[Ill.u.s.tration: Fig. 80.]

[Ill.u.s.tration: 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 pa.s.sing through the major and minor axis of the ellipse, of which _a_ is the axis or centre, _b c_ is the major axis, and _a e_ half the minor axis. Draw the rectangle _b f g c_, and then the diagonal line _b e_; at a right angle to _b e_ draw line _f h_, cutting B B at _i_. With radius _a e_ and from _a_ as a centre draw the dotted arc _e j_, giving the point _j_ on line B B. From centre _k_, which is on the line B B and central between _b_ and _j_, draw the semicircle _b m j_, cutting A A at _l_. Draw the radius of the semicircle _b m j_, cutting it at _m_, and cutting _f g_ at _n_. With the radius _m n_ mark on A A at and from _a_ as a centre the point _o_. With radius _h o_ and from centre _h_ draw the arc _p o q_. With radius _a l_ and from _b_ and _c_ as centres, draw arcs cutting _p o q_ at the points _p q_. Draw the lines _h p r_ and _h q s_ and also the lines _p i t_ and _q v w_. From _h_ as a centre draw that part of the ellipse lying between _r_ and _s_, with radius _p r_; from _p_ as a centre draw that part of the ellipse lying between _r_ and _t_, with radius _q s_, and from _q_ as a centre draw the ellipse from _s_ to _w_, with radius _i t_; and from _i_ as a centre draw the ellipse from _t_ to _b_ and with radius _v w_, and from _v_ as a centre draw the ellipse from _w_ to _c_, and one-half of the ellipse will be drawn. It will be seen that the whole construction has been performed to find the centres _h_, _p_, _q_, _i_ and _v_, and that while _v_ and _i_ may be used to carry the curve around on the other side of the ellipse, new centres must be provided for _h_ _p_ and _q_, these new centres corresponding in position to _h_ _p_ _q_. Divesting the drawing of all the lines 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 centres _v_, _p_, _q_, _h_ correspond to the same centres in Figure 81, while _v'_, _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, from _h_ the part from _r_ to _s_ is drawn; from _h'_ that part from _r'_ to _s'_. At the ends the respective centres _v_ are used for the parts from _w_ to _w'_ and from _t_ to _t'_ respectively.

[Ill.u.s.tration: Fig. 82.]

[Ill.u.s.tration: 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 the operator, 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 pa.s.sing 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.

[Ill.u.s.tration: 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, pa.s.s 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.

[Ill.u.s.tration: 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 of A 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.

TO DRAW A HEART CAM.

[Ill.u.s.tration: 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 circ.u.mference 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, and through 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.

CHAPTER IV.

_SHADOW LINES AND LINE SHADING._

SECTION LINING OR CROSS-HATCHING.

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 recesses would 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.

[Ill.u.s.tration: 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.

[Ill.u.s.tration: Fig. 88.]

[Ill.u.s.tration: 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 being filled 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 s.p.a.cing 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 s.p.a.cing of the cross-hatching.

[Ill.u.s.tration: Fig. 90.]

In Figure 91 are shown three pieces put together and having slots or keyways through them. The outer sh.e.l.l is shown to be in one piece from end to end, because the cross-hatching is not only equally s.p.a.ced, 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 recesses or 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 s.p.a.cing and direction of the cross-hatching are equal on each side of the slots, which they would not be if they were separate pieces.

[Ill.u.s.tration: Fig. 91.]

[Ill.u.s.tration: 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 left the 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.

[Ill.u.s.tration: Fig. 93]

[Ill.u.s.tration: 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 the piece 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.

[Ill.u.s.tration: Fig. 95.]

[Ill.u.s.tration: Fig. 96.]

LINE SHADING.

Mechanical drawings are made to look better and to show more distinctly by being line shaded or shaded by lines. The simplest form of line shading is by the use of the shade or shadow line.

In a mechanical drawing the light is supposed, for the purposes of line shading or of coloring, to come in from the upper left-hand corner of the drawing paper; hence it falls directly upon the upper and left-hand lines of each piece, which are therefore represented by fine lines, while the right hand and lower edges of the piece being on the shadow side may therefore, with propriety, be represented by broader lines, which are called shadow or shade lines. These lines will often serve to indicate the shape of some part of the piece represented, as will be seen from the following examples. In Figure 97 is a piece that contains a hole, the fact being shown by the circle being thickened at A. If the circle were thickened on the other side as at B, in Figure 98, it would show that it represented a cylindrical stem instead of a hole.

[Ill.u.s.tration: Fig. 97.]

[Ill.u.s.tration: Fig. 98.]

[Ill.u.s.tration: Fig. 99.]

In Figure 99 is represented a washer, the surfaces that are in the shadow side being shown in a shade line or shadow line, as it is often called.

In Figure 100 is a key drawn with a shade line, while in Figure 101 the shade line is shown applied to a nut. The shade line may be produced in straight lines by drawing the line twice over, and slightly inclining the pen, or by opening the pen points a little. For circles, however, it may be produced either by slightly moving the centre from which the circle is drawn, or by going over the shade part twice, and slightly pressing the instrument as it moves, so as to gradually spring the legs farther apart, the latter plan being generally preferable.

[Ill.u.s.tration: Fig. 100.]

[Ill.u.s.tration: Fig. 101.]

[Ill.u.s.tration: Fig. 102.]