5SUUMVEM/A ^ g. SUIBRARYQr \Vtf-UNIVER$y/) ME UNIVERS//, ^LIBRARY/?/ \\\[ -I'NIVERS/A ^> :lOSANCElFj> f ^\\E UNIVfRS^ ' v |7^ C3 irfs i Lrmv^i 3 1 n)i a A COURSE . IN . . . ELEMENTARY MECHANICAL DRAWING . .. REVISED BY WM. A. PIKE, UNIVERSITY OF MINNESOTA, MINNEAPOLIS, 1891. TRIBUNE JOB PTG. CO. Start Hfinex r PREFACE TO REVISED EDITION. THIS edition has been revised and corrected to meet objections found in the first. The number of plates of geometrical and projection problems have been reduced and more problems put on each plate, as experience showed that too much time was taken 'in the lettering and other routine work of so many plates; also, as some of the problems are but modifications of others and some seemed of hardly enough value to be retained, a number of them have been dropped in order to give more time to the application to practical drawings. The plates have been redrawn and printed with black lines on a white ground for the sake of greater plain- ness. The example of a rough sketch from which to make a fin- ished drawing has been corrected and a more modern stub end substituted for the old one. Additional plates illustrating line shading have been incorporated in the text, as most work in actual practice is finished in this way instead of by tinting and shading with the brush. Finally the text has been changed and added to wherever it seemed desirable. MINNEAPOLIS, Aug. 30th, '91. 2068014 COURSE IN Elementary JVleehanieal Drawing. DRAWING MATERIALS AND INSTRUMENTS. Each Student will require the following Instruments on begin- ning the course, viz : Half-a-dozen Sheets of Drawing Paper, a Drawing-board, a T-square, a Pair of Triangles, a Hard Pencil, a Right Line Pen, a Pair of Compasses with Pen, Pencil and Needle Points, a Pair of Plain Dividers, an accurate and finely divided Scale, a piece of India Ink, a Rubber, an Irregular Curve, and half-a-dozen Thumb-tacks. These instruments and materials are all that are absolutely required up to the time of commencing tinting and shading, when a few other articles will be needed, which will be spoken of in their proper places. Before purchasing, the following directions about the different instruments and materials should be noticed. PAPER. For all the drawings of this course, use Whatman's Imperial drawing paper. It comes in sheets of convenient size, and is well adapted to the work of the course. Six sheets will be enough up to the time of tinting. DRAWING-BOARD. Great care should be taken to secure a good drawing-board. The best boards are those made of thoroughly seasoned white pine, one inch thick, with cleats at the ends flush with the surface of the board. The most convenient size is twenty-three by thirty-one inches. This gives a small margin out- side of a whole sheet of Imperial paper, which is twenty-two by thirtv inches. One surface of the drawing-board must be plane, and the edge from which the T-square is used must be perfectly straight. T-SQI-ARE. All horizontal lines in the drawings are made by the use of the T-square. The T-square should be used from the left-hand edge of the board, unless the person is left-handed, in which case it should be used from the right hand edge. The upper edge of the blade only is to be used in drawing lines. The blade should be at least thirty inches long, and about two-and-one-half inches wide. The thickness should not be over an eighth of an inch. The head should be twelve or fourteen inches long, at least, in order that the blade may never be thrown out of line. By slid- ing the head up and down on the straight edge of the board, any number of parallel horizontal lines may be drawn. It is very important that the upper edge of the T-square be perfectly straight. TRIANGLES. For making all vertical" lines, and all lines mak- ing the angles of thirty, forty-five and sixty degrees with the hori- zontal and vertical lines, triangles are used, sliding on the upper edge of the T-square. Two triangles are necessary, one forty-five degree and one thirty and sixty degree, as they are called from their angles. Bach of these triangles has one right angle, and either can l>e used for drawing verticals. It is often convenient to have one triangle large enough for drawing quite long verticals, like border lines ; but in lettering and in other small work a small- er one is much more convenient. It is therefore advisable to get a thirty and sixty degree triangle that has one of its rectangular edges about ten inches long, and to get a forty-five degree triangle much smaller. PENCILS. Ail lines are to be made first with a hard pencil, and afterwards to be inked. It is very important that the pencil lines be very fine and even, though they need not be very dark. Ink will not run well over a soft pencil line, and it is impossible to do good work without making the lines fine. The best pencils for this work are Faber's H H H H and H H H H H H or some kind equally hard and even. The H H H H is recommended for beginners who are not accustomed to using a very hard pencil, but the H H H H H H is harder, and better adapted for nice work. One of each kind will be amply sufficient for the work of the whole course. The pencil should be sharpened at both ends, at one end with a com in on sharp round point, and at the other with the lead of about the shape of the end of a table knife. The round * By vertical lines are meant lines parallel to the edge of the board from which the T-square is tised, by horizontal those parallel to the up- per edge of the T-square. point is to be used in lettering and in other small work, and the flat point in making long lines. The flat point will keep sharp much longer than a round point. Both points should be sharpen- ed often by rubbing them on a piece of fine sand paper or on a very fine file. The flat point should always be used in the compasses with the edge perpendicular to the radius of the circle. RIGHT-LINE PEN. In selecting a right-line pen care should be taken to get one with stiff nibs, curved but little above the points. If the nibs are too slender they may bend when used against the T-square or triangles, and the result will be an uneven line. If the nibs are too open there is danger of the ink dropping out and mak- ing a blot. If too little curved the pen will not hold ink enough. The nibs are apt to be too open, than otherwise. The medium sized pens are best adapted for this work. The pen must have a good adjustment screw to regulate the width of the lines. The pens, as they are bought, are generally sharpened ready for use; but, after being used for a time, the ends of the nibs get worn down, so that it is impossible to make a smooth, fine line. When this occurs they should be sharpened very carefully on a fine stone. In order to have a pen run w r ell, two things are necessary, first the points must be exactly the same shape aud length, and both nibs must be sharp. In sharpening a pen, therefore, the first thing to be done is to even the points. This may be done 1n r moving the pen with a rocking motion from right to left in a plane perpen- dicular to the surface of the stone while the nibs are screwed together. After the nibs are evened in this way the points should be opened and each nib sharpened on the outside, only, by holding the pen at an angle of about thirty degrees with the surface of the stone, while it is moved in about the same manner as in sharpen- ing a gouge. The point should be examined often with a lens. COMPASSES. The compasses must have needle points, with shoulders to prevent them from going into the paper below: a certain depth. The needle point, when properly used, leaves a very slight hole in the center of each circle; while the triangular point with which the poorer instruments are provided, leaves a very large, unsightly hole, unless used with more than ordinary care. The pencil point should be one made to contain a small piece of lead only. All that has been said in regard to the right- line pen applies equally well to the pen point of the compass. In using the pen point be sure that both nibs press equally on the paper, otherwise it will be impossible to make an even line. Both nibs may be made to bear equally by adjusting the points in the legs of compasses. DIVIDERS. The dividers should be separate from the com- passes, as it is very inconvenient to be obliged to change the points whenever the dividers are needed. The dividers have tri- . insular points, which should be very fine, and of the same length. The legs of the dividers should move smoothly in the joint, and not hard enough to cause them to spring while being moved. The dividers are used principally for spacing oft" equal distances on lines, but are often used for taking measurements from the scale, especially when the same measurement is to be used on several different parts of a drawing. SCALE. A ver3' good scale, for this course, is one with inches divided into fourths, eighths, sixteenths. etc., on one edge; and into twelfths, twenty -fourths, etc., on the other. The first edge is verv convenient for taking measurements, and for making drawings to a scale of one-half, one-fourth, etc.; but the second is better for drawing to a scale of a certain number of inches to the foot. Triangular scales are still better, but more expensive. INK. India ink, which comes in sticks, is the best ink tor gen- eral uses ; but there are several kinds of ink in bottles which are much more convenient for making line drawings. None of the ink that comes in bottles, however, is good for shading. If the India' ink is used, an ink slab or saucer will be needed in addition to the instruments mentioned in the list. In grinding India ink, a small quantity of water is sufficient, and the ink should be ground until a very fine line can be made very black with one stroke of the pen. Ink will look black in the slab long before it is fit to use on adraw- ing. Ink should not be ground, however, so thick that it will not run well in the pen. The ink must be kept covered up or it will soon evaporate so much as to be too thick to run well. Mi HBER. Get a soft piece of rubber so as not to injure the surface of the paper in rubbing; what is known as velvet rubber is well adapted to the draughtsman's use. IRREGTLAR CURVE. In selecting an irregular curve, one should be obtained which has very different curvature in different parts, in order to fit curves which cannot be drawn with compasses. THUMB TACKS. Thumb tacks should have good large heads, so firmly fastened on that they cannot get loose. One can do much better in buying instruments, to buy them in separate pieces, each carefully selected, than to buy them in sets. It is very difficult to buy a set of instruments that will con- tain just what is required for this work, without buying many unnecessary pieces. '03 CX3 O -QD / V fc V. I ^^ I a &* GENERAL DIRECTIONS FOR COMMENCING THE WORK. Each plate of geometrical problems is to be made on a half sheet of the Imperial paper. The sheet should be folded over and cut with a sharp knife, but before cutting find out which is the right side of the paper. The right side of Whatman's paper may always be found by holding the sheet up to the light. When the name of the manufacturer can be read from left to right, the right side is the one toward the holder. The half that has not the name on it should be used first, while its right side is known ; the right side of the other piece can be found in the way described, when it is to be used. Place the paper on the drawing board so that two of its edges will be parallel to the upper edge of the T-square when in position on the edge of the board ; and fasten it down with three thumb- tacks in each of the long sides, placing each thumb tack within a cjuarter of an inch of the edge, in order that the holes may be cut off when the plate is trimmed. For convenience in working on the upper part of the plate, it is best to have the paper as near the bottom of the board as possible. Begin each plate by drawing a horizontal line, with the use of the T-square, as near the thumb tacks at the top as possible. Fourteen inches below the first line, if the longest dimension is to be horizontal, draw another parallel to it, at the bottom of the paper, and by means of the larger triangle, draw vertical lines at the right and left of the paper, twenty-one inches apart. The lines are the limits of the plate, and are the ones that the plate is to be trimmed by, when finished. All the plates that are to be drawn on a half-sheet must be of this size, twenty-one by fourteen inches, unless the paper is to be shrunk down, in which case the plates must be made somewhat smaller, as will be afterwards noticed. All the plates are to have a border line one inch from the fin- ished edge, except on the top, where the border is to be one and a quarter inches from the edge. This border should next be drawn by spacing off the proper distances from the lines just drawn, and drawing the border with T-square and triangles. There are to be fifteen geometrical problems to each plate, and for convenience in locating them, the space inside of the border lines, in the first five plates, should be divided into fifteen equal 10 rectangles, in three rows of five each. These last lines are not to be inked, but must 1>e erased when the plate is completed. The first two plates in the course are of geometrical problems. The problems that have been selected have many applications in subsequent work and, moreover, the exact construction of them -ivc< the best of practice for beginners in handling the different instruments. The construction of each problem is described in the text, with references to the plates; and each must be constructed according to the directions. The reasons for the different con- structions, though necessarily omitted in the text, will l>e evident to every one who has a knowledge of plane geometry. The geometrical problems are not to be drawn to scale, but they should be so proportioned that they will occupy about the same amount of space in the center of each rectangle. All of the lines must be made very fine and even ; and great care must be taken to get good intersections and tangencies. DIRECTIONS FOR LETTERING. Alter the problems are pencilled they must be lettered to correspond to the plates in this pamphlet. Make all the letters on the plates of geometrical problems and elementary projections like those given in Plate A. These skeleton letters are the sim- plest of all mechanical letters to construct, and, when well made, they are more appropriate for such work than if more elaborate. Make the small letters, in every case, two-thirds as high as capi- tals. Before making a letter draw a small rectangle that will just contain the letter, and then construct the letter within the rec- tangle, as shown in plate A, using instruments wherever possible. The height of all the capital letters in the problems and in the general title at the top, is to be one quarter of an inch. The widths vary, and may l)est be found in each case, until practice renders it unnecessary, by consulting plate A. Great care must be taken in lettering to make all the lines of the letters of the same si/e, and in joining the curves and straight lines. TITLES. The title of each plate of geometrical problems must corre- spond to that given in plate 1, except as to number. The title of the projection plates will correspond to that of plate III, and the titles of all other drawings will be as indicated in the text. In constructing a title always work both ways from the central letter of the title, in order that the title may be symmetrical, and over the center of the plate. In order to find 11 the middle letter of the title, count the number of letters, consider- ing the space between words as equal to that of a letter, and di- vide the number of spaces thus found by two ; this will give the number of the middle letter from either end of the title. Con- struct this letter .over the centre of the plate, and then work both ways from this in the wa3 r just indicated. Make the letters in a word about an eighth of an inch apart, though the space will vary with the shape of the letter ; and the space between words equal to that of an average letter with its spaces. It is best, in all cases, to have the title before you in rough let- ters, to avoid making mistakes in working backwards from the middle letter. The titles at the top are to be made in capitals. The letters in the general title are to be a quarter of an inch high and a quarter of an inch above the border, and those in number of the plate of letters three-sixteenths of an inch high and the same distance above the general title. The name of the draughtsman should be in the first three plates, at the lower left hand corner, three-sixteenths of an inch below the border and the date of completion in a corresponding position at the right. Make the date first, and commence the name as far from the edge, at the left, as the first figure of the date comes from the right-hand edge. Make the capitals in name and date three-sixteenths of an inch high. Number the problems as they are in the plates, commencing the first letter of the abreviations for problems in capitals, one- half an inch below, and half an inch to the right of the lines form- ing the upper right hand corner of the rectangle. The other letters of the abbreviations are to be small, and the numbers of the problems are to be marked with figures of the same height as the capitals. Great pains must be taken in lettering the plates, as the gen- eral appearance of a drawingis very much affected by the arrange- ment and construction of the letters and titles. The directions here given apply to the plates of geometrical problems. Some modifications will be made in lettering the problems in projection ; but the remarks on construction of the separate letters, and on the arrangement of the letters in a title, are general. After having had the practice in spacing and proportioning the skeleton letters, in the first three plates, the student will be allowed to use other styles of letters on the remaining work. Care must be taken, however, to have the titles symmetrical, and no letters on the plates of this course should be made over half an inch high. INKING. When the lettering is all done, a plate is ready to be inked. Be- fore using the pen on the plate, be sure that it is in a condition to 12 make a fine, even line, by testing it on a piece of drawing paper or on the part of your paper that is to be trimmed off. Be sure to have ink enough ground to ink the whole plate, as it is not best to change the ink while working on a plate, for the reason that it is nearly impossible to get the second lot of the same shade and thickness as the first. The arcs of circles should be inked first, for it is easier to get good intersections and tangencies by so doing, than it is if the straight lines are drawn first. Make all the given lines and all the required lines in full; but all the construction lines in fine dots. Make all the lines in the geometrical problems as fine and even as possible. The border lines should be made a little heavier than the others. All the fine lines should be made, if pos- sible, with one stroke of the pen. In order than an even line may be made, the pen must be held so that both nibs will bear on the paper equally ; and in order to do this, the T-square or triangle must be held a little way from the line, but parallel to it. The pen should be inclined slightly in the direction it is to be moved. In using the compass pen, the joints of the compass legs should lie so adjusted that the point will bear equally on both nibs. The ink should l>e placed in the pens by means of a quill or a thin sliver of wood. The pen should never be dipped into the ink, THE PLATES. The plates in this pamphlet are given to show the arrangement and construction of the problems but should not be followed as ex- amples too closely, as mechanical difficulties make it necessary to use coarser lines in proportion to the size of the plates than should appear on the drawings. GEOMETRICAL PROBLEMS. PROBLEM 1. To bisect a given line, A B, or to erect a perpen- dicular at the middle point of A B. From A and B as centres, with a radius greater than one-half of A B, described two arcs intersecting at C, and two arcs inter- secting at D. Join C and D by a straight line, it will bisect A B, and will be perpendicular to it. PROBLEM 2. To divide a given line, A B, into any number of equal parts, five for instance. Draw a line, A C, making any angle with A B, and on A C set off any five equal distances, A 1, 1 2, 2 3, 3 4 and 4 C ; join C and B, and through 1, 2, 3 and 4, draw lines paralled to C B, these lines will divide A B into eqmil parts. PROBLEM 3. To draw a perpendicular to a line B C, from a point A, without the line. From A as a centre, and with any radius, describe an arc, cut- Q in .o o Ql s CQ C\J o u CL O m LJ O D CD O U m ,CD h o> m co : ; LJ CD 13 ting B C in D and E. From D and E as centres, describe two arcs intersecting in F. Join F with A. PROBLEM 4. To draw a perpendicular to a line B C from a point A, nearly over one end, C, so that problem 3 cannot be used. From any point B, on the given line as a centre, describe an arc passing through A. From some other point D, of B C, describe another arc passing through A. Join A with the other point of intersection of the arcs. PROBLEM 5. To erect a perpendicular to a line B C, at a given point A, of the line. Set off from A, the equal distances A E and A F, on either side. From E and F as centres, with any radius greater than A E and A F; describe two arcs, intersecting at D. Join D with A. PROBLEM 6. To draw: a line parallel to a given line A B, at a given distance from A B. From two points C and D, of A B, which should not be too near together, describe two arcs, with the given distance as a radius. Draw a line E F tangent to these arcs. PROBLEM 7. To lay off an angle at a given point a, on a given line a c, equal to a given angle BAG. With A as a centre and any radius, describe an arc included Ijetween B A and A C. With a as a centre and the same radius, describe an indefinite arc. Lay off the chord b c equal B C from c on the arc b c. Join b with a. PROBLEM 8. To bisect a given angle BAG, whose vertex A is within the limits of the drawing. From A as a centre describe an arc, cutting A B and A C in b and a respectively. From b and a as centres describe two arcs in- tersecting in c. Join c with A. PROBLEM 9. To bisect an angle BA CD, whose vertex isnot within the limits of the drawing. Draw by problem 7, two parallels, ab and ac to AB and CD respective!}-, and at the same distance from A B and C D ; this dis- tance must IDC such that ab and ac shall intersect. The problem is then reduced to one of bisecting bac, which is done by prob- lem 8. PROBLEM 10. To pass the circumference of a circle through three points, A, B, C. Draw the lines A B and B C. Bisect A B and B C by perpen- diculars, by problem 1. With D, the intersection of these perpen- The reverse of this problem can be solved, i. e.. to find the centre of a given circle, by choosing the three points A, B, and C on the circumfer- ence of the circle. 14 cliculars. as a mitre, and I) A as a radius, describe a circumfer- i-iK-e. it will pass through A, H and C.* PKOI.I.:M 1 1. To draw two tangents to a circle, whose centre is ( ), from a point A, without the circle. loin A with (); on O A as diameter, describe a circle. Join the points B and C. in which the latter circle intersects the given one, with A. A B and A C will be the required tangents. PKOHI.I-: M 1 2. To draw circles, with given radii c and d. tan- gent internally and externally resj>ectively to a circle whose centre is ( , at a given point A. First, internally. Join A with O. Lay off on A O from A, A a equal to .-. From a as a centre, and Aa as a radius, describe a circle. Second, externally. Prolong OA, and lay off from A, Ab equal to d. With b as a centre and A b as a radius, describe a circle. PROBLEM 13. To draw a circle with a given radius m, tan- gent to two given lines AB and AC. Draw a 1> and ac parallel to A B and A C respectively, and at a distance from them equal to m. With the point of intersection a. of a b and a c as a centre, and m as a radius, describe a circle. PKOHI.KM 14-. To draw a circle with a given radius m, tan- gcnt to a given circle O, and to a given line, A B. With ( ) as a centre and O a equal to m plus the radius of the given circle, as a radius, describe an arc, a b. Draw the line C c parallel to A B, and at a distance m from it, by problem 7. With C, the intersection of the arc and parallel, as a centre, and m as a radius, dcscrilie a circle. PROBLEM 15. To draw a circle tangent to a given circle O, at a given point A, and to a given line BC. Join OA; at A draw the tangent AB, perpendicular to OA, and produce it till it meets BC at B. Bisect the angle A B C, by the line Ba, by problem 8. Produce B a till it meets O A pro- duced in D. With D as center, and DA. a radius, describe a circle. PROBLEM 16. To draw a circle, tangent to a given line AB, at a given point C, and to a given circle O. At C, draw DC, perpendicular to A B, by problem 5, and pro- duce DC below AB till Ca is equal to the radius of the given cir- cle. Join a with O, and by problem 1, erect a perpendicular D b at the middle point of Oa. With the intersection D, of Da and Db, as a centre, describe a circle. I'KOHI.KM 17. To draw a circle tangent to a given circle C, at a given point A, and to a given circle O. .J "5 15 Join AC, and produce it till AD is equal to the radius of the other circle. Join D with O, and bisect OD by a perpendicular Ea, by problem 1. With E, the intersection of Ea and AD produced as a centre, and E A as a radius, describe a circle. PROBLEM 18. Given two parallels, B A and CD, to draw re- versed curve which shall be tangent to them at A and C. Join AC. Bisect AC in 2, which will be the reversing point. Bisect B 2, and 2 C by perpendiculars, 1 F and 3 E. Draw AF and C E perpendicular to B A and C D, and with the intersection E of C E and 3 E, and the intersection F of 1 F and F A, as cen- tres, and radii, equal to EC, or AF, describe two arcs. PROBLEM 19. Given two parallels B A and CD, to draw a re- versed curve, whose tangents at A and C shall be perpendicular to B A and CD. Join AC. Bisect AC in 2, wiich will be the reversing point. Bisect A 2, and 2 C by perpendiculars 1 E and 3F. With the in- tersection D, of 1 E and C A. and the intersection F, of 3 F and D C, as centres, and radii equal to EA or FC, describe two arcs. PROBLEM 20. Given a circle O, to inscribe and circumscribe regular hexagons, and to inscribe a regular triangle. Lay off the radius O A, six times as a chord on the circum- ference. For the circumscribed hexagon, draw parallels to the sides of the inscribed figure, which shall be tangent to the circle. Join the alternate points of division of the circle for the in- scribed regular triangle. PROBLEM 21. Given a circle O, to inscribe and circumscribe regular octagons. Find first the sides A B and C of an inscribed square, by con- necting the ends of two diameters at right angles to each other. Bisect these chords by perpendiculars, and thus the arcs sub- tended by them. Join the points C, etc., with the vertices of the square for the inscribed octagon. For the circumscribed octa- gon, proceed as in circumscribing the regular hexagon in problem 20. PROBLEM 22. To construct a regular polygon with a given number of sides, five for instance, the sides to be of a given length AB. On A B as a radius describe a semi-circle. Divide the semi-cir- cumference into five equal parts, A 4, 4 3, and so on. Omitting one point of division 1, draw radii through the remaining points and produce them. With 2 as a centre, and A B as a radius, describe an arc cutting B 3 produced in C; B 2 and 2 C will be two sides of the polygon. With C as a centre, and A B as a radius, 16 descrilx- a., arc cutting B 4 produced in D; C D will be another side. Continue this construction; the last point should come at A. PROBLEM 23. On A B and C D as major and minor axis, to construct an ellipse. We proceed on the principle that the sum of the distances of any point of an ellipse from the foci is equal to the major axis. \\\ must first fix the position of the foci. From C as a centre, and O B as a radius, descrilx? two arcs, cutting A B in a and b, these are the foci. To apply the principle just mentioned, take the dis- tance from any point, as c of A B to A and B as radii, and a and b as centres. By describing arcs above and below A B, and using both radii from each centre, four points of the ellipse will be obtained. Other points are obtained by taking other points on A B, and proceeding in the same way. Connect the points found in this way by using the irregular curve. In using the irregular curve always l>e sure to have it pass through at least three points. I'KOHLKM 24. On A B and C D as a major and minor axis, to construct an ellipse; another method. On the straight edge of a slip of card board or paper, set off three points, o, c, a, the distance o a being equal to the given semi- major axis, and o c to the semi-minor. Place the slip in various positions such that a shall always rest on the minor, and c on the major axis. The various positions marked by the point o will be points of the ellipse. PKOIU.KM U~>.- To construct a parabola, given the focal dis- tance O E. We proceed on the principle that the distance of any point from a line A B, called the directrix, is equal to its distance from a certain point called the focus. Draw the indefinite line A B, for the directrix, and C D perpendicular to it. From C, lay off C E and E O each equal to the focal distance. The point O is the focus. Draw a number of perpendiculars to C D at various points. To find the points in which the parabola intersects any one of them as a a', describe an arc with O as a centre, and a C,the distance from that perpendicular to C as a radius. E is the point of C D through which the curve will pas>. PROBLEM 26. To construct an hyperbola, having given the distances AO and a O, on the horizontal axis, from the centre to either vertex, and from the centre to either focus. In the hyperbola, the difference of the distances of any point from the foci is equal to the distance between the vertices, as in the ellipse the sum. Lay off from O the equal distances A O and < > B to the vertices, and the equal distances O a and O b to the 17 foci. To obtain any point of the curve, take any point on the axis as c; with c A and c B as radii, and a and b as centres, descri1>e four pair of intersecting arcs, as in the ellipse ; the points of inter- section will be points of the hyperbola. By taking other points on the axis, other points of the curve will be obtained in the same manner. PROBLEM 27. To construct a curve similar to a given curve B A C, and reduced in a given ratio, one-half for instance. Draw some indefinite line, a centre-line, if possible, in the given curve, as A D. On A D, lay off a number of distances ; at the points of division, erect perpendiculars to meet the curve above and below A D. Draw an indefinite line a d, and on it lay off dis- tances bearing respectively to those laid off on A D, the given ratio. Through these points of division draw perpendiculars, and lay off on them above and below a d, distances bearing the given ratio to those on the perpendiculars to A D. PROBLEM 28. To describe a given number of circles, six, for instance, within a given circle O, tangent to each other and to the given circle. Divide the circumference of the given circle into twice as many parts as the number of circles to be describe, 1 2, 2 3, etc. To ob- tain the first circle, draw a tangent to the given circle at 1, la; produce O 2 1 a, at a. Lay off on O a, from a inwards the distance 1 a to b, since tangents to a circle are of equal length. At b draw a perpendicular to Oa, meeting O 1 in c. With c as a centre and c 1 as a radius, describe a circle. From O as a centre and O c as a radius, describe a circle intersecting the alternate radii O 3, O 5, etc.. in points which will be centres of the required circles. PROBLEM 29. To construct a mean proportional to two giv- en lines A D and D B. With the sum of these lines as a diameter, describe a semicircle A C B. At the point D, between the two lines, erect a perpendic- ular, meeting the circumference at C. DC will be the mean pro- portional required. PROBLEM 30. To divide a line a b into the same proportional parts as a given line A B is divided by the point C. Draw a b par- allel to A B, and draw lines through A and a, B and b, till they meet in d. Draw O d, the point c will divide a b into the same proportional parts as C divides A B. 18 PROJECTIONS. If we wish to represent a solid body by drawing, and, at the same time, to show the true dimensions of that body, we must have two or more views, or projections, of it on as many different planes. Take for example a cube. In order to show it in a draw- ing, we must have views of more than one face, in order to show that the body has three dimensions. We will consider the cube to be behind one plate of glass and below another, and in such a posi- tion that two of its faces are parallel to these plates, which are respectively vertical and horizontal. Now suppose that perpen- diculars are dropped from ever\- corner of the cube to each of these plates. The points where these perpendiculars pierce the surface of the plates, are called, respectively, the vertical and horizontal projections of the corners of the cube. If these points be joined bv lines, corresponding to the edges of the cube, we shall have in this case, exact figures of the two faces of the cube that are paral- lel to these plates. These two figures are called, respectively, the vertical and horizontal projections of the cube, accord- ing as they are on the vertical or horizontal plates. In this way we may get two views of any solid object, supposing it to be in such a position as that of the cube, in the case just noticed, with reference to two plates of glass, which we will now call the vertical and horizontal planes of projection. If the object has a third side very different from the two shown in this way, we may consider it to Tie projected on a third plane perpendicular to the two others, and on the side of the object to be represented. A fourth side may in the same way be represented on a fourth plane; but three projections are generally all that are needed to show even very complicated objects ; and in most cases two pro- jections, a vertical and a horizontal, are all that are necessarv, lines on the opposite faces being shown by dotted lines on the pro- jections of the faces toward the planes.* We have considered the planes of projection to be in front and above the object to to be represented, but drawings are often made with the planes behind and below. It is. however, believed that the method given is better for practical use. Details are often shown as projected on oblique planes, as planes parallel or at right angles to the axis of an ob- ject. 19 As it is not convenient to have two or more separate draw- ings of an object on different planes, as would be necessary if we were to represent the projections of the body in their true posi- tions; we may consider that the body has been projected in the manner indicated, and that the planes of projection have been revolved about their interesctions so as to bring them all into the horizontal plane, with the end views, if any, on the right and left of the vertical projections, and the horizontal projection above the vertical. In this way we may bring all the different views or projections into the plane of the top surface of the drawing paper ; and by representing the intersections of the planes of projections by lines, we may show all the projections in their true relative position in one drawing. The line that represents the intersection of the vertical and horizontal planes of projection, is called the ground line. The ground line, as well as the other lines of intersection of the planes of projection, is often omitted in actual drawings. It will appear on consideration of the method of projection that the distances of the projections of any point from the ground line show the true position of the point in space, with reference to the planes of projection. Suppose, for example, the horizontal projection of a point to be one inch above the ground line, and the vertical projections to be two inches below the same line, this shows that the true position of the point in space is one inch back of the vertical plane and two inches below the horizontal plane. Moreover, it may easily be demonstrated that the two projections of a point always lie in a common perpendicular to the ground line. As lines are determined by locating points in them, the princi- ples just given applj r in getting the projections of any figure that can be represented by lines. In the problems in projection, follow- ing, the ground lines must be drawn and the points located in the manner just indicated. An object may be in any position whatever with reference to the planes of projection ; but for convenience the body is usually considered to be in such a position that the vertical projection will show the most important view of the object, such, for example, as the front of a building. In representing an object of this kind in projection, the front of the object is usually considered parallel to the vertical plane of projection. The vertical projection of an object is called its elevation, and the horizontal projection, its plan. The other projections are called end views, or sections, according as they represent an end or some part cut by a plane passing through the object. 20 Hy the methods of projection just explained each projection represents the view of the object a person would have were the eye placed on the side of the object represented by the projection and at an infinite distance from it. When an object is viewed from a finite distance it is seen in j>erspective and not as it really is. Projections show an object as it really is, and not as it appears in perspective. Projections are therefore used to represent bodies in their, true form and are employed as working drawings, in which a body to be constructed is represented as it would appear in projection when finished. LINE SHADING. In order to give the projection of a bod\- the appearance of relief the light is supposed to come from some particular direction, and all lines that separate light faces from dark ones are made heavy. The direction of the light is generally taken for convenience at an angle of 45 degrees from over the left shoulder as the person would stand in viewing the projections, or in making the draw- ing ; and in all cases the projections of this book are to be shaded with the light so taken. It will be readily seen on considering the direction of the light, that the elevation of a solid rectangular object with plane faces in the common position, will have heavy lines at the lower and right hand sides, and that the plan will have heavy lines on the upper and right hand sides. When a body is in an oblique position with reference to the planes of projection, the heavy lines of the projections may be determined by using the forty -five degree triangle on the T-square. If we apply this triangle to the T-square, so that one of its edges inclines to the T-square at an angle of forty -five degrees upwards, and to the right this edge will represent the horizontal projection of a ray of light: and by noticing what lines in the plan of the object this line crosses, it may readily lie seen what faces in the deration will receive the light and what faces will be in the shade. By applying the triangle so that an edge will make an angle of forty-five degrees downward and to the right, this edge will represent the vertical projection of a ray of light, and by applying it to the elevation, the faces in the plan that will IK: in light and shade may lie determined. Where the limiting line of projection is an element of a curved surface as in the elevation of a vertical cylinder, that line should not be shaded. The plan of the vertical cylinder, which is a circle, should be shaded, for the circumference is an edge separating light from dark portions of the object. In this case 21 the darkest shade of the line should be where the diameter, that makes an angle of forty-five degrees to the right with the T-square cuts the circumference above and the lightest part should be where this diameter cuts the circumference below. The dark part should taper gradually into the light part. Lines that separate parts of a body that are flush with each other, as in joints, should never be shaded, and when a line that would otherwise be shaded, rests on a horizontal plane as in the first positions of the following problem, it should not be shaded. The shaded lines of a projection need not be very heavy if the light lines are made, as they should be, as fine as possible. THE PLATE OF PROJECTIONS. The plate of projections is to be of the same size as those of geometrical problems, and will contain twelve problems in the position shown in the cut. The border should be drawn first, leaving a margin of one and a quarter inches at the top as in the previous plates. Divide next the space within the border line in- to twelve equal rectangles, by drawing five vertical and one hori- zontal lines. Draw a ground line in each rectangle two inches and a half below the top, making the ends of the ground lines within a quarter of an inch of the vertical lines dividing the space. At a distance of two inches and three-quarters below the ground line draw a broken horizontal line, as shown in the plate. The objects projected in the following problems are all sup- posed to rest on a horizontal plane below the horizontal plane of projection and behind the vertical plane of projection. The ground line, as has already been noticed, represents the intersection of the two planes of projection before the vertical plane is revolved into the plane of the horizontal. The broken line below represents the intersection of the vertical plane with the plane on which the body rests. In order to get a good conception of the position of the object suppose that the body rests on the drawing table, and that a plate glass be held above and parallel to the table, and another plate be held in front and vertical. The position of the object in relation to the planes represented by the plates of glass will be the same as that of the cube, which we considered in explaining pro- jections in general. The ground line and the broken line below will represent in this case, respectively, the intersections of the plates of glass with each other and with the top surface of the drawing table. A sheet of paper may now be put in place of the horizontal plate glass, and it will represent the revolved position of the planes precisely as they are in the drawing. 22 In the descriptions of the problems, G L refers to the ground line, or intersection of the planes of projection, and G' L' refers to the line of intersection of the vertical plane of projection with the plane on which the body rests. The plate must be lettered to show the general title, and the number of problems, as shown in Plate III. The letters used in describing the problems, however, need not be drawn. All lines that would not be seen from the position indicated by the projection in question, must be indicated by fine dots. The problems in the finished plate must be shaded according to the di- rections above. PROBLEMS IN PROJECTION. PROBLEM 1. To construct the projections of a prism one and a quarter inches square at the base and two and a quarter inches in height, of whose laces one rests on a horizontal plane, and one is parallel to the vertical plane of projection. Draw the square, A B C D, equal to the top face of the prism, above G L, with C D one-quarter of an inch from G L and parallel to it. Draw from C and D lines perpendicular to G L, and prolong them lielow until they intersect G' L'. Measure off the height of the prism from G' L', and draw a horizontal line for the top line of the elevation. The rectangle, E F G H, formed below the line last drawn will be the elevation of the prism, and the square above G L will be the plan. No dotted lines will api>ear in this problem, as all the lines on the opposite sides of the object will 1)e covered by the full lines in front. Shade according to the directions above. PROBLEM 2. To revolve the prism of Problem 1 through a given angle about an edge through H, so that the planes parallel to the vertical plane shall remain so. Locate H on G L as far to the right in its rectangle, as H, in Problem 1 , is from the border line. As the revolution has been parallel to the vertical plane, the elevation will be unchanged in form and dimensions, but will be inclined to G L. Lay off G H, making the given angle of revolution with G' L.' Complete E F G H, on G H as base. Since the body has revolved parallel to the vertical plane, the horizontal projections of lines perpendicular to the vertical plane as A C and B D, have not changed in length, but those of lines parallel to the vertical plane, as A B and C D, will be shortened. Consider these facts, and that the two projections of a point are in the same line perpendicular to G L, the following is seen to be the construction of the plan : Prolong A B and C D of Problem 1, indefinitely to the right. As E F G H represent the same points in both elevations, erect M 100 65 CVJ. Or 23 perpendiculars from each of these points in Problem 2, intersecting the indefinite lines just drawn, for the plans of the same points. A B C D and I J K L will be the bases of the prism in its revolved position. K L is to be dotted because not seen. Shade as directed above.* PROBLEM 3. To revolve the prism, as seen in its last position, through a horizontal angle, that is about a line through H, per- pendicular to the horizontal plane. As the revolution is parallel to the horizontal plane, the plan is changed only in position, not in form or dimensions. Therefore, draw the plan of Problem 2 inclined to G L at the angle of revolution, taking L in L D produced from Problem 2. Now as each point of the body revolves in a horizontal plane, its vertical projection will move in a straight line parallel to G L. Hence we make the following construction for the elevation: In the case of any point, as B, in the plan, draw a perpendicular from this point to G L, and from F, which is the elevation of B in Problem 2, draw an indefinite line parallel to G. L. The intersec- tion of these lines gives the elevation of the point in its revolved position. Proceed in the same way with all the other points. It will be noticed that in the three positions of the body just taken the plan is drawn first in Problem 1, the elevation first in Problem 2, and the plan first in Problem 3. The reasons for so proceeding are evident from the constructions. This order will hold true in all the problems following. PROBLEM 4. To construct the projections of a regular hexa- gonal pyramid, and the projections of a section of that pyramid made by a plane which is perpendicular to the vertical plane. The height of the pyramid is to be the same as that of the prism in Problem 1, and the diameter ol the circumscribing circle of the base is to be two inches. The part of the pyramid above the section is to be represented by dotted lines and the lower part, or frustum, in full lines. To find the projections of the pyramid, draw a regular hex- agon, A B C D E P, above G L, with the lines joining the opposite vertices for the plan of the p3'ramid ; draw perpendiculars from the vertices G L. The intersections of these perpendiculars with G L will be the elevations of the corners of the base. Erect a per- pendicular from the center of the elevation of the base, and on it Nothing more will be said about shading, but it is to be understood that each projection is to be shaded, with the light taken from over the left shoulder at an angle of forty-five degress with the horizontal plane. The shading is a very important point in the problems. 24 measure off the vertical height to H, join Hwith the points on the base for the elevations of the edges. To draw the projections of the section, assume L K making an angle with G L,and cutting the pyramid as shown in the plate. Draw M N perpendicular to G L, and vertically above L. K L represents the intersection of the cutting plane with the vertical plane; and M N, its intersection, with the horizontal plane. The lines are called the traces of the cutting plane, and must be represented by broken lines like G L. The elevation of the sec- tion will be the part of K L included between the limiting edges of the pyramid, and is to be shown a lull line. The plan of the section is found by erecting perpendiculars to G L from the points where K L cuts the elevations of the edges of the pyramid, and by finding where these perpendiculars intersect the plans of the same edges. T represents the plan and elevation of one point in the section. PROBLEM 5. To revolve the frustum of the pyramid in Prob- lem 4 through a vertical angle about J. Draw S I equal to the same line in Problem 4, and making a given angle with G L. Construct, on S I, an elevation like the one in Problem 4. Each point in the plan may be found, as in the second poblem of the prisms, by erecting perpendiculars from the points on the elevation, and finding their intersections with horizontals drawn from the plans of the same points in Problem 4. T shows the plan and elevation of the point T in Problem 4. PROBLEM 6. To revolve the frustum of Problem 5 through a given horizontal angle. Draw the plan like that of Problem 5, but making the given angle of revolution with G L. Each point in the elevation may be found by drawing perpen- diculars and horizontals respectively, from points in the plan, and trom corresponding points in the elevation of Problem 5. The revolution of the point T is indicated by the dotted lines. PROBLEM 7. To construct the projections of a regular octa- gonal prism, one of whose bases is in a horizontal plane. Lay out a regular octagon, one inch and three-quarters be- tween parallel sides, for the plan ; and from its vertices draw ver- ticals to G L, and produce them below G L, until they intersect G L, for the vertical edges of the prism. Makefile top line of the elevation three-eighths of an inch below G L. PROBLEM 8. To revolve the prism of Problem 7, parallel to the vertical plane, through a given angle. 25 Construct, as usual, the vertical projection, differing only in position from that in Problem 7. In the case of any point as T, to find its plan, erect a perpendicular to G L from the elevation of the point and draw a horizontal from the corresponding point in the plan of Problem 7. The bases of the prism in this position will be equal octagons though not regular. PROBLEM 9. To revolve the prism of Problem 8, parallel to the vertical plane, through a given angle. Draw the plan like that in Problem 8, making the given angle of revolution with G L. In the case of each point, to obtain the elevation, drop a vertical from the plan of the point, and draw a horizontal from the corresponding point in the elevation of Prob- lem 8. The bases of the elevation will be equal octagons, in which the parallel sides are equal. The point T is the same point that is marked in Problems 7 and 8. PROBLEM 11. To construct the projections of the frustum of a right cone, whose base is in a horizontal plane. Draw a circle of a radius of seven-eighths of an inch for the plan of the base. Drop verticals from the right and lefthand limits of this circle intersecting G'L/, for the elevation of the base. Drop another vertical from the center of the circle, and on it measure the same length from G' L/ as that of the prism of Prob. 7 above. This will give the elevation of the apex of the complete cone, the plan of which is the center of the circle above. Join this point with the two ends of the elevation of the base for the limit- ing elements in the elevation. The upper base of the frustum in this case, is formed by a plane, cutting the cylinder, perpendicular to the vertical plane of projection, and making an angle with the horizontal plane. The cutting plane is given by its trace on the plate. This upper base will be an ellipse, as is every section of cone made by a plane that does not cut the base of the cone. The elevation of this upper base will evidently be that part of the vertical trace of the cutting plane included between the limit- ing elements. To get the plan of this base, proceed as follows: Divide the straight line, representing the elevation of this base, into any number of equal parts, and through these points of division draw horizontals as shown on the plate. The distances of these points from the axis of the cone are evi- dently equal to the lengths of the horizontals drawn through these points in the elevation, and limited by the axis and the limit- ing elements. Hence, to get any point, like A, in the plan, erect a 26 vertical from the elevation of that point ; and with O the plan of any point in the axis, as a centre, and the horizontal M X through the point in the elevation as a radius, describe arcs intersecting the vertical. A and the point above it, are both found by using the same radius, getting the intersections above and below with the vertical from the point in the elevation. PROBLEM 11. To revolve the frustum of Problem 10 through a given angle parallel to the vertical plane. Construct, as in all the preceding similar cases, the elevation of the preceding problem, making the given angle with G L. Get the plan of the lower base by locating a sufficient num- ber of points in the same way that the corners of the polygon are located in Problem 5. To get the plan of the upper base, erect verticals from points in the elevation corresponding to those marked in Problem 10, and find the intersections of horizontals from the plans of the same points in Problem 10. In case the frustum were turned through a larger angle than that shown in the plate, limiting elements would show tangent to the two ellipses in the plan. PROBLEM 12. To revolve the frustum of Problem 11 through a given horizontal angle. Construct the plan of Problem 11; making the given angle with G L, in a manner similar to that employed in .constructing the plan of pyramid in Problem 9. Get the points in the two bases b\" dropping verticals from the points in the plan, and finding the intersections of horizontals from corresponding points in the elevation of Problem 11. Draw the limiting tangent to the ellipses found in this way. The orthographic projections of an ellipse are always ellipses, the circle and the straight line being special cases of ellipses. PRACTICAL APPLICATIONS OF PROJECTIONS. Roof Truss. ''Scale one half an inch to the foot. In this case, the two sides being symmetrical, an elevation of one half and a section through A B, will show every part, and are therefore chosen as the best views for the working drawing. First, lay off a dotted horizontal line B C, which is t wen ty- *The scale of a drawing is the ratio that the lines on the drawing bear to the actual lengths of the lines on the object. The scale should always be stated on a drawing; and may be given as a fraction like \ y , 1/4, etc., or it may be stated as a certain number of inches to the foot. ..n . 27 five feet in length. Then at E erect the vertical D E making E B eight feet and E D nine feet, reducing eacli to the proper scale. Join C and D. These lines are the center lines of the principal timbers in the truss and of the iron rod D E. Draw next the timber of which C B is the centre, fifteen inches deep, leaving the end near C unfinished until the rafter is drawn. The other two timbers of which the centre lines have been drawn are twelve inches deep. Draw the lines parallel to the center lines. The line forming the joint at D is found by joining the intersec- tions of the lines of the timbers. The joint above C is formed by cutting in three inches in a direction perpendicular to the upper edge of C D and joining the end of this perpendicular to the inter- section of the lower line of C D \vith the upper line of C B. The hatched pieces near D, H and C, are sections of the pur- lins, long pieces resting on the truss and supporting the rafters. These purlins are cut into the rafters and into the truss one inch, with the exception that the one near C is not cut into the truss. These should next be drawn, rectangles ten inches by six inches. The center line of the one near C is a prolongation of the short line of the joint at C, the one near D is three inches below the joint at the top, and the third is half way between the other two. The center line of H E should start from the lower part of the center line of the middle purlin, and the top edge should meet the top edge of C B in D E. With one half the depth of H E, four inches, as a radius, describe an arc with the point where D E meets the upper line of C B as a center. Draw the center line through the point indicated above, and tangent to this arc. Cut in at E one third the depth of H E, or two and two-thirds inches and, at H, one inch, in the way shown in the plate. Draw next the rafter, F A, twelve inches deep and eight inches from the top of C D. Cut into the rafter a horizontal distance of six inches for the end of the beam C B, and make the end of the rafter in line with the bottom of C B. Make a short end of the rafter on the other side as shown at A, and show the horizontal pieces broken off as shown in the plate. Make the rod, D E, an inch and a half in diameter with washer and nut at the lower end, and with a head and an angle plate running to the purlin at the upper. Make the washer and angle plate one inch thick, the washer four inches in diameter and the nut and head according to the standard. The short bolt near E is one inch in diameter and the head, nut and washer of two- thirds the size those on in D E. The short bolt near C is precisely like the one just described, make the angular washer at the bottom of the same diameter and at right angles to the bolts as in the other washers. 28 The sectional view at the right is formed by projecting lines across from the elevation just drawn and measuring off the prop- er widths. The timbers of the truss proper are all twelve inches wide, the rafter is three inches wide and the purlins are broken off so as to show about two feet in length of each. Great care must be taken in inking not to cross lines, and those nearest the observer in any view should be full lines, and those hidden by them should be broken off or dotted. Hatching, as it is called, is a method of representing a surface cut, as in a section, and it is done by drawing fine parallel lines at an angle of forty-five degrees with the vertical and very near together. In case two different pieces joining are cut, the lines, should be at right angles to distinguish the two sections. There are no such cases in this example however. It is very important that a hatched surface shall look even, and this can only be effected by making all lines of the same width, and the same distance apart. The title, which is given in italics at the beginning of this de- scription, may be placed within the border as indicated. The student will be allowed to choose a.ny mechanical letters for the title, but the heights rmist be three-eighths of an inch in the words Roof Truss, which must be in capitals, and the letters in the words indicating the scale are to be one-half as high. The scale should be put on thus: Scale Vi" ~ 1', one dash indicating feet and two dashes inches. Stub End of a Connecting Rod. (One-half Size.) The projections chosen to represent this object are a front ele- vation, and a side elevation. In this case these projections show the different parts much more clearly than the}' could be shown in plan and elevation. The shaded portion on the front elevation shows what would be seen if the brasses were cut away on the line D E of the side ele- vation, this is done to show the lining of babbit or white metal. This example illustrates the necessity of hatching to distin- guish the cut portions from those beyond. It also shows the proper method of representing the different pieces, shown in sec- tion, by lines running in different directions on adjacent pieces. The dimensions to be used on the full sized drawing are marked in inches on the cut. Remember that the drawing is to be one- half size. The arrowheads on either side of the dimension marked LJ I- J d 29 represent the limits of the dimension. It will be noticed that some of the dimensions at the top are diameters, whilst others are radii. Divide the space within the border on a half-sheet of Imperial by two vertical lines making three equal spaces. Use these lines as centre lines of the two projections. Draw the front elevation first. Commence by assuming on the line A B, the centre C of the inner circle, near the top, and describe the inner circle, about this center with the given radius. The center of this circle should be taken far enough below the border at the top to leave about the same amount of margin above and below when the elevation is completed. The second pair of half circles it will be seen, are 1 /4" at top and bottom from the inner circle and %" at either side, by trial a centre for the upper one can easily be found on the line A B, below C, and for the lower one, above C. All other arcs are drawn with C as centre. The front elevation should be drawn first, as most of the lines of the side elevation are obtained by projection from it. The curves at the bottom of each elevation are arbitrary and only need to be tangent to the lower lines in about the propor- tions shown. The horizontal lines in the side elevation may be projected from corresponding points in the front elevation. All the horizontal distances are indicated on the side elevation, but the vertical distances, being the same as in front elevation, are purposely omitted. The most difficult part of the work in this drawing is to make the hatching even. Use a sharp pen and make all the lines of the same width and the same distance apart. This drawing should be line shaded according to previous directions. It will be seen that the side elevation which represents the left side of the object is placed at the right. This is according to the method of projection in which the glass or vertical plane is behind the object instead of in front. The latter is generally considered better. Projections of Screws. The thread of a screw may be considered to be generated by a section moving uniformly around a cylinder, and at the same time uniformly in a direction parallel to the axis of the cylinder. Plate VI shows the true projections of a V-threaded screw at the left, and of a square threaded screw at the right. V-TuKEADED SCREW. Commence by describing a semi-circle with a radius of one inch and a half, as shown in the outer dotted circle in the plan. This will l>e the half plan of the tfuter part of the thread. Drop verticals from the outer limits of the semi- circle for the limiting lines of the V threads in the elevation. The 30 projections of the head and an outline of a section of the nut should next be drawn. The standard dimensions of the heads and nuts are expressed by the following formulae, in which d is the out- side diameter of the screw, h the thickness of the head or nut and D the distance between the parallel sides of the head or nut: D= Construct the projections of the heads and nuts according to this standard, and show the hexagonal head finished as in the plate. The short arcs that cut off the corners are described with the middle of the lower line of the nut as a center, and the longer arcs bounding the top faces of the head are described with the middle point of the lower line of each face as a center. The top of the head is a circle, as shown in the plan. A section of a standard V-thread is an equilateral triangle all the angles of which are sixty degrees, hence the outlines of the sides of the elevation may be drawn by means of the thirty de- gree triangle used on the T-square. Before drawing these triangles, however, the pitch must be de- termined. The pitch of a screw is the distance from any point on a thread to another point on the same thread on a line parallel to the axis. The pitch is usuall}' expressed by stating the number of threads to the inch. This screw has two threads to the inch, therefore the pitch is one-half an inch. This i-epresents the ad- vance in the direction of the axis during one revolution. Lay off, then, on the limiting line at the left, distances of one- half an inch, commencing at the bottom of the head. Through these points draw lines as indicated above, making a series of triangles. The inner intersections of these lines will be in a vertical line, which, projected up, gives the radius of the inner dotted semicircle in the plan. The semi-circle is a half plan of the bases of the threads. As the thread advances a distance equal to the pitch in a whole revolution, is evident that in a half revolution the advance will be equal to half the pitch; therefore commence on the right hand limiting line with the first space a quarter of an inch, and from this point on, make the spaces equal to the pitch. Describes a series of triangles on this side in the same way as be- fore. Every point in the generating triangle describes a helix as it revolves about and at the same time moves in the direction of the axis of the screw. It is evident that the helices described by the vertices of the triangle will be the edges of the intersections of the threads. The manner of getting the projections of these lines will be described. The plans of these helices will be the circles which have just been obtained and which are shown in the plate in dotted 31 lines. Draw from the outer vertex of one of the triangles repre- senting the edges of the threads, an indefinite line toward the left as shown in the plate. Divide the semi-circle above into a number of equal spaces, eight at least, and draw radii to these points of division. Lay off the same number of equal divisions on the in- definite line, and at the last point erect a perpendicular equal in length to one-half the pitch. Join the end of this line with the right hand end of the horizontal line, forming a triangle. Erect verticals from each point of the division of the horizontal line. To find any point, like A, in the helix forming the edge of the threads, drop a vertical from one of the divisions of the semi-circle, and find where it intersects a horizontal drawn from a corresponding point on the diagonal line of the triangle at the left, counting the same number of spaces from the right on the diagonal line as the point taken on the semi-circles is from the left. As many points may be found this way as there are on the semi- circle. Join these points by using the irregular curve. The points in the helices at the bases of the threads may be found in the same way as shown by the dotted lines, the equal divisions of the semi-circle in this case being where the radii of the center circle cut this one. The reason of this construction will be plain on considering that the equal spaces on the arc, represent equal angles of revolution of the generating triangle; and the dis- tances between the horizontals drawn from the points of division of the diagonal line, represent the equal rates of advance in the direction of the axis. As the curves at the edges of the different threads are all alike, a pattern should be made, from thin wood, of the one constructed, and this should be used to mark all the long curves of the screw and nut. Another curve should be made for the inner helices. The helices will evidently be continuous from one end of the screw to the other, but the dotted lines which would show the parts on the back side are left out in order that the drawing may not be confused by too many lines. In the plate the screw is shown as entering only a short dis- tance into the nut which is shown in section below. The threads of the nut are the exact counter parts of the threads of the screw; but as the threads on the back side of the nut are shown in this section, the curves run in the opposite direc- tion. A small cylindrical end is shown on the bottom of the screw. This represents the end of the cylinder on which the thread is wound. Square Threaded Screws. The square threaded screw is gen- erated by a square revolving about the cylinder and at the same 32 time moving in a direction parallel to the axis. In the square single threaded screw the pitch is equal to the width of a space and the thickness of a thread, measured in a direction parallel to the axis. Draw the projections of the head and nut of the same dimen- sions as in the V-threaded screw. Lay off a series of squares, the sides of which are equal to one-half the pitch, on the two edges of the screw, and find the points in the helices as in the example pre- ceding. It should be observed that the long curves show in their full lengths, and the short ones only show to the center in the screws, whilst in the nut the opposite is true. The smaller screws near the center of the plate, show how V and square threaded screws are after represented when so small that the construction of the helices is impracticable. The con- struction only varies from the larger ones inasmuch as the curves are replaced by straight lines. Below there is shown still another method of representing very small screws, either V or square threaded, and the projection of a hexagonal head with face parallel to the plane of projection. It is customary among draughtsmen to represent all threaded screws, unless very large, by fine lines across the bolt representing the points, and shorter, heavier lines between, representing the hollows of the threads. Below is given a table of the Franklin Institute, or United States standard proportions for screw threads. This table is given here that it may be conveniently referred to whenever screws and nuts are to be drawn. A real V thread is often used, but a thread very similar, having a small flat part, in section, at the outside of each thread and a similar flat part between the threads, is becoming more common. The dimensions of such a thread are given in the following table, where diameter of screw means the outer diameter, diameter of core, the diameter of the cylinder on which the thread is wound, and width of flat, the width of the flat part just described. The four columns at the right relate to the nuts and bolt heads. PLATE VII. 33 PROPORTION OF SCREW THREADS, NUTS AND BOLT HEADS. SCRE NUTS AND HEADS U'S. HEXAGONAL. SQfARE. , "3 ! ^ Diameter Sere \\- . ? t "2 r-' 5 v.' -M . ' * .8 " % -t; 2 t; S= s s S 5O C ft s c 2 S a o Diagonal. Height t 4-~ Head. 1 i 20 .185 .0062 ,,.. 5 1 18 .240 .0070 },\ jy i i; 4 1 16 .294 .0078 sjj 11. :! 1 3 > 81 ,Y, 14 .344 ! .0089 \ |i -1 1 L 1 li Si i 13 .400 .0096 1 M r II 12 .454 .0104 1,.^ I;' 1 ft ill s 11 .507 .0113 I,-, I/,. 11 JI I 10 .620 .0125 I,',. 1] 1J s i 9 .731 ; .0140 Igi I,-,. 2A ii 1 8 .837 .0156 1J 1J 2 A 1 :t 1 i; n 7 .940 .0180 i 2 8 :l 1 } ;{ 21 I* VI 7 1.065 .0180 2, 5 ,, 2 9r -S-J 1 13 6 1.160 .0210 21 2, :! , ; 3 A IA n 6 1.284 .0210 2 : l 22 31 IA M 51 1.389 .0227 2};: L',", 3| 1A n 5 1.490 .0250 3, :! ,., 2:J 3;!! 12 ij 5 1.615 j .0250 3J g 2}^ *A US 2 *i 1.712 .0280 3g 3i 4lV. 1/6- -M 41 1.9<52 .0280 4, 1 ,., 31 4^1 I-- 21 4 2.175 .0310 41 3- r>i i>? 1 4 2.425 .0310 4|| 4J 6 21 3 31 2.628 .0357 5j{ 4^ fi !l 2 s i c; *^ ,-, 34 DRAWING FROM ROUGH SKETCHES. Plate VII is given to illustrate the method of making rough sketches of an object from which a finished drawing is to be made. The rough sketches here shown are of a large valve such as is used on large water pipes.* This example has been chosen because it is symmetrical with reference to the center line. In such a case as this, it is obviously unnecessary to make complete sketches of the whole object. Enough of the plan of the object is given above to make a complete plan from, in the drawing. The sketch of the elevation below shows all that is necessary for that. In making a rough sketch, decide what projections will best represent the object, and get in such a position as to see the object as nearly as possible as it will appear in the projection, changing the position of observation for the sketches of the different pro- jections. It must be borne in mind that the view a person has of an object while sketching is a perspective view and allowance must be made for the way it will appear in projection. Sketches similar to the projections are better than perspective sketches to work from. The sketches should be made in the same relative position that they will appear in the projection drawing. Be sure to represent every line of the object in the sketches, excepting the cases where symmetrical parts maybe drawn from a sketch of one part, atid indicate all the dimensions by plain figures and arrow heads, taking all the dimensions possible from some well defined line like a center line or a bottom line. If any of the details on the principal sketch are too small to contain the figures of the dimensions, make enlarged sketches aside from the other as indicated in the plate. All that is necessary to be known about a nut is the diameter of the bolt; the nut may then be constructed according to the standard. Often a few words of description written on the sketch as, in the case of a bolt, four threads to the inch, will describe a part sufficiently to one acquainted with the standard proportions of such common pieces as screw bolts, etc. One unaccustomed to making sketches, is apt to omit some di- mensions, and too great care cannot be taken to have everv part of the object clearly indicated in some way on the sketch. The student should make it a point to endeavor to get all necessarv measurements on his sketch before beginning work on his draw- ing, as in practice one is often sent many miles to obtain measure- ments to be worked up at home, and an omission may prove very expensive. *A complete drawing of such an object should show the internal parts, but as the object of this plate is simply to illustrate the method of sketching, the internal arrangement is not shown. 35 TINTING AND SHADING. At this point of the work the following additional materials will be needed : A set of water colors, a nest of cabinet saucers, a camels hair brush, a bottle of mucilage and brush, and a small glass for water. WATER COLORS. Winsor and Newton's water colors in " half pans" are recommended. They should contain the following colors Burnt Sienna, Raw Sienna, Crimson Lake, Gamboge, Burnt Umber, Indian Red and Prussian Blue. If the bottled ink has been used for the previous work, a stick of India ink will also need to be purchased. All the conventional colors used to represent the different materials may be mixed from the simple ones given in this list. SAUCERS. The "nest" should contain six medium sized sau- cers. BRUSH. The camel hair brush should be double ended, and should be of medium size. MUCILAGE. The mucilage needs to be very thick, as it is used in shrinking down the heavy drawing paper. The ordinary muci- lage in bottles is not fit for this use, and it is recommended that each person buy the Gum Arabic, and dissolve it in a bottle of water, using it as thick as it will run. WATER GLASS. This glass is for holding clean water with which the colors are mixed. Any small vessel will answer this purpose, but a small sized tumbler is the most convenient. DIRECTIONS FOR SHRINKING DOWN PAPER. Whenever a drawing is to be tinted it must be shrunk down in order that it may not wrinkle after tinting. To shrink down the paper proceed as follows : Lay the edge of the T-square paral- lel to an edge of the paper, and about five-eighths of an inch from it : and turn up the paper at right angles, making a sharp edge where the paper is bent up by pressing it hard against the edge of the T-square with the thumb nail or knife blade. Turn up all the edges in this wa}" so that the paper will resemble a shallow paper box. The corners need not be cut, through many draughtsmen cut 36 out a V shaped notch from each corner to save trouble in folding, but must be doubled over so that all the edges of the paper will stand nearly perpendicular. After this is done the paper should l>e turned over so as to rest on the upturned edges, and dampened ver3* slightly with a sponge on the back. Ever\ r part of the paper must be dampened except the upturned edges which must lie kept dry in order that the mucilage ma}- stick. No water should be left standing on the sheet when it is turned over. The paper should next be turned over and placed so that two edges at right angles may correspond, when turned down, to two edges of the drawing board. The other side should then be thor- oughly wet. The mucilage should next be applied to the dr- edges as rapidly as possible. The two edges that correspond to the edges of the drawing board should first be turned down, great care being taken to leave no wrinkles in these edges nor in the cor- ner between them. The other edges should then be turned down. the same care being taken to leave no wrinkles either in the edges or corners. The edges must be kept straight, and, if there are no wrinkles left in the edges, the paper will come down smooth when dry, no matter how much wrinkled while wet. The natural shrink- age of the paper is sufficient without stretching. Theedges should be pressed down smooth with the back of a knife or the thumb nail, and the paper should be allowed to drj* slowly. Paper should never be dried in the sun or in artificial heat as it will get too dry and afterwards become loose when exposed to ordinao* temper- ature. Considerable practice may be necessary before the pa]>er can be shrunk down successfully, but if the directions above are follow- ed closely there need be no difficulty. The paper must be dampen- ed evenly, and the mucilage must lie put on evenly and abundant- ly. Great care must be taken not to drop any mucilage on the middle of the drawing board, and not to get any beyond the dry edge of the paper. Otherwise the paper may be stuck down so as to make trouble in cutting the plate off when finished. THE PLATING IN TINTING AND SHADING. Plates A and B, which should l>e prepared by the Instructor and hung on the walls of the Drawing Room, contain the most common forms that are brought out by shading in ordinary me- chanical drawings and the most common conventional colors used in working drawings.* "These colored plates could not conveniently be placed in this pamphlet, and in cases where access cannot be had to the wall plates here mentioned, it is recommended that the instructor make similar ones for the use of the students. Plate A contains five rectangles in the upper ro\\, the first three of which are to be plain shades, and the other two 37 Plate A, which is shaded altogether with India ink is to be done first. Shrink down a half sheet of Imperial paper, and mark it in- side of the edges so that it may be, when cut off, twenty by thirteen and a half inches. Lay out a border one inch inside of the lines just drawn and draw the outlines of the figures with a very sharp pen, using the best of ink, and njaking the lines as fine as possible. The figures must be drawn of the same size, and arranged the same way as in the wall plates. The border and the letters, which are to correspond with those in the wall plates, should not be drawn until the plate is shaded. The dimensions of the figures need not be put on to the finish plates at all. After inking in the figures, the plate should be washed over with clean water to take out any surplus ink and to leave the paper in better condition for the water shades. The paper should be sopped very lightly with a sponge and a large quantity of water should be used. After washing the paper allow it to dry slowly. If the paper is dried in the sun it will get so warm that the shades will dry too rapidly. When the paper is down smooth and dry, it should be placed on the drawing table slightly in- clined in one direction in order that the ink or water color may always flow in one direction. Take a saucer half full of clean water and by rubbing the wet brush on the end of a stick of ink mix enough India ink to make a shade no darker than that in (c), on wall plate. A small piece of paper should be kept to try the shades on before putting them on the plate. Mix the ink thoroughly with one end of the brush before applying to the paper. One end of the brush should always be used in the ink or tint while the other end is kept clean for blending. With considerable ink in the brush but not nearlj' all it will hold, commence at the top line of (a), and follow it carefully with the first stroke. Before the ink dries at the top, lay on the ink below by moving the brush back and forth, using enough ink in the brush so that it will flow gradually, with the help of the brush, toward the bottom. The lines must be followed carefully at first, and the brush should not be used twice over the same place. In following a line w5th the brush get in such a position that the forearm will be perpendicu'ar to the direction of the line. are to be blended. The lower row of figures in this plate contains plans and elevations of the following figures in the order named, a prism, pyramid, cylinder, cone and sphere. Plate B contains, in the upper row, circular figures tinted to represent the conventions for cast-iron, wrought-iron, steel and brass , and in lower row four square figures with the conventional colors for copper, bricl<, stone and wood. 38 Do not paint the shades on but allow them to flow quite freely after the brush. In shading or tinting there is great danger of making clouded places and "water lines" unless the greatest of care is taken in using the brush. If the brush is used over a shade that is parti}- dry it will make it clouded. And if the edge of the shade is allowed to dry before finishing, a "water line" is produced where the new shade is joined to the old. In finishing up a figure the ink should be taken up with the brush so that it will not spread beyond the lines. The sun should never be allowed to shine on the paper, as it will dry it too fast. A damp day is better for tinting or shading than a dry one for the reason that the drying is then very slow. The shades of (a ), (b) and (c) are all plain. Commence on (a), and while it is dry- ing put a coat on (b). To determine when a shade is dry, look at it very obliquely, and if it does not glisten it is ready for another coat. Put four coats on( a), two on (b) and one on (c). BLENDING. A varying shade, such as is noticed in viewing a cylindrical object, may be obtained by blending with India ink. This operation of blending is emploj'ed in bringing out the forms of objects, as seen in the lower figures of plate A. The figures (d) and (e) are for practice in blending before ap- plying to the solid object below. Begin (d) by laying on a flat shade about an eight of an inch wide, using but little ink ; and when nearly dry take the other end of the brush, slightly moisten- ed in clean water, and run it along the lower edge of the shade blending downward. When this is entirely dry, lay on another plain shade a little wider than the first, and blend it downward in the same way. Use but little water and lay on the shade in strips, always commencing at the top line. When finished the lower part will have had but one coat whilst the upper part will have had several. Blend (c) in the same way as (d), but use nar- rower strips of tint in order to make more contrast between the top and bottom. SHADING SOLIDS. When a solid object is placed in a strong light coming principally from one direction, a strong contrast will be noticed between the shades of the different portions, and these shades serve to reveal the shape of the object much more clearly than when it is placed in diffused light only. For this reason, as well as from the fact that the laws of the shades of an object in light from one direction are very simple, the shades in a drawing are usually made to correspond to those of a body where the light comes from a single window. In all cases, however, it is assumed that there is a certain amount of diffused light, such as is always present in a room lighted by a single 39 window, aside from the strong beam of light that conies directly through the window. 1. The shades of an object arc always in greater contrast when the object is near the eye than far away. 2. The lightest portion of a cylinder, cone or sphere is where the direct light strikes the object perpendicularly, and the darkest portion of the same is where the light strikes tangent to the object, the shade varying gradually between these parts. The facts just given may easily be proved by holding a body in the light and noticing the shades. These facts we will assume as the principles that govern the shading of the following objects. In view of the above principles the first thing to be deter- mined, after assuming the direction of the light, is where the lightest and darkest parts will be, and what parts are near to the observer and what parts are farthest away. In all the fol- lowing cases we shall assume the light to come from over the left, shoulder, make the angle of forty-five degrees with both the vertical and horizontal planes of projection. THE PRISM. By the use of the forty-five degree triangle on the T square, draw the arrows as shown on the plan. The points where these touch the plan show where the direct light will strike by the prism. By dropping verticals from these points we see that one will fall behind the elevation and one in front, showing that the vertical edge near the right separates the light from the dark portions of the prism. The light will nowhere strike the prism perpendicularly, but it will strike that face near- est the left the most directly of any, and it will, of course, be the lightest face of the prism. The front face will be a little darker, and the right-hand face, being lighted only by diffused light, will lie much darker than either of the other two. The plan showing only the upper base, receives light at the same angle as the front face, and will have the same shade, which should be about the same as on the plate, and not blended. Considering the principle that the contrast is less between light and shade at a distance, we know that the outside parts of the faces on the right and left will tend to assume nearly the same shade as they recede from the observer, consequently the light face should be blended slight!}- toward the right, and the dark face on the right should also be blended toward the right, making the former darker toward the outside and the latter lighter toward the outside. THE PYRAMID. The light and dark portions are found in the same way, and the elevation is shaded almost precisely like the 40 prism. The top recedes slightly and the contrast there should be slighth- less than at the bottom where it is nearer the eye. The faces in the plan recede very rapidly, and the greatest contrast must be at the top. The upper right hand part receives only dif- fused light. THK CYLINDER. The dotted lines show 011 the plate the method of finding the lightest and darkest portions. Use the forty -five degree triangle on the T square so as to draw the two diagonal radii as shown. Where the one on the right cuts the lower semi-circumferences is the darkest point and where the other cuts the same on the left is the lightest point. These points projected down will give the lightest and darkest lines on the elevation. Blend quite rapidly both ways from the dark line and toward the right from the left hand limiting elements. The shades near the limiting elements should be about alike on the two sides. The shade of the plan should be the same as that of the plan of the prism. THE CONE. The instructions given for shading the cylinder with those given for shading the pyramid apply to this figure. Great care must* be taken to bring out the vertex in the plan. THE SPHERE. It will be evident on consideration that the darkest portion of the sphere is a great circle, the plane of which is perpendicular to the direction of the light ; but, as this great circle is not parallel to the planes of projection, its projections are both ellipses. There can evidently be but one point where the light can strike the sphere perpendicularly, and that is where the radius parallel to the direction of the light meets the surface. To find the lightest points in plan and elevation, join the cen- ters of the plan and elevation by a vertical, draw a diameter of the plan upward and to the right at an angle of fortA'-five degrees and of the elevation downward and to the right. Draw a line using the forty-five degree triangle from the point where the line joining the centers, cuts the circumference of the plan* perpendicu- lar to the diameter drawn in the same. Where it intersects the same will be the lightest point in the plan; a line similarly drawn gives the lightest point in the elevation as shown in the plate. The ellipse, which is the dark line of the object, crosses the two di- ameters drawn on the plan and elevation, just as far from the cen- ters of each as the light points are from the same. These points may be laid off from the centers by means of the dividers. The shade of the plan and elevation of the sphere are exactly alike, but the position of the light and dark portions are different, as seen in the plate. 41 Commence by laying on a narrow strip of shade in the form of an half ellipse over the dark point. Add other and wider strips of the same general form, and blend each toward the light point toward the outside. Great pains must be taken with this to get the correct shades and the two exactly alike. Put on all the lines and letters shown on the plate, making the dotted lines and arrows very fine. TINTING. Plate B contains the conventional tints for the fol- lowing materials : Cast Iron, Wrought Iron, Steel, Brass, Copper, Brick, Stone and Wood. The square figures are three inches on a side, and the circular figures have diameters of the same length. These colors are more difficult to lay on evenly than the India ink shades, but what has been said about the application of ink shades applies to them. Great pains must be taken to have the paperingood condition, and to keep the colors well mixed. Enough color should be mixed to finish the figure as it is almost impossi- ble to match the colors exactl3 r . Wash the brush thoroughly be- fore commencing a new figure. The plates are to be lettered like the wall plate, the initials at the bottom standing for the colors used. Below are given the materials to be used in each convention, the exact proportions of these can best be found by experiment, comparing the colors with those on the wall plate. A number of thin coats, well laid on, generally look more even than when the tints are laid on in single coats. For Cast Iron, use India ink, Prussian Blue and Crimson Lake; for Wrought Iron, Prussian Blue and India ink; for Steel, Prussian Blue ; for Brass, Gamboge, Burnt Umber and Crim- son Lake ; for Copper, Crimson Lake and Burnt Umber ; for Brick, Indian Red; for Stone, India ink and Prussian Blue; for Wood, Raw and Burnt Sienna. The convention for the body of wood is made by laying on a light coat of Raw Sienna, and the Grain is made by applying the Burnt Sienna, after the first is dry, with the point of the brush, blending slightly in one direction. Other com- binations requiring: different colors are often used but as they are all conventional the above will serve as an illustration. 42 SHADE LINING. Since the Blue Print method of copying drawings has come in- to use, very little tinting and shading with the brush is done, but instead the form is often indicated by line shading. In many cases therefore the plates of tinting and shading may be omitted, but in that case the student should study carefully the directions on pages 39 and 40, as to the method of finding light and dark places and apply the principles to shade lining. Shade lining is a method of representing the shades of an ob- ject by a series of lines drawn on the projections so as to produce the same general effect as when blended with Indiaink. This effect is produced by making the lines very fine and at a considerable dis- tance apart on the light portions, and quite heav3 r and close to- gether on the dark portions. This method of shading is often employed in uncolored drawings to bring out the forms of parts that might not otherwise be clearly understood, and often to give a drawing a fine finished appearance. Plates C and D contain the figures that generally require to be shaded in ordinary drawings. Each of these plates is made on a quarter sheet of Imperial paper, and of the size shown. These plates are made smaller than the others for the reason that each figure contains a great many lines. It is recommended that a half sheet of Imperial paper be shrunk down, and that this be divided when the plates are finished. The dimensions of each figure are marked on the plates but need not be marked on those drawn by students. The figures should all be penciled in of the dimensions indi- cated, and in the positions shown in the plates. The limiting lines of the figures to be shade lined should be drawn as fine as possible, and, on the practice figures, had better not be inked until the shade lines are drawn. The light and dark portions are found as in Plate A, prev- iously referred to, and as the shades to be represented are the same as in that plate, reference is made to the remarks on shades in the description of the same. Xo description can lie of so much value as a thorough study of 43 the plates. Notice carefully the gradation of the shade lines on each projection. In Plate C, the shade lines are all parallel; but in Plate D, they are neither parallel nor of the same width through- out. In the drawing of the sphere the light lines are made full circles, with the lightest point in each projection as a center. The middle portions only of dark lines are made with the compasses, the ends being finished free hand. In shade lining the pen must be kept very sharp and the ink must run well. Put on all the arrows and dotted lines shown on the plates. In practice, except in very fine drawings, it is customary to use much fewer lines than are here shown, putting on just enough to indicate the form rather than to fully show it. As, however, one able to do the work here shown can easily modify the method for himself, it is thought best to show how to do the best work. rS University of California SOUTHERN REGIONAL LIBRARY FACILITY 305 De Neve Drive - Parking Lot 17 Box 951388 LOS ANGELES, CALIFORNIA 90095-1388 Return this material to the library from which it was borrowed. O 0= ~r o i 1 -n t-> O ' ?3 CP CO so I Si '