Fig. 41.
Fig. 42.
Fig. 43.
Fig. 44.
Fig. 45.
Fig. 46.
A line is said to beproducedwhen it is extended beyond its natural limits: thus, in Figure 44, lines A and B areproducedin the point C.
A line is bisected when the centre of its length is marked: thus, line A in Figure 45 is bisected, at or in, as it is termed,e.
The line bounding a circle is termed its circumference or periphery and sometimes the perimeter.
A part of this circumference is termed an arc of a circle or an arc; thus Figure 46 represents an arc. When this arc has breadth it is termed a segment; thus Figures 47 and 48 are segments of a circle. A straight line cutting off an arc is termed the chord of the arc; thus, in Figure 48, line A is the chord of the arc.
Fig. 47.
Fig. 48.
Fig. 49.
Fig. 50.
Fig. 51.
A quadrant of a circle is one quarter of the same,being bounded on two of its sides by two radial lines, as in Figure 49.
When the area of a circle that is enclosed within two radial lines is either less or more than one quarter of the whole area of the circle the figure is termed a sector; thus, in Figure 50, A and B are both sectors of a circle.
A straight line touching the perimeter of a circle is said to be tangent to that circle, and the point at which it touches is that to which it is tangent; thus, in Figure 51, line A is tangent to the circle at point B. The half of a circle is termed a semicircle; thus, in Figure 52, A B and C are each a semicircle.
Fig. 52.
Fig. 53.
The point from which a circle or arc of a circle is drawn is termed its centre. The line representing the centre of a cylinder is termed its axis; thus, in Figure 53, dotdrepresents the centre of the circle, and lineb bthe axial line of the cylinder.
To draw a circle that shall pass through any three given points: Let A B and C in Figure 54 be the points through which the circumference of a circle is to pass. Draw line D connecting A to C, and line E connecting B to C. Bisect D in F and E in G. From F as a centre draw the semicircle O, and from G as a centre draw the semicircle P; these two semicircles meeting the two ends of the respective lines D E.From B as a centre draw arc H, and from C the arc I, bisecting P in J. From A as a centre draw arc K, and from C the arc L, bisecting the semicircle O in M. Draw a line passing through M and F, and a line passing through J and Q, and where these two lines intersect, as at Q, is the centre of a circle R that will pass through all three of the points A B and C.
Fig. 54.
Fig. 55.
To find the centre from which an arc of a circle has been struck: Let A A in Figure 55 be the arc whose centre is to be found. From the extreme ends of the arc bisect it in B. From end A draw the arc C, and from B the arc D. Then from the end A draw arc G, and from B the arc F. Draw line H passing through the two points of intersections of arcs C D, and line I passing through the two points of intersection of F G, and where H and I meet, as at J, is the centre from which the arc was drawn.
A degree of a circle is the 1/360 part of its circumference. The whole circumference is supposed to be divided into 360 equal divisions, which are called thedegrees of a circle; but, as one-half of the circle is simply a repetition of the other half, it is not necessary for mechanical purposes to deal with more than one-half, as is done in Figure 56. As the whole circle contains 360 degrees, half of it will contain one-half of that number, or 180; a quarter will contain 90, and an eighth will contain 45 degrees. In the protractors (as the instruments having the degrees of a circle marked on them are termed) made for sale the edges of the half-circle are marked off into degrees and half-degrees; but it is sufficient for the purpose of this explanation to divide off one quarter by lines 10 degrees apart, and the other by lines 5 degrees apart. The diameter of the circle obviously makes no difference in the number of decrees contained in any portion of it. Thus, in the quarter from 0 to 90, there are 90 degrees, as marked; but suppose the diameter of the circle were that of inner circled, and one-quarter of it would still contain 90 degrees.
Fig. 56.
So, likewise, the degrees of one line to another are not always taken from one point, as from the point O, but from any one line to another. Thus the line marked 120 is 60 degrees from line 180, or line 90 is 60 degrees from line 150. Similarly in the other quarter of the circle 60 degrees are marked. This may be explained further by stating that the point O or zero may be situated at the point from which the degrees of angle are to be taken. Here it may be remarked that, to save writing the word "degrees," it is usual to place on the right and above the figures a small °, as is done in Figure 56, the 60° meaning sixty degrees, the °, of course, standing for degrees.
Fig. 57.
Suppose, then, we are given two lines, asaandbin Figure 57, and are required to find their angle one to the other. Then, if we have a protractor, we may apply it to the lines and see how many degrees of angle they contain. This word "contain" means how many degrees of angle there are between the lines,which, in the absence of a protractor, we may find by prolonging the lines until they meet in a point as atc. From this point as a centre we draw a circle D, passing through both linesa,b. All we now have to do is to find what part, or how much of the circumference, of the circle is enclosed within the two lines. In the example we find it is the one-twelfth part; hence the lines are 30 degrees apart, for, as the whole circle contains 360, then one-twelfth must contain 30, because 360÷12 = 30.
Fig. 58.
If we have three lines, as lines A B and C in Figure 58, we may find their angles one to the other by projecting or prolonging the lines until they meet as at points D, E, and F, and use these points as the centres wherefrom to mark circles as G, H, and I. Then, from circle H, we may, by dividing it, obtain the angleof A to B or of B to A. By dividing circle I we may obtain the angle of A to C or of C to A, and by dividing circle G we may obtain the angle of B to C or of C to B.
Fig. 59.
Fig. 60.
It may happen, and, indeed, generally will do so, that the first attempt will not succeed, because the distance between the lines measured, or the arc of the circle, will not divide the circle without having the last division either too long or too short, in which case the circle may be divided as follows: The compasses set to its radius, or half its diameter, will divide the circle into 6 equal divisions, and each of these divisions will contain 60 degrees of angle, because 360 (the number of degrees in the whole circle) ÷6 (the number of divisions) = 60, the number of degrees in each division. We may, therefore, subdivide as many of the divisions as are necessary for the two lines whose degrees of angle are to be found. Thus, in Figure 59, are two lines, C, D, and it is required to find their angle one to the other. The circle is divided into six divisions, marked respectively from 1 to 6, the division being made fromthe intersection of line C with the circle. As both lines fall within less than a division, we subdivide that division as by arcsa,b, which divide it into three equal divisions, of which the lines occupy one division. Hence, it is clear that they are at an angle of 20 degrees, because twenty is one-third of sixty. When the number of degrees of angle between two lines is less than 90, the lines are said to form an acute angle one to the other, but when they are at more than 90 degrees of angle they are said to form an obtuse angle. Thus, in Figure 60, A and C are at an acute angle, while B and C are at an obtuse angle. F and G form an acute angle one to the other, as also do G and B, while H and A are at an obtuse angle. Between I and J there are 90 degrees of angle; hence they form neither an acute nor an obtuse angle, but what is termed a right-angle, or an angle of 90 degrees. E and B are at an obtuse angle. Thus it will be perceived that it is the amount of inclination of one line to another that determines itsangle, irrespective of the positions of the lines, with respect to the circle.
TRIANGLES.
A right-angled triangle is one in which two of the sides are at a right angle one to the other. Figure 61 represents a right-angled triangle, A and B forming a right angle. The side opposite, as C, is called the hypothenuse. The other sides, A and B, are called respectively the base and the perpendicular.
Fig. 61.
Fig. 62.
Fig. 63.
Fig. 64.
An acute-angled triangle has all its angles acute, as in Figure 63.
An obtuse-angled triangle has one obtuse angle, as A, Figure 62.
When all the sides of a triangle are equal in length and the angles are all equal, as in Figure 63, it is termed an equilateral triangle, and either of its sides may be called the base. When two only of the sides and two only of the angles are equal, as in Figure 64, it is termed an isosceles triangle, and the side that is unequal, as A in the figure, is termed the base.
Fig. 65.
Fig. 66.
When all the sides and angles are unequal, as in Figure 65, it is termed a scalene triangle, and either of its sides may be called the base.
The angle opposite the base of a triangle is called the vertex.
Fig. 67.
Fig. 68.
A figure that is bounded by four straight lines is termed a quadrangle, quadrilateral or tetragon. When opposite sides of the figure are parallel to eachother it is termed a parallelogram, no matter what the angle of the adjoining lines in the figure may be. When all the angles are right angles, as in Figure 66, the figure is called a rectangle. If the sides of a rectangle are of equal length, as in Figure 67, the figure is called a square. If two of the parallel sides of a rectangle are longer than the other two sides, as in Figure 66, it is called an oblong. If the length of the sides of a parallelogram are all equal and the angles are not right angles, as in Figure 68, it is called a rhomb, rhombus or diamond. If two of the parallel sides of a parallelogram are longer than the other two, and the angles are not right angles, as in Figure 69, it is called a rhomboid. If two of the parallel sides of a quadrilateral are of unequal lengths and the angles of the other two sides are not equal, as in Figure 70, it is termed a trapezoid.
Fig. 69.
Fig. 70.
Fig. 71.
If none of the sides of a quadrangle are parallel, as in Figure 71, it is termed a trapezium.
THE CONSTRUCTION OF POLYGONS.
Fig. 71 a.
Fig. 72.
The term polygon is applied to figures having flat sides equidistant from a common centre. From this centre a circle may be struck that will touch all the corners of the sides of the polygon, or the point of each side that is central in the length of the side. In drawing a polygon, one of these circles is used upon which to divide the figure into the requisite number of divisions for the sides. When the dimension of the polygon across its corners is given, the circle drawn to that dimension circumscribes the polygon, because the circle is without or outside of the polygon and touches it at its corners only. When the dimension across the flats of the polygon is given, or when the dimension given is that of a circle that can be inscribed or marked within the polygon, touching its sides but not passing through them, then the polygon circumscribes or envelops the circle, and the circle is inscribed or marked within the polygon. Thus, in Figure 71a, the circle is inscribed within the polygon, while in Figure 72 the polygon is circumscribed by the circle; the first is therefore a circumscribed andthe second an inscribed polygon. A regular polygon is one the sides of which are all of an equal length.
NAMES OF REGULAR POLYGONS.
A figure of3 sides iscalled aTrigon."4"Tetragon.polygon5"Pentagon."6"Hexagon."7"Heptaagon."8"Octagon."9"Enneagon or Nonagon.
Fig. 73.
Fig. 74.
The angles of regular polygons are designated by their degrees of angle, "at the centre" and "at the circumference." By the angle at the centre is meant the angle of a side to a radial line; thus in Figure 73 is a hexagon, and at C is a radial line; thus the angle of the side D to C is 60 degrees. Or if at the two ends of a side, as A, two radial lines be drawn, as B, C, then the angles of these two lines, one to the other, will be the "angle at the centre." The angle at the circumference is the angle of one side to its next neighbor; thus the angle at the circumference in a hexagon is 120 degrees, as shown in the figure forthe sides E, F. It is obvious that as all the sides are of equal length, they are all at the same angle both to the centre and to one another. In Figure 74 is a trigon, the angles at its centre being 120, and the angle at the circumference being 60, as marked.
The angles of regular polygons:
Trigon, atthe centre,120°,at thecircumference,60°.Tetragon,"90°,""90°.Pentagon,"72°,""108°.Hexagon,"60°,""120°.Octagon,"45°,""135°.Enneagon,"40°,""140°.Decagon,"36°,""144°.Dodecagon,"30°,""150°.
THE ELLIPSE.
An ellipse is a figure bounded by a continuous curve, whose nature will be shown presently.
The dimensions of an ellipse are taken at its extreme length and narrowest width, and they are designated in three ways, as by the length and breadth, by the major and minor axis (the major axis meaning the length, and the minor the breadth of the figure), and the conjugate and transverse diameters, the transverse meaning the shortest, and the conjugate the longest diameter of the figure.
In this book the terms major and minor axis will be used to designate the dimensions.
The minor and major axes are at a right angle one to the other, and their point of intersection is termed the axis of the ellipse.
In an ellipse there are two points situated upon theline representing the major axis, and which are termed the foci when both are spoken of, and a focus when one only is referred to, foci simply being the plural of focus. These foci are equidistant from the centre of the ellipse, which is formed as follows: Two pins are driven in on the major axis to represent the foci A and B, Figure 75, and around these pins a loop of fine twine is passed; a pencil point, C, is then placed in the loop and pulled outwards, to take up the slack of the twine. The pencil is held vertical and moved around, tracing an ellipse as shown.
Fig. 75.
Now it is obvious, from this method of construction, that there will be at every point in the pencil's path a length of twine from the final point to each of the foci, and a length from one foci to the other, and the length of twine in the loop remaining constant, it is demonstrated that if in a true ellipse we take any number of points in its curve, and for each point add together its distance to each focus, and to this add the distance apart of the foci, the total sum obtained will be the same for each point taken.
Fig. 76.
Fig. 77.
In Figures 76 and 77 are a series of ellipses marked with pins and a piece of twine, as already described. The corresponding ellipses, as A in both figures, were marked with the same loop, the difference in the two forms being due to the difference in distance apart of the foci. Again, the same loop was used for ellipses B in both figures, as also for C and D. From these figures we perceive that—
1st. With a given width or distance apart of foci, the larger the dimensions are the nearer the form of the figure will approach to that of a circle.
2d. The nearer the foci are together in an ellipse, having any given dimensions, the nearer the form of the figure will approach that of a circle.
3d. That the proportion of length to width in an ellipse is determined by the distance apart of the foci.
4th. That the area enclosed within an ellipse of a given circumference is greater in proportion as the distance apart of the foci is diminished; and,
5th. That an ellipse may be given any required proportion of width to length by locating the foci at the requisite distance apart.
The form of a true ellipse may be very nearly approached by means of the arcs of circles, if the centres from which those arcs are struck are located in the most desirable positions for the form of ellipse to be drawn.
Fig. 78.
Thus in Figure 78 are three ellipses whose forms were pencilled in by means of pins and a loop of twine, as already described, but which were inked in by finding four arcs of circles of a radius that would most closely approach the pencilled line;a bare the foci of all three ellipses A, B, and C; the centre for the end curves ofaare atcandd, and those for its side arcs are ateandf. For B the end centres are atgandh, and the side centres atiandj. For C the end centres are atk,l, and the side centres atmandn.It will be noted that, first, all the centres for the end curves fall on the line of the length or major axis, while all those for the sides fall on the line of width or the minor axis; and, second, that as the dimensions of the ellipses increase, the centres for the arcs fall nearer to the axis of the ellipse. Now in proportion as a greater number of arcs of circles are employed to form the figure, the nearer it will approach the form of a true ellipse; but in practice it is not usual to employ more than eight, while it is obvious that not less than four can be used. When four are used they will always fall somewhere on the lines on the major and minor axis; but if eight are used, two will fall on the line of the major axis, two on the line of the minor axis, and the remaining four elsewhere.
Fig. 79.
In Figure 79 is a construction wherein four arcs are used. Draw the linea b, the major axis, and at aright angle to it the linec d, the minor axis of the figure. Now find the difference between the length of half the two axes as shown below the figure, the length of linef(fromgtoi) representing half the length of the figure (as fromatoe), and the length or radius fromgtohequalling that frometod; hence fromhtoiis the difference between half the major and half the minor axis. With the radius (h i), mark fromeas a centre the arcsj k, and joinj kby linel. Take half the length of lineland fromjas a centre mark a line onato the arcm. Now the radius ofmfromewill be the radius of all the centres from which to draw the figure; hence we may draw in the circlemand draw lines, cutting the circle. Then draw lineo, passing throughm, and giving the centrep. Frompwe draw the lineq, cutting the intersection of the circle with lineaand giving the centrer. Fromrwe draw lines, meeting the circle and the linec, d, giving us the centret. Fromtwe draw lineu, passing through the centrem. These four lineso,q,s,uare prolonged past the centres, because they define what part of the curve is to be drawn from each centre: thus from centremthe curve fromvtowis drawn, from centretthe curve fromwtoxis drawn. From centrerthe curve fromxtoyis drawn, and from centrepthe curve fromytovis drawn. It is to be noted, however, that after the pointmis found, the remaining lines may be drawn very quickly, because the lineofrommtopmay be drawn with the triangle of 45 degrees resting on the square blade. The triangle may be turned over, set to pointpand lineqdrawn, and by turning the triangle again thelinesmay be drawn from pointr; finally the triangle may be again turned over and lineudrawn, which renders the drawing of the circlemunnecessary.
To draw an elliptical figure whose proportion of width to breadth shall remain the same, whatever the length of the major axis may be: Take any square figure and bisect it by the line A in Figure 80. Draw, in each half of the square, the diagonals E F, G H. From P as a centre with the radius P R draw the arc S E R. With the same radius draw from O as a centre the arc T D V. With radius L C draw arc R C V, and from K as a centre draw arc S B T.
Fig. 80.
Fig. 81.
A very near approach to the true form of a true ellipse may be drawn by the construction given in Figure 81, in which A A and B B are centre lines passing through the major and minor axis of the ellipse, of whichais the axis or centre,b cis the major axis, anda ehalf the minor axis. Draw the rectangleb f g c, and then the diagonal lineb e; at a right angle tob edraw linef h, cutting B B ati. With radiusa eand fromaas a centre draw the dotted arce j, givingthe pointjon line B B. From centrek, which is on the line B B and central betweenbandj, draw the semicircleb m j, cutting A A atl. Draw the radius of the semicircleb m j, cutting it atm, and cuttingf gatn. With the radiusm nmark on A A at and fromaas a centre the pointo. With radiush oand fromcentrehdraw the arcp o q. With radiusa land frombandcas centres, draw arcs cuttingp o qat the pointsp q. Draw the linesh p randh q sand also the linesp i tandq v w. Fromhas a centre draw that part of the ellipse lying betweenrands, with radiusp r; frompas a centre draw that part of the ellipse lying betweenrandt, with radiusq s, and fromqas a centre draw the ellipse fromstow, with radiusi t; and fromias a centre draw the ellipse fromttoband with radiusv w, and fromvas a centre draw the ellipse fromwtoc, and one-half of the ellipse will be drawn. It will be seen that the whole construction has been performed to find the centresh,p,q,iandv, and that whilevandimay be used to carry the curve around on the other side of the ellipse, new centres must be provided forhpandq, these new centres corresponding in position tohpq. Divesting the drawing of all thelines except those determining its dimensions and the centres from which the ellipse is struck, we have in Figure 82 the same ellipse drawn half as large. The centresv,p,q,hcorrespond to the same centres in Figure 81, whilev',p',q',h'are in corresponding positions to draw in the other half of the ellipse. The length of curve drawn from each centre is denoted by the dotted lines radiating from that centre; thus, fromhthe part fromrtosis drawn; fromh'that part fromr'tos'. At the ends the respective centresvare used for the parts fromwtow'and fromttot'respectively.
Fig. 82.
Fig. 83.
The most correct method of drawing an ellipse is by means of an instrument termed a trammel, which is shown in Figure 83. It consists of a cross frame in which are two grooves, represented by the broad black lines, one of which is at a right angle to the other. In these grooves are closely fitted two sliding blocks, carrying pivots E F, which may be fastened to the sliding blocks, while leaving them free to slide in the grooves at any adjusted distance apart. These blocks carry an arm or rod having a tracing point (as pen or pencil) at G. When this arm is swept around by theoperator, the blocks slide in the grooves and the pen-point describes an ellipse whose proportion of width to length is determined by the distance apart of the sliding blocks, and whose dimensions are determined by the distance of the pen-point from the sliding block. To set the instrument, draw lines representing the major and minor axes of the required ellipse, and set off on these lines (equidistant from their intersection), to mark the required length and width of ellipse. Place the trammel so that the centre of its slots is directly over the point or centre from which the axes are marked (which may be done by setting the centres of the slots true to the lines passing through the axis) and set the pivots as follows: Place the pencil-point G so that it coincides with one of the points as C, and place the pivot E so that it comes directly at the point of intersection of the two slots, and fasten it there. Then turn the arm so that the pencil-point G coincides with one of the points of the minor axis as D, the arm lying parallel to B D, and place the pivot F over the centre of the trammel and fasten it there, and the setting is complete.
Fig. 84.
To draw a parabola mechanically: In Figure 84 C D is the width and H J the height of the curve.Bisect H D in K. Draw the diagonal line J K and draw K E, cutting K at a right angle to J K, and produce it in E. With the radius H E, and from J as a centre, mark point F, which will be the focus of the curve. At any convenient distance above J fasten a straight-edge A B, setting it parallel to the base C D of the parabola. Place a square S with its back against the straight-edge, setting the edge O N coincident with the line J H. Place a pin in the focus F, and tie to it one end of a piece of twine. Place a tracing-point at J, pass the twine around the tracing-point, bringing down along the square-blade and fasten it at N, with the tracing-point kept against the edge of the square and the twine kept taut; slide the square along the straight-edge, and the tracing-point will mark the half J C of the parabola. Turn the square over and repeat the operation to trace the other half J D. This method corresponds to the method of drawing an ellipse by the twine and pins, as already described.
Fig. 85.
To draw a parabola by lines: Bisect the width A B in Figure 85, and divide each half into any convenient number of equal divisions; and through these points of division draw vertical lines, as 1, 2, 3, etc. (in each half). Divide the height A D at one end and B E at the other into as many equal divisions as the half ofA B is divided into. From the points of divisions 1, 2, 3, etc., on lines A D and B E, draw lines pointing to C, and where these lines intersect the corresponding vertical lines are points through which the curve may be drawn. Thus on the side A D of the curve, the intersection of the two lines marked 1 is a point in the curve; the intersection of the two lines marked 2 is another point in the curve, and so on.
Fig. 86.
Draw the line A B, Figure 86, equal to the length of stroke required. Divide it into any number of equal parts, and from C as a centre draw circles through the points of division. Draw the outer circle and divide its circumference into twice as many equal divisions as the line A B was divided into. Draw radial lines from each point of division on the circle, and the points of intersection of the radial lines with the circles are points for the outline of the cam, andthrough these points a curved line may be drawn giving the shape of the cam. It is obvious that the greater the number of divisions on A B, the more points and the more perfect the curve may be drawn.
When the interior of a piece is to be shown as a piece cut in half, or when a piece is broken away, as is done to make more of the parts show, or show more clearly, the surface so broken away or cut off is section-lined or cross-hatched; that is to say, diagonal lines are drawn across it, and to distinguish one piece from another these lines are drawn at varying angles and of varying widths apart. In Figure 87 is given a view of three cylindrical pieces. It may be known to be a sectional view by the cross-hatching or section lines. It would be a difficult matter to represent the three pieces put together without showing them in section, because, in an outline view, the collars and recesseswould not appear. Each piece could of course be drawn separately, but this would not show how they were placed when put together. They could be shown in one view if they were shaded by lines and a piece shown broken out where the collars and, recesses are, but line shading is too tedious for detail drawings, beside involving too much labor in their production.
Fig. 87.
Figure 88 represents a case in which there are three cylindrical pieces one within the other, the two inner ones being fastened together by a screw which is shown dotted in in the end view, and whose position along the pieces is shown in the side view. The edges of the fracture in the outer piece are in this case cross-hatched, to show the line of fracture.
Fig. 88.
Fig. 89.
In cross-hatching it is better that the diagonal lines do not quite meet the edges of the piece, than that they should in the least overrun, as is shown in Figure 89, where in the top half the diagonals slightly overrun, while in the lower half they do not quite meet the outlines of the piece.
In Figure 90 are shown in section a number of pieces one within the other, the central bore beingfilled with short plugs. All the cross-hatching was done with the triangle of 60 degrees and that of 90 degrees. It is here shown that with these two triangles only, and a judicious arrangement of the diagonals, an almost infinite number of pieces may be shown in cross section without any liability of mistaking one for the other, or any doubt as to the form and arrangement of the pieces; for, beside the difference in spacing in the cross-hatching, there are no two adjoining pieces with the diagonals running in the same direction. It will be seen that the narrow pieces are most clearly defined by a close spacing of the cross-hatching.
Fig. 90.
In Figure 91 are shown three pieces put together and having slots or keyways through them. The outer shell is shown to be in one piece from end to end, because the cross-hatching is not only equally spaced, but the diagonals are in the same direction; hence it would be known that D, F, H, and E were slots or recesses through the piece. The same remarks apply to piece B, wherein G, J, K are recessesor slots. Piece C is shown to have in its bore a recess at L. In the case of B, as of A, there would be no question as to the piece being all one from end to end, notwithstanding that the two ends are completely severed where the slots G, I, come, because the spacing and direction of the cross-hatching are equal on each side of the slots, which they would not be if they were separate pieces.
Fig. 91.
Fig. 92.
Section shading or cross-hatching may sometimes cause the lines of the drawing to appear crooked to the eye. Thus, in Figure 92, the key edge on the right appears curved inwards, while on the leftthe key edge appears curved outwards, although such is not actually the case. The same effect is produced in Figure 93 on the right-hand edge of the key, but not on the left-hand edge.
Fig. 93
Fig. 94.
A remarkable instance of this kind is shown in Figure 94, when the vertical lines appear to the eye to be at a considerable angle one to the other, although they are parallel.
The lines in sectional shading or cross-hatching may be made to denote the material of which thepiece is to be composed. Thus Professor Unwin has proposed the system shown in the Figures 95 and 96. This may be of service in some cases, but it would involve very much more labor than it is worth in ordinary machine shop drawings, except in the case of cast iron and wood, these two being shown in the simplest and the usual manner. It is much better to write the name of the material beneath the piece in a detail drawing.