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 5 
 
 Wza 
 
 SECOND BOOK. 
 
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I 
 
WALTERS' ELEMENTARY GRAPHICS. 
 
 SECOND BOOK. 
 
 PKOJECTION DKAWING. 
 
 BY 50HH DANIEL Of A LIBELS, ffl. SG. 
 
 PROFESSOR OF INDUSTRIAL, ART AND DESIGNING IN THE KANSAS STATE 
 AGRICULTURAL, COLLEGE. 
 
 MERCURY PUBLISHING HOUSE, 
 Manhattan, Kanwis. 
 
 1894. 
 
"A place should also be found in the school or college course for at least the elements of 
 the modern synthetic or protective geometry. It is astonishing that the subject should be so 
 generally ignored, for mathematics offers nothing more attractive. It posesses the con- 
 creteness of the ancient geometry without the tedious particularity, and the power of analyt- 
 ical geometry without the reckoning, and by the beauty of its ideas and methods, illustrates 
 the esthetic quality which is the char-m of the higher mathematics, but which the elementary 
 mathematics in general lacks. "From the report of the committee on Secondary Studies, 
 appointed at the meeting of the National Educational Association, July 9, 1892. 
 
 COPYRIGHTED, 1895. 
 
PREFACE. 
 
 This course in Elementary Graphics consists of four separate text- 
 books, each of which is intended to furnish daily work for a term of from 
 six to twelve weeks, or its equivalent. It includes the subjects of 
 
 1. Geometrical Drawing, 
 
 2. Projection Drawing, 
 
 3. Elements of Descriptive Geometry, 
 
 4. Linear Perspective, 
 
 and was originally prepared for the classes of the Kansas State Agricul- 
 tural (and Mechanical) College, but has been arranged here so that the 
 different volumes may be used separately and under varying conditions. 
 It is intended, however, that Geometrical Drawing should precede, and 
 Projection Drawing should follow the study of Plane and Solid Geometry. 
 
 J. D. WALTERS. 
 
 Manhattan, Kansas, October, 1894. 
 
 M319099 
 
GLOSSARY 
 
 OF MATHEMATICAL AND TECHNICAL TERMS. 
 
 The Numbers Refer to the Paragraphs Containing Definitions or Descriptive 
 
 Statements. 
 
 Axial, non-axial 33 
 
 Axis of rotation 13, 14, 15 
 
 Axonometry 3, 31 
 
 Auxiliary plane 12 
 
 Auxiliary projection 12 
 
 Black printing 48 
 
 Blue printing 48 
 
 Brilliant point 51 
 
 Brilliant line 49, 50, 51 
 
 Cabinet perspective .32 
 
 Conic section 17, 18, 19, 20, 21 
 
 Co-ordinate 19, 20 
 
 Co-ordinate planes 12 
 
 Descriptive geometry 1, 2, 14 
 
 Dip 13 
 
 Directrix 19, 20 
 
 Double curved surface 52 
 
 Development 24, 25, 26, 27 
 
 Elevation 7 
 
 Ellipse 17, 18 
 
 Ellipsoid 29 
 
 False perspective 32 
 
 Focus 17, 18, 19, 20 
 
 Generatrix 18, 19, 20 
 
 Ground line 9 
 
 Helical 29 
 
 Helical flange. 29 
 
 Helicoid 29 
 
 Helix 28, 29, 30 
 
 Hyperbola 17, 20 
 
 Inking ...47 
 
 Isometric 3, 31, 35, 36, 37, 38 
 
 Light 39 
 
 Locus 17, 18, 19, 20 
 
 Major axis 18 
 
 Minor axis 18 
 
 Monodimetric 31, 32, 33, 34 
 
 Normal 18, 19, 20 
 
 Ordinate 18, 19, 20 
 
 Orthogonal 3, 12 
 
 Orthographic 3 
 
 Parabola 17, 17 
 
 Plan 9 
 
 Planes of projection 6, 12 
 
 Plane of rotation 13 
 
 Pitch 28 
 
 Projection 6 
 
 Rabattement 15, 22 
 
 Radius vector 17, 18, 19, 20 
 
 Rotation 13, 17 
 
 Scotia 54 
 
 Screw line 28, 29, 32 
 
 Secant plane 15, 10 
 
 Section 15, 16, 17 
 
 Serpentine 37 
 
 Shades 49, 50, 51, 52 
 
 Shadow 39, 40, 41, 42 
 
 Single curved surface 49 
 
 Stippling 51 
 
 Torus 29 
 
 Trace 12, 15, 12 
 
 Zincetching 48 
 
PART I. 
 Definitions, Problems and Exercises. 
 
 INTRODUCTION. 
 
 1. Projective Geometry. 
 
 The objects of projective geometry are: 
 
 (I.) To represent, by drawings, geometrical magnitudes in space. 
 
 (2 ) To solve problems on forms in space by construction in a plane, 
 and to demonstrate by the method of projections the properties of form 
 and position. 
 
 Accordingly the subject matter of the science may be divided into 
 
 (1.) Projection Drawing 
 
 (2.) Descriptive Geometry. 
 
 2. Projection Drawing. 
 
 Projection drawing is the art of representing space forms upon plain 
 surfaces, so as to show their real dimensions and relations. 
 
 Conceptions of the form of solids or objects having three dimensions, 
 are at first obtained with difficulty from drawings. Especially is this the 
 case when the drawings are not perspectives and lack shading. ( Work- 
 ing drawings.) The systematic study of projection drawing is therefore 
 a matter of great importance to everyone intending to follow an indus- 
 trial, engineering or artistic pursuit. For the same reason, the study of 
 descriptive geometry is usually preceded by a course upon the methods of 
 rep resenting objects having three dimensions. It is evident that the 
 student must learn to read and draw the language of space forms before 
 he can expect to make progress in the analysis or such forms, just as he 
 had to learn reading and writing before he could commence the study of 
 rhetoric or literature. 
 
 3. Different Methods of Projection. 
 
 There are at least four scientific methods of projection drawing: 
 (V) Orthographic projection. 
 (2 ) Axonometric projection. 
 (3 ) Conical projection. 
 (4 ) Spherical projection. 
 
WALTERS' ELEMENTARY GRAPHICS. 
 
 Each of these has characteristics that make it suitable for certain 
 kinds of scientific or practical work. In mathematical and engineering 
 drawing, the first, or orthographic method, is commonly used. This book 
 treats orthographic and axonometric projection. 
 
 4. Method of Work. 
 
 Experience has proved that a series of problems of progressive diffi- 
 culty, which, taken in their logical order, the student can master alone, 
 or with little assistance on the part of the teacher, will accomplish the 
 desired ends better than any other. 
 
 Some of the work in projection drawing offers better opportunity for 
 artistic efforts, requires larger sheets of paper, and should be done under 
 the eye of the teacher, while a majority of the problems may be solved on 
 smaller sheets, simply with triangle and compasses. For this reason, 
 this book, like Book 1, has been divided into two parts: 
 
 Part I. Definitions, problems and exercises. 
 
 Part II. Shading and draughting. 
 
 The first part contains the subject matter of the home work, and the 
 second part that for the class room work. The study of both parts should 
 be commenced at the same time, and should be carried on together. One 
 day should be given to discussions and illustrations of new principles and 
 to blackboard recitations by the class; the next day should be devoted to 
 draughting in the class room. 
 
 FIG. 1 
 
 All home work in projection drawing should be done in ink and 
 should be finished uniformly, so as to permit binding at the close of the 
 term. Each plate should be drawn with a heavy borler line inclosing a 
 surface of 6i by 9 inches. Only one side of the paper should be used. 
 The sheet should not be rolled, but ought to be carried "flat" in a large 
 book or a portfolio. "What is worth doing at all is worth doing well !" 
 5. Tools and Materials. 
 
 The tools required for the home work in projection drawing are: 
 
 (1.) A pencil of rich quality and hardness that it will take and hold 
 a fine point. 
 
 (2.) An eraser. 
 
 (3.) A pair of solid white metal compasses with pencil and pen at- 
 tachment. 
 
 (4.) A drawing pen. 
 
 (5.) A drawing board, 12 to 14 inches bv 20 to 22 inches, (see illustra- 
 tion.) 
 
PROJECTION DRAWING. 
 
 (6.) A small T-square. 
 
 (7.) A right angled isosceles triangle having sides of about 6 inches. 
 
 (8.) Four small white metal thumb tarks. 
 
 Of materials, will be needed a bottle of jet black writing ink not 
 writing fluid, as this would corrode the drawing pen and about three 
 dozen sheets of best American drawing paper, size 8i by 11 inches. This 
 paper is usually sold in sheets measuring 17 by 22 inches and has to be 
 folded and cut. If the finished work is to be bound, larger sheets should 
 be provided and the additional margin Should be arranged for on the left 
 side of each plate. Good grades of French linen paper will also be ser- 
 viceable. 
 
 The additional tools and materials needed in shading and draughting 
 are described in the Introduction of Part II of this book. 
 
 DIVISION B. 
 
 GENERAL PRINCIPLES. 
 
 6. The Planes of Projection. 
 
 The solid or object to be drawn is supposed to be placed above a hori- 
 zontal and in front or a vertical plane. These planes are called the planes 
 of projection. 
 
 FIG. 2. 
 
 Perpendiculars, named projectors, are then supposed to be dropped 
 from every corner or conspicuous point of the solid to both planes of pro- 
 jection. By connecting the ends of those projectors upon the planes of 
 projections by lines that represent the edges or outlines of the solid, two 
 pictures are obtained. 
 
WALTERS' ELEMENTARY GRAPHICS. 
 
 It will be seen that these pictures, named the horizontal projection and 
 the vertical projection, form a perfect record of the solid or object, a record 
 that enables the reader to imagine and measure the represented form. 
 
 In practical draughting, the horizontal projection is called plan 
 and the vertical projection elevation. 
 
 A plan is a drawing made on a flat surface, which describes the 
 length and breadth of a solid or object on a surface that is considered as 
 always lying horizontal. 
 
 An elevation is a drawing made on a flat surface that is regarded as 
 always standing vertical perpendicular to the plan. 
 7. Plan and Elevation. 
 
 In figure 3,the student will recognize two vertical and two horizontal 
 projections.!. e.,the plans and elevations.of a dog house. A careful study 
 of all the features of this drawing will convince him of the facts stated in 
 paragraph 6. 
 
 FIG. 8. 
 FRONT ELEVATION. SIDE. ELEVATION. 
 
 FLOOR PLAN 
 
 ROOF PLAN. 
 
 SCALE 1 INCH TO 2 FEET. 
 
PROJECTION DRAWING. 
 
 A carpenter, asked so build a dog house like this, could go to work at 
 once and construct it without asking a single question regarding its form. 
 Understanding how to use the scale of one inch to two feet, which has been 
 used in making this draught, he could measure the height of the whole 
 object, the height of the sides and the height of the opening, in the two 
 elevations, while he could obtain the horizontal measure, the thickness 
 of the boards, etc., in the plans. 
 
 To read simple plans and elevations, like these, is not difficult, but 
 there are many questions arising in connection with the drawing and 
 reading of forms of three dimensions that will tax the student more 
 severely. Everyone can read an item in a book or newspaper without 
 much of an effort, yet few are able to write an article grammatically and 
 rhetorically correct. In order to acquire the ability to do this, language 
 must be studied systematically. The student must look for general 
 principles. 
 
 8. Seven Principles. 
 
 From the given figures, and from blackboard drawings and models 
 exhibited by the teacher, the following principles may readily be deduced: 
 
 (1 ) Two planes, at right angles to each other, are necessary to fully 
 represent the three dimensions, length, width and height, of a solid. 
 
 (2) These two planes may really consist -of one surface only. By 
 dividing it, i. e., the sheet of paper, the slate, or the blackboard, by a 
 horizontal line, we can readily imagine the upper part to be vertical and 
 the lower part to be horizontal. 
 
 This horizontal division line is called the ground line. 
 
 (3.) In order that dimensions shall be seen in the projections in their 
 true size and relative position, they must be parallel to that plane on 
 which they are shown. 
 
 (4.) Each plane shows two of the dimensions of the solid the two 
 which are parallel to it. 
 
 (5.) The dimension which is shown twice, is the one which is parallel 
 to both of the planes. 
 
 (6 ) The height of the vertical projection of a point above the ground 
 line, is equal to the height of the point itself, in space, above the horizon- 
 tal plane. 
 
 (7.) The perpendicular distance of the horizontal projection of a 
 point from the ground line, is equal to the perpendicular distance of the 
 point itself, in front of the vertical plane. 
 
 9. Examples and Problems. 
 
 Having established these principles, it should not be difficult to read 
 the following figures and to construct the projections required in this 
 paragraph. 
 
 Note that the ground line is marked G L and is drawn in full line, 
 while the projectors, and the edges not seen from above and from the front, 
 are drawn in dotted line. This should be done in all required work. 
 
10 
 
 WALTERS ELEMENTARY GRAPHICS. 
 
 In projection drawing, to draw means to tind the horizontal and verti- 
 cal projection. All solids should be drawn with their axis in vertical 
 position and standing upon H, unless otherwise specified. 
 
 PROBLEM 1. To draw the projections of a cube having edges of 2f 
 inches. 
 
 PROBLEM 2. To draw a hexagonal prism having a base diagonal of 
 2f inches and an altitude of 3 inches. 
 
 PROBLEM 3. To draw a pentagonal pyramid, 4 inches high; its base 
 having sides of H inches. 
 
 PROBLEM/! To draw a triangular pyramid, 4 inches high, with a 
 base having sides of 2 inches. Represent it as standing upon its apex. 
 
 PROBLEM 5. To draw a cone with its axis horizontal and parallel to 
 V; altitude 3 inches; diameter of base 3 inches. 
 
 FIG. 4. 
 
 FIG 5. 
 
 FIG. 6. 
 
 FIG. 7. 
 
 FIG. 8. 
 
 PROBLEM 6. To draw a sphere having a radius of If inches, and hav- 
 ing its center 2 inches from both planes of projection. 
 
 PROBLEM?. To draw the projections of a common thread spool, 
 consisting of a cylindric shaft and two truncated cones, the whole pierced 
 by a cylindric opening Length of spool 3 inches, length of shaft 2 
 inches, diameter of shaft 1 inch, diameter of ends 2 inches, diameter of 
 opening inch. 
 
 10. The Projection of the Straight Line. 
 
 From the last exercise the student has learned that a line in space can 
 have four radically different positions with regard to the planes of pro- 
 jection It may be 
 
 (1.) Parallel to both. 
 
 (2.) Parallel to one and perpendicular to the other. 
 
 (3.) Parallel to one and oblique to the other. 
 
 (4.) Oblique to both. 
 
 If the student will read figures 9-14 by illustrating them with his 
 pencil as representing the line in space, using the table to represent H 
 
PROJECTION DRAWING- 
 
 11 
 
 and a standing book to represent V, he will notice that the projection of 
 a line in space is either 
 
 (a) equal to the real line, 
 
 (b) shorter than the real line, or 
 
 (c) a point. 
 
 FIG. 9. FIG. 10. FIG. 11. FIG. 12. FIG. 13. FIG. 14. 
 
 PROBLEM 8. To draw Mnee lines parallel to H and V, two lines par- 
 allel to V and oblique to H, three lines oblique to H and V. 
 
 n. Mensuration of the Line in Space. 
 
 In the first three cases of paragraph 10, the line in space may be 
 measured directly by measuring one of its projections. In the fourth case 
 both projections are shorter than the real line and this can therefore not 
 be measured directly. The length must be tomulindirectly, which may be 
 done in several ways. 
 
 From the following figure it will appear that the line in space is the 
 hypotenuse of a triangle of which in every case the two sides are shown. 
 A FIG. 15. B 
 
 AB* = ta'b'). 2 
 A B = = VjiTbT 
 
 (m.b) 
 
 (m b) 
 
 In construction, the real length of the line in space may be found by 
 rotation, i. e., by revolving it about a known axis until it becomes parallel 
 to either of the planes of projection. 
 
 In Fig. B, the line A B has been revolved about the vertical axis A 
 M into V, where it now appears at full length. The student will see, that 
 
12 
 
 WALTERS' ELEMENTARY GRAPHICS. 
 
 the line might have been revolved about a vertical axis through B, or a 
 horizontal axis through A, or a horizontal axis through B, or in fact, 
 about any vertical or horizontal axis in space. In every case there would 
 have been the same result 
 
 The ability of revolving or rotating any line in space about any verti- 
 cal or horizontal axis into a position parallel to V or H is very valuable 
 and the student should execute such rotations until he becomes proficient, 
 not only in measuring lines of all possible original positions, but also in 
 imagining such rotations in space, without any drawing or experimenting. 
 
 DIVISION C. 
 
 AUXILIARY AND SECANT PLANES, ROTATION. 
 
 12. The Auxiliary Plane. 
 
 Figure 2 gives two elevations of a dog house 
 resents the front and the second the side 
 
 Fio 16. 
 
 The first elevation rep- 
 Most objects require two eleva- 
 
 tions on two plans, i. e., they require an additional projection upon an 
 auxiliary plane of projection. Sucn a plane is usually, though not always, 
 a vertical plane placed perpendicular to the two main (co-ordinate) planes 
 of projection to right or left of the object. 
 
 An examination of the above figure will shovv how an auxiliary 
 projection may be obtained from the given main projections. 
 
PROJECTION DRAWING. 
 
 13 
 
 The dot and dash lines represent auxiliary planes of projection, i. e., 
 they are the intersections (truces) of auxiliary planes with H or V. (These 
 two capital letters are common aoreviations for vertical plane of projec- 
 tion and horizontal plane of projection.) 
 
 After having drawn the projectors orthogonal (rectangular) to the 
 auxiliary planes, these are rotated about their vertical traces into V (or H, 
 as the case may be) and become a part of V. The arcs show the move- 
 ment of the projections pending the rotation. 
 
 PROBLEM 10. To draw the horizontal and vertical projection of a 
 vertical heptagonal prism and to find two auxiliary vertical projections. 
 
 Place the right auxiliary plane perpendicular to V and H,and the left 
 auxiliary plane at 60 to V and 90- to 1 . Draw the figures so as to reason- 
 ably fill the space within the border line of the page (6| by 9 inches.) Draw 
 all projectors in dotted line or full red ink; the auxiliary planes in black 
 dot and dash line,tbe G L and the visible edges in the projection sin strong 
 black line; the rear or invisible edges in dotted black line 
 13. Vertical Rotation. 
 
 Instead of using the auxiliary plane to obtain additional projections, 
 the object itself may be revolved or rotated so as to produce different pro- 
 jections upon V or H. 
 
 FIG. 17. 
 
 Figure 17 shows the rotation of a three-sided pyramid about an 
 imaginary axis The axis is a line perpendicular to Y through point A. 
 The plane of rotation is parallel to V. All points of the pyramid move in 
 arcs which record as arcs in V, and as straight lines parallel to G L in H. 
 As a result a second vertical projection is obtained which is equal to the 
 first, but has a different dip to H: and a second horizontal projection 
 which is entirely different from the first, but shows every point at an un- 
 changed distance from V. 
 
14 
 
 WALTERS' ELEMENTARY GRAPHICS. 
 
 Solve the following problem: 
 
 PROBLEM 11. To rotate a vertical pentagonal pyramid through an 
 angle of 30; the rotation to be parallel to V and the axis of rotation to be 
 i inch above the apex. 
 
 14. Horizontal Rotation. 
 
 In the last paragraph the solids were rotated about horizontal axes. 
 This paragraph treats the rotation of single solids about vertical axes. 
 
 FIG. 18. 
 
 PROBLEM 12. To find the projections of a three-sided pyramid hav- 
 ing a base edge of 2 inches and an altitude of 3 inches. The solid is to be 
 drawn in four positions pending a rotation about a vertical axis, as 
 indicated in figure 18. 
 
 Solve the following problem: 
 
 PROBLEM 13. To find the projections of a cube having an edge of 
 2i inches, in four positions pending a rotation about a vertical axis 
 placed in front of the solid. 
 
 FIG. 19. 
 
 PROBLEM 14. To find the projections of a right hexagonal pyramid 
 21 inches high and If inches in diameter at its base, whose axis shall be 
 oblique to both planes of projection. 
 
PROJECTION DRAWING. 
 
 15 
 
 The given illustration represents the method usually followed in" 
 placing an object in an oblique position to both planes, though descriptive 
 geometry gives additional solutions which will be explained in Book III 
 of this series. Figure A represents the pyramid standing on H and par- 
 allel to V. Figure B represents it tipped up and back to the left, and 
 moved forward parallel to V, s that every point in it remains the same 
 distance as before from V. The vertical projection remains the same, 
 except that it is oblique to G L. The horizontal projection is changed, 
 and can be found by drawing projectors and parallels. Figure C repre- 
 sents the pyramid, as it is in its second position, revolved 45 about a 
 vertical axis of rotation. This places the axis of the pyramid in the re- 
 quired position oblique to both planes of projection. The plan ia figure 
 C is the same as that of figure B, but the vertical projection is changed. 
 It can be found by drawing projectors from the third plan and parallels 
 to G L from the second elevation. 
 
 FIG. 20. 
 
 Solve the following problem: 
 
 PROBLEM 15. To draw a plinth (a low square prism), first, as lying 
 upon H; second, as having been pushed to the right a distance equal to 
 its length, and rotated 45 about a horizontal axis: third, as having been 
 pushed to the right once more and rotated 30- about a vertical axis. Di- 
 mensions 1 by 2i by 2* inches. 
 
 In Fig. 17, an auxiliary plane, U V X, has been placed oblique to H 
 and perpendicular to V, so as to coincide with one of the slanting edges 
 of the pyramid. As a result, the student will notice an additional pro- 
 jection to the right of the main vertical projection. How has it been ob- 
 tained? Has the oblique plane of projection been rotated about an axis? 
 Can you illustrate the movement by means of models, i. e., a solid and a 
 plane? 
 
16 
 
 WALTERS ELEMENTARY GRAPHICS. 
 
 It is frequently required to lay a pyramid or other object having faces 
 oblique to the axis, so upon H that it will appear to rest upon one of them. 
 The problem can be rapidly solved by this method. 
 
 15. The Secant Plane. 
 
 A secant plane is an imaginary plane passing through a solid or object 
 and cutting it into two parts. The; imaginary cut or surface is called 
 section. According to the position of the secant plane the cut may give 
 a horizontal section, a vertical section or an oblique section. 
 
 The secant plane, like the auxiliary, is usually represented by its 
 traces, i. e., its intersections with H and V. The traces are drawn in 
 lines consisting of dots and long dashes. 
 
 Figure 21 represents a hexagonal prism cut by a secant plane which 
 forms an angle of 45 with H, and of 90 with V. The section is evi- 
 dently an irregular hexagon, the real form of which is found by its 
 rotation of 45 about a horizontal axis of rotation through A and perpen- 
 dicular to G L. The study of the given figure will explain the method 
 and reveal the real form. 
 
 FIG. 21. 
 
 The rotation of a plane figure in space into a plane of projection |s 
 called rabattement. 
 
 Solve the following problem: 
 
 PROBLEM 16. To find the real form of the section of a given triangular 
 pyramid, produced by a secant plane forming an angle of *HJP to the left 
 and passing through the right base corner. ^9 
 
 16. Sections of an Irregular Pentagonal Prism. 
 
 Draw the projections of an irregular pentagonal prism resting with 
 one of its faces upon H and having its axis parallel to V. To the left 
 place an auxiliary plane perpendicular to G L. Upon this find the aux- 
 iliary vertical projection. The figure is now ready for the following 
 problem. 
 
PROJECTION DRAWING. 17 
 
 PROBLEM 17. To find the real form of three characteristic sections 
 of an irregular pentagonal prism, produced by three intersecting planes 
 which intersect each other in a line in H and perpendicular to G L. One 
 section is to be a triangle, one an irregular pentagon, and one a tra- 
 pezium. 
 
 There are really four characteristic sections of this prism, the fourth 
 being a rectangle. The student will readily see where and how the 
 fourth section may be obtained. 
 
 DIVISION D. 
 
 PROJECTIONS AND SECTIONS OR THE CYLINDER 
 AND CONE. 
 
 17. The Conic Sections. 
 
 The cylinder and the cone are closely related solids. In higher 
 mathematics the first is considered as being only a special form of the 
 second, i. e., the cylinder is defined as being a cone having an infinitely 
 extended axis. For this reason the cone will form the chief object of 
 study of this division. 
 
 FIG. 22. 
 ft 
 
 G 
 
 Figure 22 gives the vertical projection of a vertical cone. It also 
 shows the traces upon V and through the cone of a secant plane in seven 
 different positions. Every one of these sections differs from every other, 
 yet they are evidently related to each other, because it can easily be 
 shown how by rotation of the intersecting plane each becomes the other 
 six by gradual changes. 
 
 At X assume an axis of rotation, perpendicular to V. Then let the in- 
 tersecting plane rotate on this from position A through B, C, D, E, F to G. 
 
18 WALTERS' ELEMENTARY GRAPHICS. 
 
 The section at A will be a straight line. 
 
 The section at B will be an ellipse. 
 
 The section at C will be a circle. 
 
 The section at D will be an ellipse. 
 
 The section at E will be a parabola. 
 
 The section at F will be an hyperbola. 
 
 The section at G will be a straight line. 
 
 The sections at any place between A and C, and between C and E, 
 will be ellipses; and the sections between E and G, hyperbolas. 
 
 The figure shows how the line, an element of the cone at A, is trans- 
 fortned into the other 4 forms the circle, the ellipse, the parabola&nd the 
 hyperbola and how the last of these forms again becomes a line as the 
 plane is rotated out of the cone at G. 
 
 By moving point X, together with the intersecting planes A, B, C, 
 D, E, F and G, toward the apex of the cone, another interesting trans- 
 formation takes place, which culminates, when this point is reached, 
 into the following series of real or imaginary limiting forms: 
 
 The circle becomes a point. 
 
 The ellipse becomes a point. 
 
 The parabola becomes an element. 
 
 The hyperbola becomes an isoceles triangle. 
 
 By extending the axis of the cone, the parabola and hyperbola grad- 
 ually approach each other and finally merge into the same limiting form. 
 Both branches of both curves become one cylinder element. 
 
 A contraction (shortening) of the axis of the cone causes the section 
 curves in space to gradually approach their horizontal projections, until 
 the limiting form of the hyperbola, the straight line (chord), is reached. 
 
 The student should convince himself of these interesting relations of 
 the different sections of the cone by carefully experimenting with a large 
 cone, upon the curved surface of which he should draw crayon lines rep- 
 resenting the sections named in this paragraph. He should make an 
 effort to imagine the horizontal and vertical projections of these sections 
 by looking at the model from different positions, and should familiarize 
 himself with the names. 
 
 The conic section curves appear also as loci apart from the cone, but 
 the cone is the most convenient form to observe their relation to each 
 other. 
 
 In order to study the mathematical character of these figures or 
 sections, it will be well to lake them up separately, leaving out the circle, 
 which has been studied to some extent in plain and solid geometry. 
 
 18. The Ellipse. 
 
 If a point moves in such a manner that the sum of its distances from 
 two fixed points (foci) is constant, the locus traced by the moving point is 
 an ellipse. 
 
PROJECTION DRAWING. 
 
 19 
 
 Figure 23 illustrates this definition and shows how, by means of two 
 pins, a string and a pencil, an ellipse may he drawn. 
 
 The ellipse has two foci at A and B. It is a symmetrical curve about 
 two lines, one of which is called major axis and contains the foci, and 
 another, which is called the conjugate or minor axis. The axes are mutu- 
 ally perpendicular and bisect each other. The extremities of the major 
 axis are called vertices. Any line from a focus to the ellipse is called 
 radius vector. Any line perpendicular to the major axis is called ordinate. 
 A tangent to an ellipse touches but does not cut the curve. A normal is a 
 perpendicular to a tangent at the point of contact; it bisects the angle 
 formed by the radii. The sum of the radii vector of the moving point 
 in every position is equal to the major axis. 
 
 FIG. 23. FIG. 24. 
 
 x^ 
 
 PROBLEM 17. (Figs. 23 and 24.) To construct an ellipse. 
 
 (1 ) its major axis and one focus, or 
 
 (2) its major and minor axis, or 
 
 (3) its minor axis and one focus, or 
 
 (4) its two axes, or 
 
 (5) its two foci and one point in the ellipse being given. 
 
 The given definition and statements of mathematical qualities of the 
 ellipse show that the problem is really the same in all five cases. In all 
 cases it is necessary to find first the two foci and the length of the major 
 axis, i. e., the sum of the/ocaZ distances. 
 
 After having found the foci and the major axis, the problem may be 
 solved either with the string or the compasses. 
 
 In figure 23, pins are inserted at the foci and a string or a thread is 
 tied there, equal in length, between the pins, to the major axis, or, which 
 is the same, reaching from one focus to the extremity of the minor axis 
 and back to the other focus. The inserted pencil will then describe the 
 required locus. 
 
 In figure 24, the major axis is cut at random as shown. With A 1, 
 A 2, A3, as the radii and F and F' as centers, arcs are described in each 
 quadrant. With B 1, B 2, B 3, etc., as tlie radii, and F and F' as centers, 
 these arcs are intersected. A free-hand curve drawn through the points 
 of intersection will be the required ellipse. 
 
20 
 
 WALTERS' ELEMENTARY GRAPHICS. 
 
 There are many other methods of constructing the ellipse. Some of 
 these are simple, (compare figures 30, 32, 48, 49 and 59) and others are com- 
 plex, but all involve the knowledge of some branch of analytic, or pro- 
 jective geometry, which the student may not have studied. 
 
 19 The Parabola. 
 
 The parabola is a curve which at all points, is equally distant from a 
 fixed point (focus,) and from a fixed straight line (directnx.) Figure 25 
 represents a parabola with the directrix (M N,) the focus (F,) the axis, an 
 ordinate, three radii, three co-ordinates, a tangent, a normal, and a chord. 
 
 Find these lines and note that the radius vector of every point in the 
 curve is equal to the co-ordinate of the same point. 
 
 The parabola is sometimes defined as an infinitely long ellipse. Fig. 
 22 shows how by gradual change of the angle, which the intersect- 
 ing plane forms with the axis of the cone, the elliptic section becomes 
 longer until the parabola is approached. The ellipse intersects all ele- 
 ments of the cones of which it is a section. The parabola intersects all 
 elements except one the one which is parallel to it. 
 
 The circle may be considered as one limiting form of the ellipse, the 
 parabola as a second, and the straight line as a third. 
 
 FIG. 26. 
 
 PROBLEM 18. (Fig. 25.) Given, the directrix and focus, to describe 
 the curve of a parabola. 
 
 Assume a directrix M N and a focus F. Draw the axis perpendicular 
 to the directrix. Bisect F A. This is evidently a point in the curve, 
 being equidistant from directrix and focus. Next draw a series of in- 
 definite ordinates, on both sides of the axis, and find points upon them 
 in the following manner: Measure 1 A and with the focus as center, 
 cut the ordinate in H. Then the distance of F H is equal to the distance 
 from II to the directrix. It is therefore a point in the required curve. 
 In a similar manner, a series of additional points may be found, which, 
 connected by a free-hand curve, will give the required parabola. 
 
PROJECTION DRAWING. 
 
 21 
 
 PROBLEM 19. (Fig. 26 ) To inscribe a parabola within a given rect- 
 angle. 
 
 The figure shows that the parabola consists of diagonals in the trape- 
 zoids formed within the given rectangle by the two sets of connecting 
 lines. 
 
 20. The Hyperbola, 
 
 If a point moves in such a manner that its distance from a fixed point 
 is always greater than its perpendicular distance from a fixed straight 
 line, in a constant ratio, the curve traced by the moving point is an 
 hyperbola. 
 
 The fixed point is the focus, and the fixed straight line is the directrix. 
 
 In other words, an hyperbola is a curve that is equidistant at every 
 point from two unequal circles. See figure 28 
 
 The properties and functions of the hyperbola are similar to those of 
 the ellipse and the parabala. 
 
 FIG. 27. 
 
 FIG-. 28. 
 
 PROBLEM. 20. To inscribe an hyperbola within a given rectangle. 
 The solution will be plain from a study of figure 27. 
 PROBLEM 21. To draw a plane curve equidistant from two given 
 circles. 
 
 The solution will be apparent from figure 28. 
 
 21. Projection of the Circle, the Cylinder and the Cone. 
 
 A circle may assume four radically different positions to the planes 
 of projection. It may be 
 
 (1.) Perpendicular to both. 
 
 (2.) Perpendicular to one and parallel to the other. 
 
 (3. ) Perpendicular to one and oblique to the other. 
 
 (4.) Oblique to both. 
 
 In case (1) both projections are straight lines. 
 
22 
 
 WALTERS ELEMENTARY GRAPHICS. 
 
 In case (2) one projection is a line and the other a circle. 
 
 In case (3) one projection is a line and tha other an ellipse. 
 
 In case (4) both projections are ellipses. 
 
 In all cases the greatest length of the projection is equal to the di- 
 ameter of the represented circle, because in every position of the circle 
 one of its diameters must be parallel to each plane of projection. 
 
 If the projection is an ellipse, the length of the minor axis depends 
 on the angle which the plane of the circle in space forms with the plane 
 of projection. In case (4) the minor axes of the two ellipses may be, but 
 are not necessarily, equal. 
 
 FIG. 29. 
 
 Having thoroughly comprehended these elementary principles, and 
 illustrated every case by means of a circular disc and two coordinate 
 planes, the projection of the cylinder and cone should not be difficult. 
 Figure 29 represents a horizontal cylinder lying upon H, with its axis 
 oblique to V. This horizontal projection is a rectangle, and must be 
 drawn first. The vertical projection is obtained from the horizontal pro- 
 jection as follows: Project the centers of the circles into V and draw 
 the top element. The end circles of the cylinder will appear as two equal 
 ellipses in V. Find their axes by drawing the necessary projectors. 
 Having found these, the ellipses may be drawn as shown in figure 23 or 
 24. 
 
 Another method consists in rotating the rear circle into V, about a 
 
PROJECTION DRAWING. 
 
 23 
 
 vertical axis Z, marking a number of points in it as C, D, E and F.and re- 
 volving it back again to C Z. Pending this revolution, or counter rota- 
 tion, as it is called, each point, except Z, describes a horizontal arc in 
 space, which can easily be drawn in II and V Parallels to G L and pro- 
 jectors will give the positions of the marked points in the required 
 vertical projections. The outline is drawn freehand through the estab- 
 lished points. The ellipse representing the front circle can be be drawn 
 in the same manner. 
 
 The student will recognize in this work a method of constructing an 
 ellipse of given dimensions. 
 
 Solve the following problems: 
 
 PROBLEM 22. To find the projections of a cylinder oblique to H and 
 parallel to V. 
 
 PROBLEM 23. To find the projections of a cylinder of given dimen- 
 sions, oblique to both planes of projections. 
 
 The problem is to be solved like problem 15, figure 20. 
 
 22. The Elliptic Section of the Cylinder and Cone. 
 
 The study of figures 30 and 31, which illustrate two simple methods 
 of obtaining the real form of elliptic sections of the cylinder, ought to 
 be sufficient to enable the student to find the real form of elliptic sec- 
 tions of the cone also 
 
 FIG 30 
 
 FIG. 31 
 
 The first method rotates the intersecting plane into H and obtains 
 the required ellipse as as a horizontal projection. The rotation is like 
 that of figures 21 and 29 and is called rabattement. 
 
 The second method rotates the ellipse on its major axis into V- The 
 ordinates, i. e., the perpendiculars to the major axis are obtained by 
 measuring the chords in the liorizonial section which is a circle. 
 
 Solve the following problems: 
 
 PROBLEM 24. Find the horizontal projection and the real form of 
 
24 
 
 WALTERS 7 ELEMENTARY GRAPHICS. 
 
 an elliptic section of a given cone. The real form is to be found by the 
 first method. 
 
 PROBLEM 25. Find the horizontal projection and the real form of 
 an elliptic section of a given cone. The real form is to be found by the 
 second method. 
 
 23. Parabolic and Hyperbolic Sections. 
 
 Both sections are frequently met with in mechanical work; it is 
 impossible, however, to learn to draw them correctly and rapidly in all 
 different positions without the knowledge of analytic geometry, which 
 teaches their mathematical properties and relations as plain geometry 
 teaches the properties and relations of the circle. No student who would 
 master drawing can afford to neglect the study of this highly important 
 branch of mathematics. 
 
 FIG. 32. 
 
 Figure 32 shows how to find an hyperbola from its given projections. 
 The cone is posed so that both projections of the section are straight 
 lines. The drawing of a number of elements will locate points that can 
 be determined in all required positions. 
 
 Solve the following problems: 
 
 PROBLEM 26. Given the projections of a cone and the horizontal 
 projection of an hyperbolic section, to find the vertical projection and 
 the real form of the section. 
 
 PROBLEM 27. Given the vertical projection of a cone and the verti- 
 
PROJECTION DRAWING. 
 
 25 
 
 cal projection of a parabolic section, to find the horizontal projection of 
 the cone and the parabola, and to find the real form of the parabola. 
 
 PROBLEM 28 Given tha vertical projection of a cone with the verti- 
 cal projection of a parabolic section, to find the vertical projection after 
 a rotation of 4-5 about the axis of the cone. 
 
 DIVISION E. 
 
 DEVELOPMENT OR SURFACES. 
 
 34 The Development of Rectilinear Surfaces. 
 
 If a surface of a geometric solid be rolled upon a plane so as to bring 
 each consecutive face, or each consecutive element if the solid is a single 
 curved surface, in contact with that plane without stretching, folding or 
 tearing the face or element, the surface is said to be developed, and the 
 plane figure which results therefrom is termed its development. 
 
 The cube, the rectangular block, the prism, the pyramid, and all 
 polyhedrons are developable. 
 
 Figures 33 and 34 represent the developed surface of a right hex- 
 agonal prism and a right hexagonal pyramid, and explain how the 
 development has been obtained. 
 
 FIG. 33. 
 
 FIG. 34. 
 
 Solve the following problems: ' 
 
 PROBLEM 29. To find the development of a cube having a right 
 square pyramid upon two opposite faces. 
 
 PROBLEM 30. To find the development of a truncated pentagonal 
 pyramid. 
 
 PROBLEM 31. To find the development of a cube cut into halves by 
 any oblique plane, and having sides of 1 inch. 
 
 PROBLEM 32. To find the development of a tetrahedron. 
 
26 WALTERS' ELEMENTARY GRAPHICS. 
 
 PROBLEM 33. To find the development of an octahedron. 
 25. The Development of the Cylinder. 
 
 The development of a right cylinder is an oblong or square, having a 
 height equal to the axis or an element, and a base equal to the circum- 
 ference. In practical work of ordinary character the circumference 
 is usually measured by dividing the base circle into twelve equal parts 
 and measuring the chord of such an arc of 30 twelve times. The result- 
 ing perimeter is, of course, slightly less than the circumference, because 
 the chord is shorter than its arc. 
 
 In plane geometry the student has learned that the ratio of the di- 
 ameter of the circumscribed circle to the perimeter of the inscribed 
 dodecagon is 1 : 3.1058. ., while the ratio of the diameter to the circle is 
 1 : 3.1416. . The above method of approximation is, therefore, nearly cor- 
 rect. 
 
 Solve the following problem: 
 
 PROBLEM 34. To find the development of a cylinder having an axis 
 of 3 inches and a diameter of 2f inches. 
 
 PROBLEM 35. To find the development of a rectangular stove-pipe 
 elbow consisting of two equal pices of sheet iron Length of axis of each 
 part, 24 inches; diameter of pipe, 8 inches; scale, 1 inch to 4 inches. 
 
 NOTE. The tinsmith cuts both parts out of one piece of sheet iron 
 measuring about 26 by 48 inches. How can he do this V 
 
 26 The Development of the Cone. 
 
 The development of the curved surface of a co 1 e is a sector. 
 
 Solve the following problems for which no figures or solutions are 
 given: 
 
 PROBLEM 36. To draw the patterns required by the tinner in mak- 
 ing a funnel consisting of a combination of two truncated cones. Axis 
 of large cone, 4 inches; axis of small cone, 3 inches; diameters of large 
 cone, 5 inches and 1 inch; diameters of small cone, 1 inch and finch. 
 No handle. 
 
 PROBLEM 37. To find the development of a cone truncated by an 
 oblique plane. Axis of complete cone, 5 inches; axis of truncated part, 
 2J inches; diameter of base, 3i inches; slant of section, 30 to H. 
 
 27. Miscellaneous Problems Pertaining to Development. 
 
 A double curved surface is one that has no straight elements, like the 
 surface of the sphere, the torus, the ellipsoid, etc. Such a surface can not 
 be fully developed. All that can be done is to find a close approxima- 
 tion by dividing the surface by means of meridians into a number of 
 lunes and arranging these on a plane. The larger the number of lunes is 
 made, the less stretching or folding there will be. Tinners, who are 
 often required to construct such forms out of tin, sheet iron, sheet zinc, 
 or sheet copper, seldom use more than 12 lunes. 
 
 PROBLEM 38. To draw the covering for a sphere. 
 
PROJECTION DRAWING. 
 
 27 
 
 Draw the elevation or plan of the solid. Divide the circumference 
 into 12 equal parts and lay these off on a straight line. With a radius of 
 9 of these parts draw a right and left arc through each division point, 
 except the end points which require only one arc. These arcs will inter- 
 sect above and below the straight line and form the required twelve lunes 
 an approximate covering for the sphere. 
 
 PROBLEM 39. To draw the covering for a sphere having a diameter 
 of 4 inches. The covering to consist of four zones of equal width. 
 
 Tinners and coppersmiths sometimes construct globes of zones in- 
 stead of lunes. When copper or some other highly malleable metal can 
 be used, a little hammering of the four conic surfaces, which form the 
 zones will give the required sphere. 
 
 DIVISION F. 
 
 THE HELIX. 
 
 28. General Principles of the Helix, 
 
 The curves may properly be divided into plane curves, such as the 
 circle, the conic sections, the spiral, the trochoids, etc., and into space 
 curves, such as the he lix or screw line which forms the subject of investi- 
 gation of paragraphs 28, 29 and 30. 
 
 A helix is the locus of a point which moves along the surface of a 
 cylinder in such a way that a constant ratio is maintained between the 
 measure of its rotation and ascent. 
 
 FIG. 40. 
 
 It is sometimes defined as a spiral line that constantly advances in the 
 direction of a straight line called its axis. The student who has mastered 
 the principles of Division E may assist his mind in imagining this line 
 
28 
 
 WALTERS ELMENTARY GRAPHICS. 
 
 by developing the curved surface of a cylinder, drawing a diagonal or 
 any oblique line in the development, and winding this again upon the 
 solid. After the rewinding the diagonal becomes a helix. 
 
 The helix is sometimes called screwline, because it is found in the 
 thread of every screw. See figures 35, 36, 37, 38 and 39. 
 
 If the curve ascends and passes from left to right in front of the axis, 
 when the axis is vertical, the helix is said to be right-hand, and if it 
 passes from right to left it is said to be left hand. 
 
 The pitch of a helix is the distance it advances along the axis in mak- 
 ing one revolution. 
 
 FIGS. 35, 36, 37, 38, 39. 
 
 Solve the following problems: 
 
 PROBLEM 40. (Fie. 37). To draw the projections of a right-hand 
 / helical line 2 inches in diameter, 5 inches long, and 3 inches pitch. 
 
 PROBLEM 41. To draw a left-hand helix having the same dimen- 
 sions and pitch as that of the last problem. 
 
 Careful study of figure 40. which is drawn I the required scale, will 
 reveal the method of construction. The helix is draw n free-hand, through 
 the located points. 
 
PROJECTION DRAWING. 
 
 29 
 
 (2.) 
 (3.) 
 (4.) 
 (5.) 
 
 29. The Helical Band, the Helical Flange, the Helicoid. 
 
 u ^ * * - * 
 
 All surfaces may be divided into: 
 (1.) Planes, 
 
 Single curved surfaces (cylinder, cones), 
 Double curved surfaces (sphere, torus, ellipsoid), 
 Warped surfaces (helical flange, helicoid), 
 Irregular surfaces. 
 One of the surfaces named in the headline of the this paragraph be- 
 longs to the second, and two belong to the fourth class. 
 
 A helical band is the surface ota cylinder between equal and parallel 
 helices. See figures 35, 38 and 42 
 
 A helical flange is a warped surface that is between two parallel but 
 unequal helices and is perpendicular to their common axis. See figure 
 41. 
 
 FIG. 41. 
 
 A helicoid is a warped surface that is between two parallel but un- 
 equal helices and is at an invariable acute angle t > their axis. See figure 
 42. 
 
 It may be said that, as the circle and the straight line are the limit- 
 ing forms of the ellipse, the helical flange is a limiting form of the 
 helicoid. 
 
 A serpentine is a bent cylinder the axis of which is a helix. It is some 
 times defined as the locus of a sphere, the center of which moves along a 
 helix. The bed-spring and the cork screw furnish illustrations of this 
 highly interesting space form. The groove of figure 37 is a semi set-pen 
 tine. So is the groove of the drill described in problem 47 
 
30 
 
 WALTERS' ELEMENTARY GRAPHICS. 
 
 Solve the following problems: 
 
 PROBLEM 42. To find the projections of a right-hand helical band, 
 i inch wide. Diameter of cylinder, 3 inches; pitch, H inch* 
 
 PROBLEM 43 To draw the projections of a left-hand helical flange 
 4 inch wide, 2 inches in outer diameter, 4 inches long, and H inch pitch. 
 
 PROBLEM 44. To draw the projections of a left-hand helicoid screw. 
 Diameter of outer helix, 3 inches; diameter of inner helix, 2 inches; 
 pitch, 1 inch. Compare with figure 42. 
 
 PROBLEM 45. To draw a right-hand flange screw of two threads 
 (jack screw). Diameter of the four outer helices, 4 inches; diameter of 
 the four inner helices, 3 inches; pitch of each thread, 2 inches. Compare 
 figure 41. 
 
 FIG. 42. 
 
 30. Additional Problems on the Helix. 
 
 PROBLEM 46. To find the approximate projections of a three-inch 
 rope consisting of 3 equal strands. Pitch, 6 inches. 
 
 PROBLEM 47. To find the approximate projections of a drill with a 
 conical point. 
 
 A drill is simply a steel cylinder with a dull conical point, and a 
 slightly enlarged square rear end for fastening it in the brace. It has 
 one helical groove of two revolutions and about 21 inches pitch. The 
 groove extends from the square rear end to the conical point and is semi- 
 cylindric at every point. Compare figure 37. 
 
 PROBLEM 48. To.find the projections of a helical stairway winding 
 
PROJECTION DRAWING. 31 
 
 about a column of 2 feet in diameter. Distance from floor to floor 10 
 feet, 8 inches; height of riser, 8 inches; diameter of whole stairwell, 18 
 feet. Handrails, nosing and nosing moulds not required. 
 
 PROBLEM 49. To draw a helix upon a cone. 
 
 With the exception of this problem the subject of conic helices has 
 been omitted here as being beyond the reach of elementary graphics. 
 
 DIVISION G. 
 
 AXONOMETRIC PROJECTION. 
 
 31. General Principles of Axonometric Projection. 
 
 The three imaginary straight lines which represent the three dimen- 
 sions, length, breadth and height of an object are called the axes of 
 dimension. All trained geometricians imagine these axes as intersecting 
 each other at right angles at the center of the solid, object or bulk to be 
 measured. 
 
 In axonometric projection the object is drawn so that two or all axes 
 of dimension form equal angles with the vertical plane of projection, 
 while the horizontal plane of projection is dispensed with entirely. 
 
 When a solid or object is posed so that two of its axes of dimension 
 are parallell to V and the third perpendicular to V, it is said to be in a 
 monodimetric position, and when it is placed so that all three axes form 
 equal angles with V, it is said to be in an isometric position. There are, 
 therefore, two systems or methods of axonometric projection: 
 
 (1.) Monodimetric projection. 
 
 (2.) Isometric projection. 
 
 32. Monodimetric Projection. 
 
 In monodimetric projection the object is placed so that its principle 
 arid most characteristic face is parallel to the plane of projection, and 
 can be drawn as it is. All other lines that are parallel to the principal 
 face lines are also projected at their real lengths and in their real posi- 
 tions, while the lines that form right angles with the face, i. e. the lines 
 parallel with the third axis of dimension, are drawn at angles of 4-1 to 
 the horizontal and vertical lines of the front face either to right or left, 
 up or down. The length of such a retreating line is made equal to the shortest 
 side of a triangle having angles of 105^, 4-5^. 3(P, and whose longest side equals 
 he line to be represented. A few worked problems will explain this some- 
 what complex statement. 
 
32 
 
 WALTERS' ELEMENTARY GRAPHICS. 
 
 PROBLEM 50. (Fig. 43). To draw the monodimetfic projection of a 
 cube of given dimension. 
 
 Draw A B equal to the given edge. Draw the square ABED. 
 Draw C A at 3(P to A B, and B C at 45 to A B. The angle at will evi- 
 dently measure 105 Draw the remaining edges, representing those not 
 visible by dotted lines. 
 
 FIG. 43. FIG. 44. 
 
 B % 
 
 PROBLEM 51. (Fig. 44). To draw the monodimetric of a hori- 
 zontal cylinder. 
 
 Locate A and draw, full size, the face circle. Draw A B equal to the 
 length of the required axis. Draw C B at 30 and A C at 45 to A B. 
 With C as the center draw the rear circle. Add the outline elements. 
 
 Solve the following problems. 
 
 PROBLEM 52. To construct the monodimetric of a horizontal hex- 
 agonal prism having a base side of li inch and an axis of 31 inches. 
 
 It is evident that this solid could be drawn in four different posi- 
 tions. Its lateral edges could be drawn so as to retreat to right and 
 upward, to right and downward, to left and upward, to the left and down- 
 ward. Select any of the three latter positions. 
 
 PROBLEM 53. To draw the monodimetric of a grindstone ; radius, 
 2i inches ; thickness, i inch ; diameter of square hole at the center, i 
 inch. 
 
 No conditions as to position being given, it is evident that the grind- 
 stone may be drawn in any monometric position. What are these 
 positions ? Draw the object in one of the four positions stated above. 
 
 PROBLEM 54. To draw the monodimetric of a waterbucket having 
 an axis of 1 foot, abase diameter of 10 inches, atop diameter of 14 inches, 
 and a thickness in every part of 1 inch. Add two 1-inch hoops and a 
 semicircular wire handle. Finish the drawing neatly. 
 
 33. Monodimetric Projection of Non-Axial Lines. 
 
 The following worked problem is intended to answer the question of 
 how to represent and measure lines that are not parallel to axes of di- 
 mension. 
 
PROJECTION DRAWING. 
 
 33 
 
 PROBLEM 55. (Fig. 45). To find the monodimetric of a horizontal 
 pentagon. 
 
 Draw a diagram of the figure (Fig. 32 or 36, Vol. I) and in it draw 
 the lines 5 6 and 3 4. Draw A B equal to 1 2. Draw 7 D at 45 to A B, 
 and make it equal to 5 6 by ^|33. Find point F in the same names. 
 Make F E and F C equal to the halves of 34. Draw the outline. 
 
 By drawing five equal vertical lines at the angular points of the 
 monodimetric pentagon, and by connecting the upper ends of these verti- 
 cals, the monodimetric of a pentagonal prism is obtained. 
 
 Still another monodimetric projection may be obtained by drawing 
 the lateral edges of the prism downward, making the number of possible 
 cases not four, but six. 
 
 Fio. 45. 
 
 Solve the following problems: 
 
 PROBLEM 56. To find the monodimetric of a vertical heptagonal 
 pyramid. 
 
 PROBLEM 57. To find the monodimetric of a truncated triangular 
 pyramid. Height of frustrum. 3 inches;, side of base, 3 inches; side of 
 section, 2 inches 
 
 It is difficult to find the monodimetric of a horizontal circle. When 
 a circular object is to be drawn, it should be posed so that the circles will 
 be parallel to V. 
 
 34. Additional Problems in Monodimetric Projection. 
 
 Monodimetric projection is much used in drawing details of stone 
 and lumber construction, such as mortise-joints, dovetail joints, miter- 
 joints, stretcher- joints, building hardware, brick and stone arches, etc It 
 is therfore often called shop perspective, false perspective, parallel perspect- 
 ive, or cabinet perspective. In as much as a monodimetric projection shows 
 the front and two sides of the object, it resembles a true perspective pict- 
 ure; but it is not drawn in accordance with the laws of linear perspective, 
 and these terms are entirely improper and confusing The student 
 should not use them. 
 
34 
 
 WALTERS' ELMENTARY GRAPHICS. 
 
 Solve the following problems: 
 
 PROBLEM 58. To draw the monodimetric of a table, foot-stool, or 
 tool bench. 
 
 PROBLEM 59. To draw the monodimetric of any three of the Egyp- 
 tian letters of fl 47, enlarging them to four diameters and giving them a 
 thickness of i inch. 
 
 PROBLEM 60. To draw the monodimetric of a dovetailed box with a 
 half-open sliding lid. Add the scale. 
 
 PROBLEM 61. To draw the monodimetric of a saw-horse. Add the 
 scale. 
 
 These figures are to be drawn on a large scale so that each will fill a 
 page. The rear edges are to be shown in dotted lines. The edges which 
 divide front faces from side faces are to be drawn in bold black lines, 
 and all other visible edges are to be made in light black lines. 
 
 35. General Principles of Isometric Projection. 
 
 The difference of monodimetric and isometric projection has already 
 been stated in paragraph 31. It is this: 
 
 In monodimetric projection the object is posed so that two of its axes 
 of dimension are parallel, and the third axis perpendicular, to V,while in 
 isometric projection the object is represented so that all three axes form 
 equal angles with V. 
 
 An isometric drawing of an object is, therefore, a real projection, 
 while a monodimetric drawing is not. 
 
 Careful study of the following isometric projections of a cube and an 
 oblong frame will make clear these statements. 
 
 FIG. 46 FIG. 47. 
 
 Every edge of these solids is parallel to one of the three axes of di- 
 mension and all form equal angles with V. It follows that all are 
 shortened equally, i. e., according to the same ratio. They bear the 
 same relation to each other in the projection as they do in reality. 
 
 Having fully mastered this principle, the student should now try to 
 understand that, if the projection of an object is enlarged so as to make 
 
PROJECTION DRAWING. 
 
 35 
 
 every axis of dimension, and every line parallel to an axis of dimension, 
 equal to the real axis or line, no scale will be needed at all, i.e., that: 
 
 In an isometric projection every axial line (line parallel to an axis of 
 dimension) is directly measurable. 
 
 This principle, introduced by Prof. Moellinger, a noted Swiss mathe- 
 matician, is the one which has made isometric projection valuable, because 
 in almost every object of industrial art the axial lines abound. 
 
 With regard to position, it may be said, that in an isometric projec- 
 tion 
 
 (1.) The vertical lines remain vertical. 
 
 (2.) The horizontal axial lines form angles of 30, to right or left, 
 with G L. 
 
 (3.) The length and position of non-axial lines may be found by mak- 
 ing them face diagonals or space diagonals in rectangular blocks formed 
 by axial lines. 
 
 PROBLEM 62. To draw the isometric of a square box, open at the 
 top. Dimensions, 4 by 3 by 2 inches; thickness of boards, i inch. Show 
 its construction and the nail heads. 
 
 36. The Circle in Isometric. 
 
 The circle may be drawn in isometric by inclosing it in a square, 
 drawing this in isometric, and inscribing an ellipse. 
 
 FIG. 48. FIG. 49. 
 
 Thus, let A B C D (Fig. 48) be a square inclosing a circle, to be drawn 
 in isometric. The square will appear as a rhombus A B F E, having 
 angles of 60 and 120. Draw the diameters of the square and draw lines 
 representing these in the rhombus. Draw the diagonals in the square 
 and rhombus. Find the points where the ellipse crosses the rhombus 
 diagonals. The illustration shows how these points may be found. Draw 
 the ellipse freehand. 
 
 Fortunately, this special form of the ellipse can be easily imitated 
 with the compasses (oval). An examination of figure 49, representing a 
 
36 
 
 WALTERS' ELEMENTARY GRAPHICS. 
 
 cube with circles inscribed within each visible face, will explain the 
 modus operandi of this convenient approximation. 
 
 Solve the following problems: 
 
 PROBLEM 63. To draw the isometric of a vertical cylinder. Axis, 
 3i inches; diameter, 2| inches. 
 
 PROBLEM 64. To draw the isometric of a horizontal cylinder having 
 the same dimensions. 
 
 37. Isometric Projection of Non= Isometric Lines. 
 
 The following worked problem illustrates how non-isometric lines or 
 figures are drawn in isometric, and how the real length of non-isometric 
 lines may be found by construction. 
 
 PROBLEM 65. (Figs. 50 and 51). To make an isometric drawing*;of 
 an hexagonal prism. 
 
 Draw a diagram of the base of the prism inclosed by a rectangle, 
 A B C D. Draw this rectangle in isometric where it will form a rhom- 
 boid with angles of 60 and 12(P. Make E' 1' equal to A 1, 2' F' equal to 
 2 B, E' 6' equal to A 6, etc. Tiiis will give the isometric of the regular 
 hexagon. Then draw the vertical edges equal to the real height of the 
 prism. Lastly connect the upper ends of the verticals by lines forming 
 the upper isometric hexagon. 
 
 FIG 50. FIG. 51. 
 
 
 In this way all non-isometrical figures are drawn They are first in- 
 scribed within some figure that can be drawn in isometric. 
 
 Solve the following problems: 
 
 PROBLEM 67. To find the isometric of a pentagonal prism. Hori- 
 zontal edge, H inch; vertical edge, 3 inches. 
 
 PROBLEM 68. To draw the isometric of an irregular heptagonal 
 pyramid, from given orthographic projections. 
 
 PROBLEM 69. To draw the isometric of a carpenter's trestle. 
 
 PROBLEM 70. To draw the isometric of a tool bench with braces for 
 stiffening its legs. 
 
PROJECTION DRAWING. 
 
 37 
 
 38. Cone and Sphere in Isometric. 
 
 PROBLEM 71. (Fig. 52). To draw a cone in isometric. Diameter, 
 3| inches; axis, 4 inches 
 
 PROBLEM 72. (Fig. 53) To make an isometric drawing of a hemi- 
 sphere having a radius of 2 inches. 
 
 Find the isometric of the great circle, as in the former problem. 
 Then draw a semicircle over the major axis of the ellipse. Remember 
 that a sphere projects as a circle, no matter how it may be posed 
 FIG. 52. FIG. 53. 
 
 PROBLEM 73 To find the real radius of a given isometric sphere. 
 
 This problem is more difficult lhan might appear at first. Remem- 
 berthat only axial lines ca-i be measured Such a lin" might be drawn 
 through the center, but it would evidently not terminate in the outline 
 (principal meridian) of the sphere Any diameter in the circle repre- 
 senting the sphere would be too long. In isometric all figures are drawn 
 larger than in orthographic in order that their axial lines may appear as 
 they are. How will you go to wqrk to find it ? 
 
 Let this problem be a test for you. 
 
 DIVISION H. 
 
 SHAIDOWS. 
 
 39. The Light Parallel to V. 
 
 The form of the shadow of an object depends on: 
 (1.) The object casting the shadow, i. e., the outline which the ob- 
 ject presents to the light. 
 
38 
 
 WALTERS' EDEMENTARY GRAPHICS. 
 
 (2.) The surface upon which the shadow falls. 
 
 (3.) The source or direction of the light. 
 
 In mechanical drawing the rays of light are supposed to have 
 some fixed direction, and in all examples and problems of this division the 
 rays of light are assumed to be parallel, so that the direction of one ray 
 is the direction of all. 
 
 The simplest conditions prevail when the light is parallel to V and 
 falls upon H at an angle of 45. The horizontal projections of the rays 
 can then be drawn with the T-square and the vertical projections with 
 the right-angled isosceles triangle. All shadows will be on the right side 
 of the object and appear in H, the shadow of a point being where a ray 
 of light through the point would pierce the surface upon which the 
 shadow is cast. 
 
 The solution of problems under these stated conditions is so simple 
 that a study of figures 54 and 55 should be sufficient for the successful 
 working by the student of all such problems. 
 
 FIG. 54 FIG. 55. 
 
 c b 
 
 a 
 
 Figure 54 solves the problem, to find the shadow of a rectangular 
 block; and figure 55, to find the shadow of an irregular pentagonal pyra- 
 mid. 
 
 Solve the following problems for which no figures or solutions are 
 given: 
 
 PROBLEM 74. To find the shadow of an irregular, obliquely trun- 
 cated pentagonal prism. 
 
 PROBLEM 75. To find the shadow of a right cylinder. 
 
 This problem can be solved by finding the shadow of a series of 
 points in the upper circle and by connecting these by a freehand stroke 
 with the shadow of the elments tangent to the light. A better way, 
 however, is to find the shadow of the upper base, remembering that the 
 shadow of any plane form cast upon a parallel plane must be equal to the 
 form, etc. 
 
PROJECTION DRAWING/. 
 
 39 
 
 PROBLEM 76. To find the shadow of a cylinder resting upon H and 
 having its axis perpendicular to V. 
 
 PROBLEM 77. To find the shadow of a truncated oblique cone. 
 
 40. The Light Oblique to H and V, 
 
 The next step is to assume the rays ot light oblique to both planes of 
 projection. The simplest conditions of this kind prevail when the 
 descending rays are assumed so as to give horizontal and vertical projec- 
 tions of 45 to G L. The student will observe that in this case the real 
 angle of inclination and declination is not 45, but that it is less. It is 
 equal to the angle which the space diagonal of a cube form with the base 
 of the cube, i. e., an angle of about 37|V 
 
 The shadow is cast to the right and rear of the object and if this 
 should stand close to V, a part of the shadow will fall upon V. Figure 
 56 represents a triangle in space placed obliquely to V and H. The 
 shadow falls upon both planes of projection and forms a dihedral, which 
 gives a pentagonal development. The direction which the shadow of AB 
 and BC takes in crossing G L is found by obtaining th^ shadow of inter- 
 mediate points. Figure 57 shows how to find the shadow of a square 
 prism placed so that none of its lateral faces is parallel to V, and that 
 the shadow will fall upon H and V. 
 
 FIG. 56. FIG. 57. 
 
 The student who is able to read every line of these two figures 
 should have no great difficulty in solving the following problems. Let 
 him remember the following: 
 
 (1.) The shadow of a vertical line forms an angle of 45 with G L, 
 upon H, and is perpendicular to G L, upon V. 
 
 (2 ) The shadow of a line perpendicular to V forms an angle of 45 
 with G L, upon V, and is perpendicular to G L, upon H. 
 
40 
 
 WALTERS ELEMENTARY GRAPHICS. 
 
 (3.) The shadow of a line parallel to V is equal and parallel to the 
 line, upon V. 
 
 (4.) The shadow of a line parallel to H is equal and parallel to the 
 line, upon H. 
 
 Solve the following problems: 
 
 PROBLEM 78. To find the shadow cast upon II of a cube. 
 
 PROBLEM 79. To find the shadow cast upon H and V of a tall, ir- 
 regular pentagonal prism. 
 
 PROBLEM 80. To find the shadow cast upon H and V of a hexag- 
 onal pyramid standing upon its apex. 
 
 PROBLEM 81. To find the shadow of an irregular pentagonal 
 pyramid having its axis parallel to V and II. 
 
 41. Shadows of Circles and Curved Surfaces. 
 
 In problems 75, 76 and 77 the student has already solved problems per- 
 to taining'the shadows of objects having curved surfaces. In the following 
 problems the direction of the light rays is the same as in paragraph 40. 
 
 FIG. 58. 
 
 PROBLEM 82. To find the shadow upon H of a vertical circle whose 
 plane is parallel to V. 
 
 PROBLEM 83. To find the shadow upon H and V of a horizontal 
 circle. 
 
 PROBLEE 84. To find the shadows in H and V of a circle whose 
 plane is perpendicular to G L. 
 
 PROBLEM 85. To find the shadow in H and V of a vertical cone. 
 
 The method of solution (Fig. 58) is somewhat peculiar. Find the 
 shadow of the apex upon V, then imagine V as transparent and find 
 
PROJECTION DRAWING. 
 
 41 
 
 where the shadow of the apex would fall if such were the case. It would 
 evidently be cast upon the extended or rear part of H, directly behind B, 
 i. e., at C. Draw tangents from C to the base circle, and you will ob- 
 tain the shadow of the cone upon H and its rear extension. Consider 
 now V as opaque and find the shadow upon it. 
 
 PROBLEM 86. To find the shadow of a cone standing upon its apex 
 near V. 
 
 42. The Shadow of the Sphere. 
 
 PROBLEM 87. (Fig. 59.) To find the shadow in H of a sphere. 
 FIG. 59. 
 
 I 
 
 The shadow of a sphere is equal to the shadow of the one of its great 
 circles which is perpendicular to the rays of light. If this great circle is 
 found, the remaining part of the problem can present but few difficulties. 
 The work may be done as follows: 
 
 (1.) Find the shadow of the center of the sphere. 
 
 (2 ) Locate an auxiliary plane perpendicular to H and parallel to the 
 descending rays of light. 
 
 (3.) Upon this plane draw an auxiliary vertical projection of the 
 sphere. 
 
 (4.) Draw QR, the great circle, at right angles to the rays of light. 
 This is done by finding the obstructed ray of light through the center A 
 from its horizontal projection, and by drawing Q R perpendicular to it. 
 
 (5 ) Draw tangent rays at Q R, parallel to a a'. These will give 
 
42 WALTERS' ELEMENTARY GRAPHICS. 
 
 the major diameter of the ellipse which will form the shadow. The 
 minor diameter will be equal to the diameter of the sphere. 
 
 (6.) Draw the ellipse by one of the methods given in paragraph 18, 
 or find additional points by the method shown in figure 59, i. e., by using 
 planes parallel to H, like O P. 
 
 PROBLEM 88. To find the shadow of a sphere, cast upon V. 
 
 In monodimetric and isometric projection shades are often, but 
 shadows rarely, required. The rays of light are usually assumed so that 
 the shadow of every perpendicular line becomes a horizontal line equal 
 in length to the perpendicular, i. e , the rays are assumed to come from 
 the left, the front and above. In this case every obstructed ray can be 
 drawn with the right angled isosceles triangle placed against the T-square. 
 The shadows that are not cast upon parts of this object are assumed to 
 fall upon an imaginary horizontal plane supporting the object. 
 
 All shadows given in this book are assumed to be produced by 
 straight or circular outlines and to be cast upon H or V. Book III, De- 
 scriptive Geometry, continues the subject and shows how to find shadows 
 cast upon oblique and single curved surfaces by objects having more 
 complex outlines. 
 
PART II. 
 Shading and Draughting. 
 
 43. The Object. 
 
 The object of this course in shading and draughting is to teach the 
 student the methods of representing complex objects; to familiarize him 
 with the use of the scale and the common measuring tools, and to de- 
 velop his taste with regard to the disposition of details, inscriptions, etc. 
 It is intended that this work should be carried on simultaneously with 
 the work of Part I, i. e., that every alternate lesson should be given to 
 draughting in the class-room under the supervision of the .teacher. While 
 all work in graphics must be done carefully and neatly, this work, re- 
 quiring larger sheets of paper and involving more complex subjects, will 
 be especially suited for the training of the aesthetic qualities of the stu- 
 dentan educational end never to be neglected, though difficult of 
 attainment. 
 
 44. Tools and Materials. 
 
 In addition to the tools required for the home work and described in 
 Part I, each student must be provided with the following: 
 
 FIG. 60 
 
 (1.) A drawing-board. The board needed in this work should meas- 
 ure 14 to 18 inches by > to 24 inches, and should be made of good quality 
 of well-seasoned white pine lumber, glued together of about four strips. 
 One or both of the short ends should be provided with a crosspiece 
 (header) of soft mapie, cherry or some other small-grained wood. The 
 header must, of course, be straight. Fig. 1 represents such a board with 
 two headers, and the above illustration shows how a larger and 
 better, though more expensive, board may be made. 
 
44 
 
 WALTERS ELEMENTARY GRAPHICS. 
 
 (2.) A T-square , with blade extending across the board. The simple 
 construction shown in Fig. 62 makes this form preferable to any other 
 
 for the beginner. 
 
 FIG. 61. 
 
 (3.) Two triartgles, 60, 30, 90, and 45, 90, respectively. The sides 
 of these should measure not less than 6 inches. Good work may be done 
 with the right-angled isosceles triangle alone. Many practical draughts- 
 men^use the tetrangle shown in Fig. 63, in preference to two triangles. 
 
 FIG. 62. 
 
 (4.) Four white metal thumb tacks to fasten the sheet to the drawing- 
 board. 
 
 FIG. 63. 
 
 
 (5.) Apaper measure, i. e., a strip of tough, smooth paper, li by 16 
 inches, printed with lines showing inches and halves, quarters, eighths 
 and sixteenths of inches. Paper measures are better than any other 
 cheap measuring tools, and cost but few cents per piece. 
 
 45. General Remarks. 
 
 In school where a considerable number of drawing-boards are used, 
 it will be cheaper to have them made to order in some carpenter shop. 
 The other tools, however, should be bought of the instrument dealer. 
 The cheaper grade of T-squares and triangles are usually made of beech 
 or pear wood; the best are the transparent celluloid tools called amber. 
 
PROJECTION DRAWING. 45 
 
 For storing the drawing-boards between lessons, the class-room or 
 an adjoining closet- room ought to be provided with a, drawing-board rack. 
 i. e., a strong square case, open on one side, into which the boards may 
 be slit separately between cleats. The boards should lie horizontally 
 with the drawing sheet on the under side; the upper side will then serve 
 as a shelf for the deposition of the paper measure and the triangle The 
 top of the case should be of strong pine lumber, so that it may be used as 
 a table for paper cutting and the deposition of ink saucers, models, etc. 
 
 Wherever the conditions make it possible, it is well to provide the 
 drawing room with top-light, and to furnish it with drawing tables that 
 can be used either sitting or standing. There are a number of such 
 tables in the market. The best, though not the cheapest, school draw- 
 ing table known to the author of this book is the Eugene Dietzgen 
 patent drawing table I. X. L. 
 
 46. Use of Instruments and Materials. 
 
 The T-square is used to draw parallel lines by sliding its head along 
 the left side of the drawing-board and using the upper edge of the blade 
 as a straight edge. It should never be used on another side as the edges 
 of the board are seldom exactly parallel or at right- angles with each 
 other. Before ruling the line, the blade should be pressed firmly on the 
 paper, as the square is liable to move slight'y. unless the head of the 
 square and the board exactly coincide. The square should be moved 
 along the side of the board by the left hand, leaving the right hand free 
 for the use of the pencil or the pen. 
 
 Triangles are used as straight edges for drawing perpendicular and 
 oblique lines to those already drawn by the T-square. They are used 
 also for drawing parallel lines, by placing the edge of one along the line 
 to which the parallel is to be drawn, and by sliding the other triangle. 
 
 Ink. In practical draughting India ink is used, instead of writing ink. 
 The India-ink is preferable in that it can stand moisture without 
 spreading. It is also blacker and covers the paper better, though it is 
 more expensive, especially when bought j n liquid form. Many designers 
 buy India-ink sticks and prepare eac'i dav as much as they expect to use. 
 Fig. 64 represents a stick of dry India-ink. 
 
 FIG. 64. 
 
 To prepare the India- Ink. Rub a stick of the ink on a saucer or ink- 
 slab containing about a teaspoonful of water until the proper consistency 
 is obtained. To determine the latter, try thejnk from time to time 
 with a drawing-pen on a piece of paper, and if, when di~y, a perfectly black 
 
46 WALTERS' ELMENTARY GRAPHICS. 
 
 (not gray) line is left, the ink is ready for use. When ground it should 
 always be kept covered over, to prevent its drying quickly, and so becom- 
 ing thick. The end of the stick of ink should be wiped after rubbing, as 
 otherwise it is liable to crumble It is also a good plan, when a long 
 stick is used, to wrap tin-foil tightly around it, to prevent its thus 
 breaking or cracking. 
 
 Mounting the Paper Lay the pap^r upon the drawing-board smooth 
 side up. Insert one of the thumb tacks close to the edge in a corner of 
 the sheet. Square the sheet with the board by means of the T-square. 
 Insert the other thumb tacks. 
 
 Ihe Pencil Sketch All drawings should be made in pencil before 
 inking. The penciled lines should be made fine and. light with a hard 
 pencil, so as to be easily erased or inked over. To erase strong pencil 
 marks requires hard rubbing, which destroys the surface of the paper. 
 All penciling should be done carefully to avoid confusion in inking. 
 
 The tint lines and shade lines are usually drawn in ink without any 
 sketching. 
 
 Inking. Fill the drawin-g pen by means of a writing pen and keep 
 the outside of the blades clean Hold it ne irly upright and carry it along 
 the blade of the T-square from left to r ght, with the flat sides parallel to 
 the direction of the line. Use only the upper edge of the T-square. 
 Carefully clean the pen when through using In inking the horizontal 
 lines should be first drawn, commencing at the top, and then the vertical 
 lines commencing at the left. This will reduce the liability to blot the 
 work before it is dry. Never use the blotting paper. 
 
 47. Inscriptions. 
 
 It is important that the student of projection drawing should learn 
 to write and draw neat inscriptions. Well-made inscriptions greatly 
 enhance the beauty of a drawing. They serve as an indication of equal 
 care taken in the execution of the construction. Tor this reason the 
 plate of the so-called Egyptian capitals and numbers, printed in Vol I, is 
 here reprinted, not to be copied, but for reference. There is also added 
 a complete alphabet of capitals, small letters, and numbers of round wrik 
 ing, a style of letters so-called because of its predominant round form. 
 On account of its distinctness, beauty, and ease of construction, no 
 other style can surpass it for inscription purposes. 
 
 The letters and figures of this style should not be drawn, but written. 
 After some practice the writing -may be done with almost any large and 
 elastic pen. It is best, however, to procure a pen made specially for the 
 purpose, such as F. Soennecker's No. 2 or 3. Larger letters are made 
 with a good shading pen, or F. Soennecker's Parcel's pen No. 133. Stu- 
 dents who wish to practice round writing would do well to procure F. 
 Soennecker's Text Book of Round Writing, published by the Keuffel 
 Esser Co., New York, 127 Fulton Street. 
 
H 
 
 m 
 
 rn 
 o 
 
PROJECTION DRAWING. 49 
 
 For many purposes a written plain vertical letter will answer best. 
 48. Blue-printing and Black-printing. 
 
 The invention of the beautiful processes of blue or black-printing and 
 photo zinc etching has, of late, greatly changed the methods of work in 
 practical draughting. Formerly most drawings were finished in colors 
 applied with the brush, but at present nearly all ordinary work is being 
 done in black lines. 
 
 In order to reproduce a drawing by the blue or black-printing pro- 
 cess, it is first made in pencil the same as every other drawing. It is 
 then traced in strong India-ink upon a sheet of transparent paper called 
 tracing paper or patera. 
 
 The next step is to coat the blue-print paper with sensitive material. 
 The paper is simply common drawing or writing paper, usually French 
 satin. The material is a mixture in nearly equal parts of dissolved 
 cit. iron and ammonium and pot. perrocyanide. These chemicals are dis- 
 solved separately in clean water, in a ratio of about two tablespoons of 
 each compound, finely ground, to one cup of water, and allowed to stand 
 in bottles in a dark place for about forty-eight hours. The paper is then 
 tacked to a wall in an entirely dark room, the two solutions are poured 
 into a cup, and the mixture is carefully applied to the paper with a large, 
 smooth paint-brush or a sponge. It dries rapidly and will be ready for 
 use in two to three hours. 
 
 The printing is done as follows: The sheet of sensitized paper is 
 laid, coated side up, upon a drawing-board; the tracing is laid over it; 
 a plate of thick glass is laid over the latter to hold it down at every point, 
 and the whole is exposed to the bright sunlight for about five minutes, 
 when the whole surface of the coated sheet will have turned from a light 
 yellow to a dirty bluish gray. It is then immediately immersed in a bath 
 of clean water. The water will wash off the coating where it was not 
 fixed by the sunlight, i. e , under the black lines of the tracing, and these 
 will appear white, while the rest of the surface will turn a brilliant blue. 
 It is evident that as many copies can be made of a tracing as may be 
 desired. In architects' and engineers' draughting rooms more complex 
 apparatus is used than the one here described, but the process is the 
 same. 
 
 Formerly every artisan or artist prepared his own paper. For the 
 last half-dozen years, however, factories of sensitized paper have been 
 established in all large cities, and carefully prepared " blue paper" costs 
 now but little more than other paper. The difficulty with " ready blue 
 paper " is in that it must be kept in a dark place. An exposure to sun- 
 light of only a few seconds will start the chemical action and render it 
 valueless. All schools of applied mathematics have blue-printing rooms 
 connected with their classrooms in draughting, and finish their work in 
 white upon blue ground. 
 
 The black-printing process does not differ from the blue-printing 
 
50 WALTERS' ELEMENTARY GRAPHICS. 
 
 process in the manipulation of the drawing. The lines appear black 
 upon light gray or mottled ground. The paper is difficult to prepare and 
 is, therefore, bought ready for printing. The materials cost more than 
 those used in blue-printing. As a result the process is used almost only 
 by sculupturers and woodcarvers in whose drawings the blue-printing 
 process would reverse light and shade. 
 
 Photo-zinc etching requires a considerable amount of costly apparatus, 
 and the work of etching, though it is a purely chemical manipulation, 
 requires a good deal of skill. It is, therefore, done only in special art 
 establishments. The product, a relief plate, can be used in every print- 
 ing press. Nearly all line pictures in the books and newspapers of to-day 
 are printed from photo-zinc etchings made from pen drawings, and every 
 well-made pen drawing may thus be photograved. All the illustrations of 
 this book were printed from photo-zinc etching made from pen drawings 
 most of which were prepared by L. P. Brous. student assistant in the 
 Department of Industrial Art at the Kansas State Agricultural College. 
 
 49. Shades. 
 
 PLATE I. This figure represents the vertical and horizontal projec- 
 tions of a group of solids consisting of a square slab, broken off behind, an 
 octagonal prism, a cylinder and an octagonal pyramid. 
 
 The group is placed so that the source of light is to the front, the left 
 and above, and the shadow to the rear and right. Some of the faces of 
 the group, especially of the pyramid, are nearly at right angles to the 
 light rays, and receive evidently more light, proportionately, than the 
 horizontal surfaces, or some of the faces of the prism and parts of the 
 cylinder. Along two of the elements of the cylinder the light rays are 
 tangent only, i. e , they merely graze the object The parts of the solids 
 on the opposite or rear side of the group receive only reflected, but no 
 direct, light. It is these different light effects, which may be watched 
 on every object, that the shaded drawing represents. 
 
 In order to shade an object properly, the student must be able to 
 estimate approximately the quantity of light received by every part of 
 the surface, or, which is the same, the angle which every part of the sur- 
 face forms with the light rays. 
 
 There are, however, a few exceptions or modifications to this rule. 
 In the first place, there is the influence of the distance of the object from 
 the eye. Things near the eye display more contrasts of light and shade 
 than things farther away, while at any great distance all shades and 
 lights, and even colors, seem to blend into one general tone. This is due 
 to the fact that object^ appear smaller at a distance, and that the atmos- 
 phere is slightly opaque. It may be said then, that dark surfaces grow 
 lighter and light surfaces darker as they recede. In projection drawing it is 
 necessary to greatly exaggerate the difference of shade between the nearer 
 and farther parts of a surface in order to show the difference of distance 
 from the eye. 
 
PROJECTION DRAWING. 
 
 51 
 
52 WALTERS' ELEMENTARY GRAPHICS. 
 
 In the second place, there is the influence of contrast. The light parts 
 appear lighter near dark parts, and the dark parts seem darker close to 
 light parts. This fact is well-known by every one and is taken advantage 
 of in decorating and drawing. As a consequence the shading of the 
 group of solids is strongest where the shaded surfaces are adjacent to 
 light surfaces, and vice versa. 
 
 Upon objects having curved surfaces like the cylinder, the depth of 
 shade must change at every element. The brightest part is a little 
 forward of the point in H where the horizontal projection of a ray of 
 light through the axis of the cylinder would pierce the circle which forms 
 the horizontal projection of the cylinder. This part of the curved sur- 
 face is called the brilliant line. It is placed forward of the element which 
 reclines the most to the light, because the light of this element, ac- 
 cording to a well-known law in physics, would be reflected back from 
 whence it came, while the light from elements a little more to the right 
 would be reflected forward in the direction of the observer. The latter 
 elements, though receiving less light, would really appear lighter. 
 
 The darkest parts of the surface are the elements nearest the point 
 of tangency. The surface from to the outline at the right should appar- 
 ently be of the same shade, being all in the shadow. But here another 
 principle has to be observed. There is always a small amount of light 
 coming in exactly the opposite direction from the orignal one. This is 
 called reflected light, and, owing to this, the surface to the right is made 
 a little lighter as it turns back. See figure 68. 
 
 This principle influences also the shading of the pyramid. 
 
 A careful study of the given group of solids will reveal a number of 
 applications of the exceptions mentioned. The student should not draw 
 a single line, the meaning of which he does not understand. In case he 
 cannot find reasons for drawing certain lines, he should ask the eacher 
 for an explanation. 
 
 The outlines of the shadows are found by drawing at 45 to G L, 
 tangent to the outlines in H. The shadow of the pyramid is found by 
 establishing the shadow of the apex uuon the level of the base of the 
 pyramid, and drawing from there, in H, tangent to the base of this solid. 
 
 The figure should be copied twice its linear dimensions, i. e., four 
 times as large as the given original. The outline should be carefully laid 
 out in pencil and inked. The shade lines should be drawn without any 
 sketching and should be made as near as possible like the original in 
 thickness not stronger. The figure being larger, there should be an in- 
 creased number of shade lines. 
 
 All outlines separating light from dark surfaces should be drawn 
 considerably stronger than those separating faces of equal or nearly equal 
 light effects. This rule should be followed in all mechanical drawing, 
 and must not be neglected in drawing the problems of Part I, though 
 nothing is said there about shading. 
 
 Provide the plate with a uniform border line. 
 
PROJECTION DRAWING. 
 
 53 
 
 In the upper left corner of the plate write a neat inscription in round 
 writing, as described in paragraph 47. 
 
 In the lower right corner add your signature. 
 
 FIG. 68. 
 50. Shades upon Complex Surfaces. 
 
 PLATE II. This plate represents a group of solids consisting of tfcN^& 
 well-known geometric forms and a part of the shaft of a fluted column. 
 
 The direction of the light is the same as in the last plate, and there 
 are no new principles involved. The solids, however, are more complex 
 and require a more subtle grading of shade lines. The strongest shade 
 lines of the fluted shaft and the cone are made by drawing two light 
 
54 
 
 WALTERS ELEMENTARY GRAPHICS. 
 
PROJECTION DRAWING. 55 
 
 outlines for each and filling the space between the two by drawing an 
 other line or two with the drawing pen. To shade such a plate well 
 requires careful study of the principles involved and a close application 
 in the work. 
 
 Draw the figure not less than two diameters of the original. 
 
 Add a neat border line. 
 
 ILI the upper left corner inside the border line write a neat inscription 
 in round writing, as described in paragraph 47. 
 
 In the lower right corner sign your name. 
 
 51. Shades upon Double Curved Surfaces. 
 
 PLATE III shows a cavetto, a sphere and a cone, shaded by means 
 of irregularly placed points or dots. Such shading or tinting is called 
 stippling. It is done with the common writing pen, and is not difficult to 
 learn. The principles of shading are, of course, the same, no matter 
 what graphic method may be followed in producing the desired effect. 
 
 The student should take care not to place the dots in rows, but to 
 distribute them evenly over the surface, much as the farmer sows the 
 seed upon a field, i. e , irregular in position, but regular in distribution. 
 
 Note that the sphere has its brilliant point not where the light is 
 perpendicular to the surface, but somewhat nearer the center of the pro- 
 jection. The reasons for this are those given with regard to the shading 
 of the cylinder in plate 1. The lisrht rays are tangent to the sphere along 
 a great circle which projects in H and V as an ellipse. The rear surface 
 of the sphere is somewhat lighter than the region where the light is 
 tangent, for reasons already stated. 
 
 The shadows of the cone upon the sphere aiid the shadow of the 
 sphere upon the scotia is not expected to be found by the pupil, but 
 should be copied from the original plate. The proper solution of these 
 two problems must be left to higher descriptive geometry. Remember that 
 all shadows appear strongest along the outline and lighter toward the 
 middle. 
 
 Fig. 68, which represents a carpenter's glue-pot with glue brushes in 
 it, taken from Walter's Object Drawing, shows how shading of double 
 curved surfaces is being done in freehand pen work. This method o^ 
 shading is sometimes used in draughting to represent objects having 
 irregular or broken surfaces, such as rough cut stone or carved woodi 
 while stippling is used for smooth surfaces like paper, leather, 
 unpolished metal, etc. Fig. 68, belonging to the course of advanced 
 freehand drawing, is reprinted here for comparison only, and is not to be 
 copied. 
 
 Draw the plate twice as high and wide, i. e., four times as large as 
 the given original. 
 
 Draw a neat border line. 
 
 Write an inscription in the upper left corner, using the round writing 
 
 letter. 
 
 Add your signature in the lower left corner. 
 
56 
 
 WALTERS' ELEMENTARY GRAPHICS. 
 
PROJECTION DRAWING. 
 
 57 
 
 52. Shaded Isometric Projections. 
 
 PLATE IV. This plate is to be given to shading of isometric projec- 
 tions. There will be two figures, to be drawn as follows: 
 
 Figure A is to be an enlarged copy of the given cubical frame ; side, 
 2* inches. 
 
 Figure B is to represent an isometric cube having a side of H inches. 
 Upon each face draw a square pyramid having an axis of H inches and a 
 base of 1 inch. 
 
 Ink the visible outline of both figures and erase all non-visible parts. 
 
 FIG. 71. 
 
 Shade the edges dividing light from dark faces, assuming the light to 
 come from the left and above. 
 
 Shade all faces to right by strong tintlines placed close. All shaded 
 faces but two of the second figure, will be vertical and must be shaded 
 by vertical tint lines. The two faces in the second figure, which are not 
 vertical, are the right faces of the top and bottom pyramids. These are 
 to be shaded in the direction of the altitude of the face triangles. 
 
 Ink the border line. 
 
 Add an inscription. 
 
 Sign your name. 
 
 53. Plan of a Home lot. 
 
 PLATE V. This plate is to consist of an enlarged copy of figure 72 
 (adopted, with some alterations, from Prof. W. B. Ware's Examples of 
 Building Construction). 
 
 The given plan is drawn to a scale of 1 inch to 32 feet, but the copy 
 is to be drawn four times as large, i.e., to a scale of 1 : 192. Remember 
 that plane figures are to each other as the squares of homologous lines. 
 
 Turn the plan around so that north will be above and south below, 
 which is the usual position of a landscape plan. Sketch the roads and 
 the outlines of the tree-clumps in pencil, measuring all positions by tri- 
 angulation from the base line. Finish the former with hard-rubber 
 
58 
 
 WALTERS ELEMENTARY GRAPHICS. 
 
 curve and drawing pen, and draw the latter freehand, using the 
 writing pen. Notice how the shading of the clumps and ground on the 
 east gives relief and naturalness to the lawn. 
 Draw a border line. 
 
 FIG. 72 
 
PROJECTION DRAWING. 
 
 59 
 
 Add a neat inscription, "Homelot" over the plan. This may be 
 drawn in the simple and easily sketched Egyptian letter, but it should 
 not be enlarged, and the letters should be placed close together. 
 
 In the left corner, below, write in small, but neat letters, " Scale 1 
 Inch to 16 Feet." 
 
 In the right corner sign your name. 
 
 54. A Floor Plan. 
 
 PLATE VI. This plate is to consist of an enlarged copy of the given 
 floor plan (taken, with some alterations, from Soennecker's Textbook on 
 Round Writing.) 
 
 The given plan is drawn to the scale of 1 inch to 16 feet. Make the 
 copy four times as large, i. e , to the scale of 1 inch to 8 feet. 
 
 FIG. 73. 
 
60 WALTERS' ELEMENTARY GRAPHICS. 
 
 Plastered partitions on studs are assumed to be 6 inches in thick, 
 ness, half partitions on each side of a sliding door, 3 inches; and the 
 space for the sliding door, 3 inches. Brick walls are usually made 12 
 inches in thickness. The line outside the wall represents the watertable, 
 a stone band even with the floor, which projects about 2 inches. Door 
 openings are left blank, and windows are shown by 3 lines connecting 
 two squares that represent boxjambs and are partly inserted in the brick 
 walls. 
 
 After copying this floor plan it would be good practice for the 
 student to draw a set of floor plans of his home, from measurements 
 taken by himself. This would give him, in addition, an insight into the 
 work of architectural construction, which mere copying of plans and eleva- 
 tions cannot give. 
 
 All inscriptions should be horizontal, straight, plain and small. 
 There should be no attempt at fancy work of any kind. 
 
 Always add scale and signature to mechanical drawings of this kind. 
 
FOURTEEN DAY USE 
 
 RETURN TO DESK FROM WHICH BORROWED 
 
 This book is due on the last date stamped below, or 
 
 on the date to which renewed. 
 Renewed books are subject to immediate recall. 
 
 28Nov'55PTZ 
 
 
 NOV291955LU 
 
 
 
 
 -*ji$* 
 
 
 \v" 
 
 REC'D LD 
 
 
 IIA3 1 1962 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 LD 21-100m-2 '55 T .General Library / 
 (B139s22)476' University of California / 
 
YC 66,27 
 
 M319099