'gon called its base, and three or more As called its lateral faces, meeting in a common pt. called the vertex of the pyramid. The intersection of its lateral faces are its lateral edges; the others are its base edges. Figs. 36, 37. Pyramids are named from their bases; as triangular, square, etc. A pjTamid whose verte.x lies in a ± to the center of its base is a right pyramid. Fig. 36. All others are oblique. Fig. 37. A regular pyramid is a right pyramid whose base is a regular polygon. Its lateral faces are equal As. The altitude of a lateral face of a regular pyramid is the slant height of the pyramid. C-B, Fig. 36. A truncated pyramid is the portion of a pyramid included between the base and any plane section. Figs. 38 (a) and (b ) . When the section is 1 1 to the base, the included portion is a frust um of a pyramid. Fig. 38(a). 8 ESSENTIALS OF MECHANICAL DRAFTING (1) Cylinders. A cylinder is a solid bounded by a closed cylindric surface called the lateral surface, and two || plane surfaces called its bases. Figs. 39, 40, 42. A cylinder is named from its bases; as circular, elliptic, etc. The terms "right," "oblique," and "truncated" apply to a cylinder as to a prism. f B N D _^ — ""^ ^^ 'a ^ ^ ^ c Fig. 39 Fig. 40 Fig. 3e> Fig. 4 Fig. 42 A rij/ii circular cylinder may be generated by the revolution of a rectangle about one of its sides. Fig. 39. A right section of a cyUnder is a section J_ to its elements. (m) Cones. A cone is a solid bounded by a closed conic surface called the lateral surface, and a plane surface called its base. Figs. 43, 44. A cone is named from its base; as circular, elliptic, etc' The terms "right," "oblique," "truncated," and "frustum" apply to a cone as to a pyramid. A right circular cone may be generated by the revolution of a right A about one of its sides. Fig. 43. The length of an element of a right circular cone is the slant height. (n) A Sphere is a soUd bounded by a closed spheric surface every point of which is eqmdistant from a pt. within called the center. Fig. 46. A st. hne from the. center to the surface is a radius; a St. line through the center and terminated at each end by the surface is a diameter. Fig. 47 A sphere may be generated by the revolution of a semicircle about its diam. A plane section through the center is a great circle of the sphere. Any others are small circles of the sphere. Any great O divides a sphere into two equal parts called hemispheres. A spheroid (ellipsoid) is a solid which may be generated by the revolution of an ellipse about either its long or short diam. Fig. 47. INTRODUCTION 9 (o) Areas axb Voluiies. The number of times a geometric magnitude contains a given vmit of measure of the same kind is the numeric measure of the magnitude. The ratio of two magnitudes is the quotient of their numeric measures, expressed in terms of the same unit. Thus the ratio of 2" to 3" is f, or 2 : 3. The expression of the equahty of two ratios is called a proportion. As |=|. Read 2 is to 3 as 4 is to 6. The quantities compared are said to be in proportion or proportional, and are called the terms. The first and last terms are the extremes and the middle terms, the means. In any l^roportion the product of the extremes equals the product of the means; hence, if three terms are given, the fourth may be found. The area of a surface is its measure expressed in some iinit of surface, as a square inch, square foot, etc. The area (A) of a parallelogram is equal to the product of its base (b) and altitude (a) . A = ba. The area of a square is equal to the square of one of its sides (s). A = s^. The length of the side is equal to the square root of the area. s= V A. The length of the diagonal (d) is equal to the square root of 2 times the square of the side. d = r'2Xs- or d = sl 2l (,v¥=1.414). The area (A) of a A is equal to k the product of its base (b) and altitude (a). A= J ba. The altitude (a) of an equilateral A is equal to k the product of the side (s) and the square root of 3. a = ^. (V''3"= 1.732). The hjfpotenuse (h) of a right A is equal to the square root of the sum of the squares of the other two sides (b) and (c). h = T b" + c-. The area of a trapezoid is equal to J the product of its altitude (a) and the sum of its bases (b) and (b'). A=ia(b+b'). Any polygon may be divided into As. The area of the polygon is equal to the sum of the areas of its As. The length of the circumference (c) of a O is equal (very nearly) to 3.1416 times the lengtli of the diameter (d). 3.1416 is designated by the Greek letter (pi), c = Trd or 27rr. The area of a O is equal to tv times the square of its rad. (r). A = jit-. The volume of a solid is its measiu'e expressed in some unit of volume, as a cubic inch, cubic foot, etc. The volume (\) of a cube is equal to the cube of its edge (s) or third power of its dimension. V = s». The volume of a prism, or cvhnder, is equal to the product of its base (b) and altitude (a). V = ba. The lateral area (1) of a prism, or cylinder, is equal to the product of a lateral edge or element (e), and the perimeter (p) or circumference (c), of a right section. 1 = ep, or 1 = ec. The volume of a pjTamid, or cone, is equal to | the product of its base and altitude. V = Jba. The lateral area of a regular pjTamid, or right circular cone, is equal to J the product of its slant height (s) and the perimeter, or circumference, of its base. 1 = |sp, or 1 = Jsc = iiTs. The area of the surface (s) of a sphere is equal to the product of the circumference of a great O and its diam. (d), that is, 27rrd, and is equivalent to the area of 4 great Qs. s = 45n-2. The volume of a sphere is equal to the product of ^ of its rad. and the area of its surface, that is, ir X 4:^2 = J7rr3. V = i^n-s or -J7rd3. 10 ESSENTIALS OF MECHANICAL DRAFTING Decimal Equivalents of Fractions of an Inch W irV 1 3 Its A A 1 iV 8 A A 3 1 1 6T 16 7 1 3 1 M 4 1 7 fi H le 21 "6T 3 li i if 25 7, If lb 1 5 li 1 31 ¥T i 015625 .03125 .046875 .0625 .078125 .09375 .109375 .125 .140625 .15625 .171875 .1875 .203125 .21875 .234375 .25 .265625 .28125 .296875 .3125 .328125 .34375 .359375 .375 .390625 .40625 .421875 .4375 .453125 .46875 .484375 .5 T¥ 1 3 1 6 i_5 1 S" 1 7 T2 3 3 35 ¥¥ M 37 If li H If If 4.5 H 25 49 ¥¥ li li 53 II If II 59 li li II .515625 .53125 .546875 .5625 .578125 .59375 ,609375 .625 .640625 .65625 •671875 .6875 .703125 .71875 .734375 .75 .765625 .78125 .796875 .8125 .828125 .84375 .859375 .875 .890625 .90625 .921875 .9375 .953125 .96875 .984375 INTRODUCTION 11 3. General Instructions for Working Out Problems. In the solution of graphic problems clear mental images of the forms to be represented, definite ideas of the purpose of the drawings, and the orderly application of appropriate pi'inciples and working methods, are fundamental. Habits of accuracy, thoroughness, and neatness should be cultivated from the outset as the essentials of good workmanship. The value of the work lies not in the completed drawings, but in the knowledge and ability acquired by the student through his own efforts in solving problems and in striving to attain mastery of the principles and methods which will enable him to represent any form whether real or imaginary. Upon the presentation of a problem the student should first form a definite idea of what is required, the conditions and principles involved, and the method of construction to be employed. He should then start the drawing, beginning with the parts that are knoivn or given. These will suggest other parts. It is not necessary to imagine the complete solution before beginning to draw. Test and correct each stage of the solution before proceeding with the next. Upon the completion the student should make a brief notebook summary for future reference. For use of instruments and materials, see Chap. II. For general instructions in penciling and finish rendering, see Chap. III. (a) Geometric Construction Sheets. Chap. IV. Unless otherwise directed solve problems by practical methods whenever such are known or given. Within reasonable limits prove or test the constructioiis by geometric methods, using compasses and one triangle only; and retain all working lines. Make constructions as large as practicable to insure greater accuracy, and extend the working lines beyond the pts. as shown in the figures. To avoid impairing the accuracy of the constructions, it is recommended that they be left in pencil. (b) Orthographic Projection Sheets. Chaps. V-VIII. Locate first the traces of the planes, or equivalent lines of reference (base or C. Ls.). Then proceed to determine the views, beginning in general with that view or part about which most is known; in the case of an object, for example, the view which will show the largest number of the lines and surfaces of the object in their exact form and dimension. Instead of completing the views separately it is usually desirable to carry along the views of corresponding parts at the same time. Unless other- wise directed use practical rather than geometric methods for the constructions, and obtain the solutions by the aid of dimensioned freehand sketches from objects. See Art. 104. ' To determine the positions of the lines and surfaces more readily, number or letter each pt. lightly as located, marking the corresponding views to indicate that they represent the same pt. of the object. Iil locating positions of centers or other important pts., small freehand Os may be penciled about them. When the drawing has been approved, these notation marks and Os should be erased. 12 ESSENTIALS OF MECHANICAL DRAFTING (c) Isometric and Oblique Projection Sheets. Chap. IX. Locate first the axes, or reference lines, then locate the main lines or surfaces of the object. Gradually work from these to the more important details, then proceed to the smaller details. Obtain the dimensions from objects; exact orthographic views; working drawings; or from dimensioned sketches in orthographic, isometric, or oblique projection, as may be directed. (d) Working Drawings. Chap. X. Locate first the main C.Ls. of the views according to the layout sketch (Art. 105 (a)), then locate the main lines or surfaces of the object, as indicated in Fig. 209 (a), beginning with the view about which most is known, as in Art. (b). Proceed in like manner with the more important modifications or details of the main body (Fig. 209 (b)) and from these work down to the smaller details. Next, add the dimension and extension lines, arrowheads, figures, and lettering. Fig. 209 (c), (d). Finally indicate the section lines. Do not try to complete the views separately. Within practical limits those of corresponding parts should be carried along at the same time. Use practical methods for the con- structions whenever adequate. , Obtain the working data from dimensioned sketches made from objects, or as directed. For methods of representing the commoner forms of bolts, screws, etc., see Chap. XL For tables of sizes, and the construction and proportions of other machine details, see manufacturers' catalogs, books on machine design, and engineers' handbooks. For sizes, etc., of details of wood construction, see books on joinery, etc. CHAPTER II INSTRUMENTS, MATERIALS, AND THEIR USE 4. List of Equipment. The instruments and materials ordinarily needed are enumerated in the following. A good equipment is indispensable to good work. If not furnished by the school, its selection should be intrusted to an experienced draftsman. Set of Instruments, consisting of Compasses 5|" (joint in both legs preferred), with Lead Holder, Pen, and Lengthening Bar attachments; Ruling Pen, medium; Dividers, 5"; Bow Pencil; Bow Pen; and Bow Dividers. Leads, grade 4H, for compasses and bow pencil. Drawing Board, IS" -x 24", is suitable for most work. T-square, fixed head, blade sUghtlj' longer than board. Triangles, one 45°, 7", and one 30° x 60°, 9" (celluloid preferred). Scale, 12", flat, both edges divided into full size inches, halves, 4ths, 8ths, and 16ths, and the first inch, or first and last, into 32ds. Or, a 12" architect's triangular scale: one edge divided into full size inches and fractions, to 16ths, and the others to scales of 3", If", 1", f", i", f", J", ^", I", and 35" to 1 foot. (The flat scale is recommended if scales of full, half, quarter, eighth, and sixteenth sizes only are to be used.) Curve Rulers, one or two, similar to those shown in Figs. 66, 67 (celluloid preferred). Drawing Pencils, one hard, grade 4H, 5H, or 6H, and one medium hard, grade H or 2H. Erasers, one soft rubber for pencil erasing and one hard rubber for ink erasing. Needle-point, a fine needle inserted in a wooden handle about 3|" long, for use in fixing pts. Tacks, 1 oz. copper, or small thumb-tacks, for fastening paper to board. Tack-driver. A small screw-driver ground to a thin edge and sKghth' bent at the end will be suitable. Knife, for pencil sharpening (one with broad blade preferred). Lead-pointer, a strip of No. J sandpaper or No. 120 emery cloth, about f " x 4", glued upon a fiat strip of wood. Penholder and Writing Pens, for lettering, etc. (Tapering handle with cork grip preferred.) Pens should be medium and coarse. Penwiper, a piece of cloth or wash leather. Drawing Ink, one bottle waterproof black; wTiting ink is unsuitable. Drawing Paper, hard and tough, with a surface not easily roughened by erasures. For sketches a softer paper is desirable. Avoid rolling the paper. Sizes in common use are 18" x 24", 12" x 18" and 9" x 12"; also 1.5" x 22", and 11" x 15". PortfoUo or Binding Cover, to hold paper and drawings. Notebook Cover, with loose sheets, about 6" x 9". Cloths, a white cotton cloth about 14" x 20", hemmed, to place materials upon; a piece of cloth or wash leather for wiping instruments; and a small dusting cloth. Box, to hold the pencils, erasers, cloths, and other small articles. 5. Care and Arrangement of Equipment. Next in importance to having good instruments and materials is the necessity of handling them properly, keeping them clean, in good working condition, and in convenient, orderly arrange- ment. 14 ESSENTIALS OF MECHANICAL DRAFTING Keep the fingers clean, and the table and materials free from dust. The T-square and triangles especially will need frequent wiping. A paper or cloth may be fastened over a part of the drawing to protect it while working on other parts. Keep instrument case, box, and locker closed; and the ink bottle in the stand, with stopper in place. The instruments must always be carefully wiped, and properly replaced in the case when through working. Never permit ink to dry in the pens, or upon any part of the equipment. Take every precaution to insure against injury to the points, shanks, etc., of the instruments; and the edges of the scale, board, T-square, triangles, and curve rulers. Adjustments of joint-screws in compasses and dividers should be made by the instructor, unless otherwise directed. Inaccuracy, injury, or loss of any part of the equipment should be reported immediately. The table should be so placed that the light comes from the left* and, if possible, adjusted to such a height that the student may stand while at work. 6. Drawing Board. The drawing board provides a flat surface upon which to secure the paper, and a st. edge against which to guide the head of the T-square. Either short edge may be selected for this purpose, but the board must be placed so that this edge will be at the left*, and no other used as a guiding edge in elemen- tary work. See Fig. 48. For convenience in working and to insure firmness and freedom in the use of T-square and triangles, the paper should be placed about 3" from the left* and lower edges of the board. Square the paper with the board by lining up one of its edges against the ruling edge of the T-square. Art. 7. Insert tacks about A" from each corner, pressing the paper as flat as possible with the hand; then with thumb or tack-driver force the heads flush with the paper, so that they will not interfere with the use of the T-square, etc. Always remove T-square from board when using tack-driver, and avoid marring the board. After paper is fastened, the board may be inclined by means of a book or block under its farther edge, so that all parts of the drawing may be more nearly at the same distance from the eyes. Keep paper secured to the board until drawing is completed. If removed before, it should be refitted by inserting a tack in one of its upper corners and lining up the most important horizontal of the drawing against the T-square. 7. T-square. The T-square is used with its head against the left edge of the board. Fig. 48. The upper edge of the blade in this position is the ruling *In this and similar working directions, left-handed students may read "right" in place of "left."^ Fig 48 INSTRUMENTS, MATERI.\LS, AND THEIR USE 15 Fig 49 edge for all hor. lines, and guide for the triangles when drawing lines at certain Z s. Never use the lower edge of the blade as this would lead to errors difficult to trace. As the Z of the head and blade in different T-squares is apt to vary, the same T- square should be used until the drawing is com- pleted. (a) To DRAW A HORIZONTAL THROUGH A GIVEX POINT. Slide the head along the guiding edge of board until the ruling edge passes through the given pt. For preparation and use of pencil, see Art. 9. Move T-square by the head only, and while drawing the line keep head and blade securely in position by shding the fingers of the left hand along the blade and pressing towards the right. Fig. 48. 8. Triangles. The triangles (Fig. 49) are used as rulers for Hues at Zs with the hor. direction. The ruHng edges of the 4-5° triangle form an Z of 90° and two of 45° each. Those of the 30° x 60° triangle form an Z of 90°, one of 30°, and one of 60°. (a) To DRAW A LINE AT AN ANGLE OF 30°, 60°, 45°, OR 90°. Place one edge of the correspond- ing 30°, 60°, 45°, or 90° Z of the triangle against the T-square and guide the pencil along the other edge of the Z. Hold T-square and triangle firmly in position with left hand. Fig. 50. When the Une to be drawn is longer than the edge, shde the T-square until ^^■^X-^''''^ F'lG 50 the required length is obtained. Avoid ruling near the corners, as thej' are apt to be rounded; thus, in drawing a line, say at 90° with a given hor., place the T-square a httle below the hor. as shown. Never rule against the inner edges. (b) To DRAW A LINE AT AN ANGLE OF 75° OR 15°. Combine the triangles and T-square as in Fig. 51. The 30° Z added to one of 45° gives an Z of 75°. By reversing the upper triangle as shown by dotted hues, one of its edges will be at 15°. Bj' placing the 60° angle against one of 45°, a line at 75° or 15° in the opposite direction may be obtained. (c) To DRAW A PARALLEL TO ANY GIVEN LINE.* Combine the triangles with an edge of one (see dotted triangle. Fig. 52) to coincide with the nATien the given line is hor.. or at .30°, 60°, 4.5°, 90°, 7.5°, or 1.5° with the hor. du-ection, the T-square, or triangle and T-square combinations, would ordinarily be used. 16 ESSENTIALS OF MECHANICAL DRAFTING given line A-B; tlien, liolding the second securely, slide the first until the edge which originally co- incided with A-B is in the required position C-D. (d) To DRAW A LINE AT 30°, 60°, 45°, 90°, 75°, OR 15° WITH ANY GIVEN LINE.* For 30°, 60°, 45°, or 90°, Fig. 53. Combine the triangles with an edge of one || to the given line A-B, as in (c). Then holding this triangle securely, shift the second, placing it against the || edge of the first, so that one of its edges makes the re^ quired Z with A-B. For 75° or 15°, Fig. 54. It is evident that after the second triangle has been placed against the || edge of the first, the first in turn must be shifted to give the recjuired Z . 9. Pencils and Writing Pens. The hard pencil is used for ruling lines against the T-square, triangles, and curve rulers; the medium pencil for all freehand penciling, lettering, etc., and the pens for inking freehand lines, lettering, etc. (a) To SHARPEN THE PENCILS. Sharpen the end not bearing the grade stamp. Hold the inner side of both wrists firmly against the body, with the knife blade nearly flat against the upper side of the pencil and its cutting edge to the right. Cut with an outward wrist movement, removing the wood in long thin shavings, tapering it evenly down to but not cutting the lead, and so that about f " of the latter is exposed. Fig. 55. This method gives greater control of the knife and lessens the liability of soiling the fingers. The lead should be pointed by means of a lead-pointer. Hold pointer in the left hand, away from the table, and the pencil so that the entire length of lead will be tapered. Taper the lead of the freehand pencil to a fairly sharp pt. by rubbing it back and forth, at the same time rolling the pencil between thumb and fingers. Fig. (a). The lead of the ruling pencil should be tapered to a shai-p conic pt., or to a wedge pt., as the in- structor directs. The wedge pt. (Fig. b) retains its sharpness longer and fits more closely against the ruling edge. It is obtained by first tapering the lead slightly as for the freehand pt., and then rubbing the lead upon opposite sides to form a short, sharp edge at the end. Rub the leads frequently to keep them in proper condition. *When the given line is hor., or at 30°, 60°, 45°, 90°, 75°, or 15° with the hor. direction, the T-square, or triangle and T-square combinations, would ordinarily be used. Fig. 55 INSTRUMENTS, MATERIALS. AND THEIR USE 17 Fig 56 (b) Use of the Freehand Pencil. Hold pencil lightly, about I5 inches from the point. The relation of the pencil to the line, and the direction of the stroke, should usually be as in- dicated in Fig. 56. Face the paper squarely and avoid turning it while drawing. All lines should be drawn with as few strokes as possible. Draw lightly at first and correct any portion by drawing a second line before erasing. Finally strengthen the line to make it clear and even. Practice in arm and finger movements before drawing will aid in acquiring necessary freedom. (c) Use of the Ruling Pencil. Hold pen- cil as nearly upright as possible, with flat side of lead against the ruling edge. Steady pencil in this position by resting the tip of the third or fourth finger upon the ruler. Fig. 48. This per- mits the edge of the lead to wear evenly and give tmifoi-m lines. Bear lightly, — much pressure will dull the lead too rapidly, make uneven lines, or form depressions which cannot be erased. The result should be very fine, clear lines. The pencil should always be moved from left to right, the student turning his body or the board when necessary, that he maj' face the ruling edge squarely and do this more readily. Thus in drawing verts, face to the left, and draw awaj^ from the T-square. (See Fig. 50.) If ruled against a right-hand edge, the pencil is apt to glide away from the ruler and cause a break in the line. Watch the point constantlj^ as it is moved along. Never rule a line in a shadow, and never rule backward over a line. In drawing parallels move the ruxer from one position to the other in such manner that the preceding line and space will not be covered. Thus, in drawing II hors., move T-square downwards; in || verts., move triangle to the right. (d) Use of Writing Pens. Handle the pen the same as freehand pencil. Use little ink and aim to secure uniform, even lines. Exercise special care in work- ing over lines to avoid injuring the paper. 10. Needle*point. The needle-point is used to set off distances from the scale, and to fix jjts. of Kne intersections which might otherwise be erased or lost. Hold needle upright and make the smallest puncture that can be seen. 11. Scales. The scale is used for setting off measurements. It must never IS ESSENTIALS OF MECHANICAL DRAFTING be used as a ruler for drawing lines. The scales shown in Fig. 57 are graduated as described in Art. 4. (a) When the drawing is made so that each inch or fraction of an inch of measurement upon it is equal to the corresponding measurement on the object itself, the drawing is said to he full size or to a scale of 12" to afoot. Fig. 58 When for convenience or necessity the drawing is made smaller or larger than full size each unit of measurement is made smaller or larger in proportion, — • thus when drawn, say Jialf size or 6" to afoot, each half inch on the drawing repre- sents one inch on the object; each quarter inch a half, and so on. Full size measurements are obtained from a scale graduated to full size inches and fractions. Half size measurements are usually obtained from a full size scale by simply reading each half inch on the scale as one inch, each quarter inch as a half, etc. Quarter, eighth, and sixteenth size can be obtained in like manner, to 1 3" or froili scales graduated to 3", 1|", and ft. respectively. In scales graduated to feet and inches the first unit, when large enough, is divided to represent inches and fractions. For example, in the scale of IJ" to 1 ft. the first unit is divided into 12 eighth-inch parts to represent inches, and each of these subdivided to repre- sent halves and 4ths. Fig. 57. (b). When lOths, 20ths, 30ths, etc., of an inch are required, an engineer's or decimal scale is used. Any desired scale may be drawn by division of a line into the required pro- portional parts. See Art. 30. For method of determining the size or scale to be used, see Art. 100(c). (b) To SET OFF A MEASUREMENT (aS Say 2\"). Apply the scale directly to the line, with the "O" division exactly at the end of the line, and the needle-point at the 2\" division. To set off, say 2 ft. and 2|", place the di- vision representing 2|" at one end of the line and a pt. at that representing 2 ft. Never transfer measurements from the scale with compasses or dividers. Successive measurements on a INSTRUMENTS, MATERIALS, AND THEIR USE 19 line should, so far as possible, be set off without shifting the scale, so that an error in one distance may not affect all. 12. Protractor. A Protractor (Fig. 58) is a scale used in laying off and measuring Z s. Its measuring edge is graduated to degrees and fractions (usually to half degrees), and the degrees numbered to read from to 180° in both direc- tions. (a) To DRAW A LINE AT ANY GIVEN ANGLE WITH A GIVEN LINE (say 40°). Place the protractor with its 180° line 0-0 against the given line A-B and center C at the given vertex A. Now place a pt., D, at the 40° mark. Remove the protractor and draw A-D, the required line. 13. Compasses. The compass set (Fig. 59) is used for drawing circular ■curves. The needle should first be adjusted as follows: — Release clamp screw A, and remove the lead holder. Insert the pe7i in place of the latter, and clamp securely. Then set the needle so that the point of its shouldered end will be even with the pen point. Once set, the needle should not be changed. The lead only will need resetting, as it wears away. (a) To PREPARE THE COMPASSES FOR PENCILING. Place a 4H compass lead, about 1" long, in the holder, and taper it as directed for the ruling pencil. Refit the holder and adjust the lead to the length of the needle, with its edge so placed that a fine, even line will result when the compasses are revolved. (b) To DRAW A CIRCLE. Open the compasses and adjust the legs at the joints so that both lead and needle will be at right Z s to the paper while drawing. This is necessaiy to prevent the lead from wearing un- evenly and the needle from digging into the paper. The puncture should be barely visible. IG.DU Hold the compasses by the handle only, with thumb and first two fingers (see Fig. 60) and always revolve it around to the right (clockwise). Bearing lightly and evenly upon the lead point, draw the curve with one continuous motion, stopping exactly at the end of the revolution to avoid widening the line. In drawing a O or arc of given rad., first set off the rad. upon a C.L., and, in placing the compass point at the center, steady the needle with a finger of the left hand. Changes of rad. should be made, so far as possible, with the right hand only, and care taken not to enlarge the center. (c) To INK CIRCLES AND CIRCULAR ARCS. Insert the pen and clamp it securely. Clean, fill, and set the pen, as directed in Art. 17(a). Open compasses to the rad. of the penciled curve and adjust the legs, as directed in (b). To give clean-cut lines, both blades must bear evenly upon the paper. The directions for penciHng apply also to inking. Before placing the pen point upon the curve, the compasses should first be revolved over the line, in space, to make sure that the inked line will pass exactly through the desired pts. Errors caused by enlarge- ment of centers may thus be avoided. Do not go over a line a second time. 20 ESSENTIALS OF MECHANICAL DRAFTINCx Fig. 6 Fig 62 (d) Lengthening Bar. When the rad. is too great to admit of placing the points _L to the paper, the lead or pen leg should be extended by means of the lengthening bar. In this case, the needle leg may be steadied with the left hand and the drawing point moved with the other, care being taken not to change the rad. 14. Bow Compasses. The boio pencil (Fig. 61) and bow pen (Fig- 62) are used for drawing small Os and arcs for which the large com- passes are not convenient. Do not use them for radii over f". The directions for prepar-' ing and using the bow compasses are much the same as for the large compasses. The needle must project slightly beyond the pen or lead point, and the lead be tapered more neai-ty to a pt. To enable the points to be brought close together, the needle is generally flattened on its inner side. The rad. is adjusted by turning the thumb-nut on the connecting bar. Before turning, spring the points together so that the wear of the screw thread may be lessened and the adjustment made more readily. 15. Dividers. The dividers or spacers (Fig. 63) are used for transferring measurements fr'om one part of a drawing to another, and for setting off equal distances on a line when they cannot readily be laid off by means of the scale. I Handle the dividers in the same general way as the compasses. In some dividers one leg is furnished with a hairspring and nut by means of which this leg may be moved for slight changes of adjustment. (a) To DIVIDE A STRAIGHT LINE OR A CIRCULAR CURVE INTO ANY NUMBER OF EQUAL PARTS (say 3). Opeii the dividers to' a dis- tance equal (by ej'e) to ^ of the line to be divided, place one of its points upon the end of the line and revolve the dividers until the other point is exactly on the line. Proceed in this manner, revolving alternately in opposite directions, until the distance taken has been set off three times. ' Never remove both points at the same time. If the -distance taken does not apply exactly it must be increased or diminished by an amount equal to i of the difference, and the trial repeated until the line is equally divided. No pts. should be made visible until the divisions have been verified as correct. 16. Bow Dividers. Fig. 64. These are used for small distances, and in the same general manner as the large dividers. The points are adjusted as directed for bow compasses, Art. 14. 17. Ruling Pen. The ruling pen (Fig. 65) is used for inking all , lines other than circular curves. (a) To FILL AND CLEAN THE PEN. Before filling the pen, moisten a folded end of the penwiper and draw it gently between the blades. Fig. 63 When clean and dry bring the blades together at the point by means INSTRUMENTS, MATERIALS, AND THEIR USE 21 Fig. 64 Fig. 65 of the thumb-screw; then holding the pen upright, not over the drawing, insert the ink between the blades with the filler. Do not fill above J" from the point, otherwise the ink will flow out too freely. See also that there is no ink on the outside of the point, as this will widen the line, make it ragged, or cause a blot. Replace the stopper immediately to prevent the ink from thickening. Having filled the pen, set the blades to the re- quired width of line. Always try the pen on a piece of waste paper before using it on the drawing. To insure its flowing freely, the amount of ink in the pen must be kept as nearly as possible the same. Avoid having to piece out a line. As the ink dries rapidly, the pen must be cleaned and refilled frequently. In doing this, it is not necessary to open the blades. The setting should remain unchanged until all lines of the same width are inked. If the pen fails to work, it should be sharpened by an experienced person. Never put the pen aside without care- fully cleaning it. (b) To RULE A LINE. Hold and steady the pen as directed for the pencil (Art. 9 (c)), the first finger resting on the flat side of the pen above the thumb-screw and the second against the edges of the blades as shown in Fig. 50. Adjustments for width are made with thumb and second finger of the same hand. Guard against getting the pen point too close to the ruhng edge by placing the latter slightly away from the Hne, as shown. A slight downward pressure only should be necessary, but the points of both blades must bear evenly upon the paper to give a clean-cut Une. Bear lightly against the ruling edge to prevent varying the width of the line. Always steady the hand, and move the pen from left to right as directed for penciling. Just before reaching the end of the line stop the arm movement and complete the line with a finger movement, then lift the pen immediately and move the ruling edge away from the line. In ruling curves, turn the pen gradually so that the blades will not be at an Z with the ruhng edge. Art. 18. Never use the ruHng pen freehand. 18. Curve Rulers. These are used for ruhng curves that cannot be drawn with the compasses. They are made in various shapes and sizes. Two of the most serviceable are shown in Figs. 66, 67. (a) To HTJLE A CURVE, ABC. Fig. 67. First sketch the curve hghtly with freehand pencil, through previously determined pts. Now find, by trial, a portion of the ruler which will fit as much of the curve as can be ruled conveniently at onetime, as A-B, and true up that part by tracing over it with the ruling pencil. Fig. 66 22 ESSENTIALS OF MECHANICAL DRAFTING Match succeeding parts in the same manner, making the edge fit over a portion of each preceding part to insure an even, unbroken line. The freehand pencihng should not be omitted, as the tendency would be to make the ruled line curve out too much or too little. Having trued the line, rub the soft rubber lightly over it. In inking the curve, use the ruling pen as described in Art. 17(b). In ruling curves symmetrical about one or more axes, as ellipses, helices, etc., the portion of the ruler used for one part should be noted and used for corresponding parts. Sharp turns at ends of axes should first be drawn by means of compasses, the centers being taken in the axes and care taken to use the proper rad. and length of arc. (b) Non-circular curves may often be approximated throughout by tangent arcs. Thus, in inking the curve shown by the fine line in Fig. 68, beginning say at A, determine by trial the center and rad. of as much of an arc as will practically coincide with the curve. Ink this arc; then, changing the center and rad., ink the next portion; note that the centers must be on the line through the pt. of tangency. Fig. 67 Fig. 68 IQ. Erasers. (a) The soft rubber is used for pencil erasing and paper cleaning. Keep paper as clean as possible from the start. (See Art. 5.) The softest rubber is liable to roughen the paper, making it difficult to keep clean and to obtain sharp lines. In removing a line, rub lengthwise. Avoid much pressure and always remove dust before proceeding with drawing. When a drawing is finished in ink, the eraser may, if necessary, be passed lightly over the entire surface, care being taken to avoid dulling the lines. (b) The hard rubber is used for ink erasing. The part to be remoyed should first be allowed to dry. Care must be taken not to injure the surface. Never use a knife. If the surface is roughened by erasing, smooth it as well as possible with the finger nail. A thin card or piece of celluloid with narrow openings can be used to protect adjacent parts when erasing. CHAPTER III PENCILING AND FINISH RENDERING 20. Layout of the Sheet, (a) Margin Lines. To improve the general appearance of the drawing and to insure keeping all lines and figures a safe distance from the edges of the sheet, it is customary to rule border lines with uniform marginal spaces at top, bottom, and sides. See Fig. 69. Assuming the paper to be ll"xlo", the required border 10"xl3|", and marginal spaces §", proceed to lay out the border and trim lines as follows: Having tacked the paper to the board (Art.G) setoff two pts. near the left edge, I" and 10|" respectively, above the lower edge, Art. 11. Through these pts. draw Kght horizontals across the sheet, Art. 7. On the lower hor., set off a pt. I" from the left edge; |" from this place a second; Idh" from the second, a third; and |" from that a fourth. Through these pts., draw the 90° lines (verticals), Art. 8. Test distances between verts, at the top and hors. at the right with the scale. Fig. 69 When the sheet is completed and removed from the board, the J" strips con- taining the tack holes are to be cut off, thus leaving the sheet 11" x 14V'. (b) Loc.\TioN OF Name, Title, etc. On elementary drawings these maj' be lettered as shown. For titles, etc., on working drawings, see Art. 102. For instructions in lettering, see Art. 25. 21. Constructive Stage of the Drawing. All lines should be penciled first in uniform, fine full lines, as indicated in Fig. 209(a), (b), (c). Then, to avoid errors in the finishing stage (Art. 22), lines representing edges and outlines of the object should be gone over with a slightlj' firmer pressui-e, and the hidden parts 23 24 ESSENTIALS OF MECHANICAL DRAFTING dashed as in Fig.(d). The general order of pencihng different kinds of drawings is indicated in Art. 3. To avoid the necessity of piecing out, make the hnes first of indefinite length. Be sure that all measurements are set off upon definite lines and that lines intended to pass through particular pts. actually do so. No part of the penciling should be slighted. Inaccuracies can seldom be corrected in the process of finishing. Aim to do no erasing until the penciling is completed. To secure greater accuracy and economy of time, similar operations should be grouped. Thus, draw all lines that can be ruled with the T-square and triangles in one position at the same time, and ||s of one set before commencing those o£ another. When using the scale or dividers, set off all distances possible at once. When using the compasses work the circular curves in the same way, drawing those having the same rad. at one setting of the instrument, etc. As st. lines can be drawn tangent to curves more accurately than the reverse, pencil the curves first whenever possible. To enable "the method of procedure to be readily followed, all lines used in making the drawing should be left upon the sheet until it has been approved. 22. Finishing Stage of the Drawing. For greater distinctness and per- manence the drawing is usually lined-in or finished with ink, either by going over the lines on the original or by making a tracing as in Art. 106. Drawings not intended for continued use or of which no copies are needed are often finished in pencil. Do not commence the finishing stage until the constructive stage is completed and approved. See that the drawing surface is free from dust. (a) Inking. Be careful to make sharp, even lines, and see that all lines begin and end exactly where it is intended that they should. To prevent lines running together at their intersection, see that the first is thoroughly dry before inking the second. Do not use a blotter. (b) Finishing in Pencil. If the drawing is to be finished in pencil, the aim should be to secure as nearly as possible the accuracy and distinctness of an inked drawing. The medium pencil should be used, at least for strengthening object lines. In finishing dashed lines it is not necessary to era,se lines of the constructive stage between dashes as they will not be prominent if the finish lines are properly emphasized. (c) Line Conventions. The different purposes of the lines of the drawing are indicated by varying their character, width, or color. The conventions shown in Fig. 70 are commonly used on drawings finished wholly in black ink or in pencil, for the purposes stated below. They are suitable in width, length of dash, etc., for ordinary drawings. A — Visible lines of objects in all required views* and edges in developments. B — Hidden lines of objects in all required views. As a rule end dashes should touch the limiting lines. C — Visible lines of objects in auxiliary views used in determining required views. When aux. views show visible lines only, they may be finished as construction lines. *When shadow or shade lines are used, shade the curved edges as each is lined-in. (See Art. 23(c).) St. shadow lines are left in pencil until all but the border is finished. PENCILING AND FINISH RENDERING 25 D — Hidden lines of objects in auxiliary views used in determining required views. E — Traces of projection, section, and base planes; center lines, and axes. Dashes of section traces, center lines and axes should extend about |" beyond the view or part on which they are drawn. F — Construction {loorking) lines required to shoiu method of construction; pro- jectors, and extension lines. Dashes of extension lines should not touch lines of views, and should extend about -nr" beyond arrowheads of dimension lines. G — Dimension lines, and pointers. H — Break lines. Same as edges, etc., but should be drawn freehand. I — Arroivheads, figures, notes, titles, etc. These should be penciled freehand, and in inked drawings always finished in black. See Fig. 74. J — Section lining. Same as C, or as in Fig. 187. K — Line shading. As in Art. 24. L — Straight shadoiv and shade lines of all required views, and border lines. Red ink. In inked drawings, the use of full red lines for all Hues under C, E, F, and G, and dashed red for those under D is more economical of time and makes the required views more prominent. Width 2 B C D E • F I I " G h 3t 2 L Fig. 70 (d) Order of Finishing. It is desirable to complete all lines and parts of similar character before proceeding with those of another, in the general order indicated in Art.(c), in observing which, similar operations should be grouped as in the following : — Circles and arcs, beginning with the smallest; non-circular curves; horizontals, beginning with those at the top ; verticals, beginning with those at the left ; obliques, beginning with those obtainable by T-square and triangle combinations. Do not use a triangle alone, unless necessary. In finishing break lines, arrowheads, figures, and notes, etc., work from upper part of sheet downward. 23. Shadow Lining. In practical drawings, visible edges and outlines of objects are generally indicated by full lines of uniform width. It is sometimes desirable, however, to finish certain lines wider than the others for the purpose of giving an appearance of relief to the drawing and to indicate the relative positions of the surfaces more clearly. 26 ESSENTIALS OF MECHANICAL DRAFTING conventional shadow lines Fig. 71 PENCILING AND FINISH RENDERING 27 (a) If a cube placed as in Fig. 71(a) be lighted by || rays coming over the left shoulder, in the direction of diagonal A-B, it is evident that the upper, left, and front faces will be in the light, the others in the shade. Lines separating light from dark surfaces are called shade lines. The visible shade lines of the cube will, therefore, be the lower and right lines in the front view, and the upper and right lines in the top view, and may be indicated by broad lines as shown. Assuming the same direction for the light, the visible shade lines in all rectangular objects and parts, whose axes are || or J. to the planes of the views, would have the same general locations as in the cube. In such cases the shade lines can usually be determined by eye. The determination of the actual shade lines in all cases, however, would involve considerable time and labor, and their locations would frequently be such as to complicate the drawing rather than aid in explaining the form of the object; hence it is customary to disregard the actual shades and shadows and to apply the broad lines in such manner that they will indicate edges separating visible from hidden surfaces only, and so as to produce an effect of narrow shadows cast by the object. It is convenient to regard these lines as shadow lines rather than as shade lines. To further simplify the application, most draftsmen shade or shadow-line all views the same as the front. The aid given by such lines in reading a drawing will be apparent from the figures. The directions for this conventional method may be stated as follows: — (b) Shadow-line the lower and right lines of intersection between visible and hidden surfaces in all views, regardless of the actual shades or shadows of the object. This includes both sharp and slightly rounded edges. Lines not repre- senting such edges are generally shadow-lined only when surfaces extend back from them at right Zs to the plane of the drawing, in order to indicate more clearly that they do not represent edges. A sectional view is generally shadow- lined just as if the portion shown were complete. Lines representing broken surfaces are preferably not shadow-lined. Never shadow-line the line of inter- section between visible surfaces, nor dashed lines. (c) To SHADOW-LINE A DRAWING. In determining the shadow lines remember that the rays of light will be at 45° down to the right in all views and that all views are shadow-lined in the same manner. Limiting rays may be penciled as shown. Each edge to be shadow-lined should be indicated by a mark upon the line before inking is begun. Drawings are never shadow-lined in pencil. Straight Lines. A st. shadow line should be ruled the required width (3) at one stroke of the ruling pen, and its width added on the outside of the pen- cil line. Circular Curves. First ink the curve in the usual manner (width 2). Then taking a second center below and to the right of the first, on a 45° line, at a dis- tance equal to width 3 and without change of rad. or setting of pen, draw an arc on the outside or inside of the first curve according to the required location of the shadow portion. (See Fig. 71(h).) If an uninked portion remains between the arcs, spring the instrument slightly to fill it in. 28 ESSENTIALS OF MECHANICAL DRAFTING Irregular Curves. First ink the widest portion; then with the ruling pen set to width 2, blend this carefully into the rest of the curve. 24. Line Shading. This is a conventional method of producing an effect of light and shade, corresponding to that upon the object itself by means of lines, usually of graded widths and spacing. Fig. 72. It is used on drawings intended for illustrative purposes and in cases where it is necessary to indicate the direction of certain surfaces more clearly than would be possible by the mere outline or by shadow lines. Fig 72 (a) The direction of the light is assumed as for shadow-lining (Art. 23) ; the same for all views. The right and lower sides are, therefore, the dark sides on convex surfaces and the left and upper on concave surfaces. In regular curved surfaces the darkest portion is about | of the rad. from the center. It may be accurately determined as shown in top view of Fig. (a). The spacing may be indicated in pencil, as below Fig. (a). Shade first the dark portion, beginning at about j of the rad. from the center. Use fine lines only on the light side, somewhat farther apart than on the dark, and stop at about f of the rad. from the center. On small parts shade dark side only. (See Fig.(g).) Figs, (a) and (i) PENCILING AND FINISH RENDERING 29 illustrate methods of shading fillets, beads, etc., on large drawings; Figs.(e), (f); (g), for small drawings. The shading of conic surfaces by |1 lines, as shown, is less difficult and generally more satisfactor}^ than by radial lines. When desirable to shade plane surfaces, the method in Fig.(d) may be used. 25. Lettering. The names, titles, notes, and dimensions required on drawings should be carefully lettered in even, well-proportioned, and well-spaced characters. The styles shown in Fig. 73 are those most generally used. The inclined Gothic differs from the vertical only in the slant. The ^-ertical is sometimes chosen for titles and headings, and the inclined for notes and dimen- sions. A uniform style for all lettering is more common. The capitals may be used with or without lower-case (small) letters, or with small capitals in place of the latter. Titles and headings are usually more satisfactory if all capitals are used, while notes are more easily read if composed of capitals and lower-case letters. (a) Peoportions. The character, proportions, and relations of the ele- ments composing the letters and numerals should be carefuUj' noted and fixed in mind. Examination of the vertical Gothic will show that W is the widest character, M next, and A, V, and 4 next; that I, J, and 1 are narrowest; and that the others are nearly equal in width to letter 0. For general purposes this width should be from | to | of the height. These variations are designed to overcome the appearance of inequalitj^ which would result if all were made equal when used in words. For hke reason A is extended slightly above the other letters and V below. The curves of C, G, J, 0, Q, S, U, 2, 3, 5, 6, 8, and 9 also extend slightly beyond. To avoid the effect of top-heaviness the upper part of B, C, E, G, K, R, S, X, Z, 2, 3, 5, 6, and 8 is slightly narrower than the lower, and certain parts in B, E, F, H, R, S, X, 3, 5, 6, and 8 come slightly above the middle, while in A, G, K, P, Y, 4, and 9, they come below. All curves are elliptic. In practice the variations in height and width noted above should be estimated by eye. The body of the lower-case letters should be f or | of the initial capitals. The sizes of letters and numerals generally suitable are indicated in Fig. 74. (b) Spacing. The spaces between letters vary in shape with each different combination. In order to make these spaces appear equal in size and thus avoid the effect of crowding or isolating letters, it is necessary to increase the spaces in certain cases and decrease them in others. Thus, considering the vertical capitals, the spaces of I should be considerably greater than of other letters, especially when parts of adjoining letters are 1| to it. In letters whose sides are curved as in 0, C, D, etc., it is generally necessary to decrease the space. This is also true of letters having oblique sides as in A, V, etc., and letters having a greater space at one or both sides as in L, J, P, T, etc. The simplest method of obtaining good spacing is to sketch the words lightly and study the effect. The space between words should be about Ij times the M-idth of letter O; that between sentences in line about three O's; and that between || lines of letter- ing about equal to the height of the shortest letters in either. 30 ESSENTIALS OF MECHANICAL DRAFTING (c) Penciling. In penciling titles, etc., and notes, rule only the hor. guide lines, and a few vert, or slant lines to preserve the correct positions. For method of planning a title see Art. 102(c). In dimensioning drawings estimate the heights of the numerals by eye. Draw all characters freehand, using a fairly sharp pencil point. Art. 9. Before lettering a drawing, the style to be used should be practiced until it can be done reasonably well. Unless otherwise directed, begin with the vertical capitals, observing the order given by the numbers below them; then practice word-combinations; then the figures. Place Fig. 73 close to the work and analyze each character. A convenient order for drawing the strokes is indicated by the arrows. To determine the proper proportions and spacing, first point the forms and sketch lightly the main lines. Bear in mind that the value of the practice is in the carefulness and not in the amount. Do not mix the styles. (d) Inking. Use the medium-point writing pen for small letters and figures, and the coarse pen for large letters, etc. The width of the lines should be uniform and obtained at one stroke. Observe number, order, and direction of strokes as in penciling. M||l| i*i o^mo ^/if^ S3C s:(-'^o < (J > o '— m pn. ^ry ^-^ ^^^M ^^^" ^^^^ ^^ i^T*i .^^ isL^ 1=^ ^tr>tt I X R I I I [ id I I , 1 1 1 Li o f^ 1^ o o Q U ! I m L n ^1 o U I ! I , I I ! N.I w I I ! I Pfl I I ^1 "'f u. I ^ 'A fj3 Fis.73 n TITLE AND FILING INDEX. SPACE ABOUT 2X4-5: ^ SPEED LATHE HEAD STOCK DETAILS SMITH MFG CO. NEW YORK Scale: Full Size Date: 4-7-18 Dr. A.R5. Jr. V. L. Ch. H.M. App. WFR.. S.L:IO-5 BILL OF MATERIAL. :Na NAME REQ'D MAT. REMARKS I 1 Plain Bearing 1 C.I. Patt.#B-5 T L2 St'd Hex. Bolts 5x2" 2 W.I. C.H.H'ds &Nuts U-3-*- < ^1 , 1 7 r 8 ^4. h" 2 - a * t -|'0 = -l \ Y 3 \/\\ 7\ J J. f-^ /€ ^ v_ [ ^i \^ y Z X / \ / w i\L Y A 1 u T 2 Fig. 99 Fig. 97 Fig. 100 Fig. 101 (a) When a side, A-B, is given. Fig. 100. With A or B as center, rad. A-B, describe a semicircle upon A-B extended. Divide this arc by trial (Art. 15(a)) into as many equal parts as the required polygon has sides (say 7). Through the center, B, and the second pt. of division, 2, draw B-2, which will be a second side of the polygon. Describe a O through A, B, and 2. (See Art. 40, Notes 1 and 2.) Apply A-B as a chord from 2, U, V, etc., and draw 2-U, U-V, etc., to complete the polygon. Note 1. — Vertices U, V, W, and X may also be determined by radials from B, through 3, 4, 5, and 6, as shown. Note 2. — Line B-1 would be the second side of a polygon of twice as many sides. (b) When the ciecumscribing circle is given. Fig. 101. Draw a diam. 1-2, and a tangent 3-4 X to it. With 1 as center and any rad., preferably 1-C, describe a semicircle upon 3-4. Divide this arc and draw radials as in (a). Join pts. U, V, W, etc., thus found, to complete the polygon. .P. M. Divide the circumference by trial (Art. 15 (a)), and join the pts. I, U, V, W, etc. 40 ESSENTIALS OF MECHANICAL DRAFTING \^\ ■^-^//t Vw"''**^ 8 m 3 ^ M ]--:• FrG 101 (c) When the inscribed circle is given. Fig. 101. Divide the circumference as in (b). Through these pts. of division draw radials C-W, C-X, etc., and obtain one side of the inscribed polygon, as X-W. Draw a tangent S-T || to X-W, which will be one side of the required polygon. The method of obtaining the others is evident. P. M. Divide the circumference as in P., M. of (b); then proceed as above. 56. To construct a polygon similar to a givea polygon, ABCDE, upon a given side, A'-B'. Fig. 102. (a) When A'-B' is equal to the corresponding side, A-B. Draw A'-B' equal to A-B. Locate vertices C, D', and E' by drawing intersecting arcs with cen- ters A' and B' and radii equal to the distances of C, D, and E from A and from B respectively. Second Method. Divide the given polygon into As by diagonals A-C and A-D. Construct Zs at A' and B' equal to those at A and B, and at C and D', equal to those at C and D (Art. 29) ; then the cor- responding As of the given and required figures will be similar and in this case equal. P. M. Determine the vertices by Art. 57 and join them. (b) When A'-B' is greater or less than A-B. Draw A'-B' equal to the given length and proceed as in (a), second method. Then the corresponding sides of the given and required figures will be proportional. Note 1. — It is sometimes convenient to lay off A'-B' upon A-B or A-B extended, and then draw | |s to the other sides, as indicated by the dotted lines. Note 2. — When the ratio of A'-B' to A-B is given (as say 3 to 4) the length of A'-B' may be determined by a scale of proportional lengths. Fig. (a). Draw indefinite lines A-X and A-Y at any Z with each other. On one, as A-Y, set off any length, A-1, and on the other A-2, in this case equal to f of A-1. On A-Y set off the length of A-B. Draw 2-1, and B'-B || to it; then A-B' will be equal to f of A-B and thus be the required length of A'-B'. See Art. 30, Note. 57. To plot a figure similar to a means of a base line or center line, this case an irregular curve A-F). any convenient position with respect given figure by and offsets (in Fig. 103. In to the given Fig. 103 figure draw base line X-Y. From A, F, and any num- ber of intermediate pts., draw offsets J_ to X-Y. Having thus referred a sufficient number of limiting GEOMETRIC CONSTRUCTION 41 pts., draw X'-Y' as the line of reference for the re- quired figure. Upon X'-Y' set off l'-2', l'-3', etc., equal to 1-2, 1-3, etc., and draw offsets I'-A', 2'-B', 3'-C', etc., equal to 1-A, 2-B, 3-C, etc. Through pts. A', B', C, etc., thus determined, draw the required curve. (See Art. lS(a).) Note application of princi- ple in Fig. (a), also in Fig. 132(a), (b). (a) To DRAW THE FIGURE TO AN ENLARGED OR REDUCED SCALE (say enlarged in ratio 3 to 2). The co-ordinate distances would be 1 = 1^ times those of the corresponding pts. of the given figure, and may be obtained by Art. 56(b), Note 2; or proportional dividers may be used. 58. To draw a tangent, A-B, to a circle, through a given point A. (a) When A is on the circumference. Fig. 104(a). Draw rad. C-A, and tangent A-B J_ to it, Art. 34(c). P. M. Draw A-B J. to rad. C-A by Art. 8(d). (b) When A is on the circumference and the center inaccessible. Fig. 104(b). With A as center, any rad., cut the curve at 1 and 2. Draw chord 1-2. Draw A-B || to 1-2 by Art. 36(b). P. M. Obtain chord 1-2 as above. Draw A-B |1 to it by Art. 8(c). (c) When A is outside of the circumference. Fig. 105. Join A to center C. Upon A-C draw a semicircle to determine pt. of tangency B. Draw tan- gent A-B. See Art. 34(c), Note 2. Note. — A second tangent, A-B', may be drawn, as indicated. P. M. Draw A-B tangent to the O by eye. Locate pt. of tangency by a rad. C-B ± to A-B, Art. 8(d). 59. To draw a tangent, A-B, to two given circles. Fig. 106. Draw line of centers 1-2. Subtract length of rad. of smaller O from that of the larger and draw concentric arc 3-4. Obtain tangent 2-3 by Art. 58(c). Draw rad. 1-A through 3, and make Z B 2 5 equal to Z A 14, Art. 29. Draw A-B, the required tangent. Note. — To draw a tangent, C-D, passing between the centers — add length of rad. of smaller O to that of the larger, and draw con- centric arc 6-7. Draw tangent 2-7, and rad. 1-7, locating pt. C. Make / D 2 1 equal to Z 2 1 7, thus locating D. Draw C-D. P. M. Draw tangents A-B and C-D by eye. Locate pts. of tangencj' as in P. M. of Art. 58(c). Fig. 105 42 ESSENTIALS OF MECHANICAL DRAFTING 60. To draw a circular curve of given radius, D-E, tangent to a given circular curve and to a given straight line, A-B. Fig. 107. At distance D-E from A-B draw a || 1-2. Upon any rad., C-3 extended, set off 3-4 equal to D-E. Con- centric with given curve, and with rad. C-4, draw an arc cutting 1-2 at 5, the center of the required curve. Line C-5 and a J_ to A-B from 5 determine pts. of tangencj' 7 and 6. See application in Fig. (b) at G. Note. — The method of drawing a circular curve of given rad. tangent to two given circular curves is evident from Fig. (a) . 61. To draw a circular curve tangent to two given straight lines, A-B and C-D. Fig. 108. (a) When a point of tangency, E, on A-B is given. Fig. (a). Extend A-B and C-D to intersect at 1, and make 1-E' equal to 1-E. At E and E' draw ±s to intersect at 3, the center of the required curve; or bisect the included Z BID and draw _L at E to cut the bisector 1-2 at 3. Note 1. — When the vertex, 1, is inaccessible, the bisector may be obtained by Art. 28 Note. Note 2. — The construction, when A-B and C-D are ||, or at right Zs, is evident from Figs, (b) and (c). Fig. 107 (C) M Fig 108 (b) When the radius, 3-E, is given. Fig. (a). At distance 3-E draw ||s to A-B and C-D, intersecting at center 3 ; or obtain bisector 1-2 as in (a) and a 1 1 to one side intersecting 1-2 at 3. See application in Fig. 107(b) at F. Note. — ^WTien A-B and C-D are at right Zs, the rad. may be ap- plied as in Fig. (c). Fig no 62. To draw a circular curve tangent to three straight lines, A-B, A-C, and C-D. Fig. r09. The construction is identical with that of Art. 42. GEOMETRIC CONSTRUCTION 43 63. To draw circular curves tangent to two given parallel straight lines, A-B and C-D, at B and C, and to each other at any point E, in line B-C. Fig. 110. Draw ± bisectors of B-E and E-C. At B and C draw _Ls to A-B and C-D to cut the bisectors at 1 and 2, the centers of the required arcs. 64. To draw an ellipse when the axes, A-B and C-D, are given. (a) By Focal Radii. Fig. 111. Draw the axes _L to each other at their middle pts. With either end, C, of the minor axis as center and rad. equal to ^ of the major axis, cut the latter at F and F', the foci of the elhpse. Between the center, E, and either focus place any pt., 1. With the foci as centers and rad. A-1, draw arcs upon opposite sides of A-B. With same centers and rad. B-1, cut these arcs at 2, 3, 4, and 5, which will be pts. of the required curve. Assume a sufficient number of other pts. for focal radii on A-B and proceed in like manner. Draw the curve through the pts. thus found, freehand. See Art. 18(a). Note. — The method of obtaining a tangent, T-T', at a given pt., 2, in the curve is e\'ident. See Art. 2(d). (b) By Trammel Method. Fig. 111. On the St. edge of a piece of paper mark off G-I equal to | of the major axis, and I-H equal to f of the minor. Moving this trammel so that pt. G remains on the minor axis and H on the major, set off a sufficient number of pts. for the successive positions of I to determine the required curve. Note. — Having located pts. for J of the cur^'e, corre- sponding pts. could be determined by Art. 57. (c) By Revolution OF A Circle. Fig. 112. Upon the axes describe O s. Divide these O s into the same proportional parts by diams. If the large O be imagined to revolve about axis A-B, pts. 1, 2, etc., will appear to movej. to A-B. When 1 and 6 coincide with D and C, pts. 2, 3, etc., will have moved proportional distances which are determined by ||s to A-B from the corresponding pts. 2', 3', etc., of the smaller © , at 2", 3", etc., of the required curve. Note.— The figure indicates a second method of obtain- ing a tangent at a given pt. in the curve. (d) By Parallelogram Method. Fig. 113. Draw lis to A-B and C-D through A, B, C, and D, forming a parallelogram. Divide A-F into any number of equal parts and A-E into the same number of equal parts. Through the pts. of division on A-F, draw D-1, D-2, D-3. Through the corresponding pts. on A-E draw C-1, C-2, C-3 intersecting the lines from D at 1', 2', 3', Fig 1 . Ill ^ 2/\ .xJ \ //^ 3' ^ ^4 \^/ ^^^^■/ .4" Jb \/ x^ / Fig 112' 6 V Fig. 113 44 ESSENTIALS OF MECHANICAL DRAFTING which will be pts. in the required ellipse. In like manner find pts. for remainder of curve. Note 1.^ — The same construction appKes when any two conjugate diameters are given. Note 2. — The method of inscribing an elUpse in any- given parallelogram (not square) is evident. (e) By Circular Arcs. Fig. 114. The following is one of several methods of approx- imating an ellipse: Draw D-B. Make 1-2 equal to 1-D and D-3 equal to 2-B. Draw a _L bisector to 3-B, cutting C-D extended at 4, and A-B at 5. Make 1-5' equal 1-5 and 1-4' equal 1-4, and draw 4-6', 4'-6", and 4-6'". With 5 and 5' as centers, rad. 5-B, draw arcs 6-6'" and 6"-6'; then with 4 and 4' as centers, rad. 4-D, draw arcs 6'-6 and 6"'-6" to complete the curve. Fig 114- CHAPTER V ORTHOGRAPHIC PROJECTION 65. General Principles. It has been noted that mechanical drawings are made for the purpose of showing the exact facts of form, dimension, and arrange- ment of parts in objects of a structural character. (See Art. 1.) To express these facts fully and clearly it is necessary to represent the object by two or more related drawings each of which gives certain information that the others lack. These drawings, though made upon one plane, the paper, by the methods of Chap. IV, are regarded as projections of the image of the object upon different planes || to the axes or principal dimensions of the object, and imagined to be obtained by means of ±s to those planes from all pts. of the object; that is, in accordance with the principles of Orthographic (true drawing) Projection. (a) In Fig. 115(a), ABCD is a pictorial drawing of a hollow rectangular block, placed squarely in front, with the center of the opening on the level of the eye. By experiment with a similar object, it will be observed that the front sur- face, being at right Z s to the direction in which it is seen, appears in its exact form. The surfaces of the opening, although at right Z s with the front, appear fore- shortened and to incline towards each other; the farther lines appear shorter than the nearer, and the receding lines to incline and converge towards a pt. at the center. The apparent decrease in size, foreshortening, inclination, and con- vergence is due to the position of the Unes and surfaces of the object with respect to the eye of the observer, and, obviously, if the position of the ej'e be changed, the appearance will also change. Thus, if the eye be moved a certain distance upward, the object would appear as shown in Fig. (b) and if moved upward to the right, as in (c). It is evident that none of these pictorial representations shows the exact form, size, and relation of all the hues and surfaces. (b) Let LMNO (Fig. 115(a)) represent a vert, pane of glass or wire screen placed |1 to the front surface of the object. Consider this glass or screen to be merely piG. I 15 (b) (c) 46 ESSENTIALS OF MECHANICAL DRAFTING a plane, and the picture ABCD, as obtained by tracing lines upon the plane to exactly cover the outlines of the object seen through it. Since the plane is nearer to the eye than the object, and the rays of light converge from the object to the eye, it follows that all lines of the tracing must be shorter than those of the object; also, that if the object be projected forward until its front surface coincides with the plane, the tracing of that surface would be identical both in form and dimensions with the surface itself. (c) If now, instead of moving the object, lines be imagined to extend from all its pts., A, B, C, D, etc., _L to the plane, as shown pictorially in Fig. 116, these ±s would intersect the plane at A^, B^, C^, D^, etc. These pts. are called projections of the pts. of the object, and lines A^-B^, B^-C^, etc., joining them, will be projections of the lines and surfaces of the object. The plane is a plane of projection and the J_ s are the projectors of the pts. N D* , WIDTM _-j OR BREADTH F» kJ 1 L A" H» BVG* M Fig. 116 riG.117 (d) Since the projectors are || to each other, and the rear pts. of the object are perpendicularly back of the front pts., it follows that the projections of the front and rear pts. will coincide, as indicated in the figure. The projection, when viewed squarely, as in Fig. 117, thus shows what the eye would see if imagined to be directly opposite each pt. of the object at the same time; namely, the exact form and dimensions of the front and rear surfaces, and the dimensions of the object from left to right and bottom to top. (e) Although in this case the first dimension is the length and the latter the width, it is convenient in speaking of the dimensions of an object in a definite position to call the hor. dimension from left to right the width, or breadth, that from front to back the depth, and the vert, dimension the height, regardless of their extent. In the notation of this proj . and others which will be explained, the first letter in each instance indicates the pt. of the object nearer the plane, and the small letter the plane upon which the proj. of that pt. lies. To avoid confusion, the notation of the opening is omitted. (f) Since the three dimensions of an object are _L to each other and a plane has but two dimensions, it follows that when two dimensions of the object are ORTHOGRAPHIC PROJECTION 47 projected in their exact size the remaining dimension is not seen; that is, no more than two dimensions can be shown in their exact size and relation in one proj. Hence, to show the dimension from front to back (depth), another proj. upon a plane _L to the front or vertical plane must in like manner be obtained. Thus in the pictorial illustration Fig. 118, D'^C'^F^E", etc., is the proj. of the object upon a to-p or horizontal -plane ONQP, and B G F upon a side vertical or profile plane MRQN, at right Z s to the first two. Note. — The relation of the planes may be illustrated by means of a paper box, hinged panes of glass, or screens. (g) To show these projs. as they are generally arranged in a mechan- ical drawing, the planes of two of the projs. are imagined to be revolved about their hnes of intersection into the plane of the other regarded as the plane of the drawing, as in Fig. 119. (h) It should be noted that the projs. upon the top and side planes are precisely the same as would be obtained upon the front plane if the etc., its proj. FiG.118 p E" Q H" F"G" \ \ FiS.IIQ \ \ \ Q.'l D" t r 1- a o 1 "^\ A" G.L. B" 3 u N \ \ HTofP Dv Ev 1 WIDTH ^ OR BREADTH 1 1 ''" V Fv % P 1- > 2 CP 3 -DEPTH- DP' FP EP 1 1 t l- r o UJ r r 6*0" M BPAP GPHP 48 ESSENTIALS OF MECHANICAL DRAFTING object were turned from the position given so that its dimensions of width and depth, and then its height and depth, are 1 1 to that plane. The opening cannot be seen when the object is looked at squarely, either from above or from the side, but as it is necessary to show its projs. upon the top and side planes, — the lines whose projs. do not coincide with those of visible lines are represented by dashed lines, to indicate that they are invisible or hidden. Observe that the outer lines of all projs. represent visible lines of the object;' those within represent hidden lines when seen in the other projs. to lie below or behind some solid portion. (i) It is evident that the three dimensions are shown by any two of the projs.; that two ± projs. are, therefore, necessary to show the three dimensions; and that the three projs. together determine completely and clearly the exact form, size, and relation of all lines and surfaces of the object. (j) These three mutually _L planes of proj. are called the co-ordinate planes, and for brevity are denoted by V, H, and P respectively. A proj . is named from the plane upon which it is imagined to be obtained, not from the particular part seen or shown. Thus those upon V, H, and P are respec- tively the vertical, horizontal, and profile projection. In practical drafting, the first is called a front elevation ov front vieiv, the second a plan or top view and the third a side elevation or side view. The terms "plan" and "elevation" are used chiefly in architectural drafting; the term "view" is more generally used. (k) The line of intersection of V and H is called the ground line or trace of V and H, and is denoted by G L. When looking squarely at V, the G L represents H; and when looking at H, it represents V. The line of intersection of V and P is called the vertical trace of P; that of H and P is the horizontal trace of P. These are denoted bj^ V T of P and H T of P. When looking at V or H, these traces indicate the position of P. When looking at P, they represent V and H. The verts, and hors. from the views to the traces represent the projectors to the planes. (1) When H and P are revolved, the front and top views of any pt. are seen to lie in the same vert.; the front and side views in the same hor.; and its top and side views equally distant from the G L, and V T of P, respectively. Rules 1, 4, 5, and 7 (Art. 74) may here be noted. (m) Views could in like manner be obtained upon planes auxiliary to the co-ordinate or principal planes, V, H, and P; namely, upon a profile plane at the left of the object, upon a bottom plane || to H, upon a rear plane || to V, and upon planes oblique to either two of the co-ordinate planes. We may thus have front, top, right and left side, bottom, rear, and oblique views. With the exception of the latter, the relative arrangement of these views in a mechanical drawing would generally be as indicated in Fig. 120. See Art. (i). Note that the line of each view (excepting the rear) nearest the front view represents the front line or surface of the object in that view. ORTHOGRAPHIC PROJECTION 49 66. To draw the front, top, and side views of a rectangular object. Suppose it is required to draw tiiese views of the block (Fig. 119(a)), its position and dimen- sions being given. (a) First draw ONQ' and MNQ to represent the traces of V, H, and P. That portion of the paper below 0-N and to the left of M-N will represent the front plane, that above 0-N and to left of N-Q the top plane, and that to the right of M-N and below N-Q' the right side plane. The outer limiting lines of the planes are always omitted. E" H« F" G" Q — u. O ^^riG.ii9( ' i 1 1 1 1 1 1 1 UJ Q ♦ D" o A" GL. C"|B" 1 1 h H / IF" N htWp. 1 E" WIDTH OR BREADTH C F' \ p [-DEPTM»| D" D' f E" a 1 E o r > j__ Q' ^c," Ml B'-A" CK' (b) Since the distance of the object from the planes is immaterial, the views may be drawn anj- distance from the G L and traces of P. Hence, at any con- venient distance below 0-N and to the left of M-N, draw rectangle A^B^C^D^ for the main lines of the front view, making the hor. and vert, dimensions equal to the breadth and height respectively, of the object. The opening should at first be disregarded. (c) The top view of each pt. must be in a vert, through its corresponding front view. Art. 65(1). As in this case all pts. of the front face are equally distant from V, di'aw the projectors from the front view, and at any convenient distance above the G L, draw the hor. D.^-C^ for its top view. As the rear face is || to the front, its top view will be E^-F^, at a distance from D^-C^ equal to the depth of the object. As the side faces are _L to H, their top views will be the verts. D^-E^ and C^-F^, thus completing the rectangle D^C^F^E^ for the main lines of the top view of the object. (d) From Art. 65(1) it follows that the side view of each pt. will be in the hor. through its corresponding front view and at the same distance from the V T as its top view is above the G L. Hence, draw indefinite hors. from the front view, and from the top view to N-Q'. Then as N-Q and N-Q' both represent the H T of P, describe arcs with N as center, to carry the pts. from N-Q to N-Q'. From these pts. draw verts, to intersect the projectors. from the front view, thus determining B^, G^, F^, and C and forming the main lines of the side view. 50 ESSENTIALS OF MECHANICAL DRAFTING (e) Similarly, if the side view is first obtained, the reverse of the above process will determine the top view. (f) Now, returning to the front view, draw the rectangle to represent the opening, then project as before, to complete the other views. TOP VIEW LEFT SIDE VIEW FRONT VIEW RIGHT SIDE VIEW REAR VIEW BOTTOM VIEW Fig. 120 (a) Cb) Fig. 121 (g) Practical Methods. In practical drafting, only such portions of the projectors are penciled as may be necessary to locate the required pts. The traces of the planes of proj. are also omitted; center, base, or other hnes || to the positions of the traces being utilized as reference Unas in locating the views, practically as though such lines were the actual traces of the planes. Thus, in determining the views (Fig. 119(a)) the lower hor. of the top view and left vert, of the side view could have been located any convenient distance from the front view, and the depth set off from top to side or vice versa, by means of the dividers. Fig. 121(a) shows the views with the traces and projectors omitted. In this case aU measurements were set off with reference to lines representing the axes of the object, called center lines. ORTHOGRAPHIC PROJECTION 51 It is evident that a C. L. of a view may be regarded as the trace of a central plane ± to the plane of that view. Fig. 121(b) shows the same object in a different position. Working drawings of an object composed of several rectangular parts are shown in Figs. 179, 180. 67. Objects Having Surfaces Oblique to the Co-ordinate Planes. The form of an object is frequently such that some of its surfaces and lines will be oblique to the planes of one or more of the views; and, therefore, foreshortened in those views. Thus in the pyramid (Fig. 122) the front and rear faces are oblique to V and H, the left and right faces to H and P, and its slant edges to V, H, and P. These surfaces and lines are, therefore, foreshortened in all of the views. The base is 1 1 to H and has two of its edges 1 1 to V and two 1 1 to P. When thus placed the object is || to V, H, and P as much as it can be, since its axes or principal dimensions are also || to those planes, and the views enable the form of the object as a whole to be determined, as in the case of a rectangular object. Rules 2, 3 and 6 (Art. 74) may here be noted. - Fig. 122 BC BA 68. Objects xiaving Curved Surfaces. (a) Any view of a sphere is a O equal to a great O of the sphere. Fig. 123. Observe that the great O ABCD is || to V, AECF || to H, and DEBF l| to P. The view of a right circular cjdinder upon a plane to which its axis is± is a ©. Its view upon a plane || to the a.xis is a rectangle. Thus in Fig. 124 the bases are || to H and X to V; thej' are, therefore, seen upon H in their exact shape, and upon V as || hors. equal to the diam. of the base. The elements are || to V and _L to H and, therefore, seen upon V in their exact length, and upon H as pts. in the O . The views of a right circular cone whose axis is thus related to the planes are aO and a A. Fig. 125. The outer elements only being || to V, they only are seen upon it in their exact length. Intermediate elements are unequally inclined to V and, therefore, unequal upon it. As all the elements are equally inclined to H, they are equally foreshortened upon H. Note that as the third view in each case would be the same as one of the others, its representation is unnecessary. 52 ESSENTIALS OF MECHANICAL DRAFTING (b) Equidistant elements of a right circular cylinder or cone may be deter- mined by equal division of the base or other O of the surface, in the view in which that O is seen in its exact shape. Twelve are usually sufficient. (c) Fig. 126 represents a truncated, cylinder. As the elements are _L to H, the top view of the curve of the oblique surface coincides with that of the base. As the surface is oblique to P, its side view will also be a curve. To draw this view it is necessary, since there are no vertices as in rectilinear figures, tO' obtain a sufficient number of its deter- mining pts., by assuming these first in the known (front and top) views of the curve. Having located these pts. in the side view the curve is drawn through them, as shown. See Art. 18(a). In cylindric or conic surfaces these pts. maj' be regarded as the ends of elements. 69. Projections upon Oblique Auxiliary Planes. When it is necessaiy to represent an object so that some particular surface, ± to one plane but oblique to another, shall be shown in its true or exact shape, an auxiliary view upon a plane II to that surface may be obtained. (a) In Fig. 127, D^B^C'^, etc., 'represents a proj. upon an aux. plane A, II to the surface DBC, that is, upon a plane _L to V, but oblique to H. The method of obtaining this view differs from that of obtaining a side view only in that the V T in this case is 1| to D^B'^'C^, instead of ± to the G L. Having located the traces of A, draw ± s to the V T from the front view and inter- sect these by projecting from the top view as shown, or transfer the pts. with dividers as in Art. 68(g). Joining the pts. thus determined will form the required view. Note application of principle in Figs. 144, 182. (b) In Fig. 128(a) an aux. view A^B^D^, upon a plane, A, ± to H and || to a central plane, Y-Y, is shown. In this case the view was determined by Fig 127 ORTHOGRAPHIC PROJECTION 53 means of C. Ls. and base lines regarded as the traces of planes J_ to those of the views. The method differs from that of obtaining the front view in Fig. 127 only in the changed positions of the central planes X-X and Y-Y. Thus Y-Y is the H T of the given central plane; X-X is the H T, and trace upon A, of a second central plane _L to Y-Y. Z-Z is the V T of the plane of the base, and Z'-Z' drawn anj^ con- venient distance from Y-Y and || to it is the trace of the base plane upon A. Perpendiculars to Z'-Z' from A^, B", and C^ determine A^, B^, and €"*■; setting off the height of D above Z-Z fi'om the front view and joining D*, A-*-, and B^ completes the required view. (c) Fig. 128(a) also method of obtaining an jQB^BgB^ upon a plane, B, fiG 128 shows the aux. view II to a sur- face DBC, which is not ± to the plane of either of the given views, that is, not shown as a line, but ± to which a plane, Y-Y, can be directly determined. The aux. view A^B'^D^, is first ob- tained as in the preceding case. Y'-Y' II to D'^B^C^ is the trace upon B of the ± plane Y-Y. Perpendiculars to Y'-Y' from A^ and D^ determine A^ and D^; setting off the distances of B and C from Y-Y, ± to Y'-Y', and joining pts. A^, B^, C^, D^ completes the required view. (d) Fig. (b) shows the method of obtaining an aux. view upon a plane, B, A. to the axis of an object which is oblique to V, H, and P. Note that the views upon A and B determine the true dimensions of the prism. (e) In a curviUnear object, it is necessary to assume pts. in the known views of the curves and then obtain these in the aux. view, as is evident from Art. 68(c). (f) When the true shape of a particular surface only is required the same method, or that of Art. 70, maj^ be used. 54 ESSENTIALS OF MECHANICAL DRAFTING 70. Revolution of Surfaces. It is frequently desirable to determine the true shape of a surface by revolv- ing it about an axis until 1 1 to V, H, or P. (a) Fig. 129(a) represents a truncated hexagonal prism with its oblique surface ABCDEF revolved about an imaginary axis _L to V through pt. A, until it is || to H, upon which it is, therefore, seen in its true shape A'B'C'D'E'F'. In revolving a surface, each pt. not in the axis of revolution describes an arc whose plane is _L to that axis. In this case, therefore, the arcs are r IG. 129 H' G" H» seen as arcs in the front view and as lines || to the G L, or C. L., A^-D', in the top view. Hence, the front view of the surface remains a line of the same length, and the distances of the pts. of the surface from V also remain unchanged. The surface could in like manner be revolved 1 1 to H, or to P, about any other axis ± to V. Thus A"B"C"D"E"F" represents it revolved about 1-2 until 1 1 to P. Fig. (b) represents the surface revolved about its side B-C until || to V. As B-C is II to V, the front views of the arcs described by A, F, E, and D are _Ls to B-C at A^, B^, C^, and D^; and as the true distances of A, F, E, and D back of B-C are seen in the top view, these distances are set off on the ±s at A', F', E', and D' as shown. The figure also shows a lateral surface revolved about its base edge, G-H, until II to H; and, about the lateral edge, A-G, until || to V. Note application of principle in Fig. 140 to the revolution of an imaginary surface on C-D. See also Figs. 145, 153, 158, 159, 161(b). ORTHOGRAPHIC PROJECTION 55 (b) In Fig. 128(a), a surface DEC, oblique to V, H, and P, is shown revolved II to H about its base edge B-C. As B-C is jj to H, the top view of the arc described by pt. D will be a i. to B^-C^, coinciding with Y-Y. Hence, the true distance of D from B-C is determined, in this case, by an aux. view upon a plane A, || to the central plane Y-Y, to which the surface DBC is ± . Art. 69. Rule 8 (Art. 74) may here be noted. 7 1 . True Length and Position of Lines Oblique to the Co-ordinate Planes. To determine the true length of a Hne oblique to the planes, the hne may be projected upon a II aux. plane, or revolved || to a plane of proj., practically as in the case of a surface. (a) First Method. To obtain the true length of A-D (Fig. 128(a)) upon an aux. plane J_ to H. The method is evi- dent from Art. 69(b). A*-D^ is the true length, and the Z it makes with the H T of plane Y-Y containing the line is the true Z the line itself makes with H. To obtain the true length of A-D (Fig. 130) upon an aux. plane A _L to V, regard A^ -D^ as the V T of the plane containing the line; Y-Y as the H T of a central plane ± to the first; and Y'-Y' placed any distance from A^-D^ and || to it, as the trace of this second plane upon the aux. plane A. Projecting from A^ and D^ J. to Y'-Y' obtain D^, and set off the ± distance A*-D^ will be the required true length and makes with V. (b) Second Method. If the line A-D be revolved || to V about an axis ± to H through D, the line remaining at the same Z with H and pt. D fixed, pt. A will describe a hor. arc, the top view of which will be the arc A^-A', and the front, the hor. A^-A'. In this position, therefore, the top view of the line remains unchanged in length, while the front A'-D^ shows the true length of A-D and the true Z (b) that it makes with H. As all of the edges meeting in the vertex D are equal and equally inclined to H, A'-D^ is the true length and (b) the true Z of all. Again, as all pts. in the revolved line remain on the same levels, they describe similar hor. arcs. The true distance from D of any pt., as E on D-C, may, there- fore, be found by drawing a hor. from that pt. to the true length line A'-D^, as proved by the top view which shows E revolved into A'-D^. E'-D^ is, there- fore, the true length of E-D. The figure also shows the line A-D revolved 1 1 to H about an axis ± to V, through D. A"-D^is, therefore, the true length and (c) the true Z of A-D with V. The line could in like manner be revolved about an axis through A. Fig 130 that A is back of Y-Y. Then (a) the true Z that the line itself 56 ESSENTIALS OF MECHANICAL DRAFTING 72. Objects Ob- lique to the Co-ordi- nate Planes. It fre- quently occurs that the axes of one or more integral parts of an object are oblique to those of the main portion, and, there- fore, to the planes of one or more of the views. Thus, in Fig. 193 the left and right side pieces of the taboret are oblique to V and P; in Fig. 194 the braces (A) are oblique to V and H, and braces (B) to H and P (see also bolts in Fig. 182) ; in Fig. 199 the legs are obhque to V, H, and P. When the detail or part is || to V, H, or P only, its view upon that plane should, as a rule, first be determined, since that view only will show two of its dimensions in their exact size. When not || to either V, H, or P, it may, if necessary, be represented first as || to one of those planes and then as revolved to the required posi- tion; or, the method of Art. 73 may be used. (a) In Fig. 131, (a) represents a rectangular prism II to V, H, and P. (b) represents this prism II to H but oblique to V and P, that is, turned backward at the left from the position of (a), through an Z of 30° about an axis _L to H. (c) represents it 1 1 to V but oblique to H and P, that is, turned to the right from the position of (a), 30° about an axis J_ to V. (d) represents it II to P but oblique to H and V, that is, turned for- ward from the position of (a), 30° about an axis J. to P. (b) By experiment with the actual object, it will be o'bserved that the positions of its lines rela- tive to the plane to which the axis of revolution is ± remain unchanged. It follows that the object may be turned through any Z without changing the form and dimensions of the view upon that plane. Thus, in Fig. 131, the top view in (b), the front view in (c), and the side view in (d) are the same as the corresponding views in (a), save that they show the changed relation of the object to the planes to which the axis of revolution is ||. The views upon the latter planes change in outline with each position of the revolution, but as no pt. changes its distance from the ORTHOGRAPHIC PROJECTION 57 plane to which the axis is J. , the dimen- sions II to the axis also remain un- changed. Thus, corresponding pts. of the front and side views in (b) and (a) are at the same distances from the G L, and H T of P ; those of the top and side views in (c) and (a) at the same distances from the G L, and V T of P; and those of the front and top views in (d) and (a) at the same distances from the traces of P. Note Rule 8, Art. 74. (c) In determining the views of an object thus revolved, having located the G L and traces of P, or equivalent reference Hnes*, draw the main axis or axes of the unchanged view at the required Z s with the planes to which the object is oblique. (See lines A-A andB-BinFig. (b).) Measuring upon, and 1 1 to these axes, construct the view. From this view draw projectors to a second plane and set off upon them the unchanged dimensions" 1 1 to the axis of revolution, measuring from the trace of the first plane, or equivalent reference hne. Joining the pts. thus determined will complete the second view. From these, the third may be obtained as in Art. 66. (d) In Fig. 132, (a) and (b) repre- sent the same prism when obhque to V, H, and P. (a) represents it as turned to the right from the position of (b) in Fig. 131, 30° about an axis ± to V; (b) as turned backward from the position of (a), 30° about an axis J_ to P. In determining these views the same principles apply as in the preceding cases. By assuming and representing an object in a primary position, as nearly as may be like that reciuired, and then as revolved, any conceivable position may be represented. Observe that the side view in Fig. (b) was plotted from Fig. (a) by Art. 57. (e) In an object having curved surfaces, pts. must be assumed in the primary positions of the curves, as explained in Art. 68(c). Note applications of principle in Fig. 193. *In practical work the traces and projectors are omitted, as explained in Art. 66(g). shown in Fig. 131 merelj' for illustration. They are 58 ESSENTIALS OF MECHANICAL DRAFTING (f) Instead of representing each position separately, the required views may be obtained by revolution about a C. L. of a primary position as in Fig. 133(a), or about an axis || to that C. L. as in Fig. (b). 73. Partial Views, and Use of Auxiliary Views in Determining Required Views. (a) Partial views, and aux. views (views of revolved, lines or surfaces, and oblique views) may frequently be used in place of complete or regular views and thus economize space and labor. Thus in Fig. 214 half of the top view would suffice to determine the required pts. of the front; in Fig. 153 a revolved half -base is used for the same purpose; in Fig. 147 half only of the side view" of A and C is necessary; in Fig. 144 half of the oblique view would suffice; in Fig. 145 the revolved half-bases of (B) and (C) dispensed with a side view; in Fig. 146 a view _L to the axis of (B) or a revolved imaginary © is necessary to determine positions of elements. (b) In representing objects oblique to the co-ordinate planes, primary po- sitions may be omitted and the missing dimensions of the required views deter- mined by the same general methods. FfG. 133 To illustrate, Fig. 134(b) represents a pentagonal prism with its axis and lateral edges j] to H and at 30° back to the right with V. Its bottom face also is II to H. Instead of representing the prism first as || to V and H as in Fig. (a), and then as revolved about an axis _L to H until its lateral edges make the required Z of 30° with V, the representation of the first position may be omitted and the missing dimensions determined as follows: — First locate X-X, Y-Y, Z-Z, and Y'-Y' for the C.Ls. of the required views. Through the center of the top view draw V-V at 30° with Y-Y for the H T of a central plane containing the axis and W-W for the H T of a central plane ± to V-V. Equally distant from the center and at a distance apart equal to the axis of the prism, draw indefinite _Ls to V-V for the bases. Now obtain an aux. view upon a plane || to the central plane W-W. The H T of this plane would be || ORTHOGRAPHIC PROJECTION 59 to W-W, and the trace of the hor. central plane Z-Z, upon the aux. plane, would also be 1| to W-W. Hence, at any convenient distance from either base, draw this trace Z'-Z' and about its intersection with V-V construct the view, noting that as the lower surface of the prism is || to H, the hne A-B will be || to Z'-Z', as shown. Projecting perpendicularly to W-W from this view, complete the top view. Project next to V and compltte the front view, obtaining the distances of pts. above and below Z-Z by measuring from Z'-Z' in the aux. view. The missing dimension in this case could also be obtained by revolving a base || to H, then by counter revolution projecting back to the top view, as shown. In some cases a view upon an aux. plane || to the central plane V-V, as shown, would be necessary instead of, or in addition to, that of the end. 60 ESSENTIALS OF MECHANICAL DRAFTING Fig. 135 represents the same prism with its axis || to V and at 60° up to the right with H. Its rear face also is || to V. Fig. 136 represents it with its axis || to P and 30° forward with V. Its left face also is II to P. The methods of obtaining the missing dimensions are identical with those of the preceding case and are shown by the figures. (c) Fig. 137 represents the prism with its axis inclined up to right at 60° with H and in a plane at 45° back to left with V. One of rear faces also is at 45° ORTHOGRAPHIC PROJECTION 61 Fig. 137 with V. This is identical with a revolution from the position of Fig. 135, 45° about a vert. axis, but instead of represent- ing the prism first at 60° with H, as in Fig. 135, the work in this and similar cases may be shortened as follows: — First locate the C.Ls. X-X, Y-Y, and Z-Z for the required views. Through the center of top view draw U-U at 45° for the H T of a central plane con- taining the axis, and V-V ± to U-U for the trace of a central plane ± to H and to an aux. plane A 1 1 to the axis. Through any convenient point on V-V draw W-W at 60° with U-U as the C.L. of the aux. view, and about the intersection of V-V and W-W construct the aux. view as in Fig. 135. From these aux. views obtain the top view. Observe that this is identical with revolving both front and tap views of Fig. 135, through an Z of 45°. Project next from H to V, and obtain the dis- tances of pts. above and below Z-Z by measuring from Z'-Z', regarded as the trace of the hor. central plane Z-Z upon the aux. plane. Connect the proper pts. to complete the front view. 74. Rules Governing the Position of Lines and Surfaces Relative to Any Two Perpendicular Planes of Projection. (1) A line ± to a plane is 1| to the JL plane. Its view upon the first is a pt.; its view upon the second is a line || and equal to the line itself and ± to the trace of those planes. (2) A line obUque to one plane and || to a ± plane has its view upon the first a line shorter than the line itself and [ | to the trace of those planes; the Z the other view makes with the trace is equal to the Z the Une itself makes ■nath the first plane. (3) A line obhque to two _L planes has its views upon both shorter than the line itself and neither view shows the true length, nor true Zs the line itself makes with those planes. (4) The views of || hnes are || unless their views coincide, or are merel)- pts. (5) A surface || to a plane is ± to the X plane. Its view upon the first plane is a figure similar and equal to the surface itself. Its other view is a line || to the trace of the planes. (6) ^ surface obUque to one plane and ± to the ± plane is foreshortened 1 1 to the trace in its view upon the first. The Z the other view makes with the trace of those planes is equal to the Z the surface itself makes with the first plane. (7) The views of 1 1 and equal surfaces whose corresponding lines are 1 1 are similar figures whose corresponding lines are equal and ||, unless the \'iews coincide, or are merelj- lines. (8) If a line or figure be revolved from one position to another about an axis ± to a plane, its view upon that plane remains unchanged in form and dimensions. In the view upon a ± plane the distances of the pts. from the trace of the planes also remain unchanged. CHAPTER VI PLANE SECTIONS 75. Principles and Methods. If a plane be imagined to intersect an object, the portion of the object lying within it is called a plane section, and the plane a cutting or section plane, denoted by C. P. (a) If the portion of the object hiding the section be imagined as removed, the view of the remaining portion or merely that of the section itself, upon a plane || to the C. P., is called a sectional view. A sectional view thus shows the sec. in its true shape. Such views are made in place of or in addition to external views to show the form, dimension, and arrangement of hidden parts more clearly; to aid in determining the views of irregular, oblique, and inter- secting forms; and to enable the draftsman to dispense with numerous dotted lines. The figures of this chapter show sees, of geometric forms; practical applica- tions are shown in Chap. XL (b) When a sec. is not 1| to V, H, or P, its true shape may be found by projecting upon an aux. plane || to the C. P., as in Art. 69; or by revolving the sec. II to V, H, or P, as in Art. 70. (c) The position of a C. P. is indicated in the view in which it is seen edge- wise; that is, by its trace upon V, H, or P. The section is generally indicated by drawing ||s across it called section lines, usually at 45° up to the right. The spacing is dependent upon the size and shape of the sec, usually tV", judged by eye. Avoid drawing the lines 1| to the main lines of a sec. When a drawing is to be inked, it is not desirable to pencil the sec. lines, but the sectioned portion may be indicated by a few lines sketched freehand. In views which show the true shape only, sec. lines may, for economy of time, be omitted. (d) In obtaining a sectional view the trace of the C. P. is first drawn. This determines the pts. in which the C. P. cuts the edges, elements, or other lines of the object. Next project these pts. to the corresponding lines in the other views, and join them. Then proceed as in (b). 76. Objects Having Plane Surfaces. As the intersection of plane surfaces is a st. line, it follows that any plane sec. of an object bounded by plane surfaces, as a prism or pyramid, will be a polygon of 3, 4, or more sides according as the plane cuts 3, 4, or more surfaces. (a) A Prism. Fig. 138(a) represents a right square prism with its lateral faces at 45° with V, cut by a plane A-B, JL to its base and 1| to V. The pts. of intersection 1, 2, 3, and 4 with the top and bottom edges are, therefore, determined in the top view and then projected to the front. As the plane cuts four ± faces, the sec. is a rectangle; and, being || to V, is shown in its true shape in the front view. 62 PLANE SECTIONS 63 Fig. (b) represents the prism in the same position, cut by a plane _L to V, but oblique to H and P. The pts. of intersection are, therefore, determined in the front view. Fig. (c) represents the prism with its vert, faces at 30° and 60° with V, cut by a plane _L to P and oblique to V and H. This illustrates a case in which the sec. is not symmetrical with respect to a C. L. (b) A Pyramid. In Fig. 139 the C. P., D-E, cuts the lateral edges of the pyramid at 1, 2, 3, and 4. Pts. 1 and 3 may be projected to the top view in the usual manner; pts. 2 and 4, however, are in edges which are represented by vert, lines in both views. To obtain these pts. in the top view either of the follo^^^ng methods may be used. First Method. Draw a side view and transfer the distance 2^-4^ from it to the top view. 64 ESSENTIALS OF MECHANICAL DRAFTING Second Method. Imagine the edge A-B to be revolved i| to V to coincide with A-C. Then the front view of 2 will be at 2' and its top view at 2". If now the line A-B be rev6lved back to its original position, 2^ will be determined as shown; 4^ may in like manner be found. The C. P., F-G, gives a triangular sec. The method of finding pt. 7^ is pvident from the preceding. 77. Objects Having Curved Surfaces. (a) A Sphere. Any sec. of a sphere is a O whose center is in the diam. of the sphere, _L to the C. P. See Art. 2(n). In Fig. 140 the top view of sec. on A-B is thus a O whose diam. is equal to 1^-2^. The front view of sec. on C-D is an ellipse. The intersection of the Fig. 139 Fig. 140 C P. with the hor. great O of the sphere determines 3 and 4 in the top view. As the front view of that O is in the hor. E-F, these pts. will be projected at 3^ and 4^, as shown. Intermediate pts. of the curve may be found by determining pts. of intersection of the C. P. with other Os of the sphere whose planes are || to either H or V. Thus, assuming a O parallel to V, cutting the O of the sec. at 5 and 6, these points will be determined in the top view and projected at 5^ and 6^ in the front. Again, assuming a hor. O to cut the sec. at 5 and 8, these pts. will be determined in the top view, and projected at 5^ and 8^. Note that the highest pt., 7, of the curve is that in which a hor. O becomes tangent to the plane of the sec. The assumed Os may be regarded as lines of intersection given by aux. C. Ps., each of which cuts the plarie of the sec. in a chord of both ©s, as 5-6, whose end pts. are thus in the required curve. PLANE SECTIONS 65 The intermediate pts. of sec. C-D may also be found by the reverse of the method of Art. 70. Thus the half sec. revolved || to H about diam. 3-4 determines the distance of pts. 5 and 6 from diam. 3-4, to be set off above and below that line in the front view. (b) A Cylinder. Any sec. |1 to the axis of a right circular cyhnder is a rectangle, and a sec. || to the base, a O. See Art. 2(1). The sec. on A-B (Fig. 141) is an ellipse whose top view is a O coinciding with the view of the base. The sectional view may be obtained as in Art. 69(e), or by determining pts. of intersection of the C. P. with elements, as indicated. The assumed elements may be regarded as Knes of intersection given by aux. C. Ps. which cut the plane of the sec. in lines, as 2-6 and 3-5, whose ends are thus in the required curve. (c) A Cone. A sec. through the vertex and base of a right circular cone is a A; a sec. _L to the axis, a O whose center is in the axis of the cone. See Art. 2(m). The obUque sec. (Fig. 142) is an ellipse, pts.- for the top and sectional views of which may be obtained by determining the pts. of intersection with elements of the cone. A more accurate method is to assume a number of Os of the cone to cut the sec. These will be || hors. in the front view and Os concentric with the O of the base in the top. Thus, drawing the front view of a O through any pt., as 1^,- 3^, and projecting from 1^, 3^, to the top view of the O, pts. 1^ and 3^ will be determined. The vert. sec. being _L to both V and H has both front and top views a st. line, but being || to P will show its true shape, a hj^perbola, on that plane. The C. P. cuts the right element at 2 and the O of the base at 4 and 5, which pts. may be obtained in the sectional view in the usual manner. Intermediate pts. maj^ be obtained by use of elements, but it is more accurate to assume other Os of the cone, to intersect the sec. Thus a O cutting the sec. at 6 and 7 determines the distances of those pts. from the C. L. in the top view. The assumed elements and Os may be regarded as lines given by aux. C. Ps., as in the case of the cylinder, Art. (b). CHAPTER VII INTERSECTION OF SURFACES Fig 143 78. Principles and Methods. The Hne or lines in which the surface of one integral part of an object intersects that of another part is called a line of iJiter- section. This line joins the pts. in which the edges, elements, or other lines of each part meet the surface of the other, as is evident from Figs. 144(a), 145(a). The problem, therefore, in determining the line of intersection is to find the pts. of intersection of the edges, or of a sufficient number of other lines, with the surfaces, and to draw the line through the pts. thus found. (a) The pt. of intersection of a line and a surface is determined in the view in which the surface is seen edgewise or as a line, and then projected to the other view of the line precisely as in finding the pts. of a plane section. Thus, in Fig. 143, line A-B intersects the front face of the object at 1, as determined in the side view, for if it be assumed to intersect the left side face, pt. C would be seen in the side view to come above the object. A-B also intersects the right side face at 2, as determined in the front view. When the intersected surface is not shown as a line in either of the required views, an airx. view which will so represent it may sometimes be used. The pt. of intersection of any line and surface may be determined by passing a plane through the line, ± to the plane of either of its views. This C. P. cuts the given surface in a line in which the required pt. of intersection must lie, for both lines are in the same plane. Thus, in Fig. 143, a plane ± to H and P gives a sec. 3 4 5 6, and as the lines of the sec. are in the surface of the pyramid, pts. 1 and 2, in which A-B intersects them, must be the required pts. Similarly a plane ± to V, gives sec. 7 8 9 10, by which 1 and 2 are determined in the top, or side view. (b) It is evident that a plane which cuts both intersecting parts of an object would give a plane sec. of each, and the pts., if any, in which the lines of these sees, intersect, being in the surfaces of both parts, must be pts. in the required Une of intersection. By assuming a series of such C. Ps., any desired number of pts. may be found. The position of the C. P. should, when possible, be so chosen that the sec. of each part is one readily obtained. (c) In problems involving intersections, first draw each part as complete as possible in itself, that is, without regard to its intersection. Next obtain the 66 INTERSECTION OF SURFACES 67 pts. of intersection of the edges or elements which can be directly determined in either of the views; then those whose pts. must be found by means of C.Ps., or aux. views. In finishing, only such lines should be rendered as edges or outlines as represent the edges or outlines of the object as a whole. 79. Objects Having Plane Surfaces. Fig. 144 represents the intersection of a triangular prism (A) and an oblique hexagonal prism (B). The lateral edges, A-B, C-D, E-F, etc., of (B) intersect the left vert, surface of (A) at 1, 3, 5, 7, 9, 11. As this surface is seen as a line upon both V and H, the pts. are determined in both views, and since all the pts. lie in the same plane as that of the surface, the line of intersection is simply the outline of a plane sec. of (B). In this case it is an irregular hexagon which would be seen in a side view. K" I" r^l2''IO' ! Ti" Fig. 144 The edge A-B intersects the upper surface of (A) also at 2. As the surface is seen as a surface upon H and as a line upon V, pt. 2 is determined in the front view and then projected to the top. Edges C-D, E-F, I-J, and K-L intersect the front and rear surfaces of (A) also, at 4, 6, 10, and 12, respectively. As these surfaces are seen as lines upon H, the pts. are there determined and then projected to the front view. Edge G-H and the right vert, edge of (A) are in the same plane || to V and thus seen to intersect in the front view at 8. Points 13 and 14 in which the hor. edges of (A) intersect the surfaces of (B) are not directly determined in either view, for neither of the intersected surfaces is there seen as a hne. Bj' obtaining an aux. view upon a plane _L to the axis of the oblique prism, the intersected surfaces of the latter will be seen as lines D -B and B*-L^ and the pts. of intersection of the edges of (A) determined at 13 and 14^ as shown. These pts. may then be projected to the front view. It is obviously unnecessary to draw the complete aux. view. 68 ESSENTIALS OF MECHANICAL DRAFTING When the preceding method is not convenient or possible, the pts. may be found by means of a C. P.; thus, if the plane of the hor. surface containing the edges be assumed to intersect (B) it would give a sec, the intersections of whose sides with the edges of the hor. surface determine 13 and 14 in the top view as shown. Again, if the plane of the front surface of (A) be assumed to intersect (B) it will give a sec, the intersection of a side of which with the front hor. edge Fig. 145 of (A) determines pt. 13 in the front view. Pt. 14 could in like manner be found. It is evident that merely those lines or portions of the sees, which determine the required pts. need be drawn. 80. Objects Having Curved Surfaces. Fig. 146 (a) Fig. 145 represents the inter- section of two circular cylinders (A) and (B) and an equilateral triangular prism (C). The line of intersection of (A) and (B) is determined by the pts. of intersection of the elements of one cylinder with the surface of the other. Thus B, D, F, and H in which the upper, front, lower, and rear elements of (B) intersect (A) are determined in the top view, for the cylindric surface of (A) is there seen as a hne (circle). Intermediate pts. may be found by assuming intermediate elements of (A) in a side view, or of (B) as shown; the positions of the elements INTERSECTION OF SURFACES 69 of the latter being transferred from top to front view bj^ means of the revolved half-base, Art. 73. Note also that C. Ps. || to the axes of the cylinders would cut elements of both, which intersect in pts. of the required curve. The views of prism (C) may be determined by means of a side view, or a revolved half-base as shown. The pts. of intersection of the hor. edges with cyhnder (A) are determined in the top view. Note that the rear surface inter- sects the cylinder in an element. The inclined surfaces intersect (A) in elliptic curves, intermediate points in which may be found by assuming elements of the cyhnder and determining their intersection with (C) in a side view; or by assuming lines upon the prism || to its axis. Thus a hor., 5-6, intersects the cylinder at 5, determined in the top view. Again, a C. P. || to the axes of (A) and (C) would give an element of (A) *nd two lines of (C) whose pts. of intersection must be in the required curves as shown. Note that pts. 9 and 11, in which the direction of curvature changes, are at the intersection of lines 9-10 and 11-12 with the right element of (A). (b) In determining the curve of intersection of cjdinders (A) and (B) in Fig. 146, the same methods would be used as in (A) and (B) of Fig. 145. Equidistant elements of (B) may be determined by an obUque aux. view, or a revolved half-sec. taken J. to the axis, as shown. (c) In determining the intersections in Fig. 147, the C. Ps. may be taken horizontallj', or through elements, as shown. If cylinder (c) were oblique, an oblique aux. view could be used. (d) The simplest method of determining the intersection in Fig. 148 is to pass C. Ps. through the elements of (B). The sees, of (B) will thus be As and those of (A) ellipses, or other curves. (e) Fig. 149 represents a form similar to the stub end of a connecting rod. The highest pts., B and D, of the curves, in which the bell-shaped neck of the 70 ESSENTIALS OF MECHANICAL DRAFTING cylindric portion intersects the rectangular, are directly determined in the front and side views, and will be seen from the plan to be the pts. in which circular sees, become tangent to the sides of rectangular sees. The lowest pts., A, C, E, and F, lie in the O through the corners of the rectangle and, therefore, located in the front and side views by projecting from the plan to the corresponding views of that O , which are obtained as shown. Other pts. are determined by intermediate G. Ps. Thus plane G-H gives a O which cuts the rectangle at 1, 2, 3, and 4 of the required curve, which are located in the other views as in the preceding. Fig. 149 CHAPTER VIII DEVELOPMENT OF SURFACES 81. Principles and Methods. A development is the representation of the surfaces of an object as laid out, unfolded, or unrolled into the plane of the draw- ing. The operation is suggested pictorially by Figs. 150 (a), (b). A develop- ment thus shows the exact area of all surfaces of the object and the exact length of every hne of those surfaces, Fig. 151. Plane, cylindric, and conic surfaces only can thus be developed. Surfaces of double curvature and warped surfaces may be developed approximately by assuming portions to be cylindric, conic, or triangular,. Fig, 150 Developments are made to determine the shapes of surface patterns required in constructing objects of sheet metal, cardboard, etc.; to plot groove outlines for cylindric cams; and to obtain templets or patterns for irregular surfaces, etc. (a) To OBTAIN A DEVELOPMENT. First draw the views from which the measurements of the lines can be made. The surface may be imagined as opened on any lines but its different parts should, so far as practicable, be represented • as attached to each other, — each being so placed that if the dev. were cut out upon its outer line and properly bent, it would form or envelope the object, surface for surface, line for line. In obtaining lengths from the views, transfer the measurements with dividers. Remember that a line shows its true length only upon a plane to which it •is ||. If not thus shown, its true length must first be determined. The dev. may be placed in any convenient position. In some cases it is possible to place it so that one set of dimensions may be projected, as in Fig. 151. It is sometimes desirable to attach it to some line or lines of a view (Figs. 154, 159-161), prac- tically as revolved about those lines. (b) The reproduction of forms in thick paper, by cutting and bending devs., will prove an excellent aid to correct solutions. The surfaces may be held in position by providing paste laps (see Fig. 151) or by using gummed binding. To obtain neat edges the folding lines should be lightly scored with a sharp- pointed knife. 71 72 ESSENTIALS OF MECHANICAL DRAFTING (c) In practical work, allowance must be made for seams, laps, thickness of material, etc. Economy in cutting is also important. Material for double curved, or warped surfaces, is cut to patterns of assumed cylindric, conic, or triangular portions and then beaten, pressed, or stretched to the required form. In some cases the material is stamped by dies, or spun to form in a lathe. 82. Objects Having Plane Surfaces. (a) Peisms. The dev. of a right prism consists of two similar polygons for the bases and as many rectangles as the prism has sides. In obtaining the dev. of the hexagonal prism (Fig. 151), observe that the lateral edges are || to V, and the base edges to H. The front and rear base edges are also 1 1 to V. First set off the true lengths of the lower base edges upon a line, A-A. At pts. A, B, C, etc., draw _Ls. Upon these set off the lengths of the vert, edges and connect the pts. to complete the lateral surface. Attach the bases to any side. 7"r" : 6"E" Fig 151 To develop the prism after the portion above an oblique C. P. 1-4, has been removed, obtain base line A-A, lower base, and _Ls as in preceding. Now assuming the surface to be opened from edge A-1, this edge will be the outer verts. A-1, in the development. Beginning with A-1, transfer the lengths of the verts, to the corresponding lines in the dev. Similarly transfer lengths of base edges G-4 and G-5 to the hor. through G. The lines joining 1, 2, 3, 4, and 5, 6, 7, 1 will be the dev. of the line of intersection of the C. P. with the lateral surface, and must, therefore, agree in length with the corresponding lines of the sectional view. Such lines should always be compared with dividers. The method of obtaining the portion G4'5 of the upper base is evident. To complete the dev., draw a figure similar and equal to the true shape of the oblique surface by Art. 56, attaching it to any side, as B C 3 2. DE^ELOP^IE^"T of surfaces 73 (b) Pyramids. The dev. of a right pj-ramid consists of a polygon for the base and as many As as the pyramid has sides. The slant edges of the rect- angular pyramid (Fig. 152) are equal, but are not projected in their true length in either view. Assuming the surface to be opened from one of these, obtain its true length as in Art. 71, and with this as rad., describe an indefinite arc DEBCD. Upon the arc set off the successive lengths of the base edges. ' Join these pts. and each to the center A, the vertex, to complete the lateral surface. Then attach the base. Or, lay off first one side of the base as E-B, and with E and B as centers and the true length of the slant edges as rad., describe arcs to inter- sect at A; then proceed as in preceding. Since the altitude of each side is ± to a base edge, the vertex A could be determined by setting off the true length of an altitude, as A-F, ± to its corre- sponding base edge E-B, as indicated. In pyramids whose inclined edges are not equal, it is necessary to obtain the true length of each separately, and to construct the As by Art. 37, joining them on their common sides. BT'f To develop the pyramid (Fig. 152) when truncated, first obtain dev. of entire pyramid. Then obtain the true distances of pts. 1, 2, 3, and 4, from the vei'tex, or base (Art. 71). Set off these distances upon the corresponding fines in the dev. and join the pts. thus found to complete the sides. Finally, copy the true shape of the top surface from the sectional view, attaching it to any side, as BC21. 83. Objects Having CyUndric or Conic Surfaces. (a) Cylinders. Since the elements of a right circular cylinder are equal and _L to the bases, the dev. of the curved surface (Fig. 153) will be a rectangle whose height is equal to the length of the elements, and base equal to the length of the circumference. The length of the base edge may be obtained as in Art. 32, and laid off 74 ESSENTIALS OF MECHANICAL DRAFTING Fig 153 upon a line A-A. At the ends of this hne draw J_s and complete the rectangle. As each of the bases would touch the dev. of the curved surface at but one pt., it is not necessary to attach them. To develop the curved surface when cut off on line 12 3 4 1, obtain base line A-A, divide it into 12 or 24 equal parts according to the number of elements assumed upon the cylinder, and through the pts. draw _Ls. The length of the cut elements may then be transferred from the front view as in the case of the edges of a prism. The curve traced through 1, 2, 3, 4, 1 will complete the lateral surface. See Art. 18(a). When, as in (B) Fig. 159, neither end is a O, both will develop as curves. It is necessarjr, therefore, to assume some O of the surface whose development Pattern for A and A', )r one-half of E.. Fig. 1 54 Pattern for F U.5e right half for B, /left half for B' »J/ DETAIL OF PILASTER CAP Make 4 16 Oz Copper DEVELOPMENT OF SURFACES may be used as a base line upon which to set off the true distances between the ele- ments and from which to set off their lengths. Such line may be obtained by a C. P., a-E^, taken _L to the ele- ments as shown. See Fig. 160. Fig. 154 illustrates dev. of an object with lindric curved sides, true distances between elements of surfaces and (B) are determined upon A-E and L-0 in the elevation and their lengths in the plan. The distances between the elements of (E) and (F) are identical with those of (A) and (B). To determine the true distances between the elements of (C) and (D) it is necessary, since these elements are not ± to V, H, or P, to obtain the profiles of (C) and (D) by aux. views or sees, upon planes ± to the elements, as indicated. (b) Cones. Since the elements of a right circular cone are of equal length, the dev. of the curved surface (Fig. 155) will be a sector whose radial sides, A-D, are equal to the true length of an element and whose arc, D-D, equals the length of the circumference of the base. To determine the length of this arc, obtain iV or TT of the base circumference and set it off 12 or 24 times in the dev. To develop the portion below the parabolic sec, obtain the base arc D-D, divide it into 12 or 24 equal parts according to the number of elements assumed upon the cone and draw the elements. Then find their lengths when cut off, as in the case of the pj-ramid. Art. 82(b). Since the left element is 1| to V, the true distance of pt. 1 from the vertex A is seen in the front view. The true distances from A of intermediate pts., as 2 and 5, are not seen in either view, but as the elements of the cone are equal, hors. from these pts. to A-B will determine upon the latter the required distances. (See Art. 71(b).) Having transferred these to corresponding elements in the dev., trace the curve through them. The method of developing a right circular conic surface whose ends are not in planes J_ to its axis is evident from Fig. 161(b). In a conic surface other than right circular as in Fig. 156, or one whose vertex is inaccessible as in Fig. 161, it is necessary to assume the surface to be composed of plane As whose sides are elements and whose bases are short chords of the base of the cone. The method, called development by triangulation, is thus identical with that used in developing a pyramid with unequal slant edges. Art. 82(b). To develop the oblique elliptic cone (Fig. 156) obtain equidistant elements as shown. Assuming the surface to be opened upon A-D, the line A-B will be the 76 ESSENTIALS OF MECHANICAL DRAFTING C. L. of the dev. and may be drawn directly at A'-B', equal to its true length A^-B^. The true lengths of elements A-1, A-2, etc., may be found by revolving them 1 1 to V. To avoid confusing the views, however, a separate diagram of true lengths may be constructed. The true length of any element will be the hypot- enuse of a right A whose base is equal to the length of that element in the plan and its altitude equal to the vert, height of one end pt. above the other in the ele- FiG 156 Fig. 157 vation. Hence, draw A-X J. to the base line and equal to the altitude of the cone; then lay off X-1, X-2, etc., equal to A^-1^, A^-2''^, etc., and connect pts. 1, 2, etc., to A. Observe that this is equivalent to revolving the elements 1 1 to P. Now with A' as center and rad. A-1 draw an arc across A'-B'; with B' as center and rad. B^-1^ intersect this arc at 1'. Then A'-l' will be the developed position of A-1. Continue this operation until all elements have been determined and trace the curve B'1'2', etc., through the base pts. as indicated. The figure also illustrates the method of developing the cone when truncated. DEVELOPMENT OF SURFACES 77 84. Objects Having Double Curved, or Warped Surfaces. (a) Fig. 157 illustrates the method of developing a double curved surface by assuming portions, as A 7 8, to be cylindric, also by assuming portions between the O s to be conic. Fig. 158 illustrates a transition piece of piping, A, whose surface is warped, being neither cylindric nor conic. Such surfaces can be developed approxi- mately only, by triangulation. Art. 83(b). Fig 156 Half Pa+te-m for A rm f i i- • i i • . ihe surface may be divided into measurable As as shown. In the plan divide half of the base and curved end of the top edge into the same number of equal parts, at least six. Observe that as the planes of both edges are ob- lique to H, each must first be revolved II to H, or V, and then back to its oblique position. Join the corre- sponding pts. in each with full lines 1-A, 2-B, etc.; also draw lines 2-A, 3-B, etc., dotted for contl-ast. Next construct the diagram (a) for the true lengths of the full lines, noting that neither the upper nor lower pts. are on the same levels. Similarly construct diagram (b) of true lengths for the dotted lines. As the surface is symmetrical about the C. L. 1-Y, the hne 1-X may be drawn directly at I'-X' in the dev. (c), equal to its true length 1^-X^. With 1' as 78 ESSENTIALS OF MECHANICAL DRAFTING center and rad. 1-A of diagram (a) describe an arc across I'-X'. With X' as center and rad. X-A of the plan, intersect this arc at A'. Then I'-A' will be the developed position of line 1-A. With A' as center and rad. A-2 of diagram (b), describe an arc. With 1' as center and rad. 1-2 of the revolved base, intersect this arc at 2'. Then A'-2' will be the developed position of line A-2. With 2' as center and rad. 2-B of diagram (a), describe an arc. With A' as center and rad. A-B of the revolved top edge, intersect this arc at B'. Then 2'-B' will be the developed position of 2-B. Continue these operations until all points have been determined and complete the dev. as shown. 85. Intersecting Surfaces. In intersecting parts of an object, the line of intersection being common to both surfaces will be developed in each. As this line is determined by the pts. in which the edges, elements, or other lines Fig. 159 of each part intersect the surfaces of the other, and the dev. gives the true length of every line of every surface, it follows that these pts. may be found by trans- ferring the true lengths of the lines which determine them in the views, to the corresponding lines drawn in the dev. (a) The lateral surface of (B) (Fig. 159) is developed as explained in Art. 83(a). The dev. of the rear half only is shown and is attached upon element A-B for convenience in transferring lengths. The distances between the elements are seen in the revolved sec. of a-E^ and transferred to the base line E-a, as shown. (b) The method of developing the lateral surface of (C) is evident. Lines as 1-2 and 3-4, which give intermediate pts. of the line of intersection, are found by obtaining their true distances from the || edges in the revolved sec, and their true lengths, from either the front or top view. The dev. of the upper half only is shown. DE\TELOPMENT OF SURFACES 79 (c) To develop the lateral surface of (A), first obtain the dev. of the entire surface, that is, a rectangle (rear half only is shown). Assume surface to be opened from right element. This element cuts the surface of (C) in I and M. The opposite element cuts the surface of (B) in F and B and will be in the center of the dev. From these elements the positions of the other elements of (A) which cut the surfaces of (B) and (C) may be determined. The distances of these elements from the outer elements are seen in the top view, and the distances of their points of intersection from the bases of (A) are seen in the front view. (d) Fig. 160 illustrates a pipe elbow with conic and pyramidal flanges; Fig. 161, two other common forms, the methods of developing which are indicated. riG. 161 CHAPTER IX MECHANICAL PICTORIAL DRAWINGS 86. Character and Purpose of the Drawing. It has been noted that two views at least are required to show the exact form, size, and relation of all lines and surfaces of an object, and that two only of its three dimensions can be shown in their actual proportion and relation in one view. It is sometimes necessary, however, to represent an object or detail by a single oblique view having a more or less pictorial effect, while at the same time showing the relative proportions of its principal lines or dimensions to a scale. Compare Figs. 162 (a), (b), (c), etc.; 163 (a) and (b); 169 (a) and (b). (a) F D ORTHOGRAPHIC PROJECTION Fig. 162 (b) (c) (d) G ^ & B J^ ^ UN ^ ^ A i \ ' ^ F 1 ~~~^ ^-^ 1 D F C F D ISOMETRIC PROJECTION OBLIQUE PROJECTIONS f- -^ c c A 1 D TRUE PICTORIAL REPRESENTATIONS Such representations are made in place of or supplementary to the usual views for the purpose of general illustration, or to describe details of machine, furniture, and building construction and assembly more directly and clearly. They are also extensively employed for catalog illustrations and Patent Office drawings. The methods ordinarily used are those of Isometric Projection and Oblique Projection, the choice depending upon the form of the object and the particular effect desired to be given. A drawing made by either method is never a true picture, since || lines are always represented by Ijs as in orthographic proj., 80 MECHANICAL PICTORIAL DRAWINGS 81 while in a true pictorial representation the apparent convergence of receding l|s is shown. Hence, such drawings often give an unsatisfactory effect of distortion. 87. Isometric Projection. If a cube, placed as shown in Fig. 162(a), be turned 45° about a vert, axis, then forward until the diagonal of the cube through A is ± to V, all of the edges will be equally inclined to that plane, and being equal in length their projs. upon it will be equal. Such view is an isometric (equal measure) projection of the cube. (See Fig. (b).) When a rectangular object is thus related to the plane of proj., it will be seen that each of its edges will be || to one or another of three mutually _L lines, as A-B, A-C, and A-D, which correspond in direction to the axes or principal dimensions (1, b, and t) of the object. These lines are called the isometric axes, and their projs. form equal Zs of 120° about a pt. A. All lines of the object coinciding with or || to the isometric axes are called isometric lines; all others are non-isometric lines. The planes determined by the isometric axes and all planes II to these are called isometric planes. Note that isometric lines onl}', make equal Zs with the plane of proj.; hence, equal distances upon such lines only will be equally foreshortened in the view. (a) Instead of obtaining the view by the methods of Art. 72, an isometric is usually constructed by setting ofT the limiting pts. of the lines directly upon isom. lines and joining the points thus determined. Measurements must never be made upon non-isometric lines. The foreshortening of the isometric lines, which is about .81 of full size, is usually disregarded and the measurements made equal to the true lengths of the corresponding lines of the object, or to some other common scale. One of the axes is usually assumed as vert.; the others, therefore, are at 30° with the hor. direction, as in Fig. 162(b). When the lower surface of an object is to be visible the axes are reversed. Invisible lines are omitted unless they give necessary information. When dimensions are given, the dimension and extension lines must be isometric. See Fig. 163. In general, sections and breaks should be taken in isom. planes. See Figs. 164, 170, 194; also invisible section. Fig. 165. 88. To draw the isometric of a rectangular object. (a) A Cube. Fig. 162(b). From any pt., as A, draw indefinite lines A-B and A-C at 30° and a vert. A-D. Upon these set off the given dimensions of the object to the desired scale. Since each edge is || to one or another of the isom. axes, it will be || to the corresponding line in the drawing, and the rep- resentation may be completed as shown. It is evident that the isom. could be started from any assumed pt., as D or E. (b) An Object with Rectangular Details. Fig. 163. Starting at any pt., as A, draw the isom. of the bottom board (A), obtaining all measurements from the front and top views. 82 ESSENTIALS OF MECHANICAL DRAFTING n" F e JA 4- 1-14 G -lot CM tnIS LI ■0*2 1 Fig 163 F i3 B y I 1 1 1 \ i c 2 _\ Fig. 165 MECHANICAL PICTORIAL DRAWINGS 83 Details must be built up from the surface or surfaces which they intersect. Thus to draw block (B), locate a corner, as E, and draw the line of intersection E 1 2 3; then proceed practically as with (A). To draw (C), one of its lower corners, as F, must first be located upon the top face of (A) by means of co-ordinates B-4 and 4-F; that is, by measuring the distance of F from B along lines which will be || to \ / (n) two of the isom. axes. To draw the recess in the front of (C), locate a corner, as G, by co-ordinates F-5 and 5-G, and- proceed as before. Corner H is de- termined in like manner. 89. To draw the iso- metric of an object involv- ing non-isometric figures. The form and position of an integral part of an object are frequently such that some or all of its lines will be non-isom. Such lines are determined in the isom. by co-ordinates || to two or to all three of the axes. (a) Straight Lines. In Fig. 165 each of the inclined lines in (a) will be oblique to two of the axes and hence determined by co-ordinates 1| to those axes, as shown. See also oblique hues in Figs. 171, 172, 194. In Fig. 166 the hne A-B M'ill be oblique to the three axes. The end B, was located in this case by co-ordinates A-1, 1-2, 2-B II to those axes. The end pts. of C-D were de- termined in Uke manner. (b) Polygons. In Fig. 167 two sides only of the hexagon ABCDEF wiU be isom. Each pt. may be referred by co-ordinates to two of the isom. axes, as in (a), or to the sides to those axes. By placing this rectangle with one of its lines |1 to or coincident with its isom., one set of dimensions may be projected to the required figure, as shown. (c) Curves. The method of determining a curve in an isom. drawing is identical with that of a rectilinear figure save that, in the absence of vertices, pts. must be assumed in the curve and referred to the axes, as illustrated in Figs. 168(a), 169, 170. of a circumscribing rectangle whose sides will be 84 ESSENTIALS OF MECHANICAL DRAFTING The projection of a O in an isom. plane is an ellipse, in determining which it is convenient to circumscribe a square. Fig. 168(a). As the axes of the ellipse coincide with the diagonals of the isom. square, their end pts. 8, 6, 5, and 7 could be determined by co-ordinates and the curve described by Art. 64(b). See also Art. 64(d), Note 2. Fig 168 " An approximate method of drawing the ellipse by circular arcs, which is usually sufficiently exact, is shown in Fig. 168(b). The centers are the inter- sections of _Ls to the sides of the square at the middle pts. The application of this method to the rounding of corners is shown in Fig. 169. The construction at D determines the radii for all. To draw a curve not in an isom plane, as A-B (Fig. 170), determining pts. must be located by co-ordinates || to the three axes, as indicated. Screw threads are usually represented conventionally as in Fig. 173. Fig. 169 (d) Solids. The method of drawing an object by inscribing it in an isom. solid is evident from Figs. 167, 171. When it cannot be thus inscribed, pts. must be referred to the axes as indicated in Fig. 166. In order to preserve the symmetrical appearance of the piece in Fig. 172, it was necessary, either to draw the top view turned through 45°, or to take the hor. measurements for the groove centers on 45° lines instead of on hors. and verts., MECHAXICAL PICTORIAL DRAWINGS 85 'A ■+M-- FiG 171 Fig. 172 86 ESSENTIALS OF MECHANICAL DRAFTING as shown. Observe that the outer elements of conic surfaces would be tangent to the ellipses and not drawn to ends of the axes. See A and B, Fig. 173. To determine the outlines of surfaces of double curvature, obtain a series of sections and draw the required curve tangent to the isom. of these, as indicated in Figs. 174, 175, also 149. 90. Oblique Projection. An oblique projection is obtained by means of II projectors oblique to the plane, the object being so placed that two of its axes or principal dimensions are || to the plane. In the case of a rectangular object, as a cube, the view, therefore, gives the true shape and size of its front and rear faces and two dimensions of the object. All lines ± to the plane are projected as || lines whose direction and lengths depend upon the direction of the projectors. They are usually drawn at 30°, 45°, or 60°, up or down to left Fig 173 or right, and made equal to the full scale length of the corresponding lines of the object as in Fig. 176, or foreshortened, usually half, as in Fig. 177. The latter gives a better pictorial effect, but requires the use of two scales, as is evident. When the receding ±s are foreshortened one-half and inclined at 45°, the view is sometimes called a cabinet projection. The axes in oblique proj. are thus a vert. A-B, a hor. A-C, and a line A-D at 30°, 45°, or other Z . Measurements must be made only upon these axes or lines || to them. Curves, and lines not || to the axes, must be determined by co-ordinates || to the axes, as in isom. drawing. A circle not || to the plane of proj. will be projected as an ellipse, which may be obtained as in Art. 89(c). MECHANICAL PICTORIAL DRAWINGS 87 An obvious advantage of oblique proj. over isom. lies in the possibility of representing certain curved and irregular surfaces in their exact shape and size. See applications of principles in Figs. 178, 181. Fig 175 Fig. 176 Fig. 177 Fig. 178 91. Shade Lines, Shadow Lines, and Line Shading. In finishing, visible edges between light and dark surfaces are frequently shaded, as in Figs. 162(b), 164, 167, 169, 178(a). The indication of these shade lines adds relief to the drawing and increases its pictorial effect. 88 ESSENTIALS OF MECHANICAL DRAFTING In isometric drawing the rays of light are assumed 1 1 to the plane of proj . and at 30° down to the right. Hence, all rectangular objects and parts, whose lines are || to the axes, have their shade lines in the same relative positions as in the cube. Fig. 162(b). In oblique proj. the rays are assumed || to the diagonal, C-E, of the cube. Fig. 162(c). In determining the shade lines, the shadows are disregarded. Elements of curved surfaces are generally not shaded. Edges of cylindric surfaces may be shaded as in Figs. 169, 178(a). Instead of shade lines, shadow lines may be applied as in Art. 23. (See Figs. 171, 178(b).) Another method is to shade the nearest edges as in Fig. 172. Line shading may be applied as in Art. 24. CHAPTER X WORKING DRAWINGS 92. Character and Purpose of the Drawing. A working drawing is a mechan- ical drawing, containing all information as to form, dimensions, construction, material, finish, etc., necessary to the workman or mechanic in making or building the object which it represents. See Art. 1. To convey this information readily, the drawing must be accurate; as clear, simple, and direct as possible; and in accordance with shop and drafting practice. It must express the idea completely and definitely and contain noth- ing that is unnecessary, ambiguous, or misleading. In commercial drafting utility of the drawing and economy of production are the ends sought for in all cases. The nature of the views of which the drawing is composed is explained in Chap. V. 93. Types of Drawings. In complicated objects composed of different pieces or parts, certain features would inevitably be hidden or not clearly shown; hence, in such cases, two types of drawings are required — namely, general or assembly drawings and detail drawings. (a) The purpose of a general drawing (Figs. 179, 184, 185, 202) is to illustrate the design of the subject as a whole and to show the relative positions of the different pieces composing it. It may include the complete description of some or of all of the pieces, or give merely such information as may be necessary in assembling them or erecting the object. As a rule, it should be as free from representation of minor detail and hidden parts as possible. (b) A detail drawing (Figs. 180, 203-206,210-213) shows each piece by itself and gives all information necessary for making it. In simple objects, full instructions would ordinarily be given in a general drawing, sometimes called a detailed assembly. See Figs. 179, 184, 185. Note. — The table shown in Fig 179 was separately detailed (Fig. ISO) for purpose of comparison with the general drawing and to illustrate certain methods of arrangement, dimensioning, etc. Detail drawings may be shown on the same sheet with the assembly, grouped on separate sheets, or each piece shown on a sheet by itself, depending upon the character of the object, size and number of parts, etc. In general, details of the pieces of one part of an object should be grouped apart from similar groups of other parts (Fig. 205) ; and so far as possible the arrangement of details of related pieces should be such that reference may readily be made from one to the other. Figs. 180, 205. It is often desirable to show related pieces of a part as assembled. Figs. 181, 182. In drawings of machinery the special information required by the different workmen, as the pattern maker, the blacksmith, and the machinist, is often 89 90 ESSENTIALS OF MECHANICAL DRAFTING ! o a. r H z o I- o U £0 t !<■ II -IN >■ DC fll < 7^ rr o CQ AA/' 3 =12 in ////////////////////////////////, lO'" SPEED LATHE ASSEMBLY DAY MF(3 CO BOSTON Scolo 6-.|Ft Dat« 1-29-IS Dr ^i»^ Tr ^g^^ Ch.T^f-^ ^p.A^ SL-IO- WORKING DRAWINGS 109 and as an aid in the solution of constructive problems, is of great importance. To the student, careful sketching is fully as valuable, as a means of acquiring ability to make and to read working drawings, as instrumental drawing. (a) In making working sketches from objects the following order should be observed : — 1. Separate the parts, if necessary, and sketch the views of each in detail. Begin with the main C. Ls. and principal surfaces, blocking in first the larger details of the piece and proceeding in like manner down to the minor details. See Fig. 209 (a), (b), (c). Distances and directions are generally determined by eye. Co-ordinate paper, ruled in I" or I" squares, is often used and affords a more ready and accurate means of obtaining the desired proportions, etc. For use of pencil see Art. 9(b). The number and character of the views should be such as to express all required facts clearly. Leave nothing to memory. Make the sketches large enough to prevent crowding the notes and figures. The sketches should be intelligible to any one familiar with working drawings, and should enable the scale drawings to be readily made from them without having to resort to the object. Isometric or oblique views may often be used to advantage. 2. Indicate the finished surfaces and put on the necessary extension lines, dimension lines, and arrowheads. 3. Obtain the dimensions by careful measurement of the object and put on the explanatory notes. Each piece should be dimensioned independently of the others. (b) Measuring Objects. In taking measurements, a foot, or a two-foot rule may be used for ordinary work and steel rules, gages, etc., for fine work. Obtain distances from trued or finished surfaces whenever possible. In obtain- ing inside and outside diams., calipers may be used. In an object of varjang diams. (Fig. 174(a)), measure the diams. at a sufficient number of pts. and locate these diams. by measurements || to the axes. In measuring an irregular form, as the table leg (Fig. 198), it is necessary to es- tablish a base line, as A-B, by means of a triangle or a carpenter's square, and to determine the lengths of ± offsets to it with a foot rule In locating a circular hole, measure to edge of hole and add half its diam. In ordinary finished surfaces, take the nearest 32d; in rough work, the nearest 16th; in finely finished objects and parts, absolute exactness is necessary. 105. Making Scale Drawings. (a) Having completed the sketches, decide upon the number and arrange- ment of the views to be shown in the drawing, and the necessary scale. To aid in determining the scale and the locations of the chosen views upon the sheet, a rough layout sketch indicating the general outlines, main C. Ls., and margins, should be made. Thus, from the layout (Fig. 207) for the bearing shown in Fig. 208, it will be seen that the minimum space required horizontally, not including spaces between views and from margins, will be 2a + 2b. Simi- larly, that required verticalh' will be c + d + 2b. From these two sets of Fig. 203 Fig. 204 rie.205 1 1 , ; ' \r ' 1 L -> =1 f% H* /; d ' ^*r: -i,^^ < It: t < t s E iH^ ^ )-*« -HO ^ T ^, '1 Oin o ftjVy I - -1 ■ o :\\ 0- nw- J__ V V ^-I 1 I:.ij \ I-* \ 1 r o I en Fio. 206 114 ESSENTIALS OF MECHANICAL DRAFTING , 1 » -x-4- \ . J . \. a J *b-i4^b^ 7 " ) [ I 1 T 3 ^ J ; I , 1 1 1 1 1 LAYOUT SKETCH \ FOR PLAIN BEARING TITLE SPACE Fig 207 5^ Core. 5 Deep Cast Iron Patt*B-5 FiG.208 PLAIN BEARING Scale: Full Size Date- 3-7-18 Dr ^e. Tr K/E.. Ch. AP. App. CF. M-I8-I WORKING DRAWINGS 115 dimensions, with due allowance for spaces and title, the scale maybe determined, as in Art. 100(c). Having decided this, the dimensions A, B, C, and D for the locations of the C. Ls., should be estimated, proper allowance being made where necessary to preserve a well-balanced sheet. The same general directions should be observed in all drawings, as lack of provisions for the necessary views, etc., may lead to errors impossible to rectify without re-drawing the entire sheet. (b) Having completed the layout, the scale drawing (Fig. 208) may be begun. Four stages in penciling are indicated in Fig. 209. For general working instruc- tions, see Art. .3. In drawing from objects, the detail drawings are first made and the general drawing built up, piece by piece, from them. In designing four stages in penciling Fig 209 or planning objects to be built, the general drawing is usually first begun and the detail drawings worked out from it. In some cases it is necessary to carry along both detail and general drawings at the same time. 106. Tracing and Blue-printing. (a) Instead of inking the pencil drawmg, tne finished drawing is usually obtained by tracing in ink upon tracing Hnen or paper fastened over the original. This tracing is then filed as the permanent record of the construction and used for making blue-prints or other copies for shop and general use. The original is sometimes penciled directly upon the tracing material. C/) r j_ o I Fig. 210 [ -fl — -Hf - ^. KT*. y-J V* ai? X c o /^ ^) i o X u d — N X -1 \, g ll. J -1 g =12 J LO to < '■ 1 XX -?* ^ c- ^1 H> Vi :^ Tie 211 riG.212 I r 1 -1 - -4— l-N N ''J I I j/TUMQim/) 616 ( 5^r t- D Z uj D O "1 << < '(0,(0, to (O X.X I _1 c ? < 5s < a. U,V- < *i-l < lU 5 4i . r.j n <- 1.-1 i^ i^ cr- lT, o - PJ 3,0,0 O O O O O C -1 Fig. 213 120 ESSENTIALS OF MECHANICAL DRAFTING The dull side of tracing linen takes the ink better, but erasures can be made more readily upon the glazed side. Either will take the ink more readily if rubbed with a cloth and chalk powder. Penciling should be done on the dull side. Carry but little ink in the pen, and make the lines somewhat wider than ordinary as the lines print finer than those of the tracing. As the linen contracts and expands unevenly, tracings that cannot be com- pleted the same day should be inked by sections. Make ink erasures with hard eraser, rubbing gently to avoid injury to the surface. Restore the smoothness by rubbing with soapstone or a smooth piece of bone. Pencil lines may be erased with soft eraser. (b) Blue-prints are usually taken in a printing frame having a glass front and removable backboard. The tracing, or original drawing if on translucent material, is placed with the drawing side next to the glass and its under side in contact with the chemical coated surface of blue-print paper. The glazed side of the frame is then exposed to direct sunlight, which, penetrating the part of the tracing material not covered by the ink lines, causes the chemical not thus protected to change color and to adhere permanently to the paper, while that under the lines remains unchanged. After suitable exposure the paper is removed from the frame and soaked in water, which dissolves the unfixed chemical, leaving white lines on a blue ground. 107. Checking Drawings. It is customary not to permit a drawing to be worked from until it has been checked by a careful, systematic examination, and approved by the head draftsman. In checking, it is well to assume every- thing to be incorrect until proved to the contrary. The following order may be observed: — See that each piece has been represented and that its views are properly related. Check views of each piece for correct and adequate description of form and -construction. Note if C. Ls. and all necessary dimensions and notes are given. Scale every dimension, and verify overall dimensions by computation: Compare the figures on all parts that are to fit together. Check measurements in details and assembly and note if they agree. Finally, see that all items required to be recorded in the title and bill of material are complete and correct. 108. Reading Drawings. Ability to read working drawings rapidly and intelligently is quite as important as skill in making them. This ability can be acquired through the study of such drawings and comparison with the objects represented; through the execution of drawings from objects; through the making of mechanical pictorial drawings and developments, from good examples; and by making the objects, that is, working from drawings. In reading a drawing, first fix in mind the general shape of the main body of the object, observing if the outline shows it to be rectangular, cylindric, etc., or a modification of such forms. Then observe modifications of the general shape, proceeding from the more important details down to the minor details. Note carefully the conventional methods employed in the representation and complete mentally the graphic statement of what is required. WORKING DRAWINGS 121 Endeavor to visualize the object; to see in each view, not mere lines, but the object itself as if standing out of the paper. Regard the front view as the object directly in front of you; in looking at the top view imagine your- self looking down upon the object; in looking at the side view imagine yourself as viewing the object in a direction at right Zs to the front, and so on. Finally, note the dimensions, and specifications as to materials, finish, etc. All information as to sizes should be obtained from the figured dimensions and specifications; rarely by measuring the drawing itself. It is evident that full knowledge of the form, size, and relation of the lines and surfaces of each part of the object represented can be secured only through the information shown in all the views taken together. CHAPTER XI HELICAL CURVES, THREADED PARTS, AND SPRINGS lOQ. Helices. If a pt., A (Fig. 214) be imagined to move along the generating line, A-12, of a surface of revolution, while the line itself revolves about the axis, the pt. will generate a line of double curvature called a helix. The distance that the pt. advances, measured |j to the axis, during one revolution of the line, is called the pitch of the helix. If the rate of motion of the pt. and generating line of a cylindric helix be uniform, that is, if the pt. advances J, ^, or other fractional part of the pitch distance during the same fractional part of a turn, the helix is uniform or equable; if otherwise, it is variable. Again the curve is a right-hand or left-hand helix, according as the generating pt. rises to the right or to the left in the front half of a turn when the axis is vert. The helix has many applications in mechanical and architectural design, notably in screw threads, some forms of springs, winding stair rails, etc. (a) To DRAW A RIGHT-HAND EQUABLE CYLINDRIC HELIX. Fig. 214. AsSUme and represent equidistant positions of the generating element, as 1, 2, 3, etc. Divide the pitch distance, as A-A^^, into the same number of equal parts and draw hors. through the pts. of division to cut the elements at A^, A^, A^, etc. Assuming A to be the generating pt., A^, A^, A^, etc., will be twelve pts. of the desired curve, for evidently when the genei'ating ele- ment has made tV of a revolution the advancing pt. A will have moved tV of the pitch distance and will, therefore, be in the element 1 at A^. When the element has moved | a revolution the pt. A will have moved I of the pitch distance and will be in the element 6 at A®, and so on. As the curvature is more abrupt at the outer elements, additional pts. should be determined by sub-division as shown. Having fixed the pts. with the needle, trace the helix through them freehand. Fig. 214 DEVELOPMENT OF HELIX All CYLINDRl|C HELIX 122 HELICAL CURVES, THREADED PARTS, AND SPRINGS 123 Points for other turns of the curve may be located by stepping off the pitch distance upon the elements, from the pts. of the first turn. The manner of obtaining parallel or double helices is evident; also the method of drawing conic or other helices. (b) In ruling and inking the helix observe directions given in Art. 18(a). "When several cylindric helices of the same pitch are to be drawn, a templet of a half turn, as A-A®, may be made. (c) The development of a cylindric helix of one turn is the hypotenuse of a right A whose base is equal to the circumference and whose altitude equals the pitch, as shown. (d) Any desired motion of a pt. may be plotted in a development of a cylinder and then projected back to the view, as for example in designing a cam for converting circular into reciprocating motion. 110. Screw Threads. If a cylindric bar be revolved at a uniform velocity upon its axis, in a lathe, and the point of a V-shaped cutting tool be pressed against its surface and moved at a uniform rate parallel to the axis, the tool will cut a helical groove, V-shaped in section. If this groove be cut so that a similar projecting portion is left upon the bar, a V screw thread will be formed. Fig. 215(a). Similarly, if a square groove be cut and a square projecting portion left upon the bar, a square screw thread will be formed. Fig. 215(c). Fig. (b) illustrates a sec. of a V-threaded and Fig. (d) of a square-threaded hole. Observe that the helical lines of the threads correspond to the invisible lines of the screws. The V thread is commonly used on screws for fastening purposes; the square thread generally on screws for transmitting motion in the direction of the axis. All other threads are modifications of the V and square forms. A screw thread is right- or left-hand according as its helices are right- or left- hand. Thus a right-hand screw would turn around to the right (clockwise) in advancing or entering the part into which it is inserted. The diameter of the top of the thread is called the nominal diameter of the screw. The diam. of the bottom of the thread is the root diameter. The distance from the root to the top of the thread measured J_ to the axis is the depth of the thread. The distance between the centers of adjacent threads measured II to the axis is called the pitch of the thread. The term "pitch" is often used to designate the no. of threads per inch; thus "14 pitch" means 14 threads to the inch. The distance that the screw would advance in one turn is called the lead of the screw. In a single thread screw the lead equals the pitch; in a double or triple thread it is two or three times the pitch. (a) Multiple Threads. Screws are generally right-hand and single thread as shown in Fig. 215 (a), (c). If the pitch of a screw having the diam. and thread sec. shown in Fig. (c) be made say two or three times as great, the increase in the depth would obviously be such as to weaken the screw at the root. There- 124 ESSENTIALS OF MECHANICAL DRAFTING V SCREW THREAD (o) y^r^V^'"'^ (c) A SQUARE SCREW THREAD jingle ■^,t^^\\ (e) Double J-d-l— Roo-tj Dia^ I (b) U— Nomirial Dia.— -I d4— Root; Diar , - — Norn i rial DiO: — -| (d} L= Lead of Screw. P= Pitch of Thread d = Depth of Th'd Fig 215 P = .24 VD + 62S- 175 d=.65P F = | (d) ^— P-^, jSHARP V TH'D d= 866 P Tw^^jm^ 7W//m^/^ y/my. SQUARE TH'D d= sP + oi W=5P P= OS D +.04 nearly d= 64 P D = Nominal Diam of Screw P = Pitch ^y, n = No of Threads per inch d = Depth of Th'd d= 5P + 01 W= 3707 P- OOS2 Fig 216 HELICAL CURl'ES, THREADED PARTS, AND SPRINGS 125 fore, in designing a screw of which the lead shall be two or three times the pitch, instead of cutting a single thread, a second or third independent parallel thread would be cut. Such would be a double, or triple thread screw. Fig. (e) represents a right-hand double square thread of the same diam. and pitch as the single thread of Fig. (c). Observe that in a single thread screw, the top on one side is diametrically opposite the bottom on the other, while in a double thread the tops are opposite. (b) To DRAW A SCREW THREAD. The diam., pitch, and thread sec. being given, as in Fig. 215(a); first obtain the elevation and an end view, or half of revolved base of a cylinder whose diam. equals the nominal diam. of the screw. Lay off the pitch A-C and draw the sec. ADC. Then beginning at A, draw the helix A-B-C-, etc., for the top of the thread by Art. 110(a). To obtain the root helix, draw a semicircle in the base view concentric with the first, obtaining the rad. by projecting from D. Then beginning at D and using the same pitch, proceed as with the outer helix. Observe that the outlines of the thread come outside of the V sec. and are tangent to the helices. When the pitch is small, they practically coincide with the sec. outline. (c) Standard Proportions. In the preceding figures, the threads were represented with large pitch in order to show the construction more clearly. The proportions of the threads most commonly used and the formulse for obtain- ing them are given in Fig. 216. In this country the Z of the V thread is usually 60°, but for general work the tops and bottoms are flattened as shown in Fig. (a). The following table gives the U. S. St'd no. of threads per inch, for diams. from i" to 4|". U. S. Standard Screw Threads Diam. Thds. Diam. Thds. Diam. Thds. Diam. Thds. Diam. Thds. Screw Per In. Screw Per In. Screw Per In. Screw Per In. Screw Per In. 1 4 20 5 11 1 8 If 5 3 ^ A 18 1 1 1 6 11 1| 7 11 5 3i ^ f 16 3 1 10 n 7 2 4* ^ 31 TW 14 H 10 If 6 2i 4| 3i 3 1 2 13 i 9 14 6 2i 4 4 3 n T7 12 1 9 If 5| 2| 4 4i 2| On square threads the no. of threads per inch is commonly \ of the U. S. St'd. In drawing, the depth is made f • The Acme Standard or 29° thread is used for the same general purpose as the square thread. Threads per inch are likewise usually the same. In drawing, the angle is made 30°. In the Whitworth or British Standard, threads per inch are, with a few excep- tions, the same as U. S. St'd. See handbooks. 126 ESSENTIALS OF MECHANICAL DRAFTING (d) Conventional Representation of Threads. The true drawing of the thread curves involves considerable labor and in small screws would be impossible. In large threads it is customary to substitute st. lines for the helices, as in Fig. 217. In invisible threads they are often omitted altogether. V threads less than one inch diam. are usually represented as in Fig. 218 (a), (b), (c), (d). The methods (e), (f), (g), (h) are also used. The spacing is estimated by eye, without regard to the actual no. of threads per inch, but in methods (a), (b), (c), (d) the positions of the lines should indicate whether the screw is right- or left-hand. The thread of a long piece may be shown as in (i). V THREAD SQUARE THREAD ACME THREAD SINGLE R.H. V TH'D HOLE Double. Double Double Fig 217 Sect ion Small square threads are usually represented as in (j). The exact no. of threads per inch is shown unless the scale is very small. A threaded hole is generally represented in the circular view as in (k), to distinguish it from a drilled hole; a drilled and threaded hole, as in (1). The diam. of the outer O in each is equal to that of the bolt or screw; that of the smaller about equal to that of the root. Any of the methods of Fig. 218 may be used for the other views. On crowded drawings it is often best to use methods (d), (f), (h), or (m). It is better to omit the drawing of the threads beyond the screw end (see (n) ), unless method (m) is used. The point made by the drill is usually shown. Figs, (o) and (p) show methods of representing small pieces in sec. when V-threaded inside and outside. (e) Dimensioning. Give the outside diam.; the no. of threads per inch, thus: lOTh, lOThds., lOP., 10, or X: and the length of the threaded portion, from the end when chamfered, and from the curve when rounded. If the thread is other than right-hand and single, specify as indicated in Figs, (b), (c). HELICAL CUR\'ES, THREADED PARTS, AND SPRINGS 127 ^ Pipe. Tap. Fig. 218 4r" Pipe Tap. 128 ESSENTIALS OF MECHANICAL DRAFTING In a threaded hole, give the depth, the diam. of the piece to be screwed into it, and no. of threads per inch. Indicate diam. and no. of threads of a tapped hole, as in Figs, (k), (1). All parts shown V-threaded are generally understood to be U. S. St'd, unless otherwise specified; likewise when the no. of thds. is not given. 111. Pipe Threads. The threaded ends and holes of pipes and pipe fittings are tapered so that the parts may be screwed together more tightly and thus prevent leakage. The standard taper is |" per foot. The thread Z is 60°, and the tops and bottoms are slightly rounded or flattened. The thread is usually represented by the conventional methods used for V screw threads. See Fig. 218 (q), (r). The taper is commonly drawn slightly greater than the actual, to show at a glance that the threads are pipe threads. The sizes of pipes are stated by giving their nominal inside diams., which are somewhat less than the actual inside diams., as noted in the table. Pipe tapped holes are indicated by size of pipe tap required. Standard Wrought Iron Pipe Nominal Inside Diam Actual Inside Diam Actual Outside Diam Threads per In. Diam. at Top of Thread at End 1 4 f h 3 4 1 H 1| 2 91 '^2 3 ^ 4 41 5 .36 .49 .62 .82 1.05 1.38 1.61 2.07 2.47 3.07 3.55 4.03 4.51 5.05 .54 .68 .84 1.05 1.32 1.66 1.90 2.38 2.88 3.50 4.00 4.50 5.00 5.56 18 18 14 14 111 111 111 HI 8 8 8 8 8 8 .52 .62 .82 1.03 1.28 1.63 1.87 2.34 2.82 3.44 3.94 4.44 4.93 5.49 6 6.07 6.63 8 6.55 112. Bolts. The heads and nuts of machine bolts in common use are made hexagonal, or square, as in Fig. 219. The hexagonal form is more generally used in machine construction, the square in heavy work. For ordinary work, the head and nut are chamfered or beveled at the outer end. For finished machinery they are usually rounded. (a) Standard Proportions. Proportions of the U. S. St'd rough bolt- heads and nuts are given in the figure. They are generally used for the square also. There is no standard for the rad. R, nor for the bevel of the chamfers, but they are usually shown as in the figure. (b) Conventional Representations of Hexagon Heads and Nuts. In the rounded head and nut, the curves of intersection of the sides and end are circular. In the view across corners, therefore, the curve c-d is concentric with a-b, while those of the oblique sides would be elliptic. Art. 77(a). The latter, however, are always described as circular. Note that the outer curve in the nut begins at the hole instead of at the C. L. HELICAL CUR^-ES, THREADED PARTS, AND SPRINGS 129 R,= 2D R3=7-9 Rounded [ 4! 12 6 .1 D— . Jrt2Z HEXAGON HEAD BOLTS AND NUTS Rounded riiT /\ Proportions of U.S.SrD Rough Colts D= Nominal Diame+er of Bolt F = Width across Flats = l|D + 5" H = Thickness of Head=^F N= Thickness of Nut = D . Finished heads and nuts are i° less in width and thickness. /^ Chamfered square head bolts and nuvs bolt with check nut & washer. Fig. 219 130 ESSENTIALS OF MECHANICAL DRAFTING In the chamfered head and nut the curves of intersection of the sides and' end, though in reahty hyperbohc (Art. 77(c)), are likewise always described as circular. The outer Hne of the chamfer is often described by arcs concentric with a-b, as shown. HEXAGON HEAD BOLTS AND NUTS Fig 219(a) (c) To DRAW THE VIEW ACROSS CORNERS OF THE ROUNDED HEXAGON HEAD AND NUT. Upon an indefinite line 4-5 set off 1-2 equal to J F. Draw the _L 2-3 and the 30° hne from 1. Then 2-3 will represent half of a revolved TAP SCREW 1 — •a -p. Q 1 ; 1 M -B- -h- t Sq. Head On Hex. and Square I T=|L When L=4"o(- less 1 . =2L When Lis over 4" Hex.lHead CAP SCREWS Fig. 220 HELICAL CURVES, THREADED PARTS, AND SPRINGS 131 side of the hexagon and 1-3 half of its long diain. Now set off 1-4 and 1-5 equal to 1-3, and 1-6 and 1-7 equal to 2-3, and draw 4-a, 5-b, 6-c, and 7-d. Next set off H and draw arc a-b determining the length of the ±s. Finally, draw arcs c-d, a-c, and d-b. In small drawings the long diam. may be made equal to 2 D. The method of drawing the nut is evident. (d) To DRAW THE VIEW ACROSS FLATS OF THE ROUNDED HEXAGON HEAD AND NUT. Set off 7-8 equal to F, and draw 7-e and 8-f. Next set off H and draw arc e-f determining pts. e and f. Then determine pts. d, b, and g, as in Art. (c), and describe arcs d-b and b-g.. The method of drawing the nut is evident. machine: screws Flat Head Round Head \X7J CO Cone Pt. Flat Pt. Cup Pt. Round Pt. se.t screws Fig. 221 Pivot Pt. Fillistai- Head Fiat Oval ■*— ^ i ' ^ Fig. 222 (e) To DRAW THE CHAMFERED HEAD AND NUT ACROSS CORNERS OR ACROSS FLATS. The method for each is evident from Arts, (c) and (d). (f) Square Heads and Nuts. The method of drawing the square head and nut is, in general, the same as for the hexagon. (g) When drawn in connection with the parts held together, both heads and nuts should, as a rule, be represented across corners to show that proper allow- ance has been made for clearance; otherwise they should be shown aci'oss flats, as they are thus simpler to draw and to figure. (h) Fig. 219 also illustrates a st'd hexagon bolt with check nut and washer. Both nuts are often made equal in thickness, f D. (i) Dimensioning. In a st'd bolt give the diam.; length of bolt from the under side of the head to extreme end, unless the end is rounded; and the length of the threaded portion. In a special bolt give also the distance across flats, the thickness of head and nut and the no. of thds. per inch. 113. Screws. Fig. 220 represents a tap screw or tap holt; a stud holt or stud, and hexagon and square head cap screivs. A tap screw is similar to a st'd bolt without the nut; the end being screwed into a tapped hole. 132 ESSENTIALS OF MECHANICAL DRAFTING A stud is used where frequent removal is not desirable, as in cylinder heads. One end is screwed permanently into a tapped hole and a st'd nut used on the other. A cap screw is a type of tap screw used where adjustment is necessary, as on bearing caps, etc. Fig. 221 represents set screws which are used to prevent the motion of one piece by forcing the point against a second. The form of point used is dependent upon the resistance desired. Fig. 222 represents four types of machine screws which are from .06" to .45" in diam. and designated by gage number. Slots are drawn at 45° in end views for contrast with other lines. Tables of proportions of these and other forms of screws and bolts may be found in catalogs and engineers' hand- books. Fig. 223 represents wood screws. '^^^^^r^ Flat H'd R'd M'd \X7 CO wood scre.ws Fig 223 helical springs Fig. 224 114. Springs. Fig. 224 shows conventional representations of /ieh'caZ spnngfs. In small sees, the helical lines are often omitted. In dimensioning, give outside diam., gage of wire, and coils per inch when extended.