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S"t7£6 ret Le AM CHLOE AUW le T= NO Axexqrt “I JO "0 "Ss3]00q onpI9AO [Te UO spew SI ssieYD W ‘“Aojeq poduiejys ojegq 4S80,e7 2} IIOJAq IO UO YOO Sy} UN} TECHNICAL DRAWING SERIES ELEMENTS OF MECHANICAL DRAWING THE USE OF INSTRUMENTS; THEORY OF PROJECTION AND ITS APPLICATION TO PRACTICE; AND NUMEROUS PROBLEMS INVOLVING BOTH THEORY AND PRACTICE BY: GARDNER C. ANTHONY, A.M., Sc.D. PROFESSOR OF DRAWING IN TUFTS COLLEGE AND DEAN OF THE DEPARTMENT OF ENGINEERING 3 AUTHOR OF ‘‘MACHINE DRAWING’’ AND ‘‘ THE ESSENTIALS OF GEARING’’} MEMBER OF THE AMERICAN SOCIETY FOR THE PROMOTION OF ENGINEERING EDUCATION 3 MEMBER OF AMERICAN SOCIETY OF MECHANICAL ENGINEERS REVISED AND ENLARGED EDITION BOSTON, U.S.A. D. C. HEATH & CO., PUBLISHERS 1907 F OPEV \' PTV ) _ Copyricut, 1894 anp 1904, By GARDNER C. ANTHONY. = my 1o I. 28 Be [LO Man. Kyte Pr, be Apis Mec a. PRHFACE THE extended use of the “ Technical Drawing Series” has necessitated a very complete revision of the ** Elements of Mechanical Drawing,” resulting in some radical changes in the book. In adapting it to so wide a range as would include its use in the Evening Drawing School and the Technical College, it has been found necessary to separate the instruction from the problems so that it may be useful to instructors who desire a book of reference for use in connection with problems and notes of their own. The folding plates have been abandoned, and the illustrations are now printed with the text, great care having been used to make the references to the cuts either on the Same or opposite page. Many illustrations have been added, and all have been redrawn. The problems and their lay-out are printed at the end of the book, with numerous references to the text. The number and variety have been increased, and include many of a practical character suitable for elementary courses, all of which have received the test of the classroom. The student should be required to master the principles before attempting to solve the problems, receiving such instruction as his special case may demand. By this means individual instruction may be given to large classes. The method recommended for finishing drawings by leaving the construction lines in pencil, neither inking nor erasing them, has been found efficient for the following reasons: —It enables the instructor to follow the methods and reasoning of the student; it teaches neatness and care in the execution by preventing the free use of an eraser on the com- pleted drawing; it is a great saving of time. Ml 157893 IV PREFACE The Third Angle Method of projection is used exclusively, in accordance with the best modern practice. Some college instructors have objected to this because it is not commonly adopted in treatises on Descriptive Geometry; but the author has used it instead of the First Angle Method for the teaching of this subject during the past four years, and it has caused no difficulty or confusion on the part of the students. After a student has acquired the necessary skill in penmanship, the greatest stress should be laid on the subject of Projection. He should be taught to regard Graphics as a language study, the grammatical construction of which is developed in the Theory of Projection. No copying should be permitted, save in learning to use the instruments, and the subject should be taught as an art of expression rather than one of pictorial repre- sentation. Although most people recognize drawing as a medium for conveying thought, few appreciate the importance of teaching it as a language. But such it is in the fullest sense, possessing a well-defined grammatical construction, rich in varied forms of expres- sion, forcible yet simple, and truly universal. The author desires to express his thanks for the many suggestions and kindly criti- cism made by those who have found this book useful in the classroom, and wie have helped to make it what it is. GARDNER C. ANTHONY. Turts COLLEGE, July, 1904. TABLE OF CONTENTS CHAPTER I INSTRUMENTS AND THEIR USE ART, PAGE ART, PAGE 1. The Outfit 1 | 15. Use of Curves . ; ° ° ° . 0 2. Instruments and Materials 1 | 16. Compasses : : . : . ot he 3. Drawing Board 1 | 17. The Use of Compasses ° 12 4. T Square 2 | 18. Dividers : : 14 5. Use of T Square 2 | 19. Bow-pencil : 16 6. Triangles . 3 | 20. Bow-pen . : F : om Jui suelo 7. Use of T riangles 3 | 21. Bow-spacers. : 7 ‘ : : ul 8. To Test the Angles of Trian gles 5 | 22. The Ruling-pen : ; ; ° yw 9. Pencils : : 6 | 23. To Sharpen the Ruling-pen é . : ugh Es 10. To Sharpen the Pencil 6 | 24. Erasers and Erasing . ’ ; : : baer) 11. Pencilling . : Cet 20.0 meee. : ; ‘ : : ‘ - 20 12. Scales . 3 : 8 | 26. Paper ; . ; : . aD PAY 13. Use of Scales. A Nya. 8 | 27. Miscellaneous Material. : ° ° 20 14. Curves 3 : : s 10 CHAPTER II GENERAL INSTRUCTION 28. Preparation of the Paper . - : : . 21 | 33. General Instruction for Inking : . . 28 29. Character of Lines . ; Be ony ; hed at ote A Lacy st ‘ ; ‘ ‘ ; : oe eta 30. Shade Lines. : : - ; ; . 23 | 35. Lettering . 3 ; ; é : : . 30 31. Line Shading . : ; 5 : : Eee Ouh oOuL ible. : : ; ‘ ‘ ‘ : pei 32. Sections . : q 4 : : : Peers oy Peed tatoberia yea : : , : : , ey ART, 38. 39. 40. 42. 48. TABLE OF CONTENTS CHAPTER III GEOMETRICAL PROBLEMS Introduction to Geometrical Problems To bisect a right line or the arc of a circle To divide a line into any number of equal parts : . To draw a perpendicular to a line Angles To bisect an angle . To construct an angle equal to ag given angle : . To construct an angle of 45° To construct angles of 60°, 30° and 15° To construct an equilater al triangle, having given a side : : To construct an isosceles tr iangle To construct a scalene triangle Tangent, Secant and Normal . To draw a tangent to a circle . To lay off an are equal to a given tangent . On a tangent to lay off a ee equal to a given arc To draw a circle through a given ‘point and tangent to given lines To draw any number of circles tangent to each other and to two given lines. To draw a tangent to two given circles To draw a circle of a given radius aueout to two given circles . Through three given points not in the same straight line to draw a circle . ‘ To cireumscribe a circle about a given tr iangle . To inscribe a circle within a given triangle . To inscribe an equilateral triangle within a circle PAGE 39 36 36 36 38 38 38 38 38 39 39 39 40 40 41 41 42 42 42 45 43 43 4 +4 ART. 62. 63. 64. 65. 66. Gir 73. 74. 76. 77. To inscribe a square within a circle : : To inscribe a pentagon within a circle To inscribe a hexagon within a circle To circumscribe a hexagon about.a circle To draw a hexagon oes csi a long diameter ‘ To draw a hexagon having | given a short diameter . To draw a hexagon having given aside . To inscribe an octagon within a given circle To circumscribe an “octagon about a circle On a given side to construct a regular polygon having any number of sides . Within an equilateral triangle to draw ‘three equal circles tangent to each other and one side of the triangle ‘ Within an equilateral tr iangle to draw three equal circles tangent to each other and two sides of the triangle Within an equilateral triangle to draw six equal circles tangent to each other and the sides of the triangle . Within a given circle to draw three equal circles tangent to each other and the given circle Within a given circle to draw any number of equal circles tangent to each other and the given circle : About a given circle to circumscribe any. num- ber of equal circles tangent to each other and the given circle : ; . 48 48 48 48 49 49 TABLE OF CONTENTS CHAPTER IV CONIC SECTIONS ART, PAGE ART. 78. The Cone and Cutting Planes. ° . feral 87. Parabola. Second method . 79. The Ellipse. : : : : ai) 88. Given a parabola to find its axis, focus, and 80. Ellipse. First method . ‘ : : a directrix ‘ : ; ; . E $1. Ellipse. Second method : ‘ : . o4 89. The Hyperbola : 82. Ellipse. Third method . : : : . 54 90. Hyperbola. First method $3. Ellipse. Fourth method ‘ : : Bai, 91. Hyperbola. Second method . 84. Ellipse. Fifth method . : : ; OO 92. The Equilateral Hyperbola ? 85. The Parabola . : : ; 5 Peo 93. Method common to all the conic curves . 86. Parabola. First method. : é : 07 CHAPTER V ORTHOGRAPHIC PROJECTION 94. Introduction to Orthographic Projection . 61 | 100. The projection of a circle oblique to the coor- 95. Projection : Filia dinate planes : . é 96. To determine the projections of an object . 64 | 101. The projection and true length of lines . 97. Objects oblique to the planes of projection . 67 | 102. Projection of rectangular surface by aatary 98. Auxiliary planes of projection : yee LO plane , 99. Views omitted by use of auxiliary plane . 71 | 103. Rules governing the relation of lines and sur- faces to the H and V coérdinate planes CHAPTER VI ISOMETRIC AND OBLIQUE PROJECTION 104. Introduction to Isometric and Oblique Pro- 112. To make the isometric drawing of a circle jection . ; : ‘ : ; . 78 |. 113. The measurement of gree lying in isometric 105. Axonometric Pr ojection F - : - Att) planes . 106. Oblique Projection . : : : , - 78 | 114. To make an isometric dr awing of an oblique 107. Isometric Projection ‘ : : : th timber framed into a horizontal timber 108. The Isometric Axes ‘ : ‘ ‘ . 79 | 115. Suggestions for special cases . : 109. The Isometric Scale : . 80) 116. A useful case of axonometric projection 110. To make the isometric drawing ofacube . 80 | 117. Oblique, or Cabinet Projection 111. Non-isometric Lines . ‘ , ‘ tel) Vil PAGE S2 S. Ov St Si Oi Coo oO OHO ~I ~] Oo He Oo ~J =| 81 82 85 84 86 Vill ART. 118. 119. 120. 123. 124. 127. 128. 129. 130. 131. 132. 133. 140. 141. 142. 143. 144, TABLE OF CONTENTS CHAPTER VII THE DEVELOPMENT OF SURFACES PAGE ART. PASE To develop a surface ; . 88 | 121. The development of a cylinder ° . » 92 To develop a pyramid when cut by a plane . 90 | 122. To develop a cone . A . ° . . 94 The development of surfaces of revolution . 92 CHAPTER VIII THE INTERSECTION OF SURFACES The intersection of cylinders . : : - 96 | 125. The intersection of an sihas to onde a vertical The use of auxiliary planes. ‘ 4 - 98 cylinder : ; ; ‘ - 100 126. The intersection of prisms : ° ° - 101 CHAPTER IX SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS The Spiral ; ; ; ; ; . 102 | 134. The Double Thread ’ ° . 108 The Equable Spir al : . 102 | 135. United States Standard V Threads . . 109 The Equiangular or Logarithmic Spiral. . 103 | 136. Square Threads : ; ; ‘ me 8 | Involutes . ; ; ; ; : . 104 | 137. Sphere and Cutting Planes. Lae The Helix : , ; . ‘ : . 104 | 138. United States Standard Hexagonal Bolt-head Screw-Threads ; ; ° : : . 106 and Nut : eee? Conventional V Threads . : : : 108 | 139. Chamfered Head and Nut. mee Pepe os he CHAPTER X PROBLEMS General instructions for a a are the 145. Object oblique to the coérdinate er ° . 182 problems. 117 | 146. Special problems in projection . 135 The use of instruments. "Examples for 147. Isometric projection problems ett practice 5 : : : . . 117 | 148. Development problems raat 9 Geometrical pr oblems 4 ; : . 122 | 149. Intersection problems . 144 Conic section problems. : : : . 126 | 150. Spirals, screw-threads and bolt-head pr oblems 147 Orthographic projection problems . : . 128 | 151. Miscellaneous . f ; 149 HLEMEHENTS OF MECHANICAL DRAWING CHAPTER I INSTRUMENTS AND THEIR USE 1. The Outfit. It is of the first importance that the student be provided with good instruments and drawing material. The cost need not be great if they are selected by one experienced in their use. ‘The following list comprises the smallest equipment consistent with good work, and a description of these supplies may be found in the succeeding articles. 2. Instruments and Materials. Drawing Board, 16’’ x 20", page 1. . T Square, 24!’, page 2. 45° Triangle, 6'’, page 3. 30° 60° Triangle, 10’’, page 8. Scale, 12’, page 8. 4H Siberian Pencil, page 6. 6H Siberian Lead, page 6. Pencil Sharpener, page 6. Curves, page 10. A set of Drawing Instruments consisting of the following pieces: 54'’ Compasses, page 12; 5'’ Dividers, page 14; 3/’ Bow-pencil, page 16; 3/’ Bow-pen, page 16; 3’! Bow-spacers, page 16, and a 5!’ Ruling Pen, page 17. Pencil Erasing Rubber, page 19. Ink Erasing Rubber, page 19. Inks, page 20. Paper, page 20. Pens, Penholder, Penwiper, and Tacks, page 20. ) 3. Drawing Board. A pine board 7’ thick and measuring about 16/’ x 20!’ will suffice for the solution of all the problems in this book. As the upper face should be maintained a true plane, it is desirable to have two cleats on the back of the board. One of the short 1 2 INSTRUMENTS AND THEIR USE edges is chosen as the working edge, and made perfectly true. It should be tested from time to time, as any unevenness in this will impair the accuracy of a drawing. ‘Thé working edge should be placed at the left side, and is the only one which need be used in this elementary work. It is customary for draftsmen to use the edge next thg body when drawing long lines parallel to the working edge; but should this be done it itate the making of this edge true, and having the angle between the two ed y J0® ‘The upper and right- hand edges must never be used. 4. T Square. This consists of a blade securely fastened to a head by means of a clamp or screws. As it is necessary that the upper edge of the blade and the inner edge of the head be maintained true, glue should never be used in the joint. These edges, together with the work- q ing edge of the board, should be examined frequently, as the 4 — accuracy of the drawing depends primarily on them. The working edge of the blade and head should be at right angles. 9 5. Use of T Square. ‘The head should be held firmly against the left-hand edge of the board with the hand in the position shown in Fig. 1. By sliding the T square along this edge, parallel lines may be drawn. Previous to draw- ing a line at the extreme right, slide the hand along the blade with sufficient pressure to maintain it in position when drawing the line. Never move the T square by the blade or draw a line against the lower edge of the blade. Do not use the head in contact with the upper or right-hand TRIANGLES 4 edges of the board. ‘The lower edge of the board may be used if it is known to be at 90° with the working edge. 6. Triangles. Th used by draftsmen are 60°. The former has two equal angles % a right angle. The latter has angles of 30°, 60° and 90°. Cellu- loid is the best material for their construction, although wood and rubber are used also. These triangles, in combination with the T square, may be used to solve a variety of problems, and facility in rapid drawing is dependent on the skill ac- quired in using them. ngles ordinarily 7. Use of Triangles. The position in which triangles are used for drawing lines perpendicu- lar to the T square is shown in Fig. 2. As the drawing board should be placed so as to permit the light to come from the upper left-hand corner, this position of the T square and tri- angle will avoid the shadow of the blade or of the triangle being cast on the line to be drawn. Some frequent constructions involving the use of triangles are as follows :— Fig. 3. 4 INSTRUMENTS AND THEIR USE E PARALLELS. Lines perpendicular to the T square blade may #3 be drawn by sliding the triangle in contact with the blade, as in ga] \U™sveorreounes” Kio, 2, When drawn at other angles, or without the aid of a T square, slide one triangle on the long edge of another, as indicated in Fig. 8, using care to hold the second triangle firmly. The first triangle is then slid in contact with the second by means sf the first and second fingers. PERPENDICULARS. If the lines be perpendicular to the T square blade, use the triangle as in Fig. 2; but when the given line does not coincide with a position of the T square, use one of the positions indicated by Figs. 4 and 5. In Fig. 4, the 60° triangle is placed parallel to, but not touching, the given line. It is then slid on the 45° triangle to a second position parallel to the first and about 3!’ from it. The 46° triangle is then moved to the second position and the required line drawn. In Fig. 5, the 45° triangle is set parallel to the given line with a short side in contact with the 60° triangle. By turning a triangle on its right angle, as in Figs. 5 and 6, a perpendicular may be drawn. ANGLES. The method for drawing lines at angles of 45°, 30°, and 60° with the T square blade is apparent from the figures. Angles of 15° and 75° may be obtained by using the 45° and 60° triangles with the T square, as in Figs. 6 and 7. Do not draw lines within 4’ of the corners of the triangle, and never construct angles by drawing lines along adjacent sides of the triangle. TEST OF TRIANGLES 8. To Test the Angles of Triangles. The right angle may be tested as follows: Place the triangle on the T square with the vertical edge at the right, as in Fig. 8; draw a fine line, AB, in contact with this edge, then reverse the triangle so that both edge and line may be free from shadow, and move the edge toward the line. If they coincide, the angle is 90°. If they do not coincide, and the vertex of the angle formed by line and edge is at the top, as shown by A, the angle is greater than 90° by half the angle BAC. If the vertex of the angle is below, the angle is less than 90° by half the amount indicated. Test or 45° ANGLE. If the 90° angle is known to be correct, place the 45° triangle on the T square, as indicated by the dotted position, Fig. 9, and draw a line to coincide with the long edge. Reverse the triangle so as to bring the second acute angle into the position of the first, and if the edge coincides with the line drawn, the acute angles are equal and therefore 45°. If the line and edge intersect at the bottom, the angle of the triangle at this point of intersection is less than 45° by half the amount indicated by AED. If they intersect at the top, the angle of the triangle at this point is less than 45° by half the amount indicated. TEST OF 30° AND 60° ANGLES, If the 90° angle is known to be correct, draw a line to coincide with the T square blade, and from any point on this line construct an angle of 60°, as in Art. 46, es HHHH \ Fig. 10. (| femme OR SRA, CIRTENTTTST:Ti1 9D | SS Fig. If. INSTRUMENTS AND THEIR USE page 88. Test the angle by sliding the short edge on the T square until the hypothenuse coincides with or intersects the line measuring the angle. If the 90° and 60° angles are found to be correct, the third angle must be 30°, since the sum of the angles of any triangle is equal to 180°. 9. Pencils. The grade of lead commonly used by draftsmen is } 6H; but this is extremely hard, and unless used with great care will indent the paper so that the line cannot be erased. A 4H lead requires more care and frequent sharpening, but the student will thus acquire a lightness of touch which is of value. The double-end holder with movable leads has some advantages over the lead pencil; its length remains constant; a shorter lead may be used, and leads of different gerade may be used in either end. 10. To Sharpen the Pencil. Remove the wood from both ends by means of a sharp knife, exposing about 3/’ of lead. Fig. 10. One end should then be sharpened to a conical point, and the other to a chisel or wedge-shaped end. ‘This last operation should be done with a file or pencil sharpener, but never with a knife. An excellent sharp- ener may be made by mounting strips of No. 0 sandpaper, 4’/ long and 8!’ wide, on a thin piece of wood so that it may be held in the fingers without soiling them. Care must be used to prevent the fine lead dust on the sharpener from falling on the paper or board, as it will work into the surface and be difficult to remove. The chisel end should be PENCILLING 7 wedge-shaped, as indicated in Fig. 10, and the length of the edge should be reduced to about one-half the diameter of the lead. ‘This may be done by first making the end slightly conical. After finishing the edge on the file or sharpener it is well to rub it on a piece of paper, rolling the pencil slightly to remove the sharp corners. Fig. 11 represents the double-end holder with leads properly sharpened. If this type is used, a 4H lead may be used for the chisel end, and a 6H for the conical end. 11. Pencilling. Good pencilling is a prerequisite to good inking. which it bounds, otherwise the accuracy of the drawing would be impaired. Cylindrical surfaces may be outlined by fine lines only. Shade the lower right-hand quadrant | of outside circles, and the upper left-hand quadrant of Fig. 3h. Fig. 32. inside circles. One width of shade line is used for all right lines. Never shade pencilled lines or dotted lines. In general the shade line is intended to represent the surface in shadow, and the light is supposed to come from the upper left-hand corner at an angle of 45°; but if there are lines of the drawing parallel with the ray of light, the angle of the ray may be considered as greater or less than 45°. Thus, in Fig. 34, it is better to shade one of the 45° lines rather than to draw both fine. Fig. 31 represents a square block with a square hole in it. ‘This ; is indicated clearly by the top view alone, although it Fig. 33. Fig. 34. does not show the depth. Fig. 32 is a similar object, but in this case it is surmounted by a square block of smaller size. The shade lines indicate this also, but fail, of course, to give the height. Fig. 33 represents two views of a hexagonal SHADE LINES pyramid properly shaded. The lines radiating from the center of the top view, and the two inside slant lines of the front view, are division lines between visible surfaces, and therefore not shaded. | In the shading of circles and circular ares it is necessary to avoid the sudden transition from the shade to the fine line, and this is accomplished in the following manner: Having inked the circle with a fine line, remove the point of the compasses from the center, using care not to change the radius, and place it below and to the right of the first center, a distance equal to the desired width of the shade line. If it is an outside circle, draw the second arc on the outside, as in Fig. 35; and if it is an inside circle, it should be drawn as in Fig. 36. If the width of the shade line is such that the two eccentric arcs are not in contact throughout, the inter- vening space may be filled by slightly springing the instrument. Since the circular arcs are the first to be inked, care should be used to adopt a width of line that will be correct for the shaded right line. The shading of Fig. 387 differs from the two preceding in having a uniform width of shade line between points A and B, for the outside circle, and between C and JD, for the inside circle. ‘The transition from shade to fine line is made within an are of about 45° beyond the points indicated. This method is not so simple as the preceding, and the improved appear- ance will rarely justify its use. 26 GENERAL INSTRUCTION Fig. 41. Fig. 88 represents a rectangular piece with a de- pression in the center, all of the corners being rounded or filleted. Small ares, such as these, and all circles hav- ing radii less than 3!’, should be inked by means of the bow-pen, which may be used in the manner described. It is better, however, to acquire the skill necessary to spring the bow-pen, so that the shading may be done without removing the needle-point from the center. This is accomplished without changing the position of the thumb and first finger, which are used to handle the instrument. By a slight pressure of the first finger, sufficient to deflect the needle-point leg, the radius is slightly reduced when shading an inside circle, and in- creased when shading an outside circle. By this method the work may be much more rapidly executed. 31. Line Shading. When it is necessary to suggest the character of a surface without a second view, line shading may be used. Figs. 39, 40, and 41 illustrate good methods for the shading of cylindrical surfaces. The lines may be equally spaced, although the appear- ance is somewhat improved by increasing the space as one approaches the center line, and decreasing the space on the lower half, while increasing SECTIONS ek the width of the line. In shading a concave surface, as in Fig. 41, the operation is reversed. Cylindrical parts of small diameter may be shaded on the lower half only, as in Fig. 40. Conical, spherical, and other classes of surfaces may be ee by this method, but the rendering of them requires considerable skill, Big ey ats and it is beyond the scope of this treatise to A consider the various methods. an a ALAA a \ Gia 4-+ BOLTS an ZL . AX . tN NI] MX ~ \ \ 32. Sections. It is frequently necessary to make the representation of an object as it would appear if cut by a plane, and with the portion nearer the observer removed, as in Fig. 42. This cut section is indicated by a series of parallel lines, usually drawn at an angle of 45°. The width of the lines is the same as the fine line, and the distance between SQUARE gi 2 Ge N Hi = Fac el ee the lines is determined by the area and intri- a EERIE Us cacy of the section. ‘These lines are not to fl kK} 7 Seeing soccer be drawn in pencil, but only in ink. An in- exe Fig. 42. strument known as a section liner is designed to insure accuracy in the spacing, but the beginner should learn to judge this by the eye. In doing so, care must be used to avoid making too small a space, and it is desirable to try the spacing on a separate sheet before sectioning the drawing. A greater degree of uniformity may be obtained by glancing back after drawing every six or eight lines. 28 CAST IRON INSULATING MATERIAL GENERAL INSTRUCTION On KS KKK acetate oe a se 8 D> KKK SSBKK OO Sete. i eatetass SRK KS SSS LRG oer acetate ores LLL Q Q CKO OR QDR R SE eratetatctetatccansteeet SOLS Q S550 POOOOODLOO™TD Xe, OO SS © Qae> DOTTED SECTION Fig. 43. The different surfaces in the plane of the section are indicated by changing the direction of the lnes, as illus- trated by Fig. 42, which is the cross section of a piston rod packing. Seven distinct surfaces are clearly shown by changing the direction and angle of the section lines. Differences in the material are also indicated by the character of the lines; but there is no general agreement as to notation. Fig. 45 illustrates six good types of sec- tioning, together with the names of the materials which the author has chosen to indicate by them. Whenever figures or notes occur in a section, the section lines should not be drawn across them. 33. General Instruction for Inking. Never begin the inking of a drawing until the pencilling is completed. See that the sheet is free from dust. Always ink the circles and circular ares first, beginning with the small If the drawing is to have shade lines, shade each arc as drawn. ‘To omit the shading of circles until they have first been inked in fine lnes. will necessitate almost double the time otherwise required. It should be ob- arcs. served that the width of both fine and heavy lines is determined by the shading of the first circular are. Next, ink all the full and dotted ruled lines. Begin on the upper side of the TRACING 29 sheet and ink all the fine horizontal lines, omitting those lines which are to be shaded. Next, ink the vertical lines, beginning with those at the left. This method insures sufficient time for the drying of the ink. Do not dwell too long at the end of a line, especially if it be a heavy one, as the pressure of the ink in the pen will tend to widen the line. If a series of lines radiate from a point, allow sufficient time for the drying of each line, otherwise a blot may be made. Finally, ink all lines at other angles and those curved lines requiring the use of curves. ‘The same order is to be followed in the inking of the shade lines, evenness being secured by ruling them at one time. Do not shade dotted lines. Ink center and dimension lines, and put on the figures and notes. Draw the section lines and put on the title. 34. Tracing. When it is desired to reproduce a drawing, transparent cloth or paper is placed above the original and the lines of the drawing traced on the new surface, as though one were inking a pencilled drawing. ‘Tracing cloth is usually furnished with one surface glazed and the other dull. Either side may be used, but as it is difficult to erase from the dull side it is better to ink on the glazed surface. Pencilling must be done on the dull surface. Tracing cloth is used frequently in place of paper for original drawings, the pencilled paper drawing be- ing traced on the cloth in ink. Copies of this tracing may be made by the blue print process. As the cloth absorbs moisture quite rapidly, it shrinks and swells under varying atmos- pheric conditions. Because of this, large drawings which require considerable time to complete, should be inked in sections, as the cloth will require frequent adjustment in order that its surface may be smooth and in contact with the paper drawing. Only the best quality of cloth should be used, as the cheaper kinds are improperly sized and absorb ink, causing blots. If the ink fails to run freely on the glazed side, dust on the surface a little finely powdered 30 GENERAL INSTRUCTION pumice stone or chalk, rubbing it lightly across the surface with a piece of chamois skin or cloth. Use care to remove it completely from the surface. Inked lines may be removed from the glazed side by means of a sharp knife and a hard rubber, or by dusting a little finely powdered pumice stone on the lines to be erased, and briskly rubbing them with the end of the finger or a piece of medium rubber. As the pumice becomes discolored replace it with fresh powder. In erasing lines from the dull side use the hard rubber. Pencilled lines may be removed by the ordinary dite eraser, or by means of a cloth moistened with benzine. 35. Lettering. The subject of lettering is of such importance to the mechanical draftsman that he should adopt some clear type for general use, and acquire proficiency in the free-hand rendering of it. While at times it may be necessary to make use of instruments and mechanical aids for the construction of letters and figures, usually they may be written free-hand. The accompanying alphabets are such as may be recommended to students, and will be acceptable in the regular practice of drafting. Their study will afford an excellent free-hand exercise, as well as skill in the figuring and lettering of drawings. Both types should be written without the aid of instruments. The first is known as the vertical Gothic, and the second as the slant Gothic. Large and small capitals may be used in the place of capitals and lower case, as illustrated. The small capitals and lower case may be made about two-thirds the ~ height of the initial letters. This type is written quite easily by means of a hard wood stick, preferably orange or boxwood, sharpened to a point like a pencil, the size of the point being varied according to the desired width of the line. In doing this great care should be taken that the ink be black and slightly thick. The student is referred to the treatise on “ Letter- ing,” of the ‘Technical Drawing Series,” for the further consideration of this subject. LETTERING | BoCUrrGHIJIKLMNOPQO mS VX Y Z 1234567890 & 33 abcdetghijklmnopaqrstuvwx yz ABODE Fi GIH/ISKLMNOPOD LS) TESTA py (OG We P7IO0 79IDO & FF (—AbCACLGhIAITINIOLGrStUVWXYVZ Sh) bo GENERAL INSTRUCTION 86. Title. In the lay-out of the problems of this book no provision has been made for the title save in the case of applied work, such as machine drawings. It has been the custom of the author to have the name of the student printed in the right-hand lower corner just outside the margin line, and the plate number similarly placed in the upper right-hand corner. The capitals should be ;3;/’ high and the small letters 3’. In applied work the title should always be placed in the lower right-hand corner of the sheet, inside the margin line. It should designate, first, the name of the mechanism ; second, the name of the special detail ; third, the scale ; fourth, the date, which is always that of the finishing of the drawing. ‘The draftsman’s name or initials should be printed in small type. in the extreme right-hand corner within the margin. 37. Tinting. The surface of a drawing is colored or tinted for the purpose of making clear the divisions, as in map drawing; suggesting the character of the materials represented; or to indicate the character of the surface, whether plane or curved; and possibly its relation to other surfaces by the casting of shadows. THE PAPER should be of proper quality, such as Whatman’s cold pressed, and must be stretched by wetting the surface and glueing it to the board in the following manner : Having laid the paper on a flat surface, fold over about one-half inch of each edge. Thor- oughly wet all of the surface save the folded edges, using a soft sponge for this purpose, but do not rub the surface. Next apply mucilage, strong paste, or a light glue, to the underside of the folded portion and press this to the board with a slight outward pressure so as to bring the surface of the paper close to the board. As the glue should “set” before the paper begins to dry and shrink, it is necessary to have the paper very wet, but no puddles must be allowed eTENTIN CG aos to remain on the surface after the edge is glued. The paper must be allowed to dry gradually in a horizontal position, as otherwise the water would tend to moisten the lower edge and prevent the drying of the glue. If the paper should dry too rapidly, not allowing sufficient time for the glue to “set,” the surface may be moistened again. THE CoLor employed in making the wash or tinted surface may be a water color or ground India ink, but none of the prepared liquid inks are suitable for the purpose. The color should be very light, and when the desired shade is to be dark it should be obtained by applying several washes, allowing sufficient time for each to dry. The color must be as free from sediment as possible, but since some deposit is liable to take place, the brush should be dipped in the clear portion only, and not allowed to touch the bottom of the saucer. THE BrusH should be of good size, depending somewhat on the surface to be covered, and of such quality that when filled with the color or water, it will have a good point. Two classes of tinting are employed, the flat tint of uniform shade, and the graded tint for the representation of inclined or curved surfaces. THE FLAT Tint. Remove all pencilled lines which are not to be a part of the finished drawing, and do all the necessary cleaning of the surface, using the greatest care not to roughen the paper. Inking should be done after the tinting, but if for any reason it is necessary to ink the drawing first, a waterproof ink must be employed. In putting on the color, slightly incline the board to permit of the downward flow of the liquid, and, beginning at the upper portion of the drawing, pass lightly from left to right, using care just to touch the outline with the color, but not to overrun, and making a somewhat narrow horizontal band of color. Advance the color by successive bands, the brush just touch- ing the lower edge of the pool of water made by the preceding wash. This lower edge should 4 GENERAL INSTRUCTION iS) never be allowed to dry, as it would cause a streak to be made in the tinted surface. Having reached the lower edge, use less water in the brush so as to enable a better contact to be made with the outline. Finally dry the brush by squeezing it or touching it to a piece of blotting paper. It may then be used to absorb the small puddle of color at the bottom edge or corner. Avoid touching the tinted surface until it is dry, at which time any corrections that are necessary may be made by stippling. ‘This consists in using a comparatively dry brush and cross-hatching the surface to be corrected. If the surface to be covered is large, it is desirable to apply a wash of clear water before applying the color. This dampened surface will prevent the, quick drying of the color and insure a more even tint. When necessary to remove the tint from a surface, use a sponge with plenty of clean water, and by repeated wettings absorb the color, but do not rub the surface of the paper. THE GRADED TINT may be applied by several methods, the simplest being to divide the surface into narrow bands and apply successive washes, each covering an additional band. If the tint is sufficiently light, and the bands narrow, the division line between the bands will not be very noticeable, but this may be lessened by the softening of the edges with a comparatively dry brush and clean water. Another method, which requires considerable dexterity, is to put on a narrow band of the darkest tint that may be required, and, instead of removing the surplus water from the edge, touch the brush to some clean water and with this lighter tint continue the wash over the second band. Continue in this manner until the entire surface is covered. There are many modifications of this method, all of which require considerable skill and are not to be recommended to the student at this stage of his progress. CHAP TH Re ite GEOMETRICAL PROBLEMS 38. WHILE the majority of students are familiar with many of the propositions included in this chapter, the study of the methods best adapted to the draftsman is of great importance. It is not intended that these problems shall serve as copies, or that the examples relating to them, and given on page 122, shall be used for all students; but as reference data, and for the purpose of illustrating the draftsman’s methods, they are believed to be an essential part of a text-book on Technical Drawing. In most cases two methods are given, the first being the ordinary geometrical solution requiring the use of a straight-edge and compasses; and the second, the more direct method employed by draftsmen, involving the use of a T square and triangles as well as compasses and dividers. In the geometrical figures the given and required lines are shown in full heavy lines, and the construction in full fine lines. It is intended that the student shall construct the propositions by the draftsman’s method and then employ the method of the geometrician as a test. If the problems are performed with great accuracy, the technical skill acquired in the handling of the instruments will be correspondingly great. It is not well to ink geometrical problems, as the precision of the pencilling will be impaired. 35 36 Fig. 44, Cc ; Fig. 46, PROBLEMS 39. To bisect a right line, AB, or the arc of acircle, ACB. Fig. 44. With centers A and B, and any radius greater than one-half of AB, describe arcs 1 and 2. Through the points of intersection of these arcs draw a line. Its intersection with the given line AB, and the are of the circle ACB, will determine the required points. DRAFTSMAN’S METHOD. Obtain the division with the dividers as explained in Art. 18, page 14. 40. To divide a line, AB, into any number of equal parts. Fig. 45. Let the required number of divisions be five. Draw AC at any angle with AB, and lay off five equal spaces of any length. Connect the last point, 5, with B, and draw parallels through the other points intersecting AB in points 1’, 2’, 3’ and 4’ which determine the required divisions of AB. Art. 7, page 4. 41. To draw a perpendicular to a line AB. CAsE 1. Fig. 46. When the given point C is on the line, and at or near the middle of the line. From C, with any radius, draw ares 1 and 1, and from the point of intersection of these arcs with AB, with any radius greater than arc 1, draw arcs 2 and 8. The line drawn through the point of inter- section of these arcs and the given point, C, will be the required line. CAsE 2. Fig. 47. When the point is on the line, and at or near | the extremity of the line. ; oy GEOMETRICAL PROBLEMS 37 First Method. Let AB be the given line, and A the given point. From A, with any radius, describe arc 1. With center C, and same radius, describe are 2. Through C, and the intersection of arcs 1 and 2, draw CE, and with same radius as before, from intersection of arcs 1 and 2, describe arc 8. A line drawn through A, and the point of intersection of arc 3 and line CE, will be the required perpendicular. Second Method. From B, with any radius, describe are 4. From point D, with same radius, describe arc 5. From the intersection of ares 4 and 5, describe arc 6. From the intersection of ares 4 and 6, describe arc 7. The line drawn through this last point of intersec- tion, F, and the given point B, will be the required perpendicular. CASE 3. Fig. 48. When the point is outside of, and opposite, or nearly opposite, the middle of the line. — From C, with any radius, describe arcs land 1. From the point of intersection of these ares with AB, with same radius, describe arcs 2 and 3. A line drawn through this point of intersection and the given point C, will be the required line. CASE 4. Fig. 49. When the point is outside of, and at, or near, the extremity of the line. From C draw any line CD. Find E, the center of CD, by dividers, or by Art. 39. On CD as a diameter, describe a semicircle. Through the given point ©, and the intersection of semicircle with AB, draw CF, which will be the required perpendicular. DRAFTSMAN’S METHOD. See Art. 7, page 3. 38 Cc Fig. 52. PROBLEMS 42. Angles of 15°, 30°, 45°, 60° and 75°, in either quadrant, may be constructed by means of the 60° and 45° triangles used in combina- tion with the T square, as described in Art. 7, page 4. 43. To bisect an angle, ABC. Fig. 50. From B, with as large a radius as possible, describe arc 1. From,its points of intersection with AB and CD, describe arcs 2 and 8. The line drawn through their intersection and B will bisect the given angle. 44. To construct an angle, FDE, equal to a given angle ABC. Fig. 51. Draw DE. From B and D, with equal radii, describe ares land 2. From E, with radius equal to chord AC, describe are 3. Through D, and point of intersection of arcs 2 and 38, draw DF mak- ing the required angle. 45. To construct an angle of 45° with AB at point A. Fig. 52. Through the given point, A, describe a semicircle on AB; draw a perpendicular through the center C. A line drawn through the point A and intersection of the perpendicular with the semicircle will make an angle of 45° with AB. 46. To construct angles of 60°, 30° and 15° with AB. Fig. 53. From the given point A as a center, with any radius, describe are 2. From B, with the same radius, describe arc 8. A line drawn through A and this point of intersection will make an angle of 60 with AB. GEOMETRICAL PROBLEMS Through the given point A, describe a semicircle on AB. With same radius, from C describe are 1. The line drawn through A and this point of intersection will make an angle of 30° with AB. Having constructed an angle of 30°, as described, bisect the same, and FAB will be the required angle of 15°. In this case it would not be necessary to draw the 30° line. 47. To construct an equilateral triangle having given the side AB. Fig. 54. Since the sides are equal, the angles will be equal, and there- fore, 60°, the sum of the angles of any triangle being equal to 180°. With centers A and B and radius AB, describe ares 1 and 2. From the point of intersection, C, draw AC and BC. DRAFTSMAN’S MetTHop. AB being drawn with the T square, through A and B, with 60° triangle, draw AC and BC. 48. To construct an isosceles triangle. Fig. 55. Having given the base DF and the equal sides DE and EF, from centers D and F, draw arcs 1 and 2 with radius equal to the given sides. From the point of intersection, E, draw DE and EF. If the angle be given, construct FDE and EFD equal to the given angle, and draw DE and EF. 49. To construct a scalene triangle. Tig. 56. Having given the sides A’B’, A/C’ and B/C’. Draw AB equal to A’/B’. With centers A and B, and radii equal to given sides, draw arcs 2 and 1. Draw AC and CB. ae ee ee Fig. 55. 0. va Lu 8" Fig. 56. 40) PROBLEMS 50. Tangent, Secant and Normal. Fig. 57. If a line AB cuts a curve at two points, it is called a secant. Conceive the line as revoly- ing about the point A until the second point of intersection with the curve shall coincide with the first; the line will then be in the position AC, and called a tangent. AD isa line perpendicular to the curve at the point of tangency, and called a normal. 51. To draw a tangent, AB, to a circle. Case 1. When the given point A is on the circle. Fig. 58. Draw the radius AC, and erect a perpendicular, AB, at A. DRAFTSMAN’S MEerHop. Place the triangle to coincide with center C and given point A, as though to draw AC. By means of a second triangle used as a base, turn the first triangle into the second position and draw AB perpendicular to AC. CASE 2. When the point is on the circle and the center not accessible. Tig. 59. From the given point, A, with any radius, describe arcs 1 and 1. Place the edge of the triangle to coincide with points B and D. Draw-a parallel line through A. CASE 8. When the given point, B, is without the circle. Fig. 60. Two tangents may be drawn. On BC as a diameter, describe arc 1; its intersection with the circle at A and D will be the points of tangency. The angle BAC, inscribed in the semicircle, will be 90°. GEOMETRICAL PROBLEMS DRAFTSMAN’S METHOD. From the given point, B, draw BA touching the circle. Through the center, C, draw a perpendicular to AB. A will be the point of tangency. In like manner obtain BD. 52. To lay off an arc equal to a given tangent. Tig. 61. Let AB be the given tangent and AD the are. From the point B step off equal spaces with the dividers or bow-spacers until a point of the dividers is at or near the point of tangency A. Reverse the motion of the dividers, stepping off an equal number of spaces on the curve. When several arcs are tangent at the same point and it is desired to lay off the length of their common tangent on each, the following approximation may be used provided the greatest arc does not exceed 60°. On AB lay off AC equal to one-quarter of AB, and from C as a center describe the are DBEF; the arcs AD, AE and AF will closely approximate the given tangent AB. 53. On a tangent to lay off a length equal to a given arc. Tig. 62. If AD be the given arc, draw the chord of this are and continue it to C, AC being equal to one-half of AD. From C as a center, describe the are DB intersecting the tangent at B. AB will be the required length or the rectification of the are. This approximation should not be used for arcs greater than 60°. The method of spacing the distance by the dividers may be employed in this case as in the previous one, Art. 52. 4] PROBLEMS 54. Todraw a circle through a given point, A, and tangent to given lines, AB and BD. Fig. 63. Since the circle is to be tangent to AB and BD, its center must lie upon the bisector of the angle DBA; and because it is to be tangent to AB at the point A, its center must le on the perpendicular to ABat A. Bisect the angle DBA, and through the point A draw AC perpendicular to AB. C will be the center of the circle, and AC its radius. The draftsman’s method may be used for obtaining the perpendicular AC, 55. To draw any number of circles tangent to each other and to two given lines, AB and AD. Fig. 64. Bisect angle DAB, and with any radius, HK, draw a circle tangent to ABand AD. From Kk draw KE perpendicular to Als, and with radius EK describe arc 4. Through KF draw FC perpendicular to AB. C will be the center and FC the radius of the second circle. Repeat the process. If the radius of the first circle be given, draw a parallel to AD distant from it equal to the given radius. The intersection of this line, HM, with this bisector of the angle ABD will be the required center. 56. To draw a tangent to two given circles. Fig. 65. Join A and C, the centers of the given circles. From D lay off DH equal to AF. With center C and radius CH draw arc 1. From A draw a tangent to this are. Art. 51, Case 38, page 40. Through B, the point of tan- GEOMETRICAL PROBLEMS gency, draw CE, and through A draw AF parallel to CE. E and F will be the points of tangency, and EF the tangent. 57. To draw a circle of a given radius, R, tangent to two given circles having centers BandC. Fig. 66. From centers Band C draw indefinitely, in any direction, lines BF and CE. Lay off HE and KF equal to the given radius R, and through F and E, from centers B and C, describe arcs 1 and 2 intersecting at A, the required center. Since these ares intersect in a second center, there will be two solu- tions to this problem. 58. Through three given points, A, B and D, not in the same straight line, to draw a circle. Fig. 67. Bisect the imaginary chords AB and BD. The point of intersection, C, of the bisecting lines, will be the required center. 59. To circumscribe a circle about a given triangle, ABC. T[ig. 68. Bisect two of the sides, as AC and BC. The point of intersection of these lines will be the center of the required circle. Draw a circle through A, B and C. This problem is identical with the preceding. If the hypothenuse AC should pass though the center D, the angle ABC would be a right angle. See also Art. 41, case 4, page 37. 43 44 PROBLEMS 60. To inscribe a circle within a given triangle, ABC. Fig. 69. Bisect two of the angles, as CAB and ABC. The point of intersection of these lines, D, will be the center of the required circle. From this point draw a circle tangent to AC, CB and AB. 61. To inscribe an equilateral triangle within a circle. Fig. 70. From the point C draw are 1 with a radius equal to that of the circle. From its intersection with the circle, and with the same radius, draw are 2. From center C, and with chord CB as a radius, describe are 8, and connect points A, B and C, which will give the required triangle. DRAFTSMAN’S METHOD. From point C, on the vertical diameter CD, draw CA and CB with the 60° triangle. With the T square draw AB to complete the triangle. 62. To inscribe a square within a circle. Fig. 71. Draw any diameter BD. Draw a second diameter, AC, perpendicular to it. Connect points A, B, C and D to complete the square. DRAFTSMAN’S MetHop. With the 45° triangle draw perpendicu- lar diameters AC and BD, and connect points A, B, C and D. 63. To inscribe a pentagon within a circle. Fig. 72. Draw any diameter, GF, and a radius AK perpendicular to it. Bisect KF, and, with H as a center, and a radius AH, describe arc 3. With center A and radius AL describe arc 4. AB is the side of a pentagon. Obtain GEOMETRICAL PROBLEMS the remaining points by describing ares 5, 6 and 7 with same radius. Connect points A, B, C, D and E to obtain the required pentagon. DRAFTSMAN’S METHOD. Estimate an arc equal to one-fifth of the circumference, and with the dividers step off this length, dividing the circle into five parts and correcting the are as directed for the division of a line. See Art. 18, page 14. Connect the points. 64. To inscribe a hexagon within a circle. Fig. 73. Draw any diameter FC. With centers F and C and radius equal to that of the circle draw ares 1 and 2. Connect the points of intersection A, B, C, D, E and F to obtain the required hexagon. Observe that the angles at the center, as BKC, are 60°. DRAFTSMAN’S METHOD. Draw a horizontal diameter FC. With 60° triangle draw diameter EB. Draw AB and ED, and with triangle draw BC, FE, AF and CD. 65. To circumscribe a hexagon about a circle. Fig. 74. Draw any diameter, AD. With H as center and radius equal to that of the circle, describe arc 1. Bisect the are HLN, and through L draw AB parallel to HN. With center K and radius AK describe circle ACE. In this inscribe a hexagon by Art. 64. DRAFTSMAN’S METHOD. With 60° triangle draw diameters AD and EB, and with same triangle draw sides AB and ED, EF and BC, AF and CD, each tangent to the given circle. 46 PROBLEMS 66. To draw a hexagon having given a long diameter, AD. Fig. 75. Bisect AD. With K as center, describe circle ACE, and in this inscribe a hexagon by Art. 64. DRAFTSMAN’S MetHop. With dividers find center K. Through A and D with 60° triangle draw AB and DE. With same triangle draw BE, and through D and A draw CD and AF. Draw BC and FE to complete the hexagon. ‘To obtain B without finding K draw a line through D at an angle of 50° with AD intersecting a 60° line through A. 67. To draw a hexagon having given a short diameter, GH. Vig. 76. Bisect GH. From G draw a perpendicular GF. From K draw FKC at 30° with Gk. ‘Through F and with center K draw circle FBD. Inscribe a hexagon by Art. 64. DRAFTSMAN’S METHOD. With dividers find center K. With 60° triangle draw FC, and through G, K and H draw perpendiculars FE, AD and BC. Draw the sides FA and DC, AB and DE. 68. To draw a hexagon having given a side, AB. Fig. TT. With centers A and B, and radius AB, describe ares 1 and 2. From their intersection, K, with same radius describe circle AEC. Inscribe a hexagon by Art. 64. DRAFTSMAN 'S METHOD. ‘Through A and B with 60° triangle draw AD and BE, and through their intersection, K, draw FC... Draw FA, BC; FE, DC and ED: . GEOMETRICAL PROBLEMS | 47 69. To inscribe an octagon within a given circle, ACEG. Fig. 78. Draw any diameter GC. At center, and perpendicular to GC, draw AK. Bisect AKG and AKC. Connect the points of intersection with the circle. DrRAFTSMAN’S MEetHop. With 45° triangle and T square draw diameters AE, GC, FB and HD, and connect their extremities. 70. To circumscribe an octagon about a circle, ABCD. Fig. 79. Draw the perpendicular diameters AC and BD. With centers A, B, C and D, and radius AK, describe arcs 1, 2,3, 4. By connecting these points of intersection a circumscribed square will be obtained. With the centers R, 8S, V, IT, and radius RK, describe ares 5, 6, 7, 8 to obtain the points G, H, L, N, O, P, E, F, which being connected will com- plete the circumscribed octagon. DRAFTSMAN’S METHOD. With 45° triangle and T square draw tangents FE and LN, GH and PO. At 45° draw tangents FG, ON, HL and EP, completing the octagon. 71. On a given side, AB, to construct a regular polygon having any number of sides. Fig. 80. With AB as a radius describe the semi- circle D2B and divide it into as many parts as the polygon has sides; in this case five. Beginning with the second division from the left draw radial lines, A2, A3, A+. A2 will be one side of the polygon. Bisect sides AB and A2 to obtain center of circumscribing circle. The intersection of this circle with the radial lines A2, A3, and A4, will determine the vertices of the polygon. 48 PROBLEMS 72. Within an equilateral triangle, ABC, to draw three equal circles tangent to each other and one side of the triangle. Tig. 81. Bisect the angles A, Band C. Bisect the angle DCA. E is the center of one of the required circles. With center K and radius KE describe are EFG. F and G will be the remaining centers. From these centers with radius EL describe the required circles. 73. Within an equilateral triangle, ABC, to draw three equal circles tangent to each other and two sides of the triangle. I[‘ig.82. Bisect the angles A, Band C. Bisect angle DCA. E will be the center of one of the circles. With K as center and radius KE, describe are EFG to obtain the remaining centers. Draw circles tangent to the sides AB, BC and AC.* 74. Within an equilateral triangle, ABC, to draw six equal circles tangent to each other and the sides of the triangle. Tig. 83. Bisect the angles and obtain E as in Art. 72. Through E draw HN parallel to AC. Draw HM parallel to AB, and MN parallel to BC. With E, H, F, M, G and N as centers, and with radius EL, describe the required circles. 75. Within a given circle, ACE, to draw three equal circles tangent to each other and the given circle. Fig. 84. Divide the circle into six * Instead of bisecting the angle DCA, to obtain one of the centers, describe a semicircum- ference, AFC, on one of the sides, as AC, thus obtaining the center F. GEOMETRICAL PROBLEMS equal parts by diameters AD, BE, CF. Produce AD indefinitely, and from E draw the tangent EG. Bisect KGE. With K as center, and radius HK, describe are HLM, and with radius HE, from centers H, L and M, describe the required circles. 76. Within a given circle to draw any number of equal circles tangent to each other and the given circle. Fig. 85. Divide the circle by diameters into twice as many equal parts as circles required; in this case eight. Suppose the center of one of these circles to lie on AK; then the circle must be tangent to both FK and KB. Draw tangent at A intersecting KB. Since the required circle must be tangent to the given circle at A, it will also be tangent to AB, and as it must lie in the angles FKB and ABK, its center must be at D, the intersection of their bisectors. With center K draw circle through D; its intersection with EK, CK, etc., will determine the required centers. Irom these centers describe the required circles with radius AD. 77. About a given circle to circumscribe any number of equal circles tangent to each other and the given circle. Fig. 86. Divide the circle by diameters into twice as many equal parts as circles required; in this case six. From A, the extremity of any diameter, draw tangent AB. Produce KB making BC equal to AB. At C, perpendicular to BC, draw CD intersecting AK produced, at D. This will be the center and AD the radius of one of the required circles. With center K and radius DK obtain other centers. 49 SECTIONS CONIC Fig. 88. CHAPTER IV CONIC SECTIONS 78. The Cone and Cutting Planes. Figs. 87 and 88 are illustrations of a right cone with a circular base cut by planes making several angles with the axis. It is a complete cone in that it extends as much above the vertex A as below it, the two parts being known as the upper and lower nappe. It is called a right cone because the axis is perpendicular to the base. The curves of intersection between the planes and the surfaces of the cone are known as conic sections. ‘They are four in number: the CIRCLE, ELLIPSE, PARABOLA and HYPERBOLA. An edge view of the planes is illustrated by Fig. 87, which shows the relation they bear to the axis AC and an element AB. The circle is obtained by a cutting plane, N, perpendicular to the axis. The ellipse is obtained by a cutting plane, R, oblique to the axis and making a greater angle with the axis than the elements do. ‘The parabola is obtained by a cutting plane, 5, making the same angle with the axis as the elements do. ‘The hyperbola is obtained by a cutting plane, T, making a smaller angle with the axis than the elements do. All planes cutting hyperbolic curves will cut both nappes of the cone. In Figs. 87 and 88, if we conceive the plane T as revolving about the line EF as an axis, it will cut the cone in hyperbolas from T to S. At 5, parallel to AB, the curve will be a parab- ola. From S to N it will cut ellipses, and at N, a circle. The latter is not shown in Fig. 88. 51 52 CONIC SECTIONS These curves may be obtained in two ways: First, by determining the curve of inter- section between the planes and the cone, as in Fig. 88; second, by known data and a knowl- edge of the characteristics of the curve. Only the latter is considered in this chapter. 79. The Ellipse is a curve generated by a point moving in a plane so that the sum of the distances from this point to two fixed points shall be constant. If, in Fig. 89, we conceive EKF to be a cord fastened at its extremities, E and F, and held taut by a pencil-point at K, it may be seen that as motion is given to the point it will be constrained to move in a fixed path dependent on the length of the cord. When the pencil-point is at B, one segment of the cord will equal BE and the other BF, their sum being the same as KE plus KF, and also equal to AB. The fixed points E and F are called the Foct. ‘They he on the longest line that can be drawn terminating in the curve of the ellipse. ‘The line is known as the MAJgor Axis, and the perpendicular to it at its middle point, also terminating in the ellipse, is the Mrvor Axis. Their intersection is called the center of the ellipse, and lines drawn through this point and terminating in the ellipse are known as diameters. When two such diameters are so related that a tangent to the ellipse at the extremity of one is parallel to the second, they are called CoNJUGATE DIAMETERS. KL and MN are two such diameters. In order to construct an ellipse it is generally necessary that either of the following be given: The major and minor axes; either axis and the foci; two conjugate diameters; a chord and axis. 80. Ellipse. First Method. Fig. 89. By definition it may be seen that a series of points must be so chosen that the sum of the distances from either of them to the foci must equal the major axis. Thus, HE+ HF must equal CE+CF, or KF + KE, each being equal to AB. THE ELLIPSE 3 If the major axis and the foci be given to draw the curve, points may be determined as. fol- lows: From E, with any radius greater than AE and less than EB, describe an arc. From F, with a radius equal to the difference between the major axis and the first radius, describe a second are cutting the first. The points of intersection of these arcs will be points, the sum of whose distances from the foci will equal the major axis, and therefore points of an ellipse. Similarly find as many points as may be neces- sary to enable the curve to be drawn free-hand. Lightly pencil a line through these points. For inking see Art. 15, page 11. Having given the major and minor axes, we can find the foci by describing, from C as a center, an are with a radius equal to one-half the major axis. The points of intersection with the major axis will be the foci; and this must be so since the sum of these distances is equal to the major axis; and the point C being Fig. 89. ELLIPSE FIRST METHOD midway between A and B the two lines CE and CF must be equal. Again, if the major axis and foci are given, with a radius equal to one- half this axis describe arcs from the foci cutting the perpendicular drawn at the middle point of the major axis and thus obtain the minor axis. Having the two axes proceed as before. A tangent to an ellipse may be drawn at any point, K, by connecting this point with the foci, and bisecting the exterior angle SKE. KT will be the required TANGENT. b4 CONIC SECTIONS 81. Ellipse. Second Method. Fig. 90. Let AB and CD be the major and minor axes of an ellipse. Lay off on a piece of paper having a clean-cut edge the distance RT equal to one-half the major axis, and RS equal to one-half the minor axis. If point T be placed upon the minor axis and point S upon the major axis, and the paper constrained to move always under these conditions, the point R will describe an ellipse. Points may be laid off on the drawing to correspond with the different positions of R, and through these the required ellipse will be drawn. ‘This is an excellent method, as construction lines are not required. It is known as the method by trammels, since an instrument called the elliptographic trammel is constructed on this principle. 82. Ellipse. Third Method. Fig. 91. Having the major axis AB and the minor axis CD, describe circles on these as diameters. Draw any radial line, as MG. From its intersection with the outer circle draw MO perpendicular to the major axis, and from its intersection with the inner circle draw NO perpendicular to the minor axis. The intersection of these lines at O will be a point in the ellipse. Similarly obtain other points. THE ELLIPSE A tangent at the point O may be obtained by drawing a tangent to the outer circle at M and from its intersection with the major axis at B, drawing the required tangent through O. 83. Ellipse. Fourth Method. Figs. 92, 93, 94. This is a very general method and may be used when we have given either the major and minor axes, one of the axes and a chord of the ellipse, or any two con- jugate diameters. CAsE 1. Fig. 92. Having given the major and minor axes. From the extremity of the major axis, draw B6 parallel and equal to half the minor axis; di- vide it into any number of equal parts; in this case six. Divide BG into the same number of equal parts. Through points 1, 2, 3, etc., on B6, draw lines to ex- tremity C of the minor axis. From D, the other ex- tremity of the minor axis, draw lines through points 1, 2, 3, etc., on BG, intersecting the above lines in points which will lie in the required ellipse. Construct the remainder of the ellipse in the same manner. CAsE2. Fig. 93. Having given an axis CD and chord FH. From F draw F4 parallel to CD; divide it’into any number of equal parts; in this case four. Divide the half chord FE Fig. 93. Cc ELLIPSE eee METHOD 56 3 CONIC SECTIONS into the saine number of equal parts; through these points and extremities of given axis draw intersecting lines as before, thereby obtaining the elliptical are FD. Construct opposite side in the same manner. Case 38. Having given the conjugate diameters AB and CD, Fig. 94. From A and B draw lines A6 and B6 parallel to the diameter CD and equal to CG. Divide these into any number of equal parts, and, hav- ing divided BG and AG into the same number of equal 5 parts, draw lines from these points to the extremities Fig. 94. 00 es" Of diameter CD. .'The intersection of these lines with Cc Oo | the former, will determine points in the eilipse. In like . manner describe the opposite side. : 3 BS Je 84. Ellipse. Fifth Method. Fig. 95. To describe g an approximate ellipse, the major and minor axes being | 4 ‘ given. For many purposes in drawing it is sufficiently | : accurate to describe the ellipse by means of four circu- lar ares of two different radii. ‘The following is one of H K several methods: On the minor axis lay off GF and | Fig. 95. - ELLIPSE FIFTH MetHoD «6 «equal to the difference between the major and | | minor axes. On.the major axis lay off GE and GL equal to three-quarters of GF. Connect points F, E, O, L, and produce the lines. ‘With M THE PARABOLA 3 eye 57 center E and radius AE describe-are HAM. With center F and radius FD describe are KDH. In like manner describe MCR and RBK from centers.O and L. Do not use this method when the major axis is moré than twice the minor. 85. The Parabola is a curve generated by a point moving in a plane so that its distance from a fixed point shall be constantly equal to its distance from a given right line. Point F, Fig. 96, is the Focus, CD is the given right line called the Drrecrrrx, and AB, a perpendic- ular to CD through F, is the Axis. V, the intersection of the axis with the curve, is the VERTEX, and by the definition of a parabola it must be equidistant from the focus and directrix. 86. Parabola. First Method. Fig. 96. Having given the foeus F and the directrix CD. Bisect FA to find the vertex V. Through any point on the axis, as L, draw MN parallel to the directrix and with radius LA describe arc 1 from focus F as center, intersecting line MN at points Mand N. These are points in the parab- ola. Similarly obtain other points and draw the required curve. <}- => a but by making successive revolutions it is -~_.__.__. bgp A : possible to represent the object in any con- | | ceivable position. Having given the projec- | ! tions of the pyramid, as in Fig. 113,, the | following may be determined : — /\ |

~ ~ ~ ~ Qa &o N ~ Fig. 172. ment of acone. ‘The pitch is measured parallel to the axis, as in the preceding case. 106 SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS 132. Screw-Threads. In order to make the drawing of a screw-thread, it is necessary to know the diameter, pitch and section of the thread. If the thread to be drawn has a V section, as in Fig. 174, and the diameter and pitch are given, begin by drawing the section of the screw as shown in dotted lines. Next, construct a templet as follows: Draw on a rather light piece of cardboard two helices of the same pitch as the required thread, one being of a diameter equal to the outside of the thread and the other equal to the diameter at the root of the thread. These may be drawn separately, or together, as in Fig. 178, and are to be carefully cut out so as to be used as a pattern in pencilling the curves. Having drawn the helices, mark the points of tangency B, G, and A, F, as well as the centers Fig. 173, L and K, in the manner indicated. Also repeat | these marks on the opposite side after cutting out. This will enable the templet to be used either side up, and be readily set to the drawing. Observe that the curve is not to be cut off abruptly at its termination, but continued a little beyond, so that in tracing the outline the pencil-point may not injure the extreme point of the templet curve. This may now be used for the drawing of all the helices on this screw, as from A to F, B to G, C to H, etc., Fig. 174. If the pitch is small in proportion to the diameter, the drawing of the screw may now be considered finished; but the contour line does not coincide with that of the section of the thread, and in order to illustrate correctly the projection of a V thread we must consider the L— | K SCREW-THREADS character of the surface and apply a correction to the drawing. The surface which is being drawn is a helicoid, and is generated by the motion of a line, AB, which is made to revolve about the axis of the screw and at the same time move in the direction of the axis. ‘This will generate the upper half of the surface of the screw. Every point of the line will describe a helix, the diameters differing, but the pitch remaining constant. The helices generated by the extremities of the line have already been drawn and the curves 1 2 3 and 4 5 6, described by two other points, 1 and 4, are shown by the fine dotted lines. The curved line M5 2N, drawn tangent to these helices, will be the visible outline of the surface . instead of the dotted line which is concealed. As the labor of describing these helices would be great, it is customary to draw the outlines of the screw as follows: Having described the helices AF, BG, etc., reverse the templet and draw a small portion of the continuation of the helix on the opposite side of the screw, as at BP and CO. Then, draw the line MN tangent to the two helices, and in the other direction the tangent OP, a part of which is invisible. 107 108 SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS Be sD Fig. 176. 133. Conventional V Threads. In order to facilitate the drawing of screw-threads it is customary to omit the drawing of the helix, substituting therefor a right line, as in Fig. 175. In most cases, however, even this would involve too much labor as well as complication on the drawing, and the V’s are likewise omitted, the representation shown in Fig. 177 being adopted. When this is done, no care is taken to make the screw of the required pitch, and the spaces between the fine lines which represent the outer helices are estimated by the eye, as in section lining. But it is imperative for the proper representa- tion of a single thread, that the point C be over the middle of the space BD. Having drawn the line AB, make the space AC double that of EB and draw parallels. Afterwards, draw the heavy lines to represent the root of the thread. As it is difficult to make these of equal length without the aid of special lines, the method illustrated in Fig. 178 is frequently used. 134. The Double Thread. As the use and character of the double thread are generally misunderstood, Figs. 175 and 176 have been drawn to explain this problem more clearly. Fig. 175 illustrates a screw, the diameter and pitch of which are supposed to have been given; but, as the pitch is excessive for a screw of this diameter, the diameter at the root of the thread is small, and the screw propor- U.S. STANDARD V THREADS 109 tionally weak. The only way to strengthen the thread at this point without changing the angle of the V’s is by partly filing the V’s at the root, as shown by the dotted line in Fig. 175 and the left-hand portion of the complete screw in Fig. 176, thus increasing the diameter at the root. While overcoming one weakness we have introduced a second by lessening the section of the thread, so that with a nut of a given length the tendency of the thread to be stripped from the body is doubled. This last difficulty may be overcome by supposing an inter- mediate thread wound between the present threads, as shown by the right-hand portion of Fig. 176, which is a representa- tion of a double thread having the same diameter and pitch as the single thread of Fig. 175, but of increased strength. It must be noted that the threads indicated by AB and CD, of Fig. 176, are entirely independent of each other and that the point C of one is diametrically opposite a point B in the par- allel thread. This must be carefully observed-in the practical representation of a double thread, as shown in Fig. 179. Fig. 180 represents a left-hand single thread. _ 135. U.S. Standard V Threads. ‘The form of thread com- monly used is that of the U.S. Standard, also known as the Franklin Institute Standard and illustrated by Fig. 181. RIGHT HAND SINGLE THREADS Fig. (77. Fig. 178. ANY 0H DOUBLE THREAD LEFT HAND THREAD Fig. 179. Fig. 180. P= 0.24VD + 0.625 — 0.175. S = 0.65 P. 110 SPIRALS, HELICES, SCREW-THREADS AND BOLT-HEADS Although the pitch in a single-threaded screw is the: distance between consecutive threads,. the term is often applied to the number of threads per inch. Thus a screw having eight threads to an inch is frequently spoken of as 8 pitch. This is obviously wrong, but leads to no confu- sion, since the pitch and the number of threads per inch are reciprocals of each other. The flattening of the thread, as indicated in Fig. 181, is for the purpose of preventing injury by the bruising of the otherwise sharp V. 136. Square Threads. Fig. 184 represents a thread of square section. ‘The size of this square is equal to one-half the pitch in a single thread, and one-quarter of the pitch in a double thread. The construction is similar to that described for the V thread. The outline of the section is drawn first, and a templet is prepared for the inner and outer helices. The threads are drawn as in Fig. 184 for a single right-hand thread, and as in Fig. 185 for a nut in section. Figs. 186 and 187 illustrate a double square thread and nut. The conventional representation of the square thread is similar to that of the V thread, the right line being substituted for the helix. Fig. 182 is a correct drawing of the conventional square thread, and Fig. 183 a modification more commonly used. < Ek = (ea Corea Pars OF 30 WITH G.L. of GL, and the traces of P. Art. 96, page 66. The drawing of the projection lines may be SEEN CHORD Akron Vine omitted also. Use care to project all of the invisible lines of the object. | Pros. 20. It is required to represent three views of the preceding object when turned around on its base. The construction is similar to that of Prob. 18. In this case the center lines will no longer serve for the GL and traces of P. Use care to project all of the invisible lines of the object. 132 DRAW THREE VIEWS | DRAW THREE VIEWS OF THE OBJECT OF THE OBJECT WHEN REVOLVED WHEN REVOLVED PROBLEMS 21 TO 32 FROM THE POSITION | FROM THE POSITION OF FIG. 1, 30° TO THE | OF FIG. ee 25° ABOUT LEFT ABOUT AN AXIS|AN AXIS PERPENDIC- DRAW THREE VIEWS |PERPENDICULAR TO V ULAR TOH FIG. 3. DRAW THREE VIEWS | DRAW THREE VIEWS | DRAW THREE VIEWS OF THE OBJECT OF THE OBJECT OF THE OBJECT WHEN REVOLVED WHEN REVOLVED WHEN REVOLVED FROM THE POSITION |FROM THE POSITION FROM THE POSITION OF FIG.|I, 20 FORWARD OF FIG.2, 15 FORWARD] OF FIG. 5, 35° ABOUT ABOUT AN AXIS PER- | ABOUT AN AXIS PER- | AN AXIS PERPENDIC- PENDICULAR TO P PENDICULAR TO P ULAR TO H FIG. 4. FIG. 5. FIG. 6. ka Peor. 24% altitude of Tie a PROB. 22: altitude of 13!’. TOP VIEW OF FIG. Is ob RE PROB? ?23. altitude of 17! if TOP VIEW OF. FIG. I. 1 1 F Pros. 24. altitude of 13". TOP VIEW OF FIG. |. PROBLEMS 145. Objects oblique to the coordinate planes. A careful study of Art. 97, page 67, must be made previous to the solution of Problems 21 to 32 inclusive. The plate will be divided into six rectangles as in the accompanying figure, the division lines being drawn in pencil only. Six positions of three views each will be re- quired in each case. Omit the drawing of pro- jection and shade lines. If difficulty is found with the problems, number the points as in Fig. 107, page 65, but use care to retain the same number for each point throughout the problem. In the required positions draw a rectangular pyramid having an In the required positions draw a rectangular prism having an In the required positions draw a triangular pyramid having an The base is an equilateral triangle. In the required positions draw a triangular prism having an The bases are equilateral triangles. ORTHOGRAPHIC PROJECTION Prop. 25. In the required positions draw a pentagonal pyramid having an altitude of 13’. Diameter of circumscribing circle of base 13’. Pros. 26. In the required positions draw a pentagonal pyramid having an altitude of 13’... Diameter of circumscribing circle of base 13'’. Pros. 27. In the required positions draw a pentagonal pyramid having an altitude of 13’. Diameter of circumscribing circle of base 12”. Pros. 28. In the required: positions draw a hexagonal pyramid having an altitude of 12’. Pros. 29. In the required positions draw a hexagonal pyramid having an altitude of 17’. Pros. 80. In the required positions draw a wedge having an altitude rot 12. Pros. 31. In the required positions draw a wedge having an altitude of 14”. Pros. 32. In the required positions draw the frustum of. a rectangular pyramid having an altitude of 12”. 4 138 TOP VIEW OF FIQ. I, ‘TOP VIEW OF FIQ. |. TOP VIEW OF FIG. | TOP VIEW OF FIG. I. a Fs ora i TOP VIEW OF FIG. |, oe ae A TOP VIEW OF FIG, I. k- 13 Das) i TOP VIEW OF FIG. I. ca be yanmar, ee] + 5 aE 7 TOP VIEW OF FIQ. I. 134 PROBLEMS PROB. 33. OBTAIN THE PROJECTION ON THE AUXILIARY PLANE PROB. 35 HEIGHT OF, 3” PRISM 8 REVOLVE CONE 30 ABOUT VERTICAL AXIS OBTAIN 3 VIEWS PROBLEMS 33 to 37 require a knowledge of the use of auxiliary planes. Art. 98, page 70. Pros. 33. The auxiliary plane being par- allel to the plane of the right-hand end of the object, the projection on that plane will be a true representation of that surface. Other lines and surfaces will be foreshortened. Pros. 34. In this problem the auxiliary plane makes an angle of 30° with H, and the projection on this plane is to be made in place © of the top view. Pros. 35. Observe that the prism has a triangular hole extending through it. Pros. 36. This is similar to the preceding, save that the prism is hexagonal instead of tri- angular. A circular hole, 14!’ diameter, extends from base to base. Pros. 87. Study Arts. 99 and 100, pages 72 and 73, before solving this problem. Having obtained the base of the cone in the manner directed, locate the vertex of the cone. The tangents to the base drawn from the vertex will be the contour lines of the cone. ORTHOGRAPHIC PROJECTION Veh 146. Special problems in projection. ‘These problems require a space of 5!’ x 7!’. Three views are required in each case, and all invisible as well as visible lines should be shown on each view. Leave all construction lines in pencil. All polygons are regular polygons. Pros. 388. Draw the frustum of an octagonal pyramid having its base parallel to H and two of its edges making an angle of 30° with V. The diameter of the circumscribing circle of its lower base is 14’, and of the upper base, 14". The altitude is 17’. Pros. 39. Revolve the pyramid of Prob. 88, 30° to right about an axis perpendicular to V. Pros. 40. Draw a pentagonal prism resting on one of its faces and having its lateral edges at an angle of 223° with V. Diameter of circumscribing circle of base 1}/’.. Length of prism 24’. Pros. 41. Draw an equilateral triangular prism resting on one of its faces, and its lateral edges making an angle of 15° with V. The edges of the base are 13’’, and the length of the prism is 24’. There is a triangular hole extending through the bases and making the thickness of the sides 4’. Pros. 42. Draw a cylinder with its axis parallel to V and at an angle of 60° with H. The diameter of the base is 13’’, and the length of cylinder, 2}/’.. Obtain the ellipses by the method of trammels. Pros. 43. Draw an equilateral triangular pyramid having an altitude of 24’, and the edges of the base 17/’.. The base makes an angle of 30° with H and one of its edges is per- pendicular to V. Pros. 44. Revolve the pyramid of Prob. 43, 45° forward about an axis perpendicular toRk: 136 PROBLEMS Pros. 45. Draw a box having the following outside dimensions. Length 2!, width 13”, depth, including cover, 1/’.. Thickness of material 4’... The long edges of the box are parallel to H and make an angle of 30° with V. The cover is hinged on long edge and opened 30°. Pros. 46. Draw a pyramid formed of four equilateral triangles having 22” sides. The base is parallel to H and one of its edges makes an angle of 30° with V. Pros. 47.. Draw a rectangular surface, 14!’ x 28’, in the following positions : The short edges parallel to H and making an angle of 75° with V; the long edges making angles of 15°, 80° and 45° with H. Art. 102, page 76. Pros. 48. Revolve the surface from the positions required in Prob. 47, 15° forward. Pros. 49. Draw an isosceles triangle in three positions as follows : The base lying on V and inclined at an angle of 30° with H. ‘The altitude making angles of 90°, 30° and 15° with V. The base of the triangle is 14’’, and the altitude 23. Art. 102, page 76. Pros. 50. Draw the same triangle revolved from the positions in Prob. 49, 30° in either direction about a vertical axis. | Pros. 51. Draw an isosceles triangle in the following positions: The base parallel to H and making an angle of 60° with V. The altitude making angles of 45° and 60° with H. The base of triangle is 2’, and the altitude 21’’.. Art. 102, page 76. Pros. 52. Revolve the same triangle from the positions in Prob. 51, 30° backward. Pros. 53. Draw an octagonal surface inclined at an angle of 60° with H, two of its edges being parallel to H and making angles of 15° with V. ‘The diameter of circumscribing circle is: 24"! Arte 102; page 76. Pros. 54. Draw a hexagonal surface inclined at an angle of 45° with V, two of its edges being parallel to V and making angles of 30° with H. The long diameter of the hexagon is 24". Art. 102, page 76. ISOMETRIC PROJECTION s¥6 Pros. 55. Draw the projections of a line located as follows: The left-hand extremity of the line is 3 behind V and 7’ below H. The right-hand extremity is 14/’ behind. V, and 13" below H. The H projection of the line makes an angle of 30° with V. Find its length by revolving it parallel to H, Vand P. Art. 101, page 74. Pros. 56. Draw the projections of a lne of which the left-hand extremity is 1'’ behind V and 12” below H. The right-hand extremity is 3’’ behind V and 3"’ below H. The H pro- jection makes an angle of 15° with V. Find the length of the line by revolving it into the planes of projection by the second method. Art. 101, page 75. 147. Isometric projection problems. Study Chapter VI, page 78. The problems require a space of 5’ x 7'’.. Omit the invisible lines in inking. : Pros. 1. Make the isometric drawing of a 2’ cube. Art. 110, page 80. Inscribe circles on the upper and right-hand faces, the former by the exact method and the latter by the approximate method. Art. 112, page 81. From the left-hand lower corner of the left-hand. face draw lines making angles of 30°, 45° and 75° with the lower edge. Art. 113, page 82. Pros. 2. Make the isometric drawing of the frustum of a pyramid, the lower base being 2”, and the upper base 13/’ square. Height 14’’. Inscribe a circle on the upper base using the ap- proximate method. Locate the front lower corner in the center and 1'’ from lower margin. Pros. 3. Make the isometric drawing of a pentagonal plinth surmounted by a cylinder. The sides of the pentagon are 2" and the height of the plinth is 3’’.. Art. 71, page 47. The cylinder is 2!’ in diameter and 1” high. Art. 111, page 80. Pros. 4. Make the isometric drawing of a box with cover opened through an angle of 120°. The outside dimensions are: length 21/’, width 13/’, depth #/’.. Thickness of material is 4’. Locate the front lower corner in the middle of the space and 4/’ from lower margin. 138 PROBLEMS or icpan ska ea! PROB. 6 Pros. 5. Make the isometric drawing of the bearing illustrated, locating the upper corner at point A. Pros. 6.. Make the isometric drawing of a hexagonal bolt. ‘The center line may be parallel with either of the isometric axes. Art. 115, p. 83. Pros. 7. Make the isometric drawing of the pieces illustrated, and a second isometric drawing of the upright block showing the cuts neces- sary for making the required fits. The lowest portion of the upright block will be located at A in the first ease and B in the second. Pros. 8. Make the isometric drawing of the connecting-rod strap, the scale to be 3/’=1 ft. In drawing the curves observe the directions of Art. 115, page 83. Pros. 9. Make the isometric drawing of the framing details illustrated. The dimensions of the materials are given below. The space required is 10’ x 14’’. Draw toa scale of 13/’=1 ft. Sills, 613". Post, AS Bll Brace, AM Di, Window studs, 4x4". Studs, 2" x4", 12" on centers. Floor joists, 2x8", 12" on centers. Under flooring, 7" x10". Upper flooring, 7x 6". Grounds, #3!'x 2". Laths, 3" 11" x 48", 3" space. Plaster, 3" thick. Baseboard, 7x10", including cap. ‘ ISOMETRIC PROJECTION 139 eariieat= cial 9 md ; = Bia BASE-BOARD A, g a2 v/, Z FLOOR JOIST 2X6. - i LLLLLLL LION a Roope FLOOR ho epee te Cpoeneeneee _ UNDER FLOOR | : FLOOR JOIST FLOOR JOIST 2x8 i SILL. WINDOW STUD LAYOUT OF SHEET. Aen ET FT a LABBawaa’ DETAILS FOR ISOMETRIC DRAWING V|_ FLOOR voisT. OF FRAMING PLANS SSX y EY) 140 WE/EH7S PULLEY NS TN QQ e{} 2 a AN Sy Nt PROBLEMS Pros. 10. detailed sketches. Make the isometric drawing of a window from the The dimensions of the materials are given below. Show all that is given in the sections. Locate the drawing as in the lay-out sketch, observing that the short edge of the plate is hori- zontal. LAYOUT OF SHEET Art. 115, page 83. Draw to scale of 3! =1 ft. DIMENSIONS OF MATERIALS Studs, Apron, Ground easing, Inside architrave, Outside architrave, Outside casing, Outside boarding, Pocket, Pulley style, Parting bead, Stop bead, Laths, Plaster, Al x 4", "x33", 3" x 4a", gil x 5B 13"x 3h" se ga 4. Qi 4 i" thick. BIN yc BI, a" x14". 3" x 1i"x 48", spaced 8", 3" thick: Clapboards, 4" at thick edge, x 54!" x 48", laid 34"' to the weather. DEVELOPMENT OF SURFACES 141 148. Development problems. ter VII, page 88. plete development will be required in each case. In general, begin to develop the surface on the shortest edge. It will assist the student to a better understanding of this subject if the de- veloped surface be copied on stiff paper and afterwards cut and folded. Observe the order prescribed for performing these problems. Art. 119, page 90. Pros. 1. This isa prism having a pentag- onal base. ‘The vertical edges being parallel to V are seen in their true length on that plane. The length of the edges of the lower base can be obtained from the top view, and the true shape of the upper base will be found by projecting it on to an auxiliary plane, as in Art. 98, page 70. Pros. 2. Solve as for Prob. 1. Pros. 8. In this and the following prob- lems do not ink that portion of the surface lying above the cutting plane. Art. 119, page 90. Pros. 4. Observe that the slant edges of this pyramid are not of equal length. Study Chap- Three views and the com- BOB Me oe viEWion PROB. 2 | PENTAGON - HEXAGONAL PRISM — INSCRIBED Nat | REVOLVED | 2 CIRCL SECTION easy. aon SIDE VIEW | OBTAIN igharanannns DEVELOPMENT DEVELOPMENT PROB. 3 PROB. 4 | =| REVOLVED SECTION | SECTION REVOLVED OEVELOPMENT DEVELOPMENT 142 PROBLEMS PROB. 5 ’ PROB. 6 TOF VIEW OF TOP VIEW OF s QUARE PYRAMID SQUARE tf atl REVOLVED REVOLVED SECTION ! errr SIDE VIEW ex | | DEVELOPMENT DEVELOPMENT DEVELOPMENT Pros. 5. This is a square pyramid, the diagonal of the base being 2’’.. Observe that the base is cut by the plane, and one of the inclined edges will not appear on the completed develop- ment. In developing the surface open it on this line. Pros. 6. This differs from the preceding in that the base is not cut by the plane, and the position of the pyramid with respect to V is changed. Pros. 7. The surface cut by the plane will not be symmetrical with respect to the center line as in the preceding problems. Care must be used to take no dimensions from the top view that may not be represented by lines parallel to H, the vertical projection of which will be parallel to GL. Pros. 8. This is similar to Prob. T, only the position of the pyramid with respect to V being changed. DEVELOPMENT OF SURFACES Pros. 9. The cutting plane makes an angle of 40° with H, and the auxiliary plane must be at the same angle, the projecting lines being drawn perpendicular to it. Pros. 10. This is similar to Prob. 5, the base being cut by the cutting plane. In devel- oping the surface open it on the uncut edge. Pros. 11. Theslant edges of this pyramid being of unequal length must be obtained sepa- rately. One of these edges, AD, is shown in its revolved position at AD’. One-half of the development is also shown, and the method and order for drawing the lines indicated by the numbers. Thus, the line AB is drawn first, and then arcs 3 and 2 described from its ex- tremities, A and B, with radii equal to the true lengths of AC and BC: this determines the point C. In like manner the remaining points and lines are found. « Pros. 12. The development of a cylin- der is required. Arts. 120 and 121, page 92. Employ twenty-four elements in obtaining the curve. ° 148 PROB. 9 te PROB. 10 TOP VIEW OF , HEXAGONAL PYRAMID | PENTAGON INSCRIBED IN i 2-\"CIRCLE as REVOLVED | SECTION + H REVOLVED SECTION ee oy nN | SIDE VIEW SIDE VIEW | d lo DEVELOPMENT DEVELOPMENT PROB. I2 REVOLVED SECTION DEVELOPMENT 144 PROBLEMS PROB. |3 PROB. 14 REVOLVED aa REVOLVED SECTION . . SECTION SIDE VIEW DEVELOPMENT DEVELOPMENT PROB. I5 Pros. 4. Solve without auxiliary planes. Pros. 138. Use twenty-four elements for obtaining the development. Art. 122, page 94. Pros. 14. This being an elliptical cone, the elements will be of unequal length. Pros. 15. The cone is cut by four planes, CG, CB, CF, CE. Determine the top view of each and their projection on planes to which they are parallel. ‘Test the latter as follows: Determine the axes and foci of the ellipse, and test eight points by the first method, Art. 80, page 52. Test by trammels also. Test the parabola by the second method, Art. 87, page 58. The cutting plane, CI, is parallel to an element of the cone. Test the hyperbola by the second method, Art. 91, page 59. The ver- tex of the cone bisects the transverse axis. 149. Intersection problems. Chap. VIII. Pros. 1. Assume twenty-four equidis- tant elements on the small cylinder. Pros. 2. The cylinders are tangent. Use care in obtaining the limiting points. Pros. 8. Use auxiliary planes. Page 98. Art. 125, p. 100. Develop oblique cylinder. INTERSECTION DEVELOPMENT OF CYLINDER A SIDE VIEW DEVELOPMENT OF CYLINDER B PROB. 2 DEVELOPMENT OF CYLINDER A | SIDE VIEW ’ DEVELOPMENT OF CYLINDER B OF SURFACES TOP VIEW DEVELOPMENT OF CYLINDER A DEVELOPMENT OF CYLINDER 145 146 PROBLEMS DEVELOPMENT OF DEVELOPMENT OF EQUILATERAL TRIANGULAR EQUILATERAL TRIANGULAR PRISM TOP VIEW DEVELOPMENT OF HEXAGONAL PRISM HEXAGONAL PRISM A Pros. 5. It is required to find the inter- section of a cylinder and prism without using a side view. Use cutting planes parallel with V. The base of the prism will have to be revolved in order to complete the top view and enable the intersection of the cutting planes and prisms to be determined. Pros. 6. To determine the intersection of a hexagonal and a triangular prism. This is similar to Prob. 5, save that a prism is substi- tuted for the cylinder. In all cases of inter- section between prism and prism, it is only necessary to find the point of intersection of each edge of both prisms with a face of the other prism. Art. 126, page 101. Pros. 7. The lines of intersection on the top view are to be completed, and the front view with its lines of intersection are required. Develop the prism only. Pros. 8. Determine the intersection of the oblique and vertical hexagonal prisms and develop the latter. Art. 126, page 101. SPIRALS AND HELICES 147 150. Spirals, screw-threads and bolt-head problems. Study Chapter IX. Page 102. Problems 1 to 9 inclusive require a space of 43’. A graphic statement is made for the prob- lems on screw-threads and bolt-heads. Pros. 1. Draw an equable spiral with a pitch of 11/. Describe 13 revolutions of the radius vector. Art. 128, page 102. Sketch the curve, and ink as directed in Art. 15, page 12. Lea aad Pros. 2. Draw the involute of a 3” square. --3 }-—> j i! : SECTION OF Art. 130, page 104. 2 SQUARE THREAD 14 3 Sa es Pros. 3. Draw the involute of an equi- aoa ute at [THREAD | lateral triangle having 3!’ sides. Art. 180, page 104. Pros. 4. Draw the involute of a right Branckaghiy enon line 4’ long. Art. 130, page 104. pally ; ANGLE OF, V 90" Pros. 5. Draw the involute of a hexagon having sides of #/’. Art. 130, page 104. Pros. 6. Draw the involute of a circle having a diameter of 1’... Art. 130, page 104. Pros. 7. Draw a right-hand helix of 14/’ pitch, 2’’ diameter and 23’ length. Art. 131, page 104. | Pros. 8. - Draw a left-hand double helix of 13/’ pitch, diameter 13’, length 21". Pros. 9. Draw a right-hand conical helix. Pitch 1}!’, diameter of cone 2’, height 17/’. Pros. 10. The diameter and pitch being the same in both cases, but two templets are re- quired, one for the outer and one for the inner helix. Art. 132, page 106, and Art. 186, page 110. 148 PROBLEMS R.H. THREAD 24 aye 5 ° 3 PITCH 60 V {—_ - . R.H. DOUBLE . | . . CA . sQ! it P. Noo — gt =e a 2 : itp R.H.SINGLE y , haa S iciecae Same ; Z Ce oe oe Pros. 11. The first four examples are of conventional V threads, Art. 183, page 108. The second four are conventional square threads. Art. 186, page 110. The last four are to be drawn by the methods illustrated by Figs. 177 to 180, page 109. Make the pitches about the same as those in the illustrations, estimating the spaces by the eye. Distinguish clearly between the single and double threads. Pros. 12. Study Arts. 187, 188 and 139 before attempting this problem. The propor- tion and character of the heads and nuts should be so well understood that reference to the text will be unnecessary. ‘The diameters are given, and the sketch shows the character of the bolt, whether rounded or chamfered. Observe every detail and see that the dimensions are standard. Draw the rounded heads and nuts before the chamfered type. For the further consideration of this subject, the student is referred to the chapter on Bolts and Screws, in ‘Machine Drawing” of this series. MISCELLANEOUS PROBLEMS 151. Miscellaneous problems. These examples of a practical character are de- signed for pencilling and inking practices which may be used early in the course. Problems 1 and 2 may be substituted for some of the “examples for practice,” page 117 and following. They will afford an excellent practice in lettering and figuring. Problems 3 and 4 hkewise require very little knowledge of projection and serve as an excellent practice in penmanship. Full instruction is given with each sketch and the proper scale is specified for a 10" x 14" plate. The shading of the drawing is left to the discretion of the instructor. ‘The drawings should be cor- rectly figured, and the title in proper form. The author does not recommend the making of machine drawings, save as above, until the student has acquired a thorough working knowledge of projection, and has been taught something of the notation and idiomatic use of applied graphics. 149 PROB.| ECCENTRIC. eae DRAW TWO FULLVIEWS. SHADE AND DIMENSION LINES REQUIRED, 150 PROBLEMS TWO COMPLETE SECTIONS OF THE FIRST AND SECOND FORMS OR THE \FIRST ANDO THIRD FORMS MAY BE Ba alsen INA fA C 10/4 SPACE. | STANDARD, SECTION OF Z£OC al D SEWER Ca ae YQ: Ww 480 . ~ : AN SI RII; | 3 aX ID So: 18 | DRAW COMPLETE SECTION DRAW COMPLETE SECTION DRAW COMPLETE, SECTION SCALE /°=/F7. SCALE 22/FT. SCALE 2 =/FT. a MISCELLANEOUS PROBLEMS 15 ONE WROUGHT JPON. BoRDER £/NE HAN) LEVER FOR CONTROLLING CYL/NDER VALVES U.S. COAST DEFENSE VESSEL MONTEREY 152 PROBLEMS 2 Wa ra SO BARS % | | | | | | | | | | | | | eds | | | | | | Pros. 4. A sketch is given for the Commutator of a 10 K.W.D.C. Dynamo, and it is required to make a full-size sectional drawing of the complete commutator. te ” -URBANA 3 0112 123898675 UNIVERSITY OF ILLINOIS UICC Shoe ea PS a eee Sansa ee Sa Seteeeeae ee = Soe SUR an TEETER Ses eee Sore : : = pepaeereranmecerests Bota eeeeee teres eeagoo aes SSE TSE ae: : Saree Set Stee ine sees RSS Seats Re ee ae “SESS