Digitized by tine Internet Arciiive in 2010 witii funding from NCSU Libraries http://www.arcliive.org/details/textbookonmeclianOOinte A TEXTBOOK ON MECHANICAL AND ELECTRICAL ENGINEERING International Correspondence Schools SCRANTON, PA. GEOMETRICAL DRAWING MECHANICAL DRAWING PRACTICAL PROJECTION DEVELOPMENT OF SURFACES 14965 SCRANTON INTERNATIONAL TEXTBOOK COMPANY B-3 Copj^right, 1897, by The Colliery Engineer Company. Copj-right, iy02, by International Textbook Company, under the title of An Elementary Treatise on Mechanical Drawing. Geometrical Drawing : Copjrright, 1893. 1894, 1896, 1897, 1S9S, 1899, 1901. bv The COL- LiERV Engineer Comp.^xy. Mechanical Drawing: Cop^-right, 1893. 1S94. 1S98, by THE COLLIERY ENGINEER CoMPANV. CopjTight. 196-2. by Interx.\tional Textbook Comp.any. Entered at Stationers' Hall, London. Practical Projection : CopjTight, 1899, by THE COLLIERY ENGINEER COMPANY. Development of Surfaces : Copvright, 1899, bv THE Colliery Engineer Company. Plate, Projections— I : Copyright, 1393, 3894, 1896, 1897, 1898, bv THE COLLIERY ENGI- NEER Co.mpany. Plate, Projections— II : Copyright, 1893, 1894, 1896, 1897, 1898, by THE COLLIERY Engineer Company. Plate. Conic Sections: Copyright 1893, 1894, 1896, 1897, 1898, by THE COLUERT Engineer company. Plate, Intersections and Developments : Copvright, 1893. 1894, 1897, 1898, by THE Colliery Engineer Company. Plate, Details: Copyright. 1893, 1896, 1897, 1898, by THE COLLIERY ENGINEER COM- PANY. Plate Machine Details : Copyright, 1893, 1896, 1897, 1898, bv THE Colliery Engi- neer Company. Plate, Flange Coupling : Copyright. 1902, by INTERNATIONAL TEXTBOOK COM- P.\NY. Entered at Stationers' Hall. London. Plate, Eccentric and Brake Lever : Copyright, 1893, 1895. 1897, 1898, bv THE COL- LIERY Engineer Company. Plate. Timber Trestle : Copyright, 1902, bj- International Textbook Company. Entered at Stationers' Hall, London. Plate, Steel Columns and Connections: Cop\Tight, 1902. bv International Textbook Company. Entered at Stationers' Hall. London. Plate, Turret-Lathe Tools: Copvright, 1902. bv International TEXTBOOK Com- pany. Entered at Stationers' Hall, London. Plate, Commutator : Copyright, 1902, bv INTERNATIONAL TEXTBOOK COMPANY. Entered at Stationers' Hall, London. Plate, Shaft Hanger : Copj-right, 1893, 1895, 1897. 1898. bv The Colliery Engineer Company. Plate, Bench '\'ise : Copyright, 1893, 1895, 1897, 1898, bv THE Colliery Engineer Company. Plate. Profiles of Gear Teeth : Copyright, 1893, 1895, 1897, 1898, by The Colliery Engineer Company. Plate. Spur Gear Wheels : Copvright, 1893, 1895, 1896, 1897, 1898, by The Colliery Engineer Company. Plate. Bevel Gears : Copvright, 1893, 1895, 1897, 1898, by The Colliery Engineer Company. Plate, Brush Holder : Copvright, 1902, by INTERNATIONAL TEXTBOOK Company. Entered at Stationers' Hall. London. Plate, Compound Rest : Copvright. 1902, by INTERNATIONAL TEXTBOOK COMPANY. Entered at Stationers' Hall, London. Plate, Six Horsepower Horizontal Steam Engine : Copj-right, 1893, 1895, 1897, 1898, by The Colliery Engineer Company. Plate, Proiections— I : Copvright, 1899, bv The Colliery Engineer Company. Plate, Projections— IT : Copvrieht, 1899. bv THE COLLIERY ENGINEER COMPANY. Plate, Projections— III : Copvright. 1899. bv The COLLIERY ENGINEER COMPANY. Plate. Sections— I : Copvright, 1899. bv THE COLLIERY ENGINEER COMPANY. Plate, Sections— II : Copvright, 1899, by THE COLLIERY ENGINEER COMPANY. Plate. Intersections— I : Copvright. 1899, bv THE COLLIERY ENGINEER COMPANy, Plate. Intersections— II : Copvright. 1899. bv THE COLLIERY ENGINEER COMPANY. Plate, Developments— I : Cop\Tight. 1899. bv THE COLLIERY Engineer Company. Plate. Developments— II : Copvright. 1899. bv THE COLLIERY Engineer COMPANY. Plate, Developments— III : Copvright, 1899, bv THE Colliery Enginfkr Company. Plate, Developments— IV : Copvright, 1899, bv THE COLLIERY Engineer COMPANY. Plate, Developments— 'V^ : Copyright, 1899, bv The COLLIERY ENGINEER COMPANY. All rights reserved. . against one of the two edges, holding it fast GEOMETRICAL DRAWING § 13 \rith the left hand; then move the triangle A along the edge of B. The edge a b will be parallel to the line c d\ and when the edge a b reaches the point g, through which it is desired to draw the parallel line, hold both triangles t Fig. 6 Stationary with the left hand and draw the line efhy pass- ing the pencil along the edge a b. Should the triangle A extend too far beyond the edge of the triangle B after a number of lines have been drawn, hold A stationary with the left hand and shift B along the edge of A with the right hand and then proceed as before. 5. A line may be drawn at right angles to another line which is neither vertical nor horizontal, as illustrated in Fig. T. Let cdhQ the given line (shown at the left-hand side). Place one of the shorter edges, as a b, of the triangle B so that it will coincide with the line c d; then, keeping the triangle in this position, place the triangle A so that its long edge will come against the long edge of B. Now, holding A securely in place with the left hand, slide B along the edge of A with the right hand, when the lines // /, vi Ji, etc. may be drawn perpendicular to cd along the edge bf of the triangle B. The dotted lines show the position of the triangle B when moved along the edge of A. 6. The right-hand portion of Fig. 7 shows another method of accomplishing the same result, and illustrates §13 GEOMETRICAL DRAWING how the triangles may be used for drawing a rectangular figure, when the sides of the figure make an angle with the T square such that the latter cannot be used. Let the side cdoi the figure be given. Place the long side of the triangle B so as to coincide with the line c d, and bring the triangle A into position against the lower side of B, as shown. Now, holding the triangle A in place with the left hand, revolve B so that its other short edge will rest against the long edge oi A, as shown in the dotted position at i?'. The parallel line's ce and «?/ may now be drawn Fig. 7 through the points c and d by sliding the triangle B on the triangle A, as described in connection with Fig. 6. Meas- ure off the required width of the figure on the line c e, reverse the triangle B again to its original position, still holding the triangle ^ in a fixed position with the left hand, and slide B upon A until the long edge of .5 passes through ^.' Draw the line ef through the point e, and r/will be par- allel to cd. The student should practice with his triangles before beginning drawing. 7. The compasses, next to the T square and triangles, are used more than any other instrument. A pencil and pen point are provided, as shown in Fig. 8, either of which GEOMETRICAL DRAWING 13 may be inserted into a socket in one leg of the instrument, for the drawing of circles in pencil or ink. The other leg is fitted with a needle point, which acts as the center about which the circle is drawn. In all good instruments, the needle point itself is a separate piece of round steel wire, held in place in a socket provided at the end of the leg. The wire should have a square shoulder at its lower end, below which a fine, needle-like point projects. The length- ening bar, also shown in the figure, is used to extend the leg carrying the pen and pencil points when circles of large radii are to be drawn. The joint at the top of the compasses should hold the legs firmly in any position, and at the same time should permit their being opened or closed with one hand. The joint may be tightened or loosened by means of a screwdriver or wrench, which accompanies the compasses. It will be noticed in Fig. 8 that each leg of the compasses is jointed ; this is done so that the compass points may always be kept perpendicular to the paper when drawing circles, as in Fig. 11. The style of compasses shown in Fig. 8 have what is called a tongue joint, in which the head of one leg has a tongue, gener- ally of steel, which moves bC' tween two lugs on the other leg. Another common style of joint is the pivot joint , in which the head of each leg is shaped like a disk and the two disks are held together in a fork-shaped brace either by means of two pivot screws or by one screw penetrating both disks. Fig. 8 §13 GEOMETRICAL DRAWING The brace that forms a part of this joint is generally pro- vided with a handle, as the shape of the joint makes it rather Fig. 9 awkward to hold the compasses by the head, as is usual with instruments provided with tongue joints. In Fig. 9 is shown a common style of pivot joint. 8. The following suggestions for handling the compasses should be carefully observed by those who are beginning the subject of mechanical drawing. Any draftsman who handles his instruments awkwardly will create a bad impression, no matter how good a workman he may be. The tendency of Fig. 10 all beginners is to use both hands for operating the com- passes. This is to be avoided. The student should learn at the start to open and close them with one hand, holding them as shown in Fig. 10, with the needle-point leg resting between the thumb and fourth finger, and the other leg between the middle and forefinger. When drawing circles, GEOMETRICAL DRAWING § 13 hold the compasses lightly at the top between the thumb and forefinger, or thumb, forefinger, and middle finger, as in Fig. 11. Another case where both hands should not be used is in locating the needle point at a point on the drawing about which the circle is to be drawn, unless the left hand is used merely to steady the needle point. Hold the compasses as shown in Fig. 10, and incline them until the under side of the Fig. 11 hand rests upon the paper. This will steady the hand so that the needle point can be brought to exactly the right place on the drawing. Having placed the needle at the desired point, and with it still resting on the paper, the pen or pen- cil point may be moved out or in to any desired radius, as indicated in Fig. 10. When the lengthening bar is used, both hands must be employed. 9. The compasses must be handled in such a manner that the needle point will not dig large holes in the paper. Keep § 13 GEOMETRICAL DRAWING 9 the needle point adjusted so that it will he perpendicular to the paper, when drawing circles, and do not hear upon it. A slight pressure will be necessary on the pen or pencil point, but not on the needle point. 10. The dividers, shown in Figs. 9 and 12, are used for laying off distances upon a drawing, or for dividing straight lines or circles into parts. The points of the dividers should be very sharp, so that they will not punch holes in the paper larger than is absolutely necessary to be seen. Compasses are sometimes furnished with two steel divider points, besides the pen and pencil points, so that the instrument may be used either as compasses or dividers. This is the kind illustrated in Fig. 12. When using the Fig. 12 dividers to space a line or circle into a number of equal parts, hold them at the top between the thumb and forefinger, as when using the compasses, and step oflf the spaces, turning the instrument alternately to the right and left. If the line or circle does not space exactly, vary the distance between the divider points and try again; so continue until it is spaced equally. When spacing in this manner, great care must be exercised not to press the divider points into the paper; for, if the points enter the paper, the spacing can never be accurately done. The student should satisfy him- self of the truth of this statement by actual trial. 11. The bow-pencil and bow-i)en, shown in Fig. 13, are convenient for describing small circles. The two points of the instruments must be adjusted to the same length; otherwise, very small circles cannot be drawn. To open or close either of these instruments, support it in a vertical 10 GEOMETRICAL DRAWING 13 Fig. 13 position by resting the needle point on the paper and bear- ing slightly on the top of it with the forefinger of one hand, and turn the adjusting nut with the thumb and middle finger of the same hand. 13, Dra^^ing Paper and Pencils. — The draw- ing paper recommended for this series of lessons is T. S. Co. 's cold-pressed demy, the size of which is 15" X 20". It takes ink well and with- stands considerable erasing. The paper is secured to the drawing board by means of tliiimb tacks. Four are usually sufficient — one at each corner of the sheet (see Fig. 7). Place a piece of paper on the drawing board, and press a thumbtack through one of the corners about :^ or f of an inch from each edge. Place the T square in position for drawing a horizontal line, as before explained, and straighten the paper so that its upper edge will be parallel to the edge of the T-square blade. Pull the corner diagonally op- posite that in which the "thumbtack was placed, so as to stretch the paper slightly, and push in another thumb- tack. Do the same with the remaining two corners. For drawing in pencil, an HHHH pencil of any reputable make should be used The pencil should be sharpened as shown at A, Fig. 14. Cut the wood away so as to leave about ^ or f of an inch of the lead projecting; then sharpen it flat by rubbing it against a fine file or a piece of Fig. 14 GEOMETRICAL DRAWING 11 fine emery cloth or sandpaper that has been fastened to a flat stick. Grind it to a sharp edge like a knife blade, and round the corners very slightly, as shown in the figure. If sharpened to a round point, as shown at B, the point will wear away very quickly and make broad lines; when so sharpened it is difficult to draw a line exactly through a point. The lead for the compasses should be sharpened in the same manner as the pencil, but should have its width narrower. Be sure that the compass lead is so secured that when circles are struck in either direction, but one line ivill be draxvn with the same radius and center. 13. Inking. — For drawing ink lines other than arcs of circles, the ruling pen (or right-line pen, as it is sometimes called) is used. It should be held as nearly perpendicular to the board as possible, with the hand in the position Fig. 15 shown in Figs. 15 and 16, bearing lightly against the T square or triangle, along the edge of which the line is drawn. After a little practice, this position will become natural, and no difficulty will be experienced. 12 GEOMETRICAL DRAWING 13 1-4. The beginner will find that it is not always easy to make smooth lines. If the pen is held so that only one blade bears on the paper when drawing, the line will almost inva- riably be ragged on the edge where the blade does not bear. When held at right angles to the paper, as in Fig. 16, how- ever, both blades will rest on the paper, and if the pen is in good condition, smooth lines will result. The pen must not be pressed against the edge of the T square or triangle, as the blades will then close together, making the line uneven. The edge should serve as a guide simply. Fig. 16 In drawing circles with the compass pen. the same care should be taken to keep the blades perpendicular to the paper by means of the adjustment at the joint. In both the ruling pen and compass pen. the width of the lines can be altered by means of the screw which holds the blades together. The handles of most ruling pens can be unscrewed, and are provided with a needle point intended for use when copying maps by pricking through the original and the underlying paper, thus locating a series of points through which the outline may be drawn. § 13 GEOMETRICAL DRAWING 13 15. Di-a^wing Ink. — The ink we recommend for the work in this Course is the T. S. Co.'s superior waterproof liquid India ink. A quill is attached to the cork of every bottle of this ink, by means of which the pen may be filled. Dip the quill into the ink and then pass the end of it between the blades of the drawing pen. Do not put too much ink in the pen, not more than enough to fill it for a quarter of an inch along the blades, otherwise the ink is liable to drop. Many draftsmen prefer to use stick India ink; and for some purposes this is to be preferred to the prepared liquid ink recommended above. In case the stick ink is bought, put enough water in a shallow dish (a com- mon individual butter plate will do) to make enough ink for the drawing, then place one end of the stick in the water, and grind by giving the stick a circular motion. Do not bear hard upon the stick. Test the ink occasionally to see if it is black. Draw a fine line with the pen and hold the paper in a strong light. If it shows brown (or gray), grind a while longer, and test again. Keep grinding until a fine line shows black, which will usually take from fifteen min- utes to half an hour, depending on the quantity of water used. The ink should always be kept well covered with a flat plate of some kind, to keep out the dust and prevent evaporation. The drawing pen may be filled by dipping an ordinary writing pen into the ink and drawing it through the blades, as previously described when using the quill. If liquid ink is used, all the lines on all the drawings will be of the same color, and no time will be lost in grinding. If stick ink is used, it is poor economy to buy a cheap stick. A small stick of the best quality, costing, say, a dollar, will last as long, perhaps, as five dollars' worth of liquid ink. The only reason for using liquid ink is that all lines are then sure to be of equal blackness and time is saved in grinding, India ink will dry quickly on the drawing, which is desir- able, but it also causes trouble by drying between the blades and refusing to flow, especially when drawing fine lines. The only remedy is to wipe out the pen frequently with a cloth. Do not lay the pen down for any great length of time when 14 GEOMETRICAL DRAWING § 13 it contains ink; wipe it out first. The ink may sometimes be started by moistening the end of the finger and touching it to the point, or by drawing a slip of paper between the ends of the blade. Always keep the bottle corked. 16. To Sliai-pen tlie I>i*a^ving Pen. — "When the ruling, or compass, pen becomes badly worn, it must be sharp- ened. For this purpose a fine oilstone should be used. If an oilstone is to be purchased, a small, flat, close-grained stone should be obtained, those having a triangular section being preferable, as the narrow edge can be used on the inside of the blades in case the latter are not made to swing apart so as to permit the use of a thicker edge. The first step in sharpening is to screw the blades together, and, holding the pen perpendicular to the oilstone, to draw it back and forth over the stone, changing the slope of the pen from downwards and to the right to downwards and to the left for each movement of the pen to the right and left. The object of this is to bring the blades to exactly the same length and shape, and to round them nicely at the point. This process, of course, makes the edges even duller than before. To sharpen, separate the points by means of the screw, and rub one of the blades to and from the operator in a straight line, giving the pen a slight twisting motion at the same time, and holding it at an angle of about 15° with the face of the stone. Repeat the process for the other blade. To be in good condition, the edges should be fairly sharp and smooth, but not sharp enough to cut the paper. All the sharpening must be done on the outside of the blades. The inside of the blades should be rubbed on the stone only enough to remove any burr that may have been formed. Anything more than this will be likely to injure the pen. The whole operation must be done very carefully, bearing on lightly, as it is easy to spoil a pen in the process Examine the points frequently, and keep at work until the pen will draw hoth fine lines and smooth heavy lines. Many draftsmen prefer to send the pens to be sharoened to the §13 GEOMETRICAL DRAWING 15 dealer who sold them, and who is generally willing to do such sharpening at a trifling cost. 17. Irregular Curves. — Curves other than arcs of cir- cles are drawn with the pencil or ruling pen by means of curved or irregular-shaped rulers, called irrejyular curves (see Fig. 17). A series of points is first determined through which the curved line is to pass, 'i'he line is then drawn through these points by using such parts of the irregular curve as will pass through several of the points at once, the curve be- ing shifted from time to time as required. It is usually ditBcult to draw a smooth, continuous curve. The tendency is to make it curve out too much between the points, thus giving it a wavy appearance, or else to cause it to change its direction abruptly where the different lines join, making angles at these points. These defects may largely be avoided by always fitting the curve to at least three points, and, when moving it to a new position, by setting it so that it will coincide with part of the line already drawn. It will be found to be a great help if the line be first sketched in freehand, in pencil. It can then be penciled over neatly, or inked, without much difficulty, with the aid of the irregular curve, since the original pencil line will show the general direction in which the curve should be drawn. Whenever the given points are far apart, or fall in such positions that the irreg- ular curve cannot always be made to pass through three of them, the line must invariably be sketched in at first. As an example, let it be required to draw a curved line through the points a, b, c, d, etc.. Fig. IS. As just stated, a part of the irregular curve must be used which will pass through at least three points. With the curve set in the first position ./, its edge is found to coincide with four points Fig. 17 16 GEOMETRICAL DRAWING tf, b, c, and &rtpotrct irt th,a.t iin-e.. . — I. \ CASEH. — - j V^ 1 ,' ^x j \ ! /' I / i i / / 1 \ /' 1 ^v ' / ! ^ / 1 ^^-c' / .th.oxx,tit. \ PROBLEM' 4: Tfir-o-Lcgh. cl gtve7-i.potrLt to oLracuj IE H. 1 ex. straight lirte pocruLleL to a gFii/erLsira.igrhi line. N i \ 1 ,' ^ ; i ; /' / ' 1 ' "' ~~ j 1 jojh:^ sjniTM, class jv^ 4529. \ § 13 GEOMETRICAL DRAWING 27 and constantly falling on the drawing will stick to any spot where an erasure has been made, and it is then very difficult, if not impossible, to entirely remove it. For this reason, all construction lines that are to be removed, or that are liable to be changed, should be drawn lightly, that the finish of the paper may not be destroyed when erasing them. When it is found necessary to erase an ink blot or a line that has been inked in, only an ijik eraser or sand rubber should be used. After the erasure has been made, the roughened part of the surface of the paper can be smoothed by rubbing with some hard, smooth substance, as a piece of ivory or the handle of a knife. PliATE I 33. Take a sheet of drawing paper 15" wide and 20" long (demy size), and fasten it to the board as previously described. On this draw the outlines of the size of the plate, 14" X 18", and draw the border line all around y from the edge of the outline, leaving the space inside for the drawing 13" X 17". When the word drazving is used here- after, it refers only to the space inside the border lines and the objects drawn upon it. To understand clearly what follows, refer to Plate I. Divide the drawing into two equal parts by means of a faint horizontal line. This line is shown dotted in Plate I, above referred to. Divide each of these halves into three equal parts, as shown by the dotted lines; this divides the drawing into six rectangular spaces. These division lines are not to be inked in, but must be erased tvhen the plate is completed. On the first five plates, space for the lettering must be taken into account. For each of the six equal spaces, the lettering will take up one or two lines. The height of all capital letters on these plates will be -jV, and of the small letters f of this, or ■^". The dis- tance between any two lines of lettering will also be -jV'. The distance between the tops of the letters on the first line of lettering and the top line of the equal divisions of 28 GEOMETRICAL DRAWING § 13 the drawing is to be-J^"; and the space between the bottoms of the letters and the topmost point of the figure repre- sented on the drawing within one of these six divisions must also be not less than ^". This makes a very neat arrangement, if the figure is so placed that the outermost points of the bounding lines are equally distant from the sides of one of the equal rectangular spaces. Consequently, if there is one line of lettering, no point of the figure drawn should come nearer than ^" -\- ^" -{- ^" = 1-^' to the top line of the space within which it is represented ; or, if there are two lines of lettering, nearer than ^" + ^" + tV'+ ts"+ i"= lA"- The letter heading for each figure on the first five plates will be printed in heavy-faced type at the beginning of the directions explaining each prob- lem. The student must judge for himself by the length of the heading whether it will take up one line or two, and make due allowance for the space it takes up. This is a necessary precaution, because the lettering should never be done until the rest of the drawing is entirely finished and inked in. Problem 1. — To bisect a straigrht line= See Fig. 31; also 1 of Plate I. CoxsTRUCTiox. — Draw a straight Ime A B, 3^" long. With one extremity .rl as a center, and a radius greater than one-half of the length of /V the line, describe an arc of a circle on each side of the given line; with the other extremity ^ as a ,s center, and the same ra- dius, describe arcs inter- secting the first two in the points C and D. Join C and D by the line CD, and the point P, where it intersects A B, will be the Fig. 31 required point ; that is, A P= PB, and P\s the middle point § 13 GEOMETRICAL DRAWING 29 p Fig. 33 of A B. Since C D \s perpendicular to A B, this construc- tion also gives 2i perpendicular to a straight line at its middle point. Problem 2. — To dra>v a perpendicular to a straiglit line from a given point in that line. Note.— As there are two cases of this problem, requiring two figures on the plate, the line of letters will be run clear across both figures, as shown in Plate I. Case I. — When the point is at or near the center of the line. See Fig. 32 ; also 2, Case I, of Plate I. ^ Construction. — Draw A B 3V' long. Let P be the given point. With P as a center, and any radius, as PD., describe two short ^ arcs cutting A B in the points C and D. With C and D as centers, and any convenient radius greater than P D, describe two arcs intersecting in E. Draw PE^ and it will be perpendicular to ^ i? at the point P. Case II. — When the point is near the end of the line. See Fig. 33; also 2, Case II, of Plate I. Draw AB ZV' long. Take the given point P about f" from the end of the line. With any point (7 as a center, and a radius O P, describe an arc cutting A B in P and D. Draw D O, and prolong it until it in- tersects the arc in the point C. A line drawn through C and P will be perpen- dicular to A B at the point P. Fig. as 30 GEOMETRICAL DRAWING §13 Problem 3. — To di-a^^ a perpendicular to a straight line from a point ^vitlioiit it. As in Problem :2, there are two cases. Case I. — W/ien the point lies nearly over the center of the line. See Fig. 34; also 3, Case I, of Plate I. CoxsTRUCTiox. — Draw A B 3|" long. Let P be the given point. With /^ as a cen- ter, and any radius P D greater than the distance from P to A B, describe an arc cutting A B in C and D. With C and D as centers, and any con- venient radius, describe short arcs intersecting in E. A line drawn through P and E will be perpendicular to A Bat F. Case n. — Wlien the point lies nearly over one etid of the line. See Fig. 35; also 3, Case II, of Plate I. Draw .-i^3V' long, and % Fig. 34 let P be the given point. With any point C on the line A B as a center, and the distance C P as a radi- us, describe an arc P E D cutting J i? in £. With£ as a center, and the dis- tance E P as a radius, describe an arc cutting V / / Fig. 35 the arc PE D in D. The line joining the points P and D will be perpendicular to A B. Proble^i 4. — Througfli a given point, to di*a-sv a straight line parallel to a giAen sti-aiglit line. See Fig. 36; also 4 of Plate I. CoxsTRucTiox. — Let P be the given point, and A B the given straight line 3|^" long. With i" as a center, and any § 13 GEOMETRICAL DRAWING 31 convenient radius, describe an arc CD intersecting A B in D. With Z? as a center, and the same radius, describe the arc PE. With Z> as a center, and a radius equal to the chord of the arc P E, describe an arc intersecting C Dm C. A straight line drawn through P and C will be £^ ^l? parallel to A B. 34. These four prob- lems form Plate I. They should be carefully and a — accurately drawn in with -' Fig. 36 lead-pencil lines and then inked in. It will be noticed that on Plate I, and Figs. 31 to 36, the given lines are light, the required lines heavy, and the construction lines, which in a practical working drawing would be left out, are UgJit dotted. This system must also be followed in the four plates which are to follow. A single glance enables one to see at once the reason for drawing the figure, and the eye is directed immediately to the required line. In the first five plates, accuracy and neatness are the main things to be looked out for. The student should be certain that the lines are oi precisely the length that is specified in the description. When drawing a line through two points, be sure that the line goes through the points; if it does not pass exactly through the points, erase it and draw it over again. If a line is supposed to end at some particular point, make it end there — do not let it extend beyond or fall short. Thus, in Fig. 36, if the line PC does not pass through the points /*and C, it is not parallel to A B. By paying careful attention to these points, the student saves himself a great deal of trouble in the future. Do not hurry your zvork. First ink in all of the light lines and light dotted lines (which have the same thickness); then ink in the heavy required lines after the pen has been readjusted. Now do the lettering (first read carefully the paragraphs under the head *' Lettering "), and finally draw the heavy border lines, 35 GEOMETRICAL DRAWING § 13 which should be thicker than any other line on the drawing. The word '* Plate " and its number should be printed at the top of the sheet, outside the border lines, and midwa)- of its length, as shown. The student's name, followed b}^ the words "Class" and "No.," and after this his Course letter and class number should be printed in the lower right-hand comer below the border line, as shown. Thus, John Smith, Class No. C 4529. The date on which the drawing was completed should be placed in the lower left-hand corner, below the border line. All of this letterings is to be in capi- tals ^" high. Erase the division lines, and clean the draw- ing by rubbing ver\' gently with the eraser. Care must be exercised when doing this, or the inked lines will also be erased. It is best to use a so-called * * Sponge Rubber " for this purpose, as it will not injure the inked lines. If any part of a line Jtas been erased or weakened, it must be redrawn. Then write with the lead pencil your name and address in full on the back of 5-our drawing, after which put 5'our drawing in the empty tube which was sent you, and send it to the Schools. HEsTS FOR PLATE I 35. Do not forget to make a distinction bet'ween tlie width of tJie given and required lines^ nor forget to make t/te con- struction lines dotted. JVhen drawing dotted lines, take pains to have tlie dots and spaces uniform in length. Make the dots about ^" long and the spaces only about one-third tlie length of the dots. Try to get tlie work accurate. The constructions must be accurate, and all lines or figures sliould be drawn of tlie length or size previously stated. To this end., ivork carefully and keep tlu pencil leads I'cry sliarp, so tliat tlie lines will be fine. The lettering on tlu first few plates, as well as on tlu succeed- ing plates, is fully as important as the drawing, and should be done in tlu neatest possible manner. Drawings sent in for correction with the lettering omitted "will be returned for completion. 13 GEOMETRICAL DRAWING 33 The reference letters like A, B, C, etc., as shozvn in Fig. 31, are not to be put on the plates. Do not neglect to trim the plates to the required size. Do not punch large Iioles in the paper with the dividers or com- passes. Remember that the division lines are to be erased — )iot inked in PLATE II 36. Draw the division lines in the same manner as described for Plate I. The following five problems (5 to 9, inclusive) are to be drawn in regular order, as was done in Plate I, with problems from 1 to 4. The letter headings are given in heavy-faced type after the problem number. Problem 5. — To bisect a given angle.* Case I. — / Vhen the sides intersect within the liiftits of the drazving. See Fig. 37. Construction. — Let A B he the angle to be bisected. Draw the sides OA and OB U" long. With the vertex O as a center, and any convenient radius, describe an arc D£ inter- secting O A at D and O B at E. With D and E as centers, and a radius greater than the chord of half the arc DE, describe two arcs inter- secting at C. The line drawn through C and O will bisect the angle ; that is, A O C = C O B. Case II. — When the sides do not i)itersect zvitJiin the limits of the drawing. See Fig. 38. Construction. — Draw two lines, A B and CD., each 3i" long, and inclined towards each other as shown. With any * Since the letter heading in this problem is very short, it will be better to place it over each of the two cases separately, instead of run- ning it over the division line, as was done with the long headings of the two cases in Plate I. Put Case I and Case II under the heading, as in the previous plate. Fig. 37 M. E. I 3i GEOMETRICAL DRAWING 13 point -£ on C D a.s a. center and any convenient radius, describe arc F I G H\ with 6^ as a center and same radius, describe arc H LE F, intersecting/^/ 6^//^ in HsindF. With Z as a center and same radius, describe arc K GJ- with /as Fig. 38 a center and same radius, describe arc / E K, intersecting K GJ in K and J. Draw HE and./ K\ they intersect at O, a point on the bisecting line. With (9 as a center and the same or any convenient radius, describe an arc intersecting A B and C D xn M and A'. With M and .A' as centers and any radius greater than one-half J/ A', describe arcs inter- secting at P. A line drawn through O and P is the required bisecting line. Problem 6. — To divide a sriven stitiiarlit Hue into any required nvmiber of equal pait*. See Fig. 39 {a). f< ^' 9-' Construction. — A B is the given line ^Yt' loi^g- ^'^ is required to divide it into eight equal parts. Through one extremity A of the line, draw an indefinite straight line A C^ making any angle with A B. Set the dividers to any convenient distance, and space off eight equal divi- sions on A C, as A K, K I, IH, etc. Join C and B by the p o Fig. 39 (n) -V M 13 GEOMETRICAL DRAWING 35 \c p. E. F.- K. R O N M L Straight line C B, and through the points D, E, F, G, etc. draw lines DL, /i M, etc. parallel to CB, by using the two triangles; these parallels intersect A B in the points Z., M, N, etc. , which are equally distant apart. The spaces L M, M N, N O, etc. are each equal to^ A B. Proceed in a similar way for any number of equal parts into which A B is to be divided. Another method is shown in Fig. 39 (d). Draw A B a.s before, and erect the perpendicular B C. Now divide the length- of y^ i? by the number denoting the number of equal parts into which A B is to be divided, obtaining, in this case, 3tV' -^ 8 = tVs"- As y^ C is longer than A B, the equal divisions A K, K I, etc. are longer than A T, T R, etc. and may be made any con- venient length greater than A B ^ S. In this case, ^" is the most convenient frac- tion nearest to and greater than yVs"; hence, consider A K, K I, etc. to be each y in length, thus making the length of y^ C 8 X i" = 4". With A as a center and a radius equal to 4", describe an arc cut- ting B C in C, and draw A C. Then with a scale lay off A K= K I = etc. = ^", and project K, I, H, etc. upon A B, in T, R, P, etc., the required points. The advantage of this method over the other is that the T square and triangle can be used throughout, thus making it very much easier to draw the parallels D L, EM, etc. The student, when drawing this plate, is at liberty to use either of the two methods given in this problem. Problej[ 7. — To clraAV a straijyht line throxijyli any given point on a f?iven straight line to make any required angle with that line. Construction. — In Fig. 40, A B \^ the given line 3^' long, Pis the given point, and E O F \s the given angle. With the vertex O as a center, and any convenient radius, describe Fig. 39 (3) 36 GEOMETRICAL DRAWING Id Fig. 40 an arc EF cutting O E and O F m E and F. With Pas a center, and the same radius, describe an arc CD. With D as a center, and a radius equal to the chord of the arc E F, describe an arc cutting CD in C. A line drawn through the points P and C will make an angle with A B equal to the angle O, or C PD = E OF. Problem 8. — To di*aTr an equl- latei*al triangle, one side being given. Construction. — In Fig. 41, A B is the given side 'IV' long. With A B as a radius, and A and B as centers, describe two arcs intersecting in C Draw C A and C B, and C A Z? is an ^ equilateral triangle. Problem 9. — The altitude of an eqiiilatei-al triangle being given, to di-aw tlie triangle. Construction. — In Fig. 42, A B is the altitude 2^" long. Through the extremities of A B draw the parallel lines CD and E F perpendicular to A B. With B as a center, and any convenient radius, de- scribe the semicircle C H K D intersecting CD in C and D. With C and D as centers, and the same radius, describe arcs cutting the semicircle Fi^' -12 in H and K. Draw B H and B K, and prolong them to meet E F \n E and F. B E F is the required equilateral triangle. This problem finishes Plate II. The directions for inking in, lettering, etc. are the same as for Plate I. Fig. 41 c B ,I> I / \ \ J \ vV \ / -:/ \. §13 GEOMETRICAL DRAWING 37 PLATE III 37. This plate is to be divided up like Plates I and II, and the six following prob- lems are to be drawn in a similar manner: Problem 10. — T^vo sides and the included tingle of a triangle being given, to construct the triangle. Construction. — In Fig. 43, make the given sides MNU" long and PQ 1^" long. Let O be the given angle. Draw A B, and make it equal in length to PQ. Make the angle C B A equal to the given angle O, and make C B equal in length to the line AI N'. Draw C A, and C A B is the required triangle. Fig. 43 Problem 11. — To draAv a i>arallelograni ^vhen the sides and one of the angles are given. Construction. — In Fig. 44, make the given sides M N %\" long and PQ 1^'' long. Let O be the given angle. Draw A B equal to M N, and draw B C\ making an angle with A B equal to the given angle O. Make B C equal to PQ. With (T as a center, and a radius equal to M N., describe an arc at D. With yi as a center, and a radius equal to P Q, describe an arc inter- Draw A D and CD, and Fig. 44 secting the other arc in D A BCD is the required parallelogram. 38 GEOMETRICAL DRAWING S 13 K- FIG. 45 Problesi 1"2. — An arc and its radius being given, to find tlie center. CoxsTRUCTiox. — In Fig. 45, A C D B is the arc, and J/.V, If" long, is the radius. With MX as a radius, and any point C in the given arc as a center, describe an arc at O. With any other point D in the given arc as a center, and the same radius, describe an arc intersecting the first in O. O is the required center. Problem 13. — To pass a circumference tlirougli any tlxi'ee points not in tlie same straiglit line. Construction. — In Fig. 46, A, B, and C are the given points. With A and B as centers, and any convenient radius, describe arcs in- tersecting each other in A'and /. With B and C as centers, and any convenient radius, describe arcs intersecting each other in D and E. Through / and K and through D and E, draw lines inter- secting at O. With O as a center, and O A as a radius, describe a circle; it will pass through A , B', and C. fig. 46 Problem 14. — To inscribe a square in a given circle. CoNSTRUCTiox. — In Fig. 47, the circle A BCD is dl" in diameter. Draw two diam- eters, A C and DB, at right angles to each other. Draw the lines A B^ B C, CD, and DA, joining the points of in- tersection of these diameters with the circumference of the circle, and the}- will be the sides of the square. 13 GEOMETRICAL DRAWING 39 Problem 15 — To inscribe a regular hexagon in a given circle. Construction. — In Fig. 48, from (9 as a center, with the dividers set to If", describe the circle A B C D EF. Draw the diameter D O A, and from the points D and A, with the dividers set equal to the radius of^the circle, describe arcs in- tersecting the circle at £, C, F, and B. Join these points by straight lines, and they will form the sides of the hexa- gon. This problem completes Plate III. Fig. PLATE IV 38. The first four problems on this plate are more diffi- cult than any on the preceding plates and will require very careful construction. All the sides of each polygon must be of exactly the same length, so that they will space around evenly with the dividers. The figures should not be inked ^ in until the pencil construc- tion is done accurately. The preliminary directions for this plate are the same as for the preceding ones. '' \B Problem 16. — To inscribe a regular pentagon in a given circle. Construction. — In Fig. 49, from (? as a center, with the dividers set to If", describe the circle A BCD. Draw the two diameters A C and D B at right angles to each other. Bisect one of the radii, as B, at /. With / as a center, and /A as a radius, describe the arc A/ cutting DO at /. 40 GEOMETRICAL DRAWING §13 Wither/ as a center, and A/ as a radius, describe an arc/// cutting the circumference at H. The chord A H is, one side of the pentagon. Problem IT. — To inscribe a i*ea:iilar octaaron in a ariven circle. CoKSTRUCTiox. — In Fig. 50, from 6> as a center, with the dividers set to 1|", describe the circle ABCDEFG H. Draw the two diameters A E and G C 2lX. right angles to each other. Bisect one of the four equal arcs, as A G at H, and ' ^ draw the diameter HOD. Bi- sect another of the equal arcs, as A C at B, and draw the diameter EOF. Straight lines drawn from A to B^ from B to C, etc. will form the re- quired octagon. Problem Is. — To irLsci'i1>e a regxQar polygon of any nuBiber of sides in a given, circle. CoxsTRUCTiON-. — In Fig. 51, from (9 as a center, with the dividers set to If", describe the cir- cle ^ 7 CD. Draw the two diam- eters D 7 and A C at right angles to each other. Divide the diam- eter D 7 into as many equal parts as the poh-gon has sides (in this case seven). Prolong the diam- eter A C and make 3' A equal to three-fourths of the radius O A. Through -5' and 2^ the second di- vision from D on the diameter D 7, draw the line 31, cutting the cir- cumference at /. Draw the chord D /, and it is one side of the required polj-gon. others may be spaced off around the circumference. Fi-: The § 13 GEOMETRICAL DRAWING 41 Problem 19. — The side of a regular polygon being given, to construct the polygon. Construction.— In Fig. o-i, let .-^ T be the given side. If the polygon is to have eight sides, the line A C should be, for this plate, l^^" long. Produce A C to B. From C as center, with a radius equal to C A, de- scribe the semicircle A 123 J^ 5 6 7 B, and divide it into as many- equal parts as there are sides in the required polygon (in this case eight). From the point C, and through the second division from B, as 6, draw the straight line C6. Bisect the lines A C and C6 by perpendiculars intersecting in O. From O as a center, and with (9 T as a radius, describe the cir- cle C A HGFE D6. From C, and through the points 1, 2, 3, Jf, 5 in the semicircle, draw lines C H, C G, C F, etc., meet- ing the circumference. Joining the points 6 and D, D and E, E and F, etc., by straight lines, will complete the required polygon. Problem 20. — To find an arc of a circle having a known radius, which shall he equal in length to a given straight line. Note. — There is no exact method, but the following approximate method is close enough for all practical pur- poses, when the required arc does not exceed \ of the cir- cumference. Construction. — In Fig. 53, let AC he the given line 3|" long. At A, erect the perpendicu- lar A O, and make it 'i7 equal in length to the given radius, say 4" long. Fig. m 4:2 GEOMETRICAL DRAWING § 13 With O A as a radius, and (9 as a center, describe the arc A BE. Divide A C into four equal parts. A D being the first of these parts, counting from A. With i9 as a center, and a radius DC, describe the arc C B intersecting ABE in B. The length of the arc A B very nearly equals the length of the straight line A C. Problem 21. — An arc of a circle being: given, to find a straight line of tlie same length. ^-.^ This is also an approximate method, but close enough for practical purposes, when the arc does not exceed \ of the circumference. Construction. — In Fig. 54, let A B be the given arc; find the center O of the arc, and draw the radius O A . For this problem, choose the arc so that the radius will not exceed If". At A, draw A C perpen- dicular to the radius (and, of course, tangent to the arc). Draw the chord A B, and prolong it to D, so that A D = ^ the chord A B. With Z> as a center, and a radius D B, describe the arc B C cutting A C in C. AC will be very nearly equal to the arc A B. ^^^ A y ^\; "'^^\ / / / ( < > \ \ \ FIG. 54 PLATE V 39. On this plate there are five problems instead of six. It should be divided into six equal parts or divisions, as the previous ones. The two right-hand end divisions are used to draw in the last figure of Plate V, which is too large to put in one division. Problem 22. — To cli-aw an egg-shaped oval. Construction. — In Fig. 55, on the diameter A B. which is 2f " long, describe a circle A C B G. Through the center O, 13 GEOMETRICAL DRAWING 43 draw O C perpendicular to A B^ cutting the circumfer- ence ACBG in C. Draw the straight lines BCF and ACE. With B and A as centers, and the diameter y^ i5 as a radius, describe arcs terminating in D and //, the points of intersection with /) F and A E. With (T as a center, and CZ> as a radius, de- scribe the arc DH. The curve A D H B G is the required oval. Problem 23. — To draw an ellii>se, the diameters being as a center, and (9 C as a radius, describe a circle; with the same center, and O D as a. radius, describe another circle. Divide both circles into the same number of equal parts, as 1-2, 2-3, etc. This is best done by first dividing the larger circle into the required number of parts, beginning at the center line A C, and then drawing radial lines through the points of division on this circle, to the center O of the circles, as j^ shown in the upper right-hand quarter of the figure. The radial lines will divide the smaller circle into the same number of parts that the larger one has been divided into. Through the points of division on the smaller circle, draw horizontal lines, and, through the points of division on the larger circle, draw vertical Pig, 56 44 GEOMETRICAL DRAWING §13 lines; the points of intersection of these lines are points on the ellipse. Thus, the horizontal line Sc and the vertical line 3 c intersecting at c give the point c of the ellipse. Trace a curve through the points thus found by placing an irregular curve on the drawing in such a manner that one of its bounding lines will pass through three or more points, judging with the eye whether the curve so traced bulges out too much or is too flat. Then adjust the curve again, so that its bounding line will pass through several more points, and so on, until the curve is completed. Care should be taken to make all changes in curvature as gradual as possible, and all curves drawn in this manner should be drawn in pencil before being inked in. It requires con- siderable practice to be able to draw a good curved line in this manner by means of an irregular curve, and the general appearance of a curve thus drawn depends a great deal upon the student's taste and the accuracy of his eye. Problem 24. — To draw au ellipse by circular arcs. This is not a true ellipse, but is very convenient for many purposes. Construction. — In Fig. 57, use the same dimensions as before. On the major axis A B, set ofl: A a=: C D, the minor axis, and divide a B into three equal parts. With O as a center, and a radius equal to the length of two of these parts, describe arcs cutting A B in d and d'. Upon i^d' as a side, construct two equilateral triangles / intersecting d df a.nd d d ' g- in / and g. With the same radius, and b' as a center, describe the arc^ Ce ^ — ■ / "t — ---..^^ */>'^^^^ ' ^^■^fj f \/ 1 v ., V' ^ \ o A / Fig. 57 §13 GEOMETRICAL DRAWING 45 intersecting- b' d' c and b' d c in c and e. With A and ^ as centers, and a radius equal to the chord of the arcs A cox Be, describe arcs cutting A B very near to d' and d. From the points of intersection of these arcs with A B 2i^ centers, and the same radius, describe the arcs ^^^and e B f. Problem 25. — To clraAv a parabola, tlie axis and long- est double ordinate being given. Explanation. — The curve shown in Fig. 58 is called a parabola. This curve and the ellipse are the bounding Fig. 58 lines of certain sections of a cone. The line O A, which bisects the area included between the curve and the line B C, is called the axis. Any line, B A or A C, drawn perpendic- ular to O A, and whose length is included between O A and the curve, is called an ordinate. Any line, as B C, both of whose extremities rest on the curve, and is perpendicular to the axis, is called a double ordinate. The point O is called the A-ertex. Construction. — Make the axis O A equal to 3|", and the longest double ordinate BC equal to 3". B A, of course, equals.] C. Draw D Ii through tlie other extremity of the 46 GEOMETRICAL DRAWING §13 axis and perpendicular to it; also draw B D and C E par- allel to O A and intersecting DE'va. D and E. Divide DB and A B into the same number of equal parts, as shown (in this case six) ; through the vertex O, draw 1, O 2, etc. to the points of division on D B, and through the correspondii^ points /, 2, etc., on A B^ draw hues parallel to the axis. The points of intersection of these lines, tf, b, c, etc., are points on the curve, through which it may be traced. In a similar manner, dra?r the lower half OfghilCoi the curve. Probi-ew 'l-'-. — To draw a iLelix. tlie pitcli and llie Atainetei* l>emg glveiL. ExpjLAXATiox. — The helix is a curve formed by a point moving around the cylinder and at the same time advancing along its length a certain distance; this forms the winding curved line shown in Fig. 59. The center line A O, drawn through the cylinder, is called the axis of the helix, and any line perpendicular to the axis and terminated by the helix is of the same length, being equal to the radius of the cylin- der. The distance B 12 that the point advances lengthwise during one revolution is called the pitcli. CoxsTRUCTiox. — As mentioned before, this figure occupies two spaces of the plate. The diameter of the cylinder is 3|^", the pitch is •2", and a turn and a half of the helix is to be shown. The rectangle EB E D is a. side view of the cylin- der, and the circle /, 2", S", 4-\ etc. is a bottom view. It will be noticed that one-half of a turn of the helix is shown dotted ; this is because that part of it is on the other side of the cylinder, and cannot be seen. Lines that are hidden are drawn dotted. Draw the axis O A in the center of the space. Draw ED ^" long and 4" from the top border line ; on it construct a rectangle whose height EB = 3". Take the center O of the circle ^" below the point H on the axis A O, and describe a circle having a diameter of 3^'\ equal to the diameter of the cylinder. Lay off the pitch from B to 12 equal to 2", and divide it into a convenient number of equal parts (in this case 12), and divide the circle into the same 13 GEOMETRICAL DRAWING 47 number of equal parts, beginning at one extremity of the diameter 12' O 6', drawn parallel to B E. At the point 1' on the circle divisions, erect I'-l' perpendicular to B E\ through the point 1 of the pitch divisions, draw 1~1' parallel to B E^ intersecting the perpendicular in 1\ which is a point on the helix. Through the point 2\ erect a perpendicular ^'-^', intersecting 2-2' in ^', which is another point on the helix. So proceed until the point 6 is reached; from here on, until the point 12 of the helix is reached, the curve will be dotted. It will be noticed that the points of division 7', 8\ 9', 10', and 11' on the circle are directly opposite the points f/, 4', 3', 2', and T ; hence, it was not necessary to draw the lower half of the circle, since the point 5' could have been the 48 GEOMETRICAL DRAWING § 13 starting point, and the operation could have been conducted backwards to find the points on the dotted upper half of the helix. The other full-curved line of the helix can be drawn in exactly the same manner as the first half. This ends the subject of practical geometry. Mechan- ical drawing, or the representation of objects on plane surfaces, will now be commenced. THE EEPPiESEXTATIOX OF OBJECTS •40. There are five kinds of lines used in mechanical drawing, thus: The liglit full Ihie The dotted line. The broken-atid- dotted line. The broken line. The Jieavyf till line. ,_^^^^^^^_^^_._._^_^_^_^_ The light full line is used the most; it is used for draw- ing the outlines of figures, and all other parts that can be seen by the eye. The dotted line, consisting of a series of very short dashes, is used in showing the position and shape of that part of the object represented by the drawing which is concealed from the eye in the view shown ; for example, a hollow prism closed on all sides. The hollow part cannot be seen; hence its size, shape, and position are represented by dotted lines. The broken-and-dotted line, consisting of a long dash, and two dots or very short dashes repeated regularly, is used to indicate the center lines of the figure or parts of the figure, and also to indicate where a section has been taken when a sectional view is shown. This line is sometimes used for con- struction lines in geometrical figures. § 13 GEOMETRICAL DRAWING 49 The broken line, consisting of a series of long dashes, is used in putting in the dimensions, and serves to prevent the dimen- sion lines from being mistaken for lines of the drawing. The heavy full lines are made not less than twice as thick as the light full lines, and are used for shade lines. Further explanations in regard to these lines will be given when the necessity for using them arises. 41. The illustrations in this and the following paragraphs sliould l)e carefully studied, but the student is not required to send in drawings from same. In Fig. 60 is shown a per- spective view of a frustum of a pyramid having a rectangu- lar base and a hole passing through the center of the frustum. This figure represents the frustum as it actually appears when the eye of the observer is in a certain position. The angles at A,B, C, and D are right angles, the hole is round, and the sides A B and D C are of equal lengths; so also are A D and B C; but, if they were measured on the drawing, it would be found that their lengths are all differ- ent. The same difficulty would be met with in trying to measure the angles and edges of the sides A B F E, B F G C, etc. The real length of any line can be found only by a person perfectly familiar with perspective drawing, and then only with great difficulty. Consequently, this method of representing objects is of no use to a patternmaker, car- penter, machinist, or engineer, except to show what the object looks like. In order to represent the object in such a manner that any line or angle can be measured directly, what is termed projection dra>vinj;f, or ortJiograpJiic projection, is universally employed. In the perspective drawing shown in Fig. 60, three sides of the frustum are shown, and the other three are hidden; in a projection drawing, but one side is usually shown, the other five being hidden. .1/. E. l'\—i 50 GEOMETRICAL DRAWING 13 u F Fig. X> A line or surface is projected upon a plane, by drawing perpendicular lines from points on the line or surface to the plane, and joining them. Thus, if perpendiculars be drawn from the extremities of a line, as A B, to another line HK, as shown in Fig. 61, that portion of H K included between the feet of these perpendiculars is called the projection of A B upon H K. Thus, C D is the projection of A B upon H K, the point C is the projection of the point A upon H K, and the point D is the projection of the point B upon H K. The projection of any point of A B^ as E^ can be found by drawing a perpendicular from Eto H K, and the point where this perpendicular intersects HKis its projection. In this case, the point /^ is the projec- tion of the point E upon HK. It makes no difference whether the line is straight or curved — the method of finding the projection is exactlj' the same. See Fig. 62. In a similar way, a surface is projected upon a fiat surface. Thus, it is desired to project the irregular surface a b dc. Fig. 63, upon the flat surface A B D C. Draw the lines a a', bb' perpendicular to the flat surface; join the points a' and b\ where these perpen- diculars intersect the flat surface A B D C, by a straight line a b', and a' b' is the projection of the line a b upon A B D C. In the same way, a c' \s found to be the projection of ac\ c' d\ the projection of cd; and d'b', the pro- jection of db. Hence, the projection of the irregular H- F Fig. 62 X) § 13 GEOMETRICAL DRAWING 51 Fig. 04 surface abdc upon the flat surface AB DC is the quadri- lateral a' b' d'c'. The projection of any point, as e, is found as before, by drawing a perpendicular ^ from the point e to the sur- face ; thus, e' is the projection of the point e upon the plane ABDC. Suppose that the frustum, Fig. (30, were placed on a plane surface (a surface per- fectly flat, like a surface plate), and the outline of the J^*" bottom were traced by pass- ing a pencil along its edges, including the round hole, the result would look like Fig. 64, in which the rectcLngle £ F G N represents the bottom of the frustum and the circle repre- sents the hole. The angles and lengths of the sides are exactly the same as they are on the frustum itself; a similar drawing could be made to represent the top, but it is unnecessary, for the reason that the top can be projected on Fig. 64, and both objects accomplished in one drawing. Fig. 65 illustrates the meaning of the last statement. Here A' B' is the projection of the edge A B, Fig. 60 ; B' C, of B C, etc. ff' „* A' £' is the projection of the edge A E; B' F, oi B F, etc. This drawing shows the figure as it would look if the eye were directly over it. A drawing Avhich rep- resents the object as if it were resting on a horizontal plane, and the observer looking at it from above, is called a top view, or plan. The line of vision is thus perpendicular to the faces A B CD and E F G H oi the frustum. The lines A B, Fig. 65 52 GEOMETRICAL DRAWING I 13 / -»p BC^ etc., EF, FG, etc., and the diameter of the hole, can be measured directly. The drawing is not yet complete, since it does not show whether the ends and sides are romid- mz. bellowed out, or flat. For this purpose, two more . _ views are necessary — a verti- cal projection, OT front view. commonly called a front elevation, and a side pro- jection^ or side vie^sv. A front view (elevation) is drawn by imagining the eye to be so situated that the observer looks directly at the front of the object; in other words, the line of vision is parallel to the faces of the frustum. The side looked at is then drawn as if it were projected on a vertical plane at right angles to the horizontal plane, the vertical plane being also parallel to the edges ^/^ and H G oi the frustum shown in Fig. 60. The drawing would then look like Fig. 66. Here the trapezoid A B F E represents the side A B F E of the frustum; the altitude of the trapezoid being the same as the altitude of the frustiun, it can be measured directly. The hole cannot be seen when the observer looks at the frustum in this position; hence, it is indicated by dotted lines. The projections of the lines A B and D C (also, of E F and H G, of A E and D H, and of B /"and C G) coincide. To draw the side view (sometimes called a side eleva- tioiL), imagine the frustum to be revolved around on its axis 90° to the left, and then draw it in pre- cisely the same manner as the front elevation, by projecting the diflEerent lines upon a plane at right angles to the horizontal plane, and perpendicular to the . _ . edges EFzLndi H G^ that is, par- allel to BC 2indFG. The side elevation would then be drawn as shown in Fig. 67. In this view the lines A D §13 GEOMETRICAL DRAWING 53 and BC (also, EII and FG, DH and C G, and A E and B F) coincide. 42. In order to show clearly the different views, and to guard against one view being mistaken for another, they are always arranged on the drawing in a certain fixed and invari- able manner. Fig. 68 shows this method of arrangement. The plan is drawn first, then the two elevations. It is usu- ally immaterial which of these views is drawn first, but the general arrangement is as shown. Any departure from this method of arrangement should be distinctly specified on the drawing in writing, unless the purpose of the draftsman is so clearly evident that no explanation is needed. The broken and dotted lines are the centei* lines ; they serve to show the connection existing between the different views of the object, and to indicate axes of cylindrical surfaces of any 54 GEOMETRICAL DRAWING § 13 kind. It will be noticed that, in the plan view, the two center lines cross each other at right angles, and that their point of intersection O is the center of the circle which rep- resents the hole. Whenever a circle is drawn, two center lines should also be drawn through its center at right angles to each other; this enables any one looking at a drawing to instantly locate the center of any circle. This remark also applies to ellipses, semicircles etc. To draw the frustum as shown in the last figure, either the front elevation or the plan is drawn first — whichever happens to be more convenient. Suppose the front elevation to be drawn first. Draw the vertical center line ;« ;/ ; meas- ure the altitude of the frustum, and lay it off on this line, locating the points / and K; through these points, draw the lines AB and ^/^perpendicular to ;;/;;; make AI=z/B =■ \ A B, measured on the frustum ; also E K ^ KF=^ -i E F, measured on the frustum, and draw A E and B F. Lay off the radius of the circular hole on both sides of the center line ;// ;/, and draw the dotted lines parallel to ;// ;/ through the extremities of these radii to represent the hole. The front elevation is now complete. To draw the plan, decide where the center is to be located on in ;/, and draw the hori- zontal center line p q. With the point of intersection O of the two center lines as a center, and with a radius equal to the radius of the hole, describe a circle. Through the points A^ B, E, and /% draw indefinite straight lines parallel to tn n. On both sides of the center line/ q, lay off on these lines D S and S A, equal to \ D A^ and H R and RE, equal to i HE, both DA and //"^ being measured on the frus- tum. Through the points H, E, D, and A, draw the lines H G, E F, DC, and A B, and join the points H and D, E and A, F and B, and G and C by straight lines, as shown. The figure thus drawn will be the plan. To draw the side elevation, prolong the lines A B and E F, and draw the center line / v. Lay off, on each side of 1 2', -Ff/and (7 G equal to i EG, measured on the frus- tum, and B X a.nd A'" (T equal to ^ B C, measured on the frustum. Join 5 and F, and T and G, by the straight lines § 13 GEOMETRICAL DRAWING 55 B F and C G, and draw the hole dotted as in the front ele- vation. The drawing is now complete. The student should have by this time a good idea of how simple objects may be represented by the different views of a drawing, and can now begin on the next plate. DRA^VTN^G PliATE, TITLE: PROJECTION'S— I 43. In making actual drawings of objects when the size of the plate is limited, it is usually impossible to divide it up into a certain number of parts, as in the case of the preceding plates, for the various figures differ widely in their sizes. These drawings should be so made that no part shall come nearer than |" to the border line, and the figures should be so arranged as to present a pleasing appearance to the eye, and not be scattered aimlessly all over the drawing. Fig. 1 represents a rectangular prism 2" long, 14^" wide, and f" thick. The prism is represented as if it were standing on one of its small ends, with the broad side towards the observer. The elevation A B D C is drawn first; in this case, it will be a rectangle 3" X 14-". The top view, or plan, FEB A is next drawn; this is a rect- angle ly X I", the side A B being the projection of the front of the prism, and the side F E of its back. Lastly, the side elevation is drawn ; this is another rectangle B E H D, 2" X I", the side B D representing the projection of the front of the prism, and the side BE corresponding to the right-hand end B E oi the plan. Fig. 2 is a \vedge standing on one of its triangular ends. It is the rectangle shown in Fig. 1, cut diagonally through the corner from E to A on the plan. It will be noticed that the two elevations are exactly the same as in Fig. 1, the plan showing the difference between the two figures. Fig. 3 is another wedge, standing on one of its rectangu- lar sides, formed by cutting through the prism, in Fig. 1, from A to D. The plan and side elevation are the same as in Fig. 1. Here, the front elevation shows the difference 56 MECHANICAL DRAWING § 14 When constructing- cycloidal teeth for gear-wheels, the diameters of the describing circles are usually made equal to one-half the diameter of the pitch circle of a gear-wheel having 12 teeth of the same pitch as those of the gear- wheel about to be made. Let d' be the diameter of the describing circle; then, ^':=l^Xi, or./'=^. (4.) Addendum = .3/; root = .4/; thickness of teeth for cast gears is .48/, and for cut gears kp. DRA^^IXG FOLATE, TITLE : SPUR GEAR-AVHEEES 91. This plate shows the halves of two cast gear- Tv^heels having cycloidal teeth, which work together, a cross-section of each gear being also given. The drawing is full size, the wheels not being shown entire for want of room; to have done so it would have been necessary to make the drawing to a reduced scale. The pitch is 1 inch, the number of teeth in the large gear is 36, and in the small one 18. The pitch diameter of the large wheel is found by formula 1 to be d — '—- = 11.46 inches, nearlv. 3.1416 The pitch diameter of the small gear = 1 X 18 3.1416 = 5. 73 inches, nearly. The diameter of the describing circle is found by formula 4 to be d = ., ^ ,, , = l.'.-il inches. 3.1416 For all practical purposes, the diameter of the describing circle may be taken to the nearest 16th of an inch. For circular pitches under ^ inch, approximate the diameter of PRDJEC jy riffj. ^,^.2. ~j — £ ^- r iq.3 ^ig. 6. JA7ri,y:.J=.Y /<. ,S.9r For notice of copyright, see pagti riDN5-l. Fig.//. Fig./ 2. nediately following the title page. yOM-JV SJYT/TM. CLASS JY9 4:529. § 13 GEOMETRICAL DRAWING 57 off, on the center line /////, the distance yZ equal to 2", and through the points y and L draw the two horizontal lines S e and Rf, Project the points A', /. //, and G upon 5 r, as shown by the dotted lines; and through the points of inter- section of these dotted lines with Se, draw the vertical lines S R^ a b, c d, and ef, thus completing the front eleva- tion. To draw the side elevation, extend the lines .S'^' and Rfy and draw the center line t v. Make U ]\' equal to 1:^", which is equal to the distance between the parallel sides, and draw U X and W V; also, M Z, the point iM cor- responding to the point A.' of the plan. Fig. 7 represents a liexag-onal pyramid ; the distance between two parallel sides of the base is l^^", and the altitude is 2". As in Fig. G, the plan must be drawn first. Then, to draw the front elevation, lay off O I on the center line w ;/ equal to the altitude, and through /draw the base line A' D' . Project the points D, E, etc. of the plan upon ^\' 1 >\ as shown by the dotted lines, and join them with the point O by the straight lines A'O, F'O, E'O, and D'0\ these lines are the vertical projections of the edges of the pyramid; the horizontal projections of the edges are F O, E O, D O, etc. The side elevation can be easily drawn, and does not require a special description, the length of the base B E he'mg equal to the distance between the parallel sides, or 1^". Fig. 8 shows a rivet ^" in diameter, having a button head ly in diameter. The side elevation is not given, since it is exactly the same as the front elevation. Either of the two views may be drawn first, according to convenience. vSuppose that the elevation is first drawn. Draw the center line ;// n, and the line A B for the base of the head. On the center line lay off from the line A B, or the base of the head, a point (9, at a distance of J|-", the height of the head. With the compasses set to a radius of |-|", and from a point on the center line //i n, describe an arc A O B, taking care to pass this arc through the point O. Lay off from, and on both sides of, the center line ;// n a distance of j\", or \ of the diameter of the rivet, and draw /: 6" and /^//. Draw the other center line/'ry of the plan, and with O as a center, 58 GEOMETRICAL DRAWING § 13 and a radius equal to the radius of the button head, describe a circle. "With the same center, and a radius equal to y^", describe the dotted circle, the horizontal projection of the rivet. The irregular line C//" indicates that only a part of the rivet is shown. This is done so as not to take up too much space on the drawing. Fig. 9 shows an ordinary square-headed bolt y in diam- eter, having a head If" square and if" thick. DraAV the center lines ;;/ u and / g. Construct the rectangle A B D C, If'X \^\ the elevation of the head. Locate the points E and F 2X z. distance of -^' from each side of the center line, and draw E G and F H. With the compasses set to a radius of If" and from a point on the center line in n, describe the arc representing the chamfering of the head. Draw the plan of the head LKBA (a square whose edge meas- ures If"), and the dotted circle f" in diameter, the pro- jection of the body of the bolt, which cannot be seen in this view. Fig. 10 shows a distance piece used to separate two other parts, and to keep them a certain distance apart. The arrangement of the views of this figure is somewhat different from the preceding ones, in order to make room for it on the drawing. Draw the center line ;/ in, and con- struct the figure according to the dimensions marked on the plate. Use a radius of ^" for the fillets at A, B, C, and D, and an equal radius to round the corners at E, E, G, and //. Fig. 11 shows a square cast-iron Tvaslier. Instead of making an elevation and plan as usual, a section is taken through /) q; that is, the washer is imagined to be cut on the line/^, with all that part of the figure to the left oi p q removed, and an elevation drawn of the remaining part. In order to distinguish a sectional drawing without any possi- bility of mistake, the so-called section lines are employed. These are usually drawn by laying a 45° triangle against the edge of the T square, and drawing a series of parallel lines as nearly equally distant apart as can be judged by the eye. For cast iron, these lines are full, thin lines, all of the same § 13 GEOMETRICAL DRAWTNCr 59 thickness, and must not be drawn too near together. The method of sectioning for other materials will be given later on. It is not usual to draw the section lines in pencil, but to wait until the outlines of the drawing have been inked in, and then section directly with the drawing pen. The short- est distance apart of the section lines should rarely be less than -i^", unless the drawing is of such small dimensions as to cause a sectioning of this width to look coarse. This is the case with Figs. 11 and 12 of this plate. In these two figures make the section lines a full -^^" apart. Only that part of the figure is sectioned which is touched by the cut- ting plane, the rest of the figure being drawn as if it were projected upon the cutting plane. The corners of this figure should be rounded with a radius of yV"> the other dimensions can be obtained from the plate. Fig. 12 is a east-iron cylindrical ring. It is shown in plan and section. The dimensions given suffice for the drawing of the figure without further explanations. The inner circle of plan is the projection of the inner- most points of the ring which form a circle whose diameter 44. When inking in a drawing, it is generally best to draw the circles and other curved lines first, and the straight lines afterwards. This enables the draftsman to easily blend into one line the straight lines meeting the curves, so that their points of meeting cannot be detected; it enables the tangent lines to be drawn with better success, and also shortens the time of inking in a drawing. It will be noticed that some of the straight lines are heavy and some light, and that parts of the full-line circles are heavy and the rest of the circle light. These are the shade lines; they are described later on. The student may make all of the full lines except the border lines of this plate, and the three fol- lowing plates, of the same thickness, if he so desires. The dotted lines used to indicate those parts of the figures that are hidden must be of the same thickness as full lines, while the construction lines and center lines should be very thin 60 GEOMETRICAL DRAWING §13 r=^^-_^^-.-.&lvi^ J Fig. 69 45. Dlmensioiis. — The dimension lines and figures on this and succeeding plates are to be inked in by the student. Make the dimension figures y\" high, and of the same style as those shown in Art. 20. Fractions should be A' high over all. If there is not room for figures of this size, great care should be taken to make them clear. Until after the student has obtained sufficient practice in lettering, he should draw guide lines in pencil for the dimen- sion figures, as in Fig. 69, unless he can make them look well without. All the figures should have the same slant of 60°, and, when printing fractional dimensions, the luJiole fraction should have the same slant as the figures; that is, the denominator should be under the numerator in a i'/^;///;/^ direction, and not straight below it. Make the dividing line between the numerator and denominator horizontal, not slanting. Dimension and extension lines must be light, broken lines of the same thickness as the center and construction lines. Care should be exercised to make the arrowheads as neatly as possible and of a uniform size. They are made with a Gillott's No. 303 pen, and their points must touch the exten- sion lines, as illustrated in Fig heads too flaring. When putting in the dimensions, care should be taken to give ^. E J-C -mmc ^H PRDJEC i^^' T Z^.^. J-t^r- ^^-^ . ■ — nL.-:,. .— , - : jy : f coryrlgr.t. see pagt riDNS-ll. c \B' ^' 9 2- s-b: c' 1 1 ^ ' "^r^l J^z.y.. mediately following the title page. JO^HT-/ S7yZITJ-£. CZAS^JV? 45£9. § 13 GEOMETRICAL DRAWING 63 projection or front elevation of the prism. The method of drawing the plan and side elevation is apparent without further explanation. Fig-. 3 is the same prism shown in Figs. 1 and 2, but with the narrow sides parallel to the plane of the paper, and tipped until the base makes an angle of 17^° with the horizontal. The sizes are the same as in the two preceding figures, and it should be drawn without further explanation, the front elevation being drawn first. Fig. 4 shows a liexagonal prism having two of its par- allel sides parallel to the plane of the paper, and its axis vertical; instead of a side elevation at right angles to the horizontal, a side elevation is desired, as if the vertical prism were looked at in the direction of the arrow, or at an angle of 30" with the horizontal. Draw the plan first and then the front elevation from the dimensions given. To draw the other view, first draw the center line vi n, and then, by use of the T square and 30'' triangle, draw the lines A B, CD, E F, and G //, from the points A, C, E, and G, as shown. Also draw in a similar manner the other four dotted lines at the base of the prism; then draw the line IB at a right angle to the lines A B, CD, etc. At the point /, draw the line // parallel to the center line ;;/ n, and, Avith /as a center, and the points j5, D, F, H, K, L, etc. as radii describe arcs, as shown, cutting the vertical line //at the points/, M, N, O, P, Q, etc. Through the points/, M, N, O, P, Q, etc. draw hod- zontal lines as shown. On each side of the vertical center line mn, lay off a distance of f", or one-half the distance between the parallel sides of the prism, which is 1|", as shown in the plan, and draw the lines 7? 5 and T U. This view is then completed by drawing the lines V R, V T, IV X, and YX, as shown. The lines at the base are drawn in a similar manner. Fig. 5 represents a hexagfonal pyramid whose axis is parallel to the plane of the paper, the base making an angle of 30° with the horizontal. It is desired to find the vertical projection of the side elevation. Having drawn the plan ABCDEFam] the side elevation 6''y^'/i"6'7/, as shown 64 GEOMETRICAL DRAWING § 13 from the dimensions marked on the drawing, choose the position of the vertical center line t'v\ project (?' and (?'" upon it in the points O" and 0^^\ and, through O^ and 6^'", draw a fourth center line r s. On this, lay off O^ G and O" IT equal to O G and O H, and construct the projec- tion A" B" C" D" E" F' , as indicated b\^ the broken and dotted lines. Join O" E" , O" F', etc. b)- straight lines, and it will be the required projection. The figure thus drawn represents the pyramid as it would appear placed so that its base made an angle of 30° with the horizon, the line of vision being horizontal to the observer looking at it from the left side. Fig. 6 shows a cyltader whose axis is parallel to the plane of the paper and makes an angle of 77° with the horizontal. The vertical side projection is required. Draw the plan and front projection as shown from the dimensions given. Draw the center line t v vertical, and project the center CX upon it in (9"; also, A in A\ and H' in IF . To find the remain- ing points on the projected circle, divide the diameter A H of the plan into a convenient number of equal parts, in this case 7, as A 1, 1-2, 2S, etc. Through the points thus laid off, draw the lines 1-1'\ 2-2'\ 3~S'\ etc., parallel to the cen- ter line inn. Through the points A\ T\ ^", ^", etc., draw the horizontal lines as shown b}^ the dotted lines. From and on each side of the vertical center line / v^ lay off distances on each side of the horizontal lines just drawn equal to the length of that part of the lines 1-1'\ 2-2'\ SS', etc. included between the center line/^ and the semicircle ^-^ CH; thus, on the horizontal line drawn through the point 0\ the dis- tances O' C" and 0"D" are each equal to (9 C in the plan. The distances P^'-l'" and P^-P' are each equal to the dis- tance from 1 to the point of intersection of the semicircle on the line 1—1". The remaining distances are laid off in a similar manner. A curve traced through the points thus found will be the required projection of the upper base of the C5'linder. The projection of the lower base is found in exactly the same way. Drawing C"E' and D"F completes the required projection. § 13 GEOMETRICAL DRAWING 65 DRAAYIXG PLATE, TITLE: COIS^IC SECTIO]S^S 47. This plate shows the different forms of the curves formed by the intersection of a cone or cylinder by a plane. If the plane of intersection is perpendicular to the axis of the cone or cylinder, the curve of the intersection will be a circle; but if it is inclined to the axis, it will be an ellipse in the case of a cylinder, and an ellipse, hyperbola, or parab- ola in the case of a cone, according to the angle of inclination. Fig. 1 is a cone cut by a plane which does not intersect the base of the cone. VVhoi the cutting plane does not inter' sect the base, or the base of the cone extended, the curve of intersection is an ellipse. Draw the plan and front elevation of a right cone whose altitude is 3f inches and whose base is 3 inches in diameter. Cut this cone by a plane ab, making an angle of 52° with the base. See figure. Divide the circle which represents the base of the cone in the plan into any number of parts, in this case 24, and, through the points of division A, E, H, etc., draw the radii O A, O E, OH, etc. to the center O. Draw also from these points straight lines A A', E E', H H' , B B\ etc., par- allel to the axis of the cone O'n, and cutting the base A'B' in the points E' , H', etc. From these points, draw lines to the apex O' of the cone, and cutting the base A' B' in points E', //', etc. From these points, draw lines to the apex O' of the cone, as E'O' , H'O' , etc., cutting the plane a b in the points D' , F , etc. From these points D' , E' , etc., draw straight lines E'EE", D' D D" , etc., parallel to the axis O'n of the cone, and intersecting the radii O A, O E, OH, OB, etc., in the points C, D, F, K, F" , D", etc., and through these points of intersection draw the ellipse by aid of an irregular curve. Fig. 2 is a cone of the same size as in the preceding prob- lem ; but the cutting plane a b is, in this case, parallel to one of the elements* of the cone, and intersects the base. The *Any straight line drawn on the surface of a cone and passing through the apex (as OH', Fig. 1, or O'A', Fig. 2, etc.) is called an element. 66 GEOMETRICAL DRAWIXG § 13 curve formed by the intersection of a cone by a plane parallel to one of its elements is called a parabola. The plan and front elevations of the cone and curve of intersection are found in a manner similar to the method used in the last problem. To find the side elevation, proceed as follows: Draw the side elevation 0"A"B" of the cone with the center line / ^ as its axis. Draw the projection lines F* F" F^"^^ D'D"'D^, etc., and make KT" and K'F^"-' equal to KF and KF"x make I'D'" and I'D^''' equal to /Z) and I D'\ etc., and trace a curve through the points thus found. The result will be the side elevation of the cone when cut by a plane parallel to one of its elements and having the upper part removed. The side elevation of Fig. 1 may be drawn in a similar manner. Fig. 3 is a cone having the same dimensions as the two preceding problems, but cut by a plane a b parallel to the axis of the cone and perpendicular to the vertical plane of pro- jection. When the cutting plane intersects the base of a cone and is not parallel to any element (that is, if the acute angle included between the cutting plane and the base is greater than the angle O' A' B' included between any one element and the base), the curve of intersection is called a liyperbola. The plan and front elevation are constructed as before, the horizontal projection of the curve for this particular case, where the cutting plane is parallel to the axis of the cone, is also a straight line. The side elevation is found as in the last problem, by drawing the lines of projection F'F"'F^''\ j)-j)"Div^ etc., and making /'Z>'" and I'D'"' equal to ID and ID", K'F" and A"/"'^ equal to //"and IF", etc. The curve drawn through the points thus found will be the required hyperbola. Fig. 4 shows the intersection of a cylinder, 3f " long and 2" in diameter, by a plane a b, making an angle of 57* with the base. The plan and elevation may be drawn as shown, the horizontal projection of the curve being a circle, havnng the same diameter as the base. To construct the side elevation of the curve, divide the circle representing the ^^'LT/OAHY /4, /39P'. For notice of copyright, see pag >v c« n n in n n imediately following the title page. ^oj^jT syyr/TJ-c cz^ss yy9-f339. § 13 GEOMETRICAL DRAWING 67 base of the cylinder in the plan into any number of parts, in this case 24, and through the points of division A, B, C, etc. draw the radii OA, OB, O C, etc. to the center O. Draw also from these points straight lines A A, L B B' , K C C, I D D\ etc. parallel to the axis ;// ;/ of the cylinder, and cut- ting the base in the elevation. From the points A, B\ C, D\ etc. draw lines D' E' , C'F',B'G\ etc. at right angles to the axis nui. Make I'E and /'^'each equal to ID; K'F and K'F' each equal to KC\ L'G and L'G' each equal to LB, etc. The curve drawn through these points Avill be the side projection, or side elevation, of the curve of intersection. DRAWING PLATE, TITTLE: INTERSECTIONS AND DEVELOPMENTS 48. On this plate some dimensions are given in decimal fractions instead of common fractions. Such decimal dimen- sions should be laid off with a decimal scale, if the student has one. A decimal scale is a scale with inches divided into tenths, hundredths, etc. If the student has no decimal scale (and such a scale is not essential), he should take the nearest value of the decimal fraction in thirty-seconds of an inch. To change a decimal fraction to a common fraction, having a desired denominator, multiply the decimal by the desired denominator of the common fraction, and express the result as a whole number, which whole number will be the numera- tor of the fraction. Thus, to express .705" in fourths, we have .705 X 4 = 3.()G fourths == say, f". To express .765" in sixteenths, we have .7(55 X 16 = 12.24 sixteenths = say, |f". To express .765" in thirty-seconds, we have .765 X 32 = 24.48 thirty-seconds The length of the circumference of a circle = the diameter X 3.1416; hence, The length of circumference of a circle whose diameter is If" = 3.1416 X If" = 4.32" = i^". The length of circumference of a circle whose diameter is 1{" = 3.1416 X U" = 4.71" = 4f|\ 68 GEOMETRICAL DRAWING § 13 The length of circumference of a circle whose diameter is li" = 3.1416 X li" = 3.93" = 3||". The length of circumference of a circle whose diameter is 1^" = 3.1416 X IfV" = i-S"- 4.9" ^ 2 = 2.45" = 2r'6" (see Fig. 10). 49. This plate deals with the intersection of surfaces and their development. Fig. 1 shows the intersection of two unequal cylindrical surfaces whose axes p q and in n intersect at right angles. Their dimensions are given in the figure. For the sake of convenience, a bottom view is given, instead of a top view, as usual. First draw the front elevation, omitting, of course, the curve of intersection E Q G D C B A, which must be found. Then draw the side elevation and the bottom view, as shown. Divide the circle which repre- sents the side projection of the cylindrical surface F E A 1 into any convenient number of parts, in this case 12, and draw the projection lines 7E,6Q,5G,Ji.D,3C, 2 B, and 1 A parallel to the axis/^. Also draw the projection lines 4-4', 3-3', 2-2\ 1-1' , etc. parallel to the axis / v. Choose a con- venient point O, and through it draw two lines (9 /and O K parallel to the axes pq and inn of the cylinders. Continue the lines 4-4', 3-3\ etc. downwards, until they cut O I in 8, 7, 6, 5, etc. Now make O 8' = O 8, Ol' — O 1, etc."; this may be most conveniently done by taking 6^ as a center, and describing arcs of circles with radii equal to O 8, 7, O 6, etc., cutting O K in 8', 7', 6', etc. Through 8', 7', 6', etc., draw the lines 8'D', 7'C', 6'B', etc. parallel to the center line r s. Through the points/^', C',B', and A', draw the lines D'D, C'G, and B'Q, parallel to the center line ni n, and intersecting the lines J/. D, 5 G, 6 Q, 3 C, and ^ ^ in the points D, G, Q, etc. The curve traced through these points will be the front elevation of the curve of intersection of the two cylindrical surfaces. Fig. 2 shows the intersection of two equal cylindrical surfaces at right angles to each other, as in the case of a pipe elbow. When two cylinders having equal diameters intersect, and their axes also intersect, the front elevation of INTERSECTIONS Al np.9 Fi^.ll. J'-AJrVARV /4, /e3!^ For notice of copyright, see page Tiediately following the title page. jrO^^ 37yTITJ<. CLA^^JV945e9. § 13 GEOMETRICAL DRAWING 69 the curve of intersection is always a straight line, no matter what angle the two axes make with each other. Fig. 3 shows a symmetrical three-jointed ell)o^v formed by the intersection of three cylindrical surfaces. The diam- eter of each of the three surfaces is 1^". The center lines of the surfaces RAGS and M N P H z.x& to be at right angles to each other; then, in order that the arrangement shall be symmetrical, the center line of the third surface A M H G must make an angle of 45° with the center lines of the other two. To construct the elevation as shoAvn in the figure, draw the two center lines ;// // and /^ at right angles to each other; they intersect at 6. Lay off 6'/= 1|-" and draw an indefinite line RS through / perpendicular to mii. Make I R equal to /5 = 1^ x J = f", and draw RA and 5 G par- allel to nin. Draw (9 i'T parallel to vin and 1|^" below it. Through the point (7, where R S and O K intersect, draw O T passing through 6\ and bisect the angle RO T hy the line O A^ which intersects RA Siud • S G \n A and G. Lay oK 6 / = 2^" and draw PJ N perpendicular to pq. Make /P = JN= H Xi = f", and draw PH and NM parallel to pq. Draw OAT so as to bisect the angle T O K\ O M intersects P // and N M in //and J/. Finally, draw AM2in& GH. Fig. 4 shows the intersection of two unequal cylin- drical surfaces whose axes intersect at an angle of 65° instead of 90°, as in Fig. 1. The method of finding the curve of intersection is in all respects similar to that used in Fig. 1, and, as the corresponding points have been given the same letters or figures, the directions given for Fig. 1 can be applied to Fig. 4 also. Fig. 5 shows a cylindrical piece of iron 2f|" in diam- eter that has been gradually turned down to 1^^" diameter, and then having the larger part flattened on two sides. The large and small parts of the piece are connected by a graceful curve. The problem is to find the curve of inter- section A 123 B formed by the flattening. Draw the plan and front elevation from the dimensions given; also draw 70 GEOMETRICAL DRAWING §13 the curve C 6' 5' Jf.\ and its equal on the opposite side, sc that they look to the eye about as seen in the drawing. In order that all the work sent to us may be alike, the radius of this curve and the position of the center have been given on the drawing. To locate the center, draw an indefinite horizontal straight line 1" -\- Ijg" = 2^" above the base of the piece; and with C ^nd D as centers, and a radius of \^' , describe short arcs cutting the line just drawn. The points of intersection will be the required centers. With (9 as a center, and radii of convenient lengths, as (9^, 5, 6, etc., describe arcs cutting A' B' in 3' , 2' , 1' , etc. Through the points ^, 5, 6, etc. draw the lines 4-4'> 3-5', 6-6', etc., par- allel to the center line /// n, and intersecting the curve C Jf! in 4', 5', 6', C, etc. Through the points A' , 1' , 2' , etc. draw lines A' A, I'-l, 2'-2, etc., parallel to nni, intersecting hori- zontal lines drawn through C, 6', 5\ Jf! , etc., in A, 1, 2, 3, etc. The points A, 1, 2, 3, etc. are points on the required curve, and through them the curve may be drawn. Fig. 6 is the cylindrical surface of one section of the elbow 1 7 A G oi Fig. 2 rolled out into a flat plate ; hence, if a flat plate were cut into the same shape and size as Fig. G and bent into a cylinder so that the ends 1 G' and I'G" touch each other, the vertical projection or front elevation would be the same as shown hy 1 7 A G in Fig. 2. If a second plate were cut out in the same manner and bent into a circle, the two pieces on being brought together, as shown in Fig. 2, would touch at every point. The problem is to find the shape of the curve G' A' G". The length of the line 1-T is evidently equal to the length of the circumfer- ence of a circle whose diameter is If", or 4.32", very nearly. Produce the line 1-7, Fig. 2, and make 1-V equal in length to 4.32". Divide the circle 12 3.... 12 into a convenient number of equal parts, in this case 12, and erect the per- pendiculars 1 G, 2 F, 3 E, etc., cutting the line of intersec- tion G A oi the cylindrical surfaces in G, F, E, etc. Divide the line 1-1' into the same number of equal parts that the circle was divided into, thus making the length 1-2 equal length of arc 1-2; 2-3, length of arc 2-3, etc. Through i, § 13 GEOMETRICAL DRAWING 71 2, 3, etc., draw the perpendiculars 1 G\ 2 F\ 3 E\ etc. and project the points G, F, E, etc. upon these perpendiculars, as shown, thus locating the points G' , F\ E' , D\ C, B\ A' of the left-hand half of the required curve. The points on the right-hand half are found in the same manner, as shown, and the required curve can be drawn through these points. 50. A drawing like Fig. G is called the development of the cylindrical surface 17 A G. Fig. 7 is the development of the cylindrical surface A G H M of Fig. 3. Make 1-1'= Ux 3.1416 = 4.71", neaily, and divide it into 12 equal parts to correspond with the 12 equal parts into which the dotted circle is divided. Project the points 6, 5, etc. of the dotted circle upon O A as shown, thus locating the points B, C, etc. Through B, C, etc., draw B 6, C5, etc., perpendicular to O T. Make 1 G' = 1 G'" = 1 G,2 F =2 F'" = 2 F, 3 E' = 3 E'" = 3 E, etc. Through G' , F', E' , etc., trace the curve G' F' E' . . . . G" , and, throug^h G"\ F'". E'", etc., trace the curve G'" F'" E'" . . . . C^'. Drawing G' G'" and G" G^""' completes the figure. Fig. 8 is the development of the cylindrical surface 1 F E A, Fig. 1. The method used here is in all respects similar to the tAvo preceding problems. In this case, the distances 1 A, 7 E, and 1' A' are all equal to lA or E F, in Fig. 1; and 2 B, 6 Q, 8 Q' , and 12 B' are all equal to 2 B ov 6 Q, in Fig. 1. The development of L M P N"\s not given, for want of room, but the method will be explained in Fig. 10. Fig. 9 is the development of the cylindrical surface 1 FEA, Fig. 4. The student should have no difficulty in drawing this, after having studied the preceding problems. Fig. 10 is the development of the cylindrical surface LMPN, Fig. 4. Owing to the want of room, only that half of the development is shown which contains the part to be cut out. The length of a circle ly\" in diameter is 4.9", nearly; half of this is 2.45". Hence, the line Y'Y", Fig. 10, which equals the length of. the semicircle V'A'Y", Fig. 4, is 2.45" long. The distance X' V = X" V" equals the length of the cylinder, L X or MP. Lay off X'S equal 72 GEOMETRICAL DRAWING § 13 to the length of- the arc V U \ 5 Tv equal to the arc D' C \ i? A' equal to the arc C B' \ A^ J/ equal to the arc B' A' , etc. Find the lengths of these arcs by means of the method given in connection with Fig. 19. Draw through these points the perpendiculars SS',RR', etc. With the spa- cing dividers, set off 5Z>, equal to S /? in Fig. 4; i?G^, equal to R G; X Q^ equal to X Q; and ME^ equal to ME. Also, R'C^ equal to R'C; N'B^ equal to N'B; and M'A^ equal to PA. In exactly the same manner, find the points on the right-hand half of the curve. If a plate were cut of the same size and shape as shown in Fig. 10, and rolled into a semicylindrical surface, the diameter of which is liV") it would exactly fit the plate cut like Fig. 9 rolled into a cylindrical surface, the diameter of which is 1^", the two being placed together as shown in Fig. 4. Fig. 11 shows a conical surface cut t>y a plane, and Fig. 12 shows its development. Draw the elevation and horizontal projection of the base as shown in Fig. 11. Divide the projected circle (base of cone) into a convenient number of equal parts, in this case 12, and project the points i, 2, 3, etc. on the base l'-7', thus locating the points i', 2' , 3\ etc. Join these points with the apex O of the cone, by the lines 1' , 02', 03\ etc., cutting the plane in A^ B, C, etc. Now, choose a convenient point O, Fig. 12, and with this as a center, and a radius equal to 1\ or 7', Fig. 11, the slant height of the cone, describe an arc 1-1' of a circle. Make the length of this arc equal to the length of the circumference of a circle having the same diameter as the base of the cone. This may be conveniently done as follows : length of arc = 2 X 3.1416 = 6.28", nearly. Draw a straight line 6.28" long and divide it into, say, 4 equal parts. Describe an arc hav- ing a radius equal to O T , the slant height of the cone, and find the length of a part of this arc equal to 6.28 -f- 4 = 1.57" by means of the method described in connection with Fig. 48. With the dividers set for the chord of the arc just found, space off the chord four times on the longer arc 12 3 ....1', Fig. 12. Divide the arc into the same § 13 GEOMETRICAL DRAWING 73 niiniber of equal parts tliat the circle 1 2 3. . . .12 has been divided into, that is, 12 parts. Join the points of divi- sion i, 2, 3, etc. with the center O by the lines O 1,02, O 3, etc., as shown. Project the points B, C, D, etc., Fig. 11, upon O 1', in i),, C,, D^, etc., as shown, and lay ofi O A equal to O A' equal to OA, Fig. 11; O B equal to O B' equal to (9/>', ; O C equal to O C equal to O C^, etc., and through these points draw the curve. A plate cut of the same size and shape as shown by A G A' T 7 1 can be bent into the conical surface shown by the elevation A G 7' 1' . Particular attention must be given to the method explained above for laying out the curve of t lie development in Fig. 12. It would be entirely wrong to take the measurements from the lines O F, O E, O D, O C, etc., Fig. 11. The reason for this is that these lines, being on the surface of the cone, are inclined tow^ards the observer, and so do not appear in their true lengths. The line O D, for example, if measured on the surface of the cone itself, would evidently be of the same length as the line O D^\ but in the figure it is much shorter. The line O D^, however, appears in its true length in the figure, because it is not inclined to the observer in the position shown. The actual distance of point D from the apex (9, therefore, is O D^, which is the distance to be laid ofif for point D in the development. The same holds true for the other points. SHADE lilJ^ES 51. The use of the heavy shade line will now be ex- plained In Fig. 71, by means of the shade lines, the drafts- man knows, without looking at any other view of the object, that the rectangles 1 and Jf. represent square holes, and 2 and 3, square bosses. When he looks at the other view, it is to find the depth of the holes and the height of the bosses. This explains the use of the shade lines, viz. : to show, from that view of the drawing which is being examined, whether the part looked at is above or below the plane of the surface: that is, for example, whether 74 GEOMETRICAL DRAWING 13 J z i L>— — 3 4: Fig. 71 the rectangles 1, 2, 3, and 4- are the tops of bosses or bottoms of holes, and, consequently, whether they extend above or below the sur- face of A B DC. In order that the shading may be uniform on all drawings, the light is assumed to come in one invariable direc- tion, in such a man- ner as to be parallel to the plane of the paper, to make an angle of 45"^ with all horizontal and verti- cal lines of the draw- ing, and to come from the upper left-hand corner of the drawing. Each view of the object represented is shaded independently of any of the others; and, when shading, the object is always supposed to stand in such a position that the drawing will represent a top view. Any surface that can be touched by drawing a series of parallel straight lines, making an angle of 45° with the horizontal and vertical lines of the drawing, is called a light surface ; a surface that cannot be touched by lines having this angle is called a (lark surface. All of the edges caused by the inter- section of a light and dark surface, or two dark surfaces, are usually shaded; that is, the edges thus formed are drawn in heavy lines. Exceptions to this rule are some- times made by experienced draftsmen, when a rigid adher- ence to it will produce a bad effect or will render the drawing ambiguous. Fig. 72 shows a plan of a series of triangular wedges radi- ating from the common center O. The top is, of course, a light surface, and, in order to determine whether the per- pendicular surfaces are light or not. the 45° triangle maybe used. Take the wedge R O A. A line drawn at an angle of §13 GEOMETRICAL DRAWING 75 45°, the direction of the arrows, would strike the side of which 0^4 is the edge; hence, this side is a Hght surface, and tlie toi) being also a light surface, the line 0.1 must be light. OK, on the contrary, is a heavy line, since the light cannot strike the side of which OK is the edge without passing through the wedge. Hence, this is a dark surface, and its inter- section OK with the light surface OA K re- quires a shaded line. For the same reason, AK is also shaded. „, . FIG. 72 1 he same reasonmg as the above applies to the lines OB, O D, OG, 01, OK, and O M\ also, to QN, ML, and KJ. C B \s not shaded, because the light strikes the surface of which C B is the edge, as shown by the arrow, making C B the intersection of two light surfaces. (9 A^ makes an angle of exactly 45° with the horizontal, and is treated as if it were the edge of a light surface; this is done in every case in which the line considered makes an angle of 45° with the horizontal. In shading holes, or any parts of the drawing denoting depressions below the surface under consideration, a slightly different assumption is made. Fig. 73 shows the plan of a square block with a hexagonal hole in the center. If the light passed over the surface A BCD, parallel to the plane of the paper as previously assumed, all the inside sur- faces would be dark, and the entire outline of the hexagon EFG H I K would be shaded. In order to prevent this and make the work similar to that which has preceded, the rays of light are assumed to make an angle of 45° with the plane of the paper when shading holes and 76 GEOMETRICAL DRAWING 13 FIG. 73 depressions. Hence, the light will strike the surfaces whose edges are G H^ HI, and /A", as shown by the arrows, leav- ing the surfaces whose edges are K E^ E F, and EG dark as before. Therefore, these latter edges will be shaded, and the edges G H, HE and /A" will be light. See also Fig. 71. The conventional method of shading circles which represent the projections of cylinders, or circular holes, is as follows: A B, Fig. 74, is the projection or end view of a cylinder having for a base the circular area A B. Draw the arrows E A and E B^ making angles of 45' with the horizontal diameter, and tangent to the circle at A and B. That half of the circle in front of these two points of tangency is to be shaded, and, in order to make the drawing look well, the center point for the com- passes is shifted along the line C H parallel to EA and E B in the direction of the arrow an amount equal to the thickness of the desired line. With the same radius that was used to describe the orig- inal circle, describe part of another circle, being careful not to run over the first circle, and stop- ping when the two lines coincide. The directions for sha- ding a hole are precisely the same as for the projection of a cylinder base, except that the half B C A oi the circle in Fig. 75 is to be shaded, the center being shifted as before, but in the opposite direction, as shown by the arrow. Fig. 74 §13 GEOMETRICAL DRAWING 77 Vertical projections of cylinders are shaded as shown in the front elevation of Fig. 5, Drawing Plate, title: Projections — I. After studying the foregoing concerning shade lines, the student should be able to see the reason for the using or omitting of any shade lines on the drawings in the following plates. In the case of an object like the hex- agonal prism in Fig. G, Drawing Plate, title: Pro- jections — I, no part of the upper base or line Se is shaded, although, strictly speaking, the part ^ ^ of the line should be shaded ; but, as this would make part of the straight line S c heavy and the greater part light, the whole line is drawn light. This is one of the exceptions previously mentioned. Fig. MECHANICAL DRAWING Plew. lop I Frohk iHew. A ce:n^ter liio]S 1. Fig. 1 represents a thick wedge having a cyHndrical hole running through its entire length. The lines inii^pq^ and }- s are called center lines. Center lines are usu- ally drawn through the cen- ter of anything that is round, such as a cylinder or a cylin- drical hole. In the case of a circle, there are usually two center lines, one being at right angles to the other, as shown in view />, Fig. 1. By drawing two center lines through a circle in the man- ner just mentioned, the cen- ter of the circle is located by their intersection. The mere presence of these lines shows in most cases that part of the object through which they are drawn is round. It is very seldom that center lines appear on drawings unless they are the center lines of cylindrical surfaces. They may some- times be drawn to indicate that the surface is a regularly curved surface, such as would be formed by circular arcs, or ■ g 14 For notice of copyright, see page immediately following the title page. £'^t^^ew. Fig. 1 2 MECHANICAL DRAWING § 14 one having the shape of an ellipse. In very rare cases, for some special reason, a line that corresponds to a center line may be drawn for some particular purpose, but such a line is not, in the strict sense of the word, a center line. 2, Center lines are an extremely important feature of a drawing, since the workman is guided by them in doing the work called for by the drawing. For instance, suppose that it was required to make a wedge like that shown in Fig. 1, and that the workman was given a piece of cast iron having approximately the shape indicated by the drawing. Sup- pose, further, that it was necessary to have the hole located exactly as shown in the drawing with reference to the sides of the wedge, and that the sides and ends of the wedge were all to be "finished." The first thing that the work- man would probably do would be to drill the hole, and, if the job had to be very accurate, he would drill the hole a little smaller than the drawing calls for and then ream it out to size. He would then face the ends square with the center line of the hole and make the length of the wedge the same as shown on the drawing. The sides of the wedge would then be planed and finally finished with a file, the workman working all the time from the center line ;//;/. If the drawing shown in Fig. 1 is intended to be worked to, it would, in most shops, be supplied with proper dimensions. SECTIOIS'S A30> SECTIOX LIXTN'G 3. In order to show the interior of hollow objects, they are often drawn in section, and the kind of material is then usually indicated by certain combinations of lines. Unfor- tunately, there is no universally adopted standard ; thus, a certain combination of lines may indicate that the material is cast iron if drawn in one office; in another office this same combination may have been adopted to represent brass, and so on. As far as working drawings are con- cerned, there is usually no difficulty experienced on account of this diversity of practice, since as a general rule the material is, and should always be, distinctly specified on the §14 MECHANICAL DRAWING drawing in order to prevent any mistake on the part of the workman. 4. The most commonly used combination of h'nes for different materials is shown in Fig. 2. Steel of all kinds is ^^y'yyWyS'yZ. 'WM^/M\ f;V^""'""V'-'\ f--— ^^ ^^mrrmmmvij,. D j: G m. H Fig. 2 indicated as shown in view A ; view B shows the style of sec- tioning employed for wrought iron. Cast iron is usually sec- tioned as shown at C\ brass and other similar copper alloys are sectioned in the manner shown at D. For lead, Babbitt, and similar soft metal, the sectioning shown at E is exten- sively used. Wood, when cut across the grain, is usually sec- tioned as shoAvn in the upper half of view F^ and Avhen cut along the grain, as shown in the lower half. Wood is also- frequently indicated on a drawing by section lines, even when it is not a section. Glass and stone, when in section, are often indicated in the manner shown by the upper half of view G\ when not in section, they are frequently drawn as shown in the lower half of that view. Concrete may be indicated as in view H\ view /gives a common representa- tion of leather. Rubber and wood fiber are sectioned as in viewy, firebrick as in /v", and water as in L. 5. Instead of representing sections by lines, they are occasionally colored, the colors used indicating the different materials. While this practice is very common in Europe, it is very rarely found in the United States. M. E. v.— 7 MECHANICAL DRAWING 14 6. Sections of material that appear too thin on a draw- ing to be conveniently sectioned, or when it is desired to make the section very prominent, are often blackened in, as shown in Fig. 3. In order to separate ■i^H^B different pieces, a Fig. 3 white line is then usually left between them. Black sections are most fre- quently employed for sectional views of structures com- posed of plates and rolled sections, such as I beams, angle irons, bulb angles, rails, Z bars. Fig. 4 7. On many sectional views, it will be noticed that the section lines do not run in the same direction. This invari- ably means that there is more than one piece in the section given. Thus, referring to Fig. 4, it will be seen that the section lining shown at b, b is at a right angle to the other section lining. It is the general rule among draftsmen that all parts of the same piece shown in section must be section-lined in the same direction, irrespective of the continuity of the section. Thus, referring again to Fig. 4, the fact that all section lining marked A is in the same direction immediately establishes the fact that this part of the view is a section of the same piece. Likewise, since the sectioning shown at ^, b runs in the same direc- tion, it follows that b, b are sectional views of one piece, which is separate from A. 8. The above rule governing the direction of section lines is always adhered to when possible ; when any depar- ture is necessary, care is taken to prevent ambiguity. Where only the sectional view is given, it is often very difficult to understand the drawing, and sometimes a violation of the S U MECHANICAL DRAWING above rule will cause an erroneous conclusion to be drawn. Referring to Fig. 5 (a), cover up the front view shown at {/?). Then, since the sectioning of A and B, and also that shown (a) Rocis removed, (b) . at b and b' , are respectively in the same direction, any one would be perfectly justified in assuming that . / and B was a sectional view of a rod fitted with a solid bushing b. 6 MECHANICAL DRAWING § U Furthermore, since C^ C and C\ C are sectioned the same way, the conclusion that they were the jaws of a forked rod would be justifiable. Referring now to view {b), it is seen that b and b' are separate brass boxes; the part B is seen to be separate from the cap A, and the note '^ Rods removed'" indicates that C is separate from C\ The way the sectional view should have been section-lined to correspond to the front \"iew shown at {b) is given in Fig. 5 if). 9. When a cutting plane passes through the axis of a shaft, bolt, rod, or any other sohd piece having a curved surface and located in the plane on which the section is taken, it is the general practice not to show such solid pieces in section, but in fuU. Thus, in Fig. 5 the sectional view is taken on the plane represented by the line x y^ which passes through the axis of the pin D. This pin is shown in full, however. The practice here shown is rarely departed from by experienced draftsmen, since it makes a drawing easier to read and also saves considerable time in making the drawing. 10. Fig. 5 also shows another feature that is frequently met with in shop drawings. Referring to the illustration, it is seen that no bolt is shown in the lower half of the object, as far as the front view {b) is concerned. A center line op is drawn in, however ; this center line indicates to the workman, who reasons from the symmetry of the object in respect to the center line x }\ that the lower half of the object is to be supplied with a bolt placed in the plane given by the center line op. In case of symmetrical work, draftsmen wiU frequently complete only one half of the view and merely indicate the other half by a few lines or not at all, trusting to the judgment of the workman for a correct reading of the drawing. In the best practice, a note is made on the drawing calling attention to the fact that the indicated portion of the A-iew is a duplicate of the complete portion. § li MECHANICAL DRAWING BREAKS 11. When a long- and comparatively slender object is to be drawn, it often happens that, when drawn to a sufBciently large scale to make it intelligible, it will extend beyond the space avail- able. In such a case, part of the object is broken out and the remain- ing ends are pushed together. The fact that part of the object is broken away for the sake of convenience is indicated by a so-called break. It is always understood that the part broken away and not shown is of the same size and shape as the parts con- tiguous to the break. In some cases, one end of the object is broken awav (e) ^ (b) (c) (d) (f) (g) (It) Fig. 6 ] Z2 ID U 1'^. Breaks may be indicated in various ways; most commonly, the break is given an outline that will reveal the shape of the object. Con- ventional methods of indicating breaks are shown in Fig. G. Wood is usually shown broken in the manner illus- trated at {a), angle irons as at {b), T irons as at (r), Z bars as at {d). Cylindrical objects are occasionally broken as shown at (r), but most frequently in the manner shown '^t {/)■ Pipes and similar hollow cylindrical objects may be broken as shown at (,;') ; but, more frequently, the break is made as shown at (//). Rectangular objects may be broken in the manner shown at (z); plates and objects other than those included between views (c?) and (/) are often shown broken off by drawing a wavy freehand line as in (/) and {k). 1 MECHANICAL DRAWING ? 1-t HIDDEX SCREW THREADS 13. When the screw thread is hidden by part of the object and it is deemed necessary to show it in dotted lines, it is usually drawn in one of the four ways illustrated in Fig. 7. Of these the method shown in Fig. T (a) is probably r V - Fig. 7 the clearest; that shown in Fig. 7 (d) is fairly good; and the one shown in Fig. 7 (c) is cheap. The method illustrated in Fig. 7 (V /ro//. The abbreviation 77/(/s. or ///t/s., with a number prefixed, stands for " threads per inch " ; thus, 14 thds. means " make 14 threads per inch." The word tap, 10 MECHANICAL DRAWING § 14 with a number prefixed, always means that a hole is to be finished by tapping it with a tap of standard proportions, having the diameter given by the prefixed number. When a tap other than a standard tap is to be used, it is distinctly specified. Drill is always taken to mean that a hole is to be put through the object by drilling. Bored or Bore means ''finish a hole by boring it." Planed is always understood to mean *' this surface is to be finished by planing." Cored implies that the hole to which it is applied is to be cored out and left that Avay ; that is, it is not to be finished by machin- ing. Faced almost invariably implies that the surface to which it is applied is to be machined square with a hole in the object. Turtied \s an abbreviation for ''finish by turn- ing." Scraped implies that a surface is to be finished by scraping. Tool fiuish means that the surface, after machin- ing, is not to be finished any further. Black, on objects formed by forging, implies that the part to which it is applied is to be left as it comes from the smith. The term Ream or Reauied means that a hole is to be finished by ream- ing; when applied to a bolt, it is understood that the bolt is to be fitted to a hole that has been previously reamed. The terms Shrinking Fit, Forcing Fit, and Driving Fit written behind a dimension always imply that, in machining the part, the workman is to make the allowance necessary for the kind of fit called for. The fact that part of an object is to be finished by machining, filing, or grinding is often indicated by marking the outlines with an f written across or near it, or writing yf;/. along it. In some cases, draftsmen will draw a dotted line or a full red line at a little distance from the outline and write /"across it; it is usually under- stood, in that case, that the lengths of the supplementary lines denote the extent of the surfaces that are to be finished. KIXDS OF TTORKIXG DKAWIXGS 18. Working drawings are divided into two general classes, which are: assembly, or general, drawings, and detail drawiiisrs. § 14 MECHANICAL DRAWING 11 Assembly, or j>-eiieral, (ll•a^villy•s show the workman the relation between, and the places or positions occupied by, the different component parts of a structure, machine, device, fixture, implement, etc. If any dimensions are given, they are usually only leading dimensions. Detail dra^vinjifs show the exact shape and size of each integral part. For this purpose they are supplied with all the dimensions required by the workman and any additional explanatory notes that the draftsman may consider nec- essary. Detail drawings may be made so complete that they will answer for the patternmaker, blacksmith, and machinist, and they are usually so made in the smaller shops. In the large shops, however, separate drawings are often made for the patternmaker, blacksmith, and machinist; the detail drawing for the use of the patternmaker, then, contains only the dimensions and notes needed by him to make the pattern; that for the blacksmith contains the dimensions needed for making the forging; and, finally, that for the machinist contains all dimensions needed by him. 19. Attention is called to the fact that practice varies somewhat in different places in regard to the dimensions given on detail drawings, at least as far as drawings for the patternmaker and blacksmith are concerned. In some places, the dimensions given represent the size the object is to be when Jill ishi'd ; hence, the blacksmith or patternmaker must make necessary finishing allowances himself. In other places, again, the finishing allowance has been, and usually is, made by the draftsman ; the dimensions given are then those of the pattern or forging. If in doubt about the practice followed in a particular drawing office, it is a good plan to find out by inquiry what system is used in the shop under consideration. In the best modern practice, a note calling attention to the fact that the sizes given are those when finished is placed on the drawing; thus, " All finisJied sizes " or some similar note. 12 MECHAXICAL DRAWING 1-i \^ SCALES 20. When it is desired to make a drawing other than full size, special scales are used. Thus, suppose it is required to make the draw- ing \ size ; then, 3 inches on the drawing would represent 1 foot on the object. Hence, if 3 inches are laid off and divided into 12 equal parts, each of these parts will represent 1 inch on the object. If these parts be subdivided into •2, -i. 8, etc. parts, each will represent 3. ^. i. etc. of 1 inch on the object. A scale of this kind is called a quarter scale, or a scale of 3 iuclies to tlie foot. An eighth scale, or a scale of 1^ inclies to tlie foot, would be constructed in the same way, except that \\ inches would be laid off instead of 3 inches. These scales are written 3" = 1 ft., U' — 1 ft. 21. Fig. 9 shows a scale which is con- venient for the student, inasmuch as it com- bines eleven different systems of subdivision and may be used for all the work ordinarily done in a drafting room. This scale is tri- angular in section and 13 inches in length, and on each of its edges is laid off a scale, as shown at A^ B, and G. The scale at G is "full size"; that is, this edge of the scale is divided into inches and fractions of an inch down to sixteenths, and is used for drawings in which an object is represented in its natural size. On its opposite side, at B, is shown the quarter-sized scale of 3" = 1 ft. The first 3-inch (actual size) di\nsion, from B to C, is subdivided into 12 parts representing inches, and each inch is then divided into proportional fractions of an inch, generally eighths. From C toD, D to E, and E to F, the scale is marked § U MECHANICAL DRAWING 13 in its main divisions of 1 foot each, each foot being 3 inches long, actual size. From A to B the scale is independently divided into spaces of U inches (actual size) to form an eighth-sized scale, or lf=l ft., the divisions of the latter occurring on and between the marks for the 3-inch scale. The other sides and edges of the instrument are divided into scales of 1 inch and ^ inch, f inch and f inch, ^ inch and ^ inch, and j\ inch and ^% inch to the foot. Different makers do not always arrange their scales in the same man- ner. Thus, instead of having a full-size scale and scales of 3" = 1 ft. and IV = 1 ft. on one side, as shown in Fig. 9, some makers have the full-size scale and j\" = 1 ft. and -i/ = 1 ft. on one side. It will be observed that the num- bering of the feet on these scales does not start at the end of the instrument, but at the first division from the end. Thus, on the quarter-sized scale the zero mark is placed at C and the first foot is measured to D. This is done so that the feet and inches may be laid off independently and with one reading of the scale. The figures indicating the number of feet on this scale are placed along the extreme upper edge at D, £, and F, the numbers running in a direction away from the part containing the inches. The numbers indicating inches run in an opposite direction from those defining the feet. To lay off 2 feet 3f inches on a scale of 3" == 1 ft. and from a given point, place the scale on the point so that the 2-foot mark will be directly over it; then from the zero mark C lay off 3J inches, as shown, locating a second point. The length of the distance thus laid off between the two points represents 2 feet 3J inches. The scale of 1^" = 1 ft. is used in a similar manner to lay off the same distance. The figures indicating feet on this scale are placed nearer the edge, in order to prevent confusion in reading. To draw to half size, or d inches to the foot, use the full- size scale, and remember that every ^ inch on that scale 14 MECHAXICAL DRAWING § 14 corresponds to 1 inch on the object, that is, that every dimension is only half of the real length. To lay oflF of inches, lay oflF 5 half inches and ^ of an inch over; the result is a line 5f inches long to a scale of 6 inches to the foot. If it is desired to draw to a scale of f of an inch to the foot, or -^^ size, the scale of 1|^ inches to the foot may be used if the draftsman has no scale of f = 1 ft., halving all dimensions, as in the previous case of drawing to a scale of 6 inches to the foot with a full-size scale. It sometimes happens that a draftsman is obliged to make a scale, when the size of his plate is limited and a general drawing of some object is desired. By general drawing is meant a complete view of the object in plan and also one or two elevations. In such a case, one scale may be too large to enable the drawing to be made on a sheet of the required size ; another ^^ scale may make it too small to show up weU. For example, a \ scale may be too large and a ^ scale too small ; a ^ scale may be just right. If the draftsman has no ^ scale (that is, a scale of 1 inch to the foot), he may make one by taking a piece of heavy drawing paper and cutting out a strip about the size of an ordinary scale and laying off the inch divisions on it. Each division or part will represent 1 foot on the object. Divide one of the end parts into \% equal parts and each will represent 1 inch on the object. Lines indicating half and quarter inches may be drawn if considered necessary. Fig. 10 shows part of a scale made in this manner, giving feet, inches, and half inches — ^the quarters, eighths, etc. of an inch being judged by the eye. SU MECHANICAL DRAWING 15 COXTENTIONAL. REPRESENTATI0:N^ OF A NUT 22, Fig. 11 shows the ordinary conventional method of representing a nut. The bottom of the thread is 1^ inches in diameter and is represented by the dotted circle; this shows that it is intended for a screw li inches in diameter. The height of the nut equals the diameter of the bolt or screw on which the thread is cut. The two views on the center line m n should be drawn without difficulty. To draw the curves e a and a d, project b and c at right angles to / V in the points d, a, and c\ pass arcs of circles through e and a and through d and a tangent \.o f g, finding the cen- ters of these arcs by trial. The best way of doing this is to draw lines parallel to / v midway between c a and between a and d. Then, by trial with the compasses, find a center on these lines such that an arc struck with the compasses from this center will pass through ^and a (or a and d) and be tan- gent to f g. In the right-hand view, the radius of the arc be is the same as the height of the nut; the centers of the other two arcs are found by trial in the manner just described. 16 MECHANICAL DRAWING § 14 DRATTTNG PI^VTZ. TITLE: DETAILS 23. The first eight figures of this plate show the conven- tional methods of representing screws. The actual projec- tion of a screw thread will be similar to the projection of a helix; but in order to save the time required to locate the points and trace in the curves, the following methods are universally used, except, perhaps, in the case of screws of ven,- large diameter and pitch, drawn full size. 24. Fig. 1 represents a single square-tlii-eaded screAv Ih^ inches in diameter and f inch pitch. To draw the screw, first draw the center line /// fi and a line A B at right angles to it. Make the distance A B equal to the diameter of the screw, or \\ inches, and through the points A and B draw lines AD and BE parallel to the center line ;//;/. Also lay off on the line A B distances A F and B G equal to one-half of the pitch, and through the points F and G draw lines FH and GI parallel to the center line ;//;/. These lines show the depth of the thread. On the line A D lay off the width of the thread and of the groove, A C, C J^ J K, etc., each equal to one-half of the pitch, or f x i = ^ inch. Draw the line BC^ and through the points y, K^ L, J/, etc., draw lines parallel to BC. Draw faint pencil lines through the points C and P, J and Q^ K and R^ etc. to represent the back edges of the threads, and ::ake the parts that are seen full lines; then :raw the lines T V, UW, etc. The method of drawing the remainder of the screw and the reason for using the heav\' shade lines, as shown, should be apparent without further explanation. It will be noticed that the width of the thread and of the groove, measured parallel to the cen- ter line ///«, and the depth of the thread are all exactly the same ; that is, the\- are each equal to one-half of the pitch. If a section were taken through the center line in ;/, the thread and groove would look like Fig. 1-2, a series of S^;. Fig. 1-2 squares, hence the term square thread. DET/ JUNE 25. /693. For notice of copyright, see page it L5. '■esentzrz.Q Screu/3 m_ T ± CBAN/6. Wrnu aht Iron^ Fu.llST.ee Fintsheai Ft,y /O. diately following the title page. JOHN SMITH. CLASS N9 4-529. § 14 MECHANICAL DRAWING 17 25. Fig. 2 sliows a double square-threaded screw 1^ inches in diameter and with f of an inch pitch. The reason for using a double thread is that if the single square thread were used, the depth would be so great as to weaken the bolt or rod on which it was cut and render it unsafe for the purpose for which it was intended. To prevent this, either the diameter of the rod must be increased or the thread must be cut of the same depth and thickness as a thread of half the pitch, or, in this case, as if the pitch were f X i == f of an inch, as in the preceding problem; another thread of the same size and pitch (J of an inch) must be cut half way between these first threads, thus giving a double thread. The pitch, or distance that the screw would advance in one turn, would be f of an inch, the same as if it were a single- threaded screw of f of an inch pitch, while the depth of "the thread is only half as great. To draw it, proceed exactly as in the last figure. To get the direction of the line B C, which in this figure represents the projection of the bottom edge of the top of the thread, lay off A C equal to one-half of the pitch, or f x i = f of an inch, and draw the line B C. The width of the threads and grooves, and also the depth of the threads, is one-fourth of the pitch, or f x i = yV inch. Through the points A', Z, J/, etc. draw faint pencil lines A" A^ etc. to represent the back edges of the threads, and make the parts that are seen full lines. Through the point A' draw a faint pencil line A' A' at right angles to the center line vni, intersecting the line F H in a, and draw the line T(7, which represents the bottom of the thread. The remainder of the screw should now be drawn without any trouble. 26. Fig. 3 is a single V-tlireaded screw 1^ inches in diameter and having 7 threads to the inch; that is, the pitch is 1 of an inch. Draw a cylinder 1]- inches in diam- eter, having ;// // for the center line. Lay off A B, B C, CD, etc. each equal to the pitch, or \ inch. Do the same on the left-hand side. By the aid of the T square and 60° triangle make the angles A OB, BO' G, etc. The rest of the thread can be drawn by referring to the figure. 18 . MECHAXICAL DRAWING .§ U 27. Fig. 4 represents a screw exactly like the preceding one, except that the thread is left-handed instead of right- handed, as in the previous case. To ascertain whether a thread is left- or right-handed, hold the screw in such a position that its axis is horizontal. If the thread is right-handed, as it usually is, the angle that the edge of the thread makes with the horizontal on the right-hand side is obtuse ; if left-handed, it makes an acute angle Avith the right-hand side of the horizontal. Xo fur- ther instruction should be necessary for drawing the thread. 28. Fig. represents a double V-tlirea«5. In Fig. 2 are shown a section and end view of a shaft flangre coiiplinj?, used for connecting the ends of two shafts. In order not to take up too much room on the plate, the coupling is drawn to a reduced scale, which is marked on the drawing as 3" = 1 ft. Using the scale of 3" = 1 ft. (see Art. 21), draw the figure from the dimensions given, first drawing all of the sectional part, except the bolt. Then draw the end elevation as shown, and, lastly, the bolt in the sectional view. The part A B C is a key sunk into the shafts to keep them from turning within the coupling. The shafts themselves are made of wrought iron, as indicated by the sectioning. 36. Fig. 3 is a front and side elevation of a gland. This is also drawn to a scale of 3" = 1 ft., and should be constructed without difficulty from the dimensions given. The line OA, marked Of " r, means that the radius of the arc BAD is 6| inches long; so, also, CB, marked 11" r, means that the radius of the arc B£ is 1^ inches long. Since no dimensions are given for the other two arcs, it is understood that their radii are the same as for the first two. Whenever a dimension is given, like 0^1, specifying the length of a radius, the letter r, or the abbreviation rad., should always be placed after it. 37. Fig. 4 shows a riveted joint, or two plates riveted together. The front elevation and a cross-section through one of the rivets are given. The diameter of the rivets is ^ inch and the pitch (the distance between the centers of two consecutive rivets) is 14- inches. These and the other dimensions are obtained from the drawing. Draw the cross- section first, as shown in the figure, and after that the elevation. The scale is full size. U MECHANICAL DRAWING § U 38. Fig. 5 shows a clanii}, clog, or carrier, as it might be termed. It can be readily drawn from the dimensions given. It will be noticed that part of the pin A B is flat- tened. This is seen more clearly in the top view, shown by the dotted lines. It is to be drawn to a scale of 3' = 1 ft. 39. Fig. 6 shows a clamp box to be attached to beams for shafting to pass through. The scale to which it is drawn is half size, that is, 6" = 1 ft. The curves at A, B, etc. are not exact projections of the curve of intersection of the round and flat surfaces shown in the other view, but are drawn as circular arcs, from which they differ but little, and the time occupied in finding the different points on the curve is saved. This answers the purpose when making shop draw- ings just as well as the exact method. The curve C F D is the projection of the part D E F, shown by the dotted lines in the elevation, which is removed in order that the nut may be turned round. G is an oil hole. It will be noticed that the bolt holes are larger than the bolts; this allows the clamp box a little play, should it be " necessary. It also allows the holes to be cored in the box when the casting is made, instead of being drilled afterwards. On all drawings made for the shop or on which dimensions are given or required, the scale should invariabh' be specified. If more than one scale is used, as in the last plate, where three dif- ferent ones were used, it should be given for each figure. DRAWIXG PliATE, TITLE : FI^AJSTGE COI'PEIXG JrO. This plate shows a drawing of a flange coupling suitable for connecting two lengths of 2-inch line shafting. Fig. 1 is a section on the line A B (Fig. 2) and shows how the two parts C and D of the coupling are bolted together through their flanges, hence the name flange coupling. Each part is keyed on its shaft separately and true alinement of the shafts is insured by means of the recess in C into which is fitted a raised boss on D. The two parts of the coupling are first bored and then faced up on the surfaces EFGHIJ. FLANGE I -_7/ JLOVF 2c, J593. For notice of copyright, see page il DUPLINB 'ZSize. Fvq e ediately following the title page. JOM// SMITH, CLASS H^ 4529. § U MECHANICAL DRAWING 25 They are then clamped together and the keyway is cut. Fig. 2 is a half-end view and needs no comment. 41. To begin the drawing, draw the horizontal center line xj' G| inches from the lower border line. Draw Fig. 1 first, commencing with the vertical joint line E/ of inches from the left-hand border line. It is now well to draw the shaft. This is 2 inches in diameter; therefore, lay off vertically on each side of .vy, 2 -=- 2 = 1 inch, and through the points thus located draw the horizontal lines ^^?' and /;//. Next draw the hub of the coupling. Lay off 3^ inches hori- zontally on each side of the vertical joint line £/ and draw cc' and dd'. The diameter of the hub is 4|- inches; therefore, lay off 4^ -=- 2 = 2\ inches on each side of xj' and draw t'e' and /"_/'; then draw in the round corners c' e, c' d\ df\ and r/" with a radius of \ inch. To the left of the joint line EJ lay oft" a distance of ^ inch and draw H G\ lay oft" 3-4-2=1^ inches on each side of x y and draw 6^/^ and HI. On referring to Fig. 2, it will be found that the outside diameter of the coupling is 9 J inches; lay off half this diam- eter, or 4:|- inches, on each side of the horizontal center line X y and draw g g' and Ji k'. Each flange is If inches in width on the outer face; lay off If inches to both right and left of E J and draw the vertical lines ii' and 77', putting in the rounded corners i, i' , etc. with the compasses set to ^ inch, in each case completing the circle faintly. Now, draw faint vertical lines k k' and //' each -if inch from the vertical, joint line EJ. Make k k' and //' each equal to 8| inches; to do this lay off 4:^ inches vertically on each side of xy and draw faint construction lines /'' /', k I. Then with the compasses set to a radius of /^ inch and a center on the line /'/'' produced, draw the semicircle k' mil. Draw no just touching this semicircle and tangent to the dotted circle mentioned above; then draw the other three similar parts of the flange in the same manner. Now, by reference to Fig. 2, locate the bolt center lines //' and qq' ; draw the bolts and nuts and complete Fig. 1 from the dimensions given. 26 MECHANICAL DRAWING § 14 The reference letters printed in bold-face italics should be omitted on the drawing made by the student. I>RA^T::N^G PI^^TE, title: ECCEXTRIC AZS'D BRAKE LEA'ER 42. Fig. 1 shows an elevation of an eccentric and its strap. The strap is made in two pieces and bolted together with a small space gV inch wide between them. Locate the point O, the center of the strap, and draw the center lines A B, m ;/, C C\ and D D' . Make the offset O 0' one-half of the throw of the eccentric, or, in this case, |- inch, thus locating 0\ the center of the eccentric shaft. Con- struct the rest of the view from the dimensions given, noting that the arcs E E' and EP are concentric with O, while G G' and H H' are concentric with O'. The part F F G' G is entirely open and is made so in order to lighten the eccentric. •43. Fig. 2 is a section of the eccentric an". We will first draw the two 3" X 8" braces. From each side of the left-hand plumb post lay off a distance of 3 inches and draw vertical lines extending from the upper edge of the girts to the lower edge of the cap ; on either of these lines lay off from and below the line d d' a distance of 3 inches and draw the short horizontal lines representing the ends of the diagonal braces seen in this view. Now draw the 6" X 8" brace by laying oif on each side of the center line ff one-half the thickness of the brace, or 6-^-2 = 3 inches, and drawing vertical lines. This brace extends 1\ inches below the top of the sill and li inches above the lower edge d d' of the cap; lay off these distances and draw the short horizontal lines representing the ends of the brace. Each bent is supported upon four piles, two of which are located immediately under the plumb posts, the centers of the other two being 18 inches from the ends of the sill. The tops of the piles are tenoned and enter for a depth of G inches between the two timbers forming the sill. 32 MECHAXICAL. DRAWING ^4 The student should be able to draw these piles without special instructions. Next locate the bolt centers; begin by drawing the center lines of the batter posts and sway- braces in lead pencil, and from the drawing locate all bolts on these lines. Where the timbers cross each other and are bolted together, draw in lead pencil the short diagonal of the diamond formed by the crossing of the two timbers; the point where these diagonals meet is the center of the bolt. Now locate all bolts on the center lines /y and gg' and then the bolts on the center lines of the two outside piles. Dimensions are given which will enable the student to locate all other bolts. All the bolts are f inch diameter, with heads and nuts f inch thick and 1^ inches square. The washers are all 3 inches in diameter, with the exception of those marked 3| inches in Fig. 2, and are all f inch thick. In manj- places washers are introduced between timbers to act as separa- tors; these are all 3 inches in diameter, with the exception of those between the stringers, which are o\ inches in diameter. 4T. We are now ready to proceed with Fig. 2. Draw the vertical center line s s' 2iX. a. distance of 2^ inches from the left-hand border line. From and to the right of s s' lay off a distance of 15 feet and draw the center line ft'. Pro- jecting from the lower edge of the sill in Fig. 1 draw short horizontal lines a b and c d and also draw the lines cfz.n6.gh by projecting from the top edge of the cap. On each side of the center line t f lay off one-half the distance between the two timbers that form the sill, or 3" ^ 2 = \\ inches, and draw vertical lines between cd and g/i. To the right and left of eg and dh, respectively, lay off a distance of G inches and draw vertical lines as before. Projecting from the top edge of the brace-block in Fig. 1 draw the horizontal line J k and complete the end view of the brace-block by drawing the short vertical lines gj and // /'. The ends of the pieces forming the posts are separated by tenon blocks 3 inches thick and 2 feet 8 inches long. From and below the line sr h lav off a distance of 2 feet 8 inches and draw § U MECHANICAL DRAWING 33 the line ////. From and above the line r^/lay off a distance of 6 inches and draw a dotted horizontal line ; above this line lay off a distance of 2 feet 8 inches and draw the line )io. In a similar manner draw the sill, post, and cap, etc. about the center line s s' . Projecting from Fig. 1 draw the horizontal lines /^ and r s" representing the upper and lower edges of the stringers. These stringers are 30 feet long and their joints are staggered ; the joint on the center line 1 1' is represented by a full line, while that on the center line j-/ is a dotted line. Projecting from Fig. 1 draw in lead pencil horizontal lines representing the upper and lower edges of the ends of the ties. From and to the right of the center line s s' on the line r s" lay off a distance of 3 inches and draw a vertical line, as shown ; on each side of this vertical line and along the line r s" space off distances of 15 inches and draw vertical lines through each of the points; to the right of each of these vertical lines lay off a distance of 9 inches and draw vertical lines to complete the end views of the ties, as shown. Projecting from Fig. 1 draw the upper edges vzu and lower edges t" u of the guard rails, which are notched 1 inch over the ties, and next draw the horizontal lines xy, etc. repre- senting the longitudinal girts, and then draw the diagonal braces, as shown. Now, to the left of the vertical line eg and to the. right of dh lay off a distance of 3 inches and draw vertical lines defining the sway-braces. Then draw in the bolts from dimensions given and by projecting from Fig. 1. 48. Next draw Fig. 3. Draw the horizontal center line X x' at a distance of 3^ inches below the top border line. The student should experience no diffi- culty in drawing this figure; vertical lines should be projected from corresponding lines in Fig. 2, and all measurements may be found by reference to Figs. 1 and 2. The reference letters printed in bold-face italics should be omitted on the drawing made by the student. 34 MECHANICAL DRAWING 11 deawtxg pi^te, title : steel coeuimxs ais^d cox:nt:ctioxs 49. In this plate are shown two designs of columns used in the construction of modern office buildings, and a common method of connecting the floorbeams to the columns is shown. A side view and a front view of each design is given, and as it would manifestly be impossible to draw the columns in their full length, and since, besides, it would not serve any useful purpose to do so, a part of the column between the base and the floorbeam connection is broken away. When this is done, it is understood that the part broken away is similar to the ends of the column next to the break, which in this case is indicated by drawing a line consisting of a long dash and two dots across the column. Among mechanical draftsmen it is customary to indicate a break by a wavy freehand line, but many architectural draftsmen indicate a break in the manner shown in this plate. 50. In the design shown by Fig. 1, the column consists of two channels placed back to back and tied together by ,_ ^_^ — --, cover-plates. A cross-sec- tion of this column is shown in Fig. 13 (a). In the design shown by Fig. 2, the column is built up of four Z bars riveted to a web-plate, the column hav- ing the cross-section shown in Fig. 13 (d). The columns in both designs are made in sections, the lower section being 22' 8' long. The upper sections are spliced to the lower sections by means of horizontal plates, angle irons, and splice plates, the joint ;// w being a short distance above the floorbeams. (a) (b) Fig. 13 51. In the drawings there will be noticed several conven- tionalities. There are a number of rivet holes shown black in the side views (the views having the center lines c d STEEL COLUMNS JUNE 25. 1693. -oflce of copyright, see pace kND CONNECTIONS. lediately following the title page. JOHN SMITH. CLASS N? 4529. § 14 MECHANICAL DRAWING 35 and g- h) of both designs. This indicates to the shop men that the rivets destined for these holes are so-called field rivets; that is, they are to be driven in at the place where the structure is being erected and after the columns are in place. The position of the field rivets is indicated in the front views (the views having the center lines a //'and e f) by blackening in ihQ position of the holes. According to the rules of drawing, their position would be shown by dotted lines, but the conventionalism of blackening in is resorted to as better calling the attention of the workmen to the fact that the rivets destined for these holes are field rivets. This common usage is departed from only in the case where the parts through which the field rivets pass is hidden behind some other parts of the structure. Thus, referring to the front view of Fig. 1, field rivets are shown as passing through the plate and two angle irons at the splice, and as neither the plate nor the angle irons are hidden, their position is shown in black. In the side view, however, the horizontal plate at the splice is hidden behind the splice plate and the angle irons are hidden behind the splice plate and the legs of the channels; consequently, the field rivet positions are shown in this view by dotted lines. 52, All the rivets that are to be driven in the shop where the column is made are shown in position; in some drawing rooms, in order to save time, the rivets are omitted, their position being merely indicated by two center lines. In order to show the rivets clearly, their heads are shaded a little in the manner shown, using the bow-pen for the shading. 53. The dimensions of the channels, I beams, and Z bars are not given in detail, this not being necessary on a working drawing. All that is required is to give the size of the channel, etc. The notes giving the sizes are read as follows: On the front view of the left-hand design we find a note r>-15"-33-22'-7|" 0; this means two 15-inch channels, 33 pounds per .foot of length, 22' 7|" long. The note .1/. E. V.—g 36 MECHANICAL DRAWING S U on the front view of the right-hand design "4-6" X 3|-" ,-22.7-Z Bars" means that four Z bars having a depth of 6 inches, legs 3^ inches wide, and weighing 22.7 pounds per foot of length are to be used. The note " 2-12"-40-I Beams " means that two 12-inch I beams weighing 40 pounds per foot of length are to be used. The note "4" X 3^' X yV" X llV' Angle " means that an angle iron with legs measur- ing 4 inches and 3i inches, ^ inch thick, and 11^ inches long is to be used. T i,\^" ■^/' e 54. In order to aid the student in drawing the various parts the dimensions of which are not fully given on the plate, sectional views of them, fully dimensioned, are given in Fig. 14. The section of the 10-inch I beam is given in Fig. 14 (a), of the 12-inch I beam in Fig. 14 (/;), of the lo-inch channel in Fig. 14 (c), and of the Z bar in Fig. 14 (fue -'i — Cne of Ms^ Toe TOOL POST -20 Thds, J fofitbody One cf ff!/s, Tool Steel. ADJUSTING WASHER. Cne of Ihis, Tod Steel- BACK REST CLAMPING I '^q; oij^ One cf Ihis , Macfi/nery Steel. TOOL POST. -NuH. -2i 2 if this, tifacfi/nery Steel ADJUSTING NUT AND STUD FOR BACKREST. ! I I ! i I ^ /i'ardened on lip J 2 cf this, fiJactJinery Steel . BACK REST JAWS. THE SCRANTON TOOL WORKS SCRANTON, Pa., Turret Lathe Tools For Xo. 1 Tmret Lathe MATERIAL AS SPECIFIED Scale Full Size. Dale Oct. 25. 1901 Draivn tylA/IA/.C. a7ecke^ fpyJ.A.G. Dnawet- 157. refer No. 154.. PTTT LU Cne of tfiis. Machinery Sreel . HOLLOiVMILL HOLDER. '8Th'ds. i J- ^ One of tfiis. Tool Steel, Hardened. MILL HOLDER SCREiV. four cf this. Tool Steel, Harden RELEASING DIE\ JUNE 25, 1893. For notice of copyright, see page : ^ F/n/s^ on/y where markec/f. One of this. Steel, Casting. BODY. ROUGHING BOX TOOL ■^/ i ^f— rFlat. Finish a/I oii-er. ^ One of this, Mact7inefy Sfee/. Finish all oi^er. SLEEl/E FOR RELEASING DIE HOLDER. REI/I/S. One of this, Machine/y Steel. DIE HOLDER, FOR RELEASING DIE HOLDER. NO. 1984 lediately following the title page. JOHN SMITH, CLASS N? 4529. § U MECHANICAL DRAWING 39 view shows to be at a distance of 1 l inches from the center line of the shank, by drawing the center line j k to inter- sect eg. While all the vertical lines can be projected up from the side view and some of the horizontal lines can be drawn from the dimensions given, the location of other horizontal lines must be determined from the end view. Since it often occurs in making drawings that a third view must be con- structed by reference to one of the other two views from which points or lines cannot be projected, the operation will be explained in detail. The center line h i of the top view is imagined to be the top edge of a plane at right angles to the paper ; and in the end view, the front edge of this imaginary plane is given by the center line I vi passing through the center c of the shank. Suppose we Avanf to locate the edge ii of the end view in the top view. Then, Ave take the perpendicular distance of n from the plane Im with the dividers and lay it off in the top view at right angles from the plane h i and upA\>ards, drawing the line ;/' through the point thus laid off. In case of doubt as to which side of the imaginary plane of the view being drawn a certain point or line to be transferred from a view at right angles to it is located on, simply imagine the center lines representing the edges of the imaginary plane to be pro- duced until they intersect. Then, everything Avithin the 90° angle in one vieAv is Avithin the same angle in the other view; likeAvise, everything Avithin the 270° angle in one vicAv is Avithin the same angle in the other vicAV. Thus, if Ave imagine the lines // / and I vi to be produced until they intersect, the edge ;/ Avill lay Avithin the 270° angle and must, consequently, be laid off from // / npivards, in order to be located in the same angle in the top view. With this explanation of transferring points and lines from one view to another, the student will experience no difficulty in com- pleting the top view. 6-1:. The body having been draAvn, draAV the tool post, locating its center line 1 ,V inches from the left-hand border- line. The center of the top view should be about 1 j3^ inches 40 MECHANICAL DRAWING § U and the lower end of the tool post of inches below the upper border line. It will be noticed that the hole for the tool- post screw has no thread shown in it. the information that it is threaded being conveyed by the note " fg " Tap, ISThds." On working drawings a thread is rarely shown in a hole which is to be tapped ; the general rule is to draw two lines a distance apart equal to the outside diameter of the thread and to place a note on the drawing stating the size of tap to be used. This practice has been followed on the plate " Turret-Lathe Tools." 65. The center line of the adjusting washer may be located -iy^g inches from the left-hand border line ; the center of the top view may be 1 ^^g. inches and the lower end of the front view 3f inches below the upper border line. Locate the center line of the tool-post screw ^ inch below the upper border line and the right-hand end of the side view 8f^ inches from the left-hand border line. The center of the end view may be f inch to the right of the right-hand end. The center line of the back-rest clamping bolt should be drawn 2yg- inches below the upper border line ; the right- hand end of the side view may be placed T|- inches from the left-hand border line and the center of the end view \^ inch from the right-hand end of the bolt. Locate the center line of the adjusting stud o^^ inches below the upper border line and the left-hand end 3f inches from the left-hand border line. The adjusting nut being movable on the stud, it may be shown in any convenient position, say about in the posi- tion shown. The top line of the back-rest jaw may be drawn ^f^ inches below the upper border line and its left- hand end may be located ^ inch from the left-hand border line. A distance of f inch may be left between the two views of the jaw. 66. The center line of the hollow mill holder may be located 3f inches above the lower border line and the center of the end view 5|^ inches from the left hand border line. A space of ^ inch mav be left between the front view and the side view. § U MECHANICAL DRAWING 41 67. The center line of the releasing-die-holder sleeve should be located 4|- inches above the lower border line and the center of the end view 4:j\ inches from the right-hand border line. A space of f inch should be left between the end view and side view. The right-hand end is a helical surface having a pitch of :!■ inch ; it can be laid out by means of the method given in Geometrical Dratving, laying off helixes of \ inch pitch and | inch diameter and \ inch pitch and \ inch diameter. In the machine shop this helix would be cut by gearing the lathe to cut a thread of 4 to the inch and using a flat side tool set with its cutting edge at right angles to the center line of the sleeve. The center line of the die holder should be located 2 inches above the lower border line and the center of the end view 6|- inches from the right-hand border line. A space of y^ inch should be left between the end view and the side view. 68. The releasing-die screw and mill-holder screw should have their center lines drawn Ifi- inches above the lower border line. The center of the end view of the mill-holder screw may be placed \\\ inches and the center of the end view of the releasing-die screw 7yV inches from the left-hand border line. A space of f inch should be left between the end views and side views of the screws. 69. All the notes on the drawing, in accordance with the most general practice, are Avritten in capital and small letters, the former being ^\ inch and the latter J^- inch high. Detailed instruction for this style of lettering have been given in Geometrical Draiving. The names of the different parts are written in capitals ^ inch high. 70. This drawing plate differs from those previously given in that the title is placed in the lower left-hand corner. Different drawing offices have different rules in regard to titles for a drawing; some insist on having all titles placed in the lower left-hand corner and others place it in the lower right-hand corner; more rarely one of the upper corners is selected. In the arrangement of the title and the informa- tion contained therein, practice also greatly varies; this plate 42 MECHANICAL DRAWING § U simply shows the practice of one drawing office, giving the name and place of the firm, the title of the drawing, the mate- rial, the scale, the initials of the persons who made the drawing and checked it, the date, the number of the drawer in which the tracing is kept, and the number of the drawing. "When a number is given to a drawing, it generally, but not always by any means, gives the total number of drawings that have been made in a particular drawing office. Since practice varies so much, it can only be found by inquiry at the office where the drawing was made what a number on a drawing signifies. 71. To draw the title, construct a rectangle S^ X 2A- inches, placing it 4- inch from the left-hand and the lower border line. Divide it vertically into four equal parts and subdivide each of the two lower parts into two equal parts again; then draw the horizontal lines shown. Divide the -three lowest divisions horizontally into two equal parts and draw the vertical line shown. The firm name is printed in a style of letter very similar to a block letter, the letters being J':r inch high. The location of the firm is printed in block letters ^ inch high, excepting the first letter of each word, which is 3% inch in height. The line " Turret-Lathe Tools" is a light-face block letter 3^0 inch high, except the first letter of each word, which is y\ inch high. The line " For No. 1 Turret Lathe " is printed in italics, the capitals and numeral being 4- inch high and the small letters 3^ inch high. The next line is printed in light-face block letters -A- inch hieh, and the succeeding lines are written free- hand in the same style used in all the notes, the capitals and numerals being ^^ inch and the small letters -^ inch high. It is optional with the student, as far as the first five lines of the title is concerned, whether to write them in the free- hand style used for the last three lines or in the different styles shown on the plate. 73. The number appearing in the right-hand lower corner is to be made -^ inch high. The reference letters printed in bold-face italics should be omitted on the drawing made by the student. CDMML C/ampng Bolt and Washer. 8 oftfifs. ^ I Cne cf /-hi. Steel Cast. Spider. Scale 3 =/ 44-—, JUNE 25, 1693 For notice of copyright, see page FATOR 16-1 ft. /90 of this. Drawn Copper. Commutator Bar. Scale : half size, mediately following the title page. JOHN SMITH, CLASS N? 4529. § 14 MECHANICAL DRAWING 43 DRAWING PLATE, TITLE: COMMUTATOR 73. This plate shows the detail ilrawings and an assem- bly drawing of a dynamo eommutator. It is composed of a spider and clamping ring, which are drawn together by eight f-inch bolts and clamp 190 commutator bars sepa- rated by mica insulation .04 inch thick. The commutator bars are insulated from the spider and clamping ring by mica rings -jV inch thick. 74. Begin by drawing the clamping ring, locating its hor- izontal center line 4| inches below the upper border line and its vertical center line 2f inches from the left-hand border line. Leave a space of ^| inch between the two views. In the sectional view, which is a section taken on the vertical center line, it will be noticed that the two largest diameters are given to ten-thousandths of an incli. In a shop where many commutators are made from the same draAving, the clamping surfaces (those in contact with the insulation) of the clamping ring, spider, and commutator bars are turned to fit gauges prepared by the toolmaker, who lays out the gauges and makes them to suit the dimensions given. AVhile it cannot be expected that he will get such large gauges as are here required correct to yoio o inch, still the giving of the dimensions in such a small subdivision of the inch calls his attention to the fact that great accuracy is required. This practice of giving accurate dimensions in decimals and approximate dimensions of a machine part in halves, quar- ters, eighths, sixteenths, etc. of an inch, is now largely adopted in the better class of drafting rooms, the purpose of its adoption being to show the workman at a glance which parts of a machine part require to be very accurate and which do not require it, thus tending to prevent the waste of time incidental to needless accuracy. 75. The spider consists of an outer shell and a hub joined together by eight arms. The hub has a keyway for a \" X f key, which is let half into the shaft and half into the hub, making the depth of the keyway in the hub f inch. U MECHANICAL DRAVrlXG ^ 14 Locate the vertical center line in line with the vertical cen- ter line of the clamping ring and locate the horizontal center lire 3^ inches above the lower border line. Draw this horizontal center line clear across the sheet, as it also serves for the assembly drawing. A space of f inch should be left between the end view and the sectional view of the spider. 76, The center line of the clamping washer and clamp- ing bolt should be located 7f inches from the left-hand border line ; the top of the head should be "2^ inches from the upper border line. Owing to the small scale to which the bolt is drawn, it is difficult to draw the correct number of threads per inch (10), and hence the thread is shown exaggerated. This remark also applies to the tapped holes in the clamping ring. 77, Locate the bottom line of the commutator bar 4^ inches below the upper border line and locate the right- hand boundary line 1\^ inches from the right-hand border line. Locate the center line of the end view f inch from the right-hand border line. The commutator bar is dimen- sioned with the dimensions required by the toolmaker for making the gauges and b}' the machinist for turning it to size; consequently, some of the dimensions required for drawing it must be obtained by calculation. Thus, to obtain the depth of the bar at the center, we have 14.5 inches as the inside diameter and 21 inches as the out- side diameter of the ring of which the bar is a segment. 21 — 14 5 Then, the depth of the bar is = 3:^ inches. Other dimensions are obtained in a similar manner. In regard to laying oflE the dimensions given in decimals, those appearing on the detail drawings made to a scale of 3' = 1 ft. should be laid off to the nearest thirty-second inch (on a scale of 3" = 1 ft.), while those on the commutator bar should be laid off to the nearest sixty-fourth inch on a scale of 6" = 1 ft. 78, The assembly drawing is made up from the detail drawings, drawing first the spider in the half -sectional view ; then the commutator bar, the follower, and the clamping JUNE 25./893. For notice of copyrigist. see paee nediately following the title paire s/OffN SM/Tff, CLASS JV9 4529. § 14 MECHANICAL DRAWING 45 bolt should be drawn. The lower half of the side view can now be drawn. In order to get the assembly drawing within the space available, only half of the end view is shown, it being understood that the omitted part is similar to the part shown. In a working drawing, in order to save time, it is not customary to show all the commutator bars on the assembly drawing, only a few being indicated, as shown. The insulation between the bars being only .04 inch thick, it is not feasible to draw it to a scale of 3" = 1 ft. and define its thickness correctly by two lines; consequently the thick- ness is somewhat exaggerated in the drawing, which under the circumstances is a permissible liberty. Part of the mica insulation between the commutator bars and the spider and clamping ring projects beyond the end surfaces of the bars; it is wrapped securely with a layer of heavy twine well shellacked. In the assembly drawing, the vertical cen- ter line of the end view is to be located ^ inch from the right-hand border line and the left-hand surface of the spider 7yV inches from the same border line. DRAWING PLATE, TITLE: SHAFT HANGER 79. This plate shows a drawing of a lianger used to support shafting. There are shown a side view, half in ele- vation and half in section on the line Y Z, part of a top view, part of a bottom view, a complete longitudinal section on the center line in n at right angles to the plane of the paper, and two small sections through the frame. For convenience in describing, the different views will be numbered. Draw Fig. 1 first, which shows a half elevation and a half section. No great difficulty should be experienced in draw- ing it. It will be noticed that the journal-box through which the shaft passes has a spherical bearing; this is shown by the fact that the radius of the curve is the same, or 1^ inches, in Fig. 1 and Fig. 2, the latter being a view at right angles to Fig. 1. The object to be attained by making the bearing partly spherical is to provide for self-adjustment in any direction, in case the shaft is not perfectly straight. 46 MECHANICAL DRAWING § li Fig. 2 shows very plainly how the shaft ma)- be adjusted vertically. The setscrews A, A are released, and the large screws B and B' are screwed up or down, according as the shaft is to be raised or lowered, the spherical bearing allow- ing of adjustment at every point. The setscrews are then screwed down or set, so as to prevent B and B' from chan- ging their positions. It Avill be noticed that the journal-box is made in two parts. Fig. 2 is a section on a line ;// ;/ and shows the whole length of the hanger. The part sectioned at £ and E is shown in Figs. 1 and 4 by £ £. The dotted lines between them show the projection of the boss drawn at F, in Figs. 1 and 4, but not seen in this view. The lines forming the outline £ D £' £ show the projections of the hanger frame. The screws B and B' are made hollow to lighten them. The dotted ellipse in B is the projection of the rib shown at L in Fig. 1. Fig. 3 is a bottom view as far as the line C C, after the oil pan and screw B' have been removed. D is the projec- tion of the boss D in Figs.' 1 and 2, through which the set- screw A passes. Fig. 4 is a partial top view. This can be drawn from the dimensions given without an}- special instructions. Fig. 5 is a section through CD, Fig. 1, and G //, Fig. 2. It shows the shape of the frame, which is also shown by the dotted outline P, in Fig. 4. This section is drawn more particularly to show that the radius of the curve on each side of any section of the vertical part of the frame is the same. Fig. 6 is a section through A B, Fig. 4, and IK, Fig. 1 . BRAWrS G PI.ATE, TITI.E : BEXC H ^^SE 80. This plate shows a bencli ^-Ise and Its details. A drawing of this kind is called a detail drawrog. Fig. 1 is a complete top view and Fig. 2 is a section through the center line C£>. The remaining figures are drawings of the different parts, or details. The actual practice in the draw- ing room would be to draw Figs. 1 and 2 first and then the BENC -jijr- ~:r. F^3. JUHja BS./S9S Fcvr Boitioe of ooipyrigSat, see paue q u u±iid : FiyS ^2f—^~ sdiately following the title page. Zo22 gttudi?zal Section Throu^ h.^lJF_ JO-H^ SJi^riT^, CZASSJ^S 4529. § 14 MECHANICAL DRAWING 47 details, but in the present case the student will do well to draw the details first, as it will help him in drawing the other two views, particularly since nearly all the dimen- sions are given on the details. Following out this plan, leave space for Figs. 1 and 2 and begin by drawing Fig. 3. This consists of three views of tlie jaw, the part marked yJ in Figs. 1 and t>. The parts marked /.', C, and D in Figs. 1, 2, and 3 are circular in shape, so as to pern.it the back jaw A of the vise to swing when the pin E is removed. This allows the vise to hold tapered i)ieces with as much firmness as straight ones. Fig. 4 is a detail, with dimensions, of the pin E. Fig. 5 is a detail of that part of the vise marked E in Figs. 1 and 2, and also of the wrought-iron nut G. The reason that only a part of the nut G is sectioned in Fig. 2 is that a conventional method has been used. In reality the whole of the nut should be sectioned. It would be impos- sible to take a section of the nut in the manner shown. Fig. G shows a top view, longitudinal section, and cross- section of the front jaw. This is cored out to admit the screw and the nut. The form of the cored part is shown by the longitudinal section and cross-section through E E and A B. The front jaw moves in and out with the screw, as shown by Fig. 2, the collar L being held in place by the small setscrew. This is sufficient, since there is no stress on the collar beyond the force required to pull the jaw out- wards, the force exerted by screwing the jaws together coming entirely on the surface M oi the screw head. A separate detail drawing of the screw is not required, since all the dimensions are given in Fig. 2. The threads of the screw are shown the way they really appear, but the student can draw them in the conventional way explained in Art. 24. Figs. 1 and 2 can now be drawn, the necessary dimensions being oi)tained from the details. For ordinary shop drawings, the details are usually drawn to a larger scale than the general drawings. Figs. 1 and 2. In this case the size of the plate would not permit an enlargement. 48 MECHANICAL DRAWING | 14 BRAAVIXG PI.ATE, TITLE: PROFILES OF GEAR -TEETH 81. If a circle is rolled on a straight line without sliding, a point on the circumference of the circle will describe a curve called the cycloid. The circle is called the genera- ting circle. The shape of the curve and the manner of drawing it are shown in Fig. 1. Let O be the center of the generating circle, Avhich is If inches in diameter, P the point on the circumference of the generating circle, and A B the straight line on which the generating circle is rolled and which is equal in length to the circumference of the generating circle, or If " X 3.1416 = 5.4978, say 5| inches. The generating circle should be so placed that its center O lies over the center of the line A B, as shown. Divide the generating circle into any number of equal parts, in this case 12, or P 1, 1-3, 2-3, 3-Jf, etc. , and through these points draw lines CD, E F, G H, etc. parallel to the line AB. Through the center O of the generating circle draw the radius O 6. Divide each half of the line A B into half the number of equal parts that the generating circle is divided into, as .^ 1, 1-2, 2-3, etc.,. and through these points draw lines perpendicular to A B terminating in the line G H, as A G, 1-1' , 2-2', 3-3', etc. From the point 1', with a radius equal to the radius of the generating circle, as 06 or I'-l, describe an arc intersecting the line K L in the point /*' ; from the point 2', with the same radius, intersect the line // in the point P^ ; from the point 3', with the same radius, intersect the line G H; continue in a similar manner with the remaining points Jf.' , 5', 7', 8', etc., intersecting the lines EF and CD in the points P\ P\ P\ P\ etc. The points A, P\ P', P\etc. are points in the curve through which the cycloid may be drawn. It will be noticed that when the center O of the generating circle coincides with the point G, the point P on the circumference of the generating circle coincides with the point A ; and that when the generating circle is revolved towards the right, without sliding, until the center O coincides with the point 1\ the point PwiW coincide with the point P\ Thus it is seen how PRDFILES DF V\\\\\ ///// / J^zyJ. J^U-^Ea5./693. For notice ot copyright, see page ir 3EAR TEETH. diately following the title page. JOHN SMITH, CLASS JV°45a&. ^ 14 MECHANICAL DRAWING 49 the point P passes through all the points from A to 7?, namely, A^ P\ P\ P\ etc., when started at A and revolved towards the right to B. 82. If the generating circle is rolled, without sliding, on the outside of the circumference of an arc of a circle supposed to be at rest, instead of being rolled on a straight line, the curve described by a point P of the generating circle will be an ei)icycloid. The manner of drawing such a curve is shown in Fig. 2. A B is the arc upon which the generating circle is rolled, its center being at 5 and its radius being 3i inches. The diameter of the generating circle is in this case the same as in Fig. 1, or If inches. Make the lengths of the arcs 6 A and 6B equal to half the length of the circumference of the generating circle, by first calculating the length of half the circumference of the generating circle and drawing a straight line tangent to the arc AGB at 6, making it equal in length to half the circumference of the generating circle. Then make the arc 6 A equal to this line by means of the approximate method given in Geometrical Drinuing. Divide the arc A 6B and also the generating circle into the same number of equal parts, in this case 12, as A 1, 1-2, 2-3, etc. and Pi, 1-2, 2-8, etc., and draw radii from the center S to the points of division on the arc A6 B. During the revo- lution of the generating circle, the center O will describe an arc mOn concentric with the arc A6B and having the same number of degrees in it as A 6 B. Produce the radii just drawn to the arc of center positions ?n On, intersecting this arc in the points 7;^ i', i?', J', ^', etc. Through the points of equal divisions, i, L', 5, etc., of the generating circle pass concentric arcs having the center S, as CD,EF,GH,I/,SLn(\KL. With the points l',2',3',^', etc. as centers and radii equal to the radius of the genera- ting circle describe arcs cutting the arcs K L, I J, G H, etc. in the points P\ P\ P\ etc., which are points on the epicycloid. 83. When the generating circle rolls on the inside of the arc, the curve described by a point on the circumference is 50 MECHANICAL DRAWING § U called a liypocycloicl. The method of drawing it is similar in all respects to that just given for the epicycloid. The student should be able to construct it from the drawing without further explanation. The diameter of the generating circle is If inches, as before. 84. Suppose that a string is wound on a cylinder and that the end of the string is at the point P in Fig. 3. If this string is unwound from the cylinder, keeping it con- stantly tight, the end Pwill describe a curve known as the involute of tlie circle, or, more simply, the involute. To construct it geometrically, let O be the center of the given circle representing the cylinder, which, in Fig. 3, is 2i inches in diameter, and Pthe free end of the string when wound on the cylinder. Divide one-half of the given circle representing the cylinder into any number of equal parts, in this case 6, as Pi, 1-2, 2—3, etc., and through each of these points draw tangents to the circle, as P^, P'^, P^, etc. To draw these tangents, first draw the radii 01, 2, 03, etc. and then draw the tangents 1 P\ 2 P^, 3P^, etc. at right angles to them. By means of the approximate method given in Geometrical Drawing, find the length of the arc 1 P and make the length of the tangent 1 P^ equal to this length; of the tangent 2 P- equal to twice this length ; of the tangent 3 P^ equal to three times this length, and so on. The curve drawn through the points P\ P^, P^, P\ etc. will be the required involute. The use of these curves will now be explained. 85. On the plate entitled Spur Gear- Wheels, Fig. 1, is shown one-half of two spur gear-wlieels in mesh. The two dotted circles tangent to each other at Pare concentric to the centers of the gear-wheels and are called the pitcli circles. " The diameter of any gear-wheel is always under- stood to be the diameter of its pitch circle imless it is specified as diametei* at root or diameter over all. The length of that part of the pitch circle between the centers of any two consecutive teeth is called the circular pitcli, or simply the pitcli. Thus, in the last-mentioned figure, the length § 14 MECHANICAL DRAWING 51 of the arc a b is equal to the pitch of either, gear-wheel. When the gear-wheels are cut in a gear cutter, the width of the tooth c dow the pitch line is equal to the space df\ that is, the arc f ^/ is equal to the arc d f^ and each is equal to half the pitch. When the gear-wheels are cast, that is, when they are not cut in a gear cutter, clearance is given between the back of one tooth and the front of the tooth following, to allow for inequalities in casting. This clear- ance, or backlash, as it is usually termed, is generally made equal to 4f,$ of the pitch. This is done by making the thick- ness of the teeth c d equal to .48 of the pntch. The part C C^ of the tooth that lies beyond the pitch circle is called the addeiKlum, and the part C C^ that lies below it is called the root. The face of the tooth is the part C C^ C C, Fig. 2, of the tooth above the pitch circle, extending the whole width of the tooth. The flank is the part C C^C" C, Fig. 2, of the tooth below the pitch circle, extending the whole width of the tooth. The terms addendum and root mean distances only, while face and flank mean surfaces. The usual practice is to make the addendum equal to .3 P, and the root equal to A P. P= the circular pitch. The distance C^ C^ is called the whole depth of the tooth. The method of describing the curves of teeth shown on the plate entitled Profiles of Gear Teeth, Fig. 4, is a convenient way of drawing the cycloidal, or double-curved teeth. Cycloidal teeth are constructed by making the outline of the face a part of an epicycloid and the flanks a part of the hypocycloid, hence the name double-curved teeth. 86. In Fig. 4 of the plate entitled Profiles of Gear-Teeth, let A B be part of a pitch circle struck with a radius of, say, 5Mnches. For convenience in drawing the tooth, let the pitch be 2 inches. With 6^ as a center, which is the center of the gear-wheel, and a radius equal to 5i inches describe the arc A B, part of the pitch circle. Through O draw a straight line OS, cutting A B in /'. Take the radius of the generating circles .V /^and -S" /' ccjual to 1^ inches for this case and describe arcs having centers at i^ and 5' on the M. E. y.~ij 52 MECHANICAL DRAWING § 14 line O S. With O as center and (9 5 as radius describe the arc 5, 5,. In connection with the gear-wheel teeth, the generating circles are frequently called describing circles. Roll the outer describing circle upon A B in such a manner that the center 5 will move in the direction of the arrow along the arc 5, 5,. By means of the method given in Fig. 2, find the points P\ P', /", etc. on the epicycloid described by the point P. Trace a faint curve through the points just found and measure off on the pitch circle the thickness of the tooth. PD = .48/ = .48 X -r = .96'. Make £F=the addendum = . 3 X / = . 3 X 2' = . 6'. With O as a. center and O F sls a radius describe an arc cutting the epicycloid in G. Now roll the inner describing circle on A B, so that its center 5 ' moves in the direction of the arrow, and find the points P,, /*„ P^, etc. of the hypocycloid described by the point /*, through which trace a faint curve. Make EF' equal the flank of the tooth = . 4/ = .4 X 2' = .8', and with 6? as a center and OF' as a radius describe an arc cutting the hypocycloid in G' . PG' is the outline of the flank of the tooth and PG that of the face. Since it would be a tedious operation to draw all the tooth curves in this manner, it is usual to approxi- mate the curves by means of circular arcs; that is, to find by trial a center Q and a radius QP such that an arc described from this center and with this radius will pass through the points on the curve G P and coincide with that curve as closely as possible ; also, to do the same with regard to the curve PG', using the center Q' and the radius Q' P. To find the center Q or Q' of these circular arcs proceed as follows : With P and G as centers and any radius describe arcs intersecting in C and C. Draw a straight line through C and C ; the center O must line on C C to the left of G P. Try different points 1, 2, 3, 4, etc. on this line as centers and IG, 2 G, etc. as radii, and see if one of the arcs struck with either one of these centers and radii will coincide with the epicycloidal curve G P. Make this circular arc fit the Qurve for a short distance beyond G — as far as P^, for § 14 MECHANICAL DRAWING 53 example; this will insure the arc being- more nearly correct. This should be clone in every case when finding an approxi- mate radius of this kind. Continue in this manner until the point Q is found such that an arc struck with Q as 2i center and Q G or Q P as a radius Avill coincide as closely as possi- ble with G P. If a circle were drawn with O as a center and O C2 as a radius, the centers of "all the circular arcs of the faces of the teeth would lie in this circle, and the radii of these arcs would be equal in length to Q J\ Hence, to find the center (?, of the arc I) H forming the back of the tooth, take D as a center and QP as a radius and describe a short arc cutting, in (7,, the circle passing through Q. Then, with O^ as a center and the same radius describe the arc D H. In a similar manner find the center Q' and describe 7^6"', also D H' . Instead of letting the flank form a sharp corner at the bottom of the tooth, as shown dotted at G\ it is usual to put a small fillet there, as shown by the full line. This makes the tooth stronger and less liable to break or to crack in casting. The entire tooth outline or curve G PG' or H D H' is called the profile of the tooth. 87. A rack is a part of a gear-wheel whose pitch circle is a straight line; the tops of the teeth all lie in the same plane.* A portion of a rack and one tooth are shown in Fig. 5. Take the pitch the same as before, then the addendum and root are also the same, that is, .(3 of an inch and .8 of an inch. Take the radius of the describing circles 1| inches, as before. It is evident that the tooth profile will be formed of parts of cycloids formed by rolling the describing (gener- ating) circle upon the pitch line A B. Draw a small part of the cycloidal curves, as shown in the figure, by the method given in Fig. 1; lay off the addendum and root and find the approximate radius in the same manner as in the last figure. The centers of the curves for the faces and flanks of all the teeth of the rack will evidently lie on the straight * As the radius of a circle is increased indefinitely, any arc of the circle approaches more and more to a straight line; and when the radius becomes infinite, the arc becomes a straight line. 54 MECHANICAL DRAWING § 14 lines passing through Q and 0\ respectively, and parallel to the pitch line A B. 88. In Fig. 6 is shown the manner of drawing the involute, or single-curve tootli. The profile in this case is formed of a portion of an involute curve and a portion of the radius of the pitch circle. The circle from which the involute is constructed is called, in this case, the base circle. To find it draw the pitch circle, of which the arc A B is a part, with a radius equal to h\ inches and having its center at O. Draw any radius O W cutting the arc A B in D. Through D draw the straight line E F, making an angle of To' with O U\ With O as a. center and a radius to be found by trial, draw a circle tangent to £ F. This circle, of which the arc H G is a part, is the base circle, and cuts O W \n P. Upon this circle construct, in exactly the same manner as Avas shown in Fig. 3, a portion of an involute curve, passing through P. Lay off the addendum I K^ .G inch, and with O as a center and O I as a radius describe an arc to form the top of the tooth, intersect- ing the involute in L. That part of the flank below the base circle is straight and is a part of the radius drawn to the point P. K P is the root. The tooth has a fillet at L and R\ as in cycloidal teeth. A circular arc is passed through the points L and /*, coinciding as nearly as possible with the involute curve LP. Its center Q is found in the same manner as in Fig. 4. For involute teeth it is onlj^ necessary to find the one center 0\ the centers for all the remaining teeth lie on a circle having O as a center and passing through O. To draw the other side of the tooth, lay off on the pitch circle J/A'= .1^*6 inch, as before. With J/as a center and QX ^^QP as a radius draw an arc cutting, at ^,, the circle passing through 0\ with O^ as a center and the same radius describe the part P' R oi the tooth profile above the base circle. The part P' R below the base circle is a part of the radius OP'. 89. In drawing any of the curves previously described, the greater the number of parts into which the describing ^ 1-t MECHANICAL DRAWING 55 or base circles are divided the greater will be the accuracy- obtained. The profile of the rack tooth used for involute gears is a straight line making an angle of 15° with a line drawn perpendicular to the pitch line. Its construction is shown in Fie. 7. DEFINITIONS AND CALCULATIONS 90. When a revolving shaft transmits motion to another shaft parallel to it by means of gear- or tooth-wheels in such a manner that two corresponding points, one on each gear- wheel, always lie in the same plane, the two gears are called spui* geai'-'svlieels. When the shafts are not parallel, but their axes intersect in a point, as O in the plate entitled Bevel Gears, they are called bevel geai'-^vheels. If two bevel gear-wheels that work together have pitch diameters of the same size, they are called miter gear- wheels. From what has preceded, it is evident that the circular pitch mul- tiplied by the number of teeth equals the circumference of the pitch circle. Let / = circular pitch of gear-wheel; ;/ = number of teeth ; d = pitch diameter ; TT = 3.141G {- is pronounced pi). Then, d = ^~, (1.) TT or, the diameter of the pitch circle equals the circular pitch multiplied by the number of teeth divided by S.lJflG. or, the circular pitch equals the pitch diameter multiplied by 3.1410 divided by the number of teeth. d~ n s n = -, (3.) or, tlie Jiumbcr of teeth equals the pitch diameter multiplied by 3.1410 divided by the circular pitch. 56 MECHANICAL DRAWING § 14 When constructing cycloidal teeth for gear-wheels, the diameters of the describing circles are usually made equal to one-half the diameter of the pitch circle of a gear-wheel having 12 teeth of the same pitch as those of the gear- wheel about to be made. Let li' be the diameter of the describing circle; then, ^'^l^X^, or.r=^. (-4.) Addendum = .3/; root = .4/; thickness of teeth for cast gears is .48/, and for cut gears ^/. DRA^TXG PI.ATE, TITLE : SPUR GEAR-WHEELS 91. This plate shows the halves of two cast gear- wheels having cycloidal teeth, which work together, a cross-section of each gear being also given. The drawing is full size, the wheels not being shown entire for want of room ; to have done so it would have been necessary to make the drawing to a reduced scale. The pitch is 1 inch, the number of teeth in the large gear is 36, and in the small one 18. The pitch diameter of the large wheel is found by formula 1 to be d = ^ , .^ ^ = 11.46 inches, nearly. 3.1416 •' The pitch diameter of the small gear = 1 X IS . .^ . , , .^ , ,^ , = 0. .3 mches, nearlv. 3.1416 The diameter of the describing circle is found by formula 4 to be d' = = 1.91 inches. 3.1416 For all practical purposes, the diameter of the describing circle may be taken to the nearest 16th of an inch. For circular pitches under i inch, approximate the diameter of J'U^E 25. /393. For notice of copjrright, see page : SPUR GEAR WHEELS. Ca^st Iron^ FtclL Stze. lediately followint^ the title page. jrO:H-7Tight, see page BEVEL GEARS. Cexsilro-n Pzi.ll Size. ^•^■'^4^ ediately following the title page. i/ay>v s/if/r/f, class n9 4.529. § U MECHANICAL DRAWING 59 DRAWING PLATP:, TITLE: BEVEIi GEARS 93. To draw in section and projection two cast bevel geai"s whose axes intersect at right angles: The number of teeth in the large gear is 20, in the pinion 10. The circular pitch is 1 inch ; the teeth are to be of the cycloidal form, hav- ing a face 2 inches wide. In any kind of gearing, whether spur, bevel, or spiral, the smaller wheel is called the pinion. Calculate the pitch diameters, addenda, roots, and descri- bing circles by the same rules that were given for spur gears. r^- ^ f ■ ■ 16 X 1 inch , Diameter oi pmion = — — -— — -_ — = 5.09 . Diameter of the large gear = — — — 0.37". 3.1410 20 X 1 inch 3.1410 0x1 inch Diameter of describing circle = — = 1.91" 3.1410 Take this as l\f inches, as in the last plate. Addendum = .3 inch; root = .4 inch. The sectional view must be drawn first. 'Draw PP' and through some point P on this line draw P'P^ perpendicular to it. Lay off PP' equal to the diameter of the pinion = 5.09 inches; also P' P^ equal to the diameter of large gear = 0.37 inches. Bisect Py-" and /"/*,, and draw 6^ J/ and ON perpendicular to those lines at the point of bisection; they intersect in O. (9 J/ and ON are the axes of the two gears and intersect at right angles as required. Draw POP^ and P' O. Through /'draw. -J /'.]/ perpendicular to OP. Through/" draw M P' N perpendicular to OP' and through P^ draw /' .V perpendicular to O P^. PJ/and P' M intersect at J/ on the line C^J/; P' N and /', iV intersect at .V, on the line ON. Lay off /'' C, PC, and P, C\, each equal to 2 inches, or the width of the face of the teeth ; these lines are called the pitch lines, and the width of the face of the teeth is always measured on these lines. Lay off PA equal to .3 inch = the addendum, and /'/> equal to .4 inch = the root. Lay off P' E and /" D for the addendum and root of the other side, and P' P.' and P' D^ for the addendum and 60 MECHANICAL DRAWING § U root of the large gear. All these addenda and roots are each equal to .3 inch and .4 inch, respectively. In bevel gears, all straight lines of the tooth profiles pass through the point of intersection O of the axes; hence, draw ^-J (9, and A A will be the projection of the top of the tooth. Draw B O. and B B ' will represent the bottom of the tooth, the line A' C B' being perpendicular to OP. Make BF\ D F, D^ F^, etc. each equal to \ inch, according to dimen- sions. Join F' , F, F^, and F^ with O, intersecting the per- pendiculars through C, C, and C^ (namely, the lines .-J ' C B\ etc. produced) at G', G, 6",, and G„. G' G and G^ 6^„ will represent the bottom of the gears. The rest of the sec- tional part can be drawn from the dimensions. 93. To show the shape of the teeth, proceed as follows: For the large gear, take A' as a center, XP' as a radius, and describe an arc. Choose a point H and lay off H H' = .48 times the pitch = .48 inch, or the width of the tooth. With XF' and XD^ as radii, describe the addendum and root circles. Roll the describing circles upon the arc whose radius is XP' and construct the tooth profile in exactly the same manner as in Fig. 4 of the plate entitled Profiles of Gear-Teeth, O H and 0^1/' being the radii of the faces and flanks. * To show the shape of the same tooth at C, draw C X perpendicular to OP', or, what is the same thing, parallel to XP'. With X' C as a radius and X as a cen- ter, describe an arc. Draw A'//" and X H', and the distance between the points of intersection on the arc just drawn, measured on that arc, will be the pitch of the gear at the bottom of the tooth. With the same center and A ' E^ and X' E, as radii, describe arcs representing the addendum and root circles. Draw A'^^and X Q^, also (2 //and Q^H'. Through K draw K Q parallel to HO, and through K' draw K' O^ parallel to H' 0^\ the points of intersection Q and Q^ of these lines with A' Q and X Q^ are the centers for the face and flank of the tooth at K and K' . Circles pass- ing through these points concentric with A' contain the centers of all the circular arcs forming the tooth profiles § 14 MECHANICAL DRAWING 61 that may be laid off upon the arc whose radius is N K. The whole process is called developing the teeth of bevel goal's. In the same manner construct the tooth curves for the pinion, using- the same describing circles, Iff inches in diam- eter, and MP', J/' C as radii, instead of N P' and N' C . 9-4, To construct the other view, draw first the projec- tion of the pinion. Draw the center line /// //. Produce the lines F F\ D B, P' P, and E A across the drawing, as shown. Choose a point 5 on ;;/ ;/ as a center and draw a quadrant with a radius equal to the radius of the pinion, as S P. Project the points D and E upon MO in D^ and E^. "With 5 as a center and the distances E^ £"and D^ D as radii, describe quadrants to represent the tops and bottoms of the teeth, that is, the projection of the addendum and root circles of the pinion in Fig. 2. Since the whole pinion contains 16 teeth, the quadrant will contain 4 teeth; hence, divide the quadrant into 4 equal parts on the pitch circle to repre- sent the centers of the teeth. Lay off on each side of the points of division distances ge and gb^ each equal to one- half the thickness of the tooth. On each side of the points of division on the addendum circle lay off Jif and // r, each equal to one-half the thickness of the top of the tooth J K, Fig. 1, measured on the addendum circle. On each side of the points of division on the root circle lay off id and ia, each equal to one-half the thickness of the tooth at the root, as OP, Fig. 1, measured on the root circle. Having now three points on each side of all the teeth to the right of the center line ;//;/, project them upon the lines EA, P' P, and D B, produced as shown. For example, project /"and r upon E A in /' and c' ; e and b upon P' P in c' and // ; d and a upon DB in d' and a'. Draw a curve through these points, either by using an irregular curve or by circular arcs. This remark also applies to the other curves shown in the quadrant. 95. The tooth curves in Fig. 1 must be drawn as accu- rately as possible, but those shown in Fig. 2, being oblique 62 MECHANICAL DRAWING § 14 projections, are drawn to satisfy the eye, and no particular accuracy is required. To find the points on the tooth curves at the bottom of the pinion, describe a circle having a center O, upon /////, which shaU be tangent to PP' and have a diameter equal to 6.37' = the diameter of the large gear. Through B' and A\ Fig. 1, draw lines parallel to O O,; also draw other lines through (9, and the points d\ f\ c\ etc., cutting the lines first drawn in d'\ f", c", etc. Two points are considered enough in this case, as the curves are very short. They may be drawn in with the irregular curve in the same manner as the tops. The other teeth are drawn in a similar manner. Draw the middle tooth first. The left-hand half of the pinion is exactly the same as the right-hand half. 96. To draw the projection of the large gear, project the points E' , D^, L, and R upon the axis O X, in the points E^, Z>j, Zj, and R^, and with O^ as a center and radii equal to E^E\ D^D^, E^L, and R^R, describe circles to represent the addendum and root circles of the tops and bot- toms of the teeth in Fig. 2. Divide the pitch circle into 20 equal parts, to correspond with the number of teeth in the large gear, beginning with the point of intersection of the pitch circle with the center line /// n. Lay off on each side of these pitch-circle divisions, distances equal to one- half the thickness of the teeth = one-half of H H' in Fig. 1. By exactly the same method that was used to lay off the thickness of the teeth at the top and bottom on the quad- rant, lay off the thickness of the top and bottom of the teeth on the addendum and root circles in Fig. 2. Draw the bottoms of the teeth in exactly the same manner as the bottoms of the pinion teeth were drawn. All the teeth of the. large gear are alike in the projected view. 97. Bevel gears are always measured according to their largest pitch diameter, as P P' and P' P^. If a bevel gear were spoken of as 12 inches in diameter, it would be under- stood that the largest pitch diameter was 12 inches. § U MECHANICAL DRAWING 63 DRAWTXO PLATE, TITl^K : 1?RISII IIOr.DER 98. This drawing plate is a complete working drawing of the left-hand brush holder for a 15-horsepower motor and is designed for the use of carbon brushes, there being four brushes, two in each holder. The carbon brushes a, a are clamped by means of the setscrews /', /; to the clamps r, r, which are free to slide in rectangular holes in the body d of the brush holder. The setscrews b, b do not bear directly against the brushes, but against a brass shoe c. A thorough electric connection between the carbon brushes and the body of the holder is insured by flexible No. 12 cables/, /", which are composed of strands of copper wire covered with insula- ting material. The outside diameter of these cables is -jV inch, about. The carbon brushes are held against the commutator by hammers g, g operated by springs //, //. The hammers are pivoted to brackets d' cast in one Avith the body, and the springs are so hung that when the hammers are rotated away from the brushes, the springs will come to the other side of the center around which the hammers turn and thus hold the hammers away from the brushes. The springs are hooked over lugs on the body at one end and over arms projecting from the hammers on their other end. In order to insulate the brush holder from the frame of the machine, it is fastened to a piece of hardwood i by two 3^^-inch rivets and one No. 10 wood screw -J inch long. The piece of hardwood is fastened to the frame k of the machine by a 1-inch capscrew as shown. 99. Begin dr;iwing the plate by drawing the end view, locating the horizontal center line passing through the center of the rivet serving as a fulcrum for the hammer at a dis- tance of 3j-V inches below the upper border line and locating the vertical center line 2^- inches from the right-hand border line. The end view is to be drawn first because it is the only view in this particular instance in which everything can be drawn without having to project from another view. For several of the dimensions it is necessary to refer to the top view. The main ])art of the hammer is flat and has 64 MECHANICAL DRAWING § 14 joined to it a handle having a circular cross-section. The flat and round part coming together cause the intersection curve shown at /. The distance that the flexible cord / projects from the left-hand face of the body is not given on the drawing, as this information would be useless on a work- ing drawing. For the information of the student it is here given, being If inches. It is a general and a good rule with draftsmen not to give any dimensions on a drawing unless they serve a useful purpose ; everything superfluous is to be left off. The shoe e is marked No. 18 sheet brass; the corresponding thickness is .04 inch, nearly. It will be observed that the spiral spring Ji is not drawn the way it actually appears, but that it is drawn convention- ally. This is merely done in order to save the draftsman's time, as the note " Spiral Spring Piano Wire, No. 28, f" ^, " appearing in the top view supplies the necessary information to the mechanic. 100. The end view having been completed, the top view should be drawn next, locating the center of the |-inch capscrew fastening the hardwood strip i to the frame S^^g- inches from the left-hand border line. This view is drawn partially from the dimensions given and partially by pro- jecting over from the end view. The springs Ji being at an inclination, they appear foreshortened in the top view, and their length in the top view must be determined by project- ing over from the end view. Each spring is hooked over a horn cast on the body d^ a cylindrical ring being formed on the ends of the springs for this purpose. Owing to this ring being at an inclination, it will show elliptical in the top view. The outside diameter of this ring is -^ inch. The heads of the ■^^" X If" rivets have a radius of -3% inch and are \ inch high. The head of the No. 10-24 round-headed machine screws is |i inch diameter and -^^ inch high; the diameter of a No. 10 machine screw is .189 inch, say -^-^ inch. The head of the No. 8-32 round-headed machine screws is y5_ inch diameter and -^^ inch high; the diameter is .103 inch, say W inch. ^^lachine screws are made in § 14 MECHANICAL DRAWING 65 accordance with the standard American screw gauge adopted by all manufacturers of screws; for this reason it is only nec- essary on a drawing to specify the gauge number of the screw, the number of threads per inch, and the length, which latter is always measured under the head, except in flat- headed screws, where the length is measured over all. Wood screws are measured by the same gauge as machine screws and bear the same number. The heads of the No. 10 wood screws are |l inch diameter. The diameter across the flats of the head of the J-inch capscrew is f| inch; the height of the head is 4 inch. The ^Vinch setscrews have a head tV inch high and measuring j\ inch across the corners. 101. To draw the side view, begin by locating the edge ;//.^ which corresponds to the surface ;// in the end view, at a distance of 1^ inches above the lower border line. The view is to be drawn partially by projecting doAvn from the top view and partially by transferring measurements from the end view, as explained in connection with the drawing plate, title: Turret-Lathe Tools. In order to have a line in the end view from which to measure, the line represent- ing the surface ;// should be produced in lead pencil ; in the side view the edge ;u will then represent an edge of the same plane defined in the end view by the lead-pencil line just drawn. The shading of the flexible cords is done freehand. 102. The center line of the brass pin is to be located H inches from the right-hand borderline; the center of the top view is to be placed 5j\ inches above the lower border line and a distance of j\ inch is to be left between the two views. 10,3. The first line of the title is a block letter j\ inch high, the second line is ^-V inch high, and the third j\ inch high. The fourth and fifth line is written freehand, the capitals being j\ inch and the small letters ^ inch high. ' The sixth line is composed of capitals j\ inch high, except the first letter of each word, which is | inch high. The seventh line is composed of capitals i inch high and the eighth line is the same as the fourth and fifth. The student is advised to practice the block lettering shown in this title, but if he 66 MECHANICAL DRAWING § U desires it he may substitute the freehand letter shown in the fourth line. 104:. In the upper left-hand corner will be found a list of the different parts needed. It is the practice in many shops to have such a list on the drawing for convenience of reference. Draw the bottom line of the list 4^ inches below the upper border line and draw the upper line 3f inches above the bottom line and divide the space between the two into 17 equal spaces. Draw the left-hand line of the list ^ inch from the left-hand border line and the right-hand line 4 y- inches from the border line. To the right of the left- hand border line of the list draw vertical lines at a distance of ^ inch, -|- inch, dj\ inches, and. 3f inches, respectively. Then write in freehand, in the style of lettering shown, all the contents of the list. The student must omit on his draAving the reference letters printed in bold-face italics. READING A AVORKIXG DRAWIJs^G 105, The following general method of procedure has, by experience, been shown to be conducive to the accurate and rapid reading of a drawing made in projection. First, if the drawing is dimensioned, ignore the existence of the dimension lines and dimensions entirely until after the general shape of the object is fixed on the mind. Second, by referring to the several views, form an idea of the shape of the main body of the object ; that is, observe if its outline shows it to be a cube, a sphere, a cylinder, a cone, a pyramid, etc., or a combination of several of these elementary forms. The shape of the main body having been impressed on the mind, observe how it is modified by details, determining, by reference to the several views, whether they project from the main body or are recesses, or holes. Finally, by refer- ring to the dimensions, form an idea of the relative sizes of the component parts. Pay due regard to all conventional representations that may have been used; for instance, do § U MECHANICAL DRAWING 67 not become confused if the ^m of a pulley, or a rib, which, truly speaking, should have been in section, is shown in full. If two half sections are placed on either side of a common center line, remember that each half must usually be viewed independently of the other and must be mentally completed. 106. When reading a drawing in which the views are correctly placed, it is often a great aid to project points or edges of some part the shape of which is doubtful over to another view by the aid of a straightedge, in order to find the location of the doubtful part in another view. When the views are not placed in their correct relative positions, this cannot be done. An example of a case of this kind is given in the plate Compound Rest, and in reading a drawing with the view thus placed, the reader is supposed to constantly imagine that the views are in their correct relative positions; with a little practice this will be found to be quite easy. 107. In a case of this kind, it is manifestly impossible to project points or lines from one view to the other by means of a straightedge, and a different method must be followed. Select some surface whose projection appears in both views, or a center line; now place a pair of dividers so that one point rests on the projection of the surface or center line selected, and open them until the other point reaches the point or line whose projection it is desired to find in the other view. Then place one point of the dividers on the line representing the selected surface in the second view, and move the dividers along this line until a line, or the projection of a line, is found to coincide with the other point of the dividers. Examples of this will appear later on. In order to aid the student to read a drawing, we have selected the plate Compound Rest and will show in detail by what process of reasoning this drawing is read. 108. To find the shape of the different jiarts and also to discover, if possible, the relation between them, we must commence our investigation somewhere. Let us choose the .1/. E. v.— II 68 MECHANICAL DRAWING § U bottom of the front view. Looking at this it is noticed that a partial section is shown, from which, by reason of the section lining running in opposite directions, we conclude that A and B are separate parts. At the right and left of the front view, the full lines c, c show that some part of A is higher than the bottom of B, but we do not know whether these lines denote the top surfaces of projecting parts between which B is fitted, or if c is the top surface of a raised strip of some kind that extends clear through the inside of B. In order to settle this question, we note whether the top surface is continued somewhere. Looking at the front view it is seen that the line c is dotted clear through B, which settles conclusively that the part whose top surface is shown by the line r is a raised strip extending clear through B\ this fact immediately implies that B has a groove of some kind running through it longitudinally in order to admit the raised strip. Referring now to the sectional view, which, as previously stated, is a view taken on the line a b of the front view, and everything to the right of this line being removed, we may choose the bottom line d' of the sectional view as a base from which to make measurements. From the fact that the section is taken on the line a b, we know that the line just chosen is the projection of the intersection d of the plane represented hy a b with the bottom of A. Measuring from d' upwards to the highest line c' of A in the sectional view, and placing one point of the dividers on d in the front view, it will be seen that the other point coincides with the dotted line forming a continuation oi c c\ this shows that c' is the projection of c. In a similar manner, we determine that e' is the projection of r, and tracing the outlines of A in the sectional view, we notice that the raised strip on A has inclined sides. "We also notice that B is cut out tO'Suit the profile of A, except that on one side a steel part L is inter- posed betAveen the inclined sides of A and B; it is also seen that a screw rests with its point against L. Referring now to the front view, and knowing from inspection of the sectional view that the upper and lower § U MECHANICAL DRAWING 69 surfaces of the steel part L are flush with .' and .' or c and e in the front view, to determine the length of this part notice ,f any dotted or full lines showing its length are shown anywhere at a right angle to c and .. Non; being found the conclusion to be drawn is that either the steel part /. IS as long as A or has the same length as B A per son without any practical experience migi^t conclude that the length is the same as that of J; but any one havino- engineenng instinct or practical knowledge would imme^ diately notice that, as the steel strip has setscrews which evident y serve to push it against the inclined side of A it would be unnecessary to make the strip the leno-th of '-J and hence would immediately conclude that its length is the same as that of />'. This latter conclusion is th? one the arattsman desired to convey. 109. Looking at the sectional view of A again we notice that a groove, open on top, is cut into A. To find its length we must find lines corresponding to it in the front view. Measuring from d' upwards to the bottom of the groove and transferring the measurement to the front view we find that the dotted line,.,^ represents the bottom and end of the groove, which at the left is also shown to be open at the bottom, since the dotted line ,,^ curves around and continues to the bottom of A. This is also indicated by the dotted hues <^' that form an extension of the sides of the groove in the sectional view; measuring from ^' down- wards to the horizontal dotted line joining the ends of <^' is a part similar to A. Furthermore, the investigation will show that 6^ is a slide; this slide is movable by means of the screw //. which turns in the bearing y. 115. Referring again to the sectional view, we see that B and D are connected together by a pin £, whose purpose is unknown as yet. Examining this pin we notice that a hole is cut through its upper end and that a screw F, with a tapered shoulder to the right of its screw thread, passes through this hole. On close examination, we see that the hole in £ is so placed that the tapered part of the screw F bears against the upper side of the hole. We further notice that the screw F is not used as a fastening device to hold any parts of D together ; this conclusion is forced upon us by the fact that the sectional view shows D to be one piece. Now, we know from experience that a screw is used either as a fastening device or to transmit motion ; as it obviously is not used for the purpose first mentioned, we conclude that it probably serves for the latter purpose. To make § n MECHANICAL DRAWING 73 sure of this we trace out what will happen if the screw is rotated. We then noticed that if the screw is screwed inwards, it will raise the part A"; but as £ cannot move upwards by reason of being- confined by the collar on it, it shows to us that screwing F inwards will force D down on B. The logical inference is that /i and F iorm a clamp- ing device intended to clamp B and D together. Examining the pin F again, we do not find anything that would definitely tell whether it is round or square. Here judgment must be used. An experienced person would know upon the first glance that the clamping arrangement shown is an expensive one to make and one not likely to be adopted when it is only required to fasten two pieces rigidly together, in which case F might be either round or square. The next inference would be that it is used in order to allow D to be rotated around F and to be clamped in any position. This supposition requires the pin F to be round and is correct in this case. 116. Referring now to the ball handle /, of which only one view is shown, the question of whether it is circular or square is immediately settled by experience teaching us that a handle having the shape shown is not likely to be any- thing else but round, and in the absence of any note or indi- cation to the contrary, we would be justified in assuming it to be round. 117. As far as the part G is concerned, the sectional view shows it to be cored out in order to pass over the nut in which the screw H Avorks. The width and profile of the coring must be obtained from the front view, which it will be remembered is a view at a right angle to the sectional view. The natural assumption to make is that the lines giving the width and profile of the coring will be found directly in the vicinity of the screw // in the front view. Measuring from the center line of this screw in the sec- tional view upwards to the line showing the height of the coring, and then transferring this measurement to the front view, we find the full circle //. Now, as the coring is beyond 74 MECHANICAL DRAWING § 14 the bearing J, we know that its profile would show in dotted lines and conclude that the circle // represents some part of the bearing J. As this bearing has a conical projection, the inference is that the full circle represents the largest diameter of the cone, which is the case. Now, the absence of a dotted line showing the coring forces us to conclude that the dotted line would be directly behind the full circle ;/ and is thus hidden. This conclusion is further strengthened by finding two vertical dotted lines r, r tangent to the circle 7/, and we finally decide that the groove has straight sides with a semicircular top, as given by the dotted lines r, r and the upper semicircle of n. By measuring again in the manner previously explained, we decide that the dotted line t? is a front view of the nut in which H works. 118. At the right-hand end of the sectional view of G we notice a T-shaped opening. Referring to the front view we can easily discover, hy transferring measurements, that the dotted horizontal lines /, / show the length of the slot, which is seen to extend clear across G. 119. Referring now to the drawing of the tool post, it will be observed that only one view is, given. While this does not definitely settle that the post is circular in cross- section, common practice would justify a person in assu- ming, in the absence of any note or any other indication to the contrary, that such was the case. This view is strength- ened by the fact that some dimensions are marked d, signify- ing diameter, which term is rarely applied to any but a round object. 1*20. The two v^ews of the collar give its shape. Refer- ring to the front view, while there is no definite note to that effect, it would be inferred from the fact that a thread is shown that the lower part is separate, being, in fact, a circular nurled nut threaded to receive the upper part. 121. While, generally speaking, any one can learn to determine the shape of objects from a drawing, there are cases that arise in practice where this is very difficult •-sTTcsziz-c-n cr'Scyezvs. see fa's COMPOUND REST Fdr idx 36 "Speed Lathe. Cast IrtON, Unless Orj-iEiP.wfSE SpEcineo, OneOfThis- Scale Fuu. Size. The 3cr ANTON Tool Works, ScRAf^TCM, Pa. Or-Offn By ■ned T^i — -^j ^ -*-* — '— "^ r-i — y,« ,^/Z^.^,. -l^^•ll'i., TOOL POST. One a/d?tT.5,Tool Steel. % /O f----X:il^jiyFl£Cce o?2 cen ter lz/?zg . ^76 4^ Ta,;o -//7s -tofiitdecZ'- 0?z^ o ft^ts. ToalSieel, fH^ III 1 1 "t?V^ -r. %-.--"mm N'/S63 ediately following the title page. J-0^f7\rjMITH.CZA;5SJi945Z9. § 14 MECHANICAL DRAWING 75 without further verbal or written instructions. The cases in which this usually happens are where coring has various odd- shaped curved surfaces that curve in different directions, as occurs, for instance, with the steam ports and other pas- sages of steam-engine cylinders and other similar work. Practical experience with a certain line of work, and, frequently, a knowledge of the object of the doubtful part, will often allow the reader to form a correct idea of Avhat the draftsman is trying to convey; when this experi- ence or knowledge is lacking, consult somebody who is likely to know. Furthermore, the shape of an object does not necessarily in itself always reveal its purpose. Ability to determine at sight what an object is to be used for involves either a thorough knowledge of a particular line of work — in which case the purpose of objects coming within its range can usually be determined at sight — or a very wide general knowledge of engineering construction. DRAWING PLATE, TITLE : COMPOUJ^D REST 13!3. This drawing is at one and the same time a detail and an assembly drawing, as it not only gives all dimensions required for each and every part, but also shows how they are assembled. Drawings are frequently made in this manner in order to save some of the draftsman's time. The particular drawing shown exhibits a case where the draftsv man, on account of lack of space, has been compelled to break the rules governing the arrangements of the views. Referring to the plate, the sectional view taken on the line a b should have been placed alongside and on the right of the front view, which obviously cannot be done without making the drawing to a scale smaller than full size. 1^3. To begin locate the center line of the screw C in the front view at a distance of 4^ inches from the iq^per border line. The slide B being movable along the base A^ it can be located anywhere within its range of movement; 76 MECHANICAL DRAWING § U but in order to get a convenient arrangement, it is recom- mended to locate the line a b passing through the center of the screw of the upper slide at a distance of '\\\ inches from the left-hand border line. Locate the center line of the hole in the base marked |" tap lly\ inches from the left-hand border line. The front view can now be drawn from the dimensions given. 12-4, Locate the center line of the screw H in the sec- tional view 8^ inches from the upper border line and locate the center line of the handle /at a distance of "^f inches from the left-hand border line. The upper slide G being movable on its base D, the latter could be drawn anywhere within the range of movement ; in order to have a good arrange- ment, however, it is recommended to locate the vertical center line passing through the part ^ at a distance of VlW inches from the left-hand border line. The sectional view can now be drawn without any special instructions. 125. The center line of the collar should be located at a distance of If inches from the right-hand border line and the center of the top view of the collar is to be located at a dis- tance of 4r| inches from the upper border line. 126. The center line of the tool post may be located at a distance of Iff inches from the upper border line and the right-hand end of the tool post at a distance of If inches from the right-hand border line. The tool-post screw being movable, it could be shown in any position within its range of movement ; but in order to get the same arrangement as on the plate, the right-hand surface of its collar should be placed at a distance of ^ inch from the left-hand end of the body of the tool post. A number of reference letters printed in bold-face italics appear on this plate. The student should omit these letters on his drawing. 127. In order to emphasize the fact that different drafts- men prefer difi"erent styles of lettering and a different arrangement for titles, this plate has been given a title dif- fering from those previously given. The first and second ^ U MECHANICAL DRAWING 77 lines of lettering are easily made with tlie ruling pen, the inclination of the letters being 75^ The letters in the 'first line of lettering are ^\ inch high; in the second line all capitals and numerals are -L inch high, except the llrst letter of each word, which is -/V inch high. Tin- third to ninth line of lettering are written freehand. In the third and fourth lines, the lettering is ^\ inch high, except the first letter of each word, which is ^ inch high. In the fifth line, all letters are i inch high, except that the first letter of each word is ^% inch high. The sixth line is the same as the third and fourth. In the remaining lines, the capitals are ^\ inch and the small letters -^\ inch high. The heading "Tool Post " is ^ inch high. TRACINGS 128. In actual practice in the drawing room, it is necessary to have more than one copy of a drawing. It would be very expensive to make a finished drawing every time an extra copy was wanted and to avoid this tracings and blueprints are made. Any number of blueprint copies can be made from the same tracing. A complete pencil drawing is made first ; then, instead of inking in as hereto- fore, a piece of tracing paper or tracing cloth of the same size as the pencil drawing is fastened to the board over the original drawing. The tracing paper or cloth being almost transparent, the lines of the drawing can be readily seen through it, and the drawing is inked in on the tracing paper or cloth in the same manner as if inking in a finished drawing. 129. Ti-aeing paper is but little used in this country. It is easily .torn and cannot be preserved as well as tracing cloth. The two sides of the tmciiis cloth are known as the glazed side and the dull side; they are also known as the front and the back. The glazed side, or front, is covered with a preparation that gives it a very smooth polished surface; the back, or dull side, has very much the appear- ance of a piece of ordinary linen cloth. Either side may be 78 MECHANICAL DRAWING § U used for drawing upon, but when the glazed side is used, care must be taken to remove all dirt and grease, otherAvise the ink will not flow well from the pen. This can be done by taking a knife or a file and scraping or filing chalk upon the tracing cloth; then take a soft rag of some kind — cotton flannel or chamois skin — and rub it all over the tracing cloth, being sure to rub chalk over ever}^ spot. Finally, dust the rag and remove as much of the chalk from the cloth as can be gotten off by rubbing with the rag. The finer the chalk powder is the better. It is not usual to chalk the dull side, but it improves it to do so. The glazed side takes ink much better than the dull side, the finished drawing looks better and will not soil so easily, and it is also easier to erase a line that has been drawn on this side. Pencil lines can be more satisfactorily drawn on the dull side, and if it is desired to photograph the drawing, it is better to draw on this side. The draftsman uses either side, according to the work he is doing and to suit his individual taste, but if the glazed side is used, /'/ must be cJialkcd. The tracings are drawn in a manner similar to the finished drawings, the center lines, section lines, etc. being drawn exactly as previously described. In some oflfices it is customary to draw the center lines and dimension lines on a tracing in red ink, so that they may appear gray instead of white on the blueprint. 130. After having drawn the plate entitled Compound Rest, the student is required to make a tracing of the plate entitled Turret-Lathe Tools. BLirEPRIXTIKG 131. Blueprinting is the process of duplicating a tracing by means of the action of light upon a sensitized paper. The following solution is much used for sensitizing the paper : Dissolve 2 ounces of citrate of iron and ammonia in 8 ounces of water ; also 1^ ounces of red prussiate of potash in 8 ounces of water. Keep the solutions separate and in dark-colored bottles in a dark place where the light cannot § li MECHANICAL DRAWING 79 reach them. Better results will be obtained if ^ an ounce of gum arabic is dissolved in each solution. When ready to prepare the paper, mix equal portions of the two solutions, and be particularly careful not to allow any more light to strike the mixture than is absolutely necessary to see by. For thi^ reason, it is .necessary to have a dark room to work in. There must be in this room a tray or sink of some kind that will hold water; it should be larger than the blueprint and about 6 inches deep. There should also be a flat board large enough to cover the tray or sink. If the sink is lined with zinc or galvanized iron, so much the better. There must be an arrangement like a towel rack to hang the prints on while they are drying. For the want of a better name, this arrangement will be called a print rack. The paper used for blueprinting should be a good, smooth, white paper, and may be purchased of any dealer in drawing materials. Cut it into sheets a little larger than the tracing, so as to leave an edge around it when the tracing is placed upon it. Place eight or ten of these sheets upon the fiat board before mentioned, taking care to spread flatly one above another, so that the edges do not overlap. Secure the sheets to the board by driving a brad or small wire nail through the two upper corners sufficiently far into the board to hold the weight of the papers when the board is placed in a vertical position. Lay the board on the edges of the sink, so that one edge is against the wall and the board is inclined so as to make an angle of about 00° with the horizontal. Darken the room as much as possible and obtain what light may be necessary from a lamp or gas jet, which should be turned down very low. With a wide camel's-hair brush or a fine sponge, spread the solution just prepared over the top sheet of paper. Be sure to cover every spot and do not get too much on the paper. Distribute it as evenly as possible over the paper, in much the same manner that the finishing coat of varnish would be put on by a painter. Remove the sheet' by pulling on the lower edge, tearing it from the nail that holds it, and place it in a drawer where it can lie fiat and be 80 MECHANICAL DRAWING §1^ kept from the light. Treat the next sheet and each succeed- ing sheet in exactly the same manner, until the required number of sheets has been prepared. Unless a large number of prints is constantly used, it is cheaper to buy the paper already prepared. It can be bought in rolls of 10 -yards or more, of any width, or in sheets already cut and ready for use. There is very little, if anything, saved in preparing the paper, and better results are usually obtained from the commercial sensitized paper, since the manufacturers have machines for applying the solution and are able to distribute it very evenly. 132. In Figs. 15 and 16 are shown two views of a print- ing frame that is well adapted to sheets not over 17' X "21". The frame is placed face downwards and the back A is Fig. 15 removed by unhooking the brass spring clips ^, B and lifting it out. The tracing is laid upon the glass C, with the inked side touching the glass. A sheet of the prepared paper, per- fectly dry, is laid upon the tracing Avith the yellow (sensitized) § 1-t MECHANICAL DRAWING 81 side downwcirds. The j)aper and tracing are smoothed out so as to lie perfectly flat upon the glass, the cover A is replaced, and the brass spring clips B, B are sprung under the plates D, so that the back cannot fall out. While all this is being done, the paper should be kept from the light as much as possible. The frame is now placed where the sun can shine upon it and is adjusted, as shown in Fig. 16, so that the sun's rays will fall upon it as nearly at right angles as possible According to the conditions of the sky — whether Fig. 16 clear or cloudy — and the time of the year, the print must be exposed from 3 to 15 minutes. The tray, or sink, already mentioned, shoukl be filled to a depth of about 'I inches with clear water (rain water if possible). The print having been exposed the proper length of time, the frame is carried into a dark part of the room, the cover removed, and the print (prepared paper) taken out. Now place it on the water with the yellow side down and be sure that the water touches every part of it. Let it soak while putting the next print in the frame. Be sure that the hands are dry before 82 MECHANICAL DRAWING § U touching the next print. The first print having soaked a short time (about 10 minutes) take hold of two of its opposite corners and lift it slowly out of the water. Dip it back again and pull out as before. Repeat this a number of times, until the paper appears to get no bluer; then hang it by two of its corners to dry on the print rack previously mentioned. If there are any dark-purple or bronze-colored spots on the prints, it indicates that the prints were not washed thoroughly on those spots. If these spots are well washed before the print is dried, they will disappear. 133. It is best to judge the proper time of exposure to the light by the color of the strip of print projecting beyond the edge of the tracing. To obtain the exact shade of the projecting edge, take a strip of paper about 12 or 14 inches long and 3 or 4 inches wide. Divide it into, say, 12 equal parts by lead-pencil marks, and with the lead pencil number each part 1, 2, 3, etc. Sensitize this side of the paper and, after it has been properly dried, place it in the print frame with the sensitized side and the marks and figures against the glass. Expose the whole strip to the light for one minute; then cover the part of the strip n^arked 1 with a thin board or anything that will prevent the light from striking the part covered. At the end of the second minute, cover parts 2 and 1 ; at the end of the third minute, parts 3, 2, and 1, etc. When twelve minutes are up, part 1 will have been exposed one minute; part 2, two minutes, etc., part 12 having been exposed twelve minutes. Remove the frame to a dark part of the room and tear the strip so as to divide it into two strips of the same length and about half the original width. Wash one of the strips as before described, and Avhen it has dried, select a good rich shade of blue, neither too light nor too dark; notice the number of the part chosen, and it will indicate the length of time that the print >vas exposed. Examine carefully the corresponding part of the other strip, and the correct color of the edge of the print projecting beyond the tracing is determined. All prints should be exposed until this color is § 14 MECHANICAL DRAWING 83 reached, no matter how long or how short the time may be; then they should be immediately taken out and washed. Fig. 17 134. In Fig. 17 is shown a patented frame which can be shoved out of the window and adjusted to any angle. Wh M. E. y.—i3 en 84 MECHANICAL DRAWING § U not in use. it can be folded up against the wall and occupies but little space. It is made in different sizes from 16" X 24' to 48' X TS". It is one of the best frames in the market, and is placed in such a position relatively to the window that the window can be lowered to the top of the main arm. when it is desired to keep out the cold during the winter. DRAWTS'G PLATE, TITLE: SIX-HOKSEPOWER HORIZOXTAE STEA3I EXGI^'E 135. Instead of making a finished drawing of this engine, as in the previous plates, from an exact copy, the student is given the rough sketches of the details of a six- lioi-sepoTver liorizontal steam engine, with full dimen- sions marked upon them ; from these he is expected to make a general pencil drawing of the engine in two views — a plan and a side elevation. The details are not to be drawn. The pencil drawing should then be traced according to the directions previously given. The details are not drawn to scale, but are fully dimensioned. In order to draw the engine to as large and as convenient a scale as possible, it is necessary to make this tracing a trifle larger than the plates that have preceded it. The size over all will be 14+' X 18|', with the usual border line ^ inch from each edge all around. That the student may have a good idea of what he is expected to do, a greatly reduced cut of the general draw- ing is also given him. All dotted lines indicating parts not seen have been omitted in order to simplify the work. The scale to be used is 3 inches = 1 foot. 136, Draw the center lines inn, pq, rs, and ti: Draw the side elevation of the bedplate with the bearing caps in position, from the dimensions given on the detail sketches, taking care to make the parts that are likely to be hidden by the flywheel, eccentric rod, etc. light so that they may be easily erased before tracing. The drawing may be traced without removing the unnecessary construction lines, but it is better to do so, since it lessens the liability of inking in lines that will have to be erased from the tracinor. l\\f^^^=^ JU^E 25. 1893. jT notice cf copvT:g''-t, se ediately following the title page. J-OHJvsjyriTj^. CLA ss jsr^ Asag. § 14 MECHANICAL DRAWING 85 Draw the plan of the bedplate with the bearing caps, studs and nuts, foundation-bolt holes, etc. shown in their proper places and positions. The different curves of the bearing caps in the plan should be constructed by project- ing points from the view shown in the side elevation. In actual practice in the drawing room, three or four of the principal points (those that mark the limits) would be located and the curves sketched in freehand, they being inked in on the tracing by aid of an irregular curve. In such cases the draftsman has a good idea of the shape of the curves, owing to previous practice in drawing them. When drawing in the curves formed by the opening in the bedplate shown in this view, the student must exercise his own judgment regarding their shape, taking care not to get them too straight. The general drawing gives a good idea of their proper curvature. Returning to the elevation, draw the crank and crank end of the connecting-rod in the position shown in the general drawing. With the center of the rrankpiii as a center and a radius equal to the length of the connecting-rod between its centers (obtained from the detail sketch), describe an arc cutting the center line pq at a point that will be the center of the crosshead pin. Draw the crosshead, obtaining the dimensions from the detail sketch. Complete the con- necting-rod in both views and draw the piston rcxl 1 inch in diameter. Draw both views of the cylinder with the nuts aiid the steam pipe in their proper position, getting all dimensions from the detail sketches. Draw the center line of the valve stem in the plan view, and draw the stuffingbox. valve stem, valve-stem slide and its guide in both views. In order to determine the position of the valve-stem slide, it is necessary to locate the center of the eccentric. Referring to the general drawing, it is seen that the eccentric is on the dead center farthest from the cylinder. The offset of the eccentric is given as ^ inch in the detail sketch ; hence, when in this position, the center of the eccentric strap will be situated |^ inch to the right of the crank-shaft center on the line/f/. With this point as a S6 MECHANICAL DRATVIXG ? l-l center and a radius equal to the distance between the cen- ters of the eccentric strap and the hole in the stub end of the eccentric rod (see detail sketch), in this case 2 feet i^ inches, describe an arc cutting the center line />q in O: O will be the center of the pin on the valve-stem slide, which may be completed by aid of the detail sketch. Complete the drawing of the eccentric, eccentric strap, and eccentric rod in both views. Finally, draw in the bandwheel and flywheel (see general drawing for position). The flywheel will be of the same diameter as the bandwheel, but only 3 inches wide. The pencil drawing is now completed. Before beginning to trace, erase the lines that are not to be inked in. This is not necessary, but it is better to do so, since it avoids confusion and lessens the liability of making mistakes. Some draftsmen prefer to redraw a portion of those parts that are to be inked in with a somewhat softer pencil and leave the light construction lines on the drawing rather than erase them ; in some cases, this saves time. The preliminary directions for tracing a drawing have been given previously. First, trace the side elevation, begin- ning with the flywheel, and then as much of the connecting- rod, eccentric, and eccentric rod as can be seen. Trace all those parts of the bedplate, cylinder, valve stem, stuffing- box, etc. that are seen. Then trace the plan view, letter the drawing, and draw the border lines. There will be no plate number for this tracing, but the student's name, class num- ber, and the date of completion will be put on as before. The student should exercise particular care to have every iimension scale exactlv the size given in the detail sketches. PRACTICAL PROJECTION IXTROBUCTIOX 1. Ortliograpliie Projection. — When the mechanic is required to make any article whose form or dimensions are not previously known, it is evident that a description of the work in question should be furnished him. This description may be, and often is, given by verbal instruc- tion; but in order to enable the worker to understand definitely what is wanted, the form of the object, its dimen- sions, and the quality of the material to be used should be stated. Instruction given in this way is, however, seldom satisfactory either to the workman or to his employer, since it is difficult in such cases to place the responsibility for any errors that may occur. Written instruction, therefore, would seem to be prefer- able; but, since most objects would require an extended description, a shorter and more convenient method of con- veying the desired information is to be sought. A draAv- ing of the object is therefore made. These drawings are generally made by a process termed orthographic projection, or, as it is usually called, projec- tion druAving. Every detail of the object is correctly represented in this drawing, so that the workman knowing how to "read the drawing" may obtain his measurements therefrom for the construction of the object itself. He is also enabled by an examination of the drawing to under- stand exactly how the object will appear when completed. Hence, we have the following definition: §15 For notice of copyright, see page immediately following the title page. 2 PRACTICAL PROJECTION §15 Ortliogriiphic pio.K itioii is the process of viaking cor- rect representations of objects by iiteans of drawings. ' 3. A Working Dra\rtiig (Generally Xecessary. — Before a pattern for any article can be made, a working drawing is needed. Xo pattern, however simple or plain, can be produced until we have something definite to work from. The metal worker does not go to the trouble of preparing a drawing on paper for every piece of work he is called on to make, since many objects are so plain that a brief verbal or written description of their dimensions gives the mechanic all the information he needs to enable him to lay out his work. If, for example, a tinsmith is called on to make a box out of IX tin, -t inches long, 3 inches wide, and 1 inch deep, he immediately proceeds with the steel square to lay off the given sizes directly on the metal; but if the same mechanic is required to make a round pan having flaring sides or some article of a form not readily carried in the mind, there is one thing he must do before he can proceed with the work or even lay out the pattern — ^he must make a working dra^vlng of the object. 3. Wliat Constitutes a Working Drawing. — There are several ways in which this drawing may be made, depending altogether on how complicated the object is. In the case of the pan referred to, it may be desirable, by an application of certain principles, to omit the operation of making a drawing consisting of several views and proceed as Avith the box to '"lay it out" directly on the metal. In this case, however, it will be found that the operation differs from that of making the box referred to, since it is first necessary to mark out the sizes and outlines as they will appear when the pan is completed. These sizes may or may not form a part of the pattern, but thej' are required as preliminary lines from which to 'May off," or ''strike out," the pattern. Marking out these sizes or dimensions of an object is really making a working drawing. This drawing may be § 15 PRACTICAL PROJECTION 3 full size — in which case it is referred to as a detail dra^w- Ing — or it may be drawn to a scale, either lart^er or smaller than the object itself. 4. AVhere the l^ra^viiis' Is^ Made, — In the case of plain articles, the necessary drawings may be made directly on the metal; in the majority of cases, however, the work is of a more or less complex nature, makini;- it ]ii_L;hly impor- tant that a full-sized properly made detail drawing- be used. This is not always provided, and the mechanic frequently has to make his own detail from a small freehand sketch or possibly from a drawing made to a small scale. In the latter case, an enlarged drawing must generally be made before the work of laying out the pattern can proceed. This neces- sitates operations with drafting board and drawing imple- ments — with which the student is already familiar. The proficiency that has been acquired may now be put into practical use in the operations to follow. It is the purpose of this section to present methods by which working drawings may be made and read. These methods are presented in a practical way, that the principles laid down may be readily understood by the student. ge:n^eeal principles 5. Various Kinds of DraAvinjys. — The most common representations of objects are those used for purposes of illustration merely and known as perspeetive dra^vings. They are of little value to the mechanic to serve as working drawings, since they are not drawn to a scale in the same way as a projection drawing, and to obtain measurements therefrom is an operation both complicated and indirect. A photograph is an ideal perspective picture, but no one would think of using a photograph as a working drawing. The photograph and the perspective drawing represent the object " as we see it," or as it appears to the eye of the observer, while the working drawing — the projection drawing — repre- sents the object as it actually is, or will be when made. 4 PRACTICAL PROJECTION § 15 A photograph, however, shows only such objects as really exist, while a projection drawing often shows objects that exist only in the imagination of the draftsman or a person capable of understanding, or "reading," the drawing. By means of such drawings the imagination is aided in pictur- ing the object as already constructed — or, as we would say, it enables the mechanic to see the object "in his mind's eye." 6. "WTiat Is SlioAvn In a Drawing. — The perspec- tive drawing always shows more than one side of an object — generally three sides — while the working drawing seldom shows more than one side, that being the side towards the observer. The position of such other portions of the object as are not located on the side shown in the drawing may, however, be indicated in a projection drawing by dotted lines. Xo lines should be used in a working drawing that do not represent actual edges, or outlines, in the object itself. We sometimes find certain edges or outlines of an object represented in a drawing by heavy lines. These heavy lines are called shade lines ; but since they are not essential features of the working drawing, no description of them is necessary in this section. There is also an elaborate system of representing the effect of light and shade on curved and receding surfaces by means of lines properly disposed over the surfaces shown in the drawing. Since these lines are for effect only, and their meaning is apparent to the observer, the principles governing their use are not made a part of the subject matter of this section. 7. Position of Observer. — Another point of differ- ence between the perspective and the projection drawing is that, in taking the viev.^ of which the perspective drawing is a representation, the eye of the observer remains in a fixed position, and in the same relation to the drawing as the camera is to the photograph ; while in that view of the object of which the projection draAving is a representation, the eye of the observer is ahvays supposed to be directly over or opposite that point in the drawing which is being noted. §15 PRACTICAL PROJECTION This may be illustrated by the student for himself in a very simple way. Place a sheet of paper on the drawing- board and draw a horizontal line, say (J inches long; now lay an ordinary 2-foot pocket rule against this line, in the manner shown in Fig. 1, and proceed to mark off the line Fig. 1 to correspond with the divisions on the rule. He will find that he is obliged to get his eye directly over each mark on the rule, and to "sight" carefully down on to the rule before making each mark, very much as he would "sight" or look along a piece of work to see whether it is straight or not. It will be noticed also that he is making otic eye do all the work in this "sighting," and, further, it will be observed that in making the markings he is moving his head as he progresses towards the end of the line. He is obliged to do this to keep his eye exactly over each point on the line as it is marked. 8, liine of 8iglit. — The line first drawn on the paper is not the only one made use of in reproducing the markings on the rule. The student has unconsciously made use of another line, or, more properly, a set of lines, that arc an 6 PRACTICAL PROJECTION § 15 important feature of projection drawing. These lines are those made use of in doing the "sighting" necessary for the marks ; they are purely imaginary lines and are not rep- resented in the illustration. They are very properly called lines of sight. The lines of sight in a projection drawing are ahi^'ays pcrpendiciilar to the draiving. They extend from a point in the eye of the observer to a point on the drawing that is directly opposite, as indicated by the point of the draftsman's pencil in Fig. 1. These lines of sight — which, as stated, are only imaginary lines and are not represented in a working drawing — con- stitute one of the most important features of the projection drawing; for on these lines we are enabled to obtain the views from the object, and, by means of other lines, called projectors, bearing a certain relation to the lines of sight (as will be explained later), Ave can reproduce the views thus obtained on the drawing. 9. Xines of 8iglit Always Parallel. — When it is desired to make a drawing of any object, the lines of sight must be used in the same manner as in marking the divisions on the line in Fig. 1 ; that is, care must be taken to keep the lines of sight in any one view parallel to one another. We may take different views of the same object, or, to express it otherwise, we may take positions on different sides of the object, in order to obtain views therefrom; but in any view thus taken the above statement must be care- fully observed and the lines of sight kept exactly parallel to one another. 10. Several A'iews Xecessarj'. — AVe have already noted that a projection drawing seldom shows but one side of an object. Since there are no objects that present all their dimensions on any one side, it necessarily follows that, in order to convey a correct idea of the form of an object, it is necessary to make a drawing — or a projection — of as many sides as will enable the correct shape and dimensions to be shown. We may make these drawings from as many points of view as may be desired ; but for certain reasons, § 15 PRACTICAL PROJECTION 7 to whirh attention will l)e called later, it is t^cnerally prefer- able to view all objects from six sides, which correspond to the six sides of a cube. 11. Before we proceed with the explanation of ortho- graphic projection, it is important that the student should be informed about what are known as the aiigifs of pro- jection. That is to say, he must know something about the positions supposed to be assumed by the object when the different views are taken. Draftsmen generally recognize for projection drawing two principal angles in which the objects are placed, and the drawings made in these two angles are called, respectively, Jirst-aitgic projcctioiis and tJiird-aiigIc projections. These different positions of the object will be understood by the student from an examination of Fig. 2, in which four angles are formed by the intersection of the horizontal plane y^ B C D\n\\\\ the vertical plane E FG H. Then, of the four right angles included between the planes, the angle B PG is called the first angle, the angle GPC \'~> known as the second angle, the angle C PF is the third angle, and the angle F P B is the fourth angle. In the view shown at {a) these two planes are assumed to be opaque and the object to be transparent. The lines of sight are projected from the eye of the observer to the plane of projection, tJtrough the object. In the view at (b) the planes are assumed to be transparent and the lines of sight pass from the object to the eye of the observer, through the plane. The effect of these different positions of the object is merely to change the relative posi- tions of the different views when made on the drawing board. The drawings made in this section are first-angle pro- jections. Third-angle projections are made by processes similar to those here explained and the different positions of the views will be pointed out to the student at the proper time. 13, Plans and Elevations. — It being assumed tliat the object is in some fixed position, the various views take their names from the different positions of the observer in PRACTICAL PROJECTION 15 his vie^y of the object. Thus, a view taken from above, or looking down on the object, is called a plan ; so also is a H I view from beneath, or looking up at the object; thus we have the terms top plan and bottom plan. The two views § 15 PRACTICAL PROJECTION 9 thus obtained are frequently desio-nated by terms that vary with the chiss of objects represented and not infrequently derive their names from some portion of the object itself. Thus, a top i)lan of a house is a view of the roof taken from above and is called a roof plan ; while a ceiling plan is, as its name indicates, a view of another part of the house taken from the opposite direction. In the case of small objects generally, such views are termed top plan or bottom plan, as the case may be. These views, in certain cases, should be marked on the drawing, in order to guard against error. Here it should be noted that while the position of the object is not changed in making either the top or the bottom plan, yet the position of the observer is. When the two i)lans thus made are compared, it is found that the corresponding points of the draAvings are changed in their relation to each other in the same manner as the hands of two persons that are standing exactly in front of and facing each other— the right hand of the one being opposite to the left hand of the other. A vievv' taken from the side of an object is called an elevation. That side of an object shown in any elevation gives its name to that drawing; thus, a view of the front of an object is called a front elevation. So, also, we have the terms rear elevation and side elevation. In some cases it may be more convenient to designate the elevations by the points of the compass; for example, the north elevation of a build- ing is a projection of that part of the building which faces north, or, to state it as we have done before, that part of the building seen when looked at from the north. 13. Section Drawings. — Cases frequently occur in which the views or dimensions desired to be given on a working drawing cannot be shown in either plans or eleva- tions. Under such circumstances, recourse is often had to a class of drawings termed sections. A section drawing is a projection of an object assumed to have been cut in two in a certain direction, usually at right angles to the lines of sight. Those parts of the object between the observer and 10 PRACTICAL PROJECTION § 15 the place where the cut is made are assumed to have been removed, so as to present an entirely new surface. This surface is not seen in the object itself, since the cutting is entirely imaginary — done simply for the purpose of showing some interior construction. The cut just referred to may be made in a horizontal, a vertical, or an oblique direction, according to the way in which it is desired to show the section. Portions of surfaces that have been cut in this way are usually represented by certain conventional methods, indicating the character or composition of the material. A custom frequently adopted, and which will be followed in the drawings for this section, consists in designating surfaces exposed by the cut by a series of closely drawn parallel lines. Such lines are usually drawn at an oblique angle, as compared with the other portions of the drawing, and are called cross-section lines or cross-hatching. 14. A Set of Plans. — It is a common practice, when speaking of a set of drawings consisting of various plans, elevations, sections, etc. — as for a house or for some other object — to refer to them as a "set of plans." This is a collective phrase for the drawings and its use in this way is perfectly proper ; when used in this sense, however, it is not understood as applying simply to a plan view as explained in Art. 12. Drawings for large objects are frequently of a size such that the different views are more conveniently made on separate sheets. Architectural drawings are usually separated in this way, and it is often necessary for the one that is to read such drawings to arrange the sheets in a particular manner, in order that the relation between the views may be understood. This arrangement of the views will be considered later. 15. Foresliortene'inati()ii ii ^■^llllJl^)le Assistant to tlie Draftsman. — x\n object, or solid, of any conceivable shape may thus be resolved into its elementary parts or points. The drawing of the object, then, will consist simply of locating the positions of these points on the drawing. We may have drawings to make that will require the location of a hundred or more of these points, depending entirely on the form or shape of the object we are dealing with, but the principles are in all cases the same. If the student, after resolving an object in this imaginary way, will carefully study or imagine the proper location of these points in their relation to the object itself, defining their positions on the drawing one at a time, much that may appear complicated at first sight will resolve itself into very simple and comparatively elementary work. Complicated work is usually nothing more or less than the aggregation of a number of simjjle operations that appear complicated only because they are combined. There is no field of work to which the latter statement is more applieable than to that of the draftsman. It is in the "imaginary" way thus described that the student is directed to picture to himself each figure as pre- sented tt) him for the making of the drawings on the plates. This part of the study is, as will be noticed, almost entirely the work of the imagination ; but it should be practiced by the student for the sake of the assistance it will be to him later on. M. E. v.— IS 14 PRACTICAL PROJECTION § 15 The operations of projection drawing follow one another in a natural sequence, which we will proceed to trace out in a series of drawing plates. As the student follows these operations, keeping in mind the foregoing principles, he will have no difficulty in making or reading any drawing. PLATES 21, Seven plates are to be drawn by the student in accordance with the directions given in this section. They are to be of the same size as those drawn for Geometrical Draining, and the same general instructions regarding the preparation of the plates are to be observed; they must be drawn and sent to us for correction in the same manner. The letter heading for each problem, which has heretofore been placed on the drawing, will be omitted, and the stu- dent is required only to designate each plate with the letter heading, or title, that is printed in heavy-faced type, both in this section and on the reduced copies of the plate. For this purpose the block-letter alphabet is used. 22. The dimension lines and figures shown in the first three problems of the drawing plate, title: Projections I. are to be especially noticed by the Right Wrong— - student. They are ordinarily used in all working draw- ings, and preference is invari- ably given to a dimension ^"^ ^ figure, rather than to the scale to which a drawing is made. Dimension figures are not to be placed on the plates, since the object in requiring the student to draw these projections is rather to enable him to gain an idea of their principles than to be able to make a finished working drawing. Dimension and extension lines when used should be light broken lines. Care should be exercised to make the arrow- heads as neatly as possible and of a uniform size — not too flar- ing. They are made with a steel writing pen and their points should touch the extension lines, as illustrated in Fig. 3. VP 1 E^- ^F t£ .J_X. PRDJEfl .1 L. PROBLEMS. "T"i — r \—^ X ■-f- ' I. t- PROBLEM S. I + PROBLEM 6. JUNE 25, 1893. Copyright, 1899, by The Cc All righi riDNS-I. WLEMS. >- PROBLEMS, ,-i _ _ '. x\ i K AH' \ / 57 PROBLEM 7. \° PROBLEMS. ERV Engineer Company. iserved. JO/iA/ ^Af/T/y, CLA22 A/? 45S9. § 15 PRACTICAL PROJECTION 15 DRAAVING PLATE, IIILE: PROJECTIONS I 33. General Instructions. — This plate is divided into four equal spaces, and each of these divisions, with the exception of the upper left-hand space, is again divided, by- means of a central vertical liiie, into two equal parts. Use light pencil lines, as they are not to be inked in and are intended only to facilitate the location of the problems. These lines are not shown on the printed copies of the plates. Before attempting to draw any of the problems on the plate, the explanations accompanying each problem should first be carefully read and compared with the; reduced copy of the plate and also Avith the illustrations in this sec- tion. The principles of projection drawing will thus be better understood by the student and their application readily made when more difficult drawings are undertaken. TIic fiDidamcntal lazvs of projection are contained in the first four probleihs, and if these are thoroughly mastered by the student, the application of the lazes to the reuiaifiing prob- lems zvill be comparatively easy. 34. Why Different VieAvs Are Drawn. — The names of the different views have already been noted; in this plate it is shown how they are distinguished from one another in a drawing. The relation of the different views to one another will also be exjjlained. Some objects in certain positions may have all their dimensions represented in two views — a i)lan and an eleva- tion — but generally three views should be drawn. There are, indeed, many cases where views from each of the six sides, as well as sections and views taken from oblique positions, are necessary. It has already been observed that lines are represented in their true length only when at right angles to the lines of sight; consequently, since the posi- tion of the object in any set of views is not changed, it is necessary to change the i)osition of the observer in such a manner as to bring the lines of sight where they will be at right angles to the lines in the object, thus enabling the latter to be shown in their true. length on the drawing. 16 PRACTICAL PROJECTION § 15 The student will readily perceive that in drafting-room work it is of the highest importance that the lines which combine to make up the surfaces of any object should be shown in their true length, or at least be presented in such positions that their true lengths may be easily found. With- out these true lengths, no measurements can be obtained from which to lay out patterns, a pattern being merely a representation of the surfaces of some solid. It is neces- sary, therefore, to be prepared to take views of any object from any position; for there are many different forms, or shapes, of solids, and it is necessary to be able to show in its true length any line in the object that may be needed for a pattern. 25. The Base Line. — We shall first consider objects in positions that may be shown in two views. Draw a horizontal line through the central portion of the upper left- hand space on the drawing, as at tn-n on the plate. In the portion of the space below this line are to be drawn the top plans of each of the two simplest forms, viz. , the point and the line. The space above this line is to contain the eleva- tions of the same forms. The line thus drawn is called a Itase line and defines the boundary of the surfaces on which we are to "sight," or, as we shall say hereafter, on which we are to project the lines of sight. It is necessary to call the imagination into use again and imagine the paper to be bent up at a right angle on this line. 26. Plane< of Projection. — The drawing paper is imagined to be the surface that intercepts the lines of sight; and in the case of a plan, as seen from instruction already given, must be a horizontal surface, while in the case of an elevation, it is imagined to be a vertical surface. The dif- ferent portions of the drawing on which the projections are made are called planes of projection, and are also distin- guished by other names that designate the position they are supposed to occupy in intercepting the lines of sight. That portion of the drawing on which the elevation is drawn is called the vertical plane of projection; it is represented § 15 PRACTICAL PROJECTION 17 on this plate by the space above the base line; the portion below the base line is devoted to the plan and is called the horizontal plane of projection. We shall, for the sake of brevity, refer to these surfaces by the use of the letters V P and H P, respectively. Copy these letters into the upper and the lower left-hand portion of their respective spaces on the drawing, using for that purpose a block letter one- half the size of the title letter and leaving a distance of ^ inch from the border lines of the spaces. 27. Foot of tlie Tjine of Siglit. — Before proceeding with the drawing of this plate, it is desired to call the atten- tion of the student to the distinction to be observed between the ii)iaginative and the practical features of this subject. The imaginative feature is employed when a conception of an object is formed by the student in accordance with the instruction in previous articles, and also when the lines of sight are applied in the imaginary way, as in the " sighting " illustrated in Fig. 1. The application of the practical feature in this instance is made when the position of each division of the rule is indicated on the drawing by a pencil mark or dot. The practical part of the work is always accomplished by the aid of pencil and drawing instruments. The two features are closely associated, since we cannot have a practical representation of any object without first having an idea, or an imaginative conception, either of the object itself or of the means of projection. The practical feature of the work was introduced in the illustration (Fig. 1) when a mark or dot was made on the paper, thereby indica- ting the position of the point at which the line of sight was intercepted by the plane of projection. That point on any plane or drawing where a line of sight is intercepted is called the foot of the line of sight. When the foot of every line of siglit that can be used on the ele- mentary points of any object is thus represented on the drawing by dots, and connecting lines are drawn between such dots, the drawing is completed, and the object is said to be "■ projected." 18 PRACTICAL PROJECTION § 15 28. Projectors. — If but one view of an object were required, the use of the lines of sight as previously explained (representing the imaginative feature) and the drawing of the dots and lines referred to in the previous article (repre- senting the practical feature) would be all that is necessary for the student to understand before proceeding with the work on the drawing board. Since it has been shown that several views are required, another important practical feature must necessarily be explained. This relates to the connection usually established between the different views of a drawing and the lines that are drawn in a certain manner between corresponding points in each view. These lines are usually not represented in a finished drawing, since they are in the nature of construction lines. They are essential, however, to the work of making the drawing, and it is very important that the student should thoroughly understand the principles by which they are employed. These lines are called projectoi-s and may be defined as the trace of a line of sight, or the representation of the foot of a line of sight moving in a certain direction. Projectors are used in two ways, which are distinguished from each other for the present by the terms primary and secondary. 29. Primary Projectors. — This use of projectors is illustrated in Fig. 1, which shows the drawing bent up at a right angle along the base line vi-n. The point A is pro- jected to HP by the vertical line of sight C B\ it is also projected to V P by the horizontal line of sight D E\ B and E are dots at the foot of each line of sight. It is assumed that the first position of the observer is at C\ he then moves, in the direction of the arrow, along the dotted line to D. If. in so doing, he continues to sight through the point J. it is apparent that a line will be traced from j5 to /'on H P and from F Xo E oxi\ P. The upright portion of the drawing V P is now imagined to be bent back- wards until laid flat on the drawing board, and it is evident that E F B is represented on the flat surface of the two 15 PRACTICAL PROJECTION 19 planes of projection by the strai^lit line Jl' F P>. It may therefore be drawn as a straight line by the aid of the tri- angle and T square, the position of the point . / in each view being determined by the points B and E' at the extremities mtST. POSITION Fig. 4 of the line. These points (or dots, for points, being entirely imaginary, could not, of course, be actually represented) * B and £' are the projections of the point A — the line drawn between them(/>7^^') is called 2, projector. When * Attention has been called to the fact that jioints, lines, and surfaces are entirely imaginary geometrical forms. This is true in the sense that the student must consider such forms in the imaginative study of this subject. When their representation on the drawing paper is con- sidered, however, something that can actually be seen by the eye is required. Therefore, when a point is referred to in this section as per- taining to the drawing, it is to be represented by a neat dot in the proper place on the paper. In like manner, a line should be represented by a fine pencil mark drawn between two points marking its extrem- ities. When the line is to serve a special purjxjse, as e.xplained in this section, it is inked in in a particular manner characteristic of its use, in order that the drawing may be more easily read. A surface, therefore, would be represented by a portion of the drawing bounded by the proper lines and descrijitive of the form of surface rei)resentcd. Re- member that accurate work cannot be done unless the pencil points are in good condition. The student should provide himself with a smooth file or piece of fine sandpaper and frequently sharpen the chisel point of the leads in both pencil and compasses, in order that fine sharp lines may be readily drawn. 20 PRACTICAL PROJECTION § 15 projectors are used as in this illustration — that is, between two planes that may actually be bent up as shown in Fig. 4 — they are said to be used in a primary manner. The secondary use of the projector will be shown in con- nection with Problem 3, Case III. The practical use of the projector is clearly shown in the following problems. It is a most important factor in the projection, and, as will be seen from instruction soon to follow, is often the first line to be used in a drawing. PROBLEM 1 30. To project tlie plan and elevation of an iniag- inai^ point. There are two cases of this problem, representing differ- ent positions of the point. Definite instructions are given for drawing the first projection and the student is expected to draw the second projection without further directions. Case I. — WJien the point is located 1 inch from each of the two surfaces V V and W P. This position is illustrated in Fig. 4, referred to in Art. 29. CoxsTRUCTiox. — Fix a point B (see plate) 1 inch below the base line on the drawing. This point should be f inch from the left-hand side of the drawing and is the plan view of the point given in the problem. Bring the T square and triangle into position and draw the projector vertically upwards and across the base line. Fix a point 2i\..E' on the projector 1 inch above the base line. A projection drawing is thus made, showing two views — a plan and an elevation — of the required point, the position of which is thus defi- nitely established. Fig. 4 is an illustration of the imaginative feature and the projection drawing just made is a representation of the practical feature of the work — the part actually made by the draftsman. The intimate connection between the two features may be seen if the drawing just made is compared § 15 PRACTICAL PROJECTION 31 with the illustration in Fig. 4. Similar results are found to have been accomplished in both cases, the method last employed being the only one practicable for actual use. When inking- in this drawing, make small round dots to represent the positions of the points, and always ink in projectors as hght dot-and-dash lines, as shown on the plate. These dot-and-dash lines should be inked in in a uniform manner, as on the plate, the dashes being about i inch in length and spaced about ^V inch apart, with a light dot between each dash. Measure the distances by the eye and preserve a uniform shade for all projectors, thus giving the drawing a neat appearance. The base line should be represented by a heavy dotted line, as shown at in-n in the perspective illustration of Fig. 4. When making the preliminary drazvings, do not attempt to drazu dotted lines wit/i the peneil, sinee this is liable to affect the aeeuracy of the zvork. Keep the chisel point of the pencil sharp and draw as fine a line as can be distinctly seen. The contrast between the different lines on the draw- ing may then be clearly indicated when the work is inked in. Case H. — ]]'hen the point is located Ih inches from V P and ^ inch from H P. The student will fix the location of the point in the plan and elevation on the drawing without further instructions, bearing in mind the fact that distances from V P are meas- ured on the plan and distances from H P are shown in the elevation. Reference to Fig. 4 explains this statement. Case II should be placed on the drawing about I- inch to the right of the preceding figure. PROBLEM 3 31. To project the plan and elevation of an imagr- iuary line, the line being in a rifjfht position. The term right position is used in connection with pro- jection drawing as distinguished from the terms inclined, or oblique, position. The line, therefore, can be either in a 22 PRACTICAL PROJECTIOX §15 horizontal position or in a vertical position and still be designated as in a rigJit position. There are three figures for this problem, representing three cases Avhere the line is in a right position and yet represented differently on the drawing. The different positions, the various distances, and the length of the lines for the three cases of this prob- lem are clearly illustrated in the perspective drawings shown in Figs. 5, G, and 7. Instructions are given for the drawing of Case I on the plate, but the student is expected to be able to make the drawings for Cases II and III with- out further directions than those contained in the illustra- tions. Be careful to preserve a distance of \ inch between the drawings, so that the plate may present a neat appear- ance when completed. Case I. — WJicn the line is parallel to both H P and\ P. ExPLAXATioN. — This position of the line is illustrated in Fig. 5, and, as apparent from that figure, the drawing is Fig. 5 merely an extension of Problem 1. Each end of the line is treated as a point, projected first to the plan and after- wards to the elevation in precisely the same manner as was the point in Problem 1, the only difference being that there are two points instead of one, for we cannot have a line without establishing at least two points. § 15 PRACTICAL PROJECTION 23 This problem also illustniLes another principle of projec- tion already referred to, viz., all lines at ri;4ht angles to the lines of sight in any view are shown in that view in their true length; or, in other words, the lines that are to be made on the drawing to represent the plan and elevation of A B, Fig. 5, will be of the same length as A B is indi- cated in the figure, viz., 2 inches. The angles HA B and L A B are right angles, although shown in perspective in the figure; and, since the lines of sight in any view are always parallel to each other, the angles GBA and KB A must also be right angles; consequently, as the line A B is at right angles to the lines of sight in both views, it must be shown in its true length in both the plan and the eleva- tion on the drawing. Construction. — To make this drawing on the plate, draw a horizontal line of the given length and the proper dis- tance (i. e. , 1 inch) below the base line. This will be the plan of the line ./ />'. From each end of this line {CD in Fig. 5 and on the plate) draw projectors to the elevation; or, to use the term by which such operations are desig- nated, project the ends of the line CD to the elevation. After measuring off the proper height above the base line, draw the horizontal line E F, which is the elevation of the line A B. Note. — When a point is projected from one view to another, its projector (a straight line) is drawn from the first view to the view pro- jected, and always at right angles to the base line. Case II. — Where t lie line is in a Jiorizontal position ODui at ri^a^Jit angles to ^ P. ExPL.\N.\Tiox. — Fig. illustrates this case. It will be noticed that the plan view of the line does not differ very much from the plan of the line given in Case I, merely that it is represented by a vertical line on the drawing in the plan in place of a horizontal line, as in Case I. The line ^ .5 is at right angles to the vertical lines of sight G D and H C in both cases. The line .'/ B in this figure is in such a position that the horizontal line of sight K£ passes 24 PRACTICAL PROJECTION §15 through both points B and A. The projection of these points, therefore, on the elevation is the single point E at the foot of the line of sight K E. A single line of sight may pass through an unlimited number of points in any view, but the foot of such line of sight is always represented in that view by one point on the drawing. The student who is to read that drawing must picture to himself, or imagine, the position of these points as they are supposed to exist in the object of which any drawing is a representation. It is further to assist his imagination that other views are drawn, and by means of which the position of the different points may be definitely located. Thus, in this case, if we were Fig. 6 to consider the elevation alone, without paying any atten- tion to the plan, we Avould say that the point E represented merely some other point (imaginary, of course) that could be situated anywhere on the line of sight K E. A glance at the plan, however, shows not only where the location is, but how many points are represented. In this case there are two points represented by E, one directly in line with the other. The plan also shows how far apart these points are, and from the two views it can be further seen that the line is in a right position perpendicular to V P. If it were not, the two points would not be in the same line of sight, and would, consequently, require two positions on the ele- vation, whereas they are designated by the one position E, Fig. 6. 15 PRACTICAL PROJECTION 25 Construction.— Since the drawings for this case have been shown in the preceding- exphination to be similar to those of Case I, no definite instructions need be given. The two drawings differ in position only and are drawn as shown on the plate, tiie plan being first constructed. Case III. — JJ7/cn' the line is in a vertieal position. This is shown in Fig. 7, each feature of which has already been explained in connection with Figs. 5 and G. The Fig. 7 drawing may, therefore, be made by the student in accord- ance with tlie dimensions given in the illustration. 32. Proof of a Projection I)rawinjt»-.— The various cases of the foregoing problem represent lines indifferent positions, and, as in the case of the point in Problem 1, the student will see that these projections definitely repre'sent the position of each line; further, that for each position indicated, but one line can be placed, or can occupy that position. It is recommended that the student prove this assertion as follows: Copy the projections of this problem on another piece of paper and bend the paper at right angles along the base line, as shown in tlu- illustrations; take a piece of small wire of the given length, to represent the lines, and proceed to hold it, in turn, over the drawing for each case, at the same time " sighting," or using the 26 PRACTICAL PROJECTION §15 lines of sight, as illustrated. It will be seen that, in order to make the foot of each separate line of sight come to the proper place on the drawing, the wire must be held in the position Indicated in the statement of the case. PKOBIJEM 3 33. To draw the projections of an iniasinary line in A riglitlj- melinecl position- There are three cases of this problem, in all of which the given line is rightly inclined; that is, the angle of inclina- tion is such that the true length of the line may be shown in either a plan, a front elevation, or in some elevation that shall be at right angles to the front elevation. The differ- ent cases of this problem are presented in perspective views, from which the projection drawings are to be made by the stu- dent. They illustrate the principles of foreshortened views. Case I. — WJiere tlu line is horizontal, but inclined t: V P at an angle of ^. ExPLAKATiojf. — This is shown in Fig. 8, which gives all the dimensions and distances necessary to enable the student to Fig. 8 draw the projections on the plate. The student will note the position of each point and carefully observe the instructions § 15 PRACTICAL PROJECTION 27. for making the drawings. Bear in mind that, although the drawing is only that of a single line, careful study nurst be given to it, for the principles on which these simple pro- jections are made are the same as for any other projection drawing. These principles are shown in a more compre- hensive way in simple problems than if an object of com- plex form were presented, requiring a confusing number of points to define its outline. In Cases I and II of this prob- lem, the plan is first to be drawn and the elevation projected 'therefrom. Construction.— Since the line in Case I is in a horizontal position and therefore at right angles to the vertical lines of sight, it will be shown in its full length on the plan. The line is stated to be inclined to V P at an angle of 45°; draw the plan, therefore, at that angle to the base line and at such distance below the base line as indicated in Fig. 8 and shown at A B on the plate. Project the points A and B to the elevation, and at the given height above the base line draw a horizontal line between the projectors. This is the elevation of the line shown in the plan, and since its entire length is contained between the horizontal lines of sight F E and H G, Fig. 8, the line cannot be shown on the elevation as being any longer than the perpendicular distance between the projectors. This distance being less than the actual length of the line, the elevation is, in this case, called a foreshortened view of the line. It represents, however, the entire line, and reference to the plan is necessary in order to find its true length. C&,se. 11.— Where the line is parallel to V P but inelined to HP at an angle of 60°. Construction.— Fig. 9 shows that the plan is to be rep- resented by the horizontal line F If. It also gives the length of the line in the i)lan, which is a foreshortened view. Therefore, draw F H 1 inch long and 1 inch below the base line. Draw the projectors and fix a point at / on the pro- jector drawn from /% at the proper distance above the base line. This will be one end of the line in the elevation 28 PRACTICAL PROJECTION §15 With the compasses set to 2 inches (the length of A B) and using the point y, fixed on the projector drawn from i% as a center, describe an arc intersecting the other projector. The point that represents the other end of the Hne is located at this intersection, and the line may then be drawn. Now bring the T square into position and prove by the Fig. 9 triangle that the line y A' is at an angle of G0° with the base line. It will be seen that it would have been possible, after fixing the position of a point at either end. to have drawn the line at once with the 60° triangle. Attention is called to both methods in order to show the student the connection between them. Case III. — ]Vhc}'c the line is rigJitly inclined to V P at an angle of 6'6'° and is also inclined to H P. ExPLAXATiox. — In Cases I and II, the line has been in such positions as to enable its full length to have been shown in one of the views drawn on the plate. In this case, a position is illustrated in which the line is shown foreshort- ened in both of these views. It will therefore require another view to be projected in order that the line may'be shown in its true length. Since it is known that the line is rightly inclined, the additional view required will be at right 15 PRACTICAL PROJECTION 29 angles to the base line. As another view is to be drawn, so another base line will be required. This base line, being merely the lower boundary of a surface supposed to be in an upright position to receive the lines of sight and at right angles to the surface of the elevation previously drawn, will, consequently, be at right angles with the base line in a drawing that shows only two views — such drawings as have thus far been made. Fig. 10 contains the given dimensions, etc. for this position of the line. This perspective figure shows the interception Fig. 10 of the lines of sight from still another direction than has been shown in the preceding illustrations. In the same manner a view may be obtained from any side of an object or at any angle other than a right angle. The method of accomplishing these results by the use of the T square and triangle on the flat surface of the drawing board will now be shown, and the illustration of the bent-up surfaces will not be continued beyond this problem. Such illustrations are, however, always implied in a projection drawing, for that part of the work is the imaginative feature previously mentioned, to which the attention of the student will be directed throughout this instruction. Since the angle of inclination to the elevation is the same in this case as in the plan of Case II, the length shown on the elevation of this projection will be the same as in the plan of that case. In this drawing, the line is, however, in a M. E. v.— 14 30 PRACTICAL PROJECTION § 15 different position as related to the plan, and from Fig. 10 it will be seen that it must be represented by a vertical line in that view on the drawing. CoxsTRUCTiON. — Extend the oase line on the plate to the center of the next space, from which point draw a vertical line downwards to the division line; these lines are to be inked in the same as the base line in the first space. In the smaller space thus enclosed on the drawing is to be drawn the plan for this problem, the front elevation occupying the same relative position as before, directly above the plan,' while in the space at the right the side elevation will be projected. First, draw the elevation as at iT ' F' on the plate, fixing the point E ' at the specified distance above the base line. As the foreshortened length in the plan is given in Fig. 10 as If inches, draw the line C D oi that length, as shown on the plate, keeping the point C at its proper dis- "tance below the base line, as indicated in Fig. 10. The view to represent this line in its true length may next be drawn. It is known that this view must be one in which the line itself is represented at right angles to the lines of sight. There is a choice of two views for this projection — either to the right or to the left side. Having already utilized the space to the left on the plate, the side elevation is, in this case, projected to the right. The method employed in Case II might here be used to project the side elevation, but since it is customary, when a number of elevations are pro- jected from the same plan, to facilitate the operation by drawing between such views lines that are termed second- ary projectors, an explanation of their use is here presented. 34. Secondary Use of the Projector. — The term secondary is not applied in the case of projectors as indica- ting an unimportant or infrequent application of these lines. The name is used rather to distinguish operations in which similar principles as applied to the imaginative features are differently represented on the drawing in the application of the practical features of the work. In fact, both uses of these lines are required in most drawings. It is therefore §15 PRACTICAL PROJECTION 31 essential that the student should become familiar with the various means employed in producing them on the drawing. 35. It is evident, from an inspection of Fig. 10, that the eye of the observer at E in moving around along the broken line in the direction of the arrows to take position at K would trace a line from F, through O, to /.. The part of this line {F O) that shows on the front elevation is parallel to the base line of that surface; and, also, that part of the line {O L) shown on the side elevation is parallel to the base line of the side -elevation. It is seen that the definition of the projector, as previously given, applies equally to the lines F O and O L. If the two planes of projection repre- sented by the two upright surfaces could be bent in the same relation to each other as were the plan and the elevation in Fig. 4 — i. e., on the line P Q, Fig. 10 — the use of the pro- jectors in these views would be no different from that already described. It is customary, however, in first-angle pro- jection, to assume that such'upright surfaces are always bent downwards and away from the plan ; to accomplish this result, the secondary use of the projector is employed. Sup- pose, now, that the upright surfaces, represented in Fig. 10, were bent backwards until laid flat on the drawing board. Evidently, there would be an appearance presented similar to that shown in Fig. 11, and an open space would be shown Fig. 11 on the drawing board included between the angle /* (7/*'. The paper on which the drawing is made is not of an irregular shape, thus to be bent up at will; further, the operations 3-2 PRACTICAL PROJECTIOX § 15 performed in projection drawing are such that they can be accomplished only on the flat surface of the drawing board. 36. It is found that similar results may be obtained in two wavs, both being easily affected by the aid of the draw- ing instruments. The first is known as the angular method, and is thus accomplished: If the projectors F O and O' L, Fig. 11. or any other corresponding set of projectors, parallel to their respective base lines, are extended until they inter- sect each other, it is found that all the intersections arc on a diagonal line terminating exactly at the intersection of the base lines. It is also found that this diagonal line exactly bisects the outer angle formed by the base lines. Applying these principles, therefore, to the drawing, bisect the outer angle formed by the base lines on the plate and produce the bisector indefinitely towards the right-hand side of the space. The outer angle formed by the base lines in this case being an angle of 270', the bisector ma/ be drawn with the 45" triangle, since a line thus drawn will be at an angle of 135° with both base lines. 3T. Fig-. 1"2 is a reproduction of the projection drawing from the plate, showing the bisector drawn as previously sEcosx,AMT ivojxcrom /directed. Draw F' x and F' y ^/ i parallel to the base line in the front elevation; from their inter- sections with the bisector at x and r, draw x L and y R parallel to the base line in the side eleva- tion. Project the points C and D from the plan to the side elevation by the use of primary projectors, as previousl)' described. The side elevation of the line A B of Fig. 10, then, is a line drawn be- tween points of intersection of ^^- ^~ the primary with the secondary projectors, as shown by R L, Fig. 1-2. The lines F' x. x L. and E ' y, y R are called secondary projectors, and are used, § 15 TRACTICAL PROJECTION 33 as in this case, Avhen projectinij points between views that are related to one another in the manner shown. Project- ors are used in a similar way when, for reasons that will be shown later, the base lines are at an angle other than a right angle. In all cases, the outer angle is bisected as shown in Fig. 12, and the secondary projectors are drawn parallel to their respective base lines. Note that tJic front elevation, s//07^'u in Fig. 12, is a fore- shortened fieza, and corresponds in length with t lie perpen- dicular distance between the secondary projectors in the side elevation. 38. Readinjjf a Drawing. — The ability to read a draw- ing consists of the intelligent comparison of the different views and is well illustrated in the projections just drawn. The different views — or the different projections, as they are called — must never be considered as drawings apart from one another. Each projection is shown to be neces- sary in order to enable the position of some point or element of the object to be established in the reader's imagination. PROBLEM 4 39. To draAv the projections of an imaginary line in an obliquely inclined position. The projections of this problem are to be drawn by the student in the next space on the plate, following the instruc- tions here given. Construction-. — Draw the base lines as in the last space used for Problem 3, but place the lines \ inch higher on the plate and extend them h inch farther to the right in the space. These base lines will be used in the construction of the projections as before, but will not be inked in on this drawing ; they are construction lines onl)^ — to be erased from the plate after the drawing is completed. It has been shown that base lines are necessary for determining the position of the different points on a drawing and are 34 PRACTICAL PROTECTION S 15 essential in establishing the first few points in any projec- tion; but as the drawing progresses and other lines are produced, any right line in a view — i. e., a line at right angles to the lines of sight — may be used as a base from which to establish the position of points in a drawing. FIG. 13 Represent a foreshortened view of this line in the plan by a line li^ inches long, drawn at an angle of 45° with the base line of the front elevation, as shown at A B. Fig. 13; draw the front elevation (also a foreshortened view) at an angle of 30° with the base line. Draw the line A B in the plan § 15 PRACTICAL PROJECTION 35 and G E in the front elevation in such positions that the end of either Hne nearest to the base Hue shall be k inch from that line. Next draw the side elevation as explained in Problem 3, Case III, and it will be seen, when the side elevation is completed, that H J \s also a foreshortened view, not representing the true length of the line. An elevation will now be projected in which the line may be shown in its true length. This will be an elevation whose surface is parallel to the line. Draw the base line of this surface ^ inch from the line A B on the plan and parallel to that line, as at C D, Fig. 13. This figure is an illustration of the projection drawing, showing all the lines used in its construction, certain of which, as already explained, are not to appear in the completed drawing on the plate. Note that the oblique elevation K L is projected in the same manner as the side elevation was drawn, the only difference being that the outer angle O D C, formed by the base lines, is greater than a right angle, but is treated in the same way. This completes the problem, and in finish- ing the figure on the plate, the student will ink in only the different views and the primary projectors, erasing all other construction lines. 4:0. Finding True Ijengfths by Triangles. — It is possible to find the true lengths of lines from a plan and any elevation showing such lines obliquely ^ inclined by a shorter method than that given in Problem 4. This is accomplished by the use of the right-angled triangle. ^' If such a triangle is constructed, with its ^'*" '^ base equal to the length of the line shown in the plan and its altitude equal to the vertical height shown in the eleva- tion, t/ie Jiypotciiusc li'iH be equal to tJie ty-ue length of the line. This is shown in Fig. 14, in which A B is made the same length as A B, Fig. 13, and B C\ Fig. 14, is equal to the vertical height shown in the elevation, i. e., E F, Fig. 13. Since A B T is a right angle, the hypotenuse A C, Fig. 14, is equal to the true length of the line. This statement is of 36 PRACTICAL PROJECTION § 15 the greatest importance to the draftsman and should be proved by the student. Construct a triangle on a separate piece of paper and set off the lengths from the drawing with the dividers ; afterwards compare the length of the hypot- enuse with the length of the line shown in the oblique eleva- tion, or full view. This is an illustration of a principle of- much use in later problems, and one on which certain impor- tant principles of patterncutting depend. 41. All Projections Depend on Similar Prin- ciples. — There is no conceivable position of a line that may not be shown or its true length not be ascertained by the application of the principles contained in the foregoing simple problems. Lines have been used to illustrate these problems drawn at such angles as were conveniently made with the T square and the 45° or 60° triangles, but any angle or any position could as well have been represented, since the principles are in any and all cases the same. "We will now proceed with the representation of flat, or plane, surfaces. 42. Planes, or Plane Surfaces. — All drawings made to represent surfaces are composed of lines that bound, or limit, their borders, or sides. These drawings, therefore, will differ from those of the foregoing problems only in the fact that they are the representation of lines shown in their relation to one another. There are, however, certain prin- ciples relating to flat, or plane, surfaces that must be borne in mind, since they influence this relation of the different lines in a drawing. 43. That the student may have a thorough knowledge of the principles employed in the representation of surfaces, it is essential that he first have a clear conception of what a plane is. A plane surface, as has been stated, has only an imaginary existence, being bounded, or enclosed, by imagi- nary lines ; this surface may be in any conceivable position,' but is always a flat surface. If viewed from a certain direc- tion — viz., as if "on edge" — it would be represented by a single straight line. If the student can imagine a plane surface indefinitelv extended in everv direction bevond the § 15 PRACTICAL PROJECTION 37 boundary lines of the figure, he will have a very good con- ception of a plane; any number of points or lines, the posi- tions of which are anywhere on this surface thus extended, are said to be " in the same plane " in relation to one another. 44. To illustrate: Suppose two flat-top tables of the same height are on the floor of a room perfectly level and of indefinite extent. Here is a practical representation of two planes, both of them in a horizontal position; one plane is represented by the floor, while the other plane is parallel to the first and "passes through " the tops of the tables. The surfaces represented by the tops of the tables are said to be "in the same plane." The tables may be placed some distance apart, yet the straight edge of a ruler laid across their tops would exactly coincide Avith the upper surfaces of both tables and would remain in contact at all points for every position of the ruler. The plane surface represented by the top of one table is said to be " in the same plane " as the corresponding surface of the other table. The same could be said with reference to any other surfaces answering the same test. Any number of flat surfaces are said to be in, or to "lie in," the same plane with one another, and the same is true of any lines or points used to define any surface or position in that plane. The planes in the foregoing illustration of the floor and tables are horizontal planes, but may be imagined in any position, vertical or inclined, needed for the projections of a drawing. 45. How the Position of a Plaue Is Determined. Since any two points determine the position of a line, so any three points not in the same straight line determine the position of a plane. To illustrate: Take a square piece of 'cardboard, thick enough to remain flat, and push pins of equal length through each of the four corners so that they will resemble the legs of a chair. The object will stand firmly when placed on a level surface with the points of the pins down, for the reason that all the points represented by the ends of the pins are in the same plane. If one of the 38 PRACTICAL PROJECTION § 15 pins is withdrawn and a shorter one inserted in its place, the cardboard will not be stable when placed as before, and can be " rocked." for the point at the extremity of the short pin is not in the same plane with the other three. Two planes are thus defined — one determined by points at the extremities of the three long pins and the other by points at the ends of the short and the two adjacent pins. Both of these planes may be imagined as extended indefinitely, one plane being inclined to and intersecting the other. Again, a fiat sheet of metal may be supposed to represent a plane surface. All points that may be located on this sheet are in the same plane ; but if a sheet that is " buckled " is chosen, it is possible to locate some points on the surface of that sheet higher or lower than others, and the points would then be in different planes. The connection between the plane and the plane surface, then, is such that, to be defined as a plane surface, every point on that surface must be in the same plane. 46. In drawing different views for the illustration of the plane surface, we shall first use the octagon, requiring the projection of eight points and the intermediate lines. The use of the word "imaginary " in connection with the state- ment of the problem will henceforth be discontinued, since it has been clearly shown that all surfaces, as well as other geometrical elements, depend for their existence on the imaginative feature referred to in previous articles. It will be understood, therefore, when any geometrical element is mentioned, that the practical feature is to be employed — the imaginative, of course, being implied. PROBLEM 5 47. To project three vie^vs of an octagonal sur- face, reiDresenting it in a liorlzontal position. The three views consist of a plan, front, and side eleva- tion. A perspective view of the surface in the required position is shown in Fig. 15. §15 PRACTICAL PROJECTION 39 ExPLAXATiON. — All lines used to define this surface in the plan are at right angles to the vertical lines of sight ; and since the line*^ will thus be drawn in their full length in that view, the surface will there be shown in its full dimensions. Fig. 15 This principle also applies to any view of a plane surface in Avhich all its lines are at right angles to the lines of sight. The plan of the surface, then, will be a true octagon, and may be drawn on the plate with lines tangent to a circle 1^ inches in diameter, using the T square and 45° triangle for that purpose. CoxsTRUCTiox. — Draw the base line for the front elevation 3 inches above the lower border of the drawing, and draw the vertical base line (for the side elevation) 2f inches from the left-hand border. De- scribe the circle previously mentioned in such a position that the nearest edges of the octagon will be ^ inch from each base line ; the figure may then be completed in the plan. In this and the remaining problems to be drawn on this plate, the right views are to be drawn ^ inch from the base line in all cases. In both eleva- tions, the lines of sight in crossing the surface pass also 40 PRACTICAL PROJECTION § 15 through the points in that portion of the surface farthest from the observer; and as the eye of the observer travels from points opposite to M and X, Fig. IG, in tracing the front elevation, the foot of every line of sight would be projected on a single line on V P. The elevation x>i the surface, therefore, is represented on the drawing by the single straight line Jll JV, Fig. 16. Project the front and the side elevations in their proper places, completing the problem. Note. — The sirKgle line that constitutes each elevation of this prob- lem represents the eight lines of the octagonal surface shown on the plan. This is shown in Fig. Ifi, which is a copy of the plan and front elevation on the plate, lettered for convenience of reference. Two of the lines in the plan, A /) and /^ J^, Fig. 16, are shown in their full length by that portion of the line J/ A' included between the points P and (2 ; since P' E is directly in line with A B in the elevation, as already e.xplained, it is shown by the same line P O used to define A B. The line HA is shown foreshortened at MP; and a.?> G F i?, directly behind HA, MP represents (7 /-"also; Q A' bears the same relation to /j'Cand ED. The line G H is represented in the elev£rtion by the point J/; A\ in like manner, represents DC. Thus, the line M N represents a certain view of the eight lines A B,B C, etc. to HA, and also a view of the surface defined by those lines. The above is very important and should be carefully read, as it shows the application of principles to Problem 5. PROBLEM 6 48, To project tlie vie^vs of an octagonal surface that is in a riglitly inclined position. Note. — No perspective figure is shown for this problem, and the projections will be made on the plate from the following directions. The same figure is used for this problem as for Problem 5. These drawings are really a continuation of that problem; and since a full view of the surface is shown in the plan of Problem 5, projectors will be drawn from that view, in order to define the plan of this problem. Construction. — Draw the horizontal base line in the next space on the drawing and at the same distance from the lower edge as the corresponding base line was drawn in the space for Problem 5; draw the vertical base line If inches from the left side of the space. The front elevation of this problem will first be drawn. Draw a line inclined to the horizontal base line at an angle of G0° and equal in length to the line shown in the front § 15 PRACTICAL PROJECTION 41 elevation of Problem o; the lower end of this line should be ^ inch above the base litie, as previously explained. It should be drawn in such a position on the plate that vertical projectors from the ends of the line will pass through the central portion of the horizontal base line. Mark the posi- tion of the points indicated by the projectors, as at Jlf, P, Q, and y, Fig. 10, and from these points draw four vertical projectors to the plan. Intersect these with horizontal pro- jectors drawn from the plan in Problem 5; draw the con- necting lines between corresponding points thus projected; this produces a figure that is the plan of the octagonal sur- face in the rightly inclined position indicated by the front elevation. Project the side elevation by the use of secondary pro- jectors, as previously explained. Reference to the copy of this plate will be of assistance to the student during the pro- jection of these views. The completed drawings are there shown and the method of projecting between different views is indicated by projectors partially extended towards the left of the plan of this problem. 49. Basis of Projection. — The plan and side eleva- tion of this surface are foreshortened views. There are, however, two lines in each view shown in their true length; this may be proved by a comparison of the figures with the plan of Problem o; the other lines in each case are fore- shortened. It does not necessarily follow, however, that any of the lines in a rightly inclined view are shown in their true length. Had the angle of inclination been along the line A E, Fig. IG, every line would have been foreshortened; again, in the case of surfaces having curved or irregular outlines, the least angle of inclination in any direction would preclude the possibility of representing, in a foreshortened view, any portion of the outline in its true length. It will thus be seen that the point is the only geometrical element not sul)ject to change, or variation, in any view. It may therefore be relied on as a basis of projection. The 42 PRACTICAL PROJECTION § 15 outline of any surface in the different views is determined by first fixing the location of points at the extremities of the boundary lines of such surface, afterwards drawing the connecting lines, as in this problem. PROBLEM 7 50. To pi'Qject a full A-ie^v of a surface from a given plan and elevation slio'wing; that surface in a rightly inclined position. A full view of any surface may be projected by assuming a view to be taken at right angles to a line in which the entire surface is represented, for in such a view the lines of sight are at right angles to the outlines of the surface. CoNSTRUCTiox.— In the next adjoining space to the right on the plate, copy the plan and the front elevation of Prob- lem G, placing the projections so that they will occupy the same relative position in the space. To obtain a full view of the surface in this problem, a view must now be assumed at right angles to the line in the elevation ; in other words, the elevation must be considered as a plan and a new front elevation projected therefrom. The plan copied from Problem G is used as a base plan, secondary projectors being drawn from thence in the manner shown on the plate and described in the following article. The projectors in this case are drawn by the arc method, sometimes more con- veniently employed than the angular method previously described. 51. Arc Method of Drawing Secondary Project- ors. — Draw a base line, for the projection of the full view, from the intersection of the base lines previously drawn and parallel to the line that represents the surface of the octagon, as A B (see plate). At the point of intersec- tion of the base lines {B) erect a perpendicular to the oblique base line A B, as B C, producing it indefinitely towards the right. The positions of all points in the plan § 15 PRACTICAL PROJECTION 43 are now to be located on this line in the same relative posi- tion as they would occupy if projected horizontally to the vertical base line in the drawing. The points are accord- ingly projected horizontally to the vertical base line B I); thence, by using the compasses and describing arcs from a center B, located at the intersection of the base lines, they are projected to the line B C. The projectors are then con- tinued beyond B C, but parallel to A B; they are there intersected by primary projectors drawn from corresponding points in the elevation, as shown. Locate the various posi- tions of the corresponding points at the intersections of these projectors and produce the full view of the octagon by drawing the connecting lines. It will thus be seen that the drawing of secondary pro- jectors by this method involves Jirst, projection to the nearest base line ; second^ the describing of arcs from the center shown; and third, the continuation of the projectors parallel to the base line of the desired view. If the draw- ing has been carefully made, it will be found that the sur- face thus defined is an exact counterpart of the plan in Problem 5, and that the full view projected in this problem is in the same relation to the elevation as the plan in Prob- lem 5 is to the elevation of that problem. 53. Tie>vs Xecessary for the Projection of the Full Vie\v. — When it is desired to project a full view of any surface that is represented in a drawing in an inclined position, it is necessary to have one view that will show all the points of that surface as contained in one line. A pro- jection must also be drawn at right angles to that view, in order that such dimensions of the surface as are at right angles to those in the first view may be shown in their true length. 53. Full A'ie^vs Sometimes Obtained Without Pro- jection 3Ietlio(ls. — A comparison of the views in the projections of the last problem will prove that the vertical primary projectors included within the surface of the octa- gon shown in the plan are of the same length as the secondary 44 PRACTICAL PROJECTION § 15 projectors in the view last projected. This knowledge may be used to some advantage in producing a full view without using all the projectors employed in this problem. Thus, draw a horizontal center line through the plan, as E F (see drawing on the plate for Problem 7) and draw primary projectors from the elevation to the full view in the regular way; at a convenient distance draw a line at right angles to, and crossing, these projectors, as G H. This line will be the center line for the full view, the points of which may then be located with the dividers in the fol- lowing manner: Set the dividers to the length ac xn the plan and set off a corresponding distance at a e' in the full view; in like manner make b' f equal to b f, etc., as shown on the plate. Complete the outline of the full view, then, by drawing connecting lines as heretofore. This method is generally followed in pattern drafting, since it requires less time than to draw full projections as in the construction of the problem, and there is less liability of error. 54. If the student that does not clearly understand the principles by which these projections are made will cut a piece of cardboard to the same size and shape as the plan of Problem o and hold it in such positions that the foot of the lines of sight falls on the points designated on the drawings for the different views, he will at once see the correct position of the surface as represented in each view. oo. Surfaces Bounded, by Curved Lines. — Surfaces that are defined by curved lines do not present any points from which to make projections. In making such pro- jections, the same principles are employed, however; but it is first necessary to establish a number of points at various positions on the curved lines. The points thus established are then projected in the same way as in the foregoing problems. "When, for purposes of projection, points are located on the outline of a curved surface in any view, it should be observed that they are so placed that, when the § 15 PRACTICAL PROJECTION 45 points thus located are projected to a line that represents an edge view of that surface, each end of that line is defined by the projection of a point. PUOHLEM 8 56. To project views of a plane surface defined by a curved line, the snrfiice being in a rightly inclined position; also, to project a point located on that surface. Note.— Before making the projections of this problem, the lines to remain on the drawing for Problem 5 should be inked in, and to avoid confusion, all other lines not to be inked in on that figure should be erased; the circle drawn for Problem 5 may then be redrawn for this problem. Construction.— The surface for the projections of this problem is that of the circle to which the sides of the octa- gon in Problem 5 are tangent. After describing the circle, the next step is to locate points on its circumference. Do this by first drawing a vertical and a horizontal diameter, and then drawing, with the 45° triangle, two other diam- eters at right angles to each other, thus locating eight points at equal distances on the circumference. Those points indicated by the horizontal diameter will, when projected to the elevation, define the ends of the line in that view. Also, locate a point at the center of the circle. The plan and elevation of the surface thus projected is shown in Fig. 17, the points being denoted by numerals. Project these points to the elevation of Problem 5, using lines easily erased, since they are not to appear in that problem when the plate is finished. In the last space on the plate draw horizontal and vertical base lines in the same corresponding position as in the space for Problem G. The line that represents the front elevation of the circular plane surface is then drawn in the same position as in the ele- vation in Problem G; with the dividers, locate points thereon in the same position as the points projected from the circle to the front elevation in Problem 5, as M. E. V.—J5 46 PRACTICAL PROTECTION §15 1 3 Fig. it shown in Fig. IT. Project these points vertically to the plan and intersect these projectors with horizontal pro- jectors drawn from the circle in Problem 5. When projecting points across a drawing — as from the space occupied by Prob- lem 5 to the drawing for this problem — it is not necessarj- to draw lines the entire distance. By care- fully placing the edge of the T square on each point in turn, corresponding lines ma)' be drawn across the plan in this problem. This saves erasing unnecessary lines, but care must be taken, when making projections in this way, to observe that points thus located are at the intersections of projectors drawn from points in corresponding positions in each of the views. Find the location of each point thus projected and through these points trace the curve that represents the foreshort- ened view of the circle, using the irregular curve for this purpose. Thus a plan and a front elevation of the circular plane surface is drawn, in which the surface is represented in a rightly inclined position. Project the side elevation by the angular method of sec- ondary projectors, as previously explained, and designate the point in both views by a small dot at the center of the surface. Finally project the full view by the arc method, as in Problem T. In this problem a good test of accuracy is afforded if, after the nin'e points have been projected to the full view, a circle with a radius of f inch, described from the central point, passes through the other eight points. 57. Importance of Acciiiticy. — Next to a knowledge of the principles of projection, neatness and accuracy are the prime requisites in a drawing. The student should PRDJEC :^^V.l \ \\\ 1 I I /ft ; / / // :vV'\ \V5/^-t I \ \ \ 1 \ \ / /// \ J^^ X PRC PROBLEM 9. PROBLEM 12. CaseL PROBLEM 12. Case 2. JUNE 85,/ 893. Copyright, l^•99, by THE Col All right noNS-n. -^=^T\ ^ <\A \ \ ^ / / U./^'/^^^yyc^ ^ ^ .^ EM 10. Pf^OBLEM II. PROBLEM 12. Case 3. PROBLEM IS. Case 4. EK^• ICnginf.er Company. served. JOE/N 3MLTHj CL A55 N2 ^529, § 15 PRACTICAL PROJECTION 47 carefully observe that, when the points determined by the intersection of lines are used as centers for arcs or circles, the needle point of the compasses should be placed exactly on that position; again, drawing three or more lines that shall intersect at the same point is very commonly required in projection drawing and in pattern drafting; this is not an easy thing to do accurately unless carefully practiced by the student. It is needless to state that unless the work is accurately done it is of no value. When putting in the figures for the dimensions on draw- ings, care should be observed that they are placed on those views in which the lines and surfaces are shown in their true length. Do not designate a foreshortened view of a line or surface by a dimension figure, when another view is given in which the true length is shown. Again, do not repeat the same dimension on different views of the same drawing; thus, in Problem 2, Case I, the length of the line is given as 2 inches in the plan, and it is obviously unneces- sary to give the same dimension in the front elevation. The student may ink in all the problems on the plates, but the letters used to describe the different positions and lines are not placed on the drawing. The date, name, and class letter and number are inscribed as in the plates of Geometrical Drazvinsr. DRAAVING PIRATE, TITTLE: PROJECTIONS II 58. The problems for this and the succeeding plates should be practiced on other paper and then copied on the drawing that is to be sent in for correction. The student can thus judge better as to the relative position the figures sljould occupy and the completed plates will present a neat appearance. In making the projections on this and the succeeding plates, the views may be assumed to be \ inch from their respective base lines, as this will enable the pro- jections to be kept in closer proximity. The base lines are not to be inked in on this or the following plates. Divide this plate by a central horizontal line; the part of the 48 PRACTICAL PROJECTION 15 drawing above this line is divided into three, and the part below the line into four, equal spaces. PROBLEM 9 59. To project a side elevation and a full view of a rightly inclined plane surface defined by an irregular outline. This is a problem in which the student has an opportu- nity to use, in a practical way, the ^Elevation knowledge of projection thus far gained. The surface to be projected is shown in a rightly inclined position — -i-j-j-^i — f M ri ^^^ ^^S- 18, which is a foreshortened 1 1 j d i ^l f J<^ I i . t view of the surface. This figure is to jyf'fv^r^j'/ ^^ be copied, in the size indicated by the ^li i i I" i** I I ; iW^t dimension figures, into such a position in the upper left-hand space on the drawing that the projections when completed will occupy about the cen- ter of the space. Construction. — First draw the horizontal line A B 1^ inches long and bisect it at o by the vertical line C D. Make o D \ inch long and bisect it at x, as shown, making o C 1^ inches long. Set the compasses at a radius of Iff inches, and with A and C, respectively, as centers, describe arcs inter- secting at e; with the same radius and with B and C as centers, describe arcs similarly intersecting at e' . From these centers {e and e') describe the arcs A C and B C, thus producing the curved outline of the lower portion of the plan. Next, divide these arcs, by spacing, into six equal parts, thus locating the points 1, 2, 3, 4, and ,5; from these points draw vertical lines, as shown in Fig. 18. Complete the upper outline of the surface as represented in the figure; thus, locate a at the intersection of the vertical from 1 with a horizontal from A ; d, at the intersection of § 15 PRACTICAL PROJECTION 49 the vertical from ;? with a horizontal from D\ /", in like manner, at the intersection of a vertical from J with a hori- zontal from X. The points at the extremities of these lines and those located on the curved outline are now to be treated as in former problems and the projections made in the usual way. Caution. — When making the projections for this prob- lem, the student must observe the precautions given in regard to the taking of the same corresponding points in each view. Project the side elevation first ; it may be desir- able for the student to ink in that figure, in order to avoid the confusion arising from a number of lines crossing one another on the drawing. Use the angular method for the secondary projectors in projecting the side elevation and the arc method for the full view. Since it is often neces- sary, when developing patterns, to draw several views over one another in this way, the student should accustom him- self to drawings that have a complicated appearance from this cause, and should learn to follow each set of projectors as readily as though they were in separate drawings. During the construction of this projection, it will be noticed that the base line for the full view, in order to be drawn from the intersection of the other base lines, will fall below the front elevation of the surface. This is unimportant, how- ever, since its purpose is the same, and the result is merely that of a slight appearance of crowding on the drawing. PROBLEM 10 60. To project views of a plane surface in an obliquely inclined position. Explanation.— A full view of the surface to be projected in this problem is shown in Fig. 19, the dimension figures giving the size in which it is to be drawn by the student. The upper portion of this surface is defined by a semicircle, the lower by one-half of an octagon. The purpose in selecting a surface of this outline is to give the student 50 PRACTICAL PROJECTION 5 15 some practice in the projection of both straight and curved outline surfaces. It has been shown that, before an inclined view of a surface was projected, a right view — i. e., a right plan and elevation, as in Problem 5 — has first been drawn. These views alone are pro- jected in drawings of simple or plain 4j> objects, it being obviously unnecessary to show any object in a working drawing in a position not commonly occupied. But owing to the different shapes of objects, variously outlined surfaces are presented in a diversity of positions, and it is essential that the student should be capable of projecting any surface into any conceivable position and of drawing a full view from such a projection. CoxsTRUCTiox. — The method of drawing oblique views of surfaces is shown in detail at (a). (^). and (c). Fig. 20, the Fig. 19 -2> projections at (c) being the ones required for the plan and elevation of this problem. Lay a separate piece of paper over the drawing of Problem 9 on the plate, and reproduce thereon the projections shown at (a) and (^), Fig. 20, in accordance with principles already explained. Next, draw § 15 PRACTICAL PROJECTION 51 the plan and elevation at {c) in their proper places on the plate. The drawing shown at (a) may be seen to be similar to that of Problem 5 of the preceding plate; (d) is projected directly from (a), in the same manner as Problem 6, the angle of inclination being 60°. The plan of this surface in (/?) is then copied at (c) in such a position that the center line A B makes an angle of 60° with the base line of V P. This is accomplished by first drawing the center line A' B' at the given angle in (r), noting thereon the position of the points w', x\ and y' \ draw perpendiculars through these points, and make zv' a' in {c) equal to w a in (/;), x' D' equal to X D, etc. The outline of the plan at {c), therefore, is exactly the same as it is shown in {b), the only difference between the two views being the fact that the line A' B' in {c) is inclined to V P, while in {b) it is perpendicular to that plane. Let us consider what changes have here been represented. Cut a piece of cardboard to the outline and size shown in Fig. 19 and compare it with the different positions in the drawings just made. It will be seen that the cardboard must be held in a horizontal position to coincide with the drawing at {a); to represent the drawing at {b), the point C must be raised until the line CD is at the angle of 60° with H P. The plan at {b) is, therefore, a foreshortened view of the surface, although its elevation may still be rep- resented by the single line C D' . Now turn the cardboard to the position indicated in (^), that is, so that the line A' B' makes an angle of 60° with V P. It will be seen that the line C D '\n its relation to H P is not affected by this change, its angle with H P remaining as before; therefore, the vertical distances to be sJioivn in the elevation of {c) will be the same as in the elevation of {b), and may be projected directly to {c) from {b), as shown in Fig. 20. Draw horizontal projectors from the elevation at (b) to the elevation in (c), intersecting them, in the man- ner shown, by primary projectors drawn vertically upwards from the points in the plan at (c). Trace the outline of the surface thus indicated through the intersections of project- ors drawn from corresponding points in each view. The 52 PRACTICAL PROJECTION S$ 15 projections shown at (r) being completed on the plate, the paper on which (a) and (d) were drawn may now be removed and the side elevation required for the problem projected by the angular method previously described. Three views are thus shown, in all of which the surface is represented as inclined at an oblique angle to the lines of sight; all these views, therefore, are foreshortened. Oblique views may always be drawn in this manner; that is, a right view is first drawn; next, a rightly inclined view is pro- jected, the desired angle being represented in the elevation. The plan thus produced is then redrawn for the oblique view and its elevation projected as in this problem. 61. Position of Full Views: How Determined. To project a full view of this surface it is first necessary to determine whether any of the lines or distances in any of the views are shown in their true length, but without hav- ing recourse to the projections made on the separate paper, since projection methods are to be used. This may be done by comparing the relative position of any two points in the outline of the surface, as located in the plan and elevation. If it is found that a line drawn between any two of these points in the elevation will be parallel to the base line (and therefore at right angles to the vertical lines of sight), that line will be shown, of course, in its true length on the plan. Any other lines parallel to it will also be shown in their true length. It is found on examination that points in the elevation corresponding to the positions represented by A and B, Fig. 19, are located on the same horizontal pro- jector; therefore, a line drawn between these points as they are located on the plan will be represented in its true length in that view; and a view projected from these points in the plan by primary projectors drawn at right angles to this line, intersected by secondary projectors from the front elevation (by a modification of the method used in Prob- lem. 7), will be a full view of the surface. Draw the oblique base line in the proper position, i. e., parallel to that line shown in full length in the plan [A B on § 15 PRACTICAL PROJECTION 53 the plate), as above explained, and at such distance away from the plan as directed in the instructions for drawing this plate, producing the line indefinitely towards the upper portion of the drawing, as shown on the plate at E F. In this case, the line thus drawn defines the inclination of the surface, since the angle is the same in both plan and eleva- tion, viz., 60°. The full view is projected as follows: Draw the line G H at right angles to the base line EF and from the intersection oi E F with the horizontal base line. By the arc method, draw secondary projectors from the elevation, as shown; intersect these projectors by primary projectors drawn from the plan at right angles to E F, thus producing the full view of the surface. It will be found that this full view is an exact counterpart of the surface shown in Fig. 19, and should correspond to the preliminary drawing in the plan at (a) on the separate paper. Note that the portion of EF included between / and q cor- responds in length to that of the rightly inclined front elevation at (-z^), Fig. 20. This may be seen by comparing that portion of the line with the view on the preliminary drawing.* The student should now be able to recognize any view of a surface in any position; that is, he should be able to tell whether a view represents such a surface in a right position, a rightly inclined position, or an obliquely inclined position; and by the application of the principles illustrated in the foregoing problems and the exercise of a little judgment, he should be able to project any surface into any desired position. Or, being given a surface in a position indicated by a properly projected plan and elevation, he should be able to produce the full view and designate the angle of inclination. The following problem will serve as a test of his progress. The principles involved have already been presented and the method of application will be readily understood. The * It should be noted that this is the case only when the angle of inclination is the same in both views. 54 PRACTICAL PROJECTION § 15 angle of inclination in the plan is not the same as is shown in the elevation ; both angles are to be determined by pro- jection methods. PROBLEM 11 62. To project the fnll view of an irregularly outlined surface obliquely Inclined. The projections of this problem are to occupy the upper right-hand space of the drawing plate. The plan and front elevation are shown in Fig. -21 and are reproduced full size on the sheet opposite this page.* The outline represented is frequently used as a "stay," or profile, to which mold- — -^ ings are formed in cornice work. Since /^ \,^ the view shown is known to be obliquely 5 A gj^gr^izcvN^ inclined, its dimensions are foreshort- __y ened, and their true lengths are to be ^ found by projection methods, as fol- \ lows: 3f — I / \j CoxsTRUCTiON. — The student should I / / detach the sheet opposite this page and i ( y paste the plan and elevation in such / X a position in the third space on the plate \y that the base line JI X, Fig. •21, will F'<^ -' be 2+ inches below the top border line and exactly horizontal. Locate a number of points on the curved outline in the elevation, by equally spacing that portion of the figure with the dividers, as shown at i. J, 3. and 4 in the drawing for this problem on the plate, and project these points to the plan. To ascertain the angle of inclination in the plan, draw a horizontal line through the widest portion of the figure in the elevation, locating, if fKSSsible. one end of the line at an angle of the surface — as the line A B. Project this line to the plan at A' B' . as explained in Art. 61 ; parallel to this line erect the oblique base line C D. * This sheet is not inserted in the bound volume conta i ni n g this Paper. § 15 PRACTICAL PROJECTION 55 The angle formed by these lines {A' B' and C D) with the horizontal base line is the angle of inclination of the surface to V P, or that angle shown in the plan. The angle of incli- nation to H P, or that shown in the elevation, is most easily- found by constructing a right-angled triangle whose base and altitude are equal to certain distances found in the plan and elevation; that is, the base is equal to the extreme width of the figure in the plan, taken at right angles to the line of inclination in that view (shown by the dimension .V). The altitude is the vertical height shown in the elevation at AI. Construct this right-angled triangle on the horizontal base line extended, as shown at N' M', and locate one end of the base at D, the intersection of the base lines. Extend the hypotenuse indefinitely towards the right of the drawing. Next, intersect primary projectors drawn from the plan to the oblique view by secondary projectors drawn from the eleva- tion by the arc method, as shown on the plate. The full view of the surface is then traced through the intersections of these projectors, completing the problem. 63. Projection of Solids. — We now come to the pro- jection of solids, which, as before noted, are merely various combinations of surfaces. Projections of surfaces in a variety of positions having already been made, we shall encounter no new principles in the projection of solids, the surfaces of which are projected in the same manner as has been shown in preceding problems. Since lines intersect in a point, so surfaces intersect in a line, and in drawing projections of solids it is necessary only to find the true projections in any view, or set of views, of those lines that represent the correct intersections of the adjacent surfaces; this is a comparatively easy thing for the student to do, if he will use proper care and diligence in the application of the principles of the preceding problems. 64. Projection of the Cnbe Illustrated. — Every solid consists of a number of surfaces, each of which is differ- ently shown when the solid is projected to the various views. 56 PRACTICAL PROJECTION §15 This is due to the fact that the observer is assumed to occupy a different position in each view of the solid thus projected. In certain positions some soHds show one or more of their surfaces directly behind another surface of the same size and shape. This would be the case if the projections of the cube, referred to in Art. 18, were drawn as shown in Fig. 22. When the cube is in a right position — that is, with two sur- faces horizontal, and that surface nearest the observer in a side view in such position that the lines defining that surface will be at right angles to the lines of sight, as indicated in Fig. 22 — it is evident that the surface parallel to and behind Fig. 22 the front surface will be projected by the same lines of sight as the front surface. Therefore, in such a case, a projection of the front surface of the cube is equivalent to a projection of the entire cube. Each projection of the cube in the plan, front, and side elevations is a square, the sides of which are 1 inch long, while the views are arranged, as shown in Fig. 23, in such a way as to appear related to one another. 65. In "reading" the projections shown in Fig. 23, we merely compare the surfaces of the cube as they are shown in the different projections. Thus, the sur- face A B C D, Fig. 22, is represented in the plan of Fig. 23 15 PRACTICAL PROJECTION 57 Front Llevaiion IjT Siile Elevation by the line .4 />, antl in the front elevation by the line B C, while a full view of that surface is shown in the side elevation ; so, in like manner, the position of each surface of the cube may be determined. Note that, in each full view, two surfaces are represented ; thus, in the front elevation of Fig. '>2, the sur- faces /> F G C and A E H D are projected on V P as one surface at PQRS. It is thus shown that surfaces in their relation to one another, when combined in one view, as in the projection of a solid, partake of the same •" y ^ c a principle that has been '°' "" shown in its application to points and lines, viz., surfaces ivhose outlines are contained in the same lines of sight in any view are projected in that viezv as one sjirfacc. The difference in position of the several views due to the use of first- and third-angle projection may easily be illus- trated in connection with Fig. 23. Here the cube is repre- sented as having been projected in the first angle. If the student will now turn the book " upside down," he will see the three views of the cube in the relative positions they would occupy were the drawing made by the third-angle method. It will be seen that the effect is merely that of rendering the plan uppermost on the drawing. The front elevation then is seen below the plan, while the side eleva- tion, as in first-angle projection, is at the side of the plan. In the case of a simple solid like the cube, the use of differ- ent angles of projection has but slight effect, but when a more complicated solid is represented on the drawing, it becomes necessary for the one that would " read " the draw- ing to know which angle has been used by the draftsman. 58 PRACTICAL PROJECTION § 15 Note that in Fig. 23 the secondary projectors are described from i^ as a center, the lines of the cube being used as base Hues, as mentioned in Art. 39. When this short method is adopted in the case of secondary project- ors, the center from which they are described must be located at that intersection of the primary projectors near- est the two views between which secondary projectors are drawn. A further illustration of this will be given in con- nection with a later drawing. 66. Iliciaen Sui-faces : How Indicated. — When the form of a solid is such that in any view a smaller surface is hidden by a larger, the smaller surface is not shown in that view, although frequently its outline may be defined by dotted lines on the drawing. This applies also to projections in which two or more solids are shown in positions such that some of their surfaces are completely or partially hidden by other surfaces nearer the eye of the observer. Only such surfaces as receive the lines of sight directly from the eye of the observer are shown in a view by full lines, although, as mentioned above, the outline of such other surfaces as it may be desirable to show in a drawing may be indicated by dotted lines. 67. Facility in Reading DraMangfS Acquired Only l>y Practice. — The reading of working drawings is, there- fore, a comparatively easy matter, if the student will resolve each portion of the object represented into its respective surfaces and look for the various outlines as they are shown in the different projections. If this is found a difficult task, the surfaces may be further resolved into lines and points, whose respective positions may then be located in each view shown. It is not to be expected that the position of every surface in a complicated drawing will be seen by the beginner at a single glance — an expert seldom acquires such profi- ciency — but as "practice makes perfect," the student may easily accustom himself, by careful study of the various positions of the surfaces composing the solids that are § 15 PRACTICAL PROJECTION 59 projected in the following problems, to the more or less complicated projections found in the various mechanical and architectural journals, in shop drawings, or in such other projection drawings as are within his reach. 68. The Center Ldne. — It has been found convenient, when making projections of objects, to make use of a line that is imagined to pass through the central portion of the solid as it is shown in any plan and elevation. Such a line is called a cetiter line, and in many projections it is inked in when the drawing is finished, since it frequently affords a convenient means of indicating certain positions of the fig- ure, besides assisting in the location of the several surfaces of the solid in the different views. This line, however, is central only in its relation to the object of which the draw- ing is a representation, and not in relation to the planes of projection. This may be better understood by considering the center line as the projection of an imaginary surface (or plane) that passes through the central portion of the figure. It is generally represented in those views only in which that imaginary surface can be shown in one line, or, as we have said before, as if "on edge." Thus, in the right view pro- jected in Fig. 23, the lines zu x a.vA y s are center lines, rep- resented on the drawing by the broken-and-double-dotted lines shown in Fig. 23. The practical use of the center line will be illustrated in the succeeding problems by the pro- jection of solids into various positions. PROBLEM 12 (59. To dra>v tli in the drawing on the plate. Describe the circle shown at (///), which represents a full view of the end of the cylinder; next, draw the front eleva- tion C D E F o\\ the center line A B according to the given dimensions. Describe a circle similar to (w) at {ji) ; this is a temporary view of the end of the cylinder and corresponds to the plan of the prism at {ii). Fig. 25. Locate a conve- nient number of points at equal distances on the outline of each full view thus drawn at (;//) and («). Project the points of (;//) to the elevation C D E F, and thence draw primary projectors vertically downwards; intersect these primary projectors by other primary projectors drawn horizontally from similar points located on the outline of the full view at (;/). Trace the outline of the plan thus produced through points of intersection corresponding to those on the full views. The temporary full views {in) and (//) may then be erased from the plate. Project the side elevation by means of secondary projectors described by the arc method, thus completing the drawing. Case II. — Wlu'ii tlic cylinder is obliqncly inclined. Explanation. — The method of ]:)r()jecting the drawings required for this case is similar to that already given for oblique views of surfaces and solids, and has been fully explained in Art. 60, and also in connection with Case IV of Problem 12. A right view is first drawn [as the elevation and full view {in) of the preceding case] ; next, a rightly inclined view. The rightly inclined plan thus drawn is then recopied at the given angle, thus producing the plan of the oblique view; from this plan, in connection with the rightly inclined elevation, the obliquely inclined elevation 66 PRACTICAL PROTECTION §15 is projected. The rightly inclined plan and elevation having been drawn in Case I of this problem, the plan there shown may be redrawn for the plan of this case. CoxsTRUCTiox. — On a separate piece of paper reproduce the plan and front elevation of Case I and fasten this paper b)' thumbtacks to the drawing board, towards the left of the space used for this case. Next, redraw the plan of Case I in its proper place on the plate for this case, and in such a position that the line G H oi Case I forms an angle of 60° with the base line of the front elevation, as shown at G' H' on the plate. Then produce the front elevation by drawing primary projectors upwards from the plan and intersecting them by similar projectors drawn horizontally from the rightly inclined elevation on the attached sheet, which may then be removed. Trace curves through the points thus projected and draw the tangential lines, as previously described and as shown at C D' P E' on the plate. Project the side elevation as in preceding problems, taking special care to project fro'm similar points in each view. 3. PROBLEM 14 To dra%v tlie projections of a hexagonal pyi*amid. A pyramid is a solid whose base is a polygon and whose sides are triangles uniting at a common point called the vertex. The pyramid for the projec- tions of this problem is shown in Fig. 27, where its dimensions are clearh* indicated. Since these drawings are very easy and are constructed in a man- ner similar to those of preceding prob- lems, definite instructions are omitted, and the student is expected to be able to complete the drawings by the aid of the brief explanations that follow. Case I. — When the pyramid is in a Fig. 27 right position. §15 PRACTICAL PROJECTION 67 Explanation. — The plan of this projection is most con- veniently drawn tirst, a circle If inches in diameter being described from (9 as a center, as shown on the plate. The edges of the pyramid are then drawn : a horizontal diameter and two diameters at angles of 60° with the first repre- sent the upright edges; chords of the arcs thus designated are then drawn, and the plan of the pyramid is complete. Next draw the center lines A B and C D\ set off the height of the pyramid on the line A B and complete the front ele- vation by the aid of primary projectors, as shown on the plate, the side elevation being projected as in former problems. Case II. — When the pyramid is in an obliquely inclined position. Explanation. — The angles of inclination in the projec- tions of this case are 60° to H P and 45° to V P. Preliminary Fio. 28 drawings are required on separate paper, as shown in Fig. 28, first, as at {a), showing a right view of the pyramid; and 68 PRACTICAL PROJECTION § 15 second, as at {b), showing a rightly inclined view, the angle of inclination (of the center line) being 60° to H P. The plan produced at {b) is then copied on the plate in such a position that its axial line will make the required angle, viz., 45° to V P, as shown by the line zc ,r at (c), Fig. 28. The front elevation is then projected as in Case II of the preceding problem, that is, by vertical primary projectors drawn from the plan in (r), intersected by horizontal pri- mary projectors drawn from the elevation in (/?). The pro- jection of the side elevation by the arc method of secondary projectors is also similar to the preceding projections, as will be seen from an inspection of the plate. PROBLEM 15 74. To draw the projections of a cone. Thercwr is a solid that may be produced by the revolution of a right-angled triangle around one of its sides as an axis. Its base, therefore, is a circle, and its curved surface tapers uniformly towards a point at the top called the vertex, or apex. Like the cylinder, its entire form and dimensions are presented in a plan and a single elevation showing a right view of the cone. The cone for the projections of this prob- lem is shown in perspective in Fig. 29, which gives the dimensions that the cone is to present on the plate. The methods used are precisely similar to those used in the case of the hexagonal pyramid in the preceding problem. Case I. — /;/ a rightly inclined position. Explanation. — In order to produce the rightly inclined front elevation, a construction similar to that used in Case I of Problem 13 is here used. The drawing differs from that projection only in the form of the solid. The angle of inclination in this case is 50° to H P. Construction. — Draw the center line A B (see the plate) at the given angle, that is, 50° to the horizontal. Next, §15 PRACTICAL PROJECTION 69 pro- construct the triangle representing the elevation of the cone and describe the circle at (;//) — a temporary full view of the base. Describe a similar circle at («), also a full view of the base, and locate a convenient number of points on the outline of each full view — in this case eight — as shown at rt, b, c, etc., on the plate. Project the rightly inclined plan in a manner pre- cisely similar to that used in the view of the cylinder in Case I of Prob- lem 13. Erase the temporary views (/;/) and (;/), ject the side elevation as in preceding projections. Case II. — /// an obliquely inclined position. ExPLAXATiox. — The angle of inclination to H P is the same as in the drawing last made, and the plan of that projection may be recopied for the plan of this case, but it is to be drawn on the plate in such a position that the center line it' x will make an angle of 45° to V P. The plan and front eleva- tion of the preceding case must be redrawn on separate paper and temporarily fastened over the drawing towards the left of the space required for this case, in order that the projection of the front elevation may be drawn. As this process' is similar to that used in preceding constructions, no further explanation will be given. Complete the projections in the plan, front, and side elevations as shown on the plate. Note. — When inclined views are drawn of solids having curved surfaces (as the cylinder and the cone), the circular ends should first be projected. The outline of the curved surface is then represented as tangent to the base, or bases, of the solid, and without regard to the intersection of such outline with any given point on the base outline. 75. Self-Heliance. — ^The student that has intelligently completed the projections of the foregoing problems and has made frequent use of the imaginative feature of this subject, as previously explained and directed, should now possess a very complete knowledge of the methods of pro- jection used in representing plain solids in various positions. 70 PRACTICAL PROJECTION § 15 The projection of irregularly outlined figures has not been presented, since the methods are identical with those already shown. The student should acquire a degree of self-reliance in this work: for if he is to depend on having the projection of every conceivable form described for him, the principles governing those projections will become a secondary matter, whereas the practical draftsman requires, above all else, the faculty of recognizing the principles by which to define and project the various forms occurring in the course of his work. Note. — The student should understand that the percentage of marking adopted for these plates is based on the degree of accu- racy in which the projections are drawn to the angles of inclina- tion, as well as on the quality of neatness attained in the finish of the drawings. DRATTIXG PI^\TE, TITLE: SECTIONS I 76. I'se of Section Di-a-svings. — A section drawing, as previously explained, is a projection of a portion of a solid, in a view where the solid is intersected by a plane. This plane — sometimes called a cutting plane — may pass through the solid in any direction; that portion of the solid between the plane and the observer is assumed to have been removed. Section drawings are useful in many ways, for by such means the construction of interior parts of objects may be shown. It is desirable in many cases to show some particular form that a solid of peculiar shape pos- sesses at a place where it cannot be presented in an exterior view, and in such cases sections are projected. The section is considered simply as a surface that would appear on the cutting plane in any view of the object pro- jected. The outline of this surface, therefore, will depend on the form of the object, the number of surfaces intersected by the plane, and the angle of inclination of the cutting plane. To project the views of a solid in which a new surface is thus presented, it is necessary to consider the various sur- faces that originally composed the solid. The first view drawn is always that in which the cutting plane is repre- sented as oil edge. The intersections of the original surfaces 9 D f^J (b) PROBLEM /6. JUNE 25, /393. Copyright, 1899, by The Coli All rights ■ )N5- :/ / M J} PROBLEM /^ ■Y Engineer Company. rved. JOfYN SMFTH, CLAS3 NS 4529. §15 PRACTICAL PROJECTION 71 of the solid with the cutting plane are thus shown; and by using methods of projection already presented, any view of the section may then be drawn. Those portions of the solid assumed to have been removed are not shown in a section drawing, although their position is sometimes indi- cated by dotted lines. 77. Sections of the Sphere. — The solid that presents the most simple illustrations of sections is the sphere, or globe. This solid, also called a ball, is such as would be generated by the revolution of a circle around its diameter as an axis. The cylinder, cone, and sphere are sometimes called the "three round bodies," or the solids of revolution. It will be shown later that their imaginary formation by such revolution may be taken advantage of by practical short methods of pro- jection, the principles of which are based on this knowledge. A full view of any section of a sphere is a circle. If the cutting plane passes through the center of the sphere, the full view of the section is a circle whose diameter is the same as the diameter of the sphere — or the great circle of the sphere, as it is called. If the cutting plane intersects the sphere in any other way, the section is still a circle, but of smaller diameter, and is measured from a view in which the cutting plane is shown on edge. 78. Sections of drawing of the cube referred to in for- mer illustrations and shows a vertical section. It will be noticed that the view of the section in the elevation is the same as the view in the front elevation in Fig. 23. ' This is always the case in sections the Cube. — Fig. 30 is a projection Fig. 30 73 PRACTICAL PROJECTION 15 of regular prisms where the cutting plane is parallel to the ends of the prism. Fig. 31 represents a diagonal section of the cube, the measurements of which will be apparent to the student from an inspection of the drawing. Fig. 32 is an oblique section, in which but three sides of the cube are intersected by the cutting plane ; in this figure, the full view of the sectional surface is projected. Fig. 33 represents a section taken at a still different angle and posi- tion of the cube. ]]lien a cutting plane passes through a solid having parallel sides, in anj' direction that causes it to inter- sect both of those parallel sides, those sides are shown in any Fig. 31 viezv by parallel lines. Note that in the sectional views of Fig. 33 the opposite edges of the surfaces are defined by parallel lines; thus, since AB and CD are parallel to each other in the plan of the cube in Fig. 33, so A' B' and CD' are in the same relation to each other in the side elevation ; also, A" B" and C" D" in the full view of the section. The same is also true oi A C and B D, as may be seen from a comparison of the views. 79. Ho^v the Cutting Plane Is Represented. — When the cutting plane is shown on edge in a view, it is 15 PRACTICAL PROJECTION 73 usually indicated by the same kind of a broken-and-double- dotted line used for the center line and the axial line. The use of this line for these three purposes is somewhat puzzling to the beginner; but as the student is now able to read pro- jection drawings, he can readily determine which purpose the line is intended to serve. It is customary, however, as already mentioned, when center and axial lines are used, to mark them as such by neat lettering. A section line in a complicated drawing is usually designated by a letter placed Fig. 32 at each end of the line, to which reference is made in the following manner: If, in the view where the cutting plane appears as a line, it is lettered A-B, the full view of the section is designated as a "section on the line A-B." 80. The problems for this plate, of which there are four, consist of the projection of the section drawings indicated in tlie accompanying illustrations. They are to be reproduced on the plate by the student to the dimensions T4 PRACTICAL PROJECTION §15 given on each figure. The cutting plane is indicated by the line A-B. and the views shown may be understood by a care- ful study of the figures on the plate illustrating each problem. The direction of a certain line in each plan is changed in these views, and the cross-section is accordingly represented as foreshortened in the front and side elevations. A view is also to be projected in which a full view of the sectional Fig. 33 surface will be seen. The plate is to be divided into four equal spaces b)' horizontal and vertical lines. Problem 16 is to occupy the upper left-hand space. The student is recommended not to refer to thfe reduced copy of the plate more frequently than is necessary to enable him to fix the location of the views on the drawing; he should learn to depend on his own knowledge of projection. 15 PRACTICAL PROJECTION 75 PROBLEM 16 81. To project sectional views of an octagonal prism, the cutting plane crossing the solid at an oblique angle and leaving a portion of the upper surface intact. Explanation. — The position of the prism and the angle of the cutting plane is shown in Fig. 34. THis figure is to be drawn, as at (a) in the left-hand portion of the space, to the size required by the dimension figures. The plan is then copied to the right, as at (d), but in a relatively different position, as will be seen by the arrangement of the letters and the direction of the line CD. Thus, the edge (7, which in the plan of (a) is on the extreme left of the figure, occu- pies a position nearer the lower part of the drawing in the plan at (d). This shifting of position of the plan may be effected by describing circles circumscribing the octagons and drawing the diameter C D at an angle of 30° to V P, as shown in the plan at (a). This diameter is then drawn in a vertical position in the second plan (d), after which the arc D a, as measured on the first, may be set off with the dividers on the second, plan. The projections of the front and side elevations are then made in the regular way. The projection of the full view is accom- plished by drawing projectors at right angles to the cutting plane A B. At a convenient distance, as shown at (<•), draw the center line a" j'\ and from this line set off with the dividers the distances from the line a c as found in the first plan. As similar positions have corresponding letters in the different views on the plate, the student will have no difficulty in D' recognizing the method of transfer, it being the same as that mentioned ^"'' ** in Art. 53, Observe that the sides of the section b' c' and 76 PRACTICAL PROJECTION §15 g' o\ and ^" r" and g" o'\ are parallel in every view shown, since the sides b c and go{f) are parallel, as shown in the plan of the solid. The same is true of h g and r t? (ai-abola. § 15 PRACTICAL PROJECTION 86 Explanation'. — Fig. 43 is a right plan . and elevation, giving the dimensions of the cone. The cutting plane mn is in this case parallel to the side A B of the cone and cuts B C ^ inch from /)'. The method of drawing these pro- jections is precisely similar to that in the preceding case, and since corresponding points are similarly designated in the different views shown on the plate, the student should experience no difficulty in completing the drawing. Case III. — Wlicii t/ic cuttiiie; plane is perpendicular to tJic basc of the cone, the section being an hyperbola. Explanation. — To produce that section of the cone known as the hyperbola, the cutting plane may form with the base any angle included between a right angle and the angle formed by an element with the base (as the angle A B C, Fig. 44). The dimensions and position of the cone and the cutting plane are shown in Fig. 44, and since the method of projection is the same as in the two preceding cases, the student may complete the views without further instruction. In this case, the projection of a separate full view may be omitted, the latter being shown in the side elevation. PROBLEM 31 80. To project sectional vicAvs of a scalene cone. Explanation. — This solid, whii:li is of varied form and of frequent occurrence in the metal trades, is an irregular geometrical figure. It is a cone whose axis is inclined to its circular base. All the elements of a regular cone are of equal length, Init the elements of a scalene cone are necessarily of variable length, for, since its axis is inclined towards a portion of the base, the elements in that part of the surface must be shorter. It is to be noted in this case that the axis of the cone shown in Fig. 45 does not pass through the center of the circle that represents the base of the solid. Construction. — To reproduce this drawing on the plate, draw first the horizontal line ba in the plan 3§ inches 86 PRACTICAL PROJECTION § 15 long, as called for by the dimension figures in Fig. 45 ; next. describe the circle boc 2i inches in diameter from a center located on the line ba, its circum- ference passing through the point b. Project the elevation according to the dimensions given in Fig. 45. The axial line is next drawn ; bisect the angle B A C and draw the bisector A 0\ represent the cutting plane w ;; by a line drawn perpendicular to A O and cutting A C f inch from C. The method of projec- tion for the various views of this solid is the same is, divide the outline of the convenient number of spaces (12 in this problem), and from points thus located draw the elements to the apex a (see plate). Next, project these elements to the elevation and finalh'' project their intersections with the cutting plane to the different views, as shown in the drawings on the plate. Note that the full view of the section, which in this problem is taken at right angles to the axis of the solid, is an ellipse. If the cone were in an upright position — that is, with its base at right angles to the axis (as that part of the cone above the cut- ting plane in Fig. 45) — the solid would be termed an "elliptical" cone. This, strictly, is not a geometrical solid, but its characteristics in projection drawing are some- what similar to the regular cone, although the elements in each quarter of the base are alwaj-s of unequal length. The student should now be able to project any sections that mav be desired. Fig. 45 as in former cases; that base in the plan into a JUNEa5JS93. Copyright, 1899, by THE Co All riEfht TIDNS-I. I I II 1 1 I /1 1/ / /. '■ ■ \\ \\ III! If •-''' "■'— -N^ P/fOBLEM Sa. Ccrse 3. C B PROBLEM £3. ERY Engineer Company. served. JO^Af 3M/ T/i. CL ^SSy/^ 4529. § 15 PRACTICAL PROJECTION 87 DRAWII^^G PIRATE, TITLE: INTERSECTIONS I 90, The IVIiter Line. — To represent properly the inter- sections of the surfaces of solids — or to "draw the miter line," as it is commonly called — is the final process of pro- jection. It has already been remarked that plane surfaces intersect in a line ; the representation of the intersection of plane surfaces is therefore a very simple process, the draftsman merely having to define each surface by the application of the regular projection methods already explained. The intersection of curved surfaces is appar- ently more complicated, but only because it is necessary to locate a greater number of points than are required for the intersection of plane surfaces. The location of points for the representation of the intersection of curved surfaces is done in a manner somewhat similar to that already shown in connection with the projection of plane surfaces having curved outlines. There is, however, this important differ- ence to be observed : in the case of the surfaces mentioned, their projection is accomplished by means of points located on their outlines; while in the case of the intersection of curved surfaces, it is necessary to locate lines in such posi- tions on each surface that they will lie in the same plane, although drawn on different surfaces. It is possible to locate a number of these lines in such positions on the drawing that through their points of intersection a curve may be traced that will be the correct line of intersection of the surfaces. 91. Relation of the Miter Line to the Pattern. — No drawing of an object in which intersected solids are repre- sented is complete unless the line of intersection is accu- rately produced. This is a very important part of the drawing, and the correct "fit" of the pattern in work of this class often depends entirely on the accuracy with which the line of intersection is drawn. In fact, a development, or pattern, cannot be made until the drawing is complete in this particular. The principles governing the use of these lines are clearly shown in the explanation accompanying the problems, which the student should study carefully; for 88 PRACTICAL PROTECTION 15 if he thoroughly comprehends the principles governing their use and exercises due care to see that lines are drawn from the same corresponding points in each view, he will have no difficulty in producing the correct lines of intersec- tion for the surfaces of the solids represented on these plates, or. in fact, for the surfaces of any solid. The prob- lems for this plate consist of the projection of intersecting solids having plane surfaces. The solids are shown in per- spective in the illustrations accompanying each problem, and reference to the projections on the plate will be suffi- cient to show the method of finding the lines of intersection. PROBLEM 35 92. To project views of intersecting prisms. When the plane or curved surfaces of any solid so inter- sect as to present one continuous surface — that is, so that the surfaces meet "edge and edge" in the same plane — no line of intersection is necessary, since such surfaces are relatively in (and a part of) the same plane. When, how- ever, the surfaces of one solid intersect a central portion of the surface, or surfaces, of another solid, it is necessary that the line of intersection — that is, the boundary lines of the sur- faces of the intersecting solid — should be accurately drawn. Case I. — When the axes of the prisms intersect at an angle of 90°. ExpLAXATiox. — The solids for the projeciion of this prob- lem are shown in perspective in Fig. -16, which represents also their position in the intersection of Case I. The figure consists of an up- right shaft in the form of a quadrangu- lar prism with a horizontal octagonal ^ prism intersecting, or **mitering, " at right angles along the axial lines of the I two solids. The projection and ar- rangement of the solids is seen in the drawings for this problem on the plate. It will be noticed that the octagons § 15 PRACTICAL PROJECTION 89 drawn in dotted lines in the rigure do not form a part of the finished drawing, but are thus drawn to facilitate the pro- jection, as the edges of the solids may thus be determined before the side view is drawn. It is often convenient to place portions of views temporarily on the drawing in this manner, as much labor is thereby saved. The plan is com- pleted first and the remainder of the drawing finished in the usual way, as may be seen from the reduced copy of the plate. It is thus shown that the correct line of intersection is found by simple methods of projection, the position of the points in the line being determined by projection from the different views. Construction. — Draw the plan ABC D in its proper position, as shown on the plate, and next construct the elevation A A' C L\ thus completing the projection of the quadrangular prism as though that solid alone were to be represented. Draw a horizontal axial line (not shown on the plate) for the octagonal prism through both views, and at the left of the views thus drawn construct a full view of the end of the octagonal prism, as shown by the octagons in dotted lines on the plate. Next, draw the lines that rep- resent the edges of the octagonal prism in both views. From the points of intersection of such lines in the plan with the edges A B and A D of the quadrangular prism, draw primary projectors to corresponding lines in the eleva- tion. Draw connecting lines through the points of inter- section thus determined ifi the elevation, as shown at a' b' c' d' on the plate. This completes the projection of that view; the side elevation may then be projected by means of sec- ondary projectors, as in former problems. This plate is divided in the same manner as the preceding plate, and the projections for this case are drawn in the upper left-hand space. Case II. — WlicH the axes intersect at an oblique angle. This projection is completed as shown on the j)Iate. In this, as in the preceding case, the plan is first drawn; but 90 PRACTICAL PROJECTION § 15 before it can be completed, a portion of the elevation has to be drawn in order to determine the outline of the octagonal surface in the plan (Problem 6). CoxsTRUCTiox. — First draw the outline A BCD in the plan and then construct the octagon shown in dotted lines at {a), from the points of which draw horizontal lines to the plan of the quadrangular prism, in the manner shown. Next draw the elevation of the quadrangular prism and locate the point x midway on the line A A' : the angle of inclination of the octagonal prism is in this case 15', and a line at that angle is then to be drawn through j: for the axial line of that solid. Construct the full view of the end of the octagonal prism at (d), and from the points t\ f. g, etc. in that view draw lines of indefinite length towards the right ; intersect these lines at a\ b\ c\ and d' in the elevation by vertical primary projectors drawn from corresponding points in the plan, thus establishing the line of intersection in the elevation. The plan is then completed by drawing primary projectors vertically downwards from the edge view of the end of the octagonal solid in the elevation and tracing the outline of the inclined surface thus designated, as in Problem 6. The side elevation is next projected by the arc method of secondary projectors, as heretofore. Xote that the intersection of the upper portion of the octagonal surface is to be represented in that view by dotted lines. Case III. — Wlien the axes are at an oblique angle, but do not intersect. ExpLAXATiox. — The position of the solids is shown in the drawings on the plate, and the projections do not differ materially from those of the two preceding cases. The octagonal solid is in this case inclined at an angle of 30°, and its axial line m //, as shown in the plan, is drawn ^ inch below the center of the quadrangular prism. The order of procedure is the same as that given for the drawing of Case II and will present no difficulties to the attentive § 15 PRACTICAL PROJECTION 91 student. It is necessary, however, to use extreme care in these projections, in order that the position of points in the different views may be located i)i the same corresponding position zvith regard to one anotJier. * PROBLKM 33 93. To project vie^vs of a prism intei-sectecl by a cylinder. Explanation. — Fig. 47 is a perspective view of an octag- onal prism intersected by a cylinder at an oblique angle, the axes of the solids not intersect- ing. The arrangement of the views, the dimensions, and the angle of inclination are shown on the plate. The projection of this problem is very similar to the last case of Problem "I'l. Note that the position of the two solids is such that the axis of the cylinder intersects the edge D of the prism. The circles that represent ^'"^- ■*" the end surface of the cylinder in each view are divided into a convenient number of equal spaces, and from the points thus located lines parallel to the axis of the cylinder are drawn on its surface. These lines are then projected in the same way as the lines that represented the edges of the octagonal prism in Problem 2'-i. Since the surface of the cylinder is a curved surface, the line drawn through the points of intersection of the two solids will be a curved line. * Too much stress cannot be laid on this important statement ; as the student progresses with the projection of the views, he will see tlie im- portance of this matter. The work must be done slowly, keeping the pencil points'well sharpened; the position of the points in the drawing may then be accurately delermined, if the views are constantly com- pared. If this is done, there will be no difficulty attached to any of the problems to follow in this stection. 92 PRACTICAL PROJECTION § 15 /^ 94. General Instruction Relating to Intersections. When points are located on the full view of a surface in a plan and elevation, as on the circles at (a) and (d) in the drawings on the plate for the last problem, care must be taken that the distinction between the different views is maintained; thus, the point x, at (a) and in the plan, is located at .v' at (d) and in the elevation, both positions on the drawing representing the same position on the solid. Any other points thus located on an outline will be changed in a corresponding way with relation to one another. To project intersections of solids, all of whose surfaces are bounded by parallel lines or on whose sur- faces parallel lines may be drawn that will also be parallel 'to the axis of the solid, it is necessary first to draw a view that will show the intersectr^:/ surface, or surfaces, in one line — that is, "on edge." Such views have been drawn in the plans of the preceding problems. The lines of the intersect/';/^ solid are then represented in this view, and the points of their intersection with the upright surfaces are projected to the elevation in the manner described. It will be seen that, if all the surfaces of any intersecting solids are plane, their edges or outlines alone will suffice for finding the lines of intersection. If the surfaces are curved, it is merely necessary to locate a number of points on the outline of the full view, through which to draw parallel lines similar to those drawn on the cylinder in Problem 23. This practically changes, or reduces, the cylinder to a solid bounded by a number of plane surfaces. In the case of Problem 23, the cylinder has really been treated as though it were a prism having a number of sides equal to the num- ber of spaces into which the circles were divided. This would actually have been the case had straight lines been drawn between the points thus located on the circles at (a) and (d). Had this been done, the line of intersec- tion would also have been represented by a series of short lines drawn between the points located by projection methods. INTER5E i^r ^' \r M ! niii PROBLE/^ a4: \ \ ~~~^^^^-^\ \ \ fW)BLEM26 JUNE 25.1893. Copyright, l5S9, by THE CoiJ All rights! :TiDN5-n. PROBLEM as. t? d f hy Pf?OBLEM £7. , O (/, etc. Project these elements to the elevation and locate thereon by pro- jection methods the position of the points of intersec- tion a', h\ g\ etc. Complete the elevation by drawing the outline of the cylinder C in that view and tracing the line of intersection of the two solids through the points a\ h\ g' etc. Project the side elevation by the usual methods. 96 PRACTICAL PROJECTION §15 Fig. 51 PROBLEM 26 99. To project views of iutersecting cones. Fig. 51 is a perspective view of an object that illustrates this problem. The cones are shown in a somewhat more convenient proportion for this problem in the projections on the plate. The con- struction of this problem must be followed very carefully, as a number of the operations are necessarily made over one another on the draw- ing and the student must be careful to distinguish each process. CoxsTRUCTiox. — Describe a circle •2|- inches in diameter in the plan to represent the lower base of the larger, or intersected, cone in that view; and from the same center describe a circle f inch in diameter, to represent the upper base. Project the front elevation of this cone and define the frustum 2^ inches high, as shown on the plate, producing the outlines until they meet in the vertex. Next, through the point F, Fig. S'^jdraw the axial line of the smaller, or intersecting, cone (the line A B, Fig. 52) at an angle of 45° with the base of the larger cone; locate the point A 3 inches from „ F, and, after fixing the point C 1 inch from F, draw C D perpendicular to A B. Draw the outline of the upper base of the fig. 53 § 15 PRACTICAL PROJECTION 97 smaller cone in the elevation parallel to C D and f inch from A. Draw tlie full view of the base of the smaller cone zX J B //; divide this outline into a convenient number of equal parts, as at J, a, B,c, etc., and project these points to the base C D. Draw the elements of the smaller cone, as shown in Fig. 5"2, and produce them until they intersect the base of the large cone at /f, /% and 6^.* In the plan of this drawing, a series of sections of both cones are now drawn as the sectional curves would appear if each of the elements of the smaller cone shown in the elevation were considered as a cutting plane, as in Problem 20. The sections of the smaller cone will in each case be a triangle (Art. 86), while the sections of the larger cone will be ellip- tical, parabolic, or hyper- .A bolic curves, as the case may be. The point of intersec- tion with the side of each triangle and its correspond- ing sectional curve of the larger cone is then projected to the elevation and the line of intersection of the two cones traced through these points. The sectional tri- angles of the smaller cone are shown in the plan of Fig. 52. Draw vertical primary projectors to the '^•^^—^ plan from points a' , B\ and t', and on these pro- jectors set off distances from the horizontal center line of the plan similar to the distances from H / '\n the full view of the base; that is, make x' c" equal x c, etc. * In this case eight elements are repre.sented, since it is not desirable to- complicate the drawing by using more, although in practical work it will be found necessary to use a larger number of points in order that the line of intersection may be more accurately traced. Fig. 53 98 PRACTICAL PROJECTION 15 It is not necessary to develop the sectional curves of the larger cone in their entire length, since all that is required is to find a point on each curve that is at the intersection of the triangular sections of the smaller cone. This is better illustrated in Fig. 53, in which is shown the projection of the point o in the line of intersection. A study of this figure will show to the student that the operations are simi- lar to those of Problem "20; the process in the case of each point is merely a repetition of that here indicated and need not be further explained. The extreme upper and lower points J' and z. Fig. 5'2, are the points of intersection of the central section and may be projected directly to the plan from the elevation, since the outline of the figure in the elevation is really a section on the line vi Ji in the plan of Fig. 5'2. Fig. oi shows the three sections produced by the above method; that is, the irregular curve p q r. Fig. 54, is a section of the larger cone produced by the in- tersection of the cutting plane A E\ s s and t t are found as above de- scribed, the section lines being indicated in the figure by short cross- hatching. Project the side elevation as in former problems. The student may at the completion of the drawing on the plate erase all the construction lines except those projectors shown on the reduced copy of the plate, those lines only be- ing inked in that are neces- sary to show the outlines of the figure and the line of intersection in each view, as well as the outer projectors. §15 PRACTICAL PROJECTION 99 PROBLEM 27 100. To i>roject views of a sphere intei*sected by a cylinder. When the position of these two solids is such that the axis of the cylinder passes through the center of the sphere, as shown in perspective in Fig. 55, the line of intersection is shown in a right elevation as a straight line. In the case of the solid shown in perspective in Fig. 56, the axis of the cylinder does not pass through the center of the sphere, and Fig. 55 Fig. 56 the line of intersection is an irregular curve, which may be found by the following method. vSince the construction of this problem involves lines that must necessarily be drawn closely together in the small scale adopted for these drawings on the plate, a proportion is selected that does not admit of the entire figure being shown in the elevations; these pro- jections are therefore finished by broken lines, as indicated on the plate. Construction. — Draw the plan first and represent the view of the sphere by a circle SI inches in diameter. Draw a vertical diameter and from a point midway on the radius ad describe a circle If inches in diameter, to represent the end view of the cylinder, on the outline of which locate a number of points by spacing with the dividers, as at 1, 2, 3, 4, etc. Through these points, in the manner shown on the plate, draw vertical lines, a.s ad, c d, ef, gh, and i j\ each of which will now represent a cutting plane. Sections of the sphere 100 PRACTICAL PROJECTION § 15 and cylinder on these lines are now to be produced in the side elevation. These sections, as already stated, are circles and parallelograms, respectively, the diameter of the circles being ascertained from the view in which the cutting plane is shown on edge, as in the plan. In this problem, the side elevation is next projected. Describe arcs representing the sections in the side elevation, using the radii ab, c d, f^ f.gJi^ and i j\ the radius a b being the great circle of the sphere. By the aid of primary projectors, project the points from the cylinder in the plan to the side elevation, intersecting corresponding arcs in the side elevation. Trace the irregular curve shown in that view on the plate through these points. Next project the front elevation, thus completing the problem. 101, Recapitulation. — The problems of this plate have afforded the student an opportunity for careful study. Owing to the number of points necessary to be found in each figure, the problerhs may have had the appearance of more or less complication, but if the student, as previously cautioned, will carefully locate the points required, one at a time, not hurrying his work nor trying to grasp the entire problem at once, but keeping in mind the different principles in the order presented, and by referring, if necessary, again and again to the primary principles, he will experience no difficulty in making the drawings. He will also be able intelligently to project any view of any object; in other words, he will be able to make any working drawing what- ever, and. in addition to this, be able to read and understand any working drawing he may be called on to examine. DBVELOPMENT OF SURFACES IKTROD UC TICK 1. Defliiitiou of a Development. — A development is a drawing in which a full view of all the surfaces of a solid is represented. Whenever a development is to be drawn (except in the case of solids of very simple form), a pro- jection drawing must first be made. This projection should show the solid in a right position. Since the location of the various points in a development is dependent on their corre- sponding position in the projection drawing, the importance of the projection and the necessity for accuracy in its con- struction are thus clearly seen. If a solid is bounded entirely hy plane surfaces, its development can be accomplished by merely projecting their full views, as already explained in Practical Projection. A solid is said to be developed when all surfaces compo- sing it are represented on one plane and in such relation to one another that, if formed or bent up, they will constitute a solid similar to the one represented by the projection draw- ing from which the development was made. Such a rep- resentation is called a development, or a pattern, the process of laying out the pattern being termed developing the surfaces of the solid. 3. Relation of the Surfaces in a Pattern. — When it is desired to produce a pattern requiring a combination of several surfaces that are adjacent in a solid, such surfaces must be drawn in the same relation to one another in the §10 For notice of c<)pyri>rht, see page immediately followinvj the title page. 2 DEVELOPMENT OF SURFACES § 16 development. The surfaces of a solid when thus combined in a pattern, or development, bear the same relation to one another that they would if they were considered as being unfolded or unrolled — the same relation that a paper wrap- per would bear to the package from which it had been unfolded or unrolled. The paper wrapper is not always an apt illustration, as the metal worker seldom requires several thicknesses of his material. In the case of the familiar •'square pan," however, the ends are folded on one another in precisely the same way as in the pai>er wrapf>er. It will be seen from the foregoing that, were all solids bounded by flat, or plane, surfaces, the subject of develop- ments would present no new problems; it would be neces- sary merely to study the relation of surfaces to one another, project their full views, and carefully redraw them in the pattern in the same relative position. 3. Projection Method* I"«^<1. — It has been shown in Practical Projecv. r. ::..^: a single surface is developed, or, as stated, its full view is drawn, by a modification of the same methods that are used to produce the different views of that surface. Many of the operations attendant on the develop- ment of solids are like those used in producing full views of single surfaces: or, if not, the principles involved may be traced to their origin in other methods used in projection drawing. A thorough knowledge of projection is absolutely neces- sary that the student may understand the operations involved in developing the surfaces of a solid. The position of the several points located in a drawing and their corresponding location in an imaginary way on the object itself must be definitely fixed in the student's mind. Each line must be determined in its relation to the other lines of the drawing and its ideal, or imaginary, location definitely ascertained ; the surfaces, also, must be treated in a similar way. The student must picture to himself the completed object as it will appear when the surfaces laid out on the drawing board in the development are formed up in their final relation to § 16 DEVELOPMENT OF SURFACES 3 one another. This imaginary part of the study is of even greater importance in the case of developments than in pro- jection drawing. As the student has already had some drill in this part of the work, the subject he is now studying should be found less difficult than would otherwise be the case. In projection drawing, the surfaces of the solid are represented as being in their proper position ; in the develop- ment, the same surfaces are represented as being developed or spread out on the surface of the drawing board. GrE:N^ERAL classificatio:n^ 4. General Classification of Solids. — An accurate development may be drawn for the plane surfaces of any solid, or for surfaces having, when related to a given line on such surface, a curvature in one direction only. In general, it may be stated that any solid may be developed on whose surfaces it is possible to lay a straightedge, in continuous contact, in any one direction. To use in this connection the illustration of the cylinder, it will be seen that, if the straightedge is resting on the surface parallel to the'axis of the cylinder, it will remain in contact at all points. If, on the other hand, the straightedge is resting on the curved surface and is not parallel to the axis of the cylinder, the surface will be in contact at a single point only. However, the fact that it is possible to place the straightedge in con- tinuous contact on the surface allows the inference that such surface is capable of accurate development. The same rule applies to solids of irregular form. The methods of development, however, are not the same in cer- tain variously formed solids, as will be explained later. There are certain forms Avhose surfaces, owing to their curvature in several directions, are not capable of being thus laid out on a flat surface, i. e., not capable of being devel- oped. On the surfaces of solids of this class— the sphere, for example — it will be found impossible to lay the straightedge in contact in any direction. For, if placed on such a surface, 4 DEVELOPMENT OF SURFACES § 1(5 there will be but one point of contact — that of the tan- gential point. Tangential contact indicates that develop- ment can be accomplished only in an approximate way. For purposes of development, then, it is convenient to separate all solids into two general classes according to the result obtained in developing their surfaces. These two classes are: solids whose surfaces admit of accurate development and solids whose surfaces admit only of approximate devel- opment. Approximate developments are, however, so nearly accurate for the purposes of the sheet-metal worker that^ the kind of solid is more clearly marked by the method of developing its surface than by the result obtained by the development. Therefore, in order to distinguish the kinds of solids, both accurately and approximately developed solids are divided into three main classes according to the method used in developing their surfaces. These classes are explained later. 5. Accurate Developments. — Solids whose surfaces are capable of accurate development are of frequent occur- rence in the sheet-metal-working trades. To this class belong all prismatic, cylindrical, and conical forms, whether of regular or irregular geometrical form. It includes all articles or objects whose covering may be formed without being submitted to the operations known to trade workers as "raising," or '"bumping." Any solid whose surfaces may be unrolled or spread out on a flat surface without " buck- ling " may be accurately developed. Although it is often necessary, especially when working metal of unusual thick- ness, to take into account the stretching of the material when producing patterns for many objects, these objects belong to accurately developed solids, providing that the metal does not have to be " raised," or " bumped," in order to form the object. It is, therefore, essential that the metal worker should thoroughly understand the nature of the material and be well informed as to the best manner in which to provide for all laps and edges used in the construc- tion of the finished article. § IG DEVELOPMENT OF SURFACES 5 It is the purpose of Development of Snrfaees to define and illustrate theoretical developments and the means used by the draftsman in their production. 6, Approximate Developiiieuts. — The sphere and other solids whose surfaces have a curvature in two or more directions are examples of objects capable of only approximate development. The test by the straightedge is (with the exception of the helicoidal surface) a posi- tive indication of the class to which any solid may be assigned. Patterns for the surfaces of objects of this class may be approximated, because it is necessary for the metal to undergo the operations of "raising," or "bumping," before it will conform to the exact surface represented in the drawing. It is necessary in these cases to make allow- ance in the pattern for the stretching of the metal. Since this part of the subject does not belong to theoretical develop- ment, it is not treated here. SOIilDS THAT MAY BE ACCURATELY DEYEIiOPED 7. There are three distinct methods in common use, by means of which patterns are produced for solids whose sur- faces are capable of accurate development. It is advisable, therefore, to separate the different varieties of these forms into three general divisions, in order that their development may be studied in a systematic manner. This classifica- tion may be made by studying the manner in which the covering of these solids — to use again the illustration of a wrapper — would be unrolled or spread out if done by roll- ing the solid on a flat surface. 8. Solids Developed on Parallel Ijines. — A conve- nient illustration of the manner in which the surfaces of a solid will appear when unrolled as above indicated may be found in the following example, which serves at the same time to define a property peculiar to solids of a certain form. Let the continuously adjacent surfaces of the prism DEVELOPMENT OF SURFACES 16 shown in Fig. 1 (a) be carefully covered with thin paper, as at Fig. 1 [d). Denote each of the four surfaces by a letter, as.^, B, C, and D, and further designate the edges of the prism by the letters a b, c d, e f, and g h. As the ends of the paper covering meet at the edge a b, that edge of the surface D may be denoted by the letters a' b' , as shown in Fig. 1 {b). Assume now that the prism is laid on the drawing board, the surface A face down, and the paper covering removed by turning the prism over and over, the paper remaining on the surface of the drawing board, as shown in Fig. 2 {a) and {b). Two important principles relating to developments are demonstrated in these illustrations. First, as will be seen Fig. 2 from Fig. 2 {b), the edges a b, c d, e f, g h, and a' /^' are all parallel to one another. This is true both in the develop- ment and on the solid, as may be readily seen by the student, the only difference being that on the solid certain of the lines are in different planes, while in the development they are all in the same plane. Stro/id, it will be noted §16 DEVELOPMENT OF SURFACES that, since the letters shown in Fig 2 are reversed, the outer surface of the paper covering in Fig. 1 {U) corresponds to the under surface in Fig. 2 {b). In a similar manner, it is learned from the second principle that positions indicated on any surface of a solid, as shown in a projection drawing, are reversed when shown in the development of the solid. The same treatment of the cylinder is found to produce results closely resembling those shown in the case of the prism. The cylinder is represented in Fig. 3 (c?) as covered Fig. 3 with paper, a number of lines being ruled on the covering parallel to its axis, as shown at c d^ ef, etc. The paper is shown unrolled in Fig. 8 {b), and it will be observed that not only the outer edges a b and a' b' are parallel to each other, but that all other lines parallel to the axis of the solid appear in the development parallel to one another and to the edge lines a b and a' b' . The student that has ever flattened out a piece of straight molding, as for cornice work, probably noticed a number of straight parallel lines on the metal where it had been bent in the brake. This is an illustration similar to that of the cylinder, the different members' of the molding being considered as the various surfaces of an irregular solid. Such illustrations indicate that parallel lines bear some relation to certain forms, and it will be shown that the pat- terns for these forms are developed by a method whose principles are based on this fact. Many solids may be at once recognized as belonging to this division. In general^ 8 DEVELOPMENT OF SURFACES §16 any solid whose edges are parallel may be located here. In the case of solids having curved surfaces, it may be stated that, if it is possible to draw a series of parallel lines on such surfaces, the development of the solid may be pro- duced by the same methods given for this class. The first general division, therefore, comprises those solids whose surfaces may be developed on parallel lines. 9. Solids Developed on Radial JLiues. — When the test given to the cube and the cylinder in Figs. 1 to 3 is Fig. 4 applied to the pyramid, it is found that the lines indicated on the paper converge to a point, as shown in Fig. 4 {a) and {b). It is noticed, also, that this point o. Fig. 4, defines the position of the vertex of the pyramid. The same may be said of the cone, illustrated in Fig. 5 (<^) and {b). If lines Fig. 5 are first indicated on the surface of the cone corresponding to its elements, it will be found, when the covering is unrolled, that these lines also converge to a point, as in the case of the edges of the pyramid. § 16 DEVELOPMENT OF SURFACES 9 It was found possible to institute a system of obtaining developments based on parallel lines in the case of the prism and cylinder; in a similar manner, it is quite evident in this case that a system dealing with radial lines should produce like results. Since, in projection drawing, the ele- ments of the cone are known to be useful factors in deter- mining the position of points on its surface, it may readily be conceived that their use in a somewhat similar way may be adapted to developments. This is found to be the^ case- and a second general division of solids is thus made, con- sisting of those forms whose surfaces may be developed on radial hues. Included in this division are all regular taper- ing solids and such irregular forms as are derived from regular solids. The metal trades furnish manv examples of solids belonging to this division; in fact, the writers of sev- eral works on patterncutting confine their instruction almost entirely to the development of solids of this character. 10. Soliils Developed by Triaugnlatlon.— There are many forms of irregular surfaces to which the test of the straightedge may be applied and the conclusion thereby reached that their surfaces admit of accurate development. It may also be concluded that neither of the two former methods is applicable, for neither parallel lines nor a series of radial lines may be drawn on their surfaces. Many of these solids are not of such a shape as to admit of their being either turned or rolled on a plane surface. It is found"^" however, that on every such surface, series of two or more lines each may be drawn in certain directions, forming angles. On such irregular surfaces it may happen that no two of the angles thus drawn on the solid, or represented— either correctly or foreshortened— in the projection drawing, will he in the same plane or be equal to each other. Since' it is possible thus to project these angles, evidently thev may be reproduced on the flat surface of the drawing paper in their correct size. If this can be done, it may be reasonably assumed that the surfaces thus represented will be the same M. E. /-.- -yy 10 DEVELOPMENT OF SURFACES § 16 as the corresponding surfaces of the solid. An illustration of this principle, as pertaining to a plane surface, was given under another heading in Practical Projection. In Fig. 6 an irregular solid of this kind is shown. It is the solid whose projection was drawn in Problem 19 of Practical Projection. This figure illustrates in a general way the method used in arranging the trian- gles on the irregular surface of such solids. The triangles are represented in the figure in a perspective way, but they are, of course, always drawn in connection with the usual methods of projection. The third general division, therefore, con- sists of those solids whose surfaces are developed by trian- gulalion — that is, by means of triangles. 1 1 . IIoTv tlie Division of Solids Is Acconiplishe