%4 A^' . ,x^^ O t; \0°x /> ..^^ V ^ ;-6- ^-^ "^.^ v-^' / '^^. vSn""' ..v^^ <.s '/^ .0 ^c^ ■>' .0 c ^*' v-^^''<.\i :^'% I =^ -\' v^" . >- \v.'- \0^^. '^, C- '%- * s" --.. v^^ * r \V .S^ ,\.'^ .'V' V -i- a'' .VI., %■'•>• •- 5: ,0o. :/\ s. ov"^ -/' ' .0 -':% " . K * ,Ci -^^^^ '^^ .^Vv^' .#^ .^^'^ ^ ^^^gi"^-^ "OO^ ^ '7\ ^ ^ ^ * « / -> ■\' # .$^^^. ^ \ I ■ysy-^- s % \0°^. •^ 0> '^A .#!!:%». '.,^^c^ ^..^ .# '^- ^c. ^^^; ^ .0- .-^^ .^^' A COURSE IN MECHANICAL DRAWING. BY / JOHN S. REID, Instructor in Mechanical Drawing and Designing, Sibley College, Cornell University, Ithaca, N. V. FIRST EDITION, FIRST THOUSAND. NEW YORK. JOHN WILEY & SONS. London : CHAPMAN & HALL, Limited. 2?^aPco?Y, 1898. TWO COPIES RECEIVED. 'b >608 Copyright, 1898, BY JOHN S. REID. ROBERT DRUMMOND, ELECTROTYPER AND PRINTER. NEW YORK. PREFACE In the course of a large experience as an instructor in drawing and designing, the author of this work has often been called upon to teach the elements of mechanical drawing to students in marine, electrical, railway, and mechanical engi- neering. Having tried and failed to find a book on the sub- ject that was entirely suitable for his use as a text-book, he has found it necessary to prepare the present work. This course contains, in the author's judgment, a com- plete and concise statement, accompanied by examples, of the essential principles of mechanical drawing — all that any young man of ordinary intelligence needs to master, by care- ful study, the more advanced problems met with in machine construction and design. Such works as the author has tried, although most excellent from certain standpoints, were either incomplete in some of the divisions of the subject or too volu- minous and elementary in the treatment of details. The author does not imagine this work is perfect, but he believes that it comes nearer what is needed in teaching the elements of mechanical drawing in technical schools, high schools, evening drawing schools, and colleges than any work he has examined. The chapter on Con\-entions will be appreciated by students PREFA CE. when called upon to execute working drawings in practical work. The methods described are considered by the author to be those which have met with general approval by the experienced American draftsmen of the present time. My acknowledgments are due to E. C. Cleaves, professor of drawing, Sibley College, Cornell University, for reading the manuscript and making some valuable suggestions. The Author. April I, 1S9S. CONTENTS INTRODUCTION. PAGE The Complete Outfit, Illustrated i CHAPTER 1. Instruments 7 Use of Instruments 7 Pencil 7 Drawing Pen 9 Triangles 11 T Square 11 Drawing Board .* 11 Sibley College Scale 12 Scale Guard 12 Compasses 13 Dividers or Spacers 13 Spring Bows 14 Irregular Curves 14 Protractor 14 CHAPTER 11. Geometrical Drawing 16 CHAPTER III. Conventions 56 CHAPTER IV. Lettering and Figuring 64 CHAPTER V. Orthographic Projection 74 Shade Lines and Shading 103 Conventions 104 Shading 106 Isometrical Drawings 112 Working Drawings u8 iii MECHANICAL DRAWING. INTRODUCTION. A NEED has been felt by instructors and students, especially in technical courses, for a text-book that would illustrate the fundamental principles of mechanical drawing in such a prac- tical, lucid, direct and progressive way as to enable the instructor to teach, and the student to acquire, the greatest number of the essential principles involved, and the ability to apply them, in a draftsman-like manner, in the shortest space of time. With this in mind, the present work has been prepared from the experience of the writer, a practical draftsman and teacher for over fifteen years. THE COMPLETE OUTFIT. The complete outfit for students in mechanical drawing in Sibley College is as follows : (i) The Drawing-board for freshman work is if y^22" X f, the same as that used for free-hand drawing. The board for sophomore and junior drawing is 20" X 26" X not more than \" in thickness. The material should be soft pine and constructed as shown by Fig. i. MECHANICAL BRA WING. (2) Paper, quality and size to suit. (3) Pencils, one 6H and one 4H Koh-i-noor or Faber, also one Eagle Pilot No. 2 with rubber tip. (4) The T-Square for freshman work is furnished by the Fig. t. department ; a plain pearwood T-square with a fixed head is all that is necessary for sophomore or junior work. Length to suit drawing-board. (5) Instruments. ''The Sibley College Set," shown by Fig. 2, was compiled by the writer, and is recommended as a first-class medium-priced set of instruments. It contains* Fig. 2. A Compass, 5^'^ long, with fixed needle-point, pencil, pen and lengthening bar. A Spring Bow Pencil, 3'' long. A '' " Pen, 3'' long. A *' " Spacer, 3'' long. A Drawing-pen, medium length. A Hair-spring Divider, 5" long. A nickel-plated box with leads. IN TROD UCTION, 3 (6) A Triangular Boxwood Scale graduated as \ , =^ & -^ * Fig. 3. follows: V' and 2'^ 3'^ and if', i'' and J'', f' and f ', -^V'' and 5 • Fig. 4, Fig. 5 ME CHA NIC A L D KA WING. (7) I Triangle 30° x 60°, celluloid, \q" long. Fig. 4. I - 45°, '' 7" '' (8) "Sibley College Set" of Irregular Curves. (9) Glass-paper Pencil Sharpener. Fig. 6. (10) Ink, black waterproof, S.&H. Fig. 7. (11) " red '' Higgins. Fig. (12) '' blue Fig. 7. Fig. 8. (13) Ink Eraser, Faber's Typewriter. (14) Pencil Eraser, Tower's Multiplex Rubber. Fig. IN TRODIX: TION. 5 (15) Sponge Rubber or Faber's Kneaded Rubber. Fie. 10. '^'a.->«^sc ^ n»»n:^nw\.xm^ -e>-^^ts» Fig. 9. (16) Tacks, a small box of I oz. tacks. (17) Water-colors, \ pan each of Payne's Gray, Crim- son Lake, Prussian Blue, Burnt Sienna, and Gamboge. Wind- sor & Newton. Fig;. 11. Fig. 10. Fig. ii. (18) Tinting Brush, Camel's Hair No. lo. Fig. 12. Fig. 12. (19) Tinting Saucer. Fig. 13. (20) Water Glass. Fig. 14. (21) Arkansas Oil-stone. 2' (22) Piece of Sheet Celluloid No. 3000, dull on both sides. X \" XtV". 6 MECHANICAL DRA WING. (23) Protractor, German silver, about 5" diam. Fig. 15, (24) Scale Guard, '' Fig. 13. Fig. 14. (25) Sheet of Tracing-cloth, 18'' x 24''. (26) Writing-pen, point, ''Gillott" No. 303, Fig. 15. ,1, 1, 1, hi, I, Ih, 1, 1, 1. 1, 1, 1,1 Fig. 16. (27) Piece of SHEET BRASS, 4''X4". (28) Needles, two with handles. The following numbers of '* The Complete Outfit " are all that the student will be required to purchase for freshman mechanical drawing (No. 2 Register, '97-'98) : 2, 3, 5, 6, 7, 8, 9, 10, 13, 14, 16, 26. The remainder of the outfit may be purchased during the sophomore and junior years. CHAPTER I. INSTRUMENTS. It is a common belief among students that any kind of cheap instrument will do with which to learn mechanical drawing, and not until they have acquired the proper use of the instruments should they spend money in buying a first- class set. This is one of the greatest mistakes that can be made. Many a student has been discouraged and disgusted because, try as he would, he could not make a good drawing, using a set of instruments with which it would be difficult for even an experienced draftsman to make a creditable showing. If it is necessary to economize in this direction it is better and easier to get along with a fewer number, and have them of the best, than it is to have an elaborate outfit of question- able quality. The instruments composing the ''Sibley College Set" are made by T. Alteneder & Sons, and are certainly as good as the best. See Fig. 17. USE OF INSTRUMENTS. TJic Pencil. — Designs of all kinds are usually worked out in pencil first, and if to be finished and kept they are inked in and sometimes colored and shaded ; but if the drawing is only to be finished in pencil, then all the lines except construction, center, and dimension lines should be made broad and dark, 7 8 MECHANICAL DRAWING. SO that the drawing will stand out clear and distinct. It will be noticed that this calls for two kinds of pencil-lines, the first a thin, even line made with a hard, fine-grained lead- pencil, not less than 6H (either Koh-i-noor or Faber's), and sharpened to a knife-edge in the following manner: The lead should be carefully bared of the wood with a knife for about y , and the wood neatly tapered back from that point ; then lay the lead upon the glass-paper sharpener illustrated in the outfit, and carefully rub to and fro until the pencil assumes a long taper from the wood to the point; now turn it over and do the same with the other side, using toward the last a slightly oscillating motion on both sides until the point has assumed a sharp, thin, knife-edge endwise and an elliptical contour the other way. This point should then be polished on a piece of scrap drawing-paper until the rough burr left by the glass-paper is removed, leaving a smooth, keen, ideal pencil-point for draw- ing straight lines. With such a point but little pressure is required in the hands of the draftsman to draw the most desirable line, one that can be easily erased when necessary and inked in to much better advantage than if the line had been made with a blunt point, because, when the pencil-point is blunt the incli- nation is to press hard upon it when drawing a line. This forms a groove in the paper which makes it very difficult to draw an even inked line. The second kind of a pencil-line is the broad line, as explained above ; it should be drawn with a somewhat softer pencil, say 4H, and a thicker point. All lines not necessary to explain the drawing should be INSTRUMENTS. erased before inking or broadening the pencil-lines, so as to make a minimum of erasing and cleaning after the drawing is finished. When drawing pencil-lines, the pencil should be held in a plane passing through the edge of the T-square perpen- dicular to the plane of the paper and making an angle with the plane of the paper equal to about 60°. Lines should always be drawn from left to right. A soft conical-pointed pencil should be used for lettering, figuring, and all free-hand work. TJic Drawiiig-pcn. — The best form, in the writer's opinion, is that shown in Fig. 17. The spring on the upper blade Fig. 17. spreads the blades sufficiently apart to allow for thorough cleaning and sharpening. The hinged blade is therefore unnecessary. The pen should be held in a plane passing through the edge of the T-square at right angles to the plane of the paper, and making an angle with the plane of the paper ranging from 60° to 90°. The best of drawing-pens will in time wear dull on the point, and until the student has learned from a competent lO MECHANICAL DRA WING. teacher how to sharpen his pens it would be better to have them sharpened by the manufacturer. It is difficult to explain the method of sharpening a draw- ing-pen. If one blade has worn shorter than the other, the blades should be brought together by means of the thumb-screw, and placing the pen in an upright position draw the point to and fro on the oil-stone in a plane perpendicular to it, raising and lowering the handle of the pen at the same time, to give the proper curve to the point. The Arkansas oil-stones (No. 2 1 of '^ The Complete Outfit ") are best for this purpose. The blades should next be opened slightly, and holding the pen in the right hand in a nearly horizontal position, place the lower blade on the stone and move it quickly to and fro, slightly turning the pen with the fingers and elevating the handle a little at the end of each stroke. Having ground the lower blade a little, turn the pen completely over and grind the upper blade in a similar manner for about the same length of time ; then clean the blades and examine the extreme points, and if there are still bright spots to be seen continue the grinding until they entirely disappear, and finish the sharpening by polishing on a piece of smooth leather. The blades should not be too sharp, or they will cut the paper. The grinding should be continued only as long as the bright spots show on the points of the blades. When inking, the pen should be held in about the same position as described for holding the pencil. Many drafts- men hold the pen vertically. The position may be varied with good results as the pen wears. Lines made with the pen should only be drawn from left to right. INS TR UMEN TS. 1 1 THE TRIANGLES. The triangles shown at Fig. 4 (in '' The Complete Outfit ") are lo'^ and j" long respectively, and are made of transparent celluloid. The black rubber triangles sometimes used are but very little cheaper (about 10 cents) and soon become dirty when in use ; the rubber is brittle and more easily broken than the celluloid. Angles of 15°, 75°, 30°, 45°, 60°, and 90° can readily be drawn with the triangles and T-square. Lines parallel to oblique lines on the drawing can be drawn with the triangles by placing the edge representing the height of one of them so as to coincide with the given line, then place the edge rep- resenting the hypotenuse of the other against the corre- sponding edge of the first, and by sliding the upper on the lower when holding the lower firmly with the left hand any number of lines may be drawn parallel to the given line. The methods of drawing perpendicular lines and making angles with other lines within the scope of the triangles and T- square are so evident that further explanation is unnecessary^ THE T-SQUARE. The use of the T-square is very simple, and is accom- plished by holding the head firmly with the left hand against the left-hand end of the drawing-board, leaving the right hand free to use the pen or pencil in drawing the required lines. THE DRAWING-BOARD. If the left-hand edge of the drawing-board is straight and even and the paper is tacked down square with that edge and 12 MECHAXICAL DRAWING. the T-square, then horizontal lines parallel to the upper edge of the paper and perpendicular to the left-hand edge may be drawn with the T-square, and lines perpendicular to these can be made by means of the triangles, or set squares, as they are sometimes called. THE SIBLEY COLLEGE SCALE. This scale, illustrated in Fig. 3 (in *' The Complete Out- fit "), was arranged to suit the needs of the students in Sibley College. It is triangular and made of boxwood. The six edges are graduated as follows; yV" or full size, ■^-^'' , i" and I" = I ft., \" and \" = i ft., 3'' and ij" = i ft., and 4" and 2" = I ft. Drawings of very small objects are generally shown en- larged — e.g., if it is determined to make a drawing twice the full size of an object, then where the object measures one inch the drawing would be made 2", etc. Larger objects or small machine parts are often drawn full size — i.e., the same size as the object really is — and the draw- ing is said to be made to the scale of full size. Large machines and large details are usually made to a reduced scale — e.g., if a drawing is to be made to the scale of 2" = I ft., then 2" measured by the standard rule would be divided into 12 equal parts and each part would represent i'\ See Fig. 8i(^. THE SCALE GUARD. This instrument is shown in Fig. 16 (in "The Complete Outfit"). It is employed to prevent the scale from turning, so that the draftsman can use it without havins^ to look for INSTRUMEiVTS. 13 the particular edge he needs every time he wants to lay off a measurement. THE COMPASSES. When about to draw a circle or an arc of a circle, take hold of the compass at the joint with the thumb and two first fingers, guide the needle-point into the center" and set the pencil or pen leg to the required radius, then move the thumb and forefinger up to the small handle provided at the top of the instrument, and beginning at the lowest point draw the line clockwise. The weight of the compass will be the only down pressure required. The sharpening of the lead for the compasses is a very im- portant matter, and cannot be emphasized too much. Before commencing a drawing it pays well to take time to properly sharpen the pencil and the lead for compasses and to keep them always in good condition. The directions for sharpening the compass leads are the same as has already been given for the sharpening of the straight-line pencil. THE DIVIDERS OR SPACERS. This instrument should be held in the same manner as de- scribed for the compass. It is very useful in laying off equal distances on straight lines or circles. To divide a given line into any number of equal parts with the dividers, say 12, it is best to divide the line into three or four parts first, say 4, and then when one of these parts has been subdivided accu- rately into three equal parts, it will be a simple matter to steo off these latter divisions on the remaining three-fourths 14 MECHANICAL DRA WING. cf the given line. Care should be taken not to make holes in the paper with the spacers, as it is difficult to ink over them without blotting. THE SPRING BOWS. These instruments are valuable for drawing the small cir- cles and arcs of circles. It is very important that all the small arcs, such as fillets, round corners, etc., should be care- fully pencilled in before beginning to ink a drawing. ]\Iany good drawings are spoiled because of the bad joints between small arcs and straight lines. When commencing to ink a drawing, all small arcs and small circles should be inked first, then the larger arcs and circles, and the straight lines last. This is best, because it is much easier to know where to stop the arc line, and to draw the straight line tangent to it, than vice versa. IRREGULAR CURVES. The Sibley College Set of Irregular Curves shown in Fig. 5 are useful for drawing irregular curves through points that have already been found by construction, such as ellipses, cycloids, epicyloids, etc., as in the cases of gear-teeth, cam outlines, rotary pump wheels, etc. When using these curves, that curve should be selected that will coincide with the greatest number of points on the line required. THE PROTRACTOR. This instrument is for measuring and constructing angles. It is shown in Fig. 15. It is used as follows when measuring IXSTRLWEXTS. 1 5 an angle : Place the lower straight edge on the straight line Avhich forms one of the sides of the angle, with the nick •exactly on the point of the angle to be measured. Then the number of degrees contained in the angle may be read from the left, clockwise. In constructing an angle, place the nick at the point from which it is desired to draw the angle, and on the outer circum- ference of the protractor, find the figure corresponding to the number of degrees in the required angle, and mark a point on the paper as close as possible to the figure on the protractor; after removing the protractor, draw a line through this point to the nick, which will give the required angle. CHAPTER II. GEOMETRICAL DRAWING. The following problems are given to serve a double pur- pose : to teach the use of drawing instruments, and to point out those problems in practical geometry that are most useful in mechanical drawing, and to impress them upon the mind of the student so that he may readily apply them in practice. The drawing-paper for this work should be divided tem- porarily, with light pencil-lines, into as many squares and rec- tangles as may be directed by the instructor, and the drawings made as large as the size of the squares will permit. The average size of the squares should be not less than 4". When a sheet of drawings is finished these boundary lines may be erased. It will be noticed in the illustrations of this chapter that all construction lines are made very narrow, and given and required lines quite broad. This is sufficient to distinguish them, and employs less time than would be necessary if the construction lines were made broken, as is often the case. If time will permit, it is advisable to ink in some of these drawings toward the last. In that event, the given lines may be red, the construction lines blue, and the required lines black. But even when inked in in black, the broad and narrow 16 GEOMETRICAL DRAWING. I 7 lines would serve the purpose very well without the use of col- ored inks. The principal thing to be aimed at in making these draw- ings is accuracy of construction. All dimensions should be laid off carefully, correctly, and quickly. Straight lines join- ing arcs should be exactly tangent, so that the joints cannot be noticed. It is the little things like these that make or mar a drawing, and if attended to or neglected they will make or mar the draftsman. The constant endeavor of the student should be to make every drawing he begins more accurate, quicker and better in every way than the preceding one. A drawing should never be handed in as finished until the student is perfectly sure that he cannot improve it in any way whatever, for the act of handing in a drawing is the same, or should be the same, as saying This is the best that I can do ; I cannot improve it ; it is a true measure of my ability to make this drawing. If these suggestions are faithfully followed throughout this course, success awaits any one who earnestly desires it. Fig. 1 8. To Bisect a Finite Straight Line. — With A and^ in turn as centers, and a radius greater than the half of AB, draw arcs intersecting at E and i^. Join j5"i^ bisect- ing AB at C. An arc of a circle may be bisected in the same way. Fig. 19. To Erect a Perpendicular at the End of THE Line. — Assume the point E above the line as center and radius EB describe an arc CBD cutting the line AB in the pomt C. From C draw a line through E cutting the arc in D. Draw DB the perpendicular. Fig. 20. The Same Problem: a Second Method. — i8 ME CHA NIC A L DRA WING. With center B and any radius as BC describe an arc CDE with the same radius; measure off the arcs CD 2.Vi<\DE. With C and D as centers and any convenient radius describe arcs in- tersecting at F. FB is the required perpendicular. Fig. 21. Fig. 2 1. To Draw a Perpendicular to a Line FROM A Point above or below It. — Assume the point C above the hne. With center C and any suitable radius cut the line AB in E and F. From E and F describe arcs cutting in D. Draw CD the perpendicular required. GEOMETRICAL DRAWING, 19 * Fig. 22. To Bisect a Given Angle. — With A as center and any convenient radius describe the arc BC. With B and C as centers and any convenient radius draw arcs intersecting at D. Join AD, then angle BAD = an^rle DAC. Fig. 22. Fig. 23. To Draw a Line Parallel to a Given Line AB Through a Given Point C. — From any point on AB dis B with radius BC describe an arc cutting AB in A, From A with the same radius describe arc BD. From B with AC a.s radius cut arc BD in D. Draw CD. Line CD is paral- lel to AB. Fig. 23. Fig. 24. From a Point D on the Line DE to set OFF AN Angle equal to the given Angle BAC. — From 20 MECHAXICAL DRA WING. A v\-ith any convenient radius describe arc BC. From D with the same radius describe arc EF. With center E and radius BC cut arc EF in F. Join DF. Angle EDF is = angle BAC. Fig. 24. Fig. 25. To Divide ax Angle into two equal Parts, when the Lines do not Extend to a Meeting Point. — Draw the line CD and CE parallel and at equal dis- Fig. 25. tances from the lines AB and EG. With C as center and any radius draw arcs 1,2. With i and 2 as centers and any con- GEOMETRICAL DRAWIXG. 21 venient radius describe arcs intersecting 2XH. A line through C and H divides the angle into two equal parts. Fig. 26. To Construct a Rhomboid having Adja- cent Sides equal to two Given Lines AB and AC, and AN Angle equal to a Given Angle A. — Draw line DE equal to AB. Make D — angle A. Make DF = AC. From /^ with line AB as radius and from £ with line AC 3.s radius describe arcs cutting in G. Join FG and EG. Fig. 27. To Divide the Line AB into anv Number OF EQUAL Parts, sav 15. — Draw a line CD parallel to AB, of any convenient length. From C set off along this line the number of equal parts into which the line ^i)' is to be divided. Draw CA and FB and produce them until they intersect at E. Through each one of the points i, 2, 3, 4, etc., draw lines to the point E, dividing the line AB into the required number of equal parts. This problem is useful in dividing a line when the point required is difHcult to find accurately — e.g., in Fig. 28 AB is the /)itc/i of the spur gear, partly shown, which includes a 22 MECHANICAL DRA WING. space and a tooth and is measured on the pitch circle. In cast gears the space is made larger than the thickness of the tooth, the proportion being about 6 to 5 — i.e., if we divide the pitch into eleven equal parts the space will measure -f-^ 3^CP^ 1 -2, 3 4 5 6 7 8. 9 101112 13 lA IT Fig. 27. Fig 2S. and the tooth -^j. The yV which the space is larger than the tooth is called the backlash. Let A' B' be the pitch chord of the arc AB. Draw CD parallel to A'B' at any convenient distance and set off on it 1 1 equal spaces of any convenient length. Draw CA' and DB' intersecting at E. From point 5 draw a line to ii which w^ill divide A'B' as required; the one part -fj and the other y\-. Fig. 29. To Divide a Given Line into any Number OF Equal Parts: Another Method. — Let AB be the given line. From A draw AC d.t any angle, and lay off on it the required number of equal spaces of any convenient length. Join CB and through the divisions on AC drsiw lines parallel to CB, dividing^i5 as required in the points 1', 2', 3^ 4', etc. Fig. 30. To Divide a Line AB Proportionally to the Divided Line CD. — Draw AB parallel to CD at any GEOMETRICAL DRA WING. 23 distance from it. Draw lines through CA and /^^ and produce them till they meet at E. Draw lines from E through the divisions i, 2, 3, 4, etc., of line CD, cutting line AB in the A 1 2 3 4 5 fj 7 8 9 10 111213 U B Fig. 2q. points 5, 6, 7, 8, etc. The divisions on AB will have the same proportion to the divisions on CD that the whole line AB has to the whole line CD — i.e., the lines will be propor- tionally divided. E Fig. 31. The Same: Another Method. — Let BC, the divided line, make any angle with BA, the line to be di- 24 MECHANICAL DRAWING. vided at B. Draw line CA joining the two ends of the Hnes. Draw lines from 5, 6, 7, 8, parallel to CA^ dividing line AB in points i, 2, 3, 4, proportional to BC. Fig. 32. To Construct an Equilateral Triangle ON A Given Base AB. — From the points A and B with AB as radius describe arcs cutting in C. Draw lines AC and BC. The triangle ABC is equilateral and equiangular. Fig. 3 Fig. 33. To Construct an Equilateral Triangle OF A Given Altitude, AB, — From both ends oi AB draw lines perpendicular to it as CA and DB. From A with any radius describe a semicircle on CA and with its radius cut off arcs I, 2. Draw lines from A through i, 2, and produce them until they cut the base BD. Fig. 34. To Trisect a Right Angle ABC. — From the angular point B with any convenient radius describe an arc cutting the sides of the angle in C and A. From C and A with the same radius cut off arcs i and 2. Draw lines iB and 2B, and the right angle will be trisected. GEOMETRICAL DRAWING. 25 Fig. 35. To Construct any Triangle, its Three Sides AB and (Seeing given. — From one end of the base as A describe an arc with the Hne B as radius. From the other end with hne C as radius describe an arc, cutting the first arc in D. From D draw Hnes to the ends of Hne A, and a triangle will be constructed having its sides equal to the sides given. To construct any triangle the two shorter sides B and C must together be more than equal to the largest side A. Fig. 36. Fig. 37. Fig. 36. To Construct a Square, its Base AB BEING GIVEN. — Erect a perpendicular at B. Make BC equal 26 MECHANICAL DRA WING. to AB. From A and C with radius AB describe arcs cutting; in D. Join DC and DA. Fig. 37. To Construct a Square, given its Di- agonal AB. — Bisect AB in C. Draw DF perpendicular to- AB at C. Make CD and (T/^ each equal to CA. Join ^Z^,, Z>i5, ^5/^; and FA. Fig. 38. To Construct a Regular Polygon of any Number of Sides, the Circumscribing Circle being GIVEN. — At any point of contact, as C, draw a tangent AB to the given circle. From C with any radius describe a semi- circle cutting the given circle. Divide the semicircle into as many equal parts as the polygon is required to have sides, as I, 2, 3, 4, 5, 6. Draw lines from C through each division, cutting the circle in points which will give the angles of the polygon. D Fig. 39. Another Method. — Draw a diameter AB of the given circle. Divide AB into as many equal parts as the polygon is to have sides, say 5. From A and B with the GEOMETRICAL DRAWING. 27 line AB as radius describe arcs cutting in C, draw a line from C through the second division of the diameter and produce it cutting the circle in D. BD will be the side of the required polygon. The line C must 'always be drawn through the second division of the diameter, whatever the number of sides of the polygon. Fig. 40. To Construct any Regular Polygon ^VITII A Given Side AB. — Make BD perpendicular and equal to AB. With B as center and radius AB describe arc DA. Divide arc DA into as many equal parts as there are sides in the required polygon, as i, 2, 3, 4, 5. Draw B2, Bisect line AB and erect a perpendicular at the bisection cut- ting B2 in C. With C as center and radius CB describe a circle. With AB as a chord step off the remaining sides of the polygon. Fig. 40. Fig. 41. Fig. 41. Another Method. — Extend hne AB. With center A and any convenient radius describe a semicircle. Divide the semicircle into as many equal parts as there are sides in the required polygon, say 6. Draw lines through every division except the first. With A as center and AB as 28 MECHANICAL DRA WING. radius cut off A2 in C. From C with the same radius cut Ai in D. From Z), ^4 in ^. From B, As in F. Join AC, CD, DE, £F, and FB. Fig. 42. To Construct a Regular Heptagon, the Circumscribing Circle being given. — Draw a radius AB. With B as center and BA as radius, cut the circumference in 1,2; it will be bisected by the radius in C. Ci or C2 is equal to the side of the required heptagon. Fig. 42. Fig. 43. To Construct a Regular Octagon, the Circumscribing Circle being given. — Draw a diameter AB. Bisect the arcs AB in C and D. Bisect arcs CA and CB in I and 2. Draw lines from i and 2 through the center of the circle, cutting the circumference in 3 and 4. Join Ai, iC, C2, 2B, Bi, etc. Fig. 44. To Construct a Pentagon, the Side AB being given. — Produce AB. With B as center and BA as radius, describe arc AD2. With center A and same radius, describe an arc cutting the first arc in D. Bisect AB in E. GEOMETRICAL DRAWING. 29 [' Draw line DE, Bisect arc BD in F. Draw line EF. With ■' center C and radius EF q.v\\. off arc C\ and i, 2 on the semi- ' circle. Draw line B2 ; it will be a second side of the penta- gon. Bisect it and draw a line perpendicular to it at the bisection. The perpendiculars from the sides AB and B2 will cut in G. With G as center and radius GA describe a circle • it will contain the pentagon. Fig. 45. 30 MECHANICAL DRA IVIXG. Fig. 45. To Construct a Heptagon on a Given Line AB. — Extend line AB to C7. From B with radius AB describe a semicircle. With center A and same radius de- scribe an arc cutting the semicircle in D. Bisect AB in E. Draw line DE. With C as center and DE as radius, cut off arc I on the semicircle. Draw line Bi ; it is a second side of the heptagon. Bisect it and obtain the center of the circum- scribing circle as in the preceding problem. Fig. 46. To Inscribe an Octagon in a Given Square. — Draw diagonals AD, CB intersecting at O. From A, B, C, and D with radius equal to AO describe quadrants cutting the sides of the square in i, 2, 3, 4, 5, 6, 7, 8. Join these points and the octagon will be inscribed. Fig. 46. Fig. 47. To Construct a Regular Octagon on a Given Line AB. — Extend line ^^ in both directions. Erect perpendiculars at ^ and B. With centers A and B and radius ^^ describe the semicircle CEB and AF2. Bisect the quad- rants CE and DF in i and 2, then Ai and B2 will be two more sides of the octagon. At i and 2 erect perpendiculars I, 3 and 2, 4 equal to AB. Draw 1-2 and 3-4. Make the GEOMETRICAL DRAWING. 31 perpendiculars at A and B equal to 1-2 or 3-4 — viz., A^ and B6. Complete the octagon by drawing 3-5, 5-6, and 6-4. Fig. 48. To Draw a Right Line Equal to Half THE Circumference of a Given Circle. — Draw a diam- eter AB. Draw line AC perpendicular to AB and equal to three times the radius of the circle. Draw another perpen- dicular at B to AB. With center B and radius of the circle cut off arc BD, bisect it and draw a line from the center of the circle through the bisection, cutting line B in E. Join EC. Line EC will be equal to half the circumference of ■circle A. G A C Fig. 49. To Find a Mean Proportional to two Given Right Lines. — Extend the line AB to E making BE equal to CD. Bisect AE in F. From /^ with radius BA de- scribe a semicircle. At B where the two given lines are joined erect a perpendicular to AE cutting the semicircle in G. BG will be a mean proportional to CD and AB. Fig. 50. To Find a Third Proportional (less) to TWO Given Right Lines AB and CD. — Make EF= the given line AB. Draw EG making an angle with EF ^ DC. Join EG. From E with EG as radius cut EF in H. Draw 32 MECHANICAL DRA WING. H parallel to FG, cutting EG in /. EI is the third propor- tional (less) to the two given lines. Fig. 50. F Fig. 51. Fig. 51. To Find a Fourth Proportional to three Given Right Lines AB, CD, and .5"/^.— Make 6^//= the given line AB. Draw GI = CD, making any convenient angle to GH. Join HI. From G lay off GH = EF. From K draw a parallel to HI cutting GI in L. GL is the fourth proportional required. Fig. 52. Fig. 53- Fig. 52. To Find the Center of a Given Arc ABC. — Draw the chords AB and CD and bisect them. Extend the bisection lines to intersect in D the center required. GEOMETRICAL DRAWING. 33 Fig. 53. To Draw a Line Tangent to an Arc of a Circle. — (ist.) When the center is not accessible. Let B be the point through which the tangent is to be drawn. From B lay off equal distances as BE^ BF. Join EF and through B draw ABC parallel to EF. (2d.) When the cen- ter D is given. Draw BD and through B draw ABC perpen- dicular to BB. ABC is tangent to the circle at the point B. Fig. 54. To Draw Tangents to the Circle C from THE Points without It. — Draw ^6^ and bisect it in ^. . From E with radius EC describe an arc cutting circle C in B and B. Join CB, CD. Draw AB and AB tangent to the circle C. Fig. 54- Fig. 55. Fig. 55. To Draw a Tangent between two Cir- cles. — Join the centers A and B. Draw any radial line from A as A2 and make 1-2 — the radius of circle B. From A with radius A-2 describe a circle C2D. From center B 34 ME CHA NIC A L DRA WING. draw tangents BC and BD to circle C2D at the points C and D by preceding problem. Join AC and AD and through the points E and F draw parallels FG and EH to BD and ^C. /^6^ and EH are the tangents required. Fig. 56. To Draw Tangents to two Given Cir- cles A AND B. — Join A and B. From ^4 with a radius equal to the difference of the radii of the given circles de- FlG. 56. scribe a circle GF. From B draw the tangents BE and BG, by Prob. 37. Draw AF and AG extended to E and H. Through E and // draw EC and //i^ parallel to BF and ^6^ respectively. EC and DH z.x^ the tangents required. Fig. 57. To Draw an Arc of a Circle of Given Radius Tangent to two Straight Lines. — AB and AC are the two straight lines, and r the given radius. At a dis- tance — r draw parallels 1-2 and 3-4 to AC and AB, inter- GEOMETRICAL DRAWING. 35 seating at F. From F draw perpendiculars FD and FE. With F as center and FD or FE as radius describe the re- quired arc, which will be tangent to the two straight lines at the points D and E. Fig. 58. To Draw an Arc of a Circle Tangent TO TWO Straight Lines BC and CD when the Mid- position G IS GIVEN. — Draw CA the bisection of the angle BCD and EF at right angles to it through the given point G. Next bisect either of the angles FEB or FED. The bisection line will intersect the central line CA at A, which will be the center of the arc. From A draw perpendiculars Ai and A2, and with either as a radius and A as center describe an arc which will be tangent to the lines BC and CD at the points i and 2. Fig. 59. To Inscribe a Circle within a Triangle ABC. — Bisect the angles A and B. The bisectors will meet in D. Draw D\ perpendicular to AB. Then with center D and radius =z Di describe a circle which will be tangent to the given triangle at the points i, 2, 3. Fig. 60. To Draw an Arc of a Circle of Given Radius i^ tangent to two Given Circles A and B. — From A and B draw any radial lines as ^3, B4.. Outside the circumference of each circle cut off distances 1-3 and 2-4 36 MECHAXICAL DRA WING. each the given radius R. Then with center A and radius ^-3, and center B and radius B-^ describe arcs intersecting at C. Draw CAXB cutting the circles at 5 and 6. With centre C and radius (^5 or C6 describe an arc which will be tangent at points 5 and 6. p — ^ Fig. 60. Fig. 61. To Draw ax Arc of a Circle of Given Radius R tangent to two Given Circles A and B Fig 61. WHEN THE Arc includes the Circles. — Through A and B draw convenient diameters and extend them indefinitely. On GEOMETRICAL DRAWING. 37 these measure off the distances 1-2 and 3-4, each equal in length to the given radius R. Then with center A and radius A2, center B and radius ^4, describe arcs cutting at C. From C draw C^ and (76 through B and A. With center C and ra- dius C6 or 6^5 describe the arc 6, 5, which will be tangent to the circles at the points 6 and 5. Fig. 62. To Draw an Arc of a Circle of Given Radius R tangent to Two Given Circles A and B WHEN THE Arc includes one Circle and excludes the other. — Through A draw any diameter and make 1-2 = R. Fig. 62. From B draw any radius and extend it, making 3-4 = R. With center A and radius A2 and center B and radius Ba^ describe arcs cutting at C. With C as center and radius = C^ or (76 describe the arc 5, 6. Fig. 63. Draw an Arc of a Circle of Given Ra- dius R TANGENT TO A STRAIGHT LiNE AB AND A CIRCLE CD. — From £", the center of the given circle, draw an arc of a 38 MECHANICAL DRA WING. circle i, 2 concentric with CD at a distance R from it, and also a straight line 3, 4 parallel to AB at the same distance R from AB, Draw ^(9 intersecting CD at 5. Draw the perpen- dicular 06. With center O and radius (96 or O^ describe the required arc. ^ A e """^'^^1 5 ^ ~^^ ft^ A \ X\ .^^"^1 -^ Fig. 63. Fig. 64. To Describe an Ellipse Approximately BY means of three Radii (F. R. Honey's method). — Fig. 64. Draw straight lines RH a.nd NQ, making any convenient angle at H. With center H and radii equal to the semi-minor and GEOMETRICAL DRAWING. 39 semi-major axes respectively, describe arcs LM and NO. Join LO and draw MK and NP parallel to LO. Lay off Zi —\ of LX. Join Oi and draw M2 and ^¥3 parallel to Oi. Take //'3 for the longest radius (= Z), //2 for the shortest radius (= E\ and one-half the sum of the semi-axes for the third radius (== 5), and use these radii to describe the ellipse as follows: Let AB and CD be the major and minor axes. Lay off ^4 = Z" and ^5 = S. Then lay o'^ CG = T and C^ = S. With G as center and G6 as radius draw the arc 6, With center 4 and radius 4, 5, draw arc 5, ^, intersecting 6, ^ at ^. Draw the line G^- and produce it making GS = T. Draw ^, 4 and extend it to 7 making ^, 7 =: 5. With center G and radius GC {=T) draw the arc (78. With center ^ and radius g^ 8 (=5) draw the arc 8, 7. With center 4 and radius 4, 7 {= £) draw arc 7^. The remaining quadrants can be drawn in the same way. Fig. 65. To Draw ax Ellipse havixg given the Axes yi^ AND CD. — Draw AB and CD at right angles to and bisecting each other at E. With center C and radius EA cut AB in E and E' the foci. Divide EE or EE' into a number of parts as shown at i, 2, 3, 4, etc. Then with /^ and E' as cen- ^U ^<^ iWl *fr^ ^'^?*''A f ^r^^^\ .ifry t r 1 i77>r>I'^V 12 3 Jt 5 67 j \^ H^ Ccm-E,A'o.o i^ ^>tr ^^e^ -^^^"^ Fig. 65. Fig. 67. ters and Ai and ^i, and ^2 and B2, etc., as radii describe arcs intersecting in R, S, etc., until a sufficient number of points 40 ME CHA XICA L DRA WIXG. are found to draw the elliptic curve accurately throughout. (No. 5 of the ''Sibley College Set" of irregular curves is very useful in drawing this curve.) To draw a tangent to the ellipse at the point G\ Extend FG and draw the bisector of the angle HGF . KG is the tangent required. Fig. 6^, Another Method. — Let yJ^ and AC h& the semi axes. With A as center and radii AB and AC describe circles. Draw any radii as ^3 and ^^4, etc. Make 3 i, 42, etc., perpendicular to AB, and D2, E^, etc., parallel to AB. Then i, 2, 5, etc., are points on the curve. Fig. Gy. Another ^Method. — Place the diameters as before, and construct the rectangle CDEF. Divide AB and DB and BF into the same number of equal parts as I, 2, 3 and B. Draw from C through points i, 2, 3 on AB and BD lines to meet others drawn from E through points i, 2, 3 on A^B and FB intersecting in points GHK. GHK are points on the curve. Fig. 68. Another Method. — Place the diameters AB and CD as shown in Drawing Xo. i. Draw any convenient '1 Fig. 68. angle RHQ. Drawing No. 2. With center //"and radii equal to the semi-minor and semi-major axes describe arcs LM and GEOMETRICAL DRAWING. 4 1 NO. Join LO and draw MK and NP parallel to LO. Then from C and D with a distance = HP lay off the points i i' on the minor axis and from A and B with a distance = HK lay off the points 2 2' on the major axis. With centers 1,1', 2 and 2' and radii \—D and 2-B, respectively, draw arcs of circles. On a piece of transparent celluloid 7" lay off from the point G, GF and GE = the semi-minor and semi-major axes respec- tively. Place the point i^on the major axis and the point E on the minor axis. If the strip of celluloid is now moved over the figure, so that the point E is always in contact with the semi-minor axis and the point F with the semi major axis, the necessary number of points may be marked through a small hole in the celluloid at G with a sharp conical-pointed pencil, and thus complete the curve of the ellipse between the arcs of circles. Fig. 69. To Construct a Parabola, the Base CD AND THE Abscissa AB being given. — Draw EF through A parallel to CD and CE and DF parallel to AB. Divide AE, AF, EC, and FD into the same number of equal parts. Through the points i, 2, 3 on ^i^ and AE draw lines parallel to AB, and through A draw lines to the points 1,2, 3 on FD and EC intersecting the parallel lines in points 4, 5, 6, etc., of the curve. Fig. 70. Given the Directrix BD and the Focus C TO Draw a Parabola and a Tangent to It at the Point 3. — The parabola is a curve such that every point in the curve is equally distant from the directrix j5Z> and the focus C. The vertix E is equally distant from the directrix and the focus, i.e. CE is = EB. Any line parallel to the axis is a diameter. A straight line drawn across the figure at right angles to the 42 MECHANICAL DRA WIXG. axis is a double ordinate, and either half of it is an ordinate. The distance from C to any point upon the curve, as 2 is always equal to the horizontal distance from that point to the directrix. Thus Ci ^^ i, i' , C2 to 2, 2', etc. Through C draw ACF at right an2:les to BD, ACF is the axis of the Ai i 3 F / S ^ \ 1 I \ D A ^ G 5 I i \ 3 4 Fig. 70. curve. Draw parallels to BD through any points in AB^ and with center C and radii equal to the horizontal distances of these parallels from BD describe arcs cutting in the points i, 2, 3, 4, etc. These are points in the curve. The tangent to the curve at the point 3 may be drawn as follows: Produce AB to F. Make FF = the horizontal distance of ordinate 33 from F. Draw the tangent through 2,F. Fig. 71. To Draw an Hyperbola, having given THE Diameter AB, the Abscissa BD, and Double Ordi- nate FF. — Make F4. parallel and equal to BD. Divide DF and F^ into the same number of equal parts. From B draw lines to the points in /[F and from A draw lines to the points in DF. Draw the curve through the points where the lines correspondingly numbered intersect each other. GEOMETRICAL DRAWING. 43 Fig. ^2. To Construct an Oval the Width AB BEING GIVEN.— Bisect AB by the line CD in the point E, and with E as center and radius EA draw a circle cutting CD in Fig. 71, Fig. 72. F. From ^ and ^draw lines through F. From A and B with radius equal to AB draw arcs cutting the last two lines in G and H. From F with radius /^6^ describe the arc GH to meet the arcs AG and ^5//, which will complete the oval. Fig. 73. Given an Ellipse to Find the Axes and Foci. — Draw two parallel chords AB and CD. Bisect each of these in E and F. Draw EF touching the ellipse in i and 2. This line divides the ellipse obhquely into equal parts. Bisect I, 2 in 6^, which will be the center of the ellipse. From G with any radius draw a circle cutting the ellipse in HIJK. Join these four points and a rectangle will be formed in the ellipse. Lines LM and NO, bisecting the sides of the rectangle, will be the diameters or'axes of the ellipse. With iV or (9 as centers and radius = GL the semi-major axis, de- scribe arcs cutting the major axis in P and Q the foci. Fig. 74. To Construct a Sipral of one Revolu- tion. — Describe a circle using the widest limit of the spiral as 44 MECHANICAL DRA WING. a radius. Divide the circle into any number of equal parts as A., B, C, etc. Divide the radius into the same number of equal parts as i to 12. From the center with radius 12, i describe an arc cutting- the radial line B in i' . From the center con- tinue to draw arcs from points 2, 3,4, etc., cuttingthe corre- sponding radii C, D, E, etc. in the points 2', 3', 4', etc. From 12 trace the Archimedes Spiral of one revolution. X A H ^ ^ \^ K /I ^^ \y XK \\^ ' \ \ y\ / 1 1 % ^ K \G\ 7/ I VCi ^:^^ _i v-^ V Fig. 73. Fig. 75. To Describe a Spiral of any Number of Revolutions, e.g., 2. — Divide the circle into any num- ber of equal parts d^s A, B, C, etc., and draw radii. Divide the radius ^12 into a number of equal parts corresponding with the required number of revolutions and divide these into the same number of equal parts as there are radii, viz., I to 12. It will be evident that the figure consists of two separate spirals, one from the center of the circle to 12, and one from 12 to A. Commence as in the last problem, draw- ing arcs from i. 2, 3, etc., to the correspondingly numbered radii, thus obtaining the points marked i\ 2', 3', etc. The first revolution completed, proceed in the same manner to find the points i^\ 2" , 3'', etc. Through these points trace the spiral of two revolutions. GEOMETRICAL DRAWING. 45 Fig. 'j^. To Construct the Involute of the Chi- cle O. — Divide the circle into any number of equal parts and draw radii. Draw tangents at right angles to these radii. On the tangent to radius i la)- off a distance equal to one of the parts into which the circle is divided, and on each of A the tangents set off the number of parts corresponding to the number of the radii. Tangent 12 will then be the circumfer- ence of the circle unrolled, and the curve drawn through the extremities of the other tangents will be the involute. Fig. -jj. To Describe an Ionic Volute. — Divide the given height into seven equal parts, and through the point 3 the upper extremity of the third division draw 3, 3 perpen- dicular to AB, From any convenient point on 33 as a cen- ter, with radius equal to one-half of one of the divisions on AB, describe the eye of the volute NPNM, shown enlarged at Drawing No. 2. iVTV corresponds to line 3, 3, Drawing No. I. Make PM perpendicular to AW and inscribe the square AT^A^J/, bisect its sides and draw the square 11, 12, 46 MECHANICAL DRAWING. 13, 14. Draw the diagonals 11, 13 and 12, 14 and divide them as shown in Drawing No. 2. At the intersections of the horizontal with the perpendicular full lines locate the points I, 2, 3, 4, etc., which will be the centers of the quad- rants of the outer curve. The centers for the inner curve will be found at the intersections of the horizontal and per- FiG. 77. pendicular broken lines, drawn through the divisions on the diagonals. Then with center i and radius iPdraw arc PiV, and with center 2 and radius 2N draw arc NM, with center 3 and radius 3^1/ draw arc ML, etc. The inner curve is drawn in a similar way, by using the points on the diagonals indi- cated by the broken lines as centers. Fig. 78. To Describe the Cycloid. — AB is the di- rector, CB the generating circle, Jf a piece of thin transparent celluloid, with one side dull on which to draw the circle C. At any point on the circle C puncture a small hole with a sharp needle, and place the point C tangent to the director AB at the point from which the curve is to be drawn. Hold the celluloid at this point with a needle, and rotate it until GEOMETRICAL DRAWING. 47 the arc of the circle C intersects the director AB. Through the point of intersection stick another needle and rotate X until the circle is again tangent to AB, and through the punc- ture at C with a 4H pencil, sharpened to a fine conical point, mark the first point on the curve. So proceed until sufficient points have been found to complete the curve. (Note. — The thin celluloid was first used as a drawing instrument by Professor H. D. Williams, of Sibley College, Cornell University.) Fig. 79. To Find the Length of a Given Arc of a Circle approximately. — Let BC be the given arc. Draw its chord and produce it to A, making BA equal half the /V v^^ - G /) V A 'B Fig. 78. Fig. 79. chord. With center^ and radius ^4(7 describe arc crZ> cut- ting the tangent line BD at D, and making it equal to the arc BC. Fig. 80. To Describe the Cycloid by the Old IMethod. — Divide the director and the generating circle into the same number of equal parts. Through the center a draw ag parallel to AB for the line of centers, and divide it as AB in the points/', c, d, e,f, and ^. With centers/, e, d, etc., de- scribe arcs tangent to AB, and through the points of division on the generating circle 1,2, 3, etc., draw lines parallel to 48 MECHANICAL DRA WING. AB cutting the arcs in the points i', 2' , 3', etc. These will be points in the curve. An approximate curve may be drawn by arcs of circles. Thus, taking/"^ as center and f g' as radius, draw arc g' i' ^ Fig. 80. Produce \'f' and 2V until they meet at the center of the second arc 2'f\ etc. Fig. 81. To Describe the Epicycloid and the Hypocycloid. — Divide the generating circle into any num- ber of equal parts, i, 2, 3, etc., and set off these lengths from C on the directing circle CB as e' , d' , c' , etc. From A the cen- ter of the directing circle draw lines through e\ d' , c' , etc., cut- ting the circles of centers in e, d, c, etc. From each of these points as centers describe arcs tangent to the directing circle. From center A draw arcs through the points of division on the generating circle, cutting the arcs of the generating circles in their several positions at the points i', 2' , 3^ etc. These will be points in the curve. Fig. 82. Another Method. — Draw the generating circle on the celluloid and roll it on the outside of the gener- ating circle BC for the Epicycloid, and on the inside for the GEOMETRICAL DRAJVIXG. 49 Hypocycloid, marking the points in the curve 1,2, 3, etc., in similar manner to that described for the Cycloid. Fig. 82. Fig. 83. Fig. 83. To Draw the Cissoid. — Draw any line AB and BC perpendicular to it. On BC describe a circle. From the extremity C of the diameter draw any number of lines, at any distance apart, passing through the circle and meeting the line AB in i' , 2', 3', etc. Take the length from ^ to 9 and set it off from C on the same line to g" . Take the dis- tance from 8' to 8 and set it off from C on the same line to 8", etc., for the other divisions, and through 9", 8'', y" , 6", etc., draw the curve. 50 MECHANICAL DRA WING. Fig. 84. To Draw Schiele's Anti-friction Curve. — Let AB be the radius of the shaft and ^i, 2, 3, 4, etc., its axis. Set off the radius AB on the straight edge of a piece of stiff paper or thin celluloid and placing the point B on the division i of the axis, draw through points the line A\. Then lower the straight edge until the point B coincides with 2 and the point y4 just touches the last line drawn, and draw a2, and so proceed to find the points a, b, c, etc. Through these points draw the curve. 4 6 Fig. 84. Fig. 85. Fig. 85. To Describe an Interior Epicycloid. — Let the large circle X be the generator and the small circle Y the director. Divide circle Y into any number of equal parts, as B, H, /, /, etc. Draw radial lines and make HCy ID, JE, KF, etc., each equal to the radius of the generator X. With centers C, D, E, etc., describe arcs tangent at H, I, J, etc. Make Hi equal to one of the divisions of the di- rector as BH. Make I2 equal to two divisions, ^3, three divi- sions, etc., and draw the curve through the points i, 2, 3, 4, GE OME TRIG A L DRA WING. 51 etc. This curve may also be described with a piece of cellu- loid in a similar way to that explained for the cycloid. It may not be out of place here to describe a few of the MOULDINGS USED IX ARCHITECTURAL WORK, since they are often found applied to mechanical constructions. Fig. 86. To Describe the "Scotia." — i, i is the top line and 4, 4 the bottom line. From i drop a perpendicular I, 4; divide this into three equal parts, as i, 2, and 3. Through the point 2 draw ab parallel to I, i. With center 2 and radius 2, i describe the semicircle alb, and with center b and radius ba describe the arc ^5 tangent to 4, 4 at 5, draw the fillets I, I and 4, 4. i A :\ Q 3 . 5 L J, 1 Fig. 86. Fig. S7. Fig. 8;. To Describe the " Cyma Recta." — Join i, 3 and divide it into five equal parts, bisect i, 2 and 2, 3, and with radius equal to i, 2 and 2, 3 respectively describe arcs I, 2 and 2,3. Draw the fillets i, i and 3, 3 and complete the moulding. Fig. 88. To Describe the "Cavetto" or "Hol- low." — Divide the perpendicular i, 2 into three equal parts and make 2, 3 equal to two of these. From centers i and 3 with a radius somewhat greater than the half of i, 3, describe arcs intersecting at the center of the arc i, 3, 52 MECHANICAL DRAWING. Fig. 89. To Describe the *' Echinus, '^ ''Quarter Round," or "Ovolo." — Draw i, 2 perpendicular to 2, 3, and divide it into three equal parts. Make 2, 3 equal to two of these parts. From the points 2 and 3 with a radius greater than half 1,3, describe arcs cutting in the center of the required curve. Fig. 90. To Describe the "Apophygee." — Divide 3, 4 into four equal parts and lay off five of these parts from 3 to 2. From points 2 and 4 as centers and radius equal to 2,3, describe arcs intersecting in the center of the curve. Fig. 90. Fig. 91. To Describe the ''Cyma Reversa." — Make 4, 3 = 4, I. Join I, 3 and bisect it in the point 2. From the points I, 2 and 3 as centers and radii equal to about two-thirds of 1 , 2 draw arcs intersecting at 5 and 6. Points 5 and 6 are the centers of the reverse curves. Fig. 92. To Describe the '' Torus." — Let i, 2 be the breadth. Drop the perpendicular i, 2, and bisect it in the GEOMETRICAL DRAWIXG. 53 point 3. With 3 as center and radius 3, i, describe the semi- circle. Draw the fillets. Fig. 92. Fig. 93. Fig. 93. An Arched Window Opening. — The curves are all arcs of circles, drawn from the three points of the equi- lateral triangle, as shown in the figure. Fig. 94. To Describe the ''Trefoil." — The equi- lateral triangle is drawn first, and the angle 1,2,3 bisected by the line 2, 4, which also cuts the perpendicular line i, 6in the point 6. The center of the surrounding circles i, 2 and 3 are the centers of the trefoil curves. Fig. 95. To Describe the ''Quatre Foil." — Draw the square i, 2, 3, 4 in the position shown in the figure. The center of the surrounding circles, point 5, is at the intersection of the diagonals of the square. Points i, 2, 3, 4 of the square are the centers of the small arcs. Fig. 96. To Describe the " Cinquefoil Orna- ment." The curves of the cinquefoil are described from the corners of a pentagon i, 2, 3, 4, 5. Bisect 4, 5 in 6 and draw 2, 6, cutting the perpendicular in the point 7, the center of the large circles. Fig. 97. To Draw a Baluster. — Begin by drawing the center line, and lay off the extreme perpendicular height, 54 MECHAXICAL DRAWING. the intermediate, perpendicular, and horizontal dimensions, and finally the curves as shown in the figure. Fig. 94. Fig. 95. Fig. 96. Fig. 97. DRAWING TO SCALE. When we speak of a drawing as having been made to scale, we mean that every part of it has been drawn proportionately and accurately, Q\th.Qr full sice, reduced or enlarged. Very small and complicated details of machinery are usu- ally drawn enlarged ; larger details and small machines may be made full size, while larger machines and large details are shown reduced. When a drawing of a machine is made to a reduced or en- larged scale the figures placed upon it should ahvays give the full-size dimensions, i.e., the sizes the machine should meas- ure when finished. GEOMETRICAL DRAWING. 55 Fig. 98. To Construct a Scale of Third Size or A^'=- I Foot. — Draw upon a piece of tough white drawing- paper two parallel lines about \" apart and about 14'' long as shown by a, Fig. 98. From A lay off distances equal to 4" and divide the first space AB into 12 equal parts or inches by Prob. 12. Divide AB'm the same way into as many parts as it may be desired to subdivide the inch divisions on AB, E u .<>• a- V n' r jr 10- 8- 7- ,r Jf- 2' 1' Scale /= ^Ifoot. b 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ill III ill ill ill II ill ill ill ill ill ill ill ill ill ill ill ill ill ill 1!^ ill 11 ill k- -I'M-- . >i- IIHA 8' 7' f /l 2 1' iyiii!iiiliiiliiiliiililiiiiiliiiliiil.iili] ./// /Scale 1'^ Ifoot. B' 5'f Fig. gS. usually 8. When the divisions and subdivisions have been carefully and lightly drawn in pencil, as shown by a, in Fig. 98, then the lines denoting i'\i'\ i'\ i" , and 3'' should be carefully inked and numbered as shown by {b). By a further subdivision a scale of 2''= i foot may easily be made as shown by {c) in Fig. 98. CHAPTER III. CONVENTIONS. It is often unnecessary if not undesirable to represent cer- tain things as they would actually appear in a drawing, espe- cially when much time and labor is required to make them orthographically true. So for economic reasons draftsmen have agreed upon con- ventional methods to represent many things that would other- wise entail much extra labor and expense, and serve no par- ticular purpose. It is very necessary, however, that all draftsmen should know lioiv to draw these things correctly, for occasions will often arise when such knowledge will be demanded ; and be- sides it gives one a feeling of greater satisfaction when using conventional methods to know that he could make them artis- tically true if it was deemed necessary. STANDARD CONVENTIONAL SECTION LINES. Conventional section lines are placed on drawings to distin- guish the different kinds of materials used when such drawings are to be finished in pencil, or traced for blue printing, or to be used for a reproduction of any kind. Water-colors are nearly always used for finished drawings and sometimes for tracings and pencil drawings. The color tints can be applied in much less time than it 56 CONVENTIONS. 57 takes to hatch-line a drawing. So that the color method should be used whenever possible. Fig. 99. — This figure shows a collection of hatch-lined sections that is now the almost universal practice among draftsmen in this and other countries, and may be considered standard. No. I. To the right is shown a section of a wall made of rocks. When used without color, as in tracing for printing, the rocks are simply shaded with India ink and a 175 Gillott steel pen. For a colored drawing the ground work is made of gamboge or burnt umber. To the left is the conventional representation of water for tracings. For colored drawings a blended wash of Prussian blue is added. No. 2. Convention for Marble. — When colored, the whole section is made thoroughly wet and each stone is then streaked with Payne's gray. No. 3. Convention for Cliestnnt. — When colored, a ground wash of gamboge with a little crimson lake and burnt umber is used. The colors for graining should be mixed in a separate dish, burnt umber with a little Payne's gray and crimson lake added in equal quantities and made dark enough to form a sufficient contrast to the ground color. No. 4. General Convention for Wood. — When colored the ground work should be made with a light wash of burnt sienna. The graining should be done with a writing-pen and a dark mixture of burnt sienna and a modicum of India ink. No. 5. Convention for Black Walnut. — A mixture of Payne's gray, burnt umber and crimson lake in equal quanti- ties is used for the ground color. The same mixture is used for graining when made dark by adding more burnt umber. 58 MECHANICAL DRAWING. CONVEX TIOXS. 59* No. 6. Convention for Hard Pine, — For the ground color make a light wash of crimson lake, burnt umber, and gamboge, equal parts. For graining use a darker mixture of of crimson lake and burnt umber. No. 7. Convention for Building- stone. — The ground color is a light wash of Payne's gray and the shade lines are added mechanically with the drawing-pen or free-hand with the writing-pen. No. 8. Convention for EartJi. — Ground color, India ink and neutral tint. The irregular lines to be added with a writ- ing-pen and India ink. No. 9. Section Lining for Wrought or Malleable Iron. — When the drawing is to be tinted, the color used is Prussian blue. No. 10. Cast Iron. — These section lines should be drawn equidistant, not very far apart and narrower than the body lines of the drawing. The tint is Payne's gray. No. : I. Steel. — This section is used for all kinds of steel. The lines should be of the same width as those used for cast- iron and the spaces between the double and single lines should be uniform. The color tint is Prussian blue with enough crim- son lake added to make a warm purple. No. 12. Brass. — This section is generally used for all kinds of composition brass, such as gun-metal, yellow metal, bronze metal, Muntz metal, etc. The width of the full lines, dash lines and spaces should all be uniform. The color tint is a light wash of gamboge. Nos. 13-20. — The section lines and color tints for these numbers are so plainly given in the figure that further instruc- tion would seem to be superfluous. 6o MECHANICAL DRAWING, CONVENTIONAL LINES. Fig. 100. — There are four kinds: (i) The Hidden Line. — This Hne should be made of short dashes of uniform length and width, both depending some- what on the size of the drawing. The width should always be slii^htlv less than the body lines of the drawing, and the (■ length of the dash should never exceed \" . The spaces between the dashes should all be uniform, quite small, never exceeding yV''- This line is always inked in with black ink. (2) TJie Line of JMotion. — This line is used to indicate point paths. The dashes should be made shorter than those of the hidden line, just a trifle longer than dots. The spaces should of course be short and uniform. (3) Center Lines. — Most drawings of machines and parts of machines are symmetrical about their center lines. When penciling a drawing these lines may be drawn continuous and as fine as possible, but on drawings for reproductions the black- inked line should be a long narrow dash and two short ones alternately. When colored inks are used the center line should be made a continuous red line and as fine as it is possible to make it. (4) Dimension Lines and Line of Seetion. — These lines are made in black with a fine long dash and one short dash alternately. In color they should be continuous blue lines. CONVENTIONS. 61 Colored lines should be used wherever feasible, because they are so quickly drawn and when made fine they give the drawing- a much neater appearance than when the conventional black lines are used. Colored lines should never be broken. CONVENTIONAL BREAKS. Fig. ioi. — Breaks are used in drawings sometimes to indi- cate that the thing is actually longer than it is drawn, some- BS^ £sS9 l k^.^.^^.^^'.^^.^^^^ ■ ^'^'.^■^^■■^^^^^^^^^^^ ^.^^.^^^^^^^^^■^^■^^^^^^^^^^<^^^^^^^^^^.vy Fig. ioi. times to show the shape of the cross-section and the kind of material. Those given in Fig. loi show the usual practice. CROSS-SECTIONS. Fig. 102. — When a cross-section of a pulley, gear-wheel or other similar object is required and the cutting-plane passes through one of the spokes or arms, then only the rim and hub should be sectioned, as shown at xx No. i and zz No. 2, and the arm or spoke simply outlined. Cross-sections of the arms may be made as shown at A A No. 2. In working drawings of 62 MECHANICAL DRAWING. gear-wheels only the number of teeth included in one quadrant need be drawn ; the balance is usually shown by conventional lines, e.g., t\\Q pitch line the same as a center line, viz., a long Fig. I02. dash and two very short ones alternately or a fine continuous red line. The addeiidmn line {d) and the root or bottom line {U) the same as a dimension line, viz., one long dash and one short dash alternately or a fine continuous blue line. The end ele- vation of the gear-teeth should be made by projecting only the points of the teeth, as shown at No. 2. CONVENTIONAL METHODS OF SHOWING SCREW-THREADS IN WORKING DRAWINGS. Fig. 103. — No. i, shows the convention for a double V thread, U. S. standard; No. 2, a single V thread; No. 3, a single square thread; No. 4, a double left-hand V thread; No. 5, a double right-hand square thread; No. 6, any thread of small diameter; No. 7, any thread of very small diameter. The true methods for constructing these threads are explained on pages 99-101, Figs. 99-101. In No. 6. the short wide line is equal to the diameter of the thread at the bottom. The distance between the COXVENTIO.XS. 63 longer narrow lines is equal to the pitch, and the inclination is equal to half the pitch. The short dash lines in No. 7 should be made to corre- =r J^ ^-^ Fig. 103. spond to the diameter of the thread at the bottom. After some practice these lines can be drawn accurately enough by the eye. CHAPTER IV. LETTERING AND FIGURING. This subject has not been given the importance it deserves in connection with mechanical drawing. Many otherwise ex- cellent drawings and designs as far as their general appearance is concerned have been spoiled by poor lettering and figuring. All lettering on mechanical drawings should be plain and legible, but the letters in a title or the figures on a drawing should never be so large as to make them appear more prom- inent than the drawing itself. The best form of letter for practical use is that which gives the neatest appearance with a maximum of legibility and re- quires the least amount of time and labor in its construction. This would naturally suggest a " free-hand " letter, but be- fore a letter can be constructed '' free-hand " with any degree of efficiency, it will be necessary to spend considerable time in acquiring a knowledge of the form and proportions of the particular letter selected. It is very desirable then that after the student has care- fully constructed as many of the following plates of letters and numbers as time will permit and has acquired a sufficient knowledge of the form and proportions of at least the " Ro- man " and " Gothic" letters; he should then adopt some one 64 LETTERING AND FIGURING. 65 |l style and practice that at every opportunity, until he has at- tained some proficiency in its free-hand construction. When practicing the making of letters and numbers free- hand, they should be made quite large at first so as to train the hand. The " Roman " is the most legible letter and has the best appearance, but is also the most difficult to make well, either free-hand or mechanically. However, the methods given for its mechanical construction. Figs. 104 and 105, will materially modify the objections to its adoption for lettering mechanical drawings. The ''Gothic" letter is a favorite with mechanical drafts- men, because it is plain and neat and comparatively easy to construct. (See Fig. 106.) Among the type specimens given in the following pages the Bold-face Roman Italic on page 70 is one of the best for a good, plain, clear, free-hand letter, and is often used with good success on working drawings. Gillott's No. 303 steel pen is the best to use when making this letter free-hand. The " Yonkers " is a style of letter that is sometimes used for mechanical drawings. It is easy to construct with either F. Soennecken's Round Writing-pens, single point, or the Automatic Shading-pen. But it lacks legibility, and is therefore not a universal favorite. A good style for " Notes" on a drawing is the " Gothic Condensed " shown on page 70. When making notes on a drawing with this letter, the. only guides necessary are two parallel lines, drawn lightly in pencil. The letters should be sketched Hghtly in pencil firsts ^6 MECHANICAL DRAWING. and then carefully inked, improving spacing and proportions to satisfy the practiced eye. FIGURING. Great care should be taken in figuring or dimensioning a mechanical drawing, and especially a working drawing. To have a drawing accurately, legibly, and neatly figured is considered by practical men to be the most important part of a working drawing. There should be absolutely no doubt whatever about the character of a number representing a dimension on a drawing. Many mistakes have been made, incurring loss in time, labor, and money through a wrong reading of a dimension. Drawings should be so fully dimensioned that there will be no need for the pattern-maker or machinist to measure any part of them. Indeed, means are taken to prevent him from doing so, because of the liability of the workman to make mistakes, so drawings are often made to scales which are dif- ficult to measure with a common rule, such as 2 "and 4'^ = I ft. The following books, among the best of their kind, are recommended to all who desire to pursue further the study of '' Lettering" : Plain Lettering, by Prof. Henry S. Jacoby, Cornell University, Ithaca, N. Y. ; Lettering, by Charles W. Reinhardt, Chief Draftsman, Engineering News, New York; Free-hand Lettering, by F. T. Daniels, instructor in C. E. in Tufts College. LETTERIXG AND FICURIXG. 67 W :»T^ i°> ^^^^ ' ' \ \ -1 rm^^ ^ Irk i : i ■■ 1 1 1 1 i^ X y n . > '0 1 1 1 1 ! lol 1 1 |o1 > ^ '^^^ .-^ " ' ° 7^ '1 . _ 7=^ 1 0^ 0: "^^ / / J -^ ^° 1 :•' MM ' ' *N - ' 's^i (^ ;^ ^ 1 C j K ., A °, i 'I ' -1 L°! i l°J ^ i ! i:! i 1 ^ ■\ 4--^ i^!v ' y 'oi hi ^ 1 ' : 1 1 A \.\ \ ^\ i : 1 ; ^ >; 1 /' 1 1 ■ ! ^ i ! : . ! m r^ ^ -^: =^ 68 ME CHA NIC A L BRA IVIXG. ^; 3 ET rm m Z3: t I ^: ^^--K 15 ^ M I I I II m m ^S m "-<:l LETTERING AND FIGURING. 6g 70 MECHANICAL DRAWIXG, iS-Point Roman. ABCDEFGHIJKLM^^ OPQRSTUYAYX YZ abcdefgliijklmnopqrstuvwxyz 1234567890 iS-Point Italic. AB CDEFGHIJKLMXOP QBSTUV WXYZ ahcdefghijkimnopqrstuvwxyz 17- Point Gushing Italic. ABCDEFGHIJKLMNOPQRS TUVWXYZ abcdefghijklm nopqrstuvwxyz 1234567890 28-Point Boldface Italic. ABCDEFGHIJKL3I NOPQRSTUVWXYZ abcdefghijhlmnopqrstu vtvxyz 1234567890 Two-Line Nonpareil Gothic Gondensed. ABGDEFGHIJKLMNOPQRSTUVWXYZ 1234567890 ^ Three-Line Nonpareil Lightface Celtic. ABCDEFGHIJKLMNOPQR STUVV/XYZ abedefghijkl mnopqrstuvwxyz 1234567890 LETTERING AND FIGURING. 7 1 i8-Point Chelsea Circular. ABCDEFGHIJKLMNOPQRSTUVWX YZ abcdefg|-(ijl\lmr\opqrstuvwxyz 1234567890 iS-roint Elandkay. ABCDEFGHIJKLnNOPQRSTUVVXYZ 1 234567890 iS-Point Quaint Open. i^ic^EF^iiiyKLniii@r«STy¥ 28-Point Roman. ABCDEFGHIJKLM NOPQRSTUVWXYZ abcdefghijklmnopqrstu vwxyz 1234567890 28-Point Old-Style Italic. ABCDEFGHIJKLMNOP ORSTUVIVXYZ abcdefg h ijklm n opqrstuvwxyz i2345678go 7 2 ME CHA MCA L DKA WING. i2-Point Victoria Italic. ABCDEFCHIJKLMNOPQRSTU VWXYZ 1234567890 iS-Point DeVinne Italic. A B CDEFGH/JKLMNOPQRS TV VWXYZ abcdefghijklmnopqrst uvwxyz 1234567890 22-Point Gothic Italic. ABCDEFGHIJKLMNOPQRSTUmXYZ abcclefghijklmnopqrstuuwxyz 1234567890 Double-Pica Program. /IBCDEFGHIJKLMNO PQRSTUVWXYZ abcdef ghij klmnopqrstu v wxyz 1234567890 Nonpareil Telescopic Gothic. ABCDEFGHIJKLMNOPQRSTUVWXYZ 1234567S90 LETTERIXG AXD FIGURING. 73 -Point Gallican. ABCDEFGHIJKL MNOPQRSTUV\^^ XYZ 1234567890 Two-Line Virile Open. AiCPEreiiJiLawrQiiTywii lilc&fjliijyiiiop^r 22-Point Old-Stvle Roman. ABCDEFGHIJKLMNOPQRST UVWXYZ abcdefghijklmnopqrst uvwxyz 1234567890 36-Point Yonkers. y^ abcbcfgbijklmnopqr stuDtPxys 1(23^567890 CHAPTER V. ORTHOGRAPHIC PROJECTION. Orthographic Projection, sometimes called Descrip- tive Geometry and sometimes simply Projection, is one of the divisions of descriptive geometry; the other divisions are Spherical Projection, Isometric Projection, Shades and Shadows, and Linear Perspective. In this course we will take up only a sufficient number of the essential principles of Orthographic Projection, Isometric Projection, and Shades and Shade Lines, to enable the stu- dent to make a correct mechanical drawing of a machine or other object. Orthographic Projection is the science and the art of rep- resenting objects on different planes at right angles to each other, by projecting lines from t\\Q povit of sight through the principal points of the object perpendicular to the Planes of Projection. There are commonly three planes of projection used, viz., the H. P. or Horizontal Plane, the V. P. or Vertical PlaiiCy and the Pf. P. or Profile Plane. These planes, as will be seen by Figs. 107 and 109, inter- sect each other in a line called the /. L. or Intersecting Line, and form four angles, known as the first, second, third, and 74 GR THO GRA PHIC PR OJE C TIGX. 75 fourth DiJicdral Angles. Figs. 107 and 109 are perspective views of these angles. An object may be situated in any one of the dihedral angles, and its projections drawn on the corresponding co- ordinate planes. Problems in Descriptive Geometry are usually worked out in the first angle, and nearly all English draftsmen project their drawings in that angle, but in the United States the third angle is used almost exclusively. There is good reason for doing so, as will be shown hereafter. We will consider first a few projection problems in the first angle, after which the third angle will be used throughout. Fig. 107. H.P., Fig. 107, is the Horizontal Plane, V.P. the Vertical Plane, and I.L. the Intersecting Line. The Horizontal Projection of a point is where a perpen- dicular line drawn through the point pierces the H.P. The Vertical Projection of a point is where a per. line drawn through the point pierces the V.P. Conceive the point a, Fig. 107, to be situated in space ^' above the H.P. and 3" in front of the V.P. If a line is passed through the point a per. to H.P. and produced until 76 MECHANICAL DRAWING. it pierces the H.P. in the point a}\ <^^'will be the Hor. Proj. of the point a. If another Hne is projected through the point a per. to the V.P. until it pierces the V.P. in the point a", a" is the ver- tical projection of the point a. If now the V.P. is revolved upon its axis I.L. in the di- rection of the arrow until it coincides with the H.P. and let the H.P. be conceived to coincide with the plane of the drawing-paper, the projections of the point a will appear as shown by Fig. io8. The vertical projection a" 4" above the I.L. and the horizontal projection a^' i" below the I.L. both in the same straight line. In mechanical drawing the vertical projection a" is called the Elevation and the horizontal projection a!' the Plan. The projections of a line are found in a similar manner, by first finding Ihe projections of the two ends of the line, and joining them with a straight line. Let ab be a line in space 32'^ long, parallel to the V.P. and perpendicular to the H.P. One end is resting on the H.P. 2^" from the V.P. The points a and b will be vertically projected in the points <3^ angle. The point <3r, Fig. 109, is behind^ the V.P. Fig. 109. and below the H.P. Draw through a perpendiculars to the plane of projection. The Hor. proj. is found at a^ and the vert. proj. at d^ , Conceive again the V.P. to be revolved in the direction of the arrow until it coincides with the H. P. The hor. proj. OR THO GRA PHIC PR OJE C TION, 79 will then appear at a'' above the I.L. and the vert. proj. at a" beloiv the I.L., Fig. no. And so with the lines, the planes, and the solids. T I c T j' c r> . '^ 1 o- a- Fig. no. In order to still further explain _the use of the planes of projection, with regard to objects placed in the third angle, let us suppose a truncated pyramid surrounded by imaginary planes at right angles to each other, as shown by Fig. in. Fig. III. With a little attention it will easily be discerned that the pyramid is situated in the third dihedral angle, and that in addition to the V. and H. planes, we have passed two profile planes at right angles to the V. and H. planes, one at the right- hand and one at the left. When the pyramid is viewed orthographically through each of the surrounding planes, four separate views are had, 8o MECHANICAL BRA WIA'G. exactly as shown by the projections on the opposite planes, viz., a Front View, Elevation, or Vert. Proj. at F. ; a Right- hand View, Right-end Elevation, or Right-profile Projection at R. ; a Left-hand View, Left-end Elevation, or Left-profile Projection at L. ; a Top View, Plan or H. Proj. at P. If we now consider the V.P. and the right and left profile planes to be revolved toward the beholder until they coincide, using the front intersecting lines as axes, the projections of the pyramid will be seen as shown by Fi»g. 112, which when the / // -^^'^ p """^"""^ $x .^^^^ \ / ^^^^-^ ^^^"^ I ~"^^~^\ /""'^ / \ ~~"^\ (( ( Y ^ \ V \ \ 1 \ I \ / . L F R Fig. 112. imaginary planes and projecting lines have been removed, will be a True Drawing or Orthographic Projection of the truncated pyramid. NOTATION. In the drawings illustrating the following problems and their solutions the giveyi and required lines are shown wide and black. Hiddcii lines are shown broken into short dashes a little narrower than the visible lines. Construction ox projection lines are drawn with very narrow full or coiitijiuous black lines. ORTHOGRAPHIC PROJECTION. 8 1 When convenient very narrow, continuous blue lines are some- times used. The Horizontal Plane is known as the H.P., the Vertical Plane as V.P. and the Profile Plane as Pf.P. A point in space is designated by a small letter or figure, their projection by the same letters or figures with small Ji or V written above for the horizontal or vertical projection re- specti\-ely. In some complicated problems where points are designated by figures their projections are named by the same figures accented. Drawings should be carefully made to the dimensions given, the scale to be determined by the instructor. The student should continually endeavor to improve in inking straight lines, curves, and joints. In solving the following problems the student should have a model of the co-ordinate planes for his own use. This can be made by taking two pieces of stiff cardboard and cutting a slot in the center of one of them large enough to pass the folded half of the other through it ; when unfolding this half a model will be had like that shown by Fig. 107 or 109. All projections shall now be made from the third, dihedral angle. Prob. I. — A point a is situated in the third dihedral angle, i'' below the H.P. and 3" behind the V.P. It is required to draw its vertical and horizontal projec- tions. Draw a straight line a!'d% Fig. 113, perpendicular to I.L. and measure off the point a" \" below I.L. and the point a!'- 3" above I.L. 82 MECHANICAL DRAWING. (f is the vertical and a!' the horizontal projection in the same straight line cCa!'. The student should demonstrate this with his model. Prob. 2. — Draw two projections of a line 3'Mong parallel to both planes, |" below the H.P. and 2" behind the V.P. As the line is parallel to both planes, both projections will be parallel to the I.L. Draw d^b'' the vert. proj. of the line i" long, Fig. 1 14, par- allel to I.L. and f" below it. Draw the hor. proj. 2" above the I.L. and parallel to it, making it the same length as the a c h / \ " b *; b K / \ ^ \ ^^ 1 a y a T a a.] a bt i" b" ^<^ i a 2." h "^^^ 1/ Fig. 113, Fig. 114. Fig. 115. Fig. 116. Fig. 117. vert. proj. by drawing lines perpendicular to I.L. from the points a" and b"" to a^' and b^\ Prob. 3. — To draw the hor. and vert, projs. of a straight line 3'' long, per. to the vert, plane. Fig. 115. As the line is per. to the vert, plane the vert. proj. will be a point below the I.L. and the hor. proj. will be parallel to the horizontal plane and per. to I.L. Prob. 4. — To draw the plan and elevation of a straight line 6" long making an angle of 41;° with the vert, plane and and par. to the hor. plane, Fig. 116. ORTHOGRAPHIC PROJECTION. Z^ The plan or hor. proj. will be above the I.L. and make an angle of 45° with it. The elevation or vert. proj. will be below and par. to I.L. Draw from the point a'' at any convenient distance from I.L. a straight line cd'U' 6" long, making an angle 45° with I.L. Draw a^b" par. to I.L= at a convenient distance below it. The length of the elevation or vert. proj. is determined by dropping perpendiculars from the end of the hor. proj. a'^b^' to the points a"b'\ PrOB. 5, Fig. 117. — To find the true length of a straight line oblique to both planes of projection and the angle it makes with these planes. a^b"" and a''b^' are the projections of a straight line oblique to V.P. and H.P. Using a" as a pivot, revolve the line a"b'" until it becomes parallel to I.L. as shown by a'"b^''. From the point ^^i" erect a per. Through the point b^' draw a line par. to I.L. cutting the per. in the point b^'. ^ The broken line a^'b^' \s the true length of the line ^<^, and the angle is the true angle which the line makes with V.P. To find the angle it makes with H.P, : Using b^' as a pivot, revolve the line ^V' until it becomes par. to I.L. as shown by b^'a^^. From the point rt:/' drop a per. Through the point a" draw a line par. to I.L. intersecting the per. at the point a^'o is the angle which the line ab makes with H.P. and the broken line a^'b" is again its true length. Prob. 6, Fig. 118. — To project a plane surface of given size, situated in the third angle and par. to the V.P. Let abed be the plane surface 3'' long X 2" wide. If wx conceive lines to be projected from the four corners of the 84 MECHANICAL DRAWING. plane surface to theV.P. and join them with straight hnes we will have its V. projection a"' b'' c"' d'' and shown by Fig. ii8. And as the plane surface is par. to the V.P. it must be per to the H.P. since the planes of projection are at right angles to each other. So the plan or H. projection will be a straight line equal in length to one of the sides of the plane surface. At a convenient distance above I.L. draw a straight line, and from the points ^"^^"^ project lines at right angles to I.L., cutting the straight line in the points a^'b} The line a!'b^' is the hor. proj. of the plane surface abed. Prob. 7, Fig. ii8. — To draw the projections of a plane surface of given dimensions when situated in the third angle perpendicular to the H.P. and making an angle with the V.P. Let the plane surface be 3" X 2'' as before and let the angle it makes with V.P. be 60°. To draw the plan : At a convenient distance above I.L. and makincr an ano-le of 60° with it, draw aJ'bl\ Fig. 1 18, 2" long. From b^' drop a per. cutting d'b"' in the point b~' and t'd'' in the point d^\ then the rectangle d'b^'d^'c' will be the vert. proj. or elevation of the plane surface abed. Prob. 8, Fig. 119. — To draw the projections of the same plane surface (i) when parallel to the H.P., (2) when making an angle of 30° with H.P. and per. to V.P., (3) when mak- ing an angle of 60° with H.P. and per. to V.P., and (4) when per. to both planes. Fig. 119 shows the projections; further explanations are unnecessary. Prob. 9, Figs. 119 and 120. — To draw the projections of ORTHCGRAPHIC PROJECTION 85 the same plane surface when making compound angles with the planes of projection. Let the plane make an angle of 30° with H.P., as in the second position of Prob. 8, Fig. 119, and in addition to that, revolve it through at angle of 30°. First, draw the plane parallel to H.P., as shown by a''c''b'Ui^\ Fig. 119, the true size of the plane. r b, hj d} Fig. 118. Fig. iig. Fig. 120. Its elevation will be the straight line a'd'" parallel to I.L. Next revolve a"b'", using a" as a pivot, through an angle of 30°, to the position a"b^\ which is its vert. proj. when making an angle of 30° with H.P. Its plan is projected in a^'bl'c''d^\ Now as the plane is still to make an angle of 30° with H.P. after it has been revolved through an angle of 30° with relation to the V.P., its hor. proj. will remain unchanged. With a piece of celluloid or tracing-paper trace the hor. proj. <^''3,Wj^, lettering the points as shown, and revolve the 86 MECHANICAL DRA WING. tracing through the angle of 30°, or, which is the same things place the tracing so that the line a!'c^' will make an angle of 60° with I.L., and with a sharp conical-pointed pencil trans- fer the four points to the drawing-paper and join them b}- straight lines, as shown by Fig. 120. And as the line d'c^' retains its position relative to H.P. after the revolution, its elevation will be found at cfc"^ Fig. 120, in a straight line drawn through d"b'\ Fig. 119, intersect- ing perpendiculars from a!'d\ Fig. 120. And the vert. proj. of the points bl'dl' will be found at h^d^, Fig. 120, in a straight line drawn through b^, Fig. 119, parallel to I.L. and intersect- ing pers. from /^jV/', join with straight lines the points d"b,''c"d^. Draw the projections of the plane when making an angle of 60° wdth H.P. and revolved through an angle of 30° with relation to V.P. Draw the projections of the plane when making an angle of 60° with the V.P. and per. to the H.P., Fig. 120. Prob. 10. — To draw the projections of a plane surface of hexagonal form in the following positions: (i) When one of its diagonals is par. to the V.P. and making an angle of 45° with the H.P. (2) When still making an angle of 45° with the H.P. the same diagonal has been revolved through an angle of 60^. Draw the hexagon i^'2''3V5''6'S Fig. 121, at any con- venient distance above I.L., making the inscribed circle = 2\" , This will be its hor. proj. and 2''4''6''i'' its vert, proj., the diagonal I ''2^' being par. to both planes of proj. With I'' as an axis revolve 6''4''2'' through an angle of 45°. Through the points 2j^4,^6/ erect pers. to the points 6/*5/'4/'3/' ^^i^ 2^ ORTHOGRAPHIC PROJECTION. 87 and join them with straight Hnes. These are the projs. in the first position. Now trace the hor. proj, i^', 2/', etc., on a piece of celluloid or tracing-paper and revolve the tracing until the diagonal i''2/' makes an angle of 60° with the I.L., Fig. 122. Next draw pers. from the 6 points of the hexag- onal plane to intersect hors. from the corresponding points of the elevation in Fig. 121, join the points of intersection with Fig. 121. Fig. 122. straight lines, and so complete the projections of the second position, Fig. 122. Prob. II, Figs. 123 and 124. — Draw the projs. of a cir- cular plane (i) when its surface is par. to the vert, plane, (2) when it makes an angle of 45° with the V.P., and (3) when still making an angle of 45° with the V.P. it has been re- volved through an angle of 60°. First position: Draw the circular plane i^', 2% 3^, 4^, etc., Fig. 123, below the I.L. with a radius = \^' and divide and figure it as shown. 88 ME CHA NIC A L DRA WING. Since the plane is par. to V.P. its hor. proj. will be a straight line i'\ 2^\ etc. For the second position revolve the said hor. proj. through the required angle 0(45° to the position cd' . . . . i^^, Fig. 123, and through each division a!' . . . , \^' in points 2^'f . . . . . . a!" draw arcs cutting . This is the hor. proj. of the plane when making an angle of 45° with the V.P. The elevation is found by dropping pers. from the points in the hor. proj. a!' . . ,\^ to intersect hor. lines drawn through the correspondingly numbered points in the eleva- FiG. 123. Fig. 124. tion and through these intersections draw the elevation or vert. proj. of the second position. For the third position make a tracing of the elevation of the second position, numbering all the points as before, and place the tracing so that the diameter 7^7" makes the required angle of 60° with the I.L. and transfer to the drawing-paper. ORTHOGRAPHIC PROJECTION. 89 The result will be the elevation of the third position shown below the I.L., Fig. 124-. Its hor. proj. is found by drawing pers. through the points i, 2, 3,4 ... to intersect hors. drawn through the corresponding points in the hor. proj. of the 2d position and through these intersections draw the plan or hor. proj. of the third position, Fig. 124. Prob. 12, Fig. 125. — Draw the projs. of a regular hexag- onal prism, 3" high and having an inscribed circle of 4f" diam. : (i) When its axis is par. to the V.P. (2^ Draw the true form of a section of the prism when cut by a plane passing through it at an angle of 30"" with its base. (3) Draw the projection of a section when cut by a plane passing through XX, Fig. 125, per. to both planes of proj. The drawing of the I.L. may now be omitted. For the plan of the first part of this prob. draw a circle with a radius = to 2-f-^'\ and circumscribe a hexagon about it, as shown by a!'-, U\ U\ etc., Fig. 125. To project the elevation, draw at a convenient distance from the plan a hor. line par. to d'd\ and 3'' below it another line par. to it. From the points a!" y'' (^'- d''- , drop pers. cutting these par. lines in the points a''b''c''-'d'% thus completing the elevation of the prism. Second condition : Draw the edge view or trace of the cutting plane i'^' , making an angle of 30" with the base of the prism, locating the lower end 4' one-half inch above the base; parallel to i'^' , and at a convenient distance from it draw a straight line 1,4; at a distance of 2^^" on each side of 1,4 draw lines 3, 2 and 5, 6 parallel to 1,4, and through the points i'2'3'4' let fall pers. cutting these three par. lines in the points i, 2, 3, 4, 5, 6; join these points by straight lines 90 MECHANICAL DRA WING, as shown, and a true drawing of the section of the prism as required will result. For the third condition of the problem : Let XX be the edge view of the cutting plane and con- ceive that part of the prism to the right of XX to be removed. Fig. 125. Fig. 126. From the hor. proj. of the prism draw a right-hand elevation or profile proj., and through the points XX draw the lines en- closing the section, and hatch-line it as shown. Prob. 13.— To draw the development of the lower part of the prism in the elevation of the last problem. ORTHOGRAPHIC PROJECTION. 9 1 To the right of the elevation in Fig. 125, prolong the base line indefinitely and lay off upon it the distances ab, be, cd, etc., Fig. 126, each equal in length to a side of the hex. At these points erect pers., and through the points i'2''^'^' draw hor. lines intersecting the pers. in 4, 3, 2, i, etc. At be draw the hex. a''b^'b'\e''e^\d'' of the last prob. for the base, and at I, 2 draw the section i, 2, 3, 4, 5, 6 for the top. Prob. 14, Fig. 127. — To draw the projs. of a right cylin- der 3" diam. and 3'' long, (i) When its axis is per. to the H.P. (2) Draw the true form of a section of the cylinder, when cut by a plane per. to the V.P. making an angle of 30° with the H.P. (3) Draw a development of the upper part of the cyl. For the plan of the first condition, describe the circle i\ 2' , etc., with a radius = i\" and from it project the eleva- tion, which will be a square of 3'^ sides. For the second condition: Let i, 7 be the trace of the cutting plane, making the point 7, ^' from the top of the cyl. Divide the circle into 12 equal parts and let fall pers. through these divisions to the line of section, cutting it in the points I, 2, 3,4, etc. Parallel to the line of section I, 7 draw Vj'^ at a convenient distance from it, and through the points I, 2, 3, 4, etc., draw pers. to 1,7, intersecting and extending beyond i" f . Lay off on these pers. the distances 6"%" = 6'8', and 5^9" = S'q', etc., and through the points 2", i'\ 4", etc., describe the ellipse. For the development: In line with the top of the eleva- tion draw the line ^'^'' equal in length to the circumference of the circle, and divide it into 12 equal parts a\ b' , etc., a', b" , etc. Through these points drop pers. and through the points 92 MECHANICAL DRA WING. I, 2, 3, etc., draw hors. intersecting the pers. in the points I, 2, 3, etc., and through these points draw a curve. Tangent to any point on the straight line draw a 3'' circle for the top of the cyl. and tangent to any suitable point on the curve transfer a tracing of the ellipse. Prob. 15, Fig. 128. — Draw the projections of a right cone ']" high, with a base 6" in diam., pierced by aright cyl. 2" in g' f e Fig. 127. diam. and ^" long their axes intersecting at right angles 3" above the base of the cone and par. to V.P. Draw first the plan of the cone with a radius = Zk" • At a convenient distance below the plan draw the elevation to the dimensions required. 3'' above the base of the cone draw the center line of the cyl. CD, and about it construct the elevation of the cyl., which will appear as a rectangle 2" wide and 2^" each side of the axis of the cone. The half only appears in the figure. ORTHOGRAPHIC FROJECTIOX 93 To project the curves of intersection between the cyl. and cone in the plan and elevation: Draw to the right of the cyl. on the same center line a semicircle with a radius equal that of the cyl. Divide the semicircle into any number of parts, Fig. 128. Fig. 129. as I, 2, 3, 4, etc. Through i, i draw the per. A" \" equal in length to the height of the cone, and through A" draw the line y^ ''4" tangent to the semicircle at the point 4, and through the other divisions of the semicircle draw lines from A" to the line \" ^\ meeting it in the points '^"2" . From all points on the line i"4", viz., \"2"i"d^' , erect 94 ME CHA NIC A L DRA WING. pers. to the center line of the plan, cutting it in the points i/'2/'3/'4,'', and with i/' as the center draw the arcs 2/^-2, 3/'-3, 4/^-4 above the center line of the plan, and through the points 2, 3, 4 draw hors. to intersect the circle of the plan in the points 2'3'4', and lay off the same distances on the other side of the center line of the plan in same order, viz., 2'3'4'. Through each of these points on the circumference of the circle of the plan draw radii to its center A\ and through the same points also in the plan let fall pers. to the base of the elevation of the cone, cutting it in the points 2'3'4' ; and from the apex A of the elevation of the cone draw lines to the points 2^34' on the base. Hor. lines drawn through the points of division 2, 3, 4 on the semicircle will intersect the elements A~2\ A-j\ A— 4 of the cone in the points 2'3'4'; these will be points in the elevation of the curve of intersection between the cylinder and the cone. The plan of the curve is found by erecting pers. through the points in the elevation of the curve to intersect the radial lines of the plan in correspondingly figured points, through which trace the curve as shown. Repeat for the other half of the curve. Prob. 16, Fig. 129. — To draw the development of the half cone, showing the hole penetrated by the cyl. With center 4/', Fig. 129, and element ^i' of the cone, Fig. 128, as radius, describe an arc equal in length to the semi- circle of the base of the cone. Bisect it in the line 4/^1, and on each side of the point i lay off the distances 2, 3, 4, equal to the divisions of the arc in the plan Fig. 128, and from these points draw lines to 4", the center of the arc. Then with radii A-a, d, c, d, e, respectively, on the elevation Fig. 128, OK THOGRA PHIC PR OJE CTION. 95 and center 4," draw arcs intersecting the lines drawn from the arc XX ^.o its center 4/'. Through the points of intersection draw the curve as shown by Fig. 129. Prob. 17, Fig. 130. — To draw the development of the half of a truncated cone, given the plan and elevation of the cone. Fig. 130. Divide the semicircle of the plan into any number of parts, then with A as center and A i as radius, draw an arc and lay off upon it from the point I the divisions of the semicircle from I to 9, draw 9^. Then with' center yi and radius y^^ draw the arc BC. iBCg is the development of the half of the cone approximately. 96 MECHAXICAL DRA WIXG. Prob. 1 8, Fig. 131. — To draw the cun^e of intersection of a small cyl. with a larger. To the left of the center-line of Fig. 131 is a half cross-section, and to the right a half eleva- tion of the two cyls. Draw the half plan of the small cyl., which will be a semicircle, and divide it into any convenient number of parts, say 12. From each of these div^isions drop pers. On the half cross-section these pers. intersect the circum- ference of the large cyl. in the points i', 2', etc. Through Fig. 134. Fig. 133. 5 4- 3 t 1 .1 I i s i- 5 6 Fig. 132. these points draw hors. to intersect in corresponding points the pers. on the half elevation. Through the latter points draw the curve of intersection C. Prob. 19. — To draw the development of the smaller cyl. of the last prob. Draw a rectangle, Fig. 132, with sides equal to the circum- ORTHOGRAPHIC PROJECTION. 97 ference and length of the cyl. respectively, and divide it into 24 equal parts. ]\Iake AB, i i', 3 3', etc., Fig. 132, equal to AB, I'l" , - 2", 2)'z'\ etc., Fig. 131, and draw the developed curve of intersection. PrOB. 20. — To draw the orthographic projections of a cylindrical dome riveted to a cylindrical boiler of given dimensions. Let the dimensions of the dome and boiler be : dome 26h" diam. X ^j" high, boiler 54'' diam., plates i-" thick. Appl}- to the solution of this problem the principles ex- plained in Prob. No. 18, Fig. 131. When your drawings are completed, compare them with Figs. 133 and 134, which are the projections required in the problem. Letter or number the drawing and be prepared to explain how the different projections were found. Prob. 21. — To draw the development of the top gusset- sheets of a locomotive wagon-top boiler of given dimensions. First draw the longitudinal cross-section of the boiler to the dimensions given by Fig. 135, using the scale of i'^ = I ft. Then at any convenient point on your paper draw a straight line, and upon it lay off a distance AB 35-2'' long = the straight part of the top of the gusset-sheet G, Fig. 135. With center A and a radius = 2"/^" (the largest radius of the gusset) 4" 6" (the distance from the center of the boiler to the center of the gusset C, Fig. 135) = 33I", draw arc i. With center i5 and a radius = 26f (the smallest radius of the gusset) draw arc 2. Tangent to these arcs draw the 98 MECHANICAL DKA WING. Straight line i, 2 extended, and through the points A and draw lines i, A and 2, B per. to i, 2. Take a point on the per. i, 2, 6" from the point i as a center and through the point A draw an arc with a radius = 27i . ORTHOGRAPHIC PROJECTION, 99 vVith point 2 as a center and 2B as a radius {26^") draw an arc through B to meet the line 1,2. Divide both arcs into any number of parts, say 12, and through these divisions draw Hnes per. to and intersecting \A and 2B respectively. Through these intersections draw in- definite hors. and on these hors. step off the length of the arcs, with a distance = one of the 12 divisions as follows: On the first hors. lay off the length of the arc A\' and B\' = ^i and B\ respectively. Then from i' lay off the same distance to 2' on the second hors. etc. Through these points draw curves Ai^' and B\2' . Join points 12' and 13' with a straight line. Then AB12, 13 will be the developed half of the straight part of the gusset. On the two ends or front and back of the gusset we have now to add i'' for clearance + 3!" for lap -|- -J'^ allowance for truing up the plates, total = 5^''. And to the sides 2-|^' for lap -\- i" allowance for truing up, total = 3-g-''. The outline of the developed sheet may now be drawn to include these dimensions with as little waste as possible, as shown by Fig. 136. Extreme accuracy is necessary in mak- ing this drawing, as the final dimensions must be found by measurement. Prob. 22. — To draw the projections of a V-threaded screw and its nut of 3'' diam. and f pitch. Begin by drawing the center line C, Fig. 137, and lay off on each side of it the radius of the screw ii'\ Draw AB and 6B. Draw A6 the bottom of the screw, and on AB step off the pitch = ^" , beginning at the point A. On line 6D from the point 6 lay off a distance = half the pitch = f , because when the point of the thread has com- lOO MECHANICAL DRA WING. pleted half a revolution it will have risen perpendicularh^ a distance = half the pitch, viz., f". Then from the point 6" on 6Z> step off as many pitches as may be desired. From the points of the threads just found, B V Fig. 137. Fig. 13S. draw with the 30° triangle and T-square the V of the threads intersecting at the points b . . b . . the bottom of the threads. At the point O on line A6 draw two semicircles with radii II the top and bottom of the thread respectively. Divrde these into any number of equal parts and also the pitch Pinto the same number of equal parts. Through these divisions draw hors. and pers. intersecting each other in the points as ORTHOGRAPHIC PROJECTION. lOI shown by Fig. 137, which sliows an elevation partly in section and a section of a nut to fit the screw. Through the points of intersection draw the curves of the helices shown, using No. 3 of the " Sibley College Set" of Irregular Curves. Fig. 139. Prob. 22. — To draw the proj. of a square-threaded screw 3'' diam. and \" pitch and also a section of its nut. The method of construction is the same as for the last problem, and is illustrated by Fig. 138. Prob. 22. — To draw the projections of a square double threaded screw of 3'' diam. and 2" pitch, and also a section of its nut. ^' 102 MECHANICAL DRA WING. The solution of this problem is shown by Fig. 139, and further explanation should be unnecessary. Prob. 23. — To draw the curve of intersection that is formed by a plane cutting an irregular surface of revolution. Fig. 140. Figs. 140, 141, and 142 show examples of engine con- necting rod ends where the curve / is formed by the inter- >^ ^ B D Fig. 141. section of the flat stub end with the surface of revolution of the turned part of the rod. ORTHOGRAPHIC PROJECTION. 103 The method of finding the curves of intersection are so plainly shown by the figures that a detailed explanation • is deemed unnecessary. Fig. 142. SHADE LINES AND SHADING. Shade Lines are quite generally used on engineering work- ing drawings ; they give a relieving appearance to the projec- ting parts, improve the looks of the drawing and make it easier to read, and are quickly and easily applied. The SJiading of the curved surfaces of machine parts is sometimes practiced on specially finished drawings, but on working drawings most employers will not allow shading be- cause it takes too much time, and is not essential to a quick and correct reading of a drawing, especially if a system of shade lines is used. Some of the principles of shade lines and shading are given below, with a few problems illustrating their commonest applications. The shadows which opaque objects cast on the planes of 1 04 ME CHA NI CA L DRA WING . projection or on other objects are seldom or never shown on a working drawing, and as the students in Sibley College are taught this subject in a course on Descriptive Geometry, it is omitted here. CONVENTIONS. The Source of Light is considered to be at an infinite dis- tance from the object, therefore the Rays of Light will be rep- resented by parallel lines. The Source of Light is considered to be fixed, and the Point of Sight situated in front of the object and at an infinite dis- tance from it, so that the Visual Rays are parallel to one another and per. to the plane of projection. Shade Lines divide illuminated surfaces from dark surfaces. Dark surfaces are not necessarily to be defined by those surfaces which are darkened by the shadow cast by another part of the object, but by reason of their location in relation to the rays of light. It is the general practice to shade-line the different pro- jections of an object as if each projection was in the same plane, e.g., suppose a cube. Fig. 143, situated in space in the third angle, the point of sight in front of it, and the direction of the rays of light coinciding with the diagonal of the cube, as shown by Fig. 144. Then the edges a'^b'^ b^'c" will be shade lines, because they are the edges which separate the illumin- ated faces (the faces upon which fall the rays of light) from the shaded faces, as shown by Fig. 144. Now the source of light being fixed, let the point of sight remain in the same position, and conceive the object to be re- volved through the angle of 90° about a hor. axis so that a OJ^ THOG RA PHIC PR OJE C TJ ON. 105 plan at the top of the object is shown above the elevation, and as the projected rays of light falling in the direction of the diagonal of a cube make angles of 45° with thehor. , then with the use of the 45° triangle we can easily determine that the lower and right-hand edges of the plan as well as of the ele- vation should be shade lines. This practice then will be followed in this work, viz. : Shade lines shall be applied to all projections of an object. a b Fig. 143. ^-1 '^^. A-- / \ \ Fig. 144. considering the rays of light to fall upon each of them, from the same direction. Shade lines should have a width equal to 3 times that of the other outlines. Broken lines should never be shade lines. The outlines of surfaces of revolution should not be shade lines. The shade-lined figures which follow will assist in il- lustrating the above principles; they should be studied until understood. I06 MECHANICAL DRAWING. SHADING. The sJiade of an object is that part of the surface from which light is excluded by the object. The cine of sJiade is the line separating the shaded from the illuminated part of an object, and is found where the rays of light are tangent to the object. Brilliant Points. — " When a ray of light falls upon a sur- face which turns it from its course and gives it another direc- tion, the ray is said to be reflected. The ray as it falls upon the surface is called the incident ray, and after it leaves the surface the reflected ray. The point at which the reflection takes places is called the point of incidence. '' It is ascertained by experiment — '^ (ci) That the plane of the incident and reflected rays is always normal to the surface at the point of incidence ; " {U) That at the point of incidence the incident and re- flected rays make equal angles with the tangent plane or normal line to the surface. " If therefore we suppose a single luminous point and the light emanating from it to fall upon any surface and to be re- flected to the eye, the point at which the reflection takes place is called the brilliant point. The brilliant point of a surface is, then, the point at which a ray of light and a line drawn to the eye make equal angles with the tangent plane or normal line — the plane of the two lines being normal to the surface." — Davies : Shades and SJiadows. Considering the rays of light to be parallel and the point of sight at an infinite distance, the brilliant point on the sur- face of a sphere is found as follows: Let A^O and A^'0\ Fig. ORl 'HO G RA PHIC PR OJE C TION. 107 145, be a ray of light and A'"A^' a visual ray. Bisect the angles contained between the ray of light and the visual ray as fol- lows : Revolve A'O about the axis A" until it becomes parallel to the hor. plane at A'"C^ . At C^ erect a per. to intersect a hor. through C^ at 6^/' join Cl'Ll' {L may be any convenient Fig. 145. point on the line of vision), bisect the angle ZM'^^'^'^ with the line A^'iy\ Join C'V and through the point D^\ draw a hor. cutting C^D- at Dl\ then A^'D^' is the hor. projection of the bisecting line. A plane drawn per. to this bisecting line and tangent to 'the sphere touches the surface at the points R'^Bl'' where the bisecting lines pierce it. Therefore B"B'' are the two projections of the brilliant point. io8 ME CHA NICA L DRA WING. TJie point of shade can be found as follows: Draw A^'G, Fig, 145, making an angle of 45° with a hor. Join the points E and i^ with a straight line EF. Lay off on A''G a distance equal to EF, and join EG. Parallel to EG Fig. 146. Fig. 147. Fig. 148. draw a tangent to the sphere at the point T. Through T draw TP' per. to A^'G. From the point P^ drop a per. to P\ P" is the point of shade. Prob. 24. — To shade the elevation of a sphere with graded arcs of circles. OR THOGRA PHIC PROJECTION 109 First find the brilliant point and the point of shade, and divide the radius I, 2 into a suitable number of equal parts, and draw arcs of circles as shown by Fig. 146, grading them by moving the center a short distance on each side of the center of the sphere on the line B^-2 and varying the length of the radii to obtain a grade of line that will give a proper shade to the sphere. It is desirable to use a horn center to protect the center of the figure. Pig. 149 shows the stippling method of shading the sphere. Fig. 149. Fig. 150. PrOB. 25,— To shade a right cylinder with graded right lines. Find the line of light B'"' by the same method used to find the brilliant point on the sphere, except that the line of light is projected from the point B^' where the bisection line A^D cuts the circle of the cylinder. The line of shade is found where a plane of rays is tan- gent to the cyl. at S' and S'-. Fig. 150 shows how the shading lines are graded from the line of shade to the line of light. It will be noticed that the lines grow a little narrower to the right of the line of shade on Fig. 150; this shows where no MECHANICAL DRAWING, the reflection of the rays of light partly illumine the outline of the cylinder. Prob. 26, Fig. 148. — To shade a right cone with graded right lines tapering toward the apex of the cone. Find the elements of light and shade as shown by Fig. 148, and draw the shading-lines as shown by Fig. 151, grading their width toward the light and tapering them toward the apex of the cone. Fig. 151. Fig. 152. The mixed appearance of the lines near the apex of the cone on Fig. 151 can usually be avoided by letting each line dry before drawing another through it, or as some draftsmen do, stop the lines just before they touch. Prob. 27. — To shade the concave surface of a section of a hollow cylinder. Find the element of light and grade the shading lines from it to both edges as shown by Fig. 152. Fig. 153 shows a conventional method of shading a hex- agonal nut. Prob. 28. — To draw a front and end elevation of a rect- angular hollow box with a rectangular block on each face, each OR THO GRA PHJC PR OJE C TION. Ill block to have a rectangular opening, and all to be properly shade-lined and drawn to the dimensions given on Fig. 154. Fig. 153- Draw the hor. center line first, and then the vertical center line of the end view. About these center lines on the end el- FiG. 154. Fig. 155. evation construct the squares shown and erect the edges of the blocks. Next draw the hidden lines indicating the thickness I 1 2 MECHA NIC A L DRA WING. of the walls of the box and the openings through the blocks, measuring the sizes carefully to the given dimensions. Draw the front elevation by projecting lines from the va- rious points on the end elevation, and assuming the position of the line AB measure off the lengths of the hor. lines and erect their vert, boundaries as shown by the figure. Prob. 29. — Given the end elevation of the last prob., cut by three planes A^ B and C, Fig. 155. Draw the projections of these sections when the part to the left of the cutting plane has been removed, and what remains is viewed in the direction of the arrow, remembering that all the visual rays are parallel. These drawings and all that may follow are to be properly shade-lined in accordance with the principles given above. ISOMETRICAL DRAWING. In orthographic projection it is necessary to a correct understanding of an object to have at least two views, a front and end elevation, or an elevation and plan, and sometimes even three views are required. Isometric drawing on the other hand shows an object com- pletely with only one view. It is a very convenient system for the workshop. Davidson in his Projection calls it the " Perspective of the Workshop." It is more useful than per- spective for a working drawing, because, as its name implies (" equal measures ") it can be made to any scale and measured like an orthographic drawing. It is, however, mainly em- ployed to represent small objects, or large objects drawn to a small scale, whose main lines are at right angles to each other. The principles of isometrical drawing are founded on a cube resting on its lower front corner, i. Fig. 156, and its base OK THO G RA PHIC PR OJE C TION. 113 elevated so that its diagonal AB is parallel to the horizontal plane. Then if the cube is rotated on the corner i until the diagonal AB is at right angles to the vert, plane, i.e., through an angle of 90°, the front elevation will appear as shown at i, 2, 3, 4, Fig. 156, a regular hexagon. Now we know that in a regular hexagon, as shown by Fig. 156, the lines \A, ^3, etc., are all equal, and are easily drawn Fig. 156. with the 30° X 60° triangle. But although these lines and faces appear to be equal, yet, being inclined to the plane of projection, they are shorter than they would actually be on the cube itself. However, since they all bear the same pro- portion to the original sizes, they can all be measured with the same scale. We will now describe the method of making an isomet- rical scale. Draw the half of a square with sides = 2^" , Fig. 157. These two sides will make the angle of 45° with the horizontal. Now the sides of the corresponding isometrical square, we have seen, make the angle of 30° with the horizontal, so we will 114 ME CHANICAL D RA WING. draw 14, 3 4, making angles of 30° with i, 3. The differ- ence then between the angle 2, i, 3 and the angle 4, i, 3 is 15°, and the proportion of the isometrical projection to the actual object is as the length of the line 3, 2 to the line 3, 4. And if the line 3, 2 be divided into any number of equal parts, and lines be drawn through these divisions par. to 2, 4 to cut the line 3, 4 in corresponding divisions, these will divide 3, 4 proportionately to 3, 2. Now if the divisions on 3, 2 be taken to represent feet and those on 3, 4 to represent 2 feet, then 3, 4 would be an isometrical scale of \. Fig. 157. Since isometrical drawings may be made to any scale, we may make the isometrical lines of the object = their true size. This is a common practice and precludes the need of a special isometrical scale. The Direction of the Rays of Light. — The projection of a ray of light in isometrical drawing will make the angle of 30° with the horizontal as shown by the line 3, 2 on the front elevation of the hex.. Fig. 156. And the sJiade lines will be applied as in ordinary projection. Prob. 30. — To make the isometrical drawing of a two- armed cross standing on a square pedestal. OR THO GRA PHIC PR OJE C TIOiV. 115 Begin by drawing a center line AB, Fig. 158, and from the point A draw AC and AD, making an angle of 30° with the horizontal. Measure from A on the center line AB a dis- tance - -{\" , and draw lines par. to AC, AB; make AC and AD 2^'^ long and erect a perpendicular at D and C, complet- ing the two front sides of the base, etc. Fig. 158. Prob. 31. — To make the isometrical drawing of a hollow cube, with square block on each face and a square hole through each block, to dimensions given on Fig. 159. As before, first draw a center line, and make an isometrical drawing of a 2^'' cube, and upon each face of it build the blocks with the square holes in them, exactly as shown in Fig. 159. Prob. 32. — To project an isometrical circle. The circle is enclosed in a square, as shown by Fig. 160. Ii6 ME CHA Nl CA L DRA WING. Draw the circle with a radius =-2" and describe the square I, 2, 3, 4 about it. Draw the diagonals i, 2, 3, 4 and the diameters 5, 6, 7, 8 at right angles to each other. Now from the points i and 2 draw lines \A, \B and 2 A, 2B, making angles of 30° with the hor. diagonal 1,2. And Fig. 159 through the center O draw CD and EF at right angles to the isometrical square. The points CD, EF, and GH will be points in the curve of the projected isometrical circle, which will be an ellipse. The ellipse may be drawn sufficiently accurate as follows : With center B and radius BC describe the arc CF 3.nd ex- tend it a little beyond the points C and F, and with center A and same rad. describe a similar arc, then with a rad. which OR THO GRA PHI C PR OJECTION. 117 Fig. 160. Fig. 161. Fig. 164. Fig. 165. Il8 MECHANICAL DRAWING. may readily be found by trial, draw arcs through the points G and H and tangent to the two arcs already described. Figs. i6i, 162, 163, 164, 165, and 166 are for practice in the application of the preceding principles, and at least one of these should be drawn, or it would be better still if the stu- FlG. 166. dent would attempt to make an isometrical projection of his instrument-box, desk, or any familiar object at hand. These figures may be measured with the \\" scale and drawn with the 2" scale. WORKING DRAWINGS. Working drawings are sometimes made on brown detail- paper in pencil, traced on tracing-paper or cloth, and then blue printed. The latter process is accomplished as follows : The tracing is placed face down on the glass in the print- ing-frame, and the prepared paper is placed behind it, with the sensitized surface in contact with the back of the tracing. In printing from a negative the sensitized surface of the prepared paper is placed in contact with the film side of the negative, and the face is exposed to the light. The blue-print system for working drawings has many drawbacks, e.g., the sectioned parts of the drawing require to ORTHOGRAPHIC PROJECTION. II9 be hatch-lined, using the standard conventions already re ferred to for the different materials. This takes a great deal of time. The print has usually to be mounted on cardboard, although this is not always done, and unless it is varnished the frequent handling with dirty, oily fingers soon makes it unfit for use. Changes can be made on the prints with soda-water, it is true, but they seldom look well, and when many changes or additions require to be made it is best to make them on the tracing and take a new print. And the sunlight is not always favorable to quick printing. So taking everything into con- sideration the system of making working drawings directly on cards and varnishing them is probably the best. It is the system used by the Schenectady Locomotive Works and many other large engineering establishments. In size the cards are made 9" X 12", 12" X 18'', 18" X 24''; they are made of thick pasteboard mounted with Irish linen record- paper. The drawings are pencilled and inked on these cards in the usual way, and the sections are tinted with the conven- tional colors, which are much quicker applied than hatch- lines. The face of the drawing is protected with two coats of white shellac varnish, while the back of the card is usually given a coat of orange shellac. The white varnish can easily be removed with a little alcohol, and changes made on the drawing, and when revar- nished it is again ready for the shop. We wall now^ try to describe what a working drawing is and what it is for. In the hands of an experienced workman a working drawing is intended to convey to him all the neces- sary information as to shape, size, material, and finish to I20 MECHANICAL DRA WING. ORTHOGRAPHIC PROJECTION 121 Z^,.. 122 MECHANICAL DRAWING. enable him to properly construct it without any additional in- structions. This means that it must have a sufficient num- ber of elevations, sections, and plans to thoroughly explain and describe the object in every particular. And these views should be completely and conveniently dimensioned. The dimensions on the drawing must of course give the sizes to which the object is to be made, without reference to the scale to which it may be drawn. The title of a working drawing should be as brief as possible, and not very large — a neat, plain, free-hand printed letter is best for this purpose. Finished parts are usually indicated by the letter " f," and if it is all to be finished, then below the title it is customary to write or print " finished all over." The number of the drawing may be placed at the upper left-hand corner, and the initials of the draftsman immedi- ately below it. A second-year course, entitled Mechanical Drawing and Elementary Machine Design, is in preparation, and will shortly be published. Figs. 167 and 168 show working drawings of two shaft- couplings, fully figured, sectioned, and shade-lined. INDEX. A PAGE Angle, To bisect an 19 Angle, To construct an 15 Anti-friction curve, " Schiele's " 50 Arched window-opening, To draw an 53 Arkansas oil-stones 5 B Baluster, To draw a 53 Board, Drawing i Bow instruments 2 Brass, Sheet of 6 Breaks, Conventional 61 Brilliant points 106 C Celluloid, Sheet of thin 5 Center lines 60 Cinquefoil ornament. To draw the 53 Circle, Arc of a, To find the center of an , 32 Circle, Arc of a. To draw a line tangent to an 33 Circle, To draw a right line equal to half the circumference of a 31 Circle, To draw a tangent between two 33 Circle, To draw tangents to two 34 Circle, To draw an arc of a, tangent to two straight lines 34 Circle, To inscribe a, within a triangle 35 Circle, To draw an arc of a, tangent to two circles 36 Circle, To draw an arc of a, tangent to a straight line and a circle 37 Circle, To construct the involute of a 45 Circle, To find the length of an arc of a, approximately 47 123 124 INDEX. PACK Cissoid, To draw the 4g Compass 2 Conventions 56 Conventions, Shading 104 Conventional breaks 61 Conventional lines 60 Conventional screw-threads 62 Cross-sections 62 Curves, Irregular 3 Cycloid, To describe the 46 D Dark surfaces 104 Development of the surfaces of a hexagonal prism go Development of the surface of a right cylinder 92 Development of the surface of a cone 93 Development of the surface of a cylindrical dome 96 Development of a locomotive gusset sheet 97 Dihedral angles 75 Direction, The, of the rays of light 105 Dividers, Hair-spring 2 Drawing-board i Drawing-pen ... 2 Drawing to scale 12, 54 E Ellipse, To describe an - , 38 Ellipse, Given an, to find the axes and foci 43 Epicycloid, To describe the 48 Epicycloid, To describe an interior , . . . . 50 Equilateral triangle, To construct an 24 Examples of working drawings 120 F Figuring and lettering 66 Finished parts of working drawings. 122 G Geometrical drawing , 16 Glass-paper pencil sharpener 4 Gothic letters > 69 INDEX. 125 H PAGE Heptagon, To construct a 28 Hyperbola, To draw an 42 Hypocycloid, To describe the 48 I Ink eraser 4 Inks 4 Instruments 2 Intersection, The, of a cylinder with a cone 93 Intersection, The, of two cylinders g6 Intersection, The, of a plane with an irregular surface of revolution 102 Involute, of a circle, To construct the , 45 Isometrical cube 113 Isometrical drawing 112 Isometrical drawing, Direction of the rays of light in. . . 114 Isometrical drawing of a two-armed cross 115 Isometrical drawing of a hollow cube 116 Isometrical drawing, Examples of 117 Isometrical scale, The 114 L Leads for compass 13 Lettering and figuring. 64 Line of shade 106 Line, To draw a, parallel to another ig Line, To divide a 21 Line of motion 60 Line of section 60 M Mechanical drawing and elementary machine design 122 Model of the co-ordinate planes 81 Moulding, The "Scotia " 51 Moulding, The " Cyma Recta" 51 Moulding, The "Cavetto " or " Hollow " 51 Moulding, The " Echinus, " " Quatrefoil," or " Ovolo " 52 Moulding, The " Apophygee " 52 Moulding, The " Cyma Reversa " 52 Moulding, The "Torus" 52 126 INDEX. N PAGE Needles 6 Notation 80 O Octagon, To construct an 28 Orthographic projection, . , . 74 Oval, To construct an 43 P Paper , 2 Parabola, To construct a /. 41 Pencil 2 Pencil eraser 4 Pencil, To sharpen the 8 Pen, Drawing 9 Pen, To sharpen the drawing 10 Pentagon, To construct a 28 Perpendicular, To erect a 17 Planes of projection. The 75 Polygon, To construct a 26 Projection, The, of straight lines 82 Projection, The, of plane surfaces 84 Projection, The, of solids • go Projection, The, of the cone 93 Projection of the helix as applied to screw-threads 99 Proportional, To find a mean, to two given lines 31 Proportional, To find a thirds to two given lines 31 Proportional, To find a fourth, to three given lines 32 Protractor , 6 Q Quatrefoil, To draw the 53 R Rays of light 104 Rays, Visual 104 Rhomboid, To construct the 21 Right angle. To trisect a 24 Roman letters 67 INDEX. 127 S PAGE Scale guard 6 Scale, Drawing to 12, 54 Scale, To construct a 55 Schiele's curve, To draw 50 Screw-threads, Conventional 62 Screw-threads, Regular 100 Section lines , 56 Section lines, Standard 58 Shade lines and shading 103 Shade, To, the elevation of a sphere 108 Shade, To, a right cylinder 109 Shade, To, a right cone no Shade, To, a concave cylindrical surface no Sharpen pencil, To 8 Sharpen pen. To 10 Sheet brass ... 6 Sheet celluloid 6 *' Sibley College " set of irregular curves 3 *' Sibley College " set of instruments 2 Source of light 104 Spiral, To describe the 44 Sponge rubber 5 Square, To construct a 25 Stippling , 109 T Tacks 5 T-square 2 Third dihedral angle 75 Tinting brush 5 Tinting saucer 5 Title, The, of a working drawing 122 Tracing-ck)th. . . 6 Trefoil, To describe the 53 Triangles 3 Triangle, To construct a 25 Triangular scale 3 Type specimens 70 U Use of instruments 7 Use of pencil , 8 128 INDEX. PAGE Use of drawing-pen , g Use of triangles n Use of T-square 1 1 Use of drawing-board u Use of scale 12 Use of compasses 13 Use of dividers or spacers 13 Use of spring bows 14 Use of irregular curves 14 Use of protractor 14 V Visual rays 104 Volute, To describe the " Ionic" 45 W Water-colors 5 Water glass 5 Writing-pen 6 Working drawings 118 Working drawings, Method of making 119 Working drawing, What is a 119 Working drawings, Examples of 119 SHORT-TITLE CATALOGUE OF THE PUBLICATIONS OF JOHN WILEY & SONS, New York. London: CHAPMAX & HALL, Limited. ARRANGED UNDER SUBJECTS. Descriptive circulars sent on application. Books marked with an asterisk are sold at net prices only. All books are bound in cloth unless otherwise stated. AGRICULTURE. Cattle Feeding— Dairy Practice — Diseases of Animals — Gardening, Etc. Armsby's Manual of Cattle Feeding 12mo, |1 75 Downing's Fruit and Fruit Trees 8vo, 5 00 Grotenfelt's The Principles of Modern Dairy Practice. 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