fornj nal THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF John S.Prpll PI.YYVJTT INDUSTRIAL DRAWING COMPRISING THE DESCRIPTION AND USES DRAWING INSTRUMENTS, THE CONSTRUCTION OF PLANE FIGURES, TINTING, THE PROJECTIONS AND SECTIONS OF GEOMETRICAL SOLIDS, SHADOWS, SHADING, ISOMETRICAL DRAWING, OBLIQUE PROJECTION, PERSPECTIVE, ARCHITECTURAL ELEMENTS, MECHANICAL AND TOPOGRAPHICAL DRAWING. FOR THE USE OP HIGH SCHOOLS, ACADEMIES, AND SCIENTIFIC SCHOOLS, BY D. H. MAHAN, LL.D., LATE PROFESSOR OF CIVIL ENGINEERING, ETC., IN THE UNITED STATES MILITABY ACADEMY. REVISED AND ENLARGED BY DWINEL F. THOMPSON, B.S., PBOFESSOB OF DESCRIPTIVE GEOMETRY, STEREOTOMY, AND DRAWING IN THE RENSSELAEB POLYTECHNIC INSTITUTE, TBOY, N. Y. NEW YORK: JOHN WILEY & SONS, i 5 ASTOP PLAGE. 1888. JOHN S. PRELL Goil & Mechanical Engineer. *.U\ T FU A.M., I .SCO, arts. The spring should be pretty stiff. 5. Bow compasses. These are for drawing small circles both in pencil and ink. There are some forms made, where the pen and pencil points can be exchanged as in the larger compasses. It is more convenient, however, to have ,> separate instruments, one for the pen, and the other foi che pencil, as shown in Fig. 3. See that the springs are stroig. There are other forms of compasses, which are convenient at times, but not necessary to buy at first. Among thei,e are the proportional and beam compasses. 6. Proportional compasses. These are useful for en" urging or reducing drawings. The simplest form, shown ifc Fig. 4, is called halves and wholes ; where the shorter legs .*re half the length of the others, so that the distance between the points of the shorter legs will be one half of that bet tveen the points at the other end. Fig. 5 shows another form of these compasses, & tmilar in principle to the last, but constructed so that the position of the joint can be changed, thus giving different p. portions between the extremities at either end. To adjust the instru- ment, close it, and after unscrewing the nut, move the slide INDUSTRIAL DRAWING. along until the mark across it coincides with the required number, then clamp the nut. Four scales are sometimes engraved upon these compasses, called lines, circles, planes, and solids. The last two are omitted upon many of the instruments, as they are of little use. If the mark is brought to in the scale of lines, the distance ab will be one-third of cd. If the mark is brought to 8 in the scale of circles, ab will be the chord of the eighth part of the circumference of a circle whose radius is cd. Place the mark at 4 in the scale of planes, and ab will be the side of a square, or radius of a oircle whose area is one-fourth that of the square or circle formed with cd. In the scale of solids, with the mark at 4, ab will be the dia- meter of a sphere or edge of a cube whose solidity is one-fourth that of a sphere or cube whose dia- meter or edge is cd. The graduation of these instru- ments cannot always be relied upon, and the accuracy of the pro- portions is also affected by any change in the length of the legs, occasioned by use or breaking. It would be well to test their accuracy before purchasing. 7. Beam compasses. These are for describing larger arcs than would be possible with the other compasses. There are different forms of these. In all of them there is a beam of wood, or metal, with two collars, one carrying a pen or pencil point, and the other a needle point. Fig. 6 shows a conve- nient form of this instrument. In this the tops of the collars are left open so that they can be used on any straight edge, DRAWING INSTRUMENTS AND MATERIALS. 5 being damped to it by means of screws. The needle point is clamped at one end, while the other collar can be clamped at any distance from it. By turning the screw a the point b can be moved slightly, thus enabling one to adjust nicely the distance between 5 and c. In some forms there is a scale upon the beam, but ihis is not necessary and makes it more expensive. 8. Drawing pen. Fig. 7. This is made of two flat pointed steel blades, one of which has a hinge-joint at its base, thus allowing the blades to be sufficiently separated for cleaning or sharpening. The distance between the points is regulated by means of a screw connecting the two blades. This distance determines the width of the lines. Do not select one in which the blades are much curved. 9. Drawing board. This is an indispensable part of the draughtsman's outfit. It should be made of a thoroughly seasoned light wood; pine is generally used. A board twenty -eight inches long, twenty inches wide, and about one inch thick is a convenient size. Each end should be finished INDUSTRIAL DRAWING. with a cleat, as shown in Fig. 8, to prevent warping. Each side should be rubbed to a smooth plane surface, with fine sand-paper. The edges should be straight and at right angles to each other. 10. J square. This consists of a long thin blade, with a head at right angles to it ; fig. 9 is a good pattern, where the blade is simply laid upon the head and screwed to it. Any fine-grained hard wood will do for material, pear-wood is very DRAWING INSTRUMENTS AND MATERIALS. 7 good. Sometimes metal is used, and for accuracy it would be the best ; but as it rusts and tarnishes easily, the paper is liable to be soiled. The blade should be about the same length as the board. For a blade thirty inches in length, in order to be sufficiently stiff, the width should be at least two and a half inches, and the thickness one-eighth of an inch. The head is generally made of one piece, although sometimes of two, one on either side of the blade, as shown in fig. 10. One of these pieces is fixed while the other can be clamped at any angle with the blade ; so that, if the square is turned over and used with this side of the head against the edge of the board, parallel lines may be drawn making a correspond- ing angle with the sides of the board. This is not used very much, as it cannot be clamped sufficiently tight to prevent displacement by a slight rap, or any great pressure upon the blade. 11. Triangles. These are made of different materials, as wood, metal, rubber, etc., the last being the best ; metal tri- angles are open to the same objection that metal squares are. Triangles are made solid or open, as shown in fig. 11, the open ones being the most convenient to handle. The draughtsman requires two of these, one having angles of 90, 45, 45, while the other has angles of 90, 60, 30. A con- venient size is from six to ten inches on side. 12. Irregular curves. These are for tracing curves, other than arcs of circles, which are determined by points. Those made of thin pear-wood will be found to answer well. They have a great variety of shapes, so that one is at a loss which to select. Choose one with curves of both small and large radius. 8 INDUSTRIAL DBAWING. 13. India ink. It is important to have this material of the best quality, as it is used exclusively for all black lines. Good India ink, when broken across, should have a shining and somewhat golden lustre. If an end of it is wetted and rubbed on the thumb-nail, it should have a pasty feel, free from grains, and exhale an odor of musk ; when dried the end should present a shiny and golden-hued surface. Ink of inferior quality is of a dull bluish color, and when wetted and rubbed on the nail feels granular ; also when rubbed up in water it settles ; whereas good ink remains thoroughly dif- fused through the water. 14. Colors. Windsor and Newton's water colors are con- sidered the best. There are two sizes of cakes, called the whole and half cakes. The moist colors which come in porce- lain dishes are equally good, and are preferred by some, as the cakes are liable to crack and crumble. 15. Ink saucers. These come in nests of four or six, and are very convenient for preserving the colors after they are prepared. One forms a cover for another and thus keeps out the dust, besides keeping the colors moist for some time. 16. Pencils. These should be of the hardest and best quality. A. W. Faber's hexagonal pencils are the best. HH and HHH 9xv good numbers for mechanical drawing. 17. Brushes. The best brushes are made of sable hair, of which there are two varieties, the red and black. The red is somewhat the cheapest, but is about as good as the black. Camel's-hair brushes will answer very well, if one does not want to afford the sable. In selecting brushes see that they come to a point when moistened with water ; they should keep this point when used. 18. India rubber. Fine vulcanized rubber is the best for removing pencil marks. Sponge rubber is good for cleaning drawings. 19. Thumb tacks. These are for fastening the paper to the drawing board, when it is not necessary to stretch it. The best have the steel point riveted into the head. The top of the head should be slightly rounded, the edges being thin, so as to allow the J square to pass over readily. 20. Horn centre. This is used in case many circles are to DRAWING INSTRUMENTS AND MATERIALS. 9 . by 17 " 20 " 22 " 24 " 27 " 30 Elephant . 23 in . by 28 " 34 " 34 " 40 " 53 Columbier 23 ' Atlas, Double Elephant . . . Antiquarian. ... 26 ' ...27 * ...31 ' Emperor . . . .. 48 ' be drawn from the same centre. The needle point rests upon this instead of the paper, to prevent the wearing of a large hole. It is made of a piece of transparent horn, with three little steel points to prevent slipping. 21. Drawing paper. Watmau's drawing paper is consid- ered the best. It comes in sheets of standard sizes, as follows : Cap 13 in. by 17 Demy 15 Medium 17 Eoyal 19 Super Royal 19 Imperial 22 There are two kinds of this paper, the rough- or cold-pressed, and the smooth- or hot-pressed ; the smooth paper is better for finished line drawings, while the rough surface takes color better ; does not show erasures, and is better for general work. On holding the paper up to the light the maker's name is seen in water lines, and when it can be read, the nearest surface is considered the right side. If there is any difference between the two, it is supposed to be in favor of this side. The double elephant sheet is of a good weight and size for general work. For smaller sizes cut it in halves or quarters. 22. Tracing paper. This is a thin paper prepared so as to be sufficiently transparent to allow the lines of a drawing to show through, for the purpose of copying. It comes in sheets or by the roll. The best paper is tough, transparent, and without any greasiness. 23. Tracing cloth. This is for the same purpose as tracing paper. It is better than paper for preserving copies, or for those that are to be subjected to any wear, as in the case of working drawings. It comes in rolls of twenty-four yards, and of different widths. That which has a dull surface on one side is better for pencil marks or for tinting. 24. As it would be impossible in most oases to make a drawing the same size as the object, it is r ecessary to resort to the use of scales in order to preserve the relative positions 10 INDU8TBIAL DBA WING. of the lines, and also to be able to determine what proportion the drawing bears to the object. These scales consist of dis- tances of different lengths, divided with great accuracy. Various materials are used in their construction, such as ivory, wood, metal, paper, etc. Ivory is the best, but quite expensive, while those made of boxwood are good, and are generally used. Metal is expensive and tarnishes easily, while paper soon wears out. It is well in making a drawing to use as large a scale as convenient, so that the small parts of the object may be brought out distinctly and accurately. The scale of a drawing should always be given upon it, and in the case of a working drawing, the measurements also. By giving the measurements the time of the workman is saved, since he follows the dimensions rather than the drawing, in case they differ. The dimensions are given in figures, using one accent to in- dicate feet and two accents for inches, as 2', read two feet, and 3", three inches. Where feet and inches are to be written, the figures are separated by a comma simply, as 6', 4". Where only inches are to be written, use a cipher in place of feet, as 0', 6". This is not always necessary, but upon drawings where many of the dimensions are given in feet and fractions of a foot, it is well to use the cipher, as it may save mistakes in reading. To indicate the points between which the measurement reads, small arrow heads are used at the points; as seen upon PI. XVI. Fig. 144. When the arrow heads are some distance apart, they are connected by a fine dotted line. This line may be broken at the centre to allow the measurements to be in- serted, or the line may be continuous, and the figures be placed just above it or on it. To make the measurements more noticeable, red ink is generally used for the figures and arrow heads, and blue ink for the dotted lines. All the horizontal measurements should be written from left to right, the verti- cal being written from the bottom to the top. 25. A scale of equal parts. (PL I. Fig. 7.) This is a small flat instrument of ivory. On both sides of this instrument we shall find a number of lines drawn lengthwise, and divided crosswise by short lines, against which numbers are DRAWING INSTRUMENTS. 1] written. It is not our object, in this place, to give a full description of this scale, but simply of that portion which is termed the scale of equal parts. A scale of equal parts con- sists of a number of inches, or of any fractional part of an inch, set off along a line ; the first inch, or fractional part to the left, being subdivided into ten or twelve equal portions. When the divisions of the line are inches, the scale is some- times termed an inch scale; if the divisions are each three quarters of an inch, it is termed a three quarter inch scale; and so on, for any other fractional part of an inch. These different scales are usually placed on the same side of the ivory scale ; the inch scale being at the bottom, the three quarter inch scale next above ; next the half inch scale, and so on. The inch scale is usually marked IN. on the left ; the three quarter inch scale with the fraction f ; the others, in like manner, with the fractious $, *, and so on. The first division of each of these scales is usually subdivided, along the bottom line of the scale, into ten equal parts, by short lines; and just above these short lines will be found others, somewhat shorter, which divide the same division into twelve equal parts. It is not usual to place any number against the first division ; against the short line which marks the second division, the number 1 is written; at the third division the number 2 ; and so on to the right. 26. Manner of using the scale. Geometrical drawings of ob- jects are usually made on a smaller scale than the real size of the object. In making the geometrical drawing of a house, for example, the drawing would have to be very much smaller than the house, in order that a sheet of paper of moderate dimensions may contain it. We must then, in this case, a& in all others, select a suitable scale for the object to be represented. Let us suppose that the scale chosen for the example in question is the inch scale ; and that each inch on the drawing shall correspond to ten feet on the house. As a foot contains twelve inches, it is plain, that an inch on the drawing will correspond to 120 inches on the house ; thus any line on the drawing will be the one hundred and twentieth part of the corresponding line on the house ; and were we to make a model of the house, on the same scale aa 12 INDUSTRIAL DRAWING. the drawing, the height or the breadth of the model would be exactly the one hundred and twentieth part of the height or the breadth of the house. By bearing these remarks in mind, the manner of using this, or any other scale, will appear clearer. Suppose, for example, we wish to take off from the scale, by a pair of dividers, the distance of twenty- three feet ; we observe, in the first place, that as each inch corresponds to ten feet, two inches will correspond to twenty feet ; and as each tenth of an inch corresponds to one foot, three tenths will correspond to three feet. We then place one point of the dividers at the division marked 2, and extend the other point to the left through the two divisions of an inch each, and thence along the third division so as to take in three tenths. From the scale, here described, we cannot take off any fractional part of a foot, with accuracy. If we had wished to have taken off twenty-three feet and a half, for example, we might have extended the point of the dividers to the middle between the division of three tenths and four tenths ; but, as this middle point is not marked on the scale, the accuracy of the operation wil] depend upon the skill with which we can judge of distances by the eye. To avoid any error, from want of skill, we must resort to one of the scales of the fractional parts of an inch. Taking the half inch scale, for example, its principal divisions being halves of an inch, will, therefore, correspond to five feet ; the first of its principal divisions being subdivided, like the one on the inch scale, one tenth of it will correspond to half a foot, or six inches. So that from this scale we can take off any number of feet and half feet. The quarter inch scale being the half of the half inch scale, its small subdivisions will correspond to a quarter of a foot, or to three inches. The small subdivisions on the eighth of an inch scale, in like manner, correspond to one inch and a half. We thus observe that, by suitably using one of the above scales, we can take off any of the fractional parts of a foot on the inch scale, cor- responding to a half, a quarter, or an eighth. Let us now suppose, as a second example in the use of the inch scale, that we had to make a drawing of a house on a scale of one inch to twelve feet. We observe, in the first DRAWING INSTRUMENTS. IB place, as twelve feet contain 144 inches, and as one inch, ou the drawing, corresponds to 144 inches on the house in its true size, every line on the drawing will be the one hundred and forty-fourth part of the corresponding lines on the house. In this case, instead of using the division of the inch sub- divided into tenths, we use the divisions of twelfths, which are just above the tenths ; each twelfth corresponding to one foot. We next observe^ in using the twelfths, what we found in using the tenths ; that is, we cannot obtain, from the inch scale, any part corresponding to a fractional part of a foot ; but, taking the half inch scale, we find the first half inch subdivided in the same way as the inch scale into twelfths ; and, therefore, each twelfth on the half inch scale will corres- pond to half a foot, or six inches. In like manner the twelfths on the quarter inch scale will correspond to three inches ; and the twelfths on the eighth of an inch scale to one inch and a half. 27. Diagonal scale of equal parts. (PI. I. Fig. 8.) In the two examples of the use of an ordinary scale of equal parts, we found, that the smallest fractional part of a foot, that any of the scales examined would give, was one inch and a half. But, as all measurements of objects not larger than a house are commonly taken either in feet and inches, or in feet and the decimal divisions of a foot, it is very desirable to have a scale from which we can obtain either the tenths of a foot, if the decimal numeration is used ; or the inches, or twelfths, when the duodecimal numeration is used. To effect these purposes, a scale, called a diagonal scale of equal parts, has been imagined; and this scale is frequently found on the small ivory scale, on which the other scales of equal parts are marked. A diagonal scale of equal parts may be either an inch scale, or a scale of any fractional part of an inch. The first inch, or division, is divided into ten, or twelve equal parts ; the other inches to the right are numbered like th& ordinary scales. Below the top line, on which the inches are marked off, we find either ten, or twelve lines, drawn parallel to the top line, and at equal distances from each other. We next observe perpendicular lines, drawn from the top line, acaoss the parallel lines, to the bottom line, at the points OD 14 INDUSTRIAL DRAWING, the top line which mark the divisions of inches. But, from the subdivisions on the top line, we find oblique lines drawn to the bottom line ; these oblique lines being also parallel to each other, and at equal distances apart. The direction of these oblique lines, termed diagonal lines, and from which the scale takes its name, is now to be carefully noticed, in order to obtain a right understanding of the use of the scale. By examining, in the first place, the bottom line of the scale, we observe that it is divided, in all respects, precisely like the top line. In the next place, we observe, that the first diagonal line to the left, 10, is drawn from the end of the top line, to the point m t which marks the first subdivision to the left on the bottom line. The other diagonal lines, we observe, are drawn parallel to the first one, on the left, and from the points of the subdivisions on the top line ; the last diagonal line, on the right, being drawn from the last subdivision on the top line, to the end of the first division of the bottom line. On examining the perpendicular lines, we find, that they divide all the parallel lines, from top to bottom, into equal parts of an inch each. We find, also, that all the sub- divisions, on the same parallel lines, made by the diagonal lines, are equal between the diagonal lines, each being equal to the top subdivisions. But when we examine the small divisions, on the parallel lines, contained between the first perpendicular line, 10 w, on the left, and the diagonal line next to it, we find these small divisions increase in length from the top to the bottom. In like manner we find that the small divisions, on the parallel lines, between the dia- gonal line, 1 r, on the right, and the perpendicular, r, next to it, decrease in length from the top to bottom. Now, it is by means of these decreasing subdivisions that we get the tenths, or twelfths of one of the top subdivisions, according as we find ten or twelve parallel lines below the top line. Let us suppose, for example, that we have ten parallel lines below the top line ; then the small subdivision, on the first parallel line, below the top line, contained between the diagonal line, 1 r, on the right, and the perpendicular, r, would be nine tenths of the top subdivision ; that ia nine tenths of a foot, if the toj> subdivision corresponds to DRAWING INSTRUMENTS. 15 one foot. The next small division below this would be eight tenths; the next below seven tenths; and so on down Now, if we take the small subdivisions, between the per- pendicular line, on the left, and the first diagonal line, 10 m, the one below the top line will be one tenth of the top subdivision ; the next below this two tenths ; and so on, increasing by one tenth, as we proceed towards the bottom. Let us suppose, for an example in the use of this scale, that the drawing is made to a scale of one inch to ten feet ; and that we wish to take off from the scale sixteen feet and seven tenths. With the dividers slightly open in one hand, we run the point of the dividers down along the perpendicular, marked r, next to the diagonal line on the right, until we come to the parallel line on which the small division seven tenths is found ; this will be the third line below the top line. We place the point of the dividers on this parallel line, at a, where the perpendicular line, numbered r, crosses it, and we extend the other point to the right, to b, to the perpendicular marked 1, along the parallel line, to take in one inch, or ten feet. We then keep the right point of the dividers fixed, at b ; and extend the other point to the left, to c, taking in the small division seven tenths, and six of the subdivisions along the parallel line. The length, thus taken in between the two points of the dividers, will be the required distance on the scale. From the description of the diagonal scale, and the example of its use just explained, the arrangement of, and the manner of using any other diagonal scale will be easily learned. We have first to count the number of parallel lines below the top line, and this will show us the smallest fractional part of the equal subdivisions of the top line that we can obtain, by means of the scale. If, for example, we should find one hundred of these lines below the top line, then we could obtain from the scale, as small a fraction of the top sub- divisions as one hundredth ; and counting downwards on the left, or upwards on the right, the short lines, between the extreme diagonal lines and the perpendicular lines next tr them, we could obtain from the one hundredth to ninety nine hundredths of the top subdivisions. 1C INDUSTRIAL DRAWING. Remarks. In the preceding examples we have taken the scales as corresponding to feet, because dimensions of objects of ordinary magnitude are usually expressed in feet,. When the drawing represents objects of considerable extent, we then should use a scale of an inch to so many yards, or miles; as in the case of- a map representing a field, or an entire country. When the drawing represents objects whose principal dimensions are less than a foot we then use a scale of an inch to so many inches. In some cases where we wish to represent very minute objects, which cannot be drawn accurately of their natural size, we use a magnified scale, that is a scale which gives the dimensions on the drawing a certain number of times greater than those of the object represented. For example, in a drawing made to a scale of three inches to one inch, the dimensions on the drawing would be three times those of the corresponding lines on the object. 28. A convenient form of diagonal scale for obtaining twelfths of a foot, is shown in Fig. 12, where ab, be, represent 6 b /\5 / \* / \3 / \2 / \/ / \ Pro. 12. feet, while the horizontal distances of the points 1, 2, 3, etc., from the vertical line through b are each equal to a cor- responding number of inches. 29. The scales represented in Figs. 7 and 8, PI. I, although 1 used to some extent, are rather short and -ss@mMMM^B!~ not as convenient to use as those where the 2 graduation runs to the edge, so that by lay- j n g the scale on the paper the required dis- tance can be marked off directly without using the dividers. The frequent use of the dividers in taking distances from the scales, defaces the lines of the scale and should be avoided. In Fig. 13 are shown sections of no. is. scales in general use. These are all made DBA WING INSTRUMENTS AND MATERIALS. 17 with thin edges, and with the graduation running to the very edge. On Nos. 1 and 2 there would be one, or possibly two scales for each edge, while on No. 3 there can be at least six scales in all, two to each edge, one on either side. 30. Fig. 14 gives a side view of Nos. 1 or 2. There are four scales upon this, two upon each edge, one being double 1 "M 1 "M 1 ' 1 the other. The scales are duodeciraally divided, the divisions representing feet and inches. The following scales are the most useful : -, J, , 1, J-, f , 1^, 3. Two flat ones, or one tri- angular, will contain all of these. 31. Fig. 15 represents what is called a chain scale. Such scales are numbered 10, 20, 30, 40, 50, 60, according to the i. i 1 I M number of divisions contained in an inch, and are the scales in common use. The upper scale in Fig. 15 is numbered 20, that is, an inch is divided into 20 equal parts and might be used in practice for a scale of two or twenty chains to the inch ; two or twenty feet to the inch, or two or twenty miles to the inch. One triangular, or three flat scales, will contain all of those mentioned above. The duodecimally divided scales are most useful for mechanical and geometrical draw- ing, while the chain scales, as their name indicates, are used mostly for plotting surveys. 32. After the drawing of an object has been completed, it is necessary to state upon it the ratio between the lines of the drawing and those of the object. This is often done by the use of a fraction, as a scale of , f, i, etc. It is understood in such cases that the numerator refers to the drawing and the denominator to the object, and that the numerator is equal 2 18 INDUSTRIAL DRAWING. to the denominator ; for example in the scale \ we should understand that one inch on the drawing equals four inches on the object. In the scale 1 in. = 8 in. or 1" = 1 foot. Instead of always expressing scales by the use of fractions, it is often stated what the scale is to a foot, as for example, " Scale in. = 1 foot," this expressed fractionally would be ^f. Scale in. = 1 foot would be the same as T ^-. 33. Protractor. This is the instrument in most common use for laying off angles. It consists of a semicircle of thin metal or horn (Fig. 16), the circumference of which is divided into 180 equal parts, termed degrees, and numbered, both ways, from to 180, by numbers placed at every tenth divi- sion. As the protractor is only divided into degrees, subdivi- sions of a degree can be marked off by means of it only by judging by the eye. A little practice will enable one to set off a half, or even a quarter of a degree with considerable accuracy, when the circumference is not very small. Where greater accuracy is required, protractors of a more compli- cated form are used. Some of the ivory scales are graduated so that they can be used as protractors, but they cannot be depended upon for accurate work. Horn protractors are convenient to use as they are transparent, but they are liable to warp. Tracing paper has recently been used to print pro- tractors upon. Although not very durable, they have this advantage that they may be made of larger sizes, some being fourteen inches in diameter ; being upon tracing paper, they are transparent, and also cheap. There are other forms of scales more or less complex, and seldom used, which it is not necessary to describe here. USB AND CAEE OF INSTRUMENTS. 19 CHAPTER II. USE AND CAEE OF INSTRUMENTS. 1. Good tools are not only necessary for good work, but they should be kept in perfect order. This is particularly true in drawing. The draughtsman should keep at hand a clean old linen rag, and a piece of soft washleather, to be used for cleaning and wiping the instruments before and after using. The pen should be wiped dry before laying it aside, and the soil from perspiration cleaned with the wash- leather in like manner. With proper care a set of good in- struments will last a lifetime. 2. Compasses. In describing circles keep both legs nearly vertical. This can be accomplished by means of the joint in each leg. By keeping the needle in a vertical position, it will not make a larger hole than necessary in the paper. Usually one hand is sufficient when describing circles ; but when the lengthening bar is used, it is better to steady the needle point with one hand, while the circle is described with the other. When inking a circle, complete it before taking the pen from the paper, because without great care the place where the lines join will show. 3. Dividers. These are used for transferring distances from one part of a drawing to another, or from scales to the drawing. One hand is sufficient for their use ; but care must be taken when transferring distances not to change the posi- tion of the points by clumsy handling. A little care and practice will enable one to manage them easily. When tak- ing any distance with the dividers, open the points wider than necessary, and then bring to the required distance by pressing them together. In case it is required to lay off the same dis- tance a number of times on a straight line, it is better to step the distance off, keeping one of the points on the line all the 20 INDUSTBIAL DRAWING. time. In doing this, do not turn the instrument continuously in one direction, but reverse at each step, so that the moving point will pass alternately to the right and left of the line. A single trial will convince one, by the ease with which the instrument can be handled, of the advantage of the last method. Care should be taken when using the dividers not to injure the surface of the paper. When laying off distances, do not make holes in the paper ; a very slight prick is suffi- cient, and this can afterwards be marked with the pencil. 4. Drawing pen. This is one of the most important instru- ments, and should never be of inferior quality. It is used for inking the lines of a drawing after it has been pencilled. The ink can be placed between the blades by means of a small brush or a strip of paper ; but a more convenient way is to dip the edges of the pen into the ink, and they will generally take up enough ; in case they do not, breathe upon them, and then they will readily take the ink. After taking the ink in this way, always wipe the outside of the blades. When inking, incline the pen slightly to the right in the direction of the line, taking care that both points touch the paper. Do not press too hard against the ruler, nor try to keep the point of the pen too near the ruler. If the line is to be heavy, or broad, the pen should be moved along rather slowly, otherwise the edges of the line will appear rough. Always draw the lines from left to right. In case the ink does not flow readily from the pen, try it upon a piece of waste paper, or outside the border line ; a piece of blotting paper is also good for this purpose. If these means fail, pass between the blades the corner of a piece of firm paper. When the ink gets too thick, it dries rapidly in the pen and occasions considerable annoyance ; in such a case, add a little water to the ink, and mix well before using. 5. Sharpening pen. As the pen is used a great deal, the points are worn away, changing their form as well as thick- ness, making it impossible to do nice work. Every draughts- man should be able to sharpen his pen and keep it in good order ; with a little practice and attention to the following directions, one will soon be able to sharpen a pen. Having obtained a fine grained oilstone, first screw the points together, USE AND CARE OF INSTRUMENTS. 21 then draw the pen over the stone, keeping it in a perpendicu- lar plane, and turning it at the same time, so that the curve at the point shall take the shape shown in N\ 1, Fig. 17. This is a better shape than either N"os. 2 or 3, although 3 might do for heavy lines ; but for all lines, 1 is the best shape. It is well, however, to have two pens ; one being kept sharp for fine lines, while the other is used for heavy lines, and need not be sharpened as often. Having obtained the right shape, separate the points a little by means of the screw, and then lay each blade in turn upon the stone keeping the pen at an angle of about 30 with the face of the stone and grind off the thickness so as to bring to an edge along the point of this curve. For fine lines this edge should be pretty thin ; not a knife edge, which would cut the paper. See that the edges of both blades are of the same thickness, and the points of the same length ; then take the screw out and open- ing the blades, apply the inside of each to the stone just enough to take off any edge that may have been turned over. Next, screw up the blades, and try the points on the thumb nail to see if the edges are too sharp ; if they cut the nail, the sharp edge should be taken off by drawing the pen very lightly over the stone as in the first step. Next, try the pen with ink on paper, and the character of the line made will show whether anything more is needed. 6. Drawing board. The sides should be plane surfaces, and their accuracy may be tested by placing the edge of a ruler across the board in several positions, and observing whether it coincides throughout, in every position, with the surface. The edges of the board should be perfectly straight, and if one cannot detect any inaccuracy by the eye alone, they can be tested by applying a straight edge. It is not essential, however, that the edges should be exactly at right angles to each other, provided the head of the square is used only on one edge for any given drawing. 22 INDUSTRIAL DRAWING. 7. T square. By placing the head against any edge of the board, and moving it along, at the same time drawing lines along the edge of the blade in its successive positions, we have a series of parallel lines. If now the head is placed against either adjoining edge, and lines are drawn as before, we shall have two sets of parallel lines at right angles to each other ; that is, if the two edges of the board are exactly at right angles, and also the blade of the square exactly at right angles to the head. Now, as it is impossible to be sure that these angles are correct at all times, and as the lines drawn while the head is moved along any edge would be parallel, whether the blade be at right angles to the head or not, it is better to use the head upon only one edge, and depend upon the trian- gles for perpendicular lines. It is customary and more con- venient to use the head upon the left-hand edge of the board, controlling its movements with the left hand, while the right is free to draw the lines. In moving from one position to another, take hold of the head instead of the blade. Always see that the head is against the edge before drawing a line, and use the upper edge in ruling. 8. Triangles. As there are two triangles, having the angles 90, 45, 45, and 90, 60, 30, respectively, it is evident that, by placing either of these triangles against the blade of the square, lines may be drawn making corresponding angles. As before suggested, this is the best way to draw perpendicu- lars. In case parallel lines are to be drawn, making angles different from those of the triangles, it may be possible to accomplish it by a combination of both triangles with the square ; but it is difficult to hold so many pieces in place, se USE AND CAEE OF INSTRUMENTS. 23 that it is better to use either one with the square, moving the blade so that the edge of the triangle shall coincide with a line already drawn at the desired angle ; then, by holding the blade in place, and moving the triangle, lines may be drawn parallel to the first. To test the right angle of the triangle, place it against the square as in Fig. 18, and draw the line a b } then turn the triangle over, as in the dotted position, and draw another line along the vertical edge, taking care that the two lines start from the same point a. Now, if these lines correspond throughout, the angle at the base is a right augle ; if there is any deviation, it will indicate what change should be made to correct it. Maniier of using the triangles for drawing lines which are to be either parallel or perpendicular to another line. Let A B (PI. I. Fig. 4) be a line to which it is required to draw parallel lines which shall respectively pass through the points, (7, D, E, etc., on either side of A B. 1st. Place the longest side, a 5, of the triangle so as to coincide accurately with the given line ; and, if its other sides are unequal, taking care to have the next longest of the two, a c, towards the left hand. 2d. Keeping the triangle accu- rately in this position, with the left hand, place the edge of the ruler against the side, a full, ~brolf.en, dotted, and broken and dotted, &c. ; these again are divided into fine, medium, and heavy, according to the breadth of the line. A fine line is the one of least breadth that can be distinctly traced ' with the drawing pen ; the medium line is twice the breadth of the fine ; and the heavy is at least twice the breadth of the medium. The coarse broken line consists of short lines of about fa of an inch in length, with blank spaces of the same length between them. The fine broken lines and spaces are fa of an inch. The dotted line consists of small elongated dots with spaces of the same between. The broken and dotted consists of short lines from ^ to \ of an inch with spaces equal in length to the lines divided by one, two, or three dots at equal distances from each other and the ends of the lines. These lines may also be fine, medium, or heavy. When a, line is traced with quite pale ink it is termed a faint line. The lines of a problem which are either given or are to be found should be traced in full lines either fine or medium. The lines of construction should be broken or dotted. The outlines of an object that can be seen by a spectator from the point of view in which it is represented should be fall, and either fine, medium, or heavy, according to the par- ticular effect that the draftsman wishes to give. The portions of the outline that cannot be seen from the assumed point of view, but which are requisite to give a complete idea of the object, should be dotted or broken. The other lines are used for conventional purposes by the draftsman to show the connexion between the parts of a problem, &c., &c. PROBLEMS OF STRAIGHT LINES. 33 Prob. 2. (PI. II. Fig. 14.) To set of a given distanje, along a straigJvt line, from a given point on it. Let C D be the line, and A the given point. 1st. Mark the given point A, as in the preceding pro- blem. 2d. Take off the given distance, from the scale of equal parts, with the dividers. 3d. Set one foot of the dividers on A, and bring the other foot upon the line, and mark the point j?, either by pricking the surface with the foot of the dividers, or by a small dot made on the line with the sharp point of a lead pencil. When the distance to be set off is too small to be taken off from the scale with accuracy, proceed as follows : 1st. Take off in the dividers any convenient distance greater than the given distance, and set it off from A to b. 2d. Take off the length by which A b is greater than the given distance and set it off from b to c, towards A ; the part A c will be the required distance. Remark. A given distance, as the length of a line, or the distance between two given points, is sometimes required to be set off along some given line of a drawing. This is done by a series of operations precisely the same as just described. In using the dividers, they must be held without stiffness, care being taken not to alter the opening given to them, in taking off the distances, until the correctness of the result has been carefully verified, by going over the operation a second time. Particular care should be paid to the manner of hold- ing the dividers in pricking points with them, to avoid changing their opening, as well as making too large a hole in the drawing surface. Prob. 3. (PI. II. Fig. 15.) To set off, along a straight line, any number of given equal distances. Let G D be the straight line, and the given number of equal distances be eight. 1st. Commence by marking a point A, on the line, in the usual manner, as a starting point. 2d. The number of equal divisions being even, take off in the dividers from the scale their sum, and set it off from A tc 8, and mark the point 8. 3 34 INDUSTRIAL DRAWING. 3d. Take from the scale half the sum total, and set it off from A to 4 ; taking care to ascertain that the dividers will accurately extend from 4 to 8, before marking the point 4. 4th. Take off from the scale the fourth of the sum total, and set it off respectively from A to 2, and 4 to 6 ; taking care to verify, as in the preceding operation, the distances 24, and 68. 5th. Take off from the scale the given equal part, and set it off from A to 1 ; from 2 to 3, &c. ; taking care to verify the distances as before. Remark. When the number of equal distances is odd, commence by setting off from the starting point, as just described, an even number of equal distances, either greater or smaller than the given odd number by one, taking in pre- ference the even number which with its parts is divisible by two ; if, for example, the odd number is 7, then take 8 as the even number to be first set off; if it is 5 then take 4 as the even number. Having as in the first example, set off 8 parts we take only the seven required parts ; and in the second having set off 4 parts only we add on the remaining fifth part to complete the required whole. The reason for using the operations just given instead of setting off each equal part in succession, commencing at the starting point, is, if there should be the least error in taking off the first equal part from the scale, this error will increase in proportion to the total number of equal parts set off, so that the whole distance will be so much the longer or shorter than it ought to be, by the length of the error in the first equal distance multiplied by the number of times it has been Prob. 4. (PI. II. Fig. 16.) From a point on a right line to set off any number of successive unequal distances. Let C D be the given line, and A the point from which the first distance is to be reckoned ; and, for example, let the distances be respectively A b equal 20 feet ; b c equal 8 ; c d equal 15 feet ; and d B equal 25 feet. 1st. Commence by adding into one sum the total number of distances, which in this case is 68 feet. PROBLEMS OF STRAIGHT LINES. 35 2d. Take off from the scale of equal parts this total, and set it off from A to B. 3d. Add up the three first distances of which the total is 43 feet ; take this off from the scale, and set it off from A tod. 4th. Take off the distance d B in the dividers, and apply it to the scale to verify the accuracy of the construction. 5th. Set off successively the distances A b equal 20 feet ; and A c equal 28 feet ; and verify by the scale the distances b GJ and c d. Remark The object of performing the operations in the manner here laid down is to avoid carrying forward any inaccuracy that might be made were the respective distances set off separately. The verifications will serve to check, as well as to discover any error that may have been made in any part of the construction. Prob. 5. (PL II. Fig. 17.) To divide a given line, or the distances between two given points, into a given number of equal Let A B be the distance to be divided; and let, for example, the number of its equal parts be four. Take off the distance A B in the dividers, and apply it to the scale of equal parts, then see whether the number of equal parts that it measures on the scale is exactly divisible by 4, or the number of parts into which A B is to be divided. If this division can be performed, the quotient will be one of the required equal parts of A B. Having found the length of one of the equal parts proceed to divide A B precisely in the same way as in Prob. 3. If AB cannot be divided in this way, we shall be obliged to use the ruler and triangle, in addition to the dividers and scale of equal parts, to perform the requisite operations, and proceed as follows : 1st. Through the point B, draw with the ruler and pencil a straight line, which extend above and below the line A B, so that the whole length shall be longer than the longest side of the triangle used. The line G D should make nearly a right angle with A B. 2d. Take off from the scale of equal parts any distance 36 INDUSTRIAL DRAWING. greater than A B, which is exactly divisible by 4, or the number of parts into which A B is to be divided. 3d. Place one foot of the dividers at J., and bring the other foot upon the line CD, and mark this second point 4, in the usual way. 4th. Draw a straight line through A and 4. 5th. Divide, in the usual way, the distance A 4 into its four equal parts A 1 ; &c., and mark the points 1, 2, 3, &c. 6th. With the ruler, triangle, and pencil draw lines parallel to CD, through the points 3, 2, and 1 ; and mark the points df, c, and 6, where these parallel lines cross A B. The distances A b, b c, c d, and d B will be equal to each other, and each the one fourth of A B. Remarks. The distance A 4 may be taken any length greater than A B the line to be divided. It will generally be found most convenient to take a length over twice that of AB. The line B C is drawn so as to make nearly a right angle with A B, in order that the points where the lines parallel to it cross A B may be distinctly marked. Attention to the selection of lines of construction is of importance, as the accuracy of the solution will greatly depend on this choice. In this Prob., for example, the line CD might have been taken making any angle, however acute, with A B, without affecting the principle of the solution ; but the practical result might have been very far from accurate, had the angle been very acute y from the difficulty of ascertaining with accuracy, by the eye alone, the exact point at which two lines intersect which make a very acute angle between them, such as the lines drawn from the points 1, 2, 3, &c., parallel to CD. would have made with A B had the angle between it and CD been very acute. The same remarks apply to the selection of arcs of circles by which points are to be found as in figs. 18, 19, &c. The radii in such cases should be so chosen that the arcs will not intersect in a very acute angle. Prob. 6. (PI. II. Fig. 18.) From a point on a given line to construct a perpendicular to the line. Let CD be the given line, and A the point at which it is required to construct the perpendicular. PROBLEMS OF STRAIGHT LINES. 37 1st. Having fitted the pencil point to the dividers, open the legs to any convenient distance, and having placed the stee] point at A , mark, by describing a small arc across the given line with the pencil point, two points b and c, on either side of J., and at equal distances from it. 2d. Place the steel point at 6, and open the legs until the pencil point is brought accurately on the point c ; then from b, with the distance b c, describe with the pencil point a small arc above and below the line, and as nearly as the eye can judge just over and under the point A. 3d. Preserving carefully the same opening of the dividers, shift the steel point to the point e, and describe from it small arcs above and below the line, and mark with care the points where they cross the two described from b. 4th. With the ruler and pencil, draw a line through the points A and B, extend it above B as far as necessary ; this is the required perpendicular. Remarks. The accuracy of the preceding construction will depend in a great degree upon a judicious selection of the equal distances set off on each side of A in the first place ; in the opening of the dividers b c with which the arcs are described ; and upon the care taken in handling the instru- ments and marking the requisite points. With respect to the two equal distances A b and A c, they may be taken as has been already said of any length we please, but it will be seen that the longer they are taken, provided the whole distance be can be conveniently taken off with the dividers, the smaller will be the chances of error in the construction. Because the greater the distance b c the farther will the point J?, where the two arcs cross, be placed from J., and any error therefore that may happen to be made, in marking the point of crossing of the arcs at B, will throw the required perpendicular less out of its true position than if the same error had been made nearer to the point A ; moreover, it is easier to draw a straight line accurately through two points at some distance apart than when they are near each other ; particularly if the line is required to be extended beyond either or both of the points ; for if any error is made in the part of the line joining the two points it 38 INDUSTRIAL DRAWING. will increase the more the farther the line is extended eithei way beyond the points. With respect to the distance b c, with which the arcs are described, this might have been taken of any length provided it were greater than A b. But it will be found on trial, if a distance much less than the three fourths of, or much greater than b c, is taken, to describe the arcs with, that their point of crossing cannot be marked as accurately as they can be when the distance b c is used. Attention to a judicious selection of distances, &c., used in making a construction, where they can be taken at pleasure, is of great importance in attaining accuracy. Where points, like JB, are to be found, by the crossing of arcs, or of straight lines, we should endeavor to give the lines such a position that the point of crossing can be distinctly made out, and accu- rately marked; and this will, in all cases, be effected by avoiding to place the lines in a very oblique position to each other. A further point to secure accuracy of construction is to obtain means of proof, or verification. In the construction just made, the point d will serve as a means of verification ; for the perpendicular, if prolonged below A, should pass accurately through the point d if the construction is correct. Prob. 7. (PI. II. Fig. 19.) From a point, at or near the extremity of a given line, to construct a perpendicular to the line. Let CD be the line, and A the point. In this case the distance A C being too short to use it as in the last Prob., and there not being room to extend the line beyond C, a different process must be used. 1st. Mark any point as a above CD, and between A and D. 2d. Place the foot of the dividers at a, and open the legs until the pencil point is brought accurately on A ; then describe an arc to cross CD at >, and produce it from A so far above it that a straight line drawn through b and a will cross the arc above A. 3d. Mark the point b, and with the ruler and pencil draw a straight line through a and b, and prolong it to cross the arc at B. PROBLEMS OF STRAIGHT LINES. 39 4th. Mark the point B ; and with the ruler and pencil draw a line through A and B. This will be the required perpen dicular. Prdb. 8 (PI. II. Fig. 18.) From a given point, above or below a given line, to draw a perpendicular to the line. Let CD be the given line, and B the given point. 1st. Take any opening of the dividers with the pencil point, and placing the steel point at B describe two small arcs, crossing CDatb and c; and mark carefully these points. 2d. Without changing the opening of the dividers, place the steel points successively at b and c, from which describe two arcs below CD, and mark the point d where they cross. 3d. With the pencil and ruler draw a line through B d. This is the required perpendicular. Remark. The distance B c, taken to describe the first arcs, should as nearly as the eye can judge be equal to that be between them, unless the given point is very near the given line. Verification. If the construction is accurate, the distance A b will be found equal to A c ; and A B equal to A d. Prdb. 9. (PI. II. Fig. 19.) To construct the perpendicular when the point is nearly over the end of the given line. Let B be the given point, and CD the given line. 1st. Take off any equal number of equal distances from the scale with the dividers and pencil point. 2d. Place the steel point at B, and, with the distance taken off, describe an arc to cross CD at b ; and mark the point b. 3d. Draw a line through B b. 4th. Take off half the distance B 6, and set it off from either B, or b to a, and mark the point a. 5th. Place the steel point of the dividers at a, and stretch- ing the pencil point to b, or B, describe an arc to cross CD at A, and mark the point A. 6th. Draw a line through A and B. This is the required perpendicular. Verification. Produce B A below CD, and set off A d equal to A B; if the construction is accurate b d will be found equal to Bb. 4:0 INDUSTRIAL DRAWING. Prob. 10. (PI. II Figs. 18, 19.) From a given point of a line to set off a point at a given distance above or below t\t, line. Let A be the given point on the line CD. 1st. By Prob. 6, or 7, according to the position of A, con- struct a perpendicular at A to CD. 2d. Take off the given distance and set it off from A along the perpendicular, according as the point is required above or below the line. Remark. If the point may be set off, at pleasure, above or below CD, we may either construct a perpendicular at plea- sure, and set off the point as just described, or we may take the following method, which is more convenient and expedi- tious, and, with a little practice, will be found as accurate as either of the preceding. Take off the given distance in the dividers. Then place one foot of the dividers upon the paper, and describing an arc lightly with the other, notice whether it just touches, crosses, or does not reach the given line. If it crosses, the position taken for the point is too near the line, and the foot of the dividers must be shifted farther off; if the arc does not ttieet the line the foot of the dividers must be brought nearer to the line. If the arc just touches the line the point wheio the stationary foot of the dividers is placed being marked will be a point at the required distance from the given line. Verification. The correctness of this method may be verified by describing from a point set off by either of the other methods an arc with the given distance, which will be found just to touch the given line. Prob. 11. (PI. II. Fig. 20.) Through a given point to draw a line parallel to a given line. Let A be the given point ; CD the given line. 1st. Place one foot of the dividers at A, and bring the other foot in a position such that it will describe an arc that shall just touch CD at b. 2d. Without changing the opening of the dividers place one foot at a point J5, near the other end of CD, so that the arc described with this opening will just touch the line at c. PROBLEMS OF STRAIGHT LINES. 41 3d. Having marked the point B, draw through A B a line. This will be the required parallel. Verification. Having constructed the two perpendiculars A b, and B c, to CD, the distance b c will be found equal to A B if the construction is accurate. Prob. 12. (PI. II. Fig. 20.) To draw at a given distanct from a given line a parallel to the line. Let CD be the given line. 1st. Take off in the dividers the given distance at which the parallel line is to be drawn. 2d. Find, by either of the preceding methods, a point A t near one end of CD, and a point B near the other, at the given distance from CD. 3d. Having marked these points draw a line through them. This is the required parallel. Verification. The same proof may be used for this as in Prob. 11. Prob. 13. (PI. II. Fig. 21.) To transfer an angle ; or, from a point, on a given line, to draw a line which shall make with the given line an angle equal to one between two other lines on the drawing. Let the given angle to be transferred be the one bac between the lines a b and a c. Let DJEbe the given line, and A the point, at which a line is to be so drawn as to make with D E an angle at A equal to the given angle. 1st. With the dividers and pencil point describe, from o, with any opening, an arc, and mark the points b and c, where it crosses the lines containing the angle. 2d. Without changing this opening, shift the foot of the dividers to A, and describe from thence an arc as nearly as the eye can judge somewhat greater than the one 6c, and mark the point B where it crosses D E. 3d. Place the foot of the dividers at 6, and extend the pencil point to c. 4th. Shift the foot of the dividers to B, and, with the same opening, describe a small arc to cross the first at C. 5th. Having marked the point C, draw a line through A C. This line will make with D E the required angle. Prob. 14. (PI. II. Fig. 22.) From a point of a given line^ to draw a line making an angle of 60 with the given line. 42 INDUSTRIAL DRAWING. Let D E be the given line, and A the given point 1st. Take any distance in the dividers and pencil point, and Bet it off from A to B. 2d. From A and B, with the same opening, describe an arc, and mark the point where the arcs cross. 3d. Draw a line through A C. This line will make with the given one the required angle of 60. Prob. 15. (PI. II. Fig. 22.) From a point on a given line to draw a line making an angle 0/"45 with it. Let B be the given point on the line D E. 1st. Set off any distance Ba, along D E, from B. 2d. Construct by Prob. 6 a perpendicular to D E at a. 3d. Set off on this perpendicular a c equal to a B. 4th. Having marked the point c, draw through Be a line. This will make with D E the required angle of 45. Prob. 16. (PI. II. Fig. 23.) To divide a given angle into two equal parts. Let B A be the given angle. 1st. With any opening of the dividers and pencil point, describe an arc from the point A, and mark the points b and c, where it crosses the sides A B and A of the angle. 2d. Without changing the opening of the dividers, describe from b and c an arc, and mark the point D where the area cross. 3d. Draw a line through A D. This line will divide the given angle into equal parts. Verification. If we draw a line through b c, and mark the point d where it crosses AD; the distance b d will be found equal to c? c, and the line b c perpendicular to A D, if the construction is accurate. Remarks. Should it be found that the point of crossing at D of the arcs described from b and c is not well defined, owing to the obliquity of the arcs, a shorter or longer dis- tance than A b may be taken with which to describe them, without making any change in the points b and c first set off. Prob. 17. (PI. II. Fig. 24.) To find the line which wH\ divide into two equal parts the angle contained between two PROBLEMS OF CIRCLES, &C. 43 given lines, when the angular point, or point of meeting of fhi two lines, is not on the drawing. Let A B and CD be the two given lines. 1st. By Prob. 10 set off a point at b at any distance taken at pleasure from A B, and by Prob. 11 draw through this point a line parallel to A B. 2d. Set off a point d at the same distance from D as b is from A B, and draw through it a parallel to D C ; and mark the point c where these parallel lines cross. 3d. Divide the angle bed between the two parallels into two equal parts, by Prob. 16. The dividing line c a will also divide into two equal parts the angle between the given lines. Verification. If from any point, as o, on the line ca, a perpendicular o m be drawn to A B, and another on to CD, these two perpendiculars will be found equal if the construc- tion is accurate. CONSTRUCTION OP PROBLEMS OP ARCS OF CIRCLES, STRAIGHT LINES, AND POINTS. Prob. 18. (PI. II. Fig. 25.) Through two given points to describe an arc of a circle with a given radius. Let B and C be the two points. 1st. Take off the given distance in the dividers and pencil point, and with it describe an arc from B and C respectively, and mark the point A where the arcs cross. 2d. Without changing the opening of the dividers, describe an arc from the point A through B and C, which will be the one required. Prob. 19. (PI. II. Fig. 26.) To find the centre of a circle, or arc, the circumference of which can be described through three given points, and to describe it. Let A, B, and C be the three given points 1st. Take off the distance B A between the intermediate point and one of the exterior points, as A, with the dividers and pencil point, and with this opening describe two arcs from B, on either side of B A. 44 INDUSTRIAL DRAWING. 2d. With the same opening describe from A two like arcs, and mark the points a and b where these cross the two described from B. 3d. Draw a line through a b. 4th. With the distance B C in the dividers describe, from B and 7, arcs as in the preceding case, and mark the points c and d where these cross ; and then draw a line through c d. 5th. Having marked the point where the t yo lines thus drawn cross, place the steel point of the dividers at 0, and extending the pencil point to J., or either of the three given points, describe an arc, or a complete circle, with this opening This will be the required arc or circle. Verification. The fact that the arc or circle is found to pass accurately through the three points is the best proof of the correctness of the operations. Prob. 20. (PL II. Fig. 26.) At a point on an arc or the circumference of a circle to construct a tangent to the arc or the nrde. Let D be the given point and the centre of the circle. 1st. Draw through D a radius D, and prolong it outwards from the arc. 2d. At D construct by Prob. 6 a perpendicular ED Fto D. This is the required tangent. If the centre of the arc or the circle is not given, proceed as follows : 1st. With any convenient opening in the dividers and pencil point (Fig. 26) set off from D the same arc on each side of it, and mark the points A and B. 2d. Take off the distance A B and describe with it arcs from A and B on each side of the given arc, and mark the points a and b where they cross. 3d. Draw a line through a b. 4th. Construct a perpendicular to a b at D. This is the required tangent. Verification. Having set off from D the same distance on each side of it along a 5, and having set off also any distance from D along E F, the distance from this last point to the other two set off on a b will be found equal, if the construction is accurate. PROBLEMS OF CIRCLES, &C. 45 Remarks. It sometimes happens that the point to which a tangent is required is so near the extremity of the arc, as at A, or (7, that the method last explained cannot be applied. In such a case we must first find the centre of the arc, or circle, which will be done by marking two other points, as B and J., on the arc, and by Prob. 19 finding the centre of the circle of which this arc is a portion of the circumference. Having thus found the centre, the tangent at will be con- structed by the first method in this Prob. Prob. 21. (PL II. Fig. 27.) At a given point on the cir- cumference of a given circle, to construct a circle, or arc, of a given radius tangent to the given circle. Let B be the given point, and D the centre, which is either given, or has been found by Prob. 19. 1st. Through DB draw the radius, which extend outwards if the centres of the required circle and of the given one are to lie on opposite sides of a tangent line to the first circle at B ; or, in the contrary case, extend it, if requisite, from D in the opposite direction. 2d. From B set off along this line the length of the given radius of the required circle to (7, or to A. 3d. From (7, or A, with the distance CB, or AB, describe a circle. This is the one required. Prob. 22. (PI. II. Fig. 28.) From a given point without a given circle, to draw two tangents to the circle. Let A be the centre of the given circle, and B the given point. 1st. Through AB draw a line. 2d. Divide the distance AB into two equal parts by Prob. 5. 3d. From (7, with the radius CA, describe an arc, and mark the points D, and E, where it crosses the circumference of the given circle. 4th. From B draw lines through D and E. These lines are the required tangents. Verification. If lines are drawn from A, to D and E } they will be found perpendicular respectively to the tangents, if the construction is accurate. Remark. In this Prob., as in most geometrical construe- 46 INDUSTRIAL DRAWING. tions, many of the lines of construction need not be actually drawn, either in whole, or in part. In this case, for example, a small portion of the line A B, at its middle point, is alone necessary to determine this point. In like manner, the points L and E could have been marked, without describing the arc actually, but by simply dotting the points required. In this manner, a draftsman, by a skilful selection of his lines of construction, and using only such of them, in whole, or in part, as are indispensably requisite for the solution, may, in complicated constructions, avoid confusion from the intersec- tion of a multiplicity of lines of construction, and abridge his labor. Prob. 23. (PI. II. Fig. 29.) To draw a tangent to two given circles. Let A be the centre of one of the circles, and A Cits radius; B the centre, and BE the radius of the other. 1st. Through AB draw a line. 2d. From C set off CD equal to BE. 3d. From A, with the radius AD, equal to the difference between the radii of the given circles, describe a circle. 4th. From B by Prob. 22 draw a tangent BD to this last circle, and through the tangential point D, a radius A C to the given circle. 5th. Through C draw a line parallel to BD. This line will touch the other given circle, and is the required tangent. Verification. CE will be found equal to BD, if the con- struction is accurate. Prob. 24. (PI. II. Fig. 30.) Having two lines that make an angle, to construct within the angle, a circle with a given radius tangent to the two given lines. Let AB and A C be the two given lines containing the angle. 1st. By Prob. 16 construct the line AD which divides the given angle into two equal parts. 2d. By Prob. 10 set off a point b at a distance from AB equal to the given radius, and through this point draw the line ab parallel to AB, and mark the point a where it crosses AD. PROBLEMS OF CIRCLES, &C. 47 8d. From a, with the given radius, describe a circle. This is the one required. Verification. The distances Ac, and Ad : will be found equal, if the construction is accurate. Prob. 25. (PI. II. Fig. 31.) Having two lines containing an angle, and a given radius of a circle, to construct, as in the last case, this circle tangent to the two lines ; and then to construct another circle which shall be tangent to the last and also to the two lines. Let AB and CD be the two lines, the point of meeting of which is not on the drawing. 1st. By Prob. 17 find the line EFthzt equally divides the angle between the lines. 2d. By Prob. 24- construct the circle, with the given radius oj, tangent to these two lines. 3d. At b, where EF crosses the circumference, draw by Prob. 20 a tangent to the circle, and mark the point d where it crosses AB. 4th. From d, set off de equal to db, and mark the point e. 5th. At e construct by Prob. 6' a perpendicular ef, to AB, and mark the point/ where it crosses EF. 6th. From/ with the radius fe, describe a circle. This is tangent to the first circle, and to the two given lines. Remark. In like manner a third circle might be con- structed tangent to the second and to the two given lines ; and so on as many in succession as may be wanted. Prob. 26. (PI. III. Fig. 32.) Having a circle and right line given, to construct a circle of a given radius which shall be tangent to the given circle and right line. Let G be the centre of the given circle, and AB the given line. 1st. By Prob. 12 draw a line parallel to AB, and at a distance BG from it, equal to the given radius. 2d. Draw a radius CD through any point D of the given circle, and prolong it outwards. 3d. From D set off, along the radius, a distance DE equal to BG, or the given radius. 4th. From C, with the distance CE, describe an arc 48 INDUSTRIAL DRAWING. and mark the point F where it crosses the line parallel to AB. 5th. From the point F, with the given radius describe a circle. This is the one required. Remarks. If the construction is accurate a line drawn from F to G will pass through the point where the circles touch, and one drawn from ^perpendicular to AB will pass through the point where the circle touches the line. If from the centre (?, a perpendicular is drawn to AB, and the points a, I and d where the perpendicular crosses the line and the given circle are marked, it will be found that the given radius cannot be less than one half of ab nor greater than one half of ad. Prob. 27. (PI. III. Fig. 33.) Having a circle and right line given, to construct a circle which shall he tangent to the given circle at a given point, and also to the line. Let C be the centre of the given circle, D the given point on its circumference, and AB the given line. 1st. By Prob. 20 construct a tangent to the given circle at the point D; prolong it to cross the given line, and mark the point A where it crosses. 2d. By Prob. 16 construct the line AE which bisects the angle between the tangent and the given line. 3d. Through CD draw a radius, and prolong it to cross the bisecting line at E. 4th. Mark the point E, and with the distance ED describe a circle. This is the required circle. Prob. 28. (PI. III. Fig. 34.) Having a circle and right line, to construct a circle which shall he tangent to the given circle, and also to the line at a given point on it. Let C be the centre of the given circle ; AB the given line, and a the given point on it. 1st. By Prob. 6 construct a perpendicular at a to the given line. 2d. From a set off ah equal to the radius Cd of the given circle. 3d. Draw a line through bC, and by Prob. 5 bisect the distance bC by a perpendicular to the line bC. 4th. Mark the point c, where this perpendicular crosses the PROBLEMS OF CIRCLES, &C. 49 one at a; and with ac describe a circle. This is the one required. Prob. 29. (PI. III. Fig. 35.) Having two circles, to construct a third which shall be tangent to one of them at a given point, and also touch the other. Let C and B be the centres of the two given circles ; and D the given point on one of them, at which the required circle is to be tangent to it. 1st. Through CD draw a line, which prolong each way from C and D. 2d. From D set off towards the distance D A equal to the radius BF of the other given circle. 3d. Through AB draw a line, and by Probs. 5 and 6, bisect AB by the perpendicular GK 4th. Mark the point E, where the perpendicular crosses the line CD prolonged ; and with the distance ED describe a circle from K This is the one required. Prob. 30. (PI. III. Fig. 36.) Having a given distance, or line, and the perpendicular which bisects it, to construct three arcs of circles, the radii of two of which shall be equal, and of a given length, and their centres on the given line ; and the third shall pass through a given point on the perpendicular and be tangent to the other two circles. Let AB be the given line, and D the given point on the perpendicular to AB through its middle point C ; and let the distance CD be less than A C, the half of AB. 1st. Take any distance, less than CD, and set it off from A and B, to b and e, and mark these two points for the centres of the two arcs of the equal given radii less than CD. 2d. Set off from D the distance DC equal to Ab, and through be draw a line. 3d. Bisect be by a perpendicular, by Probs. 5 and 6, and mark the point d where it crosses the perpendicular to AB prolonged below it, the point d is the centre of the third arc. 4th. Draw a line through db, and prolong it ; and also one through de which prolong. 5th. From b, with the distance bA, describe an arc from A to m on the line db prolonged ; and one, from the othei centre e, from B to n on the line de prolonged. 4 60 INDUSTRIAL DRAWING. 6th. From d, with the distance dD, describe an arc around to the two lines db, and de prolonged. This is the third arc required, and touches the other two where they cross the fines db and de prolonged at m and n. Remark. This curve is termed a half oval, or a three centre curve. The other half of the curve, on the other side of AB, can be drawn bj setting off a distance Cg equal to Cd, and by continuing the arcs described from b and e around to lines drawn from g, through b and e ; and by connecting these arcs by another described from g, with a radius equal to Dd. Prob. 81. (PI. III. Fig. 37.) Having a given line and the perpendicular that bisects it; also two lines drawn through a given point, on the perpendicular, and each making the same angle, with it; to construct a curve formed of four arcs of circles, two of these arcs to have equal given radii, and their centres to lie on, the given line, and at equal distances from its extremities ; each of the other arcs to have equal^ radii, and to be tangent respec- tively to one of the given lines where it crosses the perpendicular and also to one of the first arcs. Let CD be the given line ; B the given point on the bisect- ing perpendicular; and Bm, Bn, the two lines, drawn through B, making the same angle with the perpendicular. 1st. From and D, set off the same distance to b and a, for the given radii of the two first arcs ; which distance must, in all cases, be taken less than the perpendicular distance from the point b or a, to one of the given lines through B. 2d. At B draw a perpendicular to the line Bm. 3d. Set off from B, along this perpendicular, a distance Bd equal Cb. 4th. Draw a line through bd, and bisect this distance by a perpendicular. 5th. Having marked the point c, where this last perpen- dicular crosses the one at B, draw through cb a line, and prolong it beyond b. 6th. From b, with the distance 1C, describe an arc, which prolong to the line through be ; this is one of the first required arcs. 7th. From c, with a distance cB, describe an arc. This if one of the second required arcs. PROBLEMS OF CIRCLES, &C. 51 8th. Through a, and the point/ where be crosses the per pendicular BA prolonged, draw a line. 9th. From a, set off ae equal to be. 10th. From a and e, with radii respectively equal to bC, and cB, describe arcs. These are the others required ; and CBD the required curve. Remark. If the construction is accurate, the perpendicular through BA will bisect the distance ec. This curve is termed a four centre obtuse or pointed curve, according as the distance AB is less or greater than A G. Prob. 32. (PL III. Fig. 38.) Having a line, and the per- pendicular which bisects it, and a given point on the perpen- dicular ; to construct a curve formed of jive arcs of circles, the consecutive arcs to be tangent ; the centres of two of the arcs to be on the given line, and at equal distances from its extremities; the radii of the two arcs, respectively tangent to these two, to be equal, and of a given length ; and the centre of the fifth arc, which is to be tangent to these two last, to lie on the given perpen dicular. Let AB be the given line, and C the point on its bisecting perpendicular LC. 1st. Take any distance, less than LC y and set it off from B to D, and from A tof. 2d. From C set off CG equal to BD, and draw a line from GtoD. 3d. Bisect the distance DG by a perpendicular ; and mark the point E, where this perpendicular crosses the perpen- dicular L C prolonged. 4th. Draw a line from E to D. 5th. Take any distance, less than CE, equal to the given radius of the second arc, and set it off from G to F. 6th. through F draw a line FH parallel to AB. 7th. Take off GF, the difference between CF and GQ, and with it describe an arc from D ; and mark the points a and b where it crosses the lines DE and FH. 8th. Take any point c on this arc, between the points a and b, and draw from it a line to F. 9th. Bisect the line cF by a perpendicular, and mark the point /where it crosses the perpendicular GL prolonged. 52 INDUSTRIAL DRAWING. 10th. From c draw a line through D and prolong it ; and another from /prolonged through c. llth. From c draw a perpendicular to CI : and from d, where it crosses CI, set off dg equal to cd, and mark the point g. 12th. From /draw a line through g and prolong it; and one from g prolonged through / 13. From D and/ with the distance BD, describe the arcs Bm, and Ap ; from c and g, with the distance cm, or gp, describe the arcs mn, and po ; and from / with the distance 1C, describe the arc no. The curve BCA is the one required. Remarks. This curve is also termed a semi oval; and, from the number of arcs of which it is composed, a curve of Jive centres. Prob. 33. (PI. III. Fig. 39.) Having two parallel lines, to construct a curve of three centres which shall be tangent to the two parallels at their extremities. Let AB and CD be the given parallels ; and B and D the points at which the required curve is to be drawn tangent. 1st. From B construct a perpendicular to AB, and mark the point b where it crosses CD; and also a perpendicular at D to CD. 2d. From B, set off any distance Be less than the half of Bb and through c draw a line parallel to AB, and mark the point d where it crosses the perpendicular to CD. 3d. From c, set off, along cd prolonged, the distance ca equal to cB. 4th. Taking ca, as the radius of the first arc, construct a quarter oval by Prob. 30 through the points a and D. 5th. Prolong the arc described from c to the point B. The curve BaD is the one required. Remark. This curve is termed a scotia of two centres. Prob. 34. (PI. III. Fig. 40.) Having two parallels, to con- struct a quarter of a curve of five centres tangent to them at their extremities. Let AB and CD be the given parallels, and B and D theiz extremities. 1st. Proceed, as in the last case, to draw the perpendiculars PROBLEMS OF CIRCLES, &C. 53 at B, and D, and a parallel to AS through a point c, taken on the first perpendicular, at a distance from B less than the half of m 2d. Set off from c a distance ca equal to cB ; and on cut and Dd describe the quarter oval by Prob. 32. 3d. Prolong the first arc from a to B, which will complete the required curve. Remark. This curve is termed a scotia of three centres. Prob. 35. (PI. III. Fig. 41.) Having two parallels and a given point on each, to construct two equal arcs which shall be tangent to each other and respectively tangent to the parallels ai the given points. Let AB and CD be the two parallels ; B and D the given points. 1st. Draw a line through BD, and bisect the distance BD. 2d. Bisect each half BE, and ED by perpendiculars. 3d. From B, and D draw perpendiculars to AB and CD, and mark the points a, and b, where they cross the bisecting perpendiculars. - 4th. From a, with the distance aE, describe an arc to B; and from b, with the same distance, an arc ED, These are the required arcs. Prob. 36. (PI. III. Fig. 42.) Having two parallels, and a point on each, to construct two equal arcs which shall be tangent to each other, have their centres respectively on the parallels, and pass through the given points. Let AB and CD be the parallels, B and D the given points. 1st. Join BD by a line and bisect it. 2d. Bisect each half BE, and ED by a perpendicular ; and mark the points a and b, where the perpendiculars cross the parallels. 3d. From a, with aB, describe the arc BE; and from b, with the same distance, the arc DE. These are the required arcs. 54 INDUSTRIAL DRAWING. CONSTRUCTION OF PROBLEMS OF CIRCLES AND RECTILINEAL FIGURES. Piob. 37. (PL III. Fig. 43.) Having the sides of a trianglt to construct the figure. Let A C, BO, and AB be the lengths of the given sides. 1st. Draw a line, and set off upon it the longest side AB. 2d. From the point J., with a radius equal to A C, one ol the remaining sides, describe an arc. 3d. From the point B, with the third side BC, describe a second arc, and mark the point G where the arcs cross. 4th. Draw lines from C, to A and B. The figure A CB is the one required. Remark. The side AC might have been set off from B, and BC from A ; this would have given an equal triangle to the one constructed, but its vertex would have been placed differently. Remark. This construction is also used to find the position of a point when its distances from two other given points are given. We proceed to make the construction in this case like the preceding. It will be seen, that the required point can take four different positions with respect to the two others. Two of them, like the vertex of the triangle, will lie on one side of the line joining the given points, and the other two on the other side of the line. Prob. 38. (PI. III. Fig. 44.) Having the side of a square to construct the figure. Let AB be the given side. 1st. Draw a line, and set off AB upon it. 2d. Construct perpendiculars at A, and J5, to AB. 3d. From A and B, set off the given side on these perpen- diculars to G and D ; and draw a line from G to D. The figure ABCD is the one required. Prob. 39. (PL III. Fig. 45.) Having the two sides of a parallelogram, and the angle contained by them, to construct the figure. Let AB, and AC be the given sides; and E the given angle. PROBLEMS OF CIRCLES, &C. 55 1st. Draw a line, and set off AB upon it. 2d Construct at A an angle equal to the given one bj Prob. 13. 3d. Set offj along the side of this angle, from A, the othei given line A 0. 4th. From C, with the distance AB describe an arc, and from B with the distance A C describe another arc. 5th. From the point D, where the arcs cross, draw lines to (7, and B. The figure ABDCis the one required. Prob. 40. (PI. III. Fig. 46.) To circumscribe a given triangle by a circle. Let ABO be the given triangle. As the circumference of the required circle must be described through the three given points A, B and (7, its centre and radius will be found precisely as in Prob. 19. Prob. 41. (PL III. Fig. 47.) In a given triangle to inscribe a circle. Let ABC be the given triangle. 1st. By Prob. 16 construct the lines bisecting the angles A, and G ; and mark the point D where these lines cross. 2d. From D by Prob. 8 construct a perpendicular DE, to AC. 3d. From D, with the distance DE, describe a circle. This is the one required. Prob. 42. (PI. IY. Fig. 48.) In a given circle to inscribe a square. Let be the centre of the given circle. 1st. Through draw a diameter AB, and a second dia- meter CD perpendicular to it. 2d. Draw the lines AC, CB, ED, and DA. The figure ADBC is the one required. Prob. 43. (PI. IV. Fig. 48.) In a given circle to inscribe a regular octagon. 1st. Having inscribed a square in the circle bisect each of its sides ; and through the bisecting points and the centre G draw radii. 2d. Draw lines from the points cZ, 5, a, c, where these radii meet the circumference, to the adjacent points D^ A, &c. The figure dAbC&c. is the one required. 56 INDUSTRIAL DRAWING. Remark. By bisecting the sides of the octagon, and draw ing radii through the points of bisection, and then drawing lines from the points where these radii meet the circumference to the adjacent points of the octagon, a figure of sixteen equa] sides can be inscribed, and in like manner one of 32 sides, &c. Prob. 44. (PL IY. Fig. 49.) To inscribe in a given circk a regular hexagon. Let be the centre of the given circle. 1st. Having taken off the radius OA, commence at A, and Bet it off from A to B t and from A to F^ on the circum- ference. 2d. From B set off the same distance to C ; and from C to D, and so on to F. 3d. Draw lines between the adjacent points. The figure AEG &c. is the one required. Remark. By a process similar to the one employed for constructing an octagon from a square, we can, from the hexagon, construct a figure of 12 sides ; then one of 24 ; and so on doubling the number of sides. Prob. 45. (PI. IV. Fig. 49.) To inscribe in a given circle an equilateral triangle. Having, as in the last problem, constructed a regular hex- agon, draw lines between the alternate angles, as A (7, CE, and EA ; the figure thus formed is the one required. Prob. 46. (PL IY. Fig. 50.) To inscribe in a given circle a regular pentagon. Let be the centre of the given circle. 1st. Draw a diameter of the circle AB, and a second one CD perpendicular to it. 2d. Bisect the radius OB, and from the point of bisection 2 set off the distance a C t to 5, along AB. 3d. From C, with the radius Cb, describe an arc, and mark the points H, and I t where it crosses the circumference of the given circle. 4th. From H, and /, set off the same distance to G and J on the circumference. 5th. Draw the lines Off, HG, GK, KI, and 1C. The figure CHGKIis the one required. PROBLEMS OF CIRCLES, &C. 57 Prob. 47. To construct a regular figure, the sides of which shall be respectively equal to a given line. Let AB be the given line. First Method. (PI. IV. Fig. 50.) 1st. Construct any circle, and inscribe within it a regular figure, by one of the preceding Probs. of the same number of sides as the one required. Let us suppose for example that the one required is a pen- tagon. 2d. Having constructed this inscribed figure, draw from the centre of the circle, through the angular points of the figure, lines ; and prolong them outwards, if the side of the inscribed figure is less than the given line. 3d. Prolong any one of the sides, as Of, of the inscribed figure, and set off along it, from the angular point C, a dis- tance Cm equal to the given line. 4th. Through m, draw a line parallel to the line drawn from through 0, and mark the point n, where it crosses the line drawn from through /. 5th. Through n, draw a line parallel to CI, and mark the point o, where it crosses the line OC prolonged. 6th. From 0, set off, along the other lines drawn from G through the other angular points, the distances Op, Oq, and Or, each equal to Om, or On. 7th. The points o, p, r, and n being joined by lines ; the figure opqrn is the one required. Second Method. (PI. IV. Fig. 51.) 1st. Draw a line and set off the given line AB upon it. 2d. At B construct a perpendicular to AB. 3d. From B, with BA, describe an arc Aa. 4th. Divide this arc into as many equal parts as number of sides in the required figure ; and mark the points of division from a, 1, 2, 3, &c. 5th. From A, with AB, describe an arc, and mark the point c where it crosses the arc Aa. 6th From B draw a line through the division point 2. 7th. From c, set off the distance c2, to b, on the arc Be. 58 INDUSTRIAL DRAWING. 8th. From A, draw a line through b, and mark the point where it crosses 2. 9th. From 0, with the distance OA, or OB, describe a circle. 10th. Set off the distance AB to C, D, &c., on the circum- ference. llth. Draw the lines 0, CD, DE, &c. This is the required figure. Remarks. The figure taken to illustrate this case is the pentagon, for the purpose of comparing the two methods. Prob. 48. (PI. IV. Fig. 51.) To circumscribe a given circle by a regular figure. 1st. Inscribe in the circle a regular figure of the same number of sides as the one to be circumscribed. 2d. At the angular points of the inscribed figure, draw tangents to the given circle, and mark the points where the tangents cross. These points are the angular points of the required figure, and the portions of the tangents between them are its sides. Remarks. The figure' taken to illustrate this, is the cir- cumscribed regular pentagon bcdef. Prob. 49. (PL IY. Fig. 49.) To inscribe a circle in a given regular figure. 1st. Bisect any two adjacent sides of the figure by perpen- diculars, and mark the point where they cross. 2d. From this point, with the distance to the side bisected, describe a circle. This is the one required. Remarks. The figure taken to illustrate this case is the regular hexagon ; mn and np are the adjacent sides bisected by the perpendiculars to them aO, and bO ; is the centre of the required circle, and Oa its radius. Prob. 50. (PL IV. Fig. 52.) To inscribe, in a given circle, a given number of equal circles which shall be tangent to the given circle, and to each other. Let be the centre of the given circle. 1st. Divide the circumference into as many equal parts, by lines drawn from 0, as the number of circles to be inscribed. Let us take, for illustration, six as the required number. PROBLEMS OF CIRCLES, &C. 59 2d. Bisect the angle, as DOB, between any two of these lines of division, and prolong out the bisecting line. 3d. Construct a tangent to the given circle at either B, or 7), and mark the point a where this tangent crosses the bisecting line. 4th. From a, set off aB to J, along the bisecting line. 5th. A't b construct a perpendicular to Oa, and mark the point c where it crosses OB. 6th. From c, with the distance cB t describe a circle. Thia is one of the required circles. 7th. From the other points of division, D, &c., set off the same distance Be, and from the points thus set off with this distance describe circles. These are the other required circles. Prob. 51. (PI. IV. Fig. 52.) To circumscribe a given circle by a given number of circles tangent to it, and to each oilier. 1st. Having divided the given circle into a number of equal parts, the same as the given number of required circles: bisect, in the same way, the angle between any two adjacent lines of division. Let us take for illustration six as the number of required circles. 2d. Prolong outwards one of the lines of division, as OD, and also the line, as Od, that bisects the angle between it and the adjacent line of division Oa. Construct a tangent aiD to the given circle ; and mark the point /where it crosses the bisecting line. 3d. From/ set off fD' to d, along the bisecting line ; and at d, construct a perpendicular to this line, and mark the point g where it crosses the line OD'- 4th. From g, with the distance gD\ or gd, describe a circle. This is one of the required circles. 5th. Prolong outwards the other lines of division ; and set off along them, from the points where they cross the circum- ference, the distance D'g ; and from these points with this distance describe circles. These are the remaining required circles. 60 INDUSTRIAL DRAWING. CONSTRUCTION OF PROPORTIONAL LINES AND FIGURES. Prob. 52. (PL IV. Fig. 53.) To divide a given line into parts which shall be proportional to two other given lines. Let AB be the given line to be divided ; ac and cb the other given lines. 1st. Through A draw any line making an angle with AB. 2d. From A set off Ac equal to ac : and from c the other line cb. 3d. Draw a line through B, b ; and through c a parallel to Bb, and mark the point where it crosses AB. This is the required point of division ; and A C is to CB as ac is to cb. Prob. 53. (PI. IV. Fig. 54.) To divide a line into any number of parts which shall be in any given proportion to each other, or to the same number of given lines. Let AB be the given line, and let the number of pro- portional parts for example into which it is to be divided be four, these parts being to each other as the numbers 3, 5, 7, and 2, or lines of these lengths. 1st. Through A draw any line making an angle with AB. 2d. From any scale of equal parts take off three divisions, and set this distance off from A to 3 ; from 3 set off five of the same divisions to 5 ; from 5 set off seven to 7; and from 7 two to 2. 3d. Draw a line through B2, and parallels to it through the points 7, 5, and 3, and mark the points d, c, and b where the parallels cross AB. The distances Ab, be, cd, and dB are those required. Remark. Any distance from a point, as A for example, on A B, to any other point as d, is to the distance from this point to any other, as Ab for example, as is the corresponding distance A7 to AB, on the line A2. Prob. 54. (PI. IV. Fig. 55.) To find a fourth proportional to three given lines. Let ab, be, and ad be the three given lines to which it is PROBLEMS OF PROPORTIONAL LINES, &C. 61 required to find a fourth proportional which shall be to ad as ab is to ac. 1st. Draw a line, and, from a point A, set off AB equal tc ab ; and BC equal to be. 2d. Through A draw any line, and set off upon it AD, equal to ad. 3d. Draw a line through DB, and a parallel to DB through C, and mark the point E where this crosses the line drawn through A. The distance DE is the required fourth pro- portional. Prob. 55. (PL IV. Fig. 56.) To find the line which is a mean proportional to two given lines. Let ab and be be the given lines. 1st. Draw a line, and set off on it AB } and BC, equal respectively to ab, and be. 2d. Bisect the distance A Cj and, from the bisecting point 0, describe a semicircle with the radius OC. 3d. At B construct a perpendicular to AC; and mark the point D where it crosses the circumference. The distance BD is the line required ; and ab is to BD as BD is to be. Prob. 56. (PI. IV. Fig. 57.) To divide a given line into two parts, such that the entire line shall be to one of the parts, as this part is to the other. Let ab be the given line. 1st. Draw a line, and set off AB equal to ab; and at B construct a perpendicular to AB. 2d. Set off on the perpendicular BD equal to the half of AB, and draw a line through AD. 3d. From D, with DB, describe an arc, and mark the point (7, where it crosses AD. 4th. From A, with AC, describe an arc, and mark the point E, where it crosses AB. The point E is the one required ; and AB is to AE, as AE is to EB. Remark. This construction is used for inscribing a regulai decagon in a given circle. To do this divide the radius of the given circle in the manner just described. The larger portion is the side of the required regular decagon. Having described the regular decagon, the regular pen 62 INDUSTEIAL DRAWING. tagoii can be formed, by drawing lines through the alternate angles of the decagon. Prob. 57, (PL IV. Fig. 58.) Having any given figure, to construct another, the angles of which shall be the same as tht, angles of the given figure, and the sides shall be in a given pro- portion to its sides. Let ABGDEF be the given figure. 1st. Prolong any two of the adjacent sides of the given figure, as AB, and AF, if the one required is to be greater than the given one ; and, from A, draw lines through the other angular points C, D, and E. 2d. From A set off a distance Ab, which is in the same proportion to AB, as the side of the required figure corres- ponding to BC, is to J3C ; or, in other words, AB must be contained as many times in Ab as JBG is in the corresponding side of the required figure. 3d. From b draw a line parallel to BC, and mark the point c, where it crosses A G prolonged ; from c draw a parallel to CD, and mark the point where it crosses AD prolonged ; and so on for each required side. The figure Abcdef is the one required. CONSTRUCTION OF EQUIVALENT FIGURES. Prob. 58. (PL IV. Fig. 59.) To construct a triangle which shall be equivalent to a given parallelogram. Let ABCD be the given parallelogram. 1st. Prolong the base AB, and set off BE equal to AB. 2d. Draw lines from C, to A and E. The triangle A CE is the one required. Prob. 59. (PL IV. Fig. 60.) To construct a triangle which shall be equivalent to a given quadrilateral. Let ABCD be the given quadrilateral. 1st. Draw a diagonal, as AC. 2d. From B the adjacent angle to C, to which the diagonal is drawn, draw a line parallel to A C, prolong the side AB opposite to BC f and mark the point F, where it crosses the parallel to AC. PROBLEMS OF EQUIVALENT FIGURES, &C. (>3 3d. Draw a line from G to F. The triangle FCD is the one required. Prob. 60. (PI. IT. Fig. 61.) To construct a triangle equiva lent to any given polygon. Let ABCDEFG be the given polygon. 1st. Take any side, as AB, as a base, and, from A and B, draw the diagonals AF and BD to the alternate angles to A and B. 2d. From G and (7, the adjacent angles, draw Oa parallel to FA, and Cb to DB. 3d. From the alternate angles F and Z>, draw the lines Fa and Db. A figure dbDEF is thus formed, which is equivalent to the given one, and having two sides less than it. 4th. From the angles a and &, at the base of this new figure, draw diagonals to the alternate angles, to a and b (in the Fig. this is the angle E\ and proceed, precisely as in the 3d operation, to form another figure equivalent to the last formed, and having two sides less than it. Proceed in this way until a quadrilateral or pentagon is formed equivalent to the given figure, and convert this last into its equivalent triangle, which will be the one required. The case taken for illustration is a heptagon, and HEX is the equivalent triangle. Prob. 61. To construct a triangle equivalent to any regular polygon. 1st. By Prob. 49 find the radius of the circle inscribed in the polygon. 2d. Set off on a right line a distance equal to half the sum of the sides of the polygon. This distance will be the base of the equivalent triangle, and the radius of the inscribed circle its perpendicular or altitude. CONSTRUCTION OF CURVED LINES BY POINTS. Prob. 62. (Pl.IY.Fig. 62.) To construct an ellipse on given transverse and conjugate diameters. Definitions. An ellipse is an oval-shaped curve. The line 64 INDUSTRIAL DRAWING. A B that divides it into two equal and symmetrical parts is termed the transverse axis. The line C- D, perpendicular to the transverse at its centre point, is termed the conjugate axis. The points A and B are termed the vertices of the curve. The points E and F, on the transverse axis, which are at a dis- tance from the points G and D, the extremities ef the con- jugate, equal to the semi-transverse A, are termed the foci of the ellipse. The ellipse has the characteristic feature that the sum of any two lines, as m E and m F, drawn from a point, as m, on the curve to the foci, is equal to the transverse axis. It is this characteristic property that is used in constructing the curve by points. First Method. Let ab be the length of the transverse, and cd that of the conjugate diameter. 1st. Set off ab, from A to B, on any line, bisect AB by a perpendicular, and set off on this perpendicular the equal distances 00, and OD, each equal to the half of cd 2d. From (7, with the radius OA, describe an arc, and mark carefully the points E and F, where it crosses AB. 3d. From A, take off any distance Ab, and mark the point b. 4th. With the distance Ab describe an arc from E, and a like one from F. 5th. Take off the remaining portion IB of AB ; and with it describe from the points E and F arcs, and mark the points m, n, o, p, where these arcs cross. These are four points of the required ellipse. 6th. To obtain other points of the curve take any other point on AB, as c; and with the distances Ac and cB, describe arcs from E and F, as before. The points where these cross are four more points ; and so on for as many aa may be required. Second Method. Having cut a narrow strip of stiff paper, so that one of its edges shall be a straight line, mark off from the end of this CONSTRUCTION OF CURVED LINES BY POINTS. '65 strip, along the straight edge, a distance rt equal to A 0, half the transverse axis of the ellipse ; and from the same point a distance rs equal to 00, half the conjugate axis. 1st. Place the strip thus prepared so as to bring the point s on the line A B of the transverse axis, and the point t on the line CD; having the strip in this position, mark on the drawing the position of the point r; this is one point of the required curve. 2d. Shift the strip of paper to a new position, to the right or left of the first, and having fixed it so that the point s is on AS, and the point t on CD, mark the second position of the point r ; this is the second point of the curve. By placing the strip so that the first point marked may be near A, and gradually shifting it towards C, as many points may be marked as may be wanted ; and so on for the remainder of the curve. 3d. Through the points thus marked draw a curved line. It will be the required ellipse. Remark. The accuracy of the curve when completed will depend upon the steadiness of hand and correctness of eye of the draftsman. When the points of the curve have been obtained by the first method, the accuracy of their position may be tested as follows : Joining the corresponding points, as m and n, or o and p, above and below the line AB, by right lines, these lines mn and op will be perpendicular to AB, and be bisected by it if the construction is cor- rect. Prob. 63. (PI. V. Fig. 63.) Having the transverse axis of an ellipse, and one point of the curve, to construct the conjugate axis. Let AB be the transverse axis ; and a the given point. 1st. Bisect AB by a perpendicular ; and from the centre point 0, with a radius OA, describe a semicircle. 2d. Construct, from a, a perpendicular to AB, and mark wne point c where it crosses the semicircle. 3d. Join and c ; and from a draw a parallel to AB, and mark the point r where the parallel crosses Oc. 4th. From 0, set off Or to C on the perpendicular. The distance OC is the required semi-conjugate axis. 5 66 INDUSTRIAL DRAWING. Remark. Having the semi-conjugate other points of the curve can be found, as in the preceding Prob. Prob. 64. (PL Y. Fig. 63.) At a point on the curve of an ellipse to construct a tangent to the curve. Let m be the point at which the tangent is to be drawn. 1st. With OA, as a radius, describe a semicircle on AB. 2d. From m construct a perpendicular mq to AB, and mark the point n, where it crosses the semicircle. 3d. At n construct a tangent to the semicircle, and prolong it to cut the transverse axis prolonged at p. 4th. Through p and m draw a line. This is the required tangent. Prob. 65. (PI. V. Fig. 63.) From a point without an ellipse to construct a ta.ngent to the curve. Let D be the given point from which the tangent is to be drawn. 1st. Join the point D with the centre of the ellipse, and mark the point e where this line cuts the ellipse. 2d. From 0, with the radius OA, describe the semicircle AhB. 3d. Through e draw a perpendicular eq to the transverse axis, and mark the point g where it cuts the semi- circle. 4th. Through the point D draw a perpendicular Df to the transverse axis, and prolong it towards d. 5th. From draw a line through g, prolong it to cut the perpendicular Df, and mark the point d of intersection. 6th. From d, by Prob. 22, construct a tangent to the semi- circle, and mark the point h of contact. 7th. From h draw a perpendicular hk to the transverse axis, and mark the point i where it cuts the ellipse. 8th. From D the given point draw a line through i. This is the required tangent. Remark. The other tangent from D to the ellipse can be readily obtained by constructing the second tangent to the circle, and from it finding the point on the ellipse which corresponds to the one on the circle, in the same manner a? the point i is found from h. CONSTRUCTION OF CURVED LINES BY POINTS. 67 Prob. 66. (PL IV. Fig. 64.) To copy a given curve by points. Let A CD be the given curve to be copied. 1st. Draw any line as AB across the curve. 2d. Commencing at A set off along AB any number of equal distances as A~L, 1 2, 2 3, &c. 3d. Through the points 1, 2, 3, &c., construct perpendicu- lars to AB, and prolong them to cut the curve at m, n, o, p, &c. 4th. Having drawn a right line, set off on it the equal distances A 1, 1 2, &c., taken off from the line AB, and through the points thus set off on the second line draw per- pendiculars to it. From the points where these perpendicu- lars cross the line, commencing at the first, set off the distances 1 m, 2 o, &c., on the portion of the perpendicu- lars above the line ; and the distances 1 n, 2 p, &c., below it. The curve drawn through the points thus set off will be a copy of the given one ; the accuracy of the copy depending on the skill of the draftsman. Prob. 67. (PL IV. Figs. 65, 66.) To make a copy of a given curve, so that the lines of the copy shall be greater or smaller than the corresponding lines of tJie given curve in any given pro- portion. 1st. Having drawn a line AB across the curve (Fig. 64) set off along it the equal distances A 1, 1 2, &c., and through the points 1, 2, &c., construct perpendiculars to AB, and prolong them to cut the curve on each side of it. 2d. Draw any line, as ab (Fig. 65), on which set off equal distances a 1, 1 2, &c., each in the given proportion, take for example that of 1 to 3, to those set off on the given figure, that is, make a 1 the one-third of A 1, &c., and through the points 1, 2, &c., construct perpendiculars to ib. ' 4th. Set off any given line cd (Fig. 66), with the distance ed describe two arcs, and join the point e, where they cross, with c, and d. 5th. From e set off cf and eg, each equal to one-third of cd, and join /and g. 6th. From c set off cm, equal to 1 M (Fig. 64) ; co equal 08 INDUSTRIAL DRAWING. 2 -o, &c. ; join e with the points m, o, &c. ; and mark the points r, s, &c., where these lines cross fg. 7th. Set off the distance fr from 1 to m (F;g. 65) ; fs from 2 to o, &<3. The points m, o, &c., are points of the required copy. In like manner the distances n, p, &c., below ab, are constructed. Remark. The method here used (Fig. 66) for constructing the proportional distances fr,fs, &c., to those cm, co, &c., can be used in all like cases, as for example in Prob. 17. It furnishes one of the most accurate methods for such cases, as the lines drawn from c cross the line fg so as to mark the points of crossing r, s, &c., with great accuracy. Prob. 68. (PL Y. Fig. 67.) Through three given points to describe an arc of a circle by points. Let A, B, and C be the given points. 1st. From A, with the radius AC, the distance between the points farthest apart, describe an arc Co; and from C with the same radius an arc Ap. 2d. From A and C, through S, draw lines, and prolong them to a and b on the arcs. 3d. From b, set off any number of equal arcs b 1, 1 2, &c., above b; and from a the same number of equal arcs below a. 4th. From (7, draw lines C 1, C 2, &c., to the points above b ; and from A lines A 1, &c., to the corresponding points below a. 5th. Mark the points, as m, &c., where the corresponding lines A 1 and C 1, &c., cross. These are points of the curve. 6th. Having set off equal arcs below 5, and like arcs above a, join "the corresponding points with A and C. The points n, &c., to the left of B, are points of the required curve. Remark. This construction is only useful when, from the position of the given points, the centre of the circle which would pass through them cannot be constructed. Prob. 69. (PL Y. Fig. 68.) Having given the axis, the vertex, and a point of a parabola, to find other points of the curve and describe it. PROBLEMS OF CIRCLES, &C. (Jf) Let AS be the axis ; A the vertex ; and the given point. 1st. From C draw a perpendicular to AB, and mark the point d where it crosses AS. 2d. At A construct a perpendicular to AS, and from C a parallel to AS, and mark the point F where these two lines 3d. Divide Cd and CF respectively into the same number of equal parts, say four for example. 4th. From the points of division 1, 2, 3, on Cd draw parallels to AS ; and from the point A lines to the points 1, 2, 3 on CF. 5th. Mark the points x, y, z, where the lines from A cross the corresponding parallels to AS. These will be the required points through which the curve is traced. 6th. Through the points x, y, z, drawing perpendiculars to AS, and from the points a, b, c, where they cross it, setting off distances oo/, by', and czf respectively equal to ax, &c. ; the points a/, ?/, z f will be the portion AD of the curve below AS. Prob. 70. (PL V. Fig. 69.) Having given the diameter of a circle, to construct a right line which shall be equal in length to its circumference. Let AM be the given diameter. 1st. Draw a right line, and having, from any convenient scale of equal parts, taken off a distance greater than the given diameter, and equal to 113 of these equal parts, set it off from a to b. 2d. From a and b, with the distance ab, describe arcs, and from the point c, where they cross each other, draw lines through a and b. 3d. Take off from the scale a distance equal to 355 equal parts, and set it off from c, to d t and e, on the lines drawn through a and b ; and join the points d and e. 4th. From or, set off on ab, the distance am, equal to the given diameter AM; and from c, draw a line through m, and prolong it to cross de at n. The distance dn is the required length of the circumference. 70 INDITSTBIAL CHAPTEK IV. TINTING AND SHADING. Line Shading. 1. Flat tints. It is often necessary to cover a surface with equi-distant parallel lines. This is good practice for the eye, as well as the hand, as the distance between the lines should be obtained by the eye alone ; use the triangles for this, sliding one along the other or, better, against the square. When the spacing is uniform and the lines smooth, it gives the effect of a flat tint. If, while doing this, you should hap- pen to make a space larger, or smaller, than the preceding, do not make the next line at the regular distance from the last, as this would make the break in the spacing more notice- able, but gradually reduce, or increase this distance until the regular space is reached, and then continue with that. These irregularities in spacing are less noticeable where the spacing is coarse ; so that it is well in beginning to make the lines at least ^ of an inch apart. With practice this can be reduced ; the fineness of the spacing should have some reference to the size of the figure. This kind of shading is used mostly for sections, as shown in Fig. 156. PL XVII. ; it being customary to run the lines 45 in either direction. Where the sections of two bodies join, as in Fig. 3. PL I.* let the lines run in opposite direc- tions ; where there are more than two bodies, try to arrange so that on no two adjacent ones the lines run in the same direction ; if necessary change the angle. For practice take four rectangles and cover with lines, aa shown in Fig. 2. PL I*. Afterwards try Figs. 3, 4, 5, using finer spacing. The student can easily vary these figures. TINTING AND SHADING. 71 2. Graduated tints. There are two methods by which a graduated tint may be obtained ; first, by varying the distances without changing the size of the line (PI. I*. Fig. 6), or sec- ond, by changing the size of the lines as well as the distances (PL I.* Fig. 7). The effect of the last method is the best ; always shade from the dark line to the li^ht. Try shading a rectangle by each method. For further practice try the shading of an hexagonal prism and cylinder ; as seen in Figs. 8 and 9, PI. I.*, the shading on these surfaces is made up of flat and graduated tints. When shading the cylinder make the darkest line first, and shade both ways from it ; the shade on the left-hand side should be made from left to right. Figs. 10 and 11, PI. I.* give practice with the compasses in producing graduated tints. In the first (fig. 10), each circle is to be completed with a uniform line, the shade being lightest towards the centre ; in the second (fig. 11), each circle is made with tapering lines. It requires some practice to make a tapering line with the compasses ; when the circle is to be complete, set the pen to the size of the finest part and describe the circle; then separate the points of the pen a little, and sweep over the heaviest part of the circle ; if the pen is brought in contact with the paper, and also taken from it while it is being turned, and the pressure upon the paper is varied, making it greatest at the darkest part of the circle, a very good taper can be given to the line. It would be well to protect the centre in making these figures. The rules for locating the dark and light parts on the solids just mentioned, are given in the chapter on shading; they are introduced here merely for practice, and the examples given will be sufficient guides. India Ink Shading. 3, Flat tints. Tinting with India ink is a quicker and easier method of shading than by the use of lines. For tinting one needs to have at hand a tumbler of clean water and two or three brushes of different sizes ; those with 72 INDUSTRIAL DBAWING. large bodies and fine points are best, as they will hold consid- erable tint. To prepare the tint, rub the cake of ink upou the tile and then take from that with a brush, and mix with the water in the- tumbler, until the desired shade is obtained. Care should be taken that the brush and tumbler are perfectly clean, and it is well to keep the tumbler covered after the tint is prepared. Do 'not make the tint as dark as you wish it upon the drawing, when finished ; it is much easier to lay a light tint smoothly, than a dark one, so that it is better to get the depth of shade required by successive washes ; let each wash dry before laying another. As a rule, it is better to go over the surface to be tinted first with clean water, as the first wash will lay smoother ; and if the first wash is spotted, it will show through all the rest ; this damping is especially necessary if the tint used is dark, or the surface large; when the surface is small, and some skill has been acquired, the damping may be omitted. Do not use India rubber upon the surface to be tinted, as it is difficult, to lay a smooth tint afterwards ; avoid also rest- ing the hands upon the surface, as any moisture from them will affect the flow of the tint. For practice try laying flat tints upon rectangles of differ- ent sizes, commencing with small ones. When doing this, commence at the top and work down, keeping the advancing edge nearly horizontal and always wet ; let the board be in- clined a, little, so that the tint will follow the brush ; upon reaching the bottom of the rectangle, if there is any surplus tint upon the paper, it should be removed with the brush, haying first wiped it dry. Try next some surface where it is necessary to watch two or more edges, to see that they do not get dry ; the space be- tween two concentric circles will do, or trace the outlines of the irregular curve upon paper, and either tint the curve or else tint around the curve, leaving that white. Let the outlines of surfaces to be tinted be fine hard pencil lines ; with care in laying the tints, they make the best finish for the edge, but where the edge has become uneven, a fine light line of ink will improve it. 4. Graduated tints. Having acquired some skill in laying TINTING AND SHADING. 73 flat tints, try next a graduated tint. There are several methods of doing this, one of which is by Flat tints. When it is desired to tint a rectangle so that it shall be darkest at the top, and shade off lighter towards the bottom, divide the sides into any number of equal parts with pencil marks (PI. I.* Fig. 12.); commencing at the top, lay a flat tint upon the first space ; when this is dry, commence at the top and lay a flat tint over the first two spaces. Pro- ceed in this way, commencing at the top each time, until the whole rectangle is covered ; by making the divisions of the rectangle quite small, the effect is more pleasing. See that the lower edges of the flat tints are as straight as possible, for they show through the succeeding tints, and detract much from the appearance when irregular ; it is not well to make the divi- sion marks across the surface, as they would be likely to show. There must be sufficient time between the tints to allow the top space to dry, else some of its tint will be washed to the spaces below, and when completed, the upper spaces of the rectangle will have the same tint. It is better to follow the order given, instead of going over the whole surface for the first tint, and reducing the surface, by one space, for each succeeding tint ; by the first method the edges are covered by the over-laying tints, and a softer appearance is given than would be obtained by the second method. Figs. 11, 12, PI. I.**, give examples of shading with flat tints. 5. Softened tints. This gives a much smoother appearance than the last method. Divide the rectangle as before, and tint the first space ; but instead of letting the edge dry. as a line, wash it out with clean water ; wash out the edge of each tint in the same way; and when finished, there will not be any abrupt changes from one tint to another, as in the last method. 6. Dry shading. First tint the rectangle by the method of flat tints ; then keeping the brush pretty dry, so that the strokes disappear a.bout as fast as made, use it as you would a pencil, and shade between the edges until they disappear ; keep the point of the brush pretty fine, and make the strokes short and parallel to the edges. 74: INDTTSTRTAT. DBA WING. This takes considerable time, but gives a very good result ; not as smooth as by softened tints, but full as pleasant to the eye. When the surface to be shaded is quite small, the flat tints might be omitted, and the tint applied with short strokes of the brush, wherever needed. For practice, try the different methods upon rectangles about 2 by 4 inches. Let the shade be darkest at the top, or bottom ; let the shade be darkest along the right or left hand edges ; let it be darkest through the centre, either horizontal- ly or vertically ; in the last case the centre space is the first one tinted, and the second wash covers the centre and two adjacent spaces. 7. Colors. The directions previously given will apply to the use of colors in forming flat or graduated tints ; another method may be used for graduated tints, and that is to shade the surface first with India ink, and then cover it with a flat tint of the color. REPRESENTING DIFFERENT MATERIALS. 75 CHAPTER Y. CONVENTIONAL MODES OF REPRESENTING DIFFERENT MATERIALS. 1. In mechanical drawing different materials may be repre- sented by appropriate conventional coloring, which is usually done in finished drawings; or else by simply drawing the outlines of the parts in projection, and drawing lines across them according to some rule agreed upon, to represent stone, wood, iron, etc. Although there is no uniformity in the use of conventional tints, the following will be found convenient for this purpose : Cast iron-, Payne's grey. Wrought iron, Prussian blue. Steel, Prussian blue and carmine. Brass, Gamboge. Copper, Gamboge and carmine. Stone, Sepia and yellow ochre. Brick, Light red. Wood, Burnt Sienna. Earth, Burnt Umber. For stone a light tint of India ink might be used instead of the colors given above ; if a little carmine be added to the light red, when used to represent brick, it will make a brighter color; raw sienna might be used instead of burnt sienna for wood. 2. Wood. When the drawing is on a small scale, the out- , lines only of a beam of timber are drawn in projection, when ' the sides are the parts projected, PI. VII. Fig. 88 ; when the end is the part projected, the two diagonals of the figure are drawn on the projection. If a longitudinal section of the beam is to be shown, PL VII. Fig. 89, fine parallel lines are drawn lengthwise. In a cross section, fine parallel lines diagonally. 76 INDUSTRIAL DRAWING. Where the scale is sufficiently large to admit of some resemblance to the actual appearance of the object being attempted, lines may be drawn on the projection of the side of the beam, PI. VII. Fig. 90, to represent the appearance of the fibres of the wood ; and the same on the ends. In longitudinal sections the appearance of the fibres may be expressed, PI. VII. Fig. 91, with fine parallel lines drawn over them lengthwise. In cross sections the grain may be shown, as in the projection of the ends, with fine parallel lines diagonally. Figs. 13, 14, 15, PI. I.*, give additional examples of the method of representing wood in line drawings. Before at- tempting to represent graining, it would be well to examine the graining of wood in the material itself, and to have a piece at hand to imitate ; in the floor of the drawing-room may often be found good examples. In beginning to grain a timber, make the knots first, and then fill in the remaining space with lines, arranging so as to enclose the knots ; until some practice has been acquired, it would be well to pencil the graining before inking. The size of the knots should have some reference to the size of the tim- ber ; let the graining lines be made with a tine steel pen, using light ink ; where the wood is to be colored, apply the tint be- fore graining. "When the timber is large, use a brush to make the grain- ing ; if it is to be colored, use a dark tint of burnt sienna for the lines, and wash over with a lighter tint of the same ; when using the brush for graining, make the points of the knots widest, as in Fig. 15, PI. L* ; it would add also to the looks of graining made with a pen to widen the ends of the knots with the brush. Fig. 14, PI. I.*, differs from Fig. 13 only in having a series of short marks introduced in the graining ; if desired to dis- tinguish between soft and hard wood, let Fig. 13 represent* soft, and Fig. 14 hard wood. In Fig. 15, PI. I.*, the graining is made up of a series of short hatches ; when done nicely, this gives a very good effect. This style may also be used to represent hard wood. The cross sections, Figs. 13, 14, 15, PI. I.*, are shown by a REPRESENTING DIFFERENT MATERIALS. 77 series of concentric circles, with a few lines radiating from the centre, representing cracks; use the bow compasses to describe the circles, from a centre either within or without the section. When the section is narrow, as at . If at the points a, and of perpendiculars should be erected to each plane, we see that these must intersect at the point A in space ; and as these perpendiculars can intersect in only one point, it follows that there is only one point in space that can be projected in a and a'. Thus we see, if the projections of a, point are found upon two planes at right angles, its position in space is fixed, and can be determined from the projections. If the projections of two or more points are found upon these same planes, we should not only be able to determine their positions respecting the planes, but also their relative position ; hence, it follows, if a solid be projected upon these two planes, we can determine its dimensions from the projec- tions. We have found, then, that two planes, at right angles, are necessary in projections; these are called, respectively, the horizontal and vertical planes of projection. The line of intersection, G L (Fig. 22), is called the ground line. The point a is called the horizontal projection of A, and a' the vertical projection. The horizontal projection of an object is often called the plan, and the vertical projection the elevation. The perpendiculars through A are called the projecting lines ; the plane of the two is perpendicular to both planes of projection, and also the ground line, and intersects both planes in right lines, perpendicular to the ground line at the. same point. Since it is impracticable to draw upon two planes at right angles, the vertical plane is considered as revolved back, about the ground line, until it forms one and the same surface with the horizontal plane. By this revolution the relative position of points in the vertical plane is not affected ; every point remains at the same distance from the ground line after revo- lution as before. This is shown in Fig. 22, where the vertical plane G P is represented as revolved back to the horizontal position Cr P' ; a' revolves to a" ; a" b is equal to a' b ; the line joining a a" is perpendicular to G L PROJECTIONS. 85 Remark. The preceding figures are pictorial representa- tions ; in the remaining figures a horizontal line is used to separate the planes of projection, the part above the line be- ing the vertical plane, and the part below the horizontal. It is important, then, to note, 1st, that the perpendicular distance from the horizontal projection of a point to the ground line shows how far the point itself is from the verti- cal plane of projection. 2d, that, in like manner ', the perpendicular distance from its vertical projection to the ground line shows its height above the horizontal plane. 3d, that the horizontal and vertical projections of a point lie on the right line drawn from one to the other, and per- pendicular to the ground line. 4th, that the distance, measured horizontally, between two points, is that between their horizontal projections. 5th, that their distance apart vertically, or the height the one is above the other, is measured by the difference between the respective distances of their vertical projections from the ground line. 6th, that the actual distance between two points, or the length of a right line connecting them, is equal to the hypothenuse of a right angled triangle, the base of which is equal to the distance between the horizontal projections of the points, and the altitude is the difference between the distances of their vertical projections from the ground line. For example (PI. V. Fig. 72), the line G L is the ground line. The point a being the horizontal, and the point a' the vertical projection of a point, these two points lie on the right line ax-a' joining them, and perpendicular to QL. The point itself is at a distance in front of the vertical plane measured by the line ax; and at a height from the horizontal plane measured by x a'. In like manner b b', and c c f , are the projections of two points, the horizontal distance between which is b c, the dis- tance apart of their horizontal projections ; and the vertical distance is c'y, equal to the difference between c'x' and b'x, their respective heights from the horizontal plane. The actual distance between these points, or the length of the line drawn 86 INDUSTRIAL DRAWING. from one to the other, may be found by constructing a right angled triangle, mon / the base of which, o m, being equal tc b c, and its altitude, o n, equal to c'y, its hy pothenuse, m n, will be the distance required. In making the projections of an object, when it is desirable to designate the projections that correspond to the same point, they are joined by a light broken line ; and if the projections are those of an isolated point, either the projections are made with a large round dot, or by a small dot surrounded by a small circle. When the projections of two points are those of the extremities of a right line, a full line is drawn on each plane of projection between the points, as t> c and Vd ; and a broken line is drawn between the projections of the corre- sponding extremities. Notation. Small letters are used to designate the projections of a point, the same letter being used for both projections ; to dis- tinguish between them, the vertical is accented. The point a of is also spoken of as the point A. Lines are similarly treated, as the line oib a'b', or the line AB. The letters H and V are used to designate the planes of projection. G L stands for the ground line. Shade lines. Shade lines upon outline drawings add very much to their appearance ; when properly placed, they give relief to the drawing, and are of assistance in reading it. In mechanical drawing the light is generally assumed to come in such a direction that its projections shall make angles of 45 with the ground line ; the arrows in Figs. 23 and 24 indicate the direction of light. Those edges should be heavy which separate light from dark surfaces. In the case of the cube (Fig. 23), we see that the top, front, and left-hand faces must be in the light, while the remaining faces would be in the shade. In the elevation the only visible PEOJECTION8. 87 edges separating light from dark faces are those upon the right and lower sides (a'V and 5V), while in the plan the shaded edges are upon the right and upper sides (0/and/e). Fi. 23. FKJ. 24. In case of a curved surface like the cylinder (Fig. 24), the line a'b' does not separate the light from the dark surface, yet it is well to make it a trifle darker than the left-hand edge, but not as dark as the bottom line c'V. In the plan the circle is made darkest upon the upper and right-hand side, tapering to the points of tangency (e, t f) of rays of light. Shade lines of sections follow the above rules, as shown in Fig. 83. PI. YI. Draw the shade lines with their breadth outside the outline. In colored drawings draw the shade lines last. In shaded drawings omit the shade lines altogether. Profiles and Sections. The projections of an object give only the forms and dimensions of its exterior, and the posi- tions of points, &c., on its surface. To show the thickness of its solid parts, and the form and dimensions of its interior, intersecting-planes are used. Taking a house as a model, let us conceive it to be cut, or sawed through, at some point between its two ends, in the direction of a vertical plane parallel to the ends. Setting aside one portion, let us imagine a pane of glass placed against the sawed surface of the other, and let an accurate outline of the parts thus cut through be traced on the pane. This outline is termed 88 INDUSTRIAL DRAWING. a profile. On it, to distinguish the solid parts cut through from the voids, or hollow parts, we cover them entirely with ink, or some other color, or else simply draw par- allel lines close together across them. If, besides tracing the outline of the parts resting against the pane, we were to trace the projections of all the parts, both within and without the outline of the profile, that could be seen through the pane by a person standing in front of it, the profile with these additional outlines is termed a section. A section more- over differs from a profile in this, that it may be made in any direction, whereas the profile is made by cutting vertically, and in objects, like a house, bounded by plane surfaces, in a direction perpendicular to the surface. To show the direction in which the section is made, it is usual to draw a broken and dotted line on the plan and eleva- tion of the object, marking the position of the saw-cut on the surface of the object ; and, to indicate the position to which the section corresponds, letters of reference are placed at the extremities of each of these lines, and the figure of the section is designated as vertical, or oblique section on A J2, C D, &c., according as the section is in a plane perpendicular, or oblique to the horizontal plane of projection. The sections in most general use are those made by vertical and horizontal planes. A horizontal section is made in the same manner as a vertical one, by conceiving the object cut through at some point above the horizontal plane of projec- tion, and parallel to it, and, having removed the portion above the plane of section, by making such a representation of the lower portion as would be represented by tracing on a pane of glass, laid on it, the outline of the parts in contact with the pane, with the outline of the projection of the parts on the pane that can be seen through it, whether on the exterior, or interior of the object. The solid parts in contact with the pane are represented in the same way as in other sections. The projected parts are represented only by their outlines. A broken and dotted line, with letters of reference at its extremities, is drawn on the elevation to show where the section is taken ; and the section is designated by a title, as, horizontal section on A , &c. PROJECTIONS. 89 As the broken and dotted lines that indicate the position of the planes of section are drawn on the planes of projection, and are in fact the lines in which the planes of section would cut these two planes, they are termed the vertical or hori- zontal traces of the planes of section, according as the lines are traced on the vertical or horizontal plane of projection. It will be well to note particularly that the planes of section are usually taken in front of, or above the object, that portion of it which is cut by the plane being supposed in contact with the plane ; whereas the planes of projection may be placed either behind, or in front of the object, and above or below it, as may best suit the purpose of the draftsman ; the position of the ground line therefore will always indicate on which side of the object, and whether above, or below it, the planes of projection are placed. The usual method is to place the horizontal plane of projection below the object represented and the vertical plane behind it. The more usual method also is to represent the object as resting on the horizontal plane ; its position with respect to the vertical plane, or thai of the vertical plane with respect to it, being so taken as to give the desired elevation to suit the views of the draftsman. In the case of the ordinary house, for example, the elevations of the four sides may be obtained either by supposing one vertical plane, and the four sides successively presented to it ; or by supposing the vertical plane shifted so as to be brought behind each of the sides in succession. Projections of Points and Right Lines. The method of projections presents two problems. The one is having given the forms and dimensions of an object, to construct its pro- jections ; the other, having the projections of an object, to construct its forms and dimensions. A correct understanding of the manner of projecting points and right lines, and determining their relative positions with respect to each other, is an indispensable foundation for the solution of these two questions. The methods of projecting a single point, and of obtaining its distance from the planes of projection, also of two pointe ; and determining their distance apart, have already been given. The same process would evidently be followed in 90 INDUSTRIAL DRAWING. projecting any number of points ; or, in determining their relative positions, having their projections. But, besides these general methods, there are some particular cases with which it will be well to become familiarized at the outset, as a knowledge of them will materially aid in showing, by a glance at the projections, the relative positions of the lines joining the points to the planes of projection ; that is, whether these fines are parallel, oblique, or perpendicular to one, or both of these planes. Case 1. (PI. V. Fig. 73.) Let aa' and bV be the projec- tions of two points, the distances of their vertical projections a'x and b'x' from the ground line being equal, those of their horizontal projections ax and bx' being unequal. The points themselves will be at the same height above the horizontal plane of projection but at unequal distances from the vertical plane. The vertical projection of the line joining the two points a'b' will be parallel to the ground line, and its hori- zontal projection ab will be oblique to it. from this we observe, that when two points of a right line are at the same height above tJie horizontal plane of projection, and at unequal distances from the vertical plane, the vertical projection of the line will be parallel to the ground line, and its horizontal projection oblique to this line. Finding then the two projections of a right line in these positions with respect to the ground line, we conclude that the line itself is at the same height throughout above the horizontal plane of projection, or parallel to this plane, but oblique to the vertical plane. Case 2. (PI. Y. Fig. 74.) In like manner, when we find the horizontal projections of two points a and b at the same distance from the ground line, and the vertical projections a' and b' at unequal heights from it, we conclude that the line joining the points is parallel to the vertical plane but oblique to the horizontal. Case 3. (PI. V. Fig. 75.) When the horizontal projections of two points are at the same distance from ground line, and the vertical projections also at equal distances from it, we conclude that the line itself is parallel to both planes of projection. When a line therefore is parallel to one plane of projection PBOJECTION8. 91 (done, its projection on the other willbe parallel to the ground line, and its projection on the plane to which it is parallel will be oblique to the ground line. When the line is parallel to loth planes its two projections wiU be parallel to the ground line. Case 4. (PL Y. Fig. 76.) Suppose two points as a and b to lie in the horizontal plane of projection, where are their vertical projections? From what has been already shown, these last projections must lie on the perpendiculars, from the horizontal projections a and b to the ground line ; but as the points are in the horizontal plane their projections cannot lie above the ground line. The vertical projections of a and b therefore must be at a' and b' on the ground line, where the perpendiculars from a and b cut it. For a like reason the vertical projection of a line as a b in the horizontal plane will be as a' b' in the ground line. In like manner the horizontal projections of points and lines lying in the vertical plane of projection will be also in the ground line. Case 5. (PL Y. Fig. 77.) If a line is vertical, orperpen dicular to the horizontal plane of projection, its projection on that plane will be a point simply, as a. For, the line be- ing vertical, if a plumb line were applied along it the two lines would coincide, and the point of the bob of the plumb line would indicate only one point as the projection of the entire line. Now as a is the horizontal projection of all the points of the line, their vertical projections must lie in the line from a perpendicular to the ground line, so that the ver- tical projections of any two points of the vertical line at the given heights b'x, and a'x above the horizontal plane of pro- jection, would be projected on the perpendicular from a to the ground line, and at the given distances b'x and a'x above the ground line. In like manner it can be shown, that a line perpendicular to the vertical plane is projected into a point, as a', and its horizontal projection will lie on the perpendicular to the ground line from a', as a b, in which the distances of the points a and b from the ground line show the distances of the ends of the line from the vertical plane. 90 INDUSTRIAL DRAWING. projecting any number of points ; or, in determining their relative positions, having their projections. But, besides these general methods, there are some particular cases with which it will be well to become familiarized at the outset, as a knowledge of them will materially aid in showing, by a glance at the projections, the relative positions of the lines joining the points to the planes of projection ; that is, whether these lines are parallel, oblique, or perpendicular to one, or both of these planes. Case 1. (PI. V. Fig. 73.) Let aa! and bb' be the projec- tions of two points, the distances of their vertical projections a'x and b'x' from the ground line being equal, those of their horizontal projections ax and bx' being unequal. The points themselves will be at the same height above the horizontal plane of projection but at unequal distances from the vertical plane. The vertical projection of the line joining the two points a'b' will be parallel to the ground line, and its hori- zontal projection db will be oblique to it. from this we observe, that when two points of a right line are at the same height above the horizontal plane of 'projection, and at unequal distances from the vertical plane, the vertical projection of the line will be parallel to the ground line, and its horizontal projection oblique to this line. Finding then the two projections of a right line in these positions with respect to the ground line, we conclude that the line itself is at the same height throughout above the horizontal plane of projection, or parallel to this plane, but oblique to the vertical plane. Case 2. (PI. Y. Fig. 74.) In like manner, when we find the horizontal projections of two points a and b at the same distance from the ground line, and the vertical projections a' and b' at unequal heights from it, we conclude that the line joining the points is parallel to the vertical plane but oblique to the horizontal. Case 3. (PI. V. Fig. 75.) When the horizontal projections of two points are at the same distance from ground line, and the vertical projections also at equal distances from it, we conclude that the line itself is parallel to both planes of projection. When a line therefore is parallel to one plane of projection PROJECTIONS. 91 alone, its projection on the other willle parallel to the ground line, and its projection on the plane to which it is parallel will be oblique to the ground line. When the line is parallel to both planes its two projections Witt be parallel to the ground line. Case 4. (PI. V. Fig. 76.) Suppose two points as a and b to lie in the horizontal plane of projection, where are their vertical projections? From what has been already shown, these last projections must lie on the perpendiculars, from the horizontal projections a and b to the ground line ; but as the points are in the horizontal plane their projections cannot lie above the ground line. The vertical projections of a and b therefore must be at a' and b' on the ground line, where the perpendiculars from a and b cut it. for a like reason the vertical projection of a line as a b in the horizontal plane will be as a' b' in the ground line. In like manner the horizontal projections of points and lines lying in the vertical plane of projection will be also in the ground Line. Case 5. (PL V. Fig. 77.) If a line is vertical, or perpen dicular to the horizontal plane of projection, its projection on that plane will be a point simply, as a. For, the line be- ing vertical, if a plumb line were applied along it the two lines would coincide, and the point of the bob of the plumb line would indicate only one point as the projection of the entire line. Now as a is the horizontal projection of all the points of the line, their vertical projections must lie in the line from a perpendicular to the ground line, so that the ver- tical projections of any two points of the vertical line at the given heights b'x, and a'x above the horizontal plane of pro- jection, would be projected on the perpendicular from a to the ground line, and at the given distances b'x and a'x above the ground line. In like manner it can be shown, that a line perpendicular to the vertical plane is projected into a point, as a', and its horizontal projection will lie on the perpendicular to the ground line from a', as a b, in which the distances of the points a and b from the ground line show the distances of the ends of the line from the vertical plane. 94 INDUSTEIAL DRAWING. 3d. Draw lines from o' to the points a'Vc'd'. Remarks. The line o'd' in vertical projection, and the line OG, according to what has been laid down, should be dotted. Having the projections of a like pyramid we would pro- ceed, as in the last case, to construct its edges and faces if re- quired for a model. Prob. 74. (PI. VI. Fig. 81.) To construct the projections of a right prism with a regular hexagonal base. Let the base of the prism be supposed to rest on the hori- zontal plane of projection. 1st. Construct at a convenient distance from the ground line the regular hexagon abc, &c., of the base ; taking two of its opposite sides, as o c, &nd.f e, parallel to this line. 2d. Construct the projections h'l'm', &c., of the points abc, &c. 3d. As the edges of the prism are vertical, their vertical projections will be drawn through the points AT, &c., and perpendicular to the ground line. 4th. Having drawn these lines, set off the equal distances h' a', I' 5', &c., upon them, and each equal to the height of the prism. Remarks. As the edges projected in b and c, and f and , that when a plane figure is parallel to one plane of projection it will be projected on that plane in a figure equal to itself , and on the other plane into a line parallel to the ground line. Moreover since the faces of the prism are plane surfaces perpendicular to the horizontal plane, and are projected respectively into the lines a b, b c, &c., we con- clude that a plane surf ace perpendicular to one plane of pro- jection is projected on that plane into a right line. The same is true of the base and top of the prism ; the base being in the horizontal plane, which is perpendicular to the vertical plane, is projected into the ground line in h' n' ; the top being parallel to the horizontal plane is likewise perpendicu- lar to the vertical plane, and is projected into the line a'd'. Traces of Planes on the Planes of Projection. The plane surfaces of the prism and pyramids in the preceding problems being of limited extent, we have only had to consider the lines in which they cut the horizontal plane of projection, as a b, b c, &c., the bounding lines of the bases of these solids These lines are therefore properly the traces of these limited plane surfaces on the horizontal plane. But when a plane is of indefinite extent, we may have to consider the lines in which it cuts, or meets both planes of projection. The most usual cases in which we have to consider these lines are in those of profile planes, and planes of section, in which the planes are perpendicular either to the horizontal, or vertical plane, and parallel, or oblique to the other. The position of the trace of a plane, when parallel to one plane of projection and perpendicular to the other, as has already been shown, is a line parallel to the ground line, and on that plane of projection to which the plane is perpendicu- lar, as the lines b c, and f- e, for example, which are the traces on the horizontal plane of the faces of the prism, which are perpendicular to this plane, and parallel to the vertical plane. The same may be said of the line a' d' y which would 96 INDUSTRIAL DRAWING. be the trace of the plane of the top of the prism, if it -were produced back to meet the vertical plane. When the plane is perpendicular to the horizontal plane^ but oblique to the vertical, as for example the face of the prism of which a -f, or d e is the horizontal trace, its verti- cal trace will be perpendicular to the ground line at the point where the horizontal trace meets this line. To show this, sup- pose the prism so placed as to have its back face against the vertical plane ; then the line a -f, for example, will be oblique to the ground line, the point f of this line being on it, at the point I', whilst the line I' b', the one in which the oblique face meets the vertical plane, or its trace on this plane, will be perpendicular to the ground line. The same illustration would hold true supposing the prism laid on one of its faces on the horizontal plane, with its base against, the vertical plane. If A B (PL VI. Fig. 82) therefore represents the horizon- tal trace of a plane perpendicular to the horizontal plane its vertical trace Witt be a line B b, drawn from the point B> where the horizontal trace cuts the ground line, perpendicu- lar to this line. In like manner, if EF is the vertical trace of a plane perpendicular to the vertical plane and oblique to the horizontal plane, the line Ff perpendicular to the ground line is its horizontal trace. Prob. 75. (PI. VI. Fig. 83.) To construct the projections and sections of a hollow cube of given dimensions. Let us suppose the cube so placed that, its base resting on the horizontal plane of projection in front of the vertical plane, its front and back faces shall be parallel to the vertical plane, and its other two ends perpendicular to this plane. Having constructed a square abed (Fig. X) of the same dimensions as the base of the given cube, and having its sides ab and cd parallel to the ground line, and at any convenient distance from it; this square may be taken as the projection of the base of the cube. But as the top of the cube is paral- lel to the base, and its four faces are also perpendicular to these two parts, the top will be also projected into the square abed, and the four sides respectively into the sides of the square. The square abed will therefore be the horizontal pro- jection of all the exterior faces of the cube. PROJECTIONS. 97 Having projected the base of the cube into the vertical plane, which projection (Fig. Y) will be a line h' Z', on the ground line, equal to a b, construct the square a'Vlh' equal to abed. This is the vertical projection of the cube. As the interior faces of the cube cannot be seen from with- out, the following method is adopted to represent their pro- jections : within the square abed construct another represented by the dotted lines, having its sides at the same distance from the exterior square as the thickness of the sides of the hollow cube. This square will be the projection of the interior faces ; and it is drawn with dotted lines, to show that these faces are not seen from without. Supposing the top and bottom of the cube of the same thickness as the sides, a like square constructed within a'b'l'h' will be the vertical projection of the two interior faces, which are perpendicular to the vertical plane, and of the interior faces of the top and base. Having completed the projections of the cube, suppose it is required to construct the figures of the sections cut from it by a horizontal plane, of which M N is the vertical trace ; and by a vertical plane of which O^-P is the horizontal trace. The horizontal plane of section will cut from the exterior faces of the cube a square mopn (Fig. Z] equal to the one abed, and from the interior faces another square equal to the one in dotted lines, and having its sides parallel to those of mopn. The solid portion of the four sides cut by the plane of section would be represented by the shading lines, as in Fig. Z. The plane of section of which O P is the trace, being oblique to the sides, will cut from the opposite exterior faces a d, and b c, and the exterior faces of the top and base, a rectangle of which r u (Figs. X, W) is the base, and r / (Fig. TF) equal to the height V Z', is the altitude ; in like manner it will cut from the corresponding interior faces a rectangle of which s t is the base, and ss', equal to the height of the interior face, is the altitude. The sides of the interior rectangle stt's' will be parallel to those of the one exterior ; the distance apart of the vertical sides being equal to the equal distances rs (Fig. X] and t u ; and that of 1 98 INDUSTRIAL DRAWING. the horizontal sides being the same as the thickness of the top and base of the cube. In other words, as has already been stated, the figure of the vertical section is the same that would be found by tracing the outline of the part of the cube, cut through by the plane of section, on the vertical plane of projection. Remarks. The manner of representing the interior faces of the hollow cube by dotted lines is generally adopted for all like cases ; that is, when it is desired to represent the pro- jections of the outlines of any part of an object which lies between some other part projected and the plane of pro- jection. Where several points, situated on the same right line, as r, , t, u, on the line O P (Fig. JT), are to be transferred to another right line, as in the construction of (Fig. TF), the shortest way of doing it, and, if care be taken, also the most accurate one, is to place the straight edge of a narrow strip of paper along the line, and confining it in this position to mark accurately on it near the edge the positions of the points. Having done this, the points can be transferred from the strip, by a like process, to any other line. The advantage of this method over that of transferring each distance by the dividers will be apparent in some of the succeeding problems. Prob. 76. (PL VI. Fig. 84.) To construct the projec- tions of a regular hollow pyramid truncated by a plane oblique to the horizontal plane and perpendicular to the ver- tical plane. Let us suppose the base of the pyramid a regular pentagon. Having constructed this base, and the projections of the different parts of the entire pyramid as in Prob. 71, draw a line, MN, oblique to the ground line, as the vertical trace of the assumed truncating plane ; the portion of the pyramid lying above this plane being supposed removed. Now, as the truncating plane cuts all the faces of the pyramid, and as it is itself perpendicular to the vertical plane of projection, all the lines which it cuts from these faces will be projected on the vertical plane of projection in the trace M- N. The points r', s', t ', v', and u', where the trace M -STcuts the projections of the edges of the pyramid, will PROJECTIONS. 99 be the projections of the points in which the truncating plane cuts these edges ; and the line r r v' for example is the projec- tion of the line cut from the exterior face projected in ~b'v'ti '. The horizontal projections of the points of which r', v\ &c., are the vertical projections, will be found on the hori- zontal projections vc, vb, &c., of the edges of which v'c', v'b', &c., are the vertical projections, and will be obtained in the usual way. Joining the corresponding points r, v, s, &c., thus obtained, the figure rstuv, will be the horizontal projec- tion of the one cut from the exterior faces of the pyramid by the truncating plane. Thus far nothing has been said of the projections and sections of the interior faces of the pyramid. To construct these let us take the thickness of the sides of the hollow pyramid to be the same, in which case the interior faces will be parallel to and all at the same distance from the exterior faces. If another pyramid therefore were so formed as to fit exactly the hollow space within the given one, its faces and edges would be parallel to the corresponding exterior faces and edges of the given hollow pyramid, and its vertex would likewise be on the perpendicular from the vertex of the given pyramid to its base. Constructing, therefore, a pentagon, mnopq, having its sides parallel to, and at the same distance from those of dbcde, this figure may be assumed as the base of the interior pyramid. The horizontal projections of its edges will be the lines vm, vn, &c. To find the vertical pro- jections of these lines, which will be parallel to the vertical projections of the corresponding exterior edges, project the points TO, n, &c., into the ground line, at ra', n', &c., and, from these last points, draw the lines m'v", n'v", parallel to the corresponding ones a'v', b'v', &c. ; the lines m'v", &c., will be the required projections. To obtain the horizontal projection of the figure cut from the interior faces by the plane of section, find the points s, y, x, &c., in horizontal projection, corresponding to the points z r , y', &c., in vertical projection, where the trace MN cuts the lines m'v", n'v", &c. ; joining these points the pentagon zyx, &c., will be the required horizontal projec- tion. 100 INDUSTEIAL DRAWING. Having constructed the projections of the portion of the pyramid below the truncating plane, Jet it now be required to obtain a section of this portion by a vertical plane of sec- tion through the vertex. For this purpose, to avoid the con- fusion of a number of lines on the same drawing, let us construct (Fig. 85) another figure of the projections of the outlines of the faces, &c. Having drawn the line O P for the trace of the vertical section through the projection of the vertes, we observe that this plane cuts the base of the hollow pyramid en the left-hand side, in the line a 3, and on the opposite slie in the line in n; setting off the distances (Fig. 86) a' b'j V n', and n' mf on the ground line, respectively equal to a J, &c., we obtain the line of section cut from the ba&e. Now the plane of section cuts the exterior line of the top (Fig. 85) on the left-hand side, in a point hori- zontally projected in c, and vertically in c", the height of which point above the base is the distance c' c" ; in like manner the plane cuts the interior line of the top, on the same aide, in the point projected horizontally in d, and vertically in d", its height above the base being d' d". The corre- gponding points of the top on the opposite side are those pro- lected in 0, o" ; and p, p" ; their corresponding distances above the base being respectively o' o" and p'^p". Having thus found the horizontal and vertical distances between these points, it is easy to construct their positions in the plane of section. To do this, set off on the ground line (Fig. 86) the distances a' c', a d', m' o', and m! -p' y respectively equal to the equal corresponding distances a c, &c. (Fig. 85), on O P. At the povnts c', d',p', and o' draw perpendiculars to the ground lino, on which set off the dis- tances c' c", &c., respectively equal to those c' c" of Fig. 85. Having drawn tLe lines a' c", t" d" y and V d", the figure a'Vd"c" is the section of the left-hand side ; in like manner m!n'p"o" is the figure cut fn.m the opposite face by the plane of section. As the (Fig. 86) represents a section, and not a profile of the pyramid, we must draw upon it the lines of the portion of the pyramid which lie behind the plane; that, is the portion of which aaedm is the buse. It will be well tc PROJECTIONS. 101 remark, in the first place, that removing the portion in front of the plane of section, and supposing this plane transparent, the interior surfaces of the pyramid would be seen, and the exterior hidden, the outlines of the former would therefore be drawn full on the plane of section, whilst those of the latter, if represented, should be in dotted lines. To construct the projections of these lines on the plane of section, we observe that this plane, being a vertical plane, and O P being its trace on the horizontal plane of projec- tion, this line O P may be considered as the ground line of these two planes ; in the same manner as G L is the ground line of the horizontal plane and the original vertical plane of projection. This being considered, it is plain that all the parts of the pyramid should be projected on this new vertical plane of projection, in the same manner as on the original one. Let us take, for example, the interior edge, of which q z is the horizontal projection. The point q, being in the horizontal plane, will be projected, in the ground line O P y into q' / and the point z would be projected by a perpendicu- lar from 2 to O P, at a height above O P, equal to the height of its vertical projection on the original vertical plane above the ground line O Z, which is z' z". To transfer these distances to the section (Fig. 86), take the distance a #', on O jP, and set it off from a' to q' on the section ; this will give the projection of q' on the section. Next take the distance a z', from O P, and set it off from a' to z' on the section, and at z' erect a perpendicular to the ground line ; take from (Fig. 85) the distance z' z", and set it off from z' to z" on the section ; the point z" will be the projection of the upper extremity of the interior edge in question on the plane of section ; joining therefore the points q' and 3", thus determined, the line q' z" is the required projection. In like manner, the projections of the other interior and exterior edges of the portion of the pyramid behind the plane of section can be determined and drawn, as shown in the section (Fig. 86). Remarks. The preceding problems contain the solutions of all cases of the projections and sections of bodies, the outlines of which are right lines, and their surfaces plane 102 INDUSTRIAL DBAWTOG. figures. As they embrace a very large class of objects in the arts, it is very important that these problems should be thoroughly understood. One of the most useful examples under this head is that of the plans, elevations, and sections of an ordinary dwelling, which we shall now proceed to give. Prob. 77. Plans, elevations, <&c., of a house. (PL VII. Fig. 87.) Let us suppose the house of two stories, with base- ment and garret rooms. The exterior walls of masonry, either of stone or brick. The interior wall, separating the hall from the rooms, of brick. The partition walls of the parlors and basement of timber frames, filled in with brick ; those of the bedrooms and garret of timber frames simply. It is necessary to observe, in the first place, that the gen- eral plans are horizontal sections, taken at some height, say one foot, above the window sills, for the purpose of showing the openings of the windows, &c. ; and that the sections are BO taken as best to show those portions not shown on the plans, as the stair- ways, roof -framing, &c. In the second place, that in the plans and sections are shown only the skele- ton, or framework of the more solid parts, as the masonry, or timber framing of the walls, flooring, roof, &c. Plans. Having drawn a ground line, G Z, across the sheet on which the drawings are to be made, in such a posi- tion as to leave sufficient space on each side of it for the plans and elevations respectively, commence, by drawing a line A parallel to G Z, and at a convenient distance from it to leave room for the plan of the first story towards the bottom of the sheet. Take the line A B, as the interior face of the wall, opposite to the one of which the elevation is to be represented. Having set off a distance on A B equal to the width between the side walls, construct the rectangle ABOD, of which the sides A D and BC shall be equal to the width within, between the front wall D C and the back A B. Parallel to these four sides, draw the four sides a , b GJ &c., at the distance of the thickness of the exterior walls from them. The figure thus constructed is the general outline of the plan of the exterior walls. Next proceed to draw the outline of the partition wall E- F separating the hall from the parlors. Next the walls PROJECTIONS. 103 of the pantries, G and H, between the parlors. Then mark out the openings of the windows w, w, and doors d',d', in the walls. Then the projections of the fire-places,^ in the par- lors. Having drawn the outline of all these parts with a fine ink line, proceed to fill it between the outlines of the solid parts cut through, either with small parallel lines, or by a uniform black tint. Then draw the heavy lines on those parts from which a shadow would be thrown. As the horizontal section will cut the stairs, it is usual to project on the plan the outlines of the steps below the plane in full lines as in S j and, sometimes, to show the position of the stairway to the story above, to project, in dotted lines, the steps above the plane. If the scale of the drawing is sufficiently great to show the parts distinctly, the sections of the upright timbers that form the framing of the partition walls of the pantries should be distinguished from the solid filling of brick between them, by lines drawn across them in a different direction from those of the brick. The plan of the second story is drawn in the same manner as that of the first, and is usually placed on one side of it. Front elevation. The figure of the elevation should be so placed that its parts will correspond with those on that part of the plan to which it belongs ; that is, the outlines of its walls, of the doors, windows, &c., should be on the perpen- diculars to the ground line, drawn from the corresponding parts of the front wall D C. In most cases the outlines of the principal lines of the cornice are put in, and those of the caps over the windows, if of stone when the wall is of brick, &c. ; also the outline of the porch and steps leading to it. Section. In drawings of a structure of a simple character like this, where the relations of the parts are easily seen, a single section is usually sufficient, and in such cases also it is usual to represent on the same figure parts of two different sections. For example, suppose O P to be the horizontal trace of the vertical plane of section as far as P, along the hall ; and Q R that of one from the point Q opposite P along the centre line of the parlors. On the first portion wiU 104: INDUSTRIAL DBAWING. be shown the arrangement of the stairways; and on the other the interior arrangements from Q towards R. With regard to the position for the figure of the section, it may, in some cases, be drawn by taking a ground line parallel to the trace, and arranging the lines of the figure on one side of this ground line in the same relation to the side B (7, of the plan, as the elevation has with respect to the side A B ; but as this method is not always convenient, it is usual to place the section as in the drawing, having the same ground line as the elevation, placing the parts beneath the level of the ground below the ground line. For the better understanding of the relations of the parts, where the sections of two parts are shown on the same figure, it is well to draw a heavy uneven line from the top to the bottom of the figure, to indicate the separation of the parts, as in T Uj the part here on the left of T ^representing the portion belonging to the hall, that on the right the portion within the parlors. In other words, the figure represents what would be seen by a person standing towards the side A D of the house, were the portion of it between him and the plane of section removed. This figure represents the section of the stairs and floors, and the portion of the roof above, and basement beneath of the hall ; with a section of the partition walls, floors, roof, &c., of the other portion. As the relations of the parts are all very simple and easily understood, the parts of the plans as well as of the elevations and vertical section being rectangular, figures of which all the dimensions are put down, the drawings will speak for them- selves better than any detailed description. Nothing further need be observed, except that the drawing of the vertical section may be commenced, as in the plans, by drawing the inner lines of the walls, and thence proceeding to put in the principal horizontal and vertical lines. Remarks. In drawings of tki's class, where the object is simply to show the general arrangement of the structure, the dimensions of the parts are not usually written on them, but are given by the scale appended to the drawing. Where the object of the drawing is to serve as a guide to the builder, PROJECTIONS. 105 who is to erect the structure, the dimensions of ever}' part should be carefully expressed in numbers, legibly written, and in such a manner that all those written crosswise the plan, for example, may read the same way ; and those length- wise in a similar manner ; in order to avoid the inconvenience of having frequently to shift the position of the sheet to read the numbers aright. Where several numbers are put down, expressing the respective distances of points on the same right line, it is usual to draw a fine broken line, and to write the numbers on the line with an arrow-head at each point, as shown in PI. XI. Fig. a, which is read 5 feet ; 7 feet 6 inches ; 10 feet 9 inches and 7 tenths of an inch. Where the whole distance is also required to be set down, it may be done either by writing the sum in numbers over the broken line, as in this case, 23 feet 3 inches and 7 tenths ; or, better still, with the numbers expressing the partial distances below the broken line, and the entire distance above it, as in PI. XI. Fig. b. Besides these precautions in writing the numbers, each figure also should be drawn with extreme accuracy to the given scale, and be accompanied by an ex- planatory heading, or reference table. On all drawings of this class the scale to which the drawing is made should be constructed below the figs., and be accom- panied by an explanatory heading ; thus, scale of inches, of feet to inches, or feet. In some cases, the draughtsman will find it more convenient to construct a scale on a strip of draw- ing paper for the drawing to be made, than to use the ivory one ; particularly, as with the paper one he can lay down his distances at once, without first taking them off with the dividers, in the same way as points are transferred by a strip of paper. In either case, the scale should be so divided as to aid in reading and setting off readily any required distance. The best mode of division for this purpose is the decimal ; and the following manner of constructing the scale the most convenient: Having drawn a right line, set off accurately from a point at its left-hand extremity, ten of the units of the required scale, and number these from the left 10, 9, &c., to 0. From the point, set off on the right, as many equal 106 INDU8TEIAJL DRAWING. distances, each of the length of the part from to 10, as may be requisite, and number these from to the right 10, 20, &c. From this scale any number of tens and units can be at once set off. This scale should be long enough to set off the longest dimension on the drawing. See, for example, the scale and its heading at bottom of PL m Preliminary Problems in Projections. Before entering upon the drawings of some of the many objects belonging to this class, it will be necessary to show the manner of making the projections of the cone, cylinder, and sphere ; that of obtaining their intersections by a plane ; and also that of representing their intersections with each other. Cylinder. (PL X. Fig. 107.) If we suppose a rectangle, ABCD, cut out of any thin inflexible material, as stiff paste- board, tin, &c., and on it a line, P, drawn through its centre, parallel to its side B O, for example ; this line being so fixed that the rectangle can be revolved, or turned about P, it is clear that the sides A D, and B C\ will, in every position given to the rectangle, be still parallel to O P, and at the same distance from it. The side B C, or A Z>, therefore, may be said to describe or generate a surface, in thus revolving about O P, on which right lines can be drawn parallel to O P. It will moreover be ob- served, that as the points A and B, with D and C are, respectively, at the same perpendicular distance (each equal to O B) from the axis O P, they, in revolving round it, will describe the circumferences of circles, the radii of which will also be equal to B. In like manner, if we draw any other line parallel to A B, or D C, as a 5, its point 5, or , will describe a circumference having for its radius o J, which is also equal to O B. The surface thus described is termed a right circular cylinder, and, from the mode of its generation, the following properties may be noted ; 1st, any line drawn on it parallel to the line O JP, which last is termed the axis, is a right line. PROJECTIONS. 107 and is termed a right line element of the cylinder ; 2d, any plane passed through the axis will cut out of the surface two right line elements opposite to each other; 3d, as the planes of the circumferences described by the points Z?, , &c., are perpendicular to the axis, any plane so passed will cut out of the surface a circumference equal to those already de- scribed. Prob. 85. (PI. X. Fig. 108.) Projections of the cylinder. Let us in the first place suppose the cylinder placed on the horizontal plane with its axis vertical. In this position all of its right line elements, being parallel to the axis, will also be vertical and be projected on the horizontal plane in points, at equal distances from the point 0, in which the axis is pro- jected. Describing therefore a circle, from the point 0, with the radius equal to the distance between the axis and the ele- ments, this circle will be the horizontal projection of the sur- face of the cylinder. To construct the vertical projection, let us suppose the generating rectangle placed parallel to the vertical plane ; in this position, it will be projected into the diameter a b of the circle on the horizontal plane, and into the equal rectan- gle a'b''(}'d f on the vertical plane ; in which last figure c' d' will be the projection of circle of the base of the cylinder; the line a' b' that of its top ; and the lines a' d', and b r c f those of the two opposite elements. As all the other elements will be projected on the vertical plane between these two last, the rectangle a'b'c'd' will represent the vertical projection of the entire surface of the cylinder. To show this more clearly, let us suppose the generating rectangle brought into a position oblique to the vertical plane, as ef for example ; in this position, the two elements projected in e, and ./on the hori- zontal plane, would be projected respectively into e' o', and f A', on the vertical plane ; and as the one projected in f, f A', lies behind the cylinder, it would be represented by a dotted line. All sections of the cylinder parallel to the base, or perpen- dicular to the axis, being circles, equal to that of the base, will be projected on the horizontal plane into the circle of the base, and on the vertical in lines parallel to c' d'. All sec- 108 INDUSTRIAL DRAWING. tions through the axis will be equal rectangles; which, in horizontal projection, will be diameters of the circle, as ef; and, in vertical projection, rectangles as e'f'h'o'. All sections, as 3f JVi parallel to the axis, will cut out two elements pro- jected respectively in m,m'p'; and n, n' u'. Note. The manner of constructing oblique sections will be given farther on. Prob. 86. (PL X. Fig. 109.) Cone. If in a triangle, having two equal sides d' -p\ and tfp', we draw a line p' 0', bisecting the base c f <#', and suppose the triangle revolved about this line as an axis, the equal sides will describe the curved surface, and the base the circular base of a body termed a right cone with a circular base. From what has been said on the cylinder, it will readily appear that, in this case, any plane passed through the axis will cut out of the curved surface two right lines like d'p', and c'p'; and, therefore, that from the pointy/ of the sur- face, termed the vertex, right lines can be drawn to every point of the circumference of the base. These right lines are termed the right line elements of the cone. It will also be equally evident that any line, as a 1>, drawn perpendicular to the axis, will describe a circle parallel to the base, of which a b is the diameter, and, therefore, that every section perpen- dicular to the axis is a circle. From these preliminaries, the manner of projecting a right cone with a circular base, and its sections either through the vertex, or perpendicular to the axis, will be readily gathered by reference to the figures. The horizontal projection of the entire surface will be within the circumference of the base, that of the vertex being at p. The vertical projection of the surface will be the triangle d'p'd / as the projection of all the elements must lie within it. Any plane of section through the axis^ as 0f, will cut from the surface two elements projected horizontally mpfandjp e; and vertically in e' p' and/' -p'. Any plane passed through the vertex, of which M ^Vis the hori- zontal trace, will cut from the surface two right line elements, of which % m and p n are the horizontal, and p' ra' and p' n' the vertical projections. Any plane as R S passed PROJECTIONS. 109 perpendicular to the axis will cut out a circle, of winch a' V is the vertical, and the circle described from p, with a diame- ter a b equal to a' b', is the horizontal projection. Prob. 87. Sphere. (PI. X. Fig. 110.) Drawing any diameter, O jP, in a circle, and revolving the figure around it, the circumference will describe the surface of a sphere ; the diameter A B perpendicular to the axis will describe a circle equal to the given one ; and any chord, as a b, will describe a circle, of which ab is the diameter. Supposing the sphere to rest on the horizontal plane, having the axis O P vertical, every section of the sphere perpendicular to this axis will be projected on the horizontal plane in a circle ; and, as the section through the centre cuts out the greatest circle, the entire surface will be projected within the circle, the diameter of which, A B, is equal to the diameter of the generating circle. In like manner the vertical projection will be a circle equal to this last. The centre of the sphere will be projected in C and C' / and the vertical axis in C and O P. Any section, as R /S, perpendicular to the axis, will cut out a circle of which a b is the diameter, and also the vertical projection ; the horizontal projection will be a circle described from C with a diameter equal to a b. Any section, as M jW, by a vertical plane parallel to the vertical plane of projection, will also be a circle projected on the horizontal plane in m n j and on the vertical plane in a circle, described from C' as a centre, and with a diameter equal to m n. Remarks. The three surfaces just described belong to a class termed surfaces of revolution^ which comprises a very large number of objects ; as for example, all those which are fashioned by the ordinary turner's lathe. They have all certain properties in common, which are 1st, that all sec- tions perpendicular to the axis of revolution are circles ; 2d, that all sections through the axis of revolution are equal figures. Prob. 88. (PI. X. Fig. 111.) To construct the section of a right circular cylinder by a plane oblique to its axis, and perpendicular to the vertical plane of projection. 1st Method. Let aobh and a'Vc'd' be the projections of 110 INDUSTRIAL DBA WING. the cylinder, and MN the vertical trace of the plane of section. This plane will cut the surface of the cylinder in the curve of an ellipse, all the points of which, since the curve is on the surface of the cylinder, will be horizontally projected in the circle aolh, and vertically in m' n', since the curve lies also in the plane of section. Now the plane of section cuts the two elements, projected in a, a' d', and t>, ~b' c', in the points projected in a, and m' ; and b, and ri. It cuts the two elements projected in o' ', and 0, and h, in the points projected vertically in r', and horizontally in o and h. Tak- ing any other element as the one projected in 0, e' a?', the plane cuts it in a point projected vertically in q', and horizon- tally in e. The same projection e and q' would also corre- spond to the point in which the element projected in^ e' a?', is cut by the plane ; and so on for any other pair of elements similarly placed, with respect to the line a b on the base. %d Method. As the plane of section is perpendicular to the vertical plane of projection, its trace on the horizontal plane (PL YI. Fig. 82) will be the line MP, perpendicular to the ground line. In like manner, if the plane of section was cut by another horizontal plane, as g I j or if the origi- nal horizontal plane was moved up into the position of the second, g I would become the new ground line, and the line pp", the trace of the plane of section on this second hori- zontal plane. From this it is clear, that the line in which the second horizontal plane cuts the plane of section is projected vertically in the point p\ and horizontally in the line p p", on the first horizontal plane. But the horizontal plane of which g I is the vertical trace, being perpendicular to the axis of revolution, cuts out of the cylinder a circle which is projected into aobfi / and as it is equally clear that since the line, projected in p',p -p", lies in the horizontal plane g I, it must cut the circle also, the points i and w, in which p -p" cuts the circle of the base, will thus be the projections of the points in question. A like reasoning and corresponding construction would hold true for any other points, as r', q', &c. Remarks. Of the two methods here given, the 2d, as will be seen in what follows, is the more generally applicable, as PEOJEOTION8. Ill requiring more simple constructions ; but the 1st is the more suitable for this particular case, as giving the results by the simplest constructions that the problem admits of. Prob. 89. (PI. X. Fig. 112.) To construct the sections of a rig ht cone with a circular base by planes perpendicular to the vertical plane of projection. This problem comprises five cases. 1st, where the plane of section is perpendicular to the axis ; 2d, where it passes through the vertex of the cone ; 3d, where its vertical trace makes a smaller angle with the ground line than the vertical projection of the adjacent element of the cone does; 4rth, where the trace makes the same angle as the projection of the adjacent element with the ground line, or, in other words, is parallel to this projection ; 5th, where the trace makes a greater angle than this line with the ground line. Cases 1st and 2d have already been given in the projections of the cone (PI. X. Fig. 109) ; the remaining three alone re- main to be treated of. Case 3d. In this case, the angle NML, between the trace M ^Tof the plane and the ground line G Z, is less than that o'a'b', or that between the same line and the element projected in o' a'/ and the curve cut from the surface of the cone will be an ellipse. From what precedes (Prob. 86), if the cone be intersected by a horizontal plane, of which g I is the trace, it will cut from the surface a circle of which c' d' is the diameter, and which will be projected on the horizontal plane in the cir- cumference described from o on this diameter; this plane will also cut from the plane of section a right line, projected vertically in r', and horizontally in r r" / and the projec- tions of the points 8 and -w, in which this line cuts the projec- tion of the circumference, will be two points in the projection of the curve cut from the surface. Constructing thus any number of horizontal sections, as g I, between the points ra' and n' (the vertical projections of the points in which the plane of section cuts the two ele- ments parallel to the vertical plane), any number of points in the horizontal projection cf the curve can be obtained like the two, s and u, already found. 112 INDUSTEIAL DEAWINQ. On examining the horizontal projection of the curve it will be seen that each pair of points, like s and u, are on lines perpendicular to m n, and at the same distance from it ; and that the points m and n are on the line a b, into which the two elements parallel to the vertical plane are projected It will also be seen that the two points s and u, determined by that horizontal plane which bisects the line m' n', are the farthest from the line m n. The line m n, which is the horizontal projection of the longest line of the curve of section, is the transverse axis of the ellipse into which this curve is projected, and the line * u the conjugate axis. Prob. 90. Cases 4 and 5. (PI. X. Fig. 113.) Although the curve cut from the surface in each of these cases is dif- ferent, Case 4: giving a parabola, and Case 5 an hyperbola, the manner of determining the projections of the curve is the same ; and but one example therefore will be requisite to illustrate the two cases. Taking the vertical trace M JV", of the plane of section, parallel to a! , and p' q' the projections of its axis. Let the semicircles ghi, be the horizontal, and g'h'i' the vertical projections of the front half of the hemisphere, being that portion alone which the cylinder enters ; o and o' the projections of its centre. Any horizontal plane will cut from the quarter of the sphere, thus projected, a semicircle, and from the cylinder a circle ; and if the plane is so taken that the semicircle and circle, cut from the two surfaces, intersect, the point or points, in which - these two curves intersect, will be points in the intersection of the two surfaces Let m n be the vertical trace of such a horizontal plane ; it will cut from the spherical surface a semicircle projected vertically in m. n, and hori- zontally in the semicircle mxyn, the diameter of which is PBOJECTIONS. 131 equal to m n. This semicircle cuts the circle adbc, which is the horizontal projection of the one cut by the same plane from the cylinder, in the two points x and y, which are the horizontal projections of the two points required; their vertical projections are at as' and y', on the line in n. By a like con- struction, any required number of points may be found. The highest point of the projection of the required curve will evidently, in this case, lie on that element of the cylinder which is farthest from the centre of the hemisphere; and the lowest point on the element nearest to the same point. Drawing a line, from the projection o of the centre of the hemisphere, through p, that of the axis of the cylinder, the points c and d, where it cuts the circle abed, will give the projections of the two elements in question. The vertical projections of these points will therefore be found at c' and d', on the vertical projections of the semicircles of which o c, and o d are the respective radii. The curve traced through the points c'x'd'y', &c., is the vertical, and the circle adbo the horizontal projection required. Prob. 111. (PI. XIII. Fig. 134.) To construct the pro- jections of the curves of intersection of a right circular cone and sphere. The processes followed, in this problem, are in all respects the same as in the one preceding. Any horizontal plane will cut from the two surfaces circles, and the points in which the two circles intersect will be points of the required curve. Let ghik and g'h'i'k' be the projections of the sphere ; o and o' those of its centre ; aybc and a'o'b the projections of the cone ; o, and o' -p' those of its axis. Let m! n' be the vertical trace of a horizontal plane. This plane will cut from the sphere a circle of which m' n', and mxny are the projections ; and from the cone one of which r' s', and rxsy are the projections ; the points x and y in horizontal projection, and x' and y' the corresponding ver- tical projections of the points where the two circles intersect, are projections of the points of the required curve. In like manner, the projections of any number of points may be obtained, both in the lower and upper curves, in which the cone penetrates the sphere. 132 INDUSTBIAL DRAWING. The highest and lowest points, in the vertical projection of the lower curve, will be on the elements of the cone which are farthest from and nearest to the centre of the sphere. These two elements are the ones in the vertical plane containing the axis of the cone and the centre of the sphere; and are, therefore, projected in the line c y, drawn through the points o and o / the line o c being the horizontal projec- tion of the element nearest the centre ; and o y that of the one farthest from it. The plane which contains these elements cuts from the sphere a circle, the same as g'h'i'k', and the points where the elements cut this circle will be the highest and lowest of the two curves in question. To find the relative positions of the elements cut from the cone and of the circle cut from the sphere, we will use the method explained in Prob. 95. For this purpose, let us revolve the plane, containing the lines in question, and of which c y is the horizontal trace, around the vertical line projected in o, and which passes through the centre of the sphere, until the plane is brought parallel to the vertical plane of projection ; in which new position its trace c y will take the position c' y', turning around the point o. In this new position, the circle contained in this plane will be projected in the original circle g'h'i'k' ; the elements cut from the cone into the lines a" o" and V o", and the points, in which the elements cut the circle, in/' and d' '; e : and c". Taking any one of these points, as the one d f , it will be hori- zontally projected in its new position, in the points/ and, when the plane is brought back to its original position, the point s will come to z' ; and its vertical projection will then be at d, the same height above the ground line as the point d'. The points z' and d are the projections of the highest point of the lower curve, in which the cone penetrates the sphere. In like manner, the point c", which is the lowest point of the vertical projection of the same curve, may be found ; also the points e and t f, the vertical projections of the highest and lowest points of the upper curve. The curves, traced through the horizontal and vertical pro- jections of the points thus obtained, will be the required projections of the curves. 133 Development of Cylindrical and Conical Surfaces. Cylinders and cones, when laid on their sides on a plane surface, touch the surface throughout a right line element If, in this position, a cylinder be rolled over upon the plane, until the element, along which it touched the plane in the first position is again brought in contact with the plane, it is evident that, in thus rolling over, the cylinder would mark out on the plane a rectangle, which would be exactly equal in surface to the convex surface of the cylinder. The base of the rectangle being exactly equal in length to the circum- ference of the circle of the cylinder's base, and its altitude equal to the height of the cylinder, or the length of ita elements. In like manner a cone, laid on its side on a plane, and having its vertex confined to the same point, if rolled over on the plane, until the element on which it first rested is brought again into contact with the plane, would mark out a surface on the plane exactly equal to its convex surface; and this surface would be the sector of a circle, the arc of which would be described from the point where the vertex rested, with a radius equal to the element of the cone ; the length of the arc of the sector being equal to the circumference of the base of the cone ; and the two sides of the sector being the same as the element in its first and last positions. Now, any points, or lines, that may have been traced on these surfaces in their primitive state, can be found on their developments, and be so traced, that, if the surfaces were restored to their original state, from the developments, these lines would occupy upon them the same position as at first. The developments of cylinders and cones are chiefly used in practical applications, to mark out upon objects, having cylindrical or conical surfaces, lines which have been ob- tained from drawings representing the developed surfaces of the object. Prob. 112. (PI. XI. Fig. 135.) To develop the surface of a right cylinder ; and to obtain, on the developed surface, the curved line cut from the cylindrical surface by a plane oblique to the axis of the cylinder. 134 INDUSTRIAL DRAWING. Turning to Prob. 88, Fig. Ill, which is the same as the one of which the development is here required. Draw a right line, on which set off a length a a, equal to the circumfer- ence abed of the cylinder's base ; through the points a, con- struct perpendiculars to a a, on which set off the distances a a', equal to the altitude a/ of of the cylinder ; join a,' a', the rectangle aaa'a' is the entire developed convex surface of the cylinder. Commencing at the point 0 on that of the 2d. As a a'b' is parallel to the front face of each step, its shadow upon these faces will be parallel to the line itself ; therefore p'%', found by projecting up from ^?, will be its shadow upon the front of 2. The shadow of A is at O on the top of 2. The shadow of an a' upon the front faces of the steps will have the same direction as the projection of light ; while the shadow upon the upper faces will be parallel to itself ; c'e' and e'o' are the shadows upon the faces of 3 and 4 ; do and he, found by projecting down from c' and e', are the shadows on the tops of the same steps. Remark. In finding the shadow of A, there can be no question as to which surface it falls upon, if we remember that both projections of the shadow must lie in the projec- tions of the same surface. 2d. Shadow of steps on Hand V. As this problem pre- sents no new principles, let the student, having first deter- mined what edges cast shadows, verify the shadow given The shadow on H is not complete in the figure. Prob. 8. (PI. XXI. Fig. 10.) Shadow of framing. (Fig. 3, PI. XXI Y., shows an isometrical projection of the same framing.) It is required to find the shadow of the brace upon the horizontal timbers and also upon H. The edges of the brace which cast shadows are cd c'd' and ae a'e'. Assume any point as M on cd c'd' and find its shadow, &, on the upper face of the horizontal timbers ; in this case k is on the plane of the top produced ; dk is the shadow of cd c'd' upon the top of the timbers ; a parallel line through e is the shadow of the other edge. To obtain the shadow on H, produce the ray through the 140 INDUSTRIAL DRAWING. point M until it pierces H in i draw a line through i paral lei to dk and it will be the shadow of cd c'd' upon H. A line through q parallel to it would be the shadow of ae a'e' upon H ; it does not show in this case. There is anothei shadow shown in the figure, the construction of which wiL be evident. Construct the shadow of the whole framing upon H and V. Prob. 9. (PI. XXI. Fig. 8.) Shadow of timber resting upon the top of a waU. It is required to find the shadow of the timber upon the wall, and also the shadow of both upon Y. The edges of the timber which cast shadows, whether upon the wall or upon Y, are ae a'e', ab a'c', b b'c', bo b'o' , oe o'h') and e h'e e . The shadow of ae a'e' upon the wall commences at n' and v'n' is its shadow ; v'g' is the shadow of ab a'e' ; g'x', of b b'c' '/ the line through as' parallel to v'n' , is the shadow of bo b'o' upon this face. The line bo b'o f also casts a shadow on the top of the wall which will be parallel to itself through the point I. The construction of the shadow on Y, is evident from the figure. Prob. 10. (PL XXII. Fig. 2.) Shadow of inclined timber upon triangular prism. The relative positior of the pieces is given in the figure, the timber makes an angle of 60 with H, and is parallel to V. Commencing with ad a'd f which casts the shadow ds upon H, we find that this meets the prism at v, which will there- fore be one point of the shadow upon the nearest inclined face of the prism. To obtain another point, find the shadow of ad a'd" upon an auxiliary horizontal plane through pq p'q' '/ p'q' is its vertical trace ; e is the shadow of A upon this plane, and the line eo, parallel to ds, is the shadow of ad a'd' upon the same plane ; this meets pq in the point 0, which is therefore another point of the shadow of ad a'd' upon the face of the prism ; join o with v, and we have the required shadow. The shadow of the diagonally opposite edge cfc'f can be found in the same way, or by drawing fn parallel to da and nz parallel to vo. Part of the shadow of the timber falls on H beyond the prism ; ob a' and bo a'e' are the onlj SHADOWS. 14:1 remaining edges of the timber which cast shadows. There will be no difficulty in finding these as well as the shadow of the prism. Prob. 11. (PI. XXII. Fig. 4.) To find the angle which a ray of 'light makes with either plane of projection. Let ab a'V be the projections of a ray of light. Now if we consider the vertical projecting plane of the ray to be revolved into V, the ray would take the position a"V ; the distance a' a" being equal to ca, the distance of the point A in front of Y ; the angle a'b'a" is the angle which the ray AB makes with Y, and is equal to 35 16'. As both projections of the ray make the same angle with the ground line, the ray must make the same angle with H that it does with Y. Prob. 12. (PL XXII. Fig. 3.) Shadow on interior of hollow hemisphere. The shadow is cast by the semicircle tas, and it will be obtained by finding where rays through differ- ent points of this semicircle pierce the interior. Suppose the hemisphere projected upon a plane xy, whicli is parallel to the direction of light and perpendicular to Y. To find a point of the shadow, intersect the hemisphere by a vertical plane of rays, av / this cuts the semicircle av a'e'v' from the hemisphere. Since the plane xy is parallel to the direction of light and perpendicular to Y, rays of light will be pro- jected upon it making the same angle with xy that rays of light make with Y ; that is, at an angle of 35 16'. Draw a ray through a' at such an angle, and project e', its intersection with a'e'v', to e / this will be one point of the shadow on the interior. Other points may be found in the same way ; the point o is found by using the plane bw. The shadow, commences at the points t and s, where the projection of a ray would be tangent to the circle asvt. Remark. The rays a'e, b'o', &c., are parallel to a"V in Fig. 4. Line of shade. This is the line that separates the light from the dark part of the surface. It 13 the line also that casts the shadow. This line can often be determined by mere inspection, as in the preceding problems, but in some cases special methods must be resorted to in order to determine it. 142 INDUSTRIAL DBAWINO., The method of finding the line of shade upon a vertical cylinder has been given in Prob. 4, the elements e e'o' and t t'f' being the lines of shade. Prob. 13. (PI. XXI., Fig. 7.) To find the line of shad* upon a cone. If a plane of rays be passed tangent to a cone, the element along which it is tangent, will be the line of shade. As every plane tangent to a cone must contain the vertex, a tangent plane of rays must contain the ray through the vertex, and the shadow of the vertex will be a point of its trace ; through c, the shadow of the vertex, draw the line ce tangent to the base of the cone ; this is the trace of the tangent plane, and ve v'e' is the element of contact with the cone ; cd is the trace of another plane tangent on the other side of the cone ; dv d'v' is its element of contact. The lines ce and cd are the shadows of ve v'e', and dv d'v' ', and the space included between them is the shadow of the cone. Prol, 14. (PI. XXII. Fig. 5.) To find the line of shade upon a sphere. Only the vertical projection of the sphere is used, and the vertical plane is supposed to pass through the centre of the sphere. The line of shade is the circle of con- tact of a tangent cylinder of rays. To find points of this curve, assume planes of rays, perpen- dicular to V, whose traces are dw, zv, &c. ; each of these planes will intersect the sphere in a circle, and the point at which a ray is tangent to this circle will be one point of the curve of shade. The plane zv intersects the sphere in a cir- cle projected in sv / revolve this circle about zv until it coin- cides with Y; it will then have the position vo'z. To get the point at which a ray will be tangent to this circle, it is necessary to find the position of a ray when revolved in a similar manner to the circle. According to Prob. 11, the re- volved position of a ray would make an angle of 3516 / with dw. Now, if a line be drawn at this angle and tangent to the circle vo'z, it will give o' as the point of tangency ; when the circle is revolved back to its original position, o' is pro- jected at o, and is one point of the curve of shade. Other points of the curve of shade can be determined in the same way. The curve commences at the points a and 5, SHADOWS. 143 where planes of rays, parallel to dw, would be tangent to the sphere. To find the point of the sphere which appears the bright- est, revolve the circle dw into the position dbw / fc is the re- volved position of a ray passing through the centre ; bisect the angle fob at m'j when the circle is revolved back m' falls at m, the lightest point INDUSTRIAL DRAWING. CHAPTER IX. SHADING. 1. Having given previously the methods for laying flat and graduated tints, let us see how, by their use, we may bring out the true form of an object. The following rules should be carefully studied and followed : I. Mat tints should be given to plane surfaces, when in the light) and parallel to the vertical plane / those nearest the eye being lightest. II. flat tints should be given to plane surfaces, when in the shade, and parallel to the vertical plane ; those nearest the eye being darkest. III. Graduated tints should be given to plane surfaces, when in the light and inclined to the vertical plane / increasing the shade as the surfaces recede from the eye ; when two such surfaces incline unequally the one on which the light falls most directly should be lightest. IV. Graduated tints should be given to plane surfaces, when in the shade, and inclined to the vertical plane ; decreasing the shade as the surfaces recede from the eye. 2. Applying these rules to the shading of an hexagonal prism (PI. I**. Fig. 11), we find by L, that the front face should have a flat tint ; by III., that the left-hand face should have a graduated tint, darkest at the left-hand edge ; by IV., the right-hand face has also a graduated tint, darkest at the left-hand edge. As the left-hand face receives the light more directly than the front face, the nearest part of it should be lighter than the front of the prism. The darkest part of the left-hand face should have about the same shade as the lightest part of the right-hand face. SHADING. 14.5 3. The preceding rules also apply to the shading of curved surfaces, as the cylinder (PI. I**. Fig. 12). The element of shade a' V separates the light from the dark part of the cylinder. The surface a n by IV. should be darkest at a and grow lighter as it approaches n. The part a -p of the illuminated surface, by III. should grow lighter as it approaches^? the part mp by III. should be darkest at ra and lightest at^>/ but by the second part of III. that part of the surface which receives the light most directly should be the lightest, which would make c the brightest point ; if then we take a point, e, half way between c and p, it will give, approximately, the point which will appear the brightest. The surface c e 19 brighter than e p as it receives the light more directly ; it is also lighter than a corresponding space to the left of c, as it is nearer the eye ; so that it is the lightest part of the cylinder. 4. Upon Figs. 13 and 14, PI. I**, are shown in dotted lines the positions of the darkest and lightest parts of the cone and sphere. The method of finding the lines of shade upon each has been given in the chapter on shadows. As it is not necessary in practice to locate these lines exactly, the eye being a sufficient guide, it will be well to notice that the dark line va of the cone is a little nearer the right-hand edge than the dark line of the cylinder ; while the lightest part, between vc and vb, has the same position as that of the cylinder, and is determined in the same way. On the sphere, the point n of the line of shade snp, is a little nearer a than the centre of the sphere ; the line of shade is symmetrical respecting the line ba, the direction of light. The lightest point m is a little nearer the centre than it is to b / it is also on the line ba. In shading these solids, commence at the dark line and shade both ways, using lighter tints for the lighter shades. The dark line of the sphere should be widest at n and taper both ways to p and 8 ; on the cone it tapers from the base to the vertex. 10 146 INDUSTRIAL DRAWIKO.. CHAPTER X. ISOMETEIOAL DRAWING. If we take a cube situated as in Fig. 6, PI. XXII., and tip it up to the left about the point a,e', until it takes the position shown in Fig. 7, the diagonal a'h' being horizontal, and then turn the cube horizontally, without changing its position with respect to H, until it takes the position shown in Fig. 8, we shall have in the vertical projection of Fig. 8, what is called an isometrical projection. In the case of the cube it is the projection made upon a plane perpendicular to a diagonal of the cube. The relative position of the eye, the cube, and the vertical plane is shown in Fig. 7, where /, upon a'h' produced, repre- sents the eye (at an infinite distance) ; xy is the position of the vertical plane ; the cube is placed, as shown by the pro- jections, so that the diagonal ah a'h' is parallel to H and perpendicular to xy or Y. Looking at this isometrical projection of the cube we see that the three visible faces of the cube appear equal, and that all the sides of these faces are equal ; this shows that these faces are similarly situated respecting V, and that their sides, or the edges of the cube, are equally inclined to Y. It will also be noticed that the isometrical projection of the cube can be inscribed in a circle, as the outer edges form a regular hexagon. The three angles formed by the edges meeting at the cen- tre are equal, each being 120. The point a' is called the isometric centre ; the three lines passing through the centre being called isometric axes. Any line parallel to one of these axes is called an isometric line, while any line not parallel is called a non-isometric line. ISOMETRICAL DRAWING. 147 The "plane of any two of the axes, or any parallel plane, is called an isometric plane. The two axes a'V and a'd' (Fig. 8, PI. XXII.) and all par- allel lines make angles of 30 with a horizontal line. It has been seen that the isometrical projection of a cube can be inscribed in a circle ; this renders it easy to construct an isometrical drawing of a cube, by inscribing a regular hexagon in a circle, whose radius is equal to an edge of the cube, and then drawing radii to the alternate angles. "While this would give an isometrical projection of a cube, it would not be the true projection of the cube whose edge was taken as a radius, because the edges of the cube are inclined to the plane of projection, consequently their projections cannot be equal to the edges themselves, but would be less. Let us see how the true isometrical projection of a cube may be obtained without making it necessary to construct the different projections shown in Figs. 6, 7, 8, PI. XXII. The only lines of the cube that are projected in their true size are the diagonals d'b', d'e', I'e' (Fig. 8, PI. XXII.), of the three visible faces. It is evident that the diagonal db d'b' (Fig. 8) -is parallel to V ; by looking at Fig. 7, where the rela- tive position of the eye, the plane of projection, and the cube is shown, it will also be evident that Ve' (corresponding to b'e', Fig. 8) is parallel to the plane xy (V). If now we draw a line db (Fig. 9, PI. XXII.), making an angle of 45 with db at the point b, and note its intersection a with the vertical through c, ab will be the side of a square whose diagonal is db. and would therefore be the true length of the edge of a cube, the diagonal of any face of which is equal to db; but cb is the isometrical projection of this edge, so that we have the means of comparing the two and forming a scale. Divide ab into any number of equal parts and project the points of division upon cb, by lines parallel to ac, and it will give the isometrical projection of these distances. To con- struct then an isometric scale draw a horizontal line be (Fig. 10, PI. XXII.) ; draw ba at an angle of 15 with it ; divide ba into any number of equal parts and project the points of division upon be by lines making an angle of 45 with ba 148 INDUSTRIAL DRAWING. [these projecting lines make an angle of 60 with be]. The distances on be will be the isometric length of the correspond- ing distances on ba. This scale is only good for isometric lines. The diagonals db, dm, bm (Fig. 9) are projected in their true lengths ; this would also be true of all lines parallel to them. As ch is the projection of a line equal to db, a scale may be constructed by projecting any distances, 1, 2, 3, etc., from db to ch; this scale will be good for all lines parallel to ch, ho, or Jin. Although it is well to understand the construction of these scales, they are seldom used in practice, as it is more conve- nient to use a common scale, if necessary, making the isome- tric lines equal to their true length. This method, as already shown, would make the drawing larger than the true projec tion of the object, but there is no objection to this. When made in this way it is called an isometrical drawing, to dis- tinguish from the isometrical projection. The advantage of isometrical drawing is that it offers a simple means of showing in one drawing several faces of an object, thus obviating the necessity of a plan and one or more elevations. It is particularly adapted to the representation of small objects, in which the principal lines are at right angles to each other. Direction of Light. In isometrical projection the light is supposed to have the same direction as the line be (Fig. 16, PI. XXI1L), the diagonal of the cube, that is, it makes an angle of 30 with a horizontal line. Lines of shade. According to a previous definition these are the lines which separate the light from the dark part. In the isometrical projection of the cube (Fig. 16, PL XXIII.) the two right-hand faces (front and back) and the bottom are in the dark, while the two left-hand faces (front and back) and the top are in the light ; consequently the heavy lines shown in the figure are the visible lines of shade. Prob. 1. (Fig. 1, PI. XXIII.) To construct the isometri- cal drawing of a cube, with a block upon one face and a recess in another. Let the edges of the cube be 4" ; the block 1SOMETKICAL DRAWING. 149 2" square and 1^-" thick ; the recess 2" square and 1" deep ; both the block and the recess to be ill the centre of the face. The drawing of the cube might be made, as previously de- scribed, by using a circle with a radius of 4", but a more convenient way is to draw the isometric axes, ca, cb, and cd, making each equal to 4", then isometric lines through the extremities will complete the cube. After completing the cube, divide the axes into four equal parts. To locate the block, draw isometric lines through 1 and 3, upon cb and cd, their intersections will give the base of the block ; through the points of intersection draw isometric lines parallel to ca, make them 1" in length, and connect their extremities. The recess is similarly located, the depth of V can be obtained by projecting from 2 on either ca or cb. Make the isometrical drawing of a cube with a square hole in each face, running through the cube. Prob. 2. (Fig. 2, PI. XXIII.) To construct the isometri- cal drawing of three pieces of timber bolted together. Draw the axes ca, cb, cd; make cb equal to 6"; ce 3" ; eo 2" ; on 4"; do equals ec / the vertical timber is 5"x8" ; the side timbers are each let into the vertical timber the same distance. The method of constructing the nut and washer is given in Prob. 9. Make a drawing with the front side timber removed. Prob. 3. (Fig. 3, PI. XXIII.) To construct the isomet- rical drawing of a portion of framing. The necessary dimensions of the parts are given in Fig. 10, PI. XXI. The edges of the brace being non-isometric lines, it is necessary to locate the extremities, which are in isometric planes, and then join ; cd and ca are each equal to 28" ; the other edges are parallel to ad; de is equal to d'e' (Fig. 10, P. XXI.). Prob. 4. (Figs. 4, 5, PI. XXIII.) To make the isometri- cal drawing of a circle. In Fig. 4 is shown a circle with an inscribed and circumscribed square. If we make an isome- trical drawing of these squares in their relative positions, we shall have at once eight points through which the isometric circle must pass ; these are the points common to the circle and squares ; this is shown in Fig. 5, the two figures being lettered the same. To locate any point, as v (Fig. 4), draw 150 INDUSTRIAL DRAWING. ym perpendicular to ad / make am and mv (Fig. 5) equal to the same distances on Fig. 4. in this way any point of the circle, or any point within the square, can be located. This method gives an exact drawing of the circle, the curve being an ellipse. Prob. 5. (Fig. 6, PI. XXIII.) To make an approximate construction of the isometrical drawing of a circle. Con- struct the isometric square abed ; let d be the centre and dn the radius of the arc nx t> is the centre of the arc pq, the radius being the same as before ; s and o are the centres of the arcs np and qx / the points n,p, #, x, are the centres of the sides. The curve, as thus constructed, approximates near enough to the true curve to answer most purposes. Make an isometrical drawing of a cube with a circle in- scribed in each face. Prob. 6. (Fig. 7, PI. XXIII.) To divide the isometrical drawing of a circle into equal parts. 1st method. At n, the centre of ab, erect the perpendicular nc', and make it equal to na ; from c' as a centre describe the arc mp and divide it into any number of equal parts ; draw lines through these points from c' and produce them until they meet ab ; join the points on ab with c, and the lines drawn will divide the isometric arc into the same number of parts that mp contains. %d method. Describe the semicircle de'h upon dh as a dia- meter ; this is the semicircle of which deh is the isometrical drawing and is in the position deh would take when revolved about dh, as an axis, until parallel to V. Divide de'h into equal parts and project to deh by vertical lines ; these will divide the isometric curve into a corresponding number of parts. Prob. 7. (Fig. 8, PL XXIII.) To make the isometrical drawing of a cube, cylinder, and sphere. Suppose the sphere to rest upon the top of the cylinder, and the cylinder upon the cube. The diameter of the sphere and cylinder is equal to an edge of the cube ; the height of the cylinder is equal to itf diameter. ISOMETEICAL DRAWING. 151 First construct a cube and then inscribe a circle in the top face ; this is the lower base of the cylinder ; to obtain the upper base, construct a second cube resting upon the first, and of the same size ; this is shown in the figure by dotted lines. Inscribe a circle in the top of this second cube and it will be the upper base of the cylinder ; ab and cd tangent to each of these curves are the extreme elements of the cylinder. The sphere rests upon the centre of the upper base ; erect a per- pendicular at ID four dihedral angles ; the one above H and in front of Y, is called the 1st ; the 2d is behind Y and above H ; the 3d is below the 2d, and the 4th below the 1st. The point of sight is in the first angle. Now, all the objects that we have pre viously considered have been placed in the first angle, and, as we have seen, the horizontal projections are always below the ground line, and the vertical above ; but if an object is placed in the second angle (as it is in perspective) when the vertical plane is revolved back, both projections will appear above the ground line, as in Fig. 1, where a' revolves to a'. The same rules that we have had for determining the posi tion of a point in space from its projections, hold good when both projections are above the ground line ; the distance ah (Figs. 1 and 2) is the distance of A from Y, and the distance a'h is the distance of A from H. Prob. 1. (Figs. 3, 4, 5, PI. XXY.) To find the perspec- tive of right lines in different positions by means of visual rays. 1st, when perpendicular to H : let a a'b' (Fig. 3) be a line perpendicular to H, and at the distance ab' behind Y ; cc' is the point of sight. Draw the ray ac a'c' / this pierces Y at 0, the perspective of the point aa' ; /is the perspective of the point ab' ; fe is the perspective of the line a a'b'; in the same way is found hg, the perspective of the line d a'b', which is parallel to a a'b'. 2d, when parallel to H and V: let a'b a'b' (Fig. 4) be the given line, situated in H and at the distance bb' behind Y ; the perspective of the point aa! is at e ; f is the perspec- tive of the point bb', whence efv& the perspective of ab a'b'; hg is the perspective of the line dm a'b', which is parallel to ab a'b'. 3d, when perpendicular to V: ab b' (Fig. 5) is the given line, situated in H, and at the distance bb' from Y ; ef is its perspective ; in ri is a line parallel to abb'; hb is its per- spective. Prob. 2. (Fig. 7, PI. XXY.) To find the perspective of a cube when placed with a face parallel to V. Let abid, and a'd'fe' be the projections of the cube, and cc' the point of 160 INDUSTRIAL DRAWING. Bight. Find first the perspective of the point df, by drawing the ray cd c'f; this gives o as the perspective ; the ray ca c'e? gives r, the perspective of the point ae' ; t is the per- spective of aa'j and s of dd f ; in the same way are found the points j>, q, v, z. Prob. 3. (Fig. 6, PI. XXV.) To find ike perspective of a cube when placed with its faces oblique to V. Let abid-, and a'i'h'e' be the projections of the cube ; s is the perspec- tive of the point dd'j o of the point df ; r of ae'j t of aa' } etc. By an examination of Figs. 3, 4, 6, 7, PI. XXV., it will be seen that the perspective of any right line is parallel to that line when the line is parallel to the perspective plane. It will also be noticed that the edges op, sq, tv, &c. (Figs. 6 and 7), when produced, meet at a point ; this is called the vanishing point of these lines, and since the lines op, sq, tv, etc., are the perspectives of parallel lines, we have the prin- ciple that the perspectives of all parallel lines have a com- mon vanishing point. To find then the vanishing point of any line, draw a line through the point of sight parallel to the given line, and where it pierces the perspective plane will be the vanishing point of this line, and all parallel lines. In Fig. 6, cw c'w' is parallel to di d'i'; it pierces Y at w', which is the vanish- ing point of di d'i', and the edges parallel to it. The van- ishing point of ro, ts, vq, etc., could be found in the same way. The vanishing point of parallel lines, parallel to the per- spective plane, is situated at an infinite distance ; hence the perspectives will be parallel. In Figs. 5 and 7, PI. XXY., the lines which are perpen- dicular to Y vanish at c r ; whence the principle that all lines perpendicular to the perspective plane vanish in the vertical projection of the point of sight ; this is called the centre of the picture. The horizontal line through the centre of the picture is called the horizon. The vanishing points of 'all horizontal lines are situated somewhere on the horizon. LINEAR PERSPECTIVE. 16t The point in which a line pierces a plane is called ita trace ; the trace of a line en the perspective plane is one point of its perspective ; the vanishing point of the line is another point of its perspective ; whence, the perspective of a right line joins its vanishing point with its trace. It is customary to use the vertical plane as the perspective plane, the object being placed in the second angle ; this brings the two projections of the object and the perspective to- gether, as in Fig. 6, PL XXV., which is objectionable. Dif- ferent methods may be used to prevent this, but the most convenient is by supposing the horizontal plane revolved 180, so as to bring the plan of the object in front of V, and then instead of using visual rays, to make use of aux- iliary lines called perpendiculars and diagonals, by which method the vertical projection of the object is not necessary. It is evident that if two lines intersect at a point in space, their perspectives will intersect in the perspective of the point / so that if we pass any two lines through a point and find their perspectives, their intersection will be the perspec- tive of the point. The two lines most convenient to use are a perpendicular and a diagonal. A perpendicular is a line perpendicular to the perspective plane, and vanishes, as we have seen, in the centre of the picture. A diagonal is a horizontal line, making an angle of 45 with the perspective plane. A diagonal being a horizontal line its vanishing point is on the horizon, and since it makes an angle of 45 with Y, the distance of the vanishing point from, the centre of the picture is equal to th-e distance of the point of sight in front of V. This is shown in Fig 2, PL XXVI., cc' being the point of sight, and c'd the horizon ; to find the vanishing point of diagonals draw through cc' the line ch c'd, parallel to H, and making an angle of 4:5 with Y ; it pierces Y at <, the vanishing point of all diagonals parallel to ch c'd. It is evident that c / d=wh=cw. As diagonals may be drawn either to the right or left, there are two vanishing points of diagonals, as at d and d l (Fig. 2). Prob. 4. (Fig. 8, PL XXY.) To construct the perspectiw 162 INDUSTRIAL DRAWING. of a regular hexagon, by means of diagonals and perpen- diculars. The hexagon is situated in H, at the distance nn' behind Y. The horizontal plane has been revolved 180, so that the plan of the hexagon comes in front of Y, while the horizontal projection of the point of sight at the same time revolves to c, behind Y ; c' is the centre of the picture, <2 and di the vanishing points of diagonals, found by making c'd and c'di equal to cw. To find the perspective of the point e, draw through it the perpendicular ee f , e'c' is its perspective ; also draw through e the diagonal ez, zd^ is its perspective ; the point o in which these intersect is the perspective of e ; g' and r are the traces of the perpendicular and diagonal through g g'c' and rd\ are their perspectives, and I is the perspective of g / p is the per- spective of f\ s of b / h of a / and m of n. Remark. In Fig. 8 diagonals are drawn in both directions, and it is seen that either diagonal with the perpendicular gives the perspective, or that two diagonals without the per- pendicular are sufficient. Those diagonals which in plan are drawn to the right, van- ish to the left of the centre of the picture, while those drawn to the left in plan, vanish to the right of the centre ; this comes from having revolved the plan 180. Prob. 5. (Fig. 9, PI. XXY.) To construct the perspective of a pavement made up of squares. Let abef represent the plan of the pavement, the squares being set with their sides diagonally to Y ; c' is the centre of the picture, d and d^ the vanishing points of diagonals. As the sides of the squares are diagonal lines, their perspec- tives will join their traces and the vanishing points of diago- nals ; produce mn to 0, od\ is the perspective of mo ; ad is the perspective of ae / the perspectives of the other edges are similarly found, and their intersections will give the perspec- tives of the squares ; af and be, being perpendiculars, vanish ate'. Prob. 6. (Fig. 1, PI. XXYI.) To find the perspective of a cube. Let abih, a'h'f'e' be the projections of the cube, which is placed with its face parallel to Y, and at the distance ae' behind Y. LINEAR PERSPECTIVE. 163 The lines e'c' and/V are the indefinite perspectives of the lower edges of the cube, which are horizontally projected in ab and hi / draw the diagonals gd^ nd\, md^ and we shall ob- tain the points r, o, p, z ; at these points erect perpendiculars and limit them by the lines a'c' and h'c'. Remark. In this figure the vertical projection of the cube is given, but it evidently is not necessary in order to construct the perspective ; it is sufficient to know the height of the cube, since perpendiculars and diagonals passed through points in the upper base of the cube would pierce V somewhere in the vertical trace of the plane of the top, as they are horizontal lines, and are in that plane. Prob. 7. (Fig. 8, PI. XXYI.) To find the perspective of a vertical hexagonal prism. The prism is placed with one face in Y ; the line rig' is the vertical trace of the plane of the top of the prism. First, construct the perspective of the lower base according to Prob. 4 ; then construct the perspec- tive of the upper base, remembering that the traces of the perpendiculars and diagonals passed through points in the upper base, will be in the line n'g'. Connect the two bases by vertical lines to complete the prism. Prob. 8. (Fig. 6, PL XXVI.) To find the perspective of a square pillar resting upon a pedestal. Let dbhe, a'e'ea be the projections of the pedestal, placed with its face inV; Tdpl is the horizontal projection of the pillar; k'l' is the ver- tical trace of the plane of the top of the pillar. First, construct the perspective of the pedestal in the same way that the perspective of the cube was found in Prob. 6. The face aa'e'e is its own perspective, as it is in V. To find the perspective of the pillar, construct first the per- spective of the lower base ; draw a perpendicular and diago- nal through the point Iri, ri and e' are the traces of these lines, and ric' and e'd\ their perspectives ; their intersection is the perspective of the point In' ; g' is the trace of the diago- nal through p, and g'd\ its perspective; the point o in which this intersects ric' is the perspective of the point pn' ; in the same way the remaining points of the base can be found. The perspective of the upper base of the pillar might be found in the same way, or by erecting perpendiculars at the 164: INDUSTRIAL DEALING. four points already found, and limiting them by the lines k'c' and I'd. Prob. 9. (Fig. 3, PI. XX VI.) To find the perspective of a square pyramid. Let abde be the plan of the pyramid ; hv' is the height of the vertex above the base. Find first the perspective of the base ; e is its own perspective, as it is in V ; m is the perspective of a ; em the perspective of ea / ep the perspective of ed, etc. To obtain the perspective of the vertex, find o, the perspec- tive of its horizontal projection ; at o erect a perpendicular until it meets v'c', this gives n, the perspective of the vertex ; joining n with the corners of the base completes the per- spective. Prob. 10. (Fig. 5, PI. XXVI.) To find the perspective of a square pyramid resting upon a pedestal. Let abeh be the plan of the pedestal ; r'w' is the vertical trace of the upper base ; the edge aa' is in V and is part of the perspective ; z is the perspective of the point W ; , m' d, etc., of the arch ; and so to divide the vertical distances D d, m' c, etc., that the distances between the intermediate horizontals shall be as nearly equal as prac- ticable. The distances between the horizontals of the face of the wall, below the arch, are usually the same, and equal to those just above. If the walls on which the arch rests are built for the sole purpose of supporting it they are termed abutments. The portions of these, here shown in front view, are termed the ends, or heads of the abutments ; the portions projected in A a, and B 5, and lying below the suffit of the arch, are termed the sides or faces. The lines projected in A, and B^ along which the soffit joins the faces of the abutments, are termed the springing AEOHTTEOTTJEAL ELEMENTS. 173 lines of the arch. The right line projected in 0, and parallel to the springing lines, is termed the axis of the arch. The right lines of the soffit projected in m, n, etc., are termed the soffit edges of the coursing joints of the arch ; the lines m w', n n' t etc., the face edges of the same. To construct a longitudinal section of the arch by a vertical plane through the axis of which the trace on the face is the line MN, commence by drawing a line V B" parallel to b B, and at any convenient distance from the front eleva- tion ; from V set off along the ground line the distance V &', equal to that between the front and back faces, or the length of the arch ; from V draw VB' parallel to V'B", and prolong upwards these two lines. The rectangle b'B'B"b" will be the projection of the face of the abutment on the plane of section ; the line b' B ', corresponding to that b B, etc. Drawing the horizontal lines B" ', m" m'", etc., at the same height above the ground line as the respective points B, m, etc., they will be the projections of the soffit edges of the coursing joints. The half of the soffit on the right of the plane of section M .ZV 7 ", is projected into the rectangle B ' G'C"B". The arch stones &, &, etc., forming the key of the arch, are represented in section, the two forming the ends of greater depth than those intermediate, as is very often done. That is, the key- stones at the ends, and the end walls of the abutments, are built up higher than the interior masonry be- tween them ; the top of this last being represented by the dotted line op in the elevation and the full line op' on the cross section. The arch stones running through from one end to the other, and projected between any two soffit edges, as B" B', and m" m"' t are termed a string course. The contiguous stones running from one springing line to the other, as those projected in &, &', &", &c., are termed ring courses. The lines, of which those r s are the projections, are the soffit edges of the joints, termed heading joints, between the stones of the string courses. These edges in one course alternate with those of the courses on either side of it. The cross section on R S requires no particular explana- tion. From its conventional lines, it will be seen that the 174 INDUSTRIAL DRAWING. intention is to represent the soffit and faces of the arch, and its abutments, as built of cut stone, and the backing of rubble stone. Remarks. In the drawings just made, it will be observed that no plan, or horizontal projections, were found to be necessary ; and that a perfectly clear idea is given of the forms and dimensions of all the parts by means of the eleva- tions and sections alone. A previous study of the particular object to be represented will very frequently lead to like re- sults, wherein a few views, judiciously chosen, will serve all the objects of a drawing. The methods of arranging the vertical and horizontal lines of the arch stones of the head, with those of the abutments, often present problems of some intricacy, demanding both skill and taste on the part of the draftsman, so to combine them as to produce a pleasing architectural appearance, and yet not interfere with other essential conditions. An exam- ple of this, being the case of a segment arch, is given in PI. VIII. Fig. 106. PROBLEMS OF MECHANISM. 175 CHAPTER XIV. Prdb. 114. (PL XIY. Fig. 137.) To construct the projec tions of a cylindrical spur wheel. The wheel work employed, in mechanism, to transmit the motion of rotation of one shaft to another (the axis of the second being parallel to that of the first), usually consists of a cylindrical disk, or ring, from the exterior surface of which projects a spur shaped combination, termed teeth, or cogs, so arranged that, the teeth on one wheel interlocking with those of the other, any motion of rotation, received by the one wheel, is communicated to the other by the mutual pressure of the sides of the teeth. There are various methods by which this is effected ; but it will be only necessary in this place to describe the one of most usual and simple construc- tion, for the object we have in view. The thickness of each tooth, and the width of the space between each pair of teeth, are set off upon the circumference of a circle, which is termed the pitch line or pitch circle. The thickness of the tooth and the width of the space taken together, as measured along the pitch line, is termed the pitch of the tooth. The pitch being divided into eleven equal parts, five of these parts are taken for the thickness of the tooth, and six for the width of the space. Having given the radius o m of the pitch circle, and described this circle, it must first be divided into as many parts, each equal to b A, the pitch of the teeth, as the number of teeth. Having made this division, the outline of each tooth may be set out as follows : From the point A, with the distance h b as a radius, describe an arc b c, outwards from the pitch circle ; having set off b e, the thickness of the tooth ; from the point k, with the same radius, describe the arc e / To obtain the apex cf, of the tooth, which may be either a right line, or an arc described from o as a centre ; place this arc three-tenths of the pitch fr h from the pitch line. The sides of the tooth, within the 176 INDUSTRIAL DRAWING. pitch circle, are in the directions of radii drawn from the points b and e. The bottom d g of each space is also an arc, described from o, and at a distance, from and within the pitch circle, of four-tenths of the pitch. From the preceding construction, the outline of each tooth will be the same as abcfed; and that of each space fdgi. The curved portions b -c. and efof each tooth are termed the faces, the straight portions a &, and d e, the flanks. The outline here described represents the profile of the parts, made by a plane perpendicular to the axis of the cylindrical surface, to which the teeth are attached ; which surface forms the bottom of the spaces. The breadth of the teeth is the same as the length of the cylindrical surface to which they are attached. The solid to which the teeth are attached or of which they form a portion is a rim of the same material as the teeth ; and this rim is attached to a central boss, either by arms, like the spokes of a carriage wheel, or else by a thin plate. The former is the method used for large wheels ; the latter for small ones. In wooden wheels, the cogs are let into the rims, by holes cat through the rims. In cast iron wheels, the rim and teeth are cast in one solid piece. When the latter material is used, the arms and central boss are also cast in one piece with the rim and teeth in medium wheels ; but in large sized ones, the wheel is cast in several separate portions, which are afterwards fastened together and to the arms, &c. In the example before us, we shall suppose, for simplifica- tion, that the rim joins directly to the boss ; the latter being a hollotv cylinder, projecting slightly beyond each face of the wheel ; the diameter of the hollow being the same as that of the cylindrical shaft on which the wheel is to be placed. Let us now suppose the axis of the wheel horizontal and perpendicular to the vertical plane. In this position of the wheel, the outline just described will be the vertical projec- tion of the wheel; the rectangle ESTUike horizontal projec- tions of the teeth and rim ; and the one OO'P'P, that of the boss, which projects the equal distances S 0, and R 0\ beyond the faces R T, and SUof the wheel. The faces of each tooth, the surface of its apex, and the PROBLEMS OF MECHANISM. 177 bottom of each space are all portions of cylindrical surfaces, the elements of which are parallel to the axis of the wheel. The horizontal projections of the edges of the teeth C" (7'and F"F', which correspond to the points projected in c, and^ as well as those of the spaces, which correspond to the points a and d, will be right lines parallel to 0'. Drawing the projections as C C', and F F', &c., of these edges, we obtain the complete horizontal projection of the wheel. Prob. 115. (PI. XIY. Fig. 138.) To construct the projec- tions of the same wheel when the axis is still horizontal but oblique to the vertical plane. As in the preceding Probs., of like character to this, the horizontal projection of all the parts will, in this position of the axis, be the same as in the preceding case ; and the vertical projections will be found as in like cases. The pitch circle and other circles on the faces of the wheel, and the ends of the boss, will be projected in ellipses ; the transverse axes of which are the vertical diameters of these circles ; and the conjugate axes the vertical projections of the correspond- ing horizontal diameters. The vertical projections of the edge? of the teeth, which correspond to the horizontal pro- jections C C', and F F', &c., will be the lines c c'", and //'", &c., parallel to the projection o' o", of the axis. Prob. 116. (PI. XY. Fig. 139.) To construct the projections of a mitre, or beveled wheel, the axis of the wheel being horizontal, and perpendicular to the vertical plane. The spur wheel, we have seen, is one in which the teeth project beyond a cylindrical rim, attached to a central boss either by arms, or by a thin connecting plate ; moreover that portions of the teeth project beyond the pitch line, or circle, whilst other portions lie within this line. Mitre, or beveled wheels, are those in which the teeth are attached to the surface of a conical rim ; the rim being connected with a central boss, either by arms, or a connecting plate. In the beveled wheel the faces of the teeth project beyond an imaginary conical surface, termed the pitch cone, whilst the flanks lie within the pitch cone. The faces and flanks are conical surfaces, which have the same vertex as the pitch cone the apex of each tooth is either a plane, or a conical 12 178 INDUSTRIAL DRAWING. surface which, if prolonged, would pass through the venes of the pitch cone ; and the bottom of the space between each tooth is also a portion of a cone, or a plane passing through the same point. The ends of the teeth and the rim lie on conical surfaces, the elements of which are perpendicular to those of the pitch cone, and have the same axis as it. Before commencing the projections, it will be necessary to explain how the teeth are set out, as well as the rim from which they project. Let F (Figs. A, B) be the vertex of the pitch cone ; V o its axis ; and Vmn its generating triangle. At n and TO, drawing perpendiculars to V- n and V- TO, let the point v, where they meet the axis prolonged, be the vertex of the cone that terminates the larger ends of the teeth and rim. Setting off the equal distances TO TO' and n w', and drawing TO' v' and n' v' respectively parallel to TO v and n v, let v' be the vertex of the cone, and v'm'n' its generating triangle that forms the other end of the teeth and rim. Having by Prob. 114 developed the cone, of which v is the vertex (Fig. B] and TO n the diameter of the circle of its base ; set off upon the circle, described with the radius v n, the width n b of the space between the teeth, together with the thickness b e of the teeth, as in the preceding Prob. 115, and construct the outline of each tooth and space as in those cases ; next set off a t, equal to a b, for the thickness of the rim at the larger end, and describe the circle limiting it, with the radius v i. If now we wrap this development back on the cone, we can mark out upon its surface the outline of the larger ends of the teeth ; and we observe that the faces of the teeth will thus project beyond the pitch cone, and the flanks lie within it. If we next suppose lines to be drawn from the vertex Fto the several points of the outline of the teeth, the spaces between them, and to the interior circle of the rim, the lines so drawn will lie on the bounding surfaces of the teeth and rim ; and the points in which they meet the surface of the cone, having the vertex v\ and which limits the smaller ends of the teeth, will mark out on this surface the outlines of the smaller ends of PROBLEMS OF MECHANISM. 179 the teeth and rim. The length of each tooth, measured along the element V n of the pitch cone, will be n ri. To construct the vertical projection of the wheel, we observe, in the first place, that the points n, 6, e, &c. (Fig. B), where the faces and flanks join, lie upon the circumference of the circle of which o n is the radius, and which is the pitch circle for the outline of the ends of the teeth ; in like manner that the points e,/ &c., of the apex of each tooth, lie on a circle of which p q is the radius ; the points s, a, rf, &c., lie on the circle of which r s is the radius ; and the interior circle of the rim has t u for its radius. The radii of the corresponding circles, on the smaller ends of the teeth and rim, are o' n'; p' q'; r' s'; and t' u'. In the second place all these circles are parallel to the vertical plane of projection, since the axis of the wheel is perpendicular to this plane, and they will therefore be projected on this plane in their true dimensions. From the point then, the vertical projection of the axis, describe in the first place the four concentric circles with the radii 0Q, 0N, 0S, and Z7 respectively equal to p q, o n, &c. ; and, from the same centre, the four others with radii Q', &C', respectively equal to p' q', &c. On the circle having the radius N, set off the points N, J3, E, &c., corresponding to n, b, e, &c. ; on the one Q, the points (7, F, &c., corresponding to c, f, &c. ; on the one S, the points , A, D, &c., corresponding to 5, a, c?, &c. From the points C and F, thus set off, draw right lines to the point 0; the portions of these lines, intercepted between the circles of which 0Q and Q' are respectively the radii, with the portions of the arcs, as C F, O F', intercepted between these lines, will form the outline of the vertical pro- jection of the figure of the apex of the tooth. The portions of the lines, drawn from B and E to 0, intercepted between the circles described with the radii JVand N', together with the portions of the lines forming the edge of the apex, and the curve lines B C, EF and the corresponding curves B' C", F E' on the smaller end, will be the projec tions of the outlines of the faces and flanks of the tooth. The outline of the projection of the bottom of the space will lie ISO INDUSTRIAL DRAWING. bet-ween the right lines drawn from 8 and A to 0, and the arcs S- A, /S' A', intercepted between these lines, on the circles described with the radii /S, and S'. The projection of the cylindrical eye of the boss is the circle described with the radius K. Having completed the vertical projection, the corresponding points in horizontal projection are found by projecting the points C, F, B, E, A t D, &c., into their respective circumferences (Fig. A) at c', f, b f , &c. The portions of the lines drawn from C and $ to 0, in vertical projection, will, in horizontal projection, be drawn from c' and/' to V; and so for the other elements of the surfaces of the faces, flanks, &c., of the teeth. The hori- zontal projection of the larger end of each tooth will be a figure like the one a'b'c'fe'd'. The boss projects beyond the rim at the larger end of the tfheel ; it is usually a hollow cylinder. Its horizontal pro- jection is the figure xyzw, &c. Prob. 117. (PL XY. Fig. 140.) To construct the projec- tions of the same wheel, when the axis is oblique to the vertical plane, and parallel as before to the horizontal. This variation of the problem requires no particular verbal explanation; as from preceding problems of the like cha- racter, and the Figs, the manner in which the vertical projections are obtained from the horizontal will be readily made out. The best manner however of commencing the vertical projection will be to draw, in the first place (Fig. Z>), all the ellipses which are the projections of the circles described with the radii Q, N, &c. (Fig. C), and next those of the vertices of the three cones, which will be the points v, 0', and 0". These being drawn the projections of the different lines, forming the outline of the projection cf any tooth, can be readily determined. Prob. 118. (PI. X. Fig. 141.) To construct the projec- tions of the screw with a square thread. As a preliminary to this problem, it will be requisite to show how a line termed a helix, can be so marked out on the surface of a right circular cylinder that, when this surface is developed out, the helix will be a right line on the develop- ment ; and the converse of this, having a right line drawn OD PROBLEMS OF MECHANiSM. 181 the developed surface of a right circular cylinder, to find the projection of this line, when the development is wrapped around the surface. Let ABCD be the horizontal projection of the cylinder: acc'a' its vertical projection ; and the line o o' the projec- tions of its axis. Let the circle of the base be divided into any number of equal parts, for example eight, and draw the vertical projections e e', ff, &c., corresponding to the points of division E, F, &c. Having found the development of this cylinder, by constructing a rectangle (PI. X. Fig. 142), of which the base a a is equal to the circumference of the cylinder's base, and the altitude a a' is that of the cylinder ; through the points e, 5, / c, &c., respectively equal to the equal parts A E, &c., of the circle, draw the lines e e', b &', &c., parallel to a a'. These lines will be the developed positions of the elements of the cylinder, pro- jected in e e f j b J', &c. Now on this development let any inclined line, as a m, be drawn ; and from the point n, at the same height above the point a, on the left, as the point m is above a on the right, let a second inclined line n m' be drawn parallel to a m ; and so on as many more equidistant inclined parallels as may be requisite. Now it will be observed, that the first line, a TO, cuts the different elements of the cylinder at the points marked 1, 2, 3, &c. ; and there- fore when the development of the cylinder is wrapped around it these points will be found on the projections of the same elements, and at the same heights above the projection of the base as they are on the development. Taking, for example, the elements projected in b &', and c c', the points 2 and 4 of the projection of the helix will be at the same heights, b 2, and c 4 on the projections, above a c, as they are on the development above a a. It will be further observed, that the helix of which a m is the development will extend entirely around the cylinder ; so that the point TO, on the projection, will coincide with the two m and n on the deve- lopment, when the latter is wrapped round ; and so on for the other points w, n', and m'; so that the inclined parallels will, in projection, form a continuous line or helix uniformly wound around the cylinder. Moreover, it will be seen, if 182 INDUSTRIAL DRAWING. through the points 1, 2, 3, &c., on the development, lines are drawn parallel to the base a a, that these lines will be equi- distant, or in other words the point 2 is at the same height above 1, as 1 is above a, &c. ; and that, in projection also, these points will be at the same heights above each other; this gives an easy method of constructing any helix on a cylinde when the height between its lowest and highest point for one turn around the cylinder is given. To show this; having divided the base of the cylinder into any number of equal parts (PI. X. Fig. 141), and drawn the vertical projections of the corresponding elements, set off from the foot of any element, as a, at which the helix commences, the height a ra, at which the helix is to end on the same element ; divide a m into the same number of equal parts as the base ; through the points of division draw lines parallel to a c, the projection of the base ; the points in which these parallels cut the projections of the elements will be the required points of the projection of the helix ; draw- ing the curved line a, 1, 2, &c., through these points it will be the required projection. Having explained the method for obtaining the projec- tions of a helix on a cylinder, that of obtaining the projectiona of the parts of a screw with a square fillet will be easily understood. Prob. 119. (PI. XVI. Fig. 143.) To construct the projections of a screw with a square fillet. Draw as before a circle, with any assumed radius A B, for the base of the solid cylinder which forms what is termed the newel of the screw, and around which the fillet is wrapped. Construct, as above, the projections of two parallel helices on the newel; the one a2ra; the other x2z; their distance apart, a x, being the height, or thickness of the fillet, estimated along the element a a' of the cylinder. From 0, with a radius A', describe another circle, such that A A' shall be the breadth of the fillet as estimated in a direction perpendicular to the axis of the newel ; and let the rectangle a"c''d"e" be the vertical projection of this cylinder. Having divided the base of the second cylinder into a like number of equal parts corresponding to the first, and drawn PROBLEMS OF MECHANISM. 183 the vertical projections of the elements corresponding to these points, as I ', &c., construct the vertical projection of a helix on this cylinder, which, commencing at the point a" shall in one turn reach the point ra", at the same heighi above a" as the point ra is above a. The helix thus found will evidently cut the elements of the outer cylinder at the same heights above the base as the corresponding one on the inner cylinder cuts the corresponding elements to those of the first ; the projections of the two will evidently cross each other at the point 2 on the line b b'. In like manner construct a second helix x"22", on the second cylinder and parallel to the first, commencing at a point x" at the same height above a" as x is above a. This, in like manner, will cross the projection x2z at the point 2. The four projections of helices thus found will be the projections of the exterior and interior lines of the fillet ; the exterior surface of which will coincide with that of the exterior cylinder, and the top and bottom surfaces of which will lie between the correspond- ing helices at top and bottom. The void space between the fillet which lies between the exterior cylinder and the surface of the newel is termed the channel; its dimensions are usually the same as those of the fillet. Prob. 120. (Pis. XYI. XVII. Figs. 144 to 157.) To construct the lines showing iJie usual combination of the working beam, the crank, and the connecting rod of a steam engine. In a drawing of the kind of which the principal object is to show the combination of the parts, no other detail is put down but what is requisite to give an idea of the general forms and dimensions of the main pieces, and their relative positions as determined by the motions of which they are susceptible. As each element of this combination is symmetrically dis- posed with respect to a central line, or axis, we commence the drawing by setting off, in the first place, these central lines in any assumed position of the parts; these are the lines o -f, the distance from the centre of motion of the working beam A to that of its connexion with the connecting rod B, and which is 3 inches and 55 hundredths of an inch 184 INDUSTRIAL DRAWING. actual measurement on the drawing, or 4 feet 43 hundredth? on the machine itself, the scale of the drawing being 1 inch tc l foot, or T ' 5 ; next the line/ e, the distance of the centre of motion / to that e of the connecting rod and crank C ; lastly the line o d, from the centre of motion e to that d of the crank and the working shaft, the actual distance being 1 inch 36 hundredths. These lines being accurately set off. the outlines of the parts which are symmetrically placed with respect to them may be then set off, such dimensions as are not written down being obtained by using the scale of the drawing, or from the more detailed Figs. 145 to 157. Having completed the outlines we next add a sufficient number of lines, termed indicating lines, to show the ampli- tude of motion of the parts, or the space passed over between the extreme positions of the axes, as well as the direction or paths in which the parts move. These are shown by the arc described from o with the radius of; the circle described with d e ; the lines o a, o c, and o 6, the extreme and mean positions of the axis of: with b h, and b g the extreme positions off- e. Besides the axes and indicating lines, others which may be termed axial lines, being lines drawn across the centre of motion of articulations, as through the point of the axis on cross sections, are requisite, for the full understanding of the combinations of the parts ; such, for example, as the lines z x and v w, on (Fig. 147), which is a cross section of the connecting rod, made at m n, m' ri, Figs. 145, 146 ; those X Y, X'Y, &c.; those ZW, Z'W on Figs. 148 to 157. Prob. 121. (PI. XVIII. Figs. 158 to 170.) To make the measurements, the sketches, and finished drawing of a machine from the machine itself. A very important part of the business of the draftsman and engineer is that of taking the measurements of industrial objects, with a view to making a finished drawing from the rough sketches made at the time of the measurements. For the purposes of this labor, the draftsman requires the usual instruments for measuring distances and determining the Horizontal and vertical distances apart of points ; as the PROBLEMS OF MECHANISM. 185 carpenter's rule, measuring rods, or tape, compasses, chalk line, an ordinary level; and a plumb line. The first three are used for ascertaining the actual distances between points, lines, &c. ; the chalk line to mark out on the parts to be measured central lines, or axes ; the two last to determine the horizontal and vertical distances between points. For sketching, paper ruled into small squares with blue, or any other colored lines, is most convenient ; such as is used, for example, by engineers in plotting sections of ground. With such paper, or lead pencil, and pen and ink, the draftsman needs nothing more to note down the relative positions of the parts with considerable accuracy. Taking for example the side of the small square to represent one or more units of the scale adopted for the sketch, he can judge, by the eye, pretty accurately, the fractional parts to be set off. In making measurements, it should be borne in mind, that it is better to lose the time of making a dozen useless ones, than to omit a single necessary one. The sketch is usually made in lead pencil, but it should be put in ink, by going over the pencil lines with a pen, as soon as possible ; otherwise the labor may be lost from the effacing of numbers or lines by wear. The lines running lengthwise and crosswise on the paper, and which divide its surface into squares, will serve, as vertical and horizontal lines on the sketch, to guide the hand and eye where projections are required. It is important to remember, that in making measurements we must not take it for granted that lines are parallel that seem so to the eye ; as, for example, in the sides of a room, house, &c. In all such cases the diagonals should be measured. These are indispensable lines in all rectilineal figures which are either regular or irregular except the square and rectangle. The mechanism selected for illustrating this Prob. is the ordinary machine termed a crab engine for raising heavy weights. It consists, 1st (Fig. 158), of a frame work com- posed of two standards of cast iron A, A, connected by wrought iron rods J, b with screws and nuts ; the frame being firmly fastened, by bolts passing through holes in the bottoms of the standards, to a solid bed of timber framing ; 2d, of the 186 INDUSTKIAL DRAWING. mechanism for raising the weights, a drum B to which is fastened a toothed wheel C that gears or works into a pinion D placed on the axle a; 3d, two crank arms E where the animal power as that of men is applied ; 4th, of a rope wound round the drum, at the end of which the resistance or weight to be raised is attached. The sketch (Figs. 159 to 168) is commenced by measuring the end view A' of the standards and other parts as shown in this view ; next that of the side view, as shown in (Fig. 160). To save room, the middle portion of the drum B", &c., is omitted here, but the distances apart of the different portions laid down. These parts should be placed in the same relative positions on the sketch as they will have in projec- tion on the finished drawing (Figs. 169, 170). The drum being, in the example chosen, of cast iron, sections of a portion of it are given in Figs. 163, 164. The other details speak for themselves. The chief point in making measurements is a judicious selection of a sufficient number of the best views, and then a selection of the best lines to commence with from which the details are to be laid in. This is an affair of practice. The draftsman will frequently find it well to use the chalk line to mark out some guiding lines on the machine to be copied, before commencing his measurements, so as to obtain central lines of beams, &c. ; and the sides of triangles formed by the meeting of these lines. TOPOGRAPHICAL DRAWING. 187 CHAPTER XY. TOPOGRAPHICAL DRAWING. THE term topographical drawing is applied to the methods adopted for representing by lines, or other processes, both the natural features of the surface of any given locality, and the fixed artificial objects which may be found on the surface. This is effected, by the means of projections, and profiles, or sections, as in the representation of other bodies, com- bined with certain conventional signs to designate more clearly either the forms, or the character of the objects of which the projections are given. As it would be very difficult and, indeed with very few- exceptions, impossible to represent, by the ordinary modea of projection, the natural features of a locality of any con- siderable extent, both on account of the irregularities of the surface, and the smallness of the scale to which drawings of objects of considerable size must necessarily be limited, a method has been resorted to by which the horizontal dis- tances apart of the various points of the surface can be laid down with great accuracy even to very small scales, and also the vertical distances be expressed with equal accuracy either upon the plan, or by profiles. To explain these methods by a familiar example which any one can readily illustrate practically, let us suppose (PL XIX. Fig. 171) a large and somewhat irregularly shaped potato, melon, or other like object selected, and after being carefully cut through its centre lengthwise, so that the section shall coincide as nearly as practicable with a plane surface, let one half of it be cut into slices of equal thickness by 188 INDUSTRIAL DRAWING. sections parallel to the one through the centre. This being done, let the slices be accurately placed on each other, so as to preserve the original shape, and then two pieces of straight stiff wire, A, B, be run through all the slices, taking care to place the wires as nearly perpendicular as practicable to the surface of the board on which the bottom slice rests, and into which they must be firmly inserted. Having marked out carefully on the surface of the board the outline of the figure of the under side of the bottom slice, take up the slices, being careful not to derange the positions of the wires, and, laying aside the bottom slice, place the one next above it on the two wires, in the position it had before being taken up, and, bringing its under side in contact with the board, mark out also its outline as in the first slice. The second slice being laid aside, proceed in the same manner to mark out the out- line on the board of each slice in its order from the bottom ; by which means supposing the number of slices to have been five a figure represented by Fig. 171 will be obtained. Now the curve first traced may be regarded as the outline of the base of the solid, on a horizontal plane ; whilst the other curves in succession, from the manner in which they have been traced, may be regarded as the horizontal projections of the different curves that bound the lower surfaces of the different slices ; but, as these surfaces are all parallel to the plane of the base, the curves themselves will be the hori- zontal curves traced upon the surface of the solid at the same vertical height above each other. With the projections of these curves therefore, and knowing their respective heights above the base, we are furnished with the means of forming some idea of the shape and dimensions of the surface in question. Finally, if to this projection of the horizontal curves we join one or more profiles, by vertical planes inter- secting the surface lengthwise and crosswise, we shall obtain as complete an idea of the surface as can be furnished of an object of this character which cannot be classed under any regular geometrical law. The projections of the horizontal curves being given as well as the uniform vertical distance between them, it will be very easy to construct a profile of the surface by any vertical TOPOGRAPHICAL DRAWING. 189 plane. Let X T be the trace of any such vertical plane, and the points marked a, a/, x", &c., be those in which it cuts the projections of the curves from the base upwards. Let G L be a ground line, parallel to X Y, above which the points horizontally projected in x. x', x", &c., are to be vertically projected. Drawing perpendiculars from these points to G L, the point x will be projected into the ground line at y ; that marked x' above the ground line at y\ at the height of the first curve next to the base above the horizontal plane ; the one marked x", will be vertically projected in y", at the same vertical height above x 1 as y' is above y; and so on for the other points. "We see therefore that if through the points y, y', y", &c., we draw lines y' y', &c., parallel to the ground line these lines will be at equal distances apart, and are the vertical projections of the lines in which the profile plane cuts the different horizontal planes that contain the curves of the surface, and that the curve traced through yy'y", &c., is the one cut from the surface. In like manner any number of profiles that might be deemed requisite to give a complete idea of the surface could be constructed. In examining the profile in connexion with the horizontal projection of the curves it will be seen that the curve of the profile is more or less steep in proportion as the horizontal projections of the curves are the nearer to or farther from each other. This fact then enables us to form a very good idea of the form of the surface from the horizontal projections alone of its curves ; as the distance apart of the curves will indicate the greater or less declivity of the surface, and their form as evidently shows where the surface would present a convex, or concave appearance to the eye. For any small object, like the one which has served for our illustration, the same scale may be used for both the hori- zontal and vertical projections. But in the delineation of large objects, which require to be drawn on a small scale, to accommodate the drawing to the usual dimensions of the paper used for the purpose, it often becomes impracticable to make the profile on the same scale as the plan, owing to the smallness of the vertical dimensions as compared with the horizontal ones. For example, let us suppose a hill of irre 190 f INDUSTRIAL DRAWING. gular shape, like the object of our preceding illustration, and that the horizontal curve of its base is three miles in its longest direction, and two in its narrowest, and that the highest point of the hill above its base is ninety feet ; and let us further suppose that we have the projections of the hori- zontal curves of the hill for every three feet estimated ver- tically. Now supposing the drawing of the plan made to a scale of one foot to one mile, the curve of the base would require for its delineation a sheet of paper at least 3 feet long and 2 feet broad. Supposing moreover the projection of the summit of the hill to be near the centre of the base and the declivity from this point in all directions sensibly uniform, it will be readily seen that the distance apart of the horizontal curves, estimated along the longest diameter of the curve of the base will be about half an inch, and along the shortest one about one-third of an inch ; so that although the linear dimensions of the horizontal projections are only the y^W of the actual dimensions of the hill yet no difficulty will be found in putting in the horizontal curves. But if it were required to make a profile on the same scale we should at once see that with our ordinary instruments it would be impracticable. For as any linear space on the drawing is only the j^V o- part of the corresponding space of the object, it follows that for a vertical height of three feet, the distance between the horizontal curves, will be represented on the drawing of the profile by the T ,V P art of a f ot > a distance too small to be laid off by our usual means. Now to meet this kind of difficulty, the method has been devised of drawing profiles by maintaining the same horizontal distances between the points as on the plan, but making the vertical distances on a scale, any multiple whatever greater than that of the plan, which may be found convenient. For example, in the case before us, by preserving the same scale as that of the plan for the horizontal distances, the total length of the profile would be 3 feet; but if we adopt for the vertical distances a scale of T l , 7 of an inch to one foot, then the vertical distance between the horizontal curves would be T 3 5 of an inch, and the summit of the profile would be 9 inches above its base. It will be readily seen that this method will not TOPOGEAPHICAL DRAWING. 191 alter the relative vertical distances of the points from each other ; for T \ of an inch, the distance between any two hc~i zontal curves on the profile, is the ^ of 9 inches the height of the projection of the summit, just as 3 feet is the ^ of 90 feet on the actual object. But it will be further seen that cne profile otherwise gives us no assistance in forming an idea of the actual shape and slopes of the object, and in fact rather gives a very erroneous and distorted view of them. Plane of Comparison, or Reference. To obviate the trouble of making profiles, and particularly when the scale of the plan is so small that a distorted and therefore erroneous view may be given by the profile made on a larger scale than that of the plan, recourse is* had to the projections alone of the horizontal curves, and to numbers written upon them which express their respective heights above some assumed hori- zontal plane, which is termed the plane of reference, or of comparison. In Fig. 171, for example, the plane of the baae may be regarded as the one from which the heights of all objects above it are estimated. If the scale of this drawing was i of an inch to % an inch, and the actual distance between the planes of the horizontal curves was equal to i an inch, then the curves would, in their order from the bottom, be h an inch vertically above each other. To express this fact by numbers, let there be written upon the projection of the curve of the base the cypher (0) ; upon the next this (1); &c. These numbers thus written will indicate that the height of each curve in its order above that of the base is , f , f , &c., of an inch. The unit of measure of the object in this case being half an inch. The numbers so written are termed the references of the curves, as they indicate their heights above the plane to which reference is made in estimating these heights. The selection of the position of the plane of comparison is at the option of the draftsman; as this position, however chosen, will in no respects change the actual heights of the points with respect to each other ; making only the references c/i each greater or smaller as the plane is assumed at a lower or higher level. Some fixed and well defined point is usually taken for the position of this plane. In the topography of 192 INDUSTRIAL DRAWING. loccilities near the sea, or where the height of any point of the locality above the lowest level of tide water is known, thig level is usually taken as that of the plane of comparison. This presents a convenient starting point when all the curves of the surface that require to be found lie in planes above this level. But if there are some below it, as those of the exten- sion of the shores below low water, then it presents a difficulty, as these last curves would require a different mode of reference from the first to distinguish them. This difficulty may be gotten over by numbering them thus (-1), (-2), &c., with the - sign before each, to indicate references belonging to points below the plane of comparison. The better method, however, in such a case, is to assume* the plane of comparison at any convenient number of units below the lowest water level, so that the references may all be written with numbers of the same kind. References. In all cases, to avoid ambiguity and to provide for references expressed in fractional parts of the unit, the references of whole numbers alone are written thus (2.0), that is the integer followed by a decimal point, and a 0; those of mixed or broken numbers, thus (2.30), (3.58), (0.37), &c., that is with the whole number followed by two decimal places to express the fractional part. Projections of the Horizontal Curves. No invariable rule can be laid down with respect to the vertical distance apart at which the horizontal curves should be taken. This distance must be dependent on the scale of the drawing, and the purpose which the drawing is intended to subserve. In drawings on a large scale, such for example as are to serve for calculating excavations and embankments, horizontal curves may be put in at distances of a foot, or even at less distances apart. In maps on a smaller scale they may be from a yard upwards apart. Taking the scale No. 8, in the Table of Scales farther on, which is one inch to 50 feet, or y 7 , as that of a detailed drawing, the horizontal curves may be put in even as close as one foot apart vertically. A con- venient rule may be adopted as a guide in such cases, which is to divide 600 by the fraction representing the ratio which designates the scale, and to take the resulting quotient to TOPOGRAPHICAL DRAWING. 193 express the number of feet vertically between the horizontal curves. Thus 600 -4- T J T gives one foot as the required distance; 600 ~ T? V gives 3 feet; 600 -j- ^ gives the half of a foot, &c., &c. But whatever may be this assumed distance the portion of the surface lying between any two adjacent curves ia supposed to be such, that a line drawn from a point on the upper curve, in a direction perpendicular to it and prolonged to meet the lower, is assumed to coincide with the real surface. This hypothesis, although not always strictly in accordance with the facts, approximates near enough to accuracy for all practical purposes; especially in drawings made to a small scale, or in those on a large one where the curves are taken one foot apart or nearer to each other. Let d (Fig. 172) for instance be a point on the curve (3.0), drawing from it a right line perpendicular to the direction of the tangent to the curve (3.0) at the point d, and prolonging it to c on the curve (2.0), the line d c is regarded as the projection of the line of the surface between the points pro- jected in d and c. In like manner a b may be regarded as the projection of a line on the portion of the surface between the same curves. It will be observed however that the line a b is quite oblique with respect to the curve (2.0), whereas dc is nearly perpendicular to (2.0) as well as to (3.0), owing to the portions of the curves where these lines are drawn being more nearly parallel to each other in the one case than in the other. This would give for the portion to which a b belongs a less approximation to accuracy than in the other portion referred to. To obtain a nearer degree of approxi- mation in such cases, portions of intermediate horizontal curves as x x, y y, &c., may be put in as follows. Suppose one of the new curves y y is to be midway between (2.0) and (3.0). Having drawn several lines as a b, bisect each of them, and through the points thus obtained draw the curve y y, which will be the one midway required. In like manner other intermediate curves as x cc, y y may be drawn. Having put in these curves, the true line of declivity, between the points e and / for example, will b 13 194 INDUSTRIAL DRAWING. the curved or broken line, ef cutting the intermediate curves at right angles to" the tangents where it crosses them. The intermediate curves are usually only marked In pencil, as they serve simply to give the position of the line that shows the direction of greatest declivity of the surface between the two given curves. Prob. 122. Having the references of a number of points on a drawing, as determined by an instrumental survey, to construct from these data the approximate projection of the equidistant horizontal curves having whole numbers for references. Engineers employ various methods for determining equi- distant horizontal curves, either directly by an instrumental process on the ground, or by constructions, based upon the considerations just explained, from data obtained by the ordinary means of leveling, &c. Let us suppose for example that ABCD (Fig. 173) repre- sents the' outline of a portion of ground which has been divided up into squares of 50 feet by the lines x x, x' x', y y, &c., run parallel to the sides A B and A (7, and that pickets having been driven at the points where these lines cut each other and the parallel sides, it has been deter- mined by the usual methods of leveling that these points have the references respectively written near them. With these data it is required to determine the projections of the equidistant horizontal curves with whole number references which lie one foot apart vertically. Having set off a line A x (Fig. 174), equal to A cc, on (Fig. 173) draw perpendiculars to it at the points A and x. From these two points set off along the perpendiculars any number of an assumed unit (say half an inch as the one taken), and divide each one into ten equal parts. Through these points of division draw lines parallel to A x. Cut from a piece of stiff paper a narrow strip like A G (Fig. 174), making the edge A accurately straight. By means of a large pin fasten this strip to the paper and draw- ing board at the point A. If we consider that for the distance of 50 feet between any two points on ground, of which the surface is uniform (as is TOPOGRAPHICAL DRAWING. 195 most generally the case), the line of the surface between the two points will not vary very materially from a right line, and that any inconsiderable difference will be still less sensible on a drawing of the usual proportions, we m;iy without any important error then assume the line in question to be a right line. Now as the reference of the point x is (26.20) the difference of level between it and A, or the height of x above A is 1.70 ft., or equal to the difference of the two references. But from what has just been laid down with respect to the line joining the points A and x drawn on the actual surface, it is plain that the point on this line having the whole reference (25.0) lies between A and x, and that as it is 0.50 ft. higher than A its projection will lie between A and x and its distance from A will be to the distance of x from A in the same proportion as its height above A is to the height of x above A, or as 0.50 ft. is to 1.70 ft. By calculating, or by constructing by (Prob. 54, Fig. 55) a fourth proportional to A x = 50 ft. ; 1.70 ft. = the height of a above A ; and 0.50 ft. = the height of the required point above A; we shall obtain the distance of the projection of this point from A. In like manner by calculation, or construction, we can obtain the distance from A of any other point between A and x of which the reference is given. But as the calculation of these fourth proportionals would require some labor the Fig. 174 is used to construct them by this simple process. Find on the perpendicular to A x on the right the division point marked 1.70 ; turn the strip of paper around its joint at A until the edge A is brought on this point, and confine it in this position. The portions of the parallels intercepted between A and the perpen- diculars at A will be the fourth proportionals required. For example, the vertical height between the point (24.50) and the one (25.0) being equal to the difference of the two references, or 0.50 foot, the horizontal distance which corres- ponds to this is at once obtained by taking off in the dividers the distance, on the parallel drawn through the point .5, between the perpendiculars at A and the edge A 0. This distance set off along the line A B (Fig. 174) from A to 196 INDUSTRIAL DRAWING. (25.0) will give the required point. In li^e manner the distance from A to (26.0) will be found, by taking off in the dividers the portion of the parallel drawn through the point 1.5 on the perpendicular at A, To find the points corresponding to the references (24.0), (25.0), and (26.0), on the line xx parallel to A D, which lie between the points marked (26.20) and (23.30), the vertical height between these points being (26.20) (23.30) = 2.90 feet, first bring the edge A to the point marked 2.90 on the perpendicular on the right, then, to obtain the distance corresponding to (24.0), take off the portion of the intercepted parallel through the point .7 and set it off from (23.30) towards (26.20), and so on for the other points (25.0) and (26.0). Having in this manner obtained all the points on the parallels to A B and A D, with entire numbers for references, the curves drawn through the points having the same references will be the projections of the corresponding horizontal curves of the surface. It may happen, owing to an abrupt change in the declivity of the ground between two adjacent angles of one of the squares, as at b between the point A and y, on the line A D, that it may be necessary to obtain on the ground the level and reference of this point, for greater accuracy in delineating the horizontal curves. Suppose the reference of b thus found to be (21.50), it will be seen that the rise from y to b is only 0.4 foot, whilst from b to A it is 3 feet. To obtain the references with whole numbers between b and A, take off the distance A b (Fig. 173) and set it off from A to b on (Fig. 174), and through b erect a perpendicular to A x, marking the point where this perpendicular cuts the parallel drawn through the point 3, and bringing the edge A of the strip of paper on this point, we can obtain as before the distances to be set off from b towards A (Fig. 173) to obtain the required references. TOPOGRAPHICAL DRAWING. 197 CONVENTIONAL METHODS OF REPRESENTING THE NATURAL AND ARTIFICIAL FEATURES OF A LOCALITY. For the purposes of an engineer, or for the information of a person acquainted with the method, that of representing the surface of 'the ground by the projections of equidistant horizontal curves is nearly all that is requisite ; but to aid persons in general to distinguish clearly and readily the various features of a locality, certain conventional means are employed to express natural features as well as artificial objects, which are termed topographical signs. Slopes of ground. The line of the slope, o declivity of the surface at any given point between any two equidistant horizontal curves, it has been shown is measured along a right line drawn from the upper to the lower curve, and perpendicular to the tangent to the upper curve at the given point. This slope may be estimated either by the number of degrees in the angle contained between the line of declivity and a horizontal line, in the usual way of measuring such angles ; or it may be expressed by the ratio between the perpendicular and base of a right angle triangle, the vertical distance between the equidistant horizontal curves being the perpendicular, and the projection of the line of declivity the base. If for example the line of declivity of which a b (Fig. 172) is the projection makes an angle of 45 with the horizontal plane, then the vertical distance between the points a and b on the two curves will be equal to a 6, and the ratio between the perpendicular and base of the right angle triangle, by which the declivity in this case is esti- mated is = since the equidistant curves are taken one ab 1 unit apart. As a general rule all slopes greater than 45 or y are regarded as too precipitous to be expressed by horizontal equidistant curves, the most that is done to represent them is to draw when practicable the top and bottom lines of the surface. In like manner all slopes less than 0..53'..43" or ? V are regarded as if the surface were horizontal; stil. 198 INDUSTRIAL DRAWING. upon such slopes the horizontal curves may when lequisite be put in ; but nothing further is added to express the declivity of the surface. Lines of declivity, &c. The lines used in topographical drawing to picture to the eye the undulations of the ground, and which are drawn in the direction of the lines of declivity of the surface, serve a double purpose, that of a popular representation OT the object expressed, and with which most intelligent persons are conversant, and that of giving the means, when they are drawn in accordance to some system agreed upon, of estimating the declivities which they figure, with all the accuracy required in many practical purposes for which accurate maps are consulted by the engineer or others. As the horizontal curves when accompanied by their references to some plane of comparison are of themselves amply sufficient to give an accurate configuration of the surface represented, it is not necessary to place on such drawings the lines used on general maps, and which to a certain extent replace the horizontal curves. The lines of declivity in question will therefore be confined to maps on a somewhat small scale, in which horizontal curves are not resorted to with any great precision, although they may have been used to some extent as a general guide in constructing the outlines of the map, such for example as from one inch to 100 feet, or r Vo> and upwards as far as such lines can serve any purpose of accuracy, say one inch to half a mile, or 3"! (- To represent therefore the form and declivities of all slopes, from | to - 6 V inclusive, in maps on these and inter- mediate scales, the following rules may be followed for pro portioning the breadth and the length of the lines oJ declivity, and the blank spaces between them. 1st, The distance between the centre lines of the lines of declivity shall be 2 hundredth* of an inch added to the % of the denomi- nator of the fraction denoting the declivity expressed in hun- dredths of an inch. Thus for example in the declivity denoted 6 -' T the rule jpves (2 4- V ) = 18 hundredths of an inch for the distance TOPOGRAPHICAL DRAWING. 199 apart of the lines. In the declivity of we obtain (2 4- I) = 2$ Imndredths of an inch. 2d. The lines should be the heavier as they are nearer tc each other, or as the declivity expressed by them is the steeper. For the most gentle slope so expressed, that of T y, the lines should be fine, for those of -J, or steeper, their breadth should be 1$ hundredths of an inch. This rule will make the blank space between the heavy strokes equal to half the breadth of the stroke. 3d. No absolute rules can be laid down with respect to the lengths of the strokes, these will depend upon the scale of the drawing, the skill of the draftsman, and the form of the surface to be denned by them. If we take for example the scale of 5-^0 or one inch to 50 feet, and suppose the horizontal curves to be put in at one foot apart vertically, which on the draw- ing corresponds to ^ or 2 hundredths of an inch, the distance between these curves on slopes of -j- would be 2 hundredths of an inch, whilst on a slope of Jy, the curves would be 2 x 64 = 128 hundredths, or 1.28 in., nearly an inch and one third apart. In the first case therefore if the strokes were limited between the two curves of each zone they would be only 2 hundredths of an inch long, whilst in the second, if a like limit were prescribed, they would be an inch and a third in length ; both of which would be inconvenient to the draftsman, and would present an awkward appearance, par- ticularly the latter, on the drawing. To obviate this difficulty then it has been found well, on gentle slopes, to limit the length of the stroke to about 6 tenths of an inch, and in steep slopes to adopt strokes of the length from 8 to 16 hun- dredths of an inch. These limits will require on steep slopes to certain scales that the strokes shall embrace the zones comprised between three or more horizontal curves, whilst on gentle slopes to some scales it will be necessary to divide up the zone com- prised by two curves into two or more by intermediate curves in pencil, so as to obtain auxiliary zones of convenient breadth for the draftsman between which the strokes are put in, according to the 1st and 2d rules, the strokes of one auxiliary zone not running into those of the other. 200 INDUSTRIAL DRAWING. Practical applications. Suppose on a zone between the curves (2.0) and (3.0) that the distance between the points f and /is 1.2 in., or 120 hundredths of an inch, it would be necessary according to the 3d rule to divide this zone into at least two by one auxiliary curve. Let us suppose it to be divided into four parts by three auxiliary curves x x, y y t z z put in according to what has been already laid down, Having done this calculate by rule 1st the distance occupied along this curve by 5 strokes or lines of declivity. Supposing the slope to be J v , the rule would give (2 + V) = 17 hundredths of an inch for the distance apart of two strokes ; and for five it would give 4 x 17 = 68 hundredths. Take now a strip of paper and set off on its edge 68 hundredths of an inch, which divide into four equal parts ; then apply this edge to the curve z z and set off from o to p the dots for the five strokes ; do the same for the curves y y, and x x, and through the points thus set off draw the strokes normal to the curve along which they are set off. Where the curves approach nearer to each other, and are less than 6 tenths of an inch apart and over 4 tenths, as at d- c, it will be well to draw an intermediate line as m n along which the strokes will be set off, and to which they will be drawn perpendicularly. Lines of declivity put in accurately in this manner, in groups of five, from distance to distance between the hori- zontal curves, will serve to guide the hand, in judging by the eye the positions of the intermediate lines between the groups ; the spaces gradually contracting, or widening, as the slope, as shown by the positions of the horizontal curves, becomes steeper, or more gentle. Scale of spaces. When the spaces between the lines of declivity have been carefully put in according to the pre- ceding system, they will serve to determine the declivity at any point ; and a scale of spaces, corresponding to the declivities, ought to be put down on the drawing, in like manner as we put down a scale for ascertaining horizontal distances. The following method may be taken to construct this scale. On a right line estimating from the point A (PI. XIX. Fig. 175) set off 64 equal parts to J5, each part TOPOGRAPHICAL DRAWING. 01 being equal say to J ff , or T L of an inch. Number the points of division from o at A, to 64 at R Construct perpendiculars to the right line at A and B, and on the one at A set off a distance to C corresponding to four spaces of the lines of declivity for the slope of }, and at B for spaces to D for the slope of v \. Draw a right line C D through the points thus set off. Through each of the equal divisions on A B, or through every fifth one, draw lines parallel to the two perpendiculars; each of these lines, intercepted between A B and C D, will represent four spaces, corresponding to the slope marked at the points on A B. To find the declivity of a zone between two horizontal curves, at any point, from the scale, we take off in the dividers the distance of four spaces of the lines of declivity at the point, then place the points of the dividers on the lines A B and C- D so that the line drawn between the points will be perpendicular to A B, the corresponding number on A B will give the slope. Suppose for example the points of the dividers when placed embrace the points ra and w, the corresponding number on A B being about 34 gives ^ for the required slope. Surfaces of water. (PI. XX. Fig. 176.) To represent water a series of wavy lines A, A are drawn parallel to the shores. The lines near the shores are heavier and nearer together than those towards the middle of the surface. No definite rule can be laid down further than to make the lines finer and to increase the distance between them as they recede from the shore. When the banks are steep the slope is represented by heavy lines of declivity. The water line is a tolerably heavy line. If islands B occur in the water course, some pains must be taken in uniting the water lines around its shores with the others. Shores. Sandy shelving shores C are represented by fine dots uniformly spread over the part they occupy on the drawing. The dots are strewn the more thickly as the shore is steeper. Gravelly shores are represented by a mixture of fine and coarse dots. 202 INDUSTRIAL DRAWING. Meadows. These are represented E by systems of very short fine lines placed in fan shape, so as to give the idea of tufts of grass. The tufts should be put in uniformly, parallel to the lower border of the drawing, so as to produce a uniform tint. Marshy ground. This feature F, F is represented by a combination of water and grass, as in the last case. The lines for the water surfaces are made straight, and varied in depth of tint, giving the idea of still water with reflections from its surface. Trees. Single trees /, / are represented either by a tuft resembling the foliage of a bush, with its shadow, a small circle, or a black dot, according to the scale of the drawing. Evergreens may be distinguished from other trees by tufts of fine short lines disposed in star shape. Forests O are represented by a collection of tufts, small circles, and points, so disposed as to cover the part uniformly. Brushwood E and clearings with undergrowth standing, with smaller and more sparse tufts, &c. Orchards as in 0. Rivulets, ravines, &c. Small water-courses of this kind K, K and their banks are represented by the shore lines or bank slopes, when the scale of the drawing is large enough to give the breadth of the stream. The lines gra- dually diverging, or else made farther apart below the junction of each affluent. On small scales a single line is used, which is gradually increased in heaviness below each affluent. Rocks This feature L is expressed by lines of more or less irregularity of shape, so disposed as to give an idea of rocky fragments interspersed over the surface, and connected by lines with the other portions intended to represent the mass of whole rock. Artificial objects. The above are the chief natural features represented conventionally. The principal conventional signs for artificial objects will be best gathered from Plates XIX. and XX. In most works for elementary instruction, and in the systems of topographical signs adopted in public services, almost every natural and artificial feature has its representa- TOPOGRAPHICAL DRAWING. 203 live sign. The copying of these is good practice for the pupil ; but for actual service those signs alone which desig- nate objects of a somewhat permanent character are strictly requisite; as in culture, for example, rice fields may be expressed by a sign, as they, for the most part, retain for a long time this destination ; whereas the ploughed field of the Spring is in grain in Summer and barren in Winter ; and the field of Indian corn of this year is in wheat the next, &c., &c. Practical methods. Finished topographical drawings form a part of the office work of the civil engineer, that require great time, skill, and care. For field duties he is obliged to resort to methods more expeditious in their results than those of the pen, and the use of the lead pencil furnishes one of the best. The draftsman should therefore accustom him- self to sketch in ground by the eye, and endeavor to give to his sketch at once, without repeated erasures and interlinea- tion, the final finish that it should receive to subserve his purposes. Hill slopes, horizontal curves, water, &c., &c., may be sketched in either by lines, according to rules already laid down, or else by uniform tints obtained by rubbing the pencil over the paper until a tint is obtained of such intensity as to represent the general effect of lines of declivity of varying grade, water lines, &c. The pencil used for this purpose should be very black and moderately hard, so as to obtain tints of any depth, from deep black to the lightest shade which will not be easily effaced. The effects that may be produced in this manner are very good, and considerable durability may be given to the drawing by pasting the paper on a coarse cotton cloth, and then wetting the surface of the drawing with a mixture of milk and water half and half. Every draftsman will do well to exercise himself at this work in the office until he finds he can imitate any given ground by tints. Colored Topography. The use of colors in topography is an effective and rapid method of indicating the features of land, and one largely employed. The colors used are indigo, Hooker's green, No. 2, yellow ochre, burnt sienna, carmine, gamboge, and sepia. 204 INDUSTRIAL DRAWING. Water (a PL XXYIII.) is indicated by a flat tint of indigo ; it is shaded out from the shore-line, when there is one. Grass-land (b PL XXVIII.) is indicated by a flat tint of green. Sand, roads, and streets (c PL XXYIII.) are indicated by a fiat tint of yellow ochre. JSuildings, bridges, and all structures (d. PL XXYIII.), are indicated by a tint of carmine. Railroads (e. PL XXYIII.) are indicated by a dark line of carmine without cross lines. Cultivated land (f. PL XXYIII.) is indicated by a flat tint of burnt sienna; sometimes parallel lines (g. PL XXYIII.) are ruled over the flat tint, using for the purpose either a darker tint of burnt sienna, green, or, more commonly, sepia. Uncultivated land (h. PL XXYIII.) is indicated by a double tint of burnt sienna arid green. To lay a double tint, prepare the two tints in separate saucers, then using a brush for each tint, carry one color for a short distance upon the surface, and then change for the other color and brush, letting the colors join and blend of themselves ; alternate the tints in this way until the whole surface is covered. Avoid any regularity in the mottled tint obtained. Hitts (i. PL XXYIII.) are indicated by a tint of sepia, the depth of the tint corresponding to the slope of the land, being darkest where the slope is greatest, and becoming lighter as the slope decreases. The sepia is laid over the land tints. Trees (L PL XXYIII.). There are a number of steps to be followed in indicating trees, l&t, lay the land tint ; 2d, pencil in fine lines the outlines of the trees ; 3d, tint them with green, making the lower right-band part the darkest ; 4th, touch up the trees with gamboge upon the light side (the upper left-hand) ; 5th, add the shadows of the trees with sepia. Marshy ground (1. P) . XXYIII.) is indicated by water and grass land so arranged that the position of the patches of land shall be horizontal; draw a shade line of sepia along the lower edges of the Jand. Light is supposed to come from the upper left-hand corner of the drawing ; hills are shaded without any reference to the direction of light ; only the shadows of such objects as houses, trees, etc., are represented ; sepia is used for shadows. TOPOGRAPHICAL DRAWING. 205 For the method of preparing and using tints, see the chap- ter on tinting. When making a colored plate, first pencil everything, then lay the flat tints, then touch up the trees, then the hills and shadows. The division lines between fields and all outlines are ruled with sepia. For pen drawings the draftsman should always have at hand a good supply of pens made of the best quills, with nebs of various sizes to suit lines of various grades, for slopes, &c. His ink should be of the best, and of a decided tint when laid on ; deep black, red, green, &c. The breadth of lines adopted for different objects must depend upon the importance of the object, and the magnitude of the scale to which the drawing is made. In drawings to small scales lines of not more than two breadths can be used, as the fine and medium. For those to larger scales, three sizes of lines may be introduced, the fine, medium, and heavy. Similar remarks may be made on lettering and the size, &c., of borders. To letter well requires much practice from good models. The draftsman should be able to sketch in by the eye letters of every character and size without resorting to rulers or dividers ; until he can do this, whatever pains he may take, his lettering will be stiff and ungainly. The size of the lettering will be dependent upon that of the drawing and the importance of the object. The character is an affair of good taste, and is best left to the skill and fancy of the draftsman ; for arbitrary rules cannot alone suffice, even were they ever rigorously attended to. As it is of some importance to obtain the best effects in drawings which demand so much time and labor as topo- graphical maps, it may be well to observe that, in pen or line drawings, it is best to put in the letters before the lines of declivity, water lines, &c. ; as it is less difficult to put in the lines without disfiguring the letters than to make clean and well defined letters over the lines. Tne border of the drawing, like the lettering, is frequently a fancy composition of the draftsman. It most generally consists of a light line on the interior and a heavy one on the 206 INDUSTRIAL DUAWINO. exterior ; the heavy line having the same breadth as that of the blank space between it and the light line. As the bordei is generally a rectangle in shape, the rule usually followed for proportioning its breadth which includes the light line, the blank space, an<3 the heavy line is to make it the one hundredth part of the length of the shorter side of the rect- angle. The title of the drawing is placed without the border at top when it takes up but one line ; when it requires several it is usually placed within it. The greatest height of the letters of the title should be three hundredths of the length of the shorter side of the border ; and when the title is with- out the border the blank space between it and the border should be from two to four hundredths of the shorter side. To every line of topographical drawing there should be two scales, one to express the horizontal distances between the points laid down ; the other a scale to express the slopes as in Fig. 175. The scales should be at the bottom of the drawing, either within or without the border, according to the space unoccupied by the drawing. Finally every drawing should receive the signature of the draftsman; the date of the drawing; and state from what authorities, or sources compiled ; and under whose direction, or supervision executed. If emanating from any recognized public office, it ought also to be stamped with the seal of the office. Scales of distances. In our corps of military engineers, for the purposes of preserving uniformity and attaining accuracy in the execution of maps and plans for official action, a system of regulations is adopted, to the requirements of which strict conformity is enjoined on all in any way connected with those corps, prescribing the manner in which all objects are to be represented, and the scales to which the drawings of them shall be made. As the last point is the result of much experience, and may save the young drafts- man much time in the selection of a suitable scale for any given object, it has been thought well to add in this place the following Table of S< ales, adopted for the guidance of TOPOGRAPHICAL DRAWING. TABLE OF SCALES. 207 PROPORTION or THE SCALE. APPLICATION or THE SCALE. 1 inch to ^ an incu, 1 inch to 1 inch, f 1 inch to 6 inches, 1 inch to 1 foot, A- 1 inch to 2 feet, 1 inch to 5 feet, irV 1 inch to 10 feet, rb- 1 inch to 50 feet, 12 inches to 200 yards, rb 1 inch to 220 feet, 24 inches to 1 mile, 1 inch to 440 feet, 12 inches to 1 mile. 1 inch to 880 feet, 6 inches to 1 mile. TOiTTT' 1 inch to 1320 feet, 4 inches to 1 mile, 1 inch to 2640 feet, 2 inches to 1 mile, 1 inch to 5280 feet, 1 inch to 1 mile, I inch to 10560 feet. an inch to 1 mile, Details of surveying instruments,